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Abstract

Predictive regressions play a pivotal role in assessing the predictability of returns for financial assets. However, the existence of a non-zero intercept in the predictive variable poses challenges for the popular IVX method, as the statistical properties of a nearly integrated predictive variable differ significantly with and without an intercept. This article presents a novel unified predictability test utilizing weighted inference and random weighted bootstrap. It addresses challenges posed by both conditional heteroscedasticity in linear predictive regression and the presence of a non-zero intercept in the predictor variable. Simulation results demonstrate the accurate size of the proposed test across various scenarios, including stationary, near unit root, unit root, mildly integrated, mildly explosive, and zero and non-zero intercepts. In an empirical application, we employ the proposed test to investigate the predictive capacity of eleven economic and financial variables on the monthly returns of the S&P 500 from 1980 to 2019. The findings reveal stronger evidence of predictability compared to the instrumental variable-based test.

Predictive regressions are widely used in empirical economics and finance to investigate the predictability of financial asset returns such as stocks and bonds. In the financial econometrics literature, the commonly used predictive linear regression, as formulated in Cai and Wang (2014), Phillips and Lee (2016), and Bauer and Hamilton (2018), is often built upon the following simultaneous structural linear model:
(1)
where yt and xt respectively denote an asset return and a predictor at time t, and {(ut,et)}t=1T is a martingale difference sequence. Empirically, Campbell and Yogo (2006) and Welch and Goyal (2008) show that some predictive variables such as dividend payout ratio (d/e) and T-bill ratio are highly persistent, ie, the autoregressive coefficient π is less than but close to unity. These highly persistent predictors can be “good measure[s] of the current risk premium if premia are persistent through time” (Campbell 1987).1 Also, the two innovations ut and et are strongly and negatively correlated, causing the so-called embedded endogeneity. The classical t-statistic tends to over reject the null hypothesis of no predictability even with a large sample size (Stambaugh 1999; Amihud and Hurvich 2004; Campbell and Yogo 2006).

A variety of econometric methods has been proposed to address the inference issues of predictive regression. Early works focus on strategies for correcting asymptotic bias, which depends on the predictive variable’s persistent level, such as the first-order bias correction (Stambaugh 1999), the second-order bias correction (Amihud and Hurvich 2004), and the conservative bias-adjusted estimator (Lewellen 2004). An alternative framework for statistical inference assumes that the predictive variable is a nearly integrated process (Elliott and Stock 1994; Jansson and Moreira 2006; Campbell and Yogo 2006; Hjalmarsson 2011; Cai and Wang 2014), where procedures in these studies rely on the prior information about the exact degrees of persistence of the predictive variables, which may not be consistently estimated. Recently, Hong et al. (2024) propose a weighted least squares estimator for β and show that the estimator, after a random normalization, has a normal mixture distribution (normal distribution) when xt in Equation (1) is a vector and ut is independent of xt−1 at all leads and lags, which is restrictive; see condition (A2) in Hong et al. (2024), where they use xt rather than xt−1 in Equation (1).

During the past decade, researchers have developed various unified tests robust to highly persistent predictors and endogeneity in predictive regressions. One attractive approach is the extended instrumental variable (dubbed IVX) based inference, which was initially proposed by Magdalinos and Phillips (2009) and further extensively studied by Kostakis, Magdalinos, and Stamatogiannis (2015). The distinctive feature of the IVX-based test is that it constructs a less persistent instrumental variable z˜t when the predictive variable xt−1 is nearly integrated. Specifically, the instrumental variable z˜t−1 is constructed as z˜t=Πzz˜t−1+Δxt, where z˜0=0, Δxt=xt−xt−1⁠, and Πz=1−cz/Tη for some constant cz with 0<η<1⁠. Kostakis, Magdalinos, and Stamatogiannis (2015) argue that the IVX methodology offers a good balance between size control and power loss by using cz=−1 and η=0.95⁠. Also, they show that the IVX-based Wald test is robust to predictors’ degrees of persistence in the presence of endogenous errors and conditional heteroscedasticity, which usually arises from clustering in the underlying news process (Clark 1973) or dynamic trading and rebalancing.2  Phillips and Lee (2016) show that this test remains valid for predictors with local unit roots in the explosive and mildly explosive roots. Demetrescu and Rodrigues (2022) conduct IVX-based estimation and inference in an augmented predictive regression context analogous to Amihud and Hurvich (2004) to reduce the bias in the slope coefficient estimation. They use both residual wild bootstrap and fixed regressor wild bootstrap under considerably weaker assumptions on the innovations and show that the bootstrap tests deliver considerably more accurate finite sample inference, especially for one-sided testing. Xu and Guo (2024) propose a modified IVX test when xt in Equation (1) is a high-dimensional vector. These aforementioned IVX-based inferences are developed for either zero intercepts in modeling predictors by xt=ρxt−1+et or non-zero intercepts in modeling predictors by
(2)
However, it remains unanswered if these IVX methods still work regardless of μ and ρ when
(3)
which is dramatically different from Equation (2). For example, when μ≠0 and ρ=1−c/T, |x[Ts]| in Equation (3) diverges with rate T for any s > 0 as T→∞⁠, while |x[Ts]| in Equation (2) diverges with rate T⁠. This discrepancy poses a challenge to formulating a unified IVX test applicable to both zero and non-zero intercept scenarios under the autoregressive structure (3). For example, the weighted least squares estimator in Hong et al. (2024) has a degenerate limit under model (3) when μ≠0⁠. Moreover, the data analysis indicates that Equation (3) with non-zero μ is possible.

In this study, we contribute to the literature on predictive regression by proposing a new unified predictability test that can handle a predictive regression with conditional heteroscedasticity in the error terms and a predictor of different persistent levels with and without an intercept. As outlined earlier, the existing literature lacks a comprehensive predictability test accommodating both conditional heteroscedasticity in the linear predictive regression and a non-zero intercept in modeling predictors by Equation (3). Notably, our simulation study in Section 2 demonstrates that the widely employed IVX-based method proposed by Kostakis, Magdalinos, and Stamatogiannis (2015) can yield negative Wald statistics and exhibit severely biased size when the highly persistent predictor incorporates a non-zero intercept. Theoretical insights suggest that the statistical properties of a near unit root process differ significantly with and without a non-zero intercept (Phillips 1987), posing challenges to the instrumental variable construction and variance estimation in Kostakis, Magdalinos, and Stamatogiannis (2015) when a non-zero intercept is present. The development of an IVX-based predictability test, regardless of the intercept’s presence in Equation (3), remains an intriguing yet unexplored area. Conversely, while a data splitting technique combined with the empirical likelihood method proposed by Zhu, Cai, and Peng (2014) accommodates a non-zero intercept, it falls short in handling conditional heteroscedasticity in the regression due to the application of differencing with a substantial lag for intercept removal.

Our estimation procedure contains two steps. The first step is to split data into two parts, then use each half to formulate a score equation for α and a weighted score equation for β, respectively. The resulting weighted estimation has either a normal limit or a conditional normal limit after random normalization under conditional heteroscedasticity; see Theorem 1 in Section 1. The second step overcomes these difficulties to achieve a unified test by employing the square of the weighted estimation for dealing with the random normalization and adopting a random weighted bootstrap (RWB) method, suggested by Zheng (1987) and Rao and Zhao (1992) for i.i.d. data and Zhu (2016) for stationary time series, to estimate the asymptotic variance or calculate critical values; see Theorem 2 in Section 1. This new method achieves a unified chi-squared limit that is robust to conditional heteroscedasticity in the regression errors and irrespective of the dynamic properties of the predictor, including stationary, near unit root, unit root, mildly integrated, mildly explosive, and zero or non-zero intercept. We remark that alternative bootstrap methods, not necessitating the inference of heteroscedastic errors, may be employed. However, the utilization of an RWB is favored for its reduced computational intensity, as opposed to other bootstrap methods that entail the generation of bootstrap samples from the fitted model.

We conduct extensive simulation studies to compare the performance of the new unified method with that of IVX in Kostakis, Magdalinos, and Stamatogiannis (2015) and the IVX-based bootstrap methods in Demetrescu et al. (2022) across various scenarios. Simulation results suggest that our unified method displays excellent size control across all scenarios of persistent levels, degrees of endogeneity, and presence of non-zero intercept. On the contrary, IVX is severely oversized when the degree of endogeneity is high, especially when the sample size is small. Moreover, we find evidence that IVX becomes more likely to yield negative Wald statistics for a highly persistent predictor even when it is not endogenous. Meanwhile, size distortion by the IVX-based bootstrap methods is also notably large even as the sample size increases. We additionally compare the local power among these methods and find that the proposed unified method delivers more attractive power profiles when a predictor is highly persistent or even mildly explosive regardless of the presence of non-zero intercept, while IVX and the IVX-based bootstrap methods are slightly more powerful when the predictor is stationary. Therefore, our data-splitting method does not sacrifice power in the non-stationary case in comparison with other methods.

We apply the proposed unified method to investigate the predictability of monthly U.S. stock returns spanning the period from 1980 to 2019. Employing the eleven financial and macroeconomic variables examined by Kostakis, Magdalinos, and Stamatogiannis (2015), our findings indicate that only dividend yield (d/y), dividend-price ratio (d/p), and earnings-price ratio (e/p) exhibit significant predictive power. In contrast, none of these predictors attain significance at conventional levels according to the IVX-based Wald statistic and bootstrap method. Through the classical t-tests and an alternative strategy to assess the significance of intercepts and utilizing popular unit root tests to evaluate the degrees of persistence in the eleven predictors, we find that all three significant predictors are highly persistent and contain a non-zero intercept. As the primary distinction between the new unified method and IVX lies in the unification of scenarios involving zero and non-zero intercepts, we attribute the undetected predictive ability of IVX to its pronounced undersizing resulting from the presence of a non-zero intercept.

The remaining structure of the article is as follows. Section 1 introduces the proposed method and develops the estimation procedures and corresponding test statistics. Section 2 reports the finite-sample simulation results. Section 3 applies the proposed unified method to investigate the predictive ability of eleven economic and financial variables on the monthly S&P 500 return between 1980 and 2019. Section 4 concludes. We relegate all technical proofs and some additional simulation results to the Supplementary Appendix.

1 Models, Methodologies, and Theoretical Results

Let yt be a predicted variable, such as the stock returns, and xt be a predictive variable, such as d/p and treasury bill rate. It has been a long history in finance to test if xt−1 can be used to predict yt. Statistically, one often tests for zero β in the following linear predictive regression model
(4)

Because of complicated financial activities, {xt} could be highly persistent, correlated, and conditional heteroscedastic in the error terms, {ut} in Equation (4) may be conditional heteroscedastic, and ut and xt−1 can be mutually dependent although they are uncorrelated. These characteristics challenge the predictability test and invalidate the classical t-test. Therefore, it is vital to develop a unified predictability test by accommodating all these features.

Recently, Kostakis, Magdalinos, and Stamatogiannis (2015) introduced a Wald test to accommodate the above features, utilizing a chi-squared limit under the null hypothesis of no predictability. However, our simulation study suggests that the IVX test in Kostakis, Magdalinos, and Stamatogiannis (2015) exhibits a distorted size when the predictor in Equation (3) is persistent with a non-zero intercept. Additionally, the Wald test statistic may assume a negative value due to the variance correction when xt−1 and ut are dependent. Furthermore, we cannot confirm that the theoretical derivations in Kostakis, Magdalinos, and Stamatogiannis (2015) apply to cases involving a non-zero intercept in Equation (3). When xt is univariate or involves a univariate variable and its lags, Zhu, Cai, and Peng (2014) develop an empirical likelihood test by allowing for various persistence of xt with or without an intercept but not permitting conditional heteroscedastic ut and xt. This article develops a new predictability test for model (4) regardless of {xt} being stationary, near unit root, unit root, mildly integrated, and mildly explosive with or without an intercept, and allowing {ut} and {xt} to be conditionally heteroscedastic.

To capture the feature above, we respectively model the error term {ut} in Equation (4) by a general GARCH sequence and the predictor {xt} by an AR(1) process with ARMA and general GARCH errors.3 Specifically, we write
(5)
 
(6)
where ft and ft,x are measurable functions. When ρ≠1 is constant, the non-zero μ in Equation (6) can be absorbed into α in Equation (4) by writing yt=α+βμ/(1−ρ)+β{xt−1−μ/(1−ρ)}+ut and xt−μ/(1−ρ)=ρ{xt−1−μ/(1−ρ)}+et,x⁠. However, when ρ equals one or depends on the sample size, model (4) becomes different with a dynamic trend if xt−1 is replaced with xt−1−μ/(1−ρ)⁠. In this case, simply demeaning xt’s does not suffice.

Some regularity conditions are as follows:

(C1) {(εt,εt,x)⊤} is a sequence of independent and identically distributed (iid) random vectors with zero means and finite variances. Note that we do not assume independence between εt and εt,x⁠.

(C2) ϕi,x’s satisfy ∑i=0∞ϕi,x2<∞⁠. ft and ft,x are measurable functions such that {ut} and {vt,x} are strictly stationary and ergodic with E(|ut|2+ζ)<∞ and E(|vt,x|2+ζ)<∞ for some ζ>0⁠. Also, {vt,x} is strong mixing with mixing coefficient α(n) satisfying that ∑n=0∞n2/ζα(n)<∞⁠. Further, assume Ω=(σu,uσu,eσu,eσe,e) is positive definite, where σu,u=E(u12), σe,e=E(e1,x2)+2∑k=1∞E(e1,xe1+k,x)⁠, and σu,e=E(u1e1,x)+∑k=1∞[E(u1e1+k,x)+E(e1,xu1+k)].

(C3) For nearly stationary and nearly explosive Cases (iv)–(vii) given in (C4) below, we employ Assumption 2.2 in Phillips and Lee (2016). That is, we assume et,x=vt,x and the following geometric moment contraction condition (see Wu 2007 or Phillips and Lee 2016): E(|vt,x−vt,x*|q)≤Krt for some constant K > 0, r∈(0,1), and the same q in (C2), with vt,x*=ft,x(εt−1,x,…,ε1,x,ε0,x*,ε−1,x*,…)εt,x, where {εi,x*:−∞<i<∞} is an iid version of {εi,x:−∞<i<∞}⁠.

(C4) Model (6) satisfies one of the following cases:

 Case (i): |ρ|<1⁠. That is, {xt} is a stationary AR(1) process.

 Case (ii): μ = 0 and ρ=1−c/T for some constant c. That is, {xt} is a nearly integrated process when c > 0, a nearly explosive process when c < 0, and a unit root process when c = 0.

 Case (iii): μ≠0 and ρ=1−c/T for some constant c.

 Case (iv): μ = 0 and ρ=1−c/Td for some constant d∈(0,1) and c > 0. That is, {xt} is a mildly integrated process.

 Case (v): μ = 0 and ρ=1−c/Td for some constant d∈(0,1) and c < 0. That is, {xt} is a mildly explosive process.

 Case (vi): μ≠0 and ρ=1−c/Td for some constant d∈(0,1) and c > 0.

 Case (vii): μ≠0 and ρ=1−c/Td for some constant d∈(0,1) and c < 0.

Following Kostakis, Magdalinos, and Stamatogiannis (2015), we assume that the initial value x0 is any fixed constant or a random process satisfying x0=op(T1/2) in Cases (i)–(iii), and x0=op(Td/2) in Cases (iv)–(vii).

 
Remark 1.

Condition (C3) includes the popular GARCH model and its variants; see  Zhang and Ling (2015,) and  Zhu and Ling (2015,). Conditions (C2) and (C3) are similar to the heteroscedastic case in Kostakis, Magdalinos, and Stamatogiannis (2015), and the geometric moment contraction condition is employed by  Phillips and Lee (2016) to deal with the mildly integrated and mildly explosive cases.

To develop a unified predictability test for H0:β=0 regardless of the properties of {xt}⁠, we split the data into two even parts and use the first part to formulate a score equation for α and the second part to construct a weighted score equation for β. In this way, the weighted estimator for β after a random normalization has either a normal limit or a conditional normal limit. As the random normalization depends on the properties of {xt}⁠, we employ an RWB method to develop a unified predictability test. Also, because the random normalization does not have a constant sign for all cases, we can only develop a unified two-sided test. More specifically, denoting m=[T/2] with [·] being the integer operator, we estimate α and β by solving the following equations:
(7)
where the weight wt is defined as
(8)

Denote the resulted estimators by α^ and β^⁠. The reason to choose the above weight is that |wt|→p1 as t→∞ and T→∞ when {xt} is non-stationary (see the proof of Theorem 1 in Supplementary Appendix), ensuring that the weighted least squares estimators have either a normal limit or a conditional normal limit after a random normalization. Furthermore, the limit would degenerate for Cases (iii), (vi), and (vii) if we did not split the data. The presence of the factor 1m∑s=1mxs in the above weight function is crucial for testing power. Without it, according to our unreported simulation investigation, the test would lack power in the stationary case, as the two equations in Equation (7) become uncorrelated. While numerous choices of wt exist such that |wt|→p1 as t→∞ and T→∞ for non-stationary xt⁠, determining the optimal one is a challenging task beyond the scope of this article. Nevertheless, our chosen approach in the simulation study has demonstrated effective performance.

Put Smh,u=1m∑s=1[mh]us and Smh,e=1m∑s=1[mh]es,x. Using conditions (C1)–(C3), we have the following functional law (see, e.g., Equation (2.7) of Phillips and Lee 2016):
(9)
where {(Wu(h),We(h))⊤:0≤h≤2} is a bivariate Gaussian process with zero mean, (Wu(h1)−Wu(h2),We(h1)−We(h2)) for h1>h2 has covariance matrix (h1−h2)Ω with Ω defined in condition (C2), and “→D” denotes the convergence in the space D2([0,2]) of càdlàg functions on [0,2] to R2⁠.
Throughout, we define Jc(s)=∫0se−c(s−s1)dWe(s1)⁠,
where Φ(·) denotes the cumulative distribution function of the standard normal. Note that σ02 above is random in Case (ii). The theorem below derives the asymptotic limit of the above estimation. 
Theorem 1.
Assume models (4), (5), and (6) satisfy conditions (C1)–(C4). Then, as T→∞⁠,

Theorem 1 above does not lead to a unified test because σ02 depends on the properties of {xt} and is random in Case (ii). To develop a unified test, we adopt a RWB method used by Zheng (1987), Rao and Zhao (1992), Jin, Ying, and Wei (2001), and Zhu (2016). The RWB method works here because it does not require modeling or resampling the residuals. Additionally, the quantity Am(β^−β) is asymptotically normally or mixed normally distributed, regardless of the stationarity of the predictor.

Step (A1): Draw a random sample with sample size 2m from the standard exponential distribution. Denote the random sample by δ1b,…,δ2mb⁠.

Step (A2): Solve the following equations to get estimators α^b and β^b for α and β:

(10)

Step (A3): Repeat the above two steps B times to get {α^b}b=1B and {β^b}b=1B⁠.

Throughout, E*, P*, →p*, →d*, Op*(1)⁠, and op*(1) denote expectation, probability, convergence in probability, convergence in distribution, bounded in probability, and smaller order in probability for {δtb:b=1,…,B, and t=1,…,2m} given {(ut,et,x):t=1,…,2m}⁠, respectively, while the corresponding notations without star are either for {(ut,et,x):t=1,…,2m} or for {(ut,et,x,δtb):t=1,…,2m and b=1,…,B} when {δtb:t=1,…,2m, and b=1,…,B} are involved. We further define
 
Theorem 2.
Assume the conditions of  Theorem 1  hold. Then, as T→∞⁠,
(11)
Furthermore, when B≤Tq  for some q > 0,
(12)
for all Cases (i)–(vii) as T→∞  and B→∞  jointly.
 
Remark 2.

Because |Amb/Am|  could be very close to zero with an exponentially small probability, we do not have E[Am2(β^b−β^)2]<∞  to prove the convergence of B−1∑b=1BAm2(β^b−β^)2, despite we can show E[(Amb)2(β^b−β^)2]<∞, which can be used to prove the convergence of B−1∑b=1B(Amb)2(β^b−β^)2. However, when we assume B≤Tq  for some q > 0, we can prove the convergence of B−1∑b=1BAm2(β^b−β^)2  by letting T→∞  and B→∞  jointly, which leads to  Equation (12). Also, since Am has a random sign in Case (ii), we cannot conduct a one-sided test.

Using Equation (11) in Theorem 2 and Theorem 1 and following the classical percentile bootstrap method, we calculate the p-value for testing H0:β=0 against H1:β≠0 by
(13)
which has the asymptotically correct size by either letting B→∞ and then T→∞ or letting both B→∞ and T→∞ jointly. Alternatively, we can use Theorem 1 and Equation (12) in Theorem 2 to construct a unified test based on the Wald statistic for H0:β=0 against H1:β≠0 as
(14)
where the null hypothesis of no predictability is rejected at level a whenever W^>χ1,1−a2 with χ1,1−a2 denoting the (1−a)-th quantile of the chi-squared distribution with one degree of freedom. This test has the asymptotically correct size when B≤Tq for some q > 0 and both B→∞ and T→∞ jointly.
Next, we generalize model (4) to multiple linear regression with one predictor and its differences being other predictors:
(15)
where
Let Γ be a known d1×d0 matrix with d1≤d0 and rank d1. Define θ=(γ⊤,β)⊤ and the (d1+1)×(d0+1) matrix Γ¯=(Γ001). Here, we use 0 to denote a matrix with all elements being zero throughout, where its dimension depends on different situations. We consider the hypothesis test
(16)
which is equivalent to
(17)
To develop a unified test, we first estimate α, β, and γ by solving the following equations:
(18)
and denote the resulted estimators by α˜,γ˜,β˜ with θ˜=(γ˜⊤,β˜)⊤⁠.
As before, we can show that the above estimator β˜ has a conditional normal limit in Case (ii) and a normal limit in other cases. Hence, to test for Equation (16) in a unified way, we use the following RWB strategy:

Step (B1): Draw a random sample with sample size 2m from the standard exponential distribution. Denote the random sample by δ1b,…,δ2mb⁠.

Step (B2): Solve the following equations to get estimators α˜b,γ˜b,β˜b with θ˜b=(γ˜b⊤,β˜b)⊤⁠:

 

Step (B3): Repeat the above two steps B times to get {θ˜b}b=1B⁠.

Like (14), we consider the following Wald statistic
(19)
to test Equation (16), that is, Equation (17). Similar to Equation (12) in Theorem 2, we assume B≤Tq for some q > 0 and take B→∞ and T→∞ jointly for deriving the limit of W˜ in the following theorem. 
Theorem 3.

Assume models (15), (5), and (6) satisfy conditions (C1)–(C4). Further, assume B≤Tq  for some q > 0. Then, W˜  converges in distribution to a chi-squared limit with (1+d1)  degrees of freedom when B→∞  and T→∞  jointly.

Using the above theorem, we reject the null hypothesis of Equation (16) at level a whenever W˜>χ1+d1,1−a2⁠, where χ1+d1,1−a2 is the (1−a)-th quantile of a chi-squared limit with (1+d1) degrees of freedom. 

Remark 3.

The above method cannot be used to test for H0:Γ0(βγ)=0  for a given matrix Γ0  because β˜  and γ˜  have different rates of convergence except in Case (i). Also, the above theorem holds when {zt}  is replaced by a vector of stationary sequences.

Finally, we can extend the proposed test above to the case that {ut} in Equation (15) is both serially dependent and conditionally heteroscedastic. Under this circumstance, we model {ut} by an ARMA process with general GARCH errors and use the ARMA model to develop a unified predictability test. Notably, our proposed test remains robust to heteroscedasticity, because we neither explicitly model the heteroscedasticity nor infer it. The testing procedure is outlined in detail below.

Replacing Equation (5) by
(20)
Define θ=(ϕ1,…,ϕs,ψ1,…,ψr)⊤ and write
for t=1,…,T⁠. As before, we can estimate α,β,γ,θ by solving
(21)

Similar to Theorem 3 above, we can develop a unified test for Equation (16). We skip the details to conserve space.

2 Numerical Studies

This section uses simulated data to evaluate the finite-sample size and power performance of the proposed unified method using Equation (14). We compare it with the IVX-based Wald statistic in Kostakis, Magdalinos, and Stamatogiannis (2015) and IVX-based bootstrap methods in Demetrescu et al. (2022). We generate data from
where α=0.0066 and the GARCH(1,1) parameters are ω0=0.0001, a1=0.2, and b1=0.75⁠. We model the predictor xt by the following AR(1)-GARCH(1,1) process:
where the intercept μ∈{0,0.1}⁠, and the autoregressive coefficient ρ=1−c/T is determined by c∈{50,20,10,5,0,−1,−3} and T∈{100,250,500} to capture differential persistence of xt, ranging from stationarity to mild explosiveness.4 The three GARCH parameters are ω0,x=0.01, a1,x=0.2⁠, and b1,x=0.7⁠.5 Further, we set ϕ∈{0,0.5} to capture possible serial correlation in et’s. Finally, to characterize heavy tail and asymmetry which are prevailing in financial data, as well as a variety of dependence structures between the two innovations ϵt and εt⁠, we consider the following eight cases, where {e1t},{e2t},{e3t} and {e4t} are sequences of random variables independently drawn from the standard exponential distribution, the standard normal distribution, the student’s t-distribution with three degrees of freedom, and the standard normal distribution, respectively.

Case 1: εt=e1t−1 and ϵt=(θεt+e2t)/θ2+1⁠.

Case 2: ϵt=e1t−1 and εt=(θϵt+e2t)/θ2+1⁠.

Case 3: ϵt=εt=e3t/3⁠.

Case 4: ϵt=e3,t+1/3 and εt=e3t/3⁠.

Case 5: ϵt=(e3,t+1/3)×(e1t/2) and εt=e3t/3⁠.

Case 6: ϵt=(e3,t+1/3) and εt=(e3t/3)×(e1t/2)⁠.

Case 7: ϵt=e1t−1 and εt=e1t−1⁠.

Case 8: ϵt=e2t and εt=(−2e2t+e4t)/5⁠.

For example, in Cases 1 and 2, θ∈{−2,0,2} imply that the correlation coefficients between ut and vt are approximately −0.9, 0, and 0.9, respectively. Each simulation is repeated 10,000 times.

We implement the Wald test under the null hypothesis H0:β=0 versus the alternative H1:β≠0 at the 5% nominal level. Results regarding the empirical sizes of both methods in Case 1 with ϕ=0 are presented in Table 1. Here, RWB denotes the unified method through the RWB strategy with B = 499 bootstrap replications described in Equation (14), while IVX1 denotes the IVX-based Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015).6 We additionally examine the rejection rate of the IVX-based Wald statistic when it is either significant at the 5% level or negative, denoted by IVX2. Furthermore, Wild and FWild, respectively, represent the rejection rates of the residual wild bootstrap and fixed regressor wild bootstrap, two IVX-based bootstrap algorithms proposed in Demetrescu et al. (2022). Results in Panels A and Panel B respectively assume xt follows an AR(1) process without (μ = 0) and with (⁠μ=0.1⁠) an intercept. We have three observations from Table 1. First, RWB displays excellent size control across all combinations of the degree of the persistence in xt (i.e., ρ=1−c/T⁠), dependence between ϵt and εt (i.e., θ), and presence of non-zero intercept (i.e., μ). Although RWB is slightly oversized when T = 100, the size distortion shrinks as T increases. Second, compared with RWB, IVX is prominently oversized when T = 100 and θ=−2 or 2. Additionally, although the rejection rates displayed by IVX2 are quite close to IVX1 in most cases, they can be remarkably larger in the unit root and mild explosive scenarios when θ = 0. This suggests that the likelihood of yielding a negative Wald statistic increases when a predictor follows a unit root or mildly explosive process, making IVX occasionally inapplicable. Third, size distortion by Wild and FWild is notably larger than RWB, even when T = 500 in both panels. More prominently, with the presence of a non-zero intercept, when T = 500 and c∈{0,−1,−3}⁠, Wild is notably undersized when if c∈{0,−1,−3}⁠, and FWild becomes undersized when c = 0.

Table 1
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Case 1: Comparison of empirical sizes with ϕ=0

xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.06480.09110.09110.08810.08330.06070.07610.07610.05670.07200.06040.08790.08790.08470.0812
200.05730.08180.08180.07760.07440.05400.06950.06950.05230.06280.05430.07950.07950.07590.0733
100.05590.08900.08900.08370.07810.05510.06970.07010.05110.05780.05430.08250.08250.07450.0714
50.05580.09260.09260.08150.07730.05390.07310.07640.05060.05230.05640.08990.08990.07740.0729
00.06230.08760.08760.07170.06320.05430.08660.15300.06410.04790.05950.08350.08380.07090.0620
−10.06310.08640.08650.07190.06910.05380.08460.18120.06010.04380.06050.07990.08040.06510.0645
−30.05740.09110.09390.05610.08580.05420.09990.22810.05500.04610.05550.09660.09980.05930.0866
250500.05270.07490.07490.07460.07180.05230.06140.06140.05610.05970.05620.07460.07460.07530.0674
200.05170.07960.07960.07920.07500.05180.06240.06240.05660.06110.04860.07760.07760.07700.0724
100.05000.08080.08080.07860.07100.05440.05610.05620.04840.05060.05220.08320.08320.07970.0746
50.05250.08800.08800.08380.07770.05060.06270.06540.05420.04860.04980.08300.08300.07880.0675
00.05180.07720.07730.06990.06140.05320.07650.13020.06230.04810.05200.07570.07580.06880.0627
−10.05110.06910.06910.06180.06520.05060.08600.16470.06350.04230.05150.07680.07690.06790.0714
−30.04860.07640.07910.05260.08260.05200.09550.18620.04970.03870.05170.07950.08120.05450.0804
500500.05350.06970.06970.07260.06580.05080.05600.05600.05330.05510.05420.07090.07090.07300.0660
200.04940.07800.07800.07950.07170.04870.05270.05280.05030.05230.05100.07190.07190.07310.0687
100.04970.08690.08690.08480.07750.04880.05580.05600.05240.05260.05230.08530.08530.08310.0750
50.05410.08280.08280.08030.06840.04940.05840.05940.05270.04930.04990.08530.08540.08360.0665
00.05130.07080.07080.06770.06010.04680.07140.10990.06320.04650.05100.07060.07060.06700.0590
−10.05100.06600.06610.06090.06660.05010.07570.13840.06050.04280.05110.07030.07050.06340.0679
−30.05010.06970.07180.05170.07160.05330.08760.14410.04560.03470.04890.06730.06880.05360.0746
Panel B: α=0.0066, μ=0.1
100500.06380.09070.09070.08860.08160.05490.07100.07100.05480.06470.06420.08720.08720.08300.0762
200.05960.08880.08880.08440.08240.05380.07230.07230.05520.06320.05670.08470.08470.08240.0755
100.05830.09010.09010.08660.07720.05580.07170.07210.05380.05590.05720.09050.09050.08340.0801
50.05500.08790.08790.08030.07590.06000.07770.08180.05700.05130.05920.09340.09340.08360.0765
00.05650.08390.08420.07150.05690.05090.08920.16360.06450.04530.05880.08390.08400.07240.0595
−10.05980.07670.07700.06130.06710.05450.08500.18750.05950.04470.05560.07800.07810.06390.0689
−30.05800.09170.09400.05240.08920.05880.10040.23430.05090.04250.05690.08950.09300.05320.0896
250500.05770.07420.07420.07490.07140.04980.05640.05640.04990.05260.05730.07040.07040.06950.0675
200.05150.07130.07130.07150.07240.05280.06080.06090.05450.05490.05570.07110.07110.07030.0733
100.04980.06850.06850.06580.07120.04590.06200.06250.05390.04940.05080.06980.06980.06980.0714
50.05170.06890.06890.06730.07080.05060.07000.07370.05710.04490.04990.06870.06870.06450.0707
00.05230.05580.05580.05130.05150.05160.08070.15180.06380.04120.05390.05750.05750.05470.0522
−10.05270.05260.05300.04810.06580.05310.08310.17100.06210.04390.05390.05690.05710.05040.0682
−30.05380.06270.06620.04170.07120.04890.09480.18170.04600.03920.05360.06640.06900.04080.0773
500500.05290.06290.06290.06510.06840.05420.05900.05900.05780.05670.05220.06430.06430.06580.0693
200.05500.06430.06430.06550.07440.04830.06050.06060.05840.05230.04920.05680.05680.05730.0661
100.05010.05550.05550.05420.06760.05230.06400.06500.05760.04990.04910.05630.05630.05620.0677
50.04880.04360.04360.04160.05520.05370.06750.07460.06270.04540.05090.04670.04670.04490.0540
00.04900.02890.02930.02830.03940.04970.08350.15220.06740.04290.04530.02870.02870.02800.0362
−10.04800.02550.02600.02330.05100.05390.08880.16920.06220.04230.05280.03010.03050.02910.0571
−30.04830.04480.04840.02750.05360.05180.09380.15130.04510.04020.05320.04620.04900.03010.0534
xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.06480.09110.09110.08810.08330.06070.07610.07610.05670.07200.06040.08790.08790.08470.0812
200.05730.08180.08180.07760.07440.05400.06950.06950.05230.06280.05430.07950.07950.07590.0733
100.05590.08900.08900.08370.07810.05510.06970.07010.05110.05780.05430.08250.08250.07450.0714
50.05580.09260.09260.08150.07730.05390.07310.07640.05060.05230.05640.08990.08990.07740.0729
00.06230.08760.08760.07170.06320.05430.08660.15300.06410.04790.05950.08350.08380.07090.0620
−10.06310.08640.08650.07190.06910.05380.08460.18120.06010.04380.06050.07990.08040.06510.0645
−30.05740.09110.09390.05610.08580.05420.09990.22810.05500.04610.05550.09660.09980.05930.0866
250500.05270.07490.07490.07460.07180.05230.06140.06140.05610.05970.05620.07460.07460.07530.0674
200.05170.07960.07960.07920.07500.05180.06240.06240.05660.06110.04860.07760.07760.07700.0724
100.05000.08080.08080.07860.07100.05440.05610.05620.04840.05060.05220.08320.08320.07970.0746
50.05250.08800.08800.08380.07770.05060.06270.06540.05420.04860.04980.08300.08300.07880.0675
00.05180.07720.07730.06990.06140.05320.07650.13020.06230.04810.05200.07570.07580.06880.0627
−10.05110.06910.06910.06180.06520.05060.08600.16470.06350.04230.05150.07680.07690.06790.0714
−30.04860.07640.07910.05260.08260.05200.09550.18620.04970.03870.05170.07950.08120.05450.0804
500500.05350.06970.06970.07260.06580.05080.05600.05600.05330.05510.05420.07090.07090.07300.0660
200.04940.07800.07800.07950.07170.04870.05270.05280.05030.05230.05100.07190.07190.07310.0687
100.04970.08690.08690.08480.07750.04880.05580.05600.05240.05260.05230.08530.08530.08310.0750
50.05410.08280.08280.08030.06840.04940.05840.05940.05270.04930.04990.08530.08540.08360.0665
00.05130.07080.07080.06770.06010.04680.07140.10990.06320.04650.05100.07060.07060.06700.0590
−10.05100.06600.06610.06090.06660.05010.07570.13840.06050.04280.05110.07030.07050.06340.0679
−30.05010.06970.07180.05170.07160.05330.08760.14410.04560.03470.04890.06730.06880.05360.0746
Panel B: α=0.0066, μ=0.1
100500.06380.09070.09070.08860.08160.05490.07100.07100.05480.06470.06420.08720.08720.08300.0762
200.05960.08880.08880.08440.08240.05380.07230.07230.05520.06320.05670.08470.08470.08240.0755
100.05830.09010.09010.08660.07720.05580.07170.07210.05380.05590.05720.09050.09050.08340.0801
50.05500.08790.08790.08030.07590.06000.07770.08180.05700.05130.05920.09340.09340.08360.0765
00.05650.08390.08420.07150.05690.05090.08920.16360.06450.04530.05880.08390.08400.07240.0595
−10.05980.07670.07700.06130.06710.05450.08500.18750.05950.04470.05560.07800.07810.06390.0689
−30.05800.09170.09400.05240.08920.05880.10040.23430.05090.04250.05690.08950.09300.05320.0896
250500.05770.07420.07420.07490.07140.04980.05640.05640.04990.05260.05730.07040.07040.06950.0675
200.05150.07130.07130.07150.07240.05280.06080.06090.05450.05490.05570.07110.07110.07030.0733
100.04980.06850.06850.06580.07120.04590.06200.06250.05390.04940.05080.06980.06980.06980.0714
50.05170.06890.06890.06730.07080.05060.07000.07370.05710.04490.04990.06870.06870.06450.0707
00.05230.05580.05580.05130.05150.05160.08070.15180.06380.04120.05390.05750.05750.05470.0522
−10.05270.05260.05300.04810.06580.05310.08310.17100.06210.04390.05390.05690.05710.05040.0682
−30.05380.06270.06620.04170.07120.04890.09480.18170.04600.03920.05360.06640.06900.04080.0773
500500.05290.06290.06290.06510.06840.05420.05900.05900.05780.05670.05220.06430.06430.06580.0693
200.05500.06430.06430.06550.07440.04830.06050.06060.05840.05230.04920.05680.05680.05730.0661
100.05010.05550.05550.05420.06760.05230.06400.06500.05760.04990.04910.05630.05630.05620.0677
50.04880.04360.04360.04160.05520.05370.06750.07460.06270.04540.05090.04670.04670.04490.0540
00.04900.02890.02930.02830.03940.04970.08350.15220.06740.04290.04530.02870.02870.02800.0362
−10.04800.02550.02600.02330.05100.05390.08880.16920.06220.04230.05280.03010.03050.02910.0571
−30.04830.04480.04840.02750.05360.05180.09380.15130.04510.04020.05320.04620.04900.03010.0534

Notes: This table presents finite-sample sizes in Case 1 with ϕ=0 for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX, and IVX-based bootstrap tests under 5% nominal level. RWB denotes the rejection rate for the Wald statistic by RWB. IVX1 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015). IVX2 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015) with possible negative Wald statistics included. Wild denotes the rejection rate for the residual wild bootstrap algorithm by Demetrescu et al. (2022). FWild denotes the rejection rate for the fixed regressor wild bootstrap algorithm by Demetrescu et al. (2022). Results are reported for different combinations of μ∈{0,0.1}, θ∈{−2,0,2}, T∈{100,250,500}⁠, and c∈{50,20,10,5,0,−1,−3}⁠. The average rejection rates are calculated over 10,000 replications.

Table 1
Open in new tab

Case 1: Comparison of empirical sizes with ϕ=0

xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.06480.09110.09110.08810.08330.06070.07610.07610.05670.07200.06040.08790.08790.08470.0812
200.05730.08180.08180.07760.07440.05400.06950.06950.05230.06280.05430.07950.07950.07590.0733
100.05590.08900.08900.08370.07810.05510.06970.07010.05110.05780.05430.08250.08250.07450.0714
50.05580.09260.09260.08150.07730.05390.07310.07640.05060.05230.05640.08990.08990.07740.0729
00.06230.08760.08760.07170.06320.05430.08660.15300.06410.04790.05950.08350.08380.07090.0620
−10.06310.08640.08650.07190.06910.05380.08460.18120.06010.04380.06050.07990.08040.06510.0645
−30.05740.09110.09390.05610.08580.05420.09990.22810.05500.04610.05550.09660.09980.05930.0866
250500.05270.07490.07490.07460.07180.05230.06140.06140.05610.05970.05620.07460.07460.07530.0674
200.05170.07960.07960.07920.07500.05180.06240.06240.05660.06110.04860.07760.07760.07700.0724
100.05000.08080.08080.07860.07100.05440.05610.05620.04840.05060.05220.08320.08320.07970.0746
50.05250.08800.08800.08380.07770.05060.06270.06540.05420.04860.04980.08300.08300.07880.0675
00.05180.07720.07730.06990.06140.05320.07650.13020.06230.04810.05200.07570.07580.06880.0627
−10.05110.06910.06910.06180.06520.05060.08600.16470.06350.04230.05150.07680.07690.06790.0714
−30.04860.07640.07910.05260.08260.05200.09550.18620.04970.03870.05170.07950.08120.05450.0804
500500.05350.06970.06970.07260.06580.05080.05600.05600.05330.05510.05420.07090.07090.07300.0660
200.04940.07800.07800.07950.07170.04870.05270.05280.05030.05230.05100.07190.07190.07310.0687
100.04970.08690.08690.08480.07750.04880.05580.05600.05240.05260.05230.08530.08530.08310.0750
50.05410.08280.08280.08030.06840.04940.05840.05940.05270.04930.04990.08530.08540.08360.0665
00.05130.07080.07080.06770.06010.04680.07140.10990.06320.04650.05100.07060.07060.06700.0590
−10.05100.06600.06610.06090.06660.05010.07570.13840.06050.04280.05110.07030.07050.06340.0679
−30.05010.06970.07180.05170.07160.05330.08760.14410.04560.03470.04890.06730.06880.05360.0746
Panel B: α=0.0066, μ=0.1
100500.06380.09070.09070.08860.08160.05490.07100.07100.05480.06470.06420.08720.08720.08300.0762
200.05960.08880.08880.08440.08240.05380.07230.07230.05520.06320.05670.08470.08470.08240.0755
100.05830.09010.09010.08660.07720.05580.07170.07210.05380.05590.05720.09050.09050.08340.0801
50.05500.08790.08790.08030.07590.06000.07770.08180.05700.05130.05920.09340.09340.08360.0765
00.05650.08390.08420.07150.05690.05090.08920.16360.06450.04530.05880.08390.08400.07240.0595
−10.05980.07670.07700.06130.06710.05450.08500.18750.05950.04470.05560.07800.07810.06390.0689
−30.05800.09170.09400.05240.08920.05880.10040.23430.05090.04250.05690.08950.09300.05320.0896
250500.05770.07420.07420.07490.07140.04980.05640.05640.04990.05260.05730.07040.07040.06950.0675
200.05150.07130.07130.07150.07240.05280.06080.06090.05450.05490.05570.07110.07110.07030.0733
100.04980.06850.06850.06580.07120.04590.06200.06250.05390.04940.05080.06980.06980.06980.0714
50.05170.06890.06890.06730.07080.05060.07000.07370.05710.04490.04990.06870.06870.06450.0707
00.05230.05580.05580.05130.05150.05160.08070.15180.06380.04120.05390.05750.05750.05470.0522
−10.05270.05260.05300.04810.06580.05310.08310.17100.06210.04390.05390.05690.05710.05040.0682
−30.05380.06270.06620.04170.07120.04890.09480.18170.04600.03920.05360.06640.06900.04080.0773
500500.05290.06290.06290.06510.06840.05420.05900.05900.05780.05670.05220.06430.06430.06580.0693
200.05500.06430.06430.06550.07440.04830.06050.06060.05840.05230.04920.05680.05680.05730.0661
100.05010.05550.05550.05420.06760.05230.06400.06500.05760.04990.04910.05630.05630.05620.0677
50.04880.04360.04360.04160.05520.05370.06750.07460.06270.04540.05090.04670.04670.04490.0540
00.04900.02890.02930.02830.03940.04970.08350.15220.06740.04290.04530.02870.02870.02800.0362
−10.04800.02550.02600.02330.05100.05390.08880.16920.06220.04230.05280.03010.03050.02910.0571
−30.04830.04480.04840.02750.05360.05180.09380.15130.04510.04020.05320.04620.04900.03010.0534
xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.06480.09110.09110.08810.08330.06070.07610.07610.05670.07200.06040.08790.08790.08470.0812
200.05730.08180.08180.07760.07440.05400.06950.06950.05230.06280.05430.07950.07950.07590.0733
100.05590.08900.08900.08370.07810.05510.06970.07010.05110.05780.05430.08250.08250.07450.0714
50.05580.09260.09260.08150.07730.05390.07310.07640.05060.05230.05640.08990.08990.07740.0729
00.06230.08760.08760.07170.06320.05430.08660.15300.06410.04790.05950.08350.08380.07090.0620
−10.06310.08640.08650.07190.06910.05380.08460.18120.06010.04380.06050.07990.08040.06510.0645
−30.05740.09110.09390.05610.08580.05420.09990.22810.05500.04610.05550.09660.09980.05930.0866
250500.05270.07490.07490.07460.07180.05230.06140.06140.05610.05970.05620.07460.07460.07530.0674
200.05170.07960.07960.07920.07500.05180.06240.06240.05660.06110.04860.07760.07760.07700.0724
100.05000.08080.08080.07860.07100.05440.05610.05620.04840.05060.05220.08320.08320.07970.0746
50.05250.08800.08800.08380.07770.05060.06270.06540.05420.04860.04980.08300.08300.07880.0675
00.05180.07720.07730.06990.06140.05320.07650.13020.06230.04810.05200.07570.07580.06880.0627
−10.05110.06910.06910.06180.06520.05060.08600.16470.06350.04230.05150.07680.07690.06790.0714
−30.04860.07640.07910.05260.08260.05200.09550.18620.04970.03870.05170.07950.08120.05450.0804
500500.05350.06970.06970.07260.06580.05080.05600.05600.05330.05510.05420.07090.07090.07300.0660
200.04940.07800.07800.07950.07170.04870.05270.05280.05030.05230.05100.07190.07190.07310.0687
100.04970.08690.08690.08480.07750.04880.05580.05600.05240.05260.05230.08530.08530.08310.0750
50.05410.08280.08280.08030.06840.04940.05840.05940.05270.04930.04990.08530.08540.08360.0665
00.05130.07080.07080.06770.06010.04680.07140.10990.06320.04650.05100.07060.07060.06700.0590
−10.05100.06600.06610.06090.06660.05010.07570.13840.06050.04280.05110.07030.07050.06340.0679
−30.05010.06970.07180.05170.07160.05330.08760.14410.04560.03470.04890.06730.06880.05360.0746
Panel B: α=0.0066, μ=0.1
100500.06380.09070.09070.08860.08160.05490.07100.07100.05480.06470.06420.08720.08720.08300.0762
200.05960.08880.08880.08440.08240.05380.07230.07230.05520.06320.05670.08470.08470.08240.0755
100.05830.09010.09010.08660.07720.05580.07170.07210.05380.05590.05720.09050.09050.08340.0801
50.05500.08790.08790.08030.07590.06000.07770.08180.05700.05130.05920.09340.09340.08360.0765
00.05650.08390.08420.07150.05690.05090.08920.16360.06450.04530.05880.08390.08400.07240.0595
−10.05980.07670.07700.06130.06710.05450.08500.18750.05950.04470.05560.07800.07810.06390.0689
−30.05800.09170.09400.05240.08920.05880.10040.23430.05090.04250.05690.08950.09300.05320.0896
250500.05770.07420.07420.07490.07140.04980.05640.05640.04990.05260.05730.07040.07040.06950.0675
200.05150.07130.07130.07150.07240.05280.06080.06090.05450.05490.05570.07110.07110.07030.0733
100.04980.06850.06850.06580.07120.04590.06200.06250.05390.04940.05080.06980.06980.06980.0714
50.05170.06890.06890.06730.07080.05060.07000.07370.05710.04490.04990.06870.06870.06450.0707
00.05230.05580.05580.05130.05150.05160.08070.15180.06380.04120.05390.05750.05750.05470.0522
−10.05270.05260.05300.04810.06580.05310.08310.17100.06210.04390.05390.05690.05710.05040.0682
−30.05380.06270.06620.04170.07120.04890.09480.18170.04600.03920.05360.06640.06900.04080.0773
500500.05290.06290.06290.06510.06840.05420.05900.05900.05780.05670.05220.06430.06430.06580.0693
200.05500.06430.06430.06550.07440.04830.06050.06060.05840.05230.04920.05680.05680.05730.0661
100.05010.05550.05550.05420.06760.05230.06400.06500.05760.04990.04910.05630.05630.05620.0677
50.04880.04360.04360.04160.05520.05370.06750.07460.06270.04540.05090.04670.04670.04490.0540
00.04900.02890.02930.02830.03940.04970.08350.15220.06740.04290.04530.02870.02870.02800.0362
−10.04800.02550.02600.02330.05100.05390.08880.16920.06220.04230.05280.03010.03050.02910.0571
−30.04830.04480.04840.02750.05360.05180.09380.15130.04510.04020.05320.04620.04900.03010.0534

Notes: This table presents finite-sample sizes in Case 1 with ϕ=0 for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX, and IVX-based bootstrap tests under 5% nominal level. RWB denotes the rejection rate for the Wald statistic by RWB. IVX1 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015). IVX2 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015) with possible negative Wald statistics included. Wild denotes the rejection rate for the residual wild bootstrap algorithm by Demetrescu et al. (2022). FWild denotes the rejection rate for the fixed regressor wild bootstrap algorithm by Demetrescu et al. (2022). Results are reported for different combinations of μ∈{0,0.1}, θ∈{−2,0,2}, T∈{100,250,500}⁠, and c∈{50,20,10,5,0,−1,−3}⁠. The average rejection rates are calculated over 10,000 replications.

In Table 2, we compare their empirical sizes when ϕ=0.5 in Case 1. RWB unifies all scenarios and still exhibits excellent performance in size control. On the other hand, IVX’s oversizing becomes even more salient, and increasing T only slightly shrinks the size distortion. When θ = 0 and c∈{0,−1,−3}⁠, the rejection rate by IVX2 is still substantially larger than that by IVX1. Wild and FWild produce reasonably good size control compared with IVX but are still less accurate than RWB in most scenarios. In particular, when c∈{1,3}⁠, FWild becomes prominently oversized when a non-zero intercept exists, although the size distortion is slightly alleviated when T = 500.

Table 2
Open in new tab

Case 1: Comparison of empirical sizes with ϕ=0.5

xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0.5
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.06020.07930.07930.07370.07340.05670.07680.07680.05670.07130.06210.07970.07970.07510.0730
200.05780.08060.08060.07090.07520.05380.06990.06990.04900.06220.05620.08470.08470.07530.0762
100.05630.08960.08960.07600.07940.05090.06700.06780.04460.05310.05380.08640.08640.07680.0759
50.05490.09410.09410.07800.07880.05140.07600.07980.05110.05290.05020.09450.09450.08020.0750
00.05410.10150.10200.07800.06710.05200.08910.14940.06010.04920.05600.09630.09690.07550.0664
−10.05910.10120.10320.07840.07500.05070.08130.18350.05060.03930.05980.10330.10510.07910.0747
−30.05320.11620.12940.05980.08880.05570.09990.23480.05190.04390.05390.11900.13270.06210.0894
250500.05260.07430.07430.07290.06910.05550.05820.05820.05110.05680.05090.07390.07390.07330.0694
200.05400.08130.08130.07790.07650.05020.06050.06050.05010.05480.05220.07930.07930.07490.0736
100.05050.08480.08480.07830.07500.05250.05930.05960.04520.05190.04880.08540.08540.07840.0741
50.04770.08970.08980.08430.07260.05350.06650.06970.05420.05380.05020.08670.08670.08060.0724
00.05570.09730.09750.08760.07350.04970.07990.12950.06230.04830.05640.08690.08710.07770.0669
−10.05130.09050.09120.07840.07940.05300.07970.15840.05950.04580.05730.08610.08740.07400.0748
−30.04960.09630.10460.06190.08570.05350.09680.18470.05110.04270.04840.09480.10180.05930.0849
500500.05060.07280.07280.07270.06850.05550.05700.05700.05470.05540.05160.06830.06830.06980.0630
200.05160.07800.07800.07670.07120.04840.05080.05080.04940.04990.05220.08020.08020.07710.0737
100.04910.08180.08180.07830.07250.05470.05550.05550.05040.05230.04790.08130.08130.07740.0717
50.04950.08770.08770.08370.07100.04540.05630.05820.04870.04940.04760.09040.09040.08530.0741
00.05520.08670.08680.07830.06750.05140.07460.10940.06000.04550.05410.07760.07780.06890.0656
−10.05380.07900.07920.07050.07050.04930.07140.12940.05420.04120.05270.07470.07540.06770.0737
−30.04690.08600.09140.05820.08340.05180.08790.14750.04390.03970.04640.08300.08820.05600.0742
Panel B: α=0.0066, μ=0.1
100500.06150.08430.08430.08080.07720.05520.07200.07200.05440.06500.06170.07970.07970.07550.0746
200.05440.08440.08440.07360.07640.05180.07120.07120.05110.06110.05470.08750.08750.07580.0779
100.05370.09030.09030.07630.08030.05230.07150.07230.04860.05420.05530.09030.09030.07820.0782
50.05090.09530.09540.08050.07960.05010.07990.08290.04970.05160.05180.09510.09510.08000.0789
00.06530.10670.10730.08430.07120.05270.08670.15390.05820.04620.05440.10730.10840.08520.0720
−10.06330.10330.10460.07840.08070.05470.08270.18160.05220.04400.05940.10480.10590.08220.0781
−30.05660.11600.13010.06030.09070.05630.09410.22140.04890.04250.05890.12590.14010.06580.1012
250500.05490.07500.07500.07450.07360.05430.06070.06070.05550.05890.05740.07170.07170.07150.0681
200.04710.07090.07090.06870.07120.04780.05640.05640.04860.05120.05360.07610.07610.07460.0774
100.05210.08550.08550.07930.08260.04950.06020.06060.05140.05130.05370.08400.08400.07890.0846
50.05000.08590.08590.08010.07940.04600.06340.06590.05020.04490.05170.08100.08100.07720.0794
00.05780.08180.08200.07230.06940.05250.08090.13490.06170.04520.05670.08370.08380.07500.0722
−10.05310.08370.08440.07390.08410.04820.08280.16730.05920.04470.05260.08250.08350.07180.0794
−30.04820.09360.10410.05630.08350.05310.09180.18210.04480.04050.04730.10050.10930.06010.0874
500500.04960.06480.06480.06580.06570.04900.05340.05340.05190.05230.05810.06550.06550.06600.0682
200.04720.06840.06840.06840.07400.05410.05970.05970.05460.05650.05040.06840.06840.06650.0730
100.05220.06840.06840.06550.07700.05110.06340.06380.05490.05530.04890.06870.06870.06610.0761
50.04470.07030.07030.06380.07540.04680.06300.06600.05400.04930.04730.06840.06840.06390.0750
00.05330.06330.06350.05880.06060.04930.07910.13120.06030.04310.04690.06250.06280.05830.0618
−10.04790.06700.06760.06160.07800.04940.07710.14370.05920.04600.05180.06190.06280.05630.0748
−30.05020.07380.07970.04660.06560.05180.09120.15090.04610.04310.04610.07810.08540.05070.0706
xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0.5
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.06020.07930.07930.07370.07340.05670.07680.07680.05670.07130.06210.07970.07970.07510.0730
200.05780.08060.08060.07090.07520.05380.06990.06990.04900.06220.05620.08470.08470.07530.0762
100.05630.08960.08960.07600.07940.05090.06700.06780.04460.05310.05380.08640.08640.07680.0759
50.05490.09410.09410.07800.07880.05140.07600.07980.05110.05290.05020.09450.09450.08020.0750
00.05410.10150.10200.07800.06710.05200.08910.14940.06010.04920.05600.09630.09690.07550.0664
−10.05910.10120.10320.07840.07500.05070.08130.18350.05060.03930.05980.10330.10510.07910.0747
−30.05320.11620.12940.05980.08880.05570.09990.23480.05190.04390.05390.11900.13270.06210.0894
250500.05260.07430.07430.07290.06910.05550.05820.05820.05110.05680.05090.07390.07390.07330.0694
200.05400.08130.08130.07790.07650.05020.06050.06050.05010.05480.05220.07930.07930.07490.0736
100.05050.08480.08480.07830.07500.05250.05930.05960.04520.05190.04880.08540.08540.07840.0741
50.04770.08970.08980.08430.07260.05350.06650.06970.05420.05380.05020.08670.08670.08060.0724
00.05570.09730.09750.08760.07350.04970.07990.12950.06230.04830.05640.08690.08710.07770.0669
−10.05130.09050.09120.07840.07940.05300.07970.15840.05950.04580.05730.08610.08740.07400.0748
−30.04960.09630.10460.06190.08570.05350.09680.18470.05110.04270.04840.09480.10180.05930.0849
500500.05060.07280.07280.07270.06850.05550.05700.05700.05470.05540.05160.06830.06830.06980.0630
200.05160.07800.07800.07670.07120.04840.05080.05080.04940.04990.05220.08020.08020.07710.0737
100.04910.08180.08180.07830.07250.05470.05550.05550.05040.05230.04790.08130.08130.07740.0717
50.04950.08770.08770.08370.07100.04540.05630.05820.04870.04940.04760.09040.09040.08530.0741
00.05520.08670.08680.07830.06750.05140.07460.10940.06000.04550.05410.07760.07780.06890.0656
−10.05380.07900.07920.07050.07050.04930.07140.12940.05420.04120.05270.07470.07540.06770.0737
−30.04690.08600.09140.05820.08340.05180.08790.14750.04390.03970.04640.08300.08820.05600.0742
Panel B: α=0.0066, μ=0.1
100500.06150.08430.08430.08080.07720.05520.07200.07200.05440.06500.06170.07970.07970.07550.0746
200.05440.08440.08440.07360.07640.05180.07120.07120.05110.06110.05470.08750.08750.07580.0779
100.05370.09030.09030.07630.08030.05230.07150.07230.04860.05420.05530.09030.09030.07820.0782
50.05090.09530.09540.08050.07960.05010.07990.08290.04970.05160.05180.09510.09510.08000.0789
00.06530.10670.10730.08430.07120.05270.08670.15390.05820.04620.05440.10730.10840.08520.0720
−10.06330.10330.10460.07840.08070.05470.08270.18160.05220.04400.05940.10480.10590.08220.0781
−30.05660.11600.13010.06030.09070.05630.09410.22140.04890.04250.05890.12590.14010.06580.1012
250500.05490.07500.07500.07450.07360.05430.06070.06070.05550.05890.05740.07170.07170.07150.0681
200.04710.07090.07090.06870.07120.04780.05640.05640.04860.05120.05360.07610.07610.07460.0774
100.05210.08550.08550.07930.08260.04950.06020.06060.05140.05130.05370.08400.08400.07890.0846
50.05000.08590.08590.08010.07940.04600.06340.06590.05020.04490.05170.08100.08100.07720.0794
00.05780.08180.08200.07230.06940.05250.08090.13490.06170.04520.05670.08370.08380.07500.0722
−10.05310.08370.08440.07390.08410.04820.08280.16730.05920.04470.05260.08250.08350.07180.0794
−30.04820.09360.10410.05630.08350.05310.09180.18210.04480.04050.04730.10050.10930.06010.0874
500500.04960.06480.06480.06580.06570.04900.05340.05340.05190.05230.05810.06550.06550.06600.0682
200.04720.06840.06840.06840.07400.05410.05970.05970.05460.05650.05040.06840.06840.06650.0730
100.05220.06840.06840.06550.07700.05110.06340.06380.05490.05530.04890.06870.06870.06610.0761
50.04470.07030.07030.06380.07540.04680.06300.06600.05400.04930.04730.06840.06840.06390.0750
00.05330.06330.06350.05880.06060.04930.07910.13120.06030.04310.04690.06250.06280.05830.0618
−10.04790.06700.06760.06160.07800.04940.07710.14370.05920.04600.05180.06190.06280.05630.0748
−30.05020.07380.07970.04660.06560.05180.09120.15090.04610.04310.04610.07810.08540.05070.0706

Notes: This table presents finite-sample sizes in Case 1 with ϕ=0.5 for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX, and IVX-based bootstrap tests under 5% nominal level. RWB denotes the rejection rate for the Wald statistic by RWB. IVX1 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015). IVX2 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015) with possible negative Wald statistics included. Wild denotes the rejection rate for the residual wild bootstrap algorithm by Demetrescu et al. (2022). FWild denotes the rejection rate for the fixed regressor wild bootstrap algorithm by Demetrescu et al. (2022). Results are reported for different combinations of μ∈{0,0.1}, θ∈{−2,0,2}, T∈{100,250,500}⁠, and c∈{50,20,10,5,0,−1,−3}⁠. The average rejection rates are calculated over 10,000 replications.

Table 2
Open in new tab

Case 1: Comparison of empirical sizes with ϕ=0.5

xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0.5
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.06020.07930.07930.07370.07340.05670.07680.07680.05670.07130.06210.07970.07970.07510.0730
200.05780.08060.08060.07090.07520.05380.06990.06990.04900.06220.05620.08470.08470.07530.0762
100.05630.08960.08960.07600.07940.05090.06700.06780.04460.05310.05380.08640.08640.07680.0759
50.05490.09410.09410.07800.07880.05140.07600.07980.05110.05290.05020.09450.09450.08020.0750
00.05410.10150.10200.07800.06710.05200.08910.14940.06010.04920.05600.09630.09690.07550.0664
−10.05910.10120.10320.07840.07500.05070.08130.18350.05060.03930.05980.10330.10510.07910.0747
−30.05320.11620.12940.05980.08880.05570.09990.23480.05190.04390.05390.11900.13270.06210.0894
250500.05260.07430.07430.07290.06910.05550.05820.05820.05110.05680.05090.07390.07390.07330.0694
200.05400.08130.08130.07790.07650.05020.06050.06050.05010.05480.05220.07930.07930.07490.0736
100.05050.08480.08480.07830.07500.05250.05930.05960.04520.05190.04880.08540.08540.07840.0741
50.04770.08970.08980.08430.07260.05350.06650.06970.05420.05380.05020.08670.08670.08060.0724
00.05570.09730.09750.08760.07350.04970.07990.12950.06230.04830.05640.08690.08710.07770.0669
−10.05130.09050.09120.07840.07940.05300.07970.15840.05950.04580.05730.08610.08740.07400.0748
−30.04960.09630.10460.06190.08570.05350.09680.18470.05110.04270.04840.09480.10180.05930.0849
500500.05060.07280.07280.07270.06850.05550.05700.05700.05470.05540.05160.06830.06830.06980.0630
200.05160.07800.07800.07670.07120.04840.05080.05080.04940.04990.05220.08020.08020.07710.0737
100.04910.08180.08180.07830.07250.05470.05550.05550.05040.05230.04790.08130.08130.07740.0717
50.04950.08770.08770.08370.07100.04540.05630.05820.04870.04940.04760.09040.09040.08530.0741
00.05520.08670.08680.07830.06750.05140.07460.10940.06000.04550.05410.07760.07780.06890.0656
−10.05380.07900.07920.07050.07050.04930.07140.12940.05420.04120.05270.07470.07540.06770.0737
−30.04690.08600.09140.05820.08340.05180.08790.14750.04390.03970.04640.08300.08820.05600.0742
Panel B: α=0.0066, μ=0.1
100500.06150.08430.08430.08080.07720.05520.07200.07200.05440.06500.06170.07970.07970.07550.0746
200.05440.08440.08440.07360.07640.05180.07120.07120.05110.06110.05470.08750.08750.07580.0779
100.05370.09030.09030.07630.08030.05230.07150.07230.04860.05420.05530.09030.09030.07820.0782
50.05090.09530.09540.08050.07960.05010.07990.08290.04970.05160.05180.09510.09510.08000.0789
00.06530.10670.10730.08430.07120.05270.08670.15390.05820.04620.05440.10730.10840.08520.0720
−10.06330.10330.10460.07840.08070.05470.08270.18160.05220.04400.05940.10480.10590.08220.0781
−30.05660.11600.13010.06030.09070.05630.09410.22140.04890.04250.05890.12590.14010.06580.1012
250500.05490.07500.07500.07450.07360.05430.06070.06070.05550.05890.05740.07170.07170.07150.0681
200.04710.07090.07090.06870.07120.04780.05640.05640.04860.05120.05360.07610.07610.07460.0774
100.05210.08550.08550.07930.08260.04950.06020.06060.05140.05130.05370.08400.08400.07890.0846
50.05000.08590.08590.08010.07940.04600.06340.06590.05020.04490.05170.08100.08100.07720.0794
00.05780.08180.08200.07230.06940.05250.08090.13490.06170.04520.05670.08370.08380.07500.0722
−10.05310.08370.08440.07390.08410.04820.08280.16730.05920.04470.05260.08250.08350.07180.0794
−30.04820.09360.10410.05630.08350.05310.09180.18210.04480.04050.04730.10050.10930.06010.0874
500500.04960.06480.06480.06580.06570.04900.05340.05340.05190.05230.05810.06550.06550.06600.0682
200.04720.06840.06840.06840.07400.05410.05970.05970.05460.05650.05040.06840.06840.06650.0730
100.05220.06840.06840.06550.07700.05110.06340.06380.05490.05530.04890.06870.06870.06610.0761
50.04470.07030.07030.06380.07540.04680.06300.06600.05400.04930.04730.06840.06840.06390.0750
00.05330.06330.06350.05880.06060.04930.07910.13120.06030.04310.04690.06250.06280.05830.0618
−10.04790.06700.06760.06160.07800.04940.07710.14370.05920.04600.05180.06190.06280.05630.0748
−30.05020.07380.07970.04660.06560.05180.09120.15090.04610.04310.04610.07810.08540.05070.0706
xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0.5
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.06020.07930.07930.07370.07340.05670.07680.07680.05670.07130.06210.07970.07970.07510.0730
200.05780.08060.08060.07090.07520.05380.06990.06990.04900.06220.05620.08470.08470.07530.0762
100.05630.08960.08960.07600.07940.05090.06700.06780.04460.05310.05380.08640.08640.07680.0759
50.05490.09410.09410.07800.07880.05140.07600.07980.05110.05290.05020.09450.09450.08020.0750
00.05410.10150.10200.07800.06710.05200.08910.14940.06010.04920.05600.09630.09690.07550.0664
−10.05910.10120.10320.07840.07500.05070.08130.18350.05060.03930.05980.10330.10510.07910.0747
−30.05320.11620.12940.05980.08880.05570.09990.23480.05190.04390.05390.11900.13270.06210.0894
250500.05260.07430.07430.07290.06910.05550.05820.05820.05110.05680.05090.07390.07390.07330.0694
200.05400.08130.08130.07790.07650.05020.06050.06050.05010.05480.05220.07930.07930.07490.0736
100.05050.08480.08480.07830.07500.05250.05930.05960.04520.05190.04880.08540.08540.07840.0741
50.04770.08970.08980.08430.07260.05350.06650.06970.05420.05380.05020.08670.08670.08060.0724
00.05570.09730.09750.08760.07350.04970.07990.12950.06230.04830.05640.08690.08710.07770.0669
−10.05130.09050.09120.07840.07940.05300.07970.15840.05950.04580.05730.08610.08740.07400.0748
−30.04960.09630.10460.06190.08570.05350.09680.18470.05110.04270.04840.09480.10180.05930.0849
500500.05060.07280.07280.07270.06850.05550.05700.05700.05470.05540.05160.06830.06830.06980.0630
200.05160.07800.07800.07670.07120.04840.05080.05080.04940.04990.05220.08020.08020.07710.0737
100.04910.08180.08180.07830.07250.05470.05550.05550.05040.05230.04790.08130.08130.07740.0717
50.04950.08770.08770.08370.07100.04540.05630.05820.04870.04940.04760.09040.09040.08530.0741
00.05520.08670.08680.07830.06750.05140.07460.10940.06000.04550.05410.07760.07780.06890.0656
−10.05380.07900.07920.07050.07050.04930.07140.12940.05420.04120.05270.07470.07540.06770.0737
−30.04690.08600.09140.05820.08340.05180.08790.14750.04390.03970.04640.08300.08820.05600.0742
Panel B: α=0.0066, μ=0.1
100500.06150.08430.08430.08080.07720.05520.07200.07200.05440.06500.06170.07970.07970.07550.0746
200.05440.08440.08440.07360.07640.05180.07120.07120.05110.06110.05470.08750.08750.07580.0779
100.05370.09030.09030.07630.08030.05230.07150.07230.04860.05420.05530.09030.09030.07820.0782
50.05090.09530.09540.08050.07960.05010.07990.08290.04970.05160.05180.09510.09510.08000.0789
00.06530.10670.10730.08430.07120.05270.08670.15390.05820.04620.05440.10730.10840.08520.0720
−10.06330.10330.10460.07840.08070.05470.08270.18160.05220.04400.05940.10480.10590.08220.0781
−30.05660.11600.13010.06030.09070.05630.09410.22140.04890.04250.05890.12590.14010.06580.1012
250500.05490.07500.07500.07450.07360.05430.06070.06070.05550.05890.05740.07170.07170.07150.0681
200.04710.07090.07090.06870.07120.04780.05640.05640.04860.05120.05360.07610.07610.07460.0774
100.05210.08550.08550.07930.08260.04950.06020.06060.05140.05130.05370.08400.08400.07890.0846
50.05000.08590.08590.08010.07940.04600.06340.06590.05020.04490.05170.08100.08100.07720.0794
00.05780.08180.08200.07230.06940.05250.08090.13490.06170.04520.05670.08370.08380.07500.0722
−10.05310.08370.08440.07390.08410.04820.08280.16730.05920.04470.05260.08250.08350.07180.0794
−30.04820.09360.10410.05630.08350.05310.09180.18210.04480.04050.04730.10050.10930.06010.0874
500500.04960.06480.06480.06580.06570.04900.05340.05340.05190.05230.05810.06550.06550.06600.0682
200.04720.06840.06840.06840.07400.05410.05970.05970.05460.05650.05040.06840.06840.06650.0730
100.05220.06840.06840.06550.07700.05110.06340.06380.05490.05530.04890.06870.06870.06610.0761
50.04470.07030.07030.06380.07540.04680.06300.06600.05400.04930.04730.06840.06840.06390.0750
00.05330.06330.06350.05880.06060.04930.07910.13120.06030.04310.04690.06250.06280.05830.0618
−10.04790.06700.06760.06160.07800.04940.07710.14370.05920.04600.05180.06190.06280.05630.0748
−30.05020.07380.07970.04660.06560.05180.09120.15090.04610.04310.04610.07810.08540.05070.0706

Notes: This table presents finite-sample sizes in Case 1 with ϕ=0.5 for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX, and IVX-based bootstrap tests under 5% nominal level. RWB denotes the rejection rate for the Wald statistic by RWB. IVX1 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015). IVX2 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015) with possible negative Wald statistics included. Wild denotes the rejection rate for the residual wild bootstrap algorithm by Demetrescu et al. (2022). FWild denotes the rejection rate for the fixed regressor wild bootstrap algorithm by Demetrescu et al. (2022). Results are reported for different combinations of μ∈{0,0.1}, θ∈{−2,0,2}, T∈{100,250,500}⁠, and c∈{50,20,10,5,0,−1,−3}⁠. The average rejection rates are calculated over 10,000 replications.

Results in Tables 3 and 4 compare RWB with IVX and the two IVX-based bootstrap methods in Case 2. We observe similar patterns as in Case 1, except that IVX and the two IVX-based bootstrap methods become remarkably undersized for highly persistent predictors with a non-zero intercept.7  Supplementary Appendix Tables A1 and A2 display results in Cases 3–8 when ϕ=0 and 0.5, respectively. In both tables, regardless of the value of intercept (μ = 0 in Panel A and μ=0.1 in Panel B), RWB exhibits excellent size control in all six cases. RWB is slightly oversized when T = 100, but the size distortion becomes smaller as the sample size increases. IVX, Wild, and FWild perform reasonably well in most cases when T = 500 and c≤0 in Panel A, but their size distortion is still notably larger than that of RWB. On the other hand, when μ=0.1⁠, Panel B shows that these competing methods become severely undersized in all six cases when the predictor is highly persistent, and increasing sample size does not meaningfully improve their size performance.

Table 3
Open in new tab

Case 2: Comparison of empirical sizes with ϕ=0

xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.05950.08250.08250.08340.07710.05050.05890.05890.04600.05530.05980.08390.08390.08610.0779
200.05310.09110.09110.08650.08410.04510.06080.06080.04470.05250.05220.08220.08220.08070.0758
100.05060.09210.09210.08430.08330.04340.06290.06420.04700.04830.05090.08980.08980.08280.0824
50.05200.09180.09180.08400.08170.05010.06630.07440.04860.04300.05360.09040.09040.08190.0768
00.05990.09280.09290.07960.06740.05430.06850.16560.05180.03770.05460.08490.08510.07240.0663
−10.05340.08610.08660.07210.07430.05540.06270.19950.04560.03590.05650.08750.08810.07110.0720
−30.04810.10140.10370.05600.09860.05680.06300.25700.03700.03040.04730.09910.10310.05780.0952
250500.05040.07340.07340.07390.07120.05410.04690.04690.04140.04780.05360.07230.07230.07450.0713
200.05090.07940.07940.07840.07770.05790.04960.04970.04330.04720.04800.07910.07910.08000.0760
100.04670.08410.08410.08100.07850.05400.04870.05000.04040.03940.04550.08460.08460.08190.0789
50.04540.08490.08490.08170.07630.04770.05150.05730.04200.03700.04510.08480.08480.08050.0778
00.05530.07670.07680.07190.06520.04990.05830.14280.04720.03000.05560.07360.07380.06820.0627
−10.05850.07330.07350.06600.06840.05100.05210.16800.04220.02980.06080.07170.07220.06380.0672
−30.04930.08610.08830.05580.08960.05450.06160.21480.03840.02890.05230.08070.08380.05160.0872
500500.05000.07210.07210.07400.07000.05200.04430.04430.04300.04420.04930.07660.07660.07590.0757
200.04680.08130.08130.08350.07510.04640.04070.04070.03890.03950.04860.08050.08050.08170.0781
100.04970.08980.08980.08830.08300.04600.04670.04780.04120.03700.04700.07870.07870.07800.0752
50.04510.08530.08530.08340.07590.04670.04800.05370.04270.03240.04420.08640.08640.08270.0786
00.05470.07640.07650.07240.06750.05220.05610.12270.04680.02880.05770.07360.07370.07070.0651
−10.05480.07080.07090.06820.06760.05230.05720.16050.04440.03000.05510.06920.06950.06150.0708
−30.04980.07280.07500.05100.07620.05540.07020.19110.04140.03160.05240.07880.08120.05650.0817
Panel B: α=0.0066, μ=0.1
100500.06140.06010.06010.06050.05840.05230.06680.06680.05020.04920.06110.07580.07580.07550.0544
200.05720.03760.03760.03580.03210.04800.07830.09000.06540.03660.05570.06160.06160.06100.0211
100.04680.02290.02310.02110.01510.04150.08380.14240.07090.03650.04820.03310.03310.03110.0074
50.04360.01490.01530.01340.01270.04700.06750.22290.05530.03610.04570.01210.01210.01120.0044
00.05270.01150.01350.00530.03700.05490.04330.29650.02930.03490.05130.00490.00600.00330.0395
−10.05580.01370.01610.00370.04200.05430.04850.29500.03100.03140.05440.01450.01540.00780.0691
−30.05340.03170.03590.00840.05270.05480.06430.28740.03190.03020.05570.04520.05150.00850.0640
250500.05360.04900.04900.05110.03290.05530.06230.06390.05580.04030.05330.04700.04700.04720.0376
200.04890.03230.03230.03270.00750.04810.08090.12030.07750.03510.04690.01950.01950.02110.0097
100.04520.01630.01630.01540.01370.04830.07150.19310.06560.03110.04610.00260.00270.00240.0023
50.05280.01110.01160.00920.05930.05290.05170.25670.04580.03350.04970.00090.00110.00090.0033
00.05240.01010.01280.00520.05720.05240.04100.27890.02970.02940.05230.00300.00410.00180.0488
−10.05540.01130.01480.00320.06030.04990.04510.26730.02760.02790.05290.01070.01230.00530.0627
−30.05130.02430.02850.00700.04850.05420.06780.24300.03110.02710.05220.03270.03720.00840.0558
500500.04680.04530.04530.04740.01330.04400.07030.07930.06920.03430.04750.02030.02030.02070.0266
200.04610.02110.02110.02170.00980.04790.07560.15070.07480.02980.04510.00270.00270.00280.0057
100.04480.01590.01630.01520.07300.04510.05730.21550.05630.03200.04460.00050.00050.00060.0019
50.04430.01380.01650.01300.14520.05030.04480.28510.04100.03480.04740.00030.00040.00020.0043
00.05140.00880.01150.00480.06620.05750.04440.26450.03090.02930.05170.00320.00390.00230.0534
−10.05310.01070.01400.00410.05540.05000.04810.23740.03060.02610.05450.00860.01020.00410.0596
−30.05330.02490.02890.00880.03870.05640.06830.20410.02830.02480.05210.02800.03200.00970.0447
xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.05950.08250.08250.08340.07710.05050.05890.05890.04600.05530.05980.08390.08390.08610.0779
200.05310.09110.09110.08650.08410.04510.06080.06080.04470.05250.05220.08220.08220.08070.0758
100.05060.09210.09210.08430.08330.04340.06290.06420.04700.04830.05090.08980.08980.08280.0824
50.05200.09180.09180.08400.08170.05010.06630.07440.04860.04300.05360.09040.09040.08190.0768
00.05990.09280.09290.07960.06740.05430.06850.16560.05180.03770.05460.08490.08510.07240.0663
−10.05340.08610.08660.07210.07430.05540.06270.19950.04560.03590.05650.08750.08810.07110.0720
−30.04810.10140.10370.05600.09860.05680.06300.25700.03700.03040.04730.09910.10310.05780.0952
250500.05040.07340.07340.07390.07120.05410.04690.04690.04140.04780.05360.07230.07230.07450.0713
200.05090.07940.07940.07840.07770.05790.04960.04970.04330.04720.04800.07910.07910.08000.0760
100.04670.08410.08410.08100.07850.05400.04870.05000.04040.03940.04550.08460.08460.08190.0789
50.04540.08490.08490.08170.07630.04770.05150.05730.04200.03700.04510.08480.08480.08050.0778
00.05530.07670.07680.07190.06520.04990.05830.14280.04720.03000.05560.07360.07380.06820.0627
−10.05850.07330.07350.06600.06840.05100.05210.16800.04220.02980.06080.07170.07220.06380.0672
−30.04930.08610.08830.05580.08960.05450.06160.21480.03840.02890.05230.08070.08380.05160.0872
500500.05000.07210.07210.07400.07000.05200.04430.04430.04300.04420.04930.07660.07660.07590.0757
200.04680.08130.08130.08350.07510.04640.04070.04070.03890.03950.04860.08050.08050.08170.0781
100.04970.08980.08980.08830.08300.04600.04670.04780.04120.03700.04700.07870.07870.07800.0752
50.04510.08530.08530.08340.07590.04670.04800.05370.04270.03240.04420.08640.08640.08270.0786
00.05470.07640.07650.07240.06750.05220.05610.12270.04680.02880.05770.07360.07370.07070.0651
−10.05480.07080.07090.06820.06760.05230.05720.16050.04440.03000.05510.06920.06950.06150.0708
−30.04980.07280.07500.05100.07620.05540.07020.19110.04140.03160.05240.07880.08120.05650.0817
Panel B: α=0.0066, μ=0.1
100500.06140.06010.06010.06050.05840.05230.06680.06680.05020.04920.06110.07580.07580.07550.0544
200.05720.03760.03760.03580.03210.04800.07830.09000.06540.03660.05570.06160.06160.06100.0211
100.04680.02290.02310.02110.01510.04150.08380.14240.07090.03650.04820.03310.03310.03110.0074
50.04360.01490.01530.01340.01270.04700.06750.22290.05530.03610.04570.01210.01210.01120.0044
00.05270.01150.01350.00530.03700.05490.04330.29650.02930.03490.05130.00490.00600.00330.0395
−10.05580.01370.01610.00370.04200.05430.04850.29500.03100.03140.05440.01450.01540.00780.0691
−30.05340.03170.03590.00840.05270.05480.06430.28740.03190.03020.05570.04520.05150.00850.0640
250500.05360.04900.04900.05110.03290.05530.06230.06390.05580.04030.05330.04700.04700.04720.0376
200.04890.03230.03230.03270.00750.04810.08090.12030.07750.03510.04690.01950.01950.02110.0097
100.04520.01630.01630.01540.01370.04830.07150.19310.06560.03110.04610.00260.00270.00240.0023
50.05280.01110.01160.00920.05930.05290.05170.25670.04580.03350.04970.00090.00110.00090.0033
00.05240.01010.01280.00520.05720.05240.04100.27890.02970.02940.05230.00300.00410.00180.0488
−10.05540.01130.01480.00320.06030.04990.04510.26730.02760.02790.05290.01070.01230.00530.0627
−30.05130.02430.02850.00700.04850.05420.06780.24300.03110.02710.05220.03270.03720.00840.0558
500500.04680.04530.04530.04740.01330.04400.07030.07930.06920.03430.04750.02030.02030.02070.0266
200.04610.02110.02110.02170.00980.04790.07560.15070.07480.02980.04510.00270.00270.00280.0057
100.04480.01590.01630.01520.07300.04510.05730.21550.05630.03200.04460.00050.00050.00060.0019
50.04430.01380.01650.01300.14520.05030.04480.28510.04100.03480.04740.00030.00040.00020.0043
00.05140.00880.01150.00480.06620.05750.04440.26450.03090.02930.05170.00320.00390.00230.0534
−10.05310.01070.01400.00410.05540.05000.04810.23740.03060.02610.05450.00860.01020.00410.0596
−30.05330.02490.02890.00880.03870.05640.06830.20410.02830.02480.05210.02800.03200.00970.0447

Notes: This table presents finite-sample sizes in Case 2 with ϕ=0 for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX, and IVX-based bootstrap tests under 5% nominal level. RWB denotes the rejection rate for the Wald statistic by RWB. IVX1 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015). IVX2 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015) with possible negative Wald statistics included. Wild denotes the rejection rate for the residual wild bootstrap algorithm by Demetrescu et al. (2022). FWild denotes the rejection rate for the fixed regressor wild bootstrap algorithm by Demetrescu et al. (2022). Results are reported for different combinations of μ∈{0,0.1}, θ∈{−2,0,2}, T∈{100,250,500}⁠, and c∈{50,20,10,5,0,−1,−3}⁠. The average rejection rates are calculated over 10,000 replications.

Table 3
Open in new tab

Case 2: Comparison of empirical sizes with ϕ=0

xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.05950.08250.08250.08340.07710.05050.05890.05890.04600.05530.05980.08390.08390.08610.0779
200.05310.09110.09110.08650.08410.04510.06080.06080.04470.05250.05220.08220.08220.08070.0758
100.05060.09210.09210.08430.08330.04340.06290.06420.04700.04830.05090.08980.08980.08280.0824
50.05200.09180.09180.08400.08170.05010.06630.07440.04860.04300.05360.09040.09040.08190.0768
00.05990.09280.09290.07960.06740.05430.06850.16560.05180.03770.05460.08490.08510.07240.0663
−10.05340.08610.08660.07210.07430.05540.06270.19950.04560.03590.05650.08750.08810.07110.0720
−30.04810.10140.10370.05600.09860.05680.06300.25700.03700.03040.04730.09910.10310.05780.0952
250500.05040.07340.07340.07390.07120.05410.04690.04690.04140.04780.05360.07230.07230.07450.0713
200.05090.07940.07940.07840.07770.05790.04960.04970.04330.04720.04800.07910.07910.08000.0760
100.04670.08410.08410.08100.07850.05400.04870.05000.04040.03940.04550.08460.08460.08190.0789
50.04540.08490.08490.08170.07630.04770.05150.05730.04200.03700.04510.08480.08480.08050.0778
00.05530.07670.07680.07190.06520.04990.05830.14280.04720.03000.05560.07360.07380.06820.0627
−10.05850.07330.07350.06600.06840.05100.05210.16800.04220.02980.06080.07170.07220.06380.0672
−30.04930.08610.08830.05580.08960.05450.06160.21480.03840.02890.05230.08070.08380.05160.0872
500500.05000.07210.07210.07400.07000.05200.04430.04430.04300.04420.04930.07660.07660.07590.0757
200.04680.08130.08130.08350.07510.04640.04070.04070.03890.03950.04860.08050.08050.08170.0781
100.04970.08980.08980.08830.08300.04600.04670.04780.04120.03700.04700.07870.07870.07800.0752
50.04510.08530.08530.08340.07590.04670.04800.05370.04270.03240.04420.08640.08640.08270.0786
00.05470.07640.07650.07240.06750.05220.05610.12270.04680.02880.05770.07360.07370.07070.0651
−10.05480.07080.07090.06820.06760.05230.05720.16050.04440.03000.05510.06920.06950.06150.0708
−30.04980.07280.07500.05100.07620.05540.07020.19110.04140.03160.05240.07880.08120.05650.0817
Panel B: α=0.0066, μ=0.1
100500.06140.06010.06010.06050.05840.05230.06680.06680.05020.04920.06110.07580.07580.07550.0544
200.05720.03760.03760.03580.03210.04800.07830.09000.06540.03660.05570.06160.06160.06100.0211
100.04680.02290.02310.02110.01510.04150.08380.14240.07090.03650.04820.03310.03310.03110.0074
50.04360.01490.01530.01340.01270.04700.06750.22290.05530.03610.04570.01210.01210.01120.0044
00.05270.01150.01350.00530.03700.05490.04330.29650.02930.03490.05130.00490.00600.00330.0395
−10.05580.01370.01610.00370.04200.05430.04850.29500.03100.03140.05440.01450.01540.00780.0691
−30.05340.03170.03590.00840.05270.05480.06430.28740.03190.03020.05570.04520.05150.00850.0640
250500.05360.04900.04900.05110.03290.05530.06230.06390.05580.04030.05330.04700.04700.04720.0376
200.04890.03230.03230.03270.00750.04810.08090.12030.07750.03510.04690.01950.01950.02110.0097
100.04520.01630.01630.01540.01370.04830.07150.19310.06560.03110.04610.00260.00270.00240.0023
50.05280.01110.01160.00920.05930.05290.05170.25670.04580.03350.04970.00090.00110.00090.0033
00.05240.01010.01280.00520.05720.05240.04100.27890.02970.02940.05230.00300.00410.00180.0488
−10.05540.01130.01480.00320.06030.04990.04510.26730.02760.02790.05290.01070.01230.00530.0627
−30.05130.02430.02850.00700.04850.05420.06780.24300.03110.02710.05220.03270.03720.00840.0558
500500.04680.04530.04530.04740.01330.04400.07030.07930.06920.03430.04750.02030.02030.02070.0266
200.04610.02110.02110.02170.00980.04790.07560.15070.07480.02980.04510.00270.00270.00280.0057
100.04480.01590.01630.01520.07300.04510.05730.21550.05630.03200.04460.00050.00050.00060.0019
50.04430.01380.01650.01300.14520.05030.04480.28510.04100.03480.04740.00030.00040.00020.0043
00.05140.00880.01150.00480.06620.05750.04440.26450.03090.02930.05170.00320.00390.00230.0534
−10.05310.01070.01400.00410.05540.05000.04810.23740.03060.02610.05450.00860.01020.00410.0596
−30.05330.02490.02890.00880.03870.05640.06830.20410.02830.02480.05210.02800.03200.00970.0447
xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.05950.08250.08250.08340.07710.05050.05890.05890.04600.05530.05980.08390.08390.08610.0779
200.05310.09110.09110.08650.08410.04510.06080.06080.04470.05250.05220.08220.08220.08070.0758
100.05060.09210.09210.08430.08330.04340.06290.06420.04700.04830.05090.08980.08980.08280.0824
50.05200.09180.09180.08400.08170.05010.06630.07440.04860.04300.05360.09040.09040.08190.0768
00.05990.09280.09290.07960.06740.05430.06850.16560.05180.03770.05460.08490.08510.07240.0663
−10.05340.08610.08660.07210.07430.05540.06270.19950.04560.03590.05650.08750.08810.07110.0720
−30.04810.10140.10370.05600.09860.05680.06300.25700.03700.03040.04730.09910.10310.05780.0952
250500.05040.07340.07340.07390.07120.05410.04690.04690.04140.04780.05360.07230.07230.07450.0713
200.05090.07940.07940.07840.07770.05790.04960.04970.04330.04720.04800.07910.07910.08000.0760
100.04670.08410.08410.08100.07850.05400.04870.05000.04040.03940.04550.08460.08460.08190.0789
50.04540.08490.08490.08170.07630.04770.05150.05730.04200.03700.04510.08480.08480.08050.0778
00.05530.07670.07680.07190.06520.04990.05830.14280.04720.03000.05560.07360.07380.06820.0627
−10.05850.07330.07350.06600.06840.05100.05210.16800.04220.02980.06080.07170.07220.06380.0672
−30.04930.08610.08830.05580.08960.05450.06160.21480.03840.02890.05230.08070.08380.05160.0872
500500.05000.07210.07210.07400.07000.05200.04430.04430.04300.04420.04930.07660.07660.07590.0757
200.04680.08130.08130.08350.07510.04640.04070.04070.03890.03950.04860.08050.08050.08170.0781
100.04970.08980.08980.08830.08300.04600.04670.04780.04120.03700.04700.07870.07870.07800.0752
50.04510.08530.08530.08340.07590.04670.04800.05370.04270.03240.04420.08640.08640.08270.0786
00.05470.07640.07650.07240.06750.05220.05610.12270.04680.02880.05770.07360.07370.07070.0651
−10.05480.07080.07090.06820.06760.05230.05720.16050.04440.03000.05510.06920.06950.06150.0708
−30.04980.07280.07500.05100.07620.05540.07020.19110.04140.03160.05240.07880.08120.05650.0817
Panel B: α=0.0066, μ=0.1
100500.06140.06010.06010.06050.05840.05230.06680.06680.05020.04920.06110.07580.07580.07550.0544
200.05720.03760.03760.03580.03210.04800.07830.09000.06540.03660.05570.06160.06160.06100.0211
100.04680.02290.02310.02110.01510.04150.08380.14240.07090.03650.04820.03310.03310.03110.0074
50.04360.01490.01530.01340.01270.04700.06750.22290.05530.03610.04570.01210.01210.01120.0044
00.05270.01150.01350.00530.03700.05490.04330.29650.02930.03490.05130.00490.00600.00330.0395
−10.05580.01370.01610.00370.04200.05430.04850.29500.03100.03140.05440.01450.01540.00780.0691
−30.05340.03170.03590.00840.05270.05480.06430.28740.03190.03020.05570.04520.05150.00850.0640
250500.05360.04900.04900.05110.03290.05530.06230.06390.05580.04030.05330.04700.04700.04720.0376
200.04890.03230.03230.03270.00750.04810.08090.12030.07750.03510.04690.01950.01950.02110.0097
100.04520.01630.01630.01540.01370.04830.07150.19310.06560.03110.04610.00260.00270.00240.0023
50.05280.01110.01160.00920.05930.05290.05170.25670.04580.03350.04970.00090.00110.00090.0033
00.05240.01010.01280.00520.05720.05240.04100.27890.02970.02940.05230.00300.00410.00180.0488
−10.05540.01130.01480.00320.06030.04990.04510.26730.02760.02790.05290.01070.01230.00530.0627
−30.05130.02430.02850.00700.04850.05420.06780.24300.03110.02710.05220.03270.03720.00840.0558
500500.04680.04530.04530.04740.01330.04400.07030.07930.06920.03430.04750.02030.02030.02070.0266
200.04610.02110.02110.02170.00980.04790.07560.15070.07480.02980.04510.00270.00270.00280.0057
100.04480.01590.01630.01520.07300.04510.05730.21550.05630.03200.04460.00050.00050.00060.0019
50.04430.01380.01650.01300.14520.05030.04480.28510.04100.03480.04740.00030.00040.00020.0043
00.05140.00880.01150.00480.06620.05750.04440.26450.03090.02930.05170.00320.00390.00230.0534
−10.05310.01070.01400.00410.05540.05000.04810.23740.03060.02610.05450.00860.01020.00410.0596
−30.05330.02490.02890.00880.03870.05640.06830.20410.02830.02480.05210.02800.03200.00970.0447

Notes: This table presents finite-sample sizes in Case 2 with ϕ=0 for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX, and IVX-based bootstrap tests under 5% nominal level. RWB denotes the rejection rate for the Wald statistic by RWB. IVX1 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015). IVX2 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015) with possible negative Wald statistics included. Wild denotes the rejection rate for the residual wild bootstrap algorithm by Demetrescu et al. (2022). FWild denotes the rejection rate for the fixed regressor wild bootstrap algorithm by Demetrescu et al. (2022). Results are reported for different combinations of μ∈{0,0.1}, θ∈{−2,0,2}, T∈{100,250,500}⁠, and c∈{50,20,10,5,0,−1,−3}⁠. The average rejection rates are calculated over 10,000 replications.

Table 4
Open in new tab

Case 2: Comparison of empirical sizes with ϕ=0.5

xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0.5
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.05660.08140.08140.07780.07660.04890.06370.06370.04800.06150.05910.07980.07980.07650.0764
200.04870.08160.08160.07190.07660.05860.05530.05560.03950.04770.05290.08920.08920.07780.0820
100.04660.09020.09020.07760.08370.05830.05870.06050.03730.04250.05010.08790.08790.07580.0825
50.04940.09130.09130.07570.07690.05170.06820.07830.04330.04330.05110.09610.09610.07940.0806
00.06020.10150.10220.07980.07530.05560.06850.17100.04760.03800.06230.10540.10580.08050.0786
−10.06350.10950.11180.08200.07810.05310.06220.20660.04210.03620.06070.10790.10990.07840.0804
−30.04450.12020.13970.06270.09480.05340.06130.25310.03790.03180.04810.13250.14820.06790.1019
250500.05560.07700.07700.07580.07290.05240.04780.04780.04260.04780.05130.07930.07930.07950.0774
200.04770.08000.08000.07690.07710.05490.05050.05060.04160.04440.04640.08180.08180.07950.0797
100.04530.08910.08910.08320.08310.05180.04660.04840.03650.03900.04520.08930.08930.08460.0827
50.04510.09030.09030.08250.07910.04720.05490.06280.03890.03700.04420.09450.09450.08620.0813
00.05190.09310.09330.08170.07430.05360.06470.15020.04810.03880.05400.09170.09210.08010.0717
−10.05270.08480.08620.07210.07670.05530.05330.17160.04050.03220.05280.09060.09180.07800.0807
−30.05060.10530.11690.06290.09080.05280.06240.22390.03610.02890.05210.10650.11640.06280.0909
500500.04680.07510.07510.07470.07360.04420.04870.04870.04730.04980.04820.07370.07370.07490.0701
200.04350.08350.08350.08360.07840.04400.03800.03810.03430.03710.04590.08110.08110.07920.0776
100.04860.08820.08820.08420.07790.04630.04540.04620.03710.03870.04420.07980.07980.07650.0747
50.04520.09490.09490.08970.08680.04490.04840.05400.03740.03270.04320.08400.08400.08000.0759
00.04930.08350.08370.07580.07270.05120.06100.13620.04700.03280.04800.07880.07900.07220.0707
−10.05020.08180.08270.07200.08070.05340.05810.15360.04510.03160.04970.08430.08540.07580.0832
−30.04810.09400.10170.05690.08310.05100.06420.18560.03410.02770.05080.08960.09710.05550.0779
Panel B: α=0.0066, μ=0.1
100500.05560.06880.06880.06400.06170.04960.07070.07080.05290.05720.05730.08080.08080.07730.0657
200.05450.05720.05720.04950.04530.04280.07680.08130.06010.04750.05100.07850.07850.07150.0495
100.05170.04710.04720.03830.03230.04150.08060.10860.06110.03340.05120.06910.06910.05900.0285
50.04680.03400.03450.02600.02170.04560.07140.16750.05260.03520.04750.05450.05450.04450.0158
00.04740.02470.03000.01380.03170.04540.05500.26980.03760.03760.04320.02580.02840.01700.0397
−10.04740.02350.03240.00910.03670.04250.05380.27400.03600.03440.04520.03870.04400.02120.0668
−30.04700.04890.06460.01800.05160.04620.06200.27730.03290.02870.04340.06890.09160.01600.0625
250500.04950.06450.06450.06500.05020.04330.06080.06150.05540.04710.05130.06160.06160.06200.0616
200.04530.05240.05240.05000.02320.04800.07080.08230.06530.03350.04600.04440.04440.04270.0355
100.04250.03720.03750.03300.01190.04750.07390.12620.06560.02990.04630.02360.02370.02280.0109
50.04670.02870.03000.02430.02590.04280.06610.20250.05600.03120.04840.01030.01060.00950.0043
00.05090.01270.01910.00590.04150.04490.04270.26210.03210.02980.04520.00790.01000.00600.0439
−10.05590.01610.02240.00430.04750.05150.04680.26290.03010.02770.04990.01730.02320.00990.0572
−30.04950.03070.04220.00900.04010.05400.06520.24110.03430.02910.04920.04530.05920.00910.0494
500500.04820.07130.07130.07210.03580.04620.05520.05760.05440.04040.04870.04010.04010.04060.0506
200.04660.05180.05190.04980.01130.04830.06870.09420.06590.03110.04460.01640.01640.01660.0215
100.04910.03280.03360.03150.02410.05330.06980.15920.06570.03200.04700.00500.00500.00490.0060
50.05000.02220.02580.01960.07450.05180.05190.22980.04650.02820.05120.00120.00130.00120.0022
00.05530.01330.02000.00530.05670.05170.03990.24230.02950.02710.05260.00620.00890.00510.0456
−10.05190.01500.02270.00440.04480.05250.04840.23020.03120.02550.05050.01640.02100.00770.0532
−30.05230.02860.03740.00680.03600.05680.07330.20470.03230.02670.05350.03820.04960.01020.0387
xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0.5
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.05660.08140.08140.07780.07660.04890.06370.06370.04800.06150.05910.07980.07980.07650.0764
200.04870.08160.08160.07190.07660.05860.05530.05560.03950.04770.05290.08920.08920.07780.0820
100.04660.09020.09020.07760.08370.05830.05870.06050.03730.04250.05010.08790.08790.07580.0825
50.04940.09130.09130.07570.07690.05170.06820.07830.04330.04330.05110.09610.09610.07940.0806
00.06020.10150.10220.07980.07530.05560.06850.17100.04760.03800.06230.10540.10580.08050.0786
−10.06350.10950.11180.08200.07810.05310.06220.20660.04210.03620.06070.10790.10990.07840.0804
−30.04450.12020.13970.06270.09480.05340.06130.25310.03790.03180.04810.13250.14820.06790.1019
250500.05560.07700.07700.07580.07290.05240.04780.04780.04260.04780.05130.07930.07930.07950.0774
200.04770.08000.08000.07690.07710.05490.05050.05060.04160.04440.04640.08180.08180.07950.0797
100.04530.08910.08910.08320.08310.05180.04660.04840.03650.03900.04520.08930.08930.08460.0827
50.04510.09030.09030.08250.07910.04720.05490.06280.03890.03700.04420.09450.09450.08620.0813
00.05190.09310.09330.08170.07430.05360.06470.15020.04810.03880.05400.09170.09210.08010.0717
−10.05270.08480.08620.07210.07670.05530.05330.17160.04050.03220.05280.09060.09180.07800.0807
−30.05060.10530.11690.06290.09080.05280.06240.22390.03610.02890.05210.10650.11640.06280.0909
500500.04680.07510.07510.07470.07360.04420.04870.04870.04730.04980.04820.07370.07370.07490.0701
200.04350.08350.08350.08360.07840.04400.03800.03810.03430.03710.04590.08110.08110.07920.0776
100.04860.08820.08820.08420.07790.04630.04540.04620.03710.03870.04420.07980.07980.07650.0747
50.04520.09490.09490.08970.08680.04490.04840.05400.03740.03270.04320.08400.08400.08000.0759
00.04930.08350.08370.07580.07270.05120.06100.13620.04700.03280.04800.07880.07900.07220.0707
−10.05020.08180.08270.07200.08070.05340.05810.15360.04510.03160.04970.08430.08540.07580.0832
−30.04810.09400.10170.05690.08310.05100.06420.18560.03410.02770.05080.08960.09710.05550.0779
Panel B: α=0.0066, μ=0.1
100500.05560.06880.06880.06400.06170.04960.07070.07080.05290.05720.05730.08080.08080.07730.0657
200.05450.05720.05720.04950.04530.04280.07680.08130.06010.04750.05100.07850.07850.07150.0495
100.05170.04710.04720.03830.03230.04150.08060.10860.06110.03340.05120.06910.06910.05900.0285
50.04680.03400.03450.02600.02170.04560.07140.16750.05260.03520.04750.05450.05450.04450.0158
00.04740.02470.03000.01380.03170.04540.05500.26980.03760.03760.04320.02580.02840.01700.0397
−10.04740.02350.03240.00910.03670.04250.05380.27400.03600.03440.04520.03870.04400.02120.0668
−30.04700.04890.06460.01800.05160.04620.06200.27730.03290.02870.04340.06890.09160.01600.0625
250500.04950.06450.06450.06500.05020.04330.06080.06150.05540.04710.05130.06160.06160.06200.0616
200.04530.05240.05240.05000.02320.04800.07080.08230.06530.03350.04600.04440.04440.04270.0355
100.04250.03720.03750.03300.01190.04750.07390.12620.06560.02990.04630.02360.02370.02280.0109
50.04670.02870.03000.02430.02590.04280.06610.20250.05600.03120.04840.01030.01060.00950.0043
00.05090.01270.01910.00590.04150.04490.04270.26210.03210.02980.04520.00790.01000.00600.0439
−10.05590.01610.02240.00430.04750.05150.04680.26290.03010.02770.04990.01730.02320.00990.0572
−30.04950.03070.04220.00900.04010.05400.06520.24110.03430.02910.04920.04530.05920.00910.0494
500500.04820.07130.07130.07210.03580.04620.05520.05760.05440.04040.04870.04010.04010.04060.0506
200.04660.05180.05190.04980.01130.04830.06870.09420.06590.03110.04460.01640.01640.01660.0215
100.04910.03280.03360.03150.02410.05330.06980.15920.06570.03200.04700.00500.00500.00490.0060
50.05000.02220.02580.01960.07450.05180.05190.22980.04650.02820.05120.00120.00130.00120.0022
00.05530.01330.02000.00530.05670.05170.03990.24230.02950.02710.05260.00620.00890.00510.0456
−10.05190.01500.02270.00440.04480.05250.04840.23020.03120.02550.05050.01640.02100.00770.0532
−30.05230.02860.03740.00680.03600.05680.07330.20470.03230.02670.05350.03820.04960.01020.0387

Notes: This table presents finite-sample sizes in Case 2 with ϕ=0.5 for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX, and IVX-based bootstrap tests under 5% nominal level. RWB denotes the rejection rate for the Wald statistic by RWB. IVX1 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015). IVX2 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015) with possible negative Wald statistics included. Wild denotes the rejection rate for the residual wild bootstrap algorithm by Demetrescu et al. (2022). FWild denotes the rejection rate for the fixed regressor wild bootstrap algorithm by Demetrescu et al. (2022). Results are reported for different combinations of μ∈{0,0.1}, θ∈{−2,0,2}, T∈{100,250,500}⁠, and c∈{50,20,10,5,0,−1,−3}⁠. The average rejection rates are calculated over 10,000 replications.

Table 4
Open in new tab

Case 2: Comparison of empirical sizes with ϕ=0.5

xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0.5
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.05660.08140.08140.07780.07660.04890.06370.06370.04800.06150.05910.07980.07980.07650.0764
200.04870.08160.08160.07190.07660.05860.05530.05560.03950.04770.05290.08920.08920.07780.0820
100.04660.09020.09020.07760.08370.05830.05870.06050.03730.04250.05010.08790.08790.07580.0825
50.04940.09130.09130.07570.07690.05170.06820.07830.04330.04330.05110.09610.09610.07940.0806
00.06020.10150.10220.07980.07530.05560.06850.17100.04760.03800.06230.10540.10580.08050.0786
−10.06350.10950.11180.08200.07810.05310.06220.20660.04210.03620.06070.10790.10990.07840.0804
−30.04450.12020.13970.06270.09480.05340.06130.25310.03790.03180.04810.13250.14820.06790.1019
250500.05560.07700.07700.07580.07290.05240.04780.04780.04260.04780.05130.07930.07930.07950.0774
200.04770.08000.08000.07690.07710.05490.05050.05060.04160.04440.04640.08180.08180.07950.0797
100.04530.08910.08910.08320.08310.05180.04660.04840.03650.03900.04520.08930.08930.08460.0827
50.04510.09030.09030.08250.07910.04720.05490.06280.03890.03700.04420.09450.09450.08620.0813
00.05190.09310.09330.08170.07430.05360.06470.15020.04810.03880.05400.09170.09210.08010.0717
−10.05270.08480.08620.07210.07670.05530.05330.17160.04050.03220.05280.09060.09180.07800.0807
−30.05060.10530.11690.06290.09080.05280.06240.22390.03610.02890.05210.10650.11640.06280.0909
500500.04680.07510.07510.07470.07360.04420.04870.04870.04730.04980.04820.07370.07370.07490.0701
200.04350.08350.08350.08360.07840.04400.03800.03810.03430.03710.04590.08110.08110.07920.0776
100.04860.08820.08820.08420.07790.04630.04540.04620.03710.03870.04420.07980.07980.07650.0747
50.04520.09490.09490.08970.08680.04490.04840.05400.03740.03270.04320.08400.08400.08000.0759
00.04930.08350.08370.07580.07270.05120.06100.13620.04700.03280.04800.07880.07900.07220.0707
−10.05020.08180.08270.07200.08070.05340.05810.15360.04510.03160.04970.08430.08540.07580.0832
−30.04810.09400.10170.05690.08310.05100.06420.18560.03410.02770.05080.08960.09710.05550.0779
Panel B: α=0.0066, μ=0.1
100500.05560.06880.06880.06400.06170.04960.07070.07080.05290.05720.05730.08080.08080.07730.0657
200.05450.05720.05720.04950.04530.04280.07680.08130.06010.04750.05100.07850.07850.07150.0495
100.05170.04710.04720.03830.03230.04150.08060.10860.06110.03340.05120.06910.06910.05900.0285
50.04680.03400.03450.02600.02170.04560.07140.16750.05260.03520.04750.05450.05450.04450.0158
00.04740.02470.03000.01380.03170.04540.05500.26980.03760.03760.04320.02580.02840.01700.0397
−10.04740.02350.03240.00910.03670.04250.05380.27400.03600.03440.04520.03870.04400.02120.0668
−30.04700.04890.06460.01800.05160.04620.06200.27730.03290.02870.04340.06890.09160.01600.0625
250500.04950.06450.06450.06500.05020.04330.06080.06150.05540.04710.05130.06160.06160.06200.0616
200.04530.05240.05240.05000.02320.04800.07080.08230.06530.03350.04600.04440.04440.04270.0355
100.04250.03720.03750.03300.01190.04750.07390.12620.06560.02990.04630.02360.02370.02280.0109
50.04670.02870.03000.02430.02590.04280.06610.20250.05600.03120.04840.01030.01060.00950.0043
00.05090.01270.01910.00590.04150.04490.04270.26210.03210.02980.04520.00790.01000.00600.0439
−10.05590.01610.02240.00430.04750.05150.04680.26290.03010.02770.04990.01730.02320.00990.0572
−30.04950.03070.04220.00900.04010.05400.06520.24110.03430.02910.04920.04530.05920.00910.0494
500500.04820.07130.07130.07210.03580.04620.05520.05760.05440.04040.04870.04010.04010.04060.0506
200.04660.05180.05190.04980.01130.04830.06870.09420.06590.03110.04460.01640.01640.01660.0215
100.04910.03280.03360.03150.02410.05330.06980.15920.06570.03200.04700.00500.00500.00490.0060
50.05000.02220.02580.01960.07450.05180.05190.22980.04650.02820.05120.00120.00130.00120.0022
00.05530.01330.02000.00530.05670.05170.03990.24230.02950.02710.05260.00620.00890.00510.0456
−10.05190.01500.02270.00440.04480.05250.04840.23020.03120.02550.05050.01640.02100.00770.0532
−30.05230.02860.03740.00680.03600.05680.07330.20470.03230.02670.05350.03820.04960.01020.0387
xt=μ+(1−c/T)xt−1+et, et=ϕvt−1+vt with ϕ=0.5
θ=−2
θ = 0
θ = 2
TcRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWildRWBIVX1IVX2WildFWild
Panel A: α=0.0066⁠, μ = 0
100500.05660.08140.08140.07780.07660.04890.06370.06370.04800.06150.05910.07980.07980.07650.0764
200.04870.08160.08160.07190.07660.05860.05530.05560.03950.04770.05290.08920.08920.07780.0820
100.04660.09020.09020.07760.08370.05830.05870.06050.03730.04250.05010.08790.08790.07580.0825
50.04940.09130.09130.07570.07690.05170.06820.07830.04330.04330.05110.09610.09610.07940.0806
00.06020.10150.10220.07980.07530.05560.06850.17100.04760.03800.06230.10540.10580.08050.0786
−10.06350.10950.11180.08200.07810.05310.06220.20660.04210.03620.06070.10790.10990.07840.0804
−30.04450.12020.13970.06270.09480.05340.06130.25310.03790.03180.04810.13250.14820.06790.1019
250500.05560.07700.07700.07580.07290.05240.04780.04780.04260.04780.05130.07930.07930.07950.0774
200.04770.08000.08000.07690.07710.05490.05050.05060.04160.04440.04640.08180.08180.07950.0797
100.04530.08910.08910.08320.08310.05180.04660.04840.03650.03900.04520.08930.08930.08460.0827
50.04510.09030.09030.08250.07910.04720.05490.06280.03890.03700.04420.09450.09450.08620.0813
00.05190.09310.09330.08170.07430.05360.06470.15020.04810.03880.05400.09170.09210.08010.0717
−10.05270.08480.08620.07210.07670.05530.05330.17160.04050.03220.05280.09060.09180.07800.0807
−30.05060.10530.11690.06290.09080.05280.06240.22390.03610.02890.05210.10650.11640.06280.0909
500500.04680.07510.07510.07470.07360.04420.04870.04870.04730.04980.04820.07370.07370.07490.0701
200.04350.08350.08350.08360.07840.04400.03800.03810.03430.03710.04590.08110.08110.07920.0776
100.04860.08820.08820.08420.07790.04630.04540.04620.03710.03870.04420.07980.07980.07650.0747
50.04520.09490.09490.08970.08680.04490.04840.05400.03740.03270.04320.08400.08400.08000.0759
00.04930.08350.08370.07580.07270.05120.06100.13620.04700.03280.04800.07880.07900.07220.0707
−10.05020.08180.08270.07200.08070.05340.05810.15360.04510.03160.04970.08430.08540.07580.0832
−30.04810.09400.10170.05690.08310.05100.06420.18560.03410.02770.05080.08960.09710.05550.0779
Panel B: α=0.0066, μ=0.1
100500.05560.06880.06880.06400.06170.04960.07070.07080.05290.05720.05730.08080.08080.07730.0657
200.05450.05720.05720.04950.04530.04280.07680.08130.06010.04750.05100.07850.07850.07150.0495
100.05170.04710.04720.03830.03230.04150.08060.10860.06110.03340.05120.06910.06910.05900.0285
50.04680.03400.03450.02600.02170.04560.07140.16750.05260.03520.04750.05450.05450.04450.0158
00.04740.02470.03000.01380.03170.04540.05500.26980.03760.03760.04320.02580.02840.01700.0397
−10.04740.02350.03240.00910.03670.04250.05380.27400.03600.03440.04520.03870.04400.02120.0668
−30.04700.04890.06460.01800.05160.04620.06200.27730.03290.02870.04340.06890.09160.01600.0625
250500.04950.06450.06450.06500.05020.04330.06080.06150.05540.04710.05130.06160.06160.06200.0616
200.04530.05240.05240.05000.02320.04800.07080.08230.06530.03350.04600.04440.04440.04270.0355
100.04250.03720.03750.03300.01190.04750.07390.12620.06560.02990.04630.02360.02370.02280.0109
50.04670.02870.03000.02430.02590.04280.06610.20250.05600.03120.04840.01030.01060.00950.0043
00.05090.01270.01910.00590.04150.04490.04270.26210.03210.02980.04520.00790.01000.00600.0439
−10.05590.01610.02240.00430.04750.05150.04680.26290.03010.02770.04990.01730.02320.00990.0572
−30.04950.03070.04220.00900.04010.05400.06520.24110.03430.02910.04920.04530.05920.00910.0494
500500.04820.07130.07130.07210.03580.04620.05520.05760.05440.04040.04870.04010.04010.04060.0506
200.04660.05180.05190.04980.01130.04830.06870.09420.06590.03110.04460.01640.01640.01660.0215
100.04910.03280.03360.03150.02410.05330.06980.15920.06570.03200.04700.00500.00500.00490.0060
50.05000.02220.02580.01960.07450.05180.05190.22980.04650.02820.05120.00120.00130.00120.0022
00.05530.01330.02000.00530.05670.05170.03990.24230.02950.02710.05260.00620.00890.00510.0456
−10.05190.01500.02270.00440.04480.05250.04840.23020.03120.02550.05050.01640.02100.00770.0532
−30.05230.02860.03740.00680.03600.05680.07330.20470.03230.02670.05350.03820.04960.01020.0387

Notes: This table presents finite-sample sizes in Case 2 with ϕ=0.5 for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX, and IVX-based bootstrap tests under 5% nominal level. RWB denotes the rejection rate for the Wald statistic by RWB. IVX1 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015). IVX2 denotes the rejection rate for the Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015) with possible negative Wald statistics included. Wild denotes the rejection rate for the residual wild bootstrap algorithm by Demetrescu et al. (2022). FWild denotes the rejection rate for the fixed regressor wild bootstrap algorithm by Demetrescu et al. (2022). Results are reported for different combinations of μ∈{0,0.1}, θ∈{−2,0,2}, T∈{100,250,500}⁠, and c∈{50,20,10,5,0,−1,−3}⁠. The average rejection rates are calculated over 10,000 replications.

Next, we investigate the local power of RWB and the competing methods and plot their power curves for Cases 1–8. We let the true values of β, demonstrated by the horizontal axis, increase from zero so that β = 0 refers to the size of the test, and other non-zero values of β represent the power of the test. We use the same GARCH parameters as in the test size study above. Rejection rates in each panel are calculated based on 10,000 repetitions with a sample size of T = 250.

The two panels in Figure 1 present the power curves with ϕ=0 and θ = 2 when μ = 0 (upper panel) and 0.1 (lower panel) in Case 1. In each panel, a separate plot represents a specific persistent level determined by the labeled value of c. A reference horizontal dashed line with y-axis value equals 0.05, the nominal size, is drawn in each plot. When μ = 0, Panel (a) suggests that the five power curves track each other closely if c∈{−50,−20,−10,−5}⁠, but the RWB curve, denoted by the solid black line, becomes slightly higher when c∈{0,1,3}⁠. RWB’s excellent power performance becomes even more salient when μ=0.1⁠: Panel (b) shows that the RWB curve is well above the other methods when the predictor is highly persistent or explosive. Meanwhile, the competing methods are severely undersized when β is close to zero. We observe similar patterns in Case 1 with ϕ=0.5 and Case 2, as displayed in Figures 2–4.

Case 1: Power plots for sample size T = 250, ϕ=0 and θ = 2.
Figure 1

Case 1: Power plots for sample size T = 250, ϕ=0 and θ = 2.

Notes: This figure displays the rejection rates for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX (IVX1 and IVX2), and IVX-based bootstrap (Wild and FWild) methods as the true value of β increases in Case 1. The horizontal dashed line denotes the 5% nominal size. Results are reported for different combinations of ϕ=0⁠, θ = 2, μ = 0 (a) and μ=0.1 (b). The average rejection rates are calculated over 10,000 replications with T = 250. RWB: Solid black; FWild: Dot-dashed blue; Wild: Yellow dashed; IVX1: Red short-dashed; IVX2: Green long-dashed.

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Case 1: Power plots for sample size T = 250, ϕ=0.5 and θ = 2.
Figure 2

Case 1: Power plots for sample size T = 250, ϕ=0.5 and θ = 2.

Notes: This figure displays the rejection rates for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX (IVX1 and IVX2), and IVX-based bootstrap (Wild and FWild) methods as the true value of β increases in Case 1. The horizontal dashed line denotes the 5% nominal size. Results are reported for different combinations of ϕ=0.5⁠, θ = 2, μ = 0 (a) and μ=0.1 (b). The average rejection rates are calculated over 10,000 replications with T = 250. RWB: Solid black; FWild: Dot-dashed blue; Wild: Yellow dashed; IVX1: Red short-dashed; IVX2: Green long-dashed.

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Case 2: Power plots for sample size T = 250, ϕ=0 and θ = 2.
Figure 3

Case 2: Power plots for sample size T = 250, ϕ=0 and θ = 2.

Notes: This figure displays the rejection rates for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX (IVX1 and IVX2), and IVX-based bootstrap (Wild and FWild) methods as the true value of β increases in Case 2. The horizontal dashed line denotes the 5% nominal size. Results are reported for different combinations of ϕ=0⁠, θ = 2, μ = 0 (a) and μ=0.1 (b). The average rejection rates are calculated over 10,000 replications with T = 250. RWB: Solid black; FWild: Dot-dashed blue; Wild: Yellow dashed; IVX1: Red short-dashed; IVX2: Green long-dashed.

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Case 2: Power plots for sample size T = 250, ϕ=0.5 and θ = 2.
Figure 4

Case 2: Power plots for sample size T = 250, ϕ=0.5 and θ = 2.

Notes: This figure displays the rejection rates for testing the null hypothesis H0: β = 0 versus the alternative H1: β≠0 by the proposed RWB in Equation (14), IVX (IVX1 and IVX2), and IVX-based bootstrap (Wild and FWild) methods as the true value of β increases in Case 2. The horizontal dashed line denotes the 5% nominal size. Results are reported for different combinations of ϕ=0.5⁠, θ = 2, μ = 0 (a) and μ=0.1 (b). The average rejection rates are calculated over 10,000 replications with T = 250. RWB: Solid black; FWild: Dot-dashed blue; Wild: Yellow dashed; IVX1: Red short-dashed; IVX2: Green long-dashed.

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Time series: Eleven predictors.
Figure 5

Time series: Eleven predictors.

Notes: This figure displays the time series plots of the eleven potential predictors from January 1980 to December 2019. The data can be retrieved from Professor Amit Goyal’s homepage: https://sites.google.com/view/agoyal145. RWB: Solid black; FWild: Dot-dashed blue; Wild: Yellow dashed; IVX1: Red short-dashed; IVX2: Green long-dashed.

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Supplementary Appendix Figures A1–A4 further compare their power performance in Cases 3–8. In Supplementary Appendix Figure A1, which denotes the case of μ = 0 and ϕ=0⁠, the five power curves track each other closely, although the RWB curve slightly surpasses the other four when the predictor becomes mildly explosive (i.e., c = 1 and 3). When a non-zero intercept indeed presents, as displayed by Supplementary Appendix Figure A2, RWB displays more salient local power than the competing methods when the degree of persistence increases, as its power curve becomes well above the other four in most cases when c∈{20,10,5,0,−1}⁠. We observe similar patterns when ϕ=0.5⁠, as demonstrated by Supplementary Appendix Figures A3 and A4.

In summary, the proposed RWB method unifies different scenarios of persistence and intercept and provides an accurate size for testing H0:β=0 in all considered cases. On the other hand, the competing IVX and IVX-based bootstrap methods exhibit severe size distortion and become remarkably undersized when the predictor is highly persistent and contains a non-zero intercept. In particular, IVX is more likely to yield negative Wald statistics in some scenarios, leading to potential problems in empirical studies. In addition, the RWB test delivers more attractive power profiles when a predictor is highly persistent or mildly explosive. In the meantime, the IVX and IVX-based bootstrap methods are slightly more powerful in stationary cases.

3 An Empirical Illustration

This section revisits the predictability of stock returns using financial and macroeconomic variables. Specifically, we analyze the predictability of the S&P 500 value-weighted log excess monthly returns from 1980 to 2019, considering eleven variables examined by Welch and Goyal (2008): d/e, long-term yield (lty), d/y, d/p, T-bill rate (tbl), e/p, book-to-market value ratio (b/m), default yield spread (dfy), net equity expansion (ntis), term spread (tms), and inflation rate (inf).8 Definitions of these predictors are provided in Welch and Goyal (2008). Supplementary Appendix Figure A5 illustrates the time series plots for the eleven variables, revealing a unit root pattern in most variables during the sample period, except for the inf.

We initiate our analysis by examining the least squares estimates of each predictor in an autoregressive regression xt=μ+ρxt−1+et⁠. Table 5 presents the OLS estimates for the intercept and slope coefficient. Two findings can be observed in Table 5. First, based on the associated t-statistics for the intercept terms, seven out of eleven predictors (d/e, d/y, d/p, e/p, dfy, tms, and inf) exhibit significance at conventional levels, indicating the presence of non-zero drifts in these time series.9 Second, except inf, the remaining 10 predictors display high persistence, with autoregressive coefficients (ρ) close to unity. To rigorously assess the existence of a unit root within these predictors, we employ four widely used tests: the augmented Dickey–Fuller (ADF) test (Said and Dickey 1984), the DF-GLS test (Elliott, Rothenberg, and Stock 1996), the Phillips–Perron (PP) test (Phillips and Perron 1988), and the Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) test (Kwiatkowski et al., 1992). ADF, DF-GLS, and PP tests hypothesize a unit root, while the KPSS test assumes no unit root.10 The four tests align on identifying a unit root for lty, d/y, and d/p, but yield conflicting results for the remaining predictors.

Table 5
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Unit root tests for eleven economic variables

μ^tμ^ρ^ADFDF-GLSPPKPSS
d/e−0.0118−1.729*0.9848−3.989***−3.328***−3.703***0.087
lty0.00020.5100.9942−1.5420.470−1.1380.727***
d/y−0.0368−2.007**0.9907−1.833−0.049−1.9201.481***
d/p−0.0350−1.896*0.9912−1.661−0.086−1.8251.469***
tbl0.00030.9780.9874−3.177**−0.318−2.2460.448***
e/p−0.0524−2.434**0.9830−3.826***−0.819−3.106**0.795***
b/m0.00321.5040.9876−2.1400.332−2.625*1.585***
dfy0.00042.674***0.9636−3.683***−2.721***−3.561***0.759***
ntis0.00000.1580.9800−2.932**−3.880***−2.732*0.483***
tms0.00133.278***0.9434−3.723***−1.556−3.910***0.180***
inf0.00116.853***0.5516−12.697***−0.085−11.502***0.181***
μ^tμ^ρ^ADFDF-GLSPPKPSS
d/e−0.0118−1.729*0.9848−3.989***−3.328***−3.703***0.087
lty0.00020.5100.9942−1.5420.470−1.1380.727***
d/y−0.0368−2.007**0.9907−1.833−0.049−1.9201.481***
d/p−0.0350−1.896*0.9912−1.661−0.086−1.8251.469***
tbl0.00030.9780.9874−3.177**−0.318−2.2460.448***
e/p−0.0524−2.434**0.9830−3.826***−0.819−3.106**0.795***
b/m0.00321.5040.9876−2.1400.332−2.625*1.585***
dfy0.00042.674***0.9636−3.683***−2.721***−3.561***0.759***
ntis0.00000.1580.9800−2.932**−3.880***−2.732*0.483***
tms0.00133.278***0.9434−3.723***−1.556−3.910***0.180***
inf0.00116.853***0.5516−12.697***−0.085−11.502***0.181***

Notes: This table documents four unit root tests’ results for eleven predictive variables: d/e, lty, d/y, d/p, tbl, e/p, b/m, dfy, ntis, tms, and inf. μ^ and ρ^ are OLS estimates in the AR(1) process: xt=μ+ρxt−1+et⁠. tμ^ is the corresponding t statistics for μ^⁠. ADF represents the statistic of the ADF test by Said and Dickey (1984) for the null that xt has a unit root. DF-GLS represents the statistic from an ADF-type test by Elliott, Rothenberg, and Stock (1996) for the null that xt has a unit root. PP refers to the statistic from the PP test by Phillips and Perron (1988) for the null that xt has a unit root. KPSS refers to the statistic from Kwiatkowski et al. (1992)’s unit root test for the null that xt is stationary. The optimal length of lag is determined by the Bayesian information criterion (BIC) for the ADF test and DF-GLS test. *, **, and *** indicate the rejection of the null hypothesis at 10%, 5%, and 1%, respectively.

Table 5
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Unit root tests for eleven economic variables

μ^tμ^ρ^ADFDF-GLSPPKPSS
d/e−0.0118−1.729*0.9848−3.989***−3.328***−3.703***0.087
lty0.00020.5100.9942−1.5420.470−1.1380.727***
d/y−0.0368−2.007**0.9907−1.833−0.049−1.9201.481***
d/p−0.0350−1.896*0.9912−1.661−0.086−1.8251.469***
tbl0.00030.9780.9874−3.177**−0.318−2.2460.448***
e/p−0.0524−2.434**0.9830−3.826***−0.819−3.106**0.795***
b/m0.00321.5040.9876−2.1400.332−2.625*1.585***
dfy0.00042.674***0.9636−3.683***−2.721***−3.561***0.759***
ntis0.00000.1580.9800−2.932**−3.880***−2.732*0.483***
tms0.00133.278***0.9434−3.723***−1.556−3.910***0.180***
inf0.00116.853***0.5516−12.697***−0.085−11.502***0.181***
μ^tμ^ρ^ADFDF-GLSPPKPSS
d/e−0.0118−1.729*0.9848−3.989***−3.328***−3.703***0.087
lty0.00020.5100.9942−1.5420.470−1.1380.727***
d/y−0.0368−2.007**0.9907−1.833−0.049−1.9201.481***
d/p−0.0350−1.896*0.9912−1.661−0.086−1.8251.469***
tbl0.00030.9780.9874−3.177**−0.318−2.2460.448***
e/p−0.0524−2.434**0.9830−3.826***−0.819−3.106**0.795***
b/m0.00321.5040.9876−2.1400.332−2.625*1.585***
dfy0.00042.674***0.9636−3.683***−2.721***−3.561***0.759***
ntis0.00000.1580.9800−2.932**−3.880***−2.732*0.483***
tms0.00133.278***0.9434−3.723***−1.556−3.910***0.180***
inf0.00116.853***0.5516−12.697***−0.085−11.502***0.181***

Notes: This table documents four unit root tests’ results for eleven predictive variables: d/e, lty, d/y, d/p, tbl, e/p, b/m, dfy, ntis, tms, and inf. μ^ and ρ^ are OLS estimates in the AR(1) process: xt=μ+ρxt−1+et⁠. tμ^ is the corresponding t statistics for μ^⁠. ADF represents the statistic of the ADF test by Said and Dickey (1984) for the null that xt has a unit root. DF-GLS represents the statistic from an ADF-type test by Elliott, Rothenberg, and Stock (1996) for the null that xt has a unit root. PP refers to the statistic from the PP test by Phillips and Perron (1988) for the null that xt has a unit root. KPSS refers to the statistic from Kwiatkowski et al. (1992)’s unit root test for the null that xt is stationary. The optimal length of lag is determined by the Bayesian information criterion (BIC) for the ADF test and DF-GLS test. *, **, and *** indicate the rejection of the null hypothesis at 10%, 5%, and 1%, respectively.

Table 6 documents some summary statistics of u^t, e^t⁠, and v^t⁠, the three estimated residual terms. The first two columns display the results of the Ljung–Box test (Ljung and Box 1978) and show that the residuals of the predictive regression are not serially correlated but exhibit conditional heteroscedasticity. We test the skewness and kurtosis of u^t in columns (3) and (4) and find all are slightly left-skewed and display larger kurtosis than the standard normal. Column (5) presents the results of the Ljung–Box test for e^t⁠, the residuals of AR(1) for the eleven predictors, and indicates that most are serially correlated. Columns (7) and (8) show that many residuals become significantly right-skewed, and their kurtosis becomes even larger. To capture such potential serial correlation within e^t⁠, we use ARMA(p, q) models to fit the residuals of the eleven predictors, respectively. Here, the Akaike information criterion (AIC) determines the optimal choice of p and q. Results in columns (8) and (9) indicate that most v^t becomes serially uncorrelated, but all display a strong conditionally heteroscedastic effect. Finally, we present the correlation coefficients between u^t and v^t and the correlation test results in column (10). As can be seen therein, d/e, lty, d/p, e/p, b/m, and dfy exhibit significant non-zero correlation with u^t⁠, indicating the potential endogeneity in the predictive regression.

Table 6
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Summary statistics for estimated ut, et, and vt

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
Ljung–Box(⁠u^t⁠)Ljung–Box(⁠u^t2⁠)Skewness(⁠u^t⁠)Kurtosis(⁠u^t⁠)Ljung–Box(⁠e^t⁠)Skewness(⁠e^t⁠)Kurtosis(⁠e^t⁠)Ljung–Box(⁠v^t⁠)Ljung–Box(⁠v^t2⁠)Corr(⁠u^t,v^t⁠)
d/e7.4128.35***−0.665.22***768.03***−1.7062.42***13.52449.11***−0.09*
lty7.3229.07***−0.635.21***24.67**−0.205.89***5.57212.16***−0.12***
d/y7.4328.47***−0.655.17***8.870.78***6.04***8.8727.48***−0.03
d/p7.4529.01***−0.665.18***8.680.77***5.99***7.6622.26**−0.99***
tbl7.3429.37***−0.635.23***190.45***−1.6632.75***61.56***373.47***−0.04
e/p7.2030.08***−0.635.15***228.77***1.02***36.85***2.45146.66***−0.61***
b/m7.3228.94***−0.655.17***21.07**0.58***13.96***2.8479.17***−0.65***
dfy7.2628.47***−0.645.20***104.34***1.48***15.75***3.8056.20***−0.16***
ntis7.3228.94***−0.655.21***112.28***0.21*9.96***27.38***61.77***0.06
tms7.3329.1***−0.655.23***63.28***0.1422.73***19.14*124.94***−0.04
inf7.1931.02***−0.655.21***61.18***−0.244.99***3.19110.44***−0.01
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
Ljung–Box(⁠u^t⁠)Ljung–Box(⁠u^t2⁠)Skewness(⁠u^t⁠)Kurtosis(⁠u^t⁠)Ljung–Box(⁠e^t⁠)Skewness(⁠e^t⁠)Kurtosis(⁠e^t⁠)Ljung–Box(⁠v^t⁠)Ljung–Box(⁠v^t2⁠)Corr(⁠u^t,v^t⁠)
d/e7.4128.35***−0.665.22***768.03***−1.7062.42***13.52449.11***−0.09*
lty7.3229.07***−0.635.21***24.67**−0.205.89***5.57212.16***−0.12***
d/y7.4328.47***−0.655.17***8.870.78***6.04***8.8727.48***−0.03
d/p7.4529.01***−0.665.18***8.680.77***5.99***7.6622.26**−0.99***
tbl7.3429.37***−0.635.23***190.45***−1.6632.75***61.56***373.47***−0.04
e/p7.2030.08***−0.635.15***228.77***1.02***36.85***2.45146.66***−0.61***
b/m7.3228.94***−0.655.17***21.07**0.58***13.96***2.8479.17***−0.65***
dfy7.2628.47***−0.645.20***104.34***1.48***15.75***3.8056.20***−0.16***
ntis7.3228.94***−0.655.21***112.28***0.21*9.96***27.38***61.77***0.06
tms7.3329.1***−0.655.23***63.28***0.1422.73***19.14*124.94***−0.04
inf7.1931.02***−0.655.21***61.18***−0.244.99***3.19110.44***−0.01

Notes: This table documents summary statistics for three estimated residuals. u^t refers to the OLS residuals from the simple linear model yt=α+βxt−1+ut⁠, where yt is the monthly S&P 500 value-weighted log excess returns between 1980 and 2019, and xt−1 is one of eleven predictive variables: d/e, lty, d/y, d/p, tbl, e/p, b/m, dfy, ntis, tms, and inf. e^t refers to the residuals from the AR(1) model: xt=μ+ρxt−1+et⁠. v^t refers to the residuals of the ARMA(p, q) model with orders p and q determined by the AIC. Ljung–Box denotes a serial correlation test proposed by Ljung and Box (1978) with 12 lags. Skewness and kurtosis represent sample skewness and sample excess kurtosis of residuals. Corr(⁠u^t,v^t⁠) represents sample correlation coefficients between u^t and v^t⁠. *, **, and *** indicate the rejection of the null hypothesis at 10%, 5%, and 1%, respectively.

Table 6
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Summary statistics for estimated ut, et, and vt

(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
Ljung–Box(⁠u^t⁠)Ljung–Box(⁠u^t2⁠)Skewness(⁠u^t⁠)Kurtosis(⁠u^t⁠)Ljung–Box(⁠e^t⁠)Skewness(⁠e^t⁠)Kurtosis(⁠e^t⁠)Ljung–Box(⁠v^t⁠)Ljung–Box(⁠v^t2⁠)Corr(⁠u^t,v^t⁠)
d/e7.4128.35***−0.665.22***768.03***−1.7062.42***13.52449.11***−0.09*
lty7.3229.07***−0.635.21***24.67**−0.205.89***5.57212.16***−0.12***
d/y7.4328.47***−0.655.17***8.870.78***6.04***8.8727.48***−0.03
d/p7.4529.01***−0.665.18***8.680.77***5.99***7.6622.26**−0.99***
tbl7.3429.37***−0.635.23***190.45***−1.6632.75***61.56***373.47***−0.04
e/p7.2030.08***−0.635.15***228.77***1.02***36.85***2.45146.66***−0.61***
b/m7.3228.94***−0.655.17***21.07**0.58***13.96***2.8479.17***−0.65***
dfy7.2628.47***−0.645.20***104.34***1.48***15.75***3.8056.20***−0.16***
ntis7.3228.94***−0.655.21***112.28***0.21*9.96***27.38***61.77***0.06
tms7.3329.1***−0.655.23***63.28***0.1422.73***19.14*124.94***−0.04
inf7.1931.02***−0.655.21***61.18***−0.244.99***3.19110.44***−0.01
(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
Ljung–Box(⁠u^t⁠)Ljung–Box(⁠u^t2⁠)Skewness(⁠u^t⁠)Kurtosis(⁠u^t⁠)Ljung–Box(⁠e^t⁠)Skewness(⁠e^t⁠)Kurtosis(⁠e^t⁠)Ljung–Box(⁠v^t⁠)Ljung–Box(⁠v^t2⁠)Corr(⁠u^t,v^t⁠)
d/e7.4128.35***−0.665.22***768.03***−1.7062.42***13.52449.11***−0.09*
lty7.3229.07***−0.635.21***24.67**−0.205.89***5.57212.16***−0.12***
d/y7.4328.47***−0.655.17***8.870.78***6.04***8.8727.48***−0.03
d/p7.4529.01***−0.665.18***8.680.77***5.99***7.6622.26**−0.99***
tbl7.3429.37***−0.635.23***190.45***−1.6632.75***61.56***373.47***−0.04
e/p7.2030.08***−0.635.15***228.77***1.02***36.85***2.45146.66***−0.61***
b/m7.3228.94***−0.655.17***21.07**0.58***13.96***2.8479.17***−0.65***
dfy7.2628.47***−0.645.20***104.34***1.48***15.75***3.8056.20***−0.16***
ntis7.3228.94***−0.655.21***112.28***0.21*9.96***27.38***61.77***0.06
tms7.3329.1***−0.655.23***63.28***0.1422.73***19.14*124.94***−0.04
inf7.1931.02***−0.655.21***61.18***−0.244.99***3.19110.44***−0.01

Notes: This table documents summary statistics for three estimated residuals. u^t refers to the OLS residuals from the simple linear model yt=α+βxt−1+ut⁠, where yt is the monthly S&P 500 value-weighted log excess returns between 1980 and 2019, and xt−1 is one of eleven predictive variables: d/e, lty, d/y, d/p, tbl, e/p, b/m, dfy, ntis, tms, and inf. e^t refers to the residuals from the AR(1) model: xt=μ+ρxt−1+et⁠. v^t refers to the residuals of the ARMA(p, q) model with orders p and q determined by the AIC. Ljung–Box denotes a serial correlation test proposed by Ljung and Box (1978) with 12 lags. Skewness and kurtosis represent sample skewness and sample excess kurtosis of residuals. Corr(⁠u^t,v^t⁠) represents sample correlation coefficients between u^t and v^t⁠. *, **, and *** indicate the rejection of the null hypothesis at 10%, 5%, and 1%, respectively.

In summary, most predictors are highly persistent with autoregressive coefficients close to unity, and many are endogenous during the sample period. According to the simulation evidence, non-zero intercept and high degrees of persistence within these predictors would induce concerns about the potential size distortion of IVX.

Table 7 contains the results for these univariate regressions by the OLS, IVX, and proposed estimate β^⁠, as well as the t-statistic, IVX-based Wald statistic by Kostakis, Magdalinos, and Stamatogiannis (2015) and Demetrescu et al. (2022), and the proposed test in Equation (14) under the null hypothesis of no predictability. As can be seen therein, the results yielded by IVX and the unified method are not precisely the same. While the IVX (WaldIV X) and IVX-based bootstrap (WaldWild and WaldFWild) methods unanimously show that none of the eleven predictors are significant at conventional levels, the unified method delivers three significant predictors (d/y, d/p, and e/p) at the 10% level. Interestingly, as documented by Table 5 and Supplementary Appendix Table A5, all three predictors that exhibit significant predicting power are likely to have non-zero intercepts, and two (d/y and d/p) follow the unit root process according to all four unit root tests. This result echoes our findings in the simulation section. When a predictor is highly persistent and contains a non-zero intercept, IVX tends to be badly undersized and thus yields fewer significant predictors than the unified method. On the other hand, our method and IVX agree on the significance of the remaining predictors. In summary, although not all eleven predictors exhibit significant predicting power on the predictability of S&P 500 monthly returns during the sample period, it is clear that our strategy provides a method to unify zero and non-zero intercept and uncovers more significant predictors than IVX.11

Table 7
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Estimates of predictive regressions on 1980–2019 S&P 500 monthly returns

β^OLStOLSβ^IVXWaldIV XWaldWildWaldFWildβ^RWBWaldRWB
d/e0.00200.35600.00130.05100.05100.05100.00230.098
lty−0.0574−0.9150−0.08251.54001.54001.54000.04550.401
d/y0.00621.27700.01011.04601.04601.04600.01182.954*
d/p0.00581.20700.00910.85500.85500.85500.01162.719*
tbl−0.0617−1.1240−0.09761.91101.91101.91100.04170.351
e/p0.00370.82400.00581.14401.14401.14400.01343.154*
b/m0.00290.34500.00690.12100.12100.12100.02562.336
dfy−0.0553−0.1310−0.11460.06800.06800.0680−0.14020.022
ntis−0.0148−0.1540−0.00240.00100.00100.00100.19171.456
tms0.11100.80900.08070.26600.26600.2660−0.03980.039
inf−0.5385−0.9540−0.55330.49300.49300.49300.80191.041
β^OLStOLSβ^IVXWaldIV XWaldWildWaldFWildβ^RWBWaldRWB
d/e0.00200.35600.00130.05100.05100.05100.00230.098
lty−0.0574−0.9150−0.08251.54001.54001.54000.04550.401
d/y0.00621.27700.01011.04601.04601.04600.01182.954*
d/p0.00581.20700.00910.85500.85500.85500.01162.719*
tbl−0.0617−1.1240−0.09761.91101.91101.91100.04170.351
e/p0.00370.82400.00581.14401.14401.14400.01343.154*
b/m0.00290.34500.00690.12100.12100.12100.02562.336
dfy−0.0553−0.1310−0.11460.06800.06800.0680−0.14020.022
ntis−0.0148−0.1540−0.00240.00100.00100.00100.19171.456
tms0.11100.80900.08070.26600.26600.2660−0.03980.039
inf−0.5385−0.9540−0.55330.49300.49300.49300.80191.041

Notes: This table documents the results of univariate predictive regressions for observations from January 1980 to December 2019. The dependent variable is the monthly S&P 500 value-weighted log excess return, and the lagged persistent predictor in each of the following variables defined in Section 3: d/e, lty, d/y, d/p, tbl, e/p, b/m, dfy, ntis, tms, and inf. β^OLS, β^IVX⁠, and β^RWB respectively denote the estimates of slope coefficients by the OLS, IVX, and RWB methods. tOLS, WaldIV X, WaldWild, WaldFWild, and WaldRWB respectively denote the t-statistic associated with the OLS estimate, the Wald statistic associated with the IVX by Kostakis, Magdalinos, and Stamatogiannis (2015), the IVX-based bootstrap method by Demetrescu et al. (2022), and the RWB method in Equation (14). “*” means significance at a 10% level.

Table 7
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Estimates of predictive regressions on 1980–2019 S&P 500 monthly returns

β^OLStOLSβ^IVXWaldIV XWaldWildWaldFWildβ^RWBWaldRWB
d/e0.00200.35600.00130.05100.05100.05100.00230.098
lty−0.0574−0.9150−0.08251.54001.54001.54000.04550.401
d/y0.00621.27700.01011.04601.04601.04600.01182.954*
d/p0.00581.20700.00910.85500.85500.85500.01162.719*
tbl−0.0617−1.1240−0.09761.91101.91101.91100.04170.351
e/p0.00370.82400.00581.14401.14401.14400.01343.154*
b/m0.00290.34500.00690.12100.12100.12100.02562.336
dfy−0.0553−0.1310−0.11460.06800.06800.0680−0.14020.022
ntis−0.0148−0.1540−0.00240.00100.00100.00100.19171.456
tms0.11100.80900.08070.26600.26600.2660−0.03980.039
inf−0.5385−0.9540−0.55330.49300.49300.49300.80191.041
β^OLStOLSβ^IVXWaldIV XWaldWildWaldFWildβ^RWBWaldRWB
d/e0.00200.35600.00130.05100.05100.05100.00230.098
lty−0.0574−0.9150−0.08251.54001.54001.54000.04550.401
d/y0.00621.27700.01011.04601.04601.04600.01182.954*
d/p0.00581.20700.00910.85500.85500.85500.01162.719*
tbl−0.0617−1.1240−0.09761.91101.91101.91100.04170.351
e/p0.00370.82400.00581.14401.14401.14400.01343.154*
b/m0.00290.34500.00690.12100.12100.12100.02562.336
dfy−0.0553−0.1310−0.11460.06800.06800.0680−0.14020.022
ntis−0.0148−0.1540−0.00240.00100.00100.00100.19171.456
tms0.11100.80900.08070.26600.26600.2660−0.03980.039
inf−0.5385−0.9540−0.55330.49300.49300.49300.80191.041

Notes: This table documents the results of univariate predictive regressions for observations from January 1980 to December 2019. The dependent variable is the monthly S&P 500 value-weighted log excess return, and the lagged persistent predictor in each of the following variables defined in Section 3: d/e, lty, d/y, d/p, tbl, e/p, b/m, dfy, ntis, tms, and inf. β^OLS, β^IVX⁠, and β^RWB respectively denote the estimates of slope coefficients by the OLS, IVX, and RWB methods. tOLS, WaldIV X, WaldWild, WaldFWild, and WaldRWB respectively denote the t-statistic associated with the OLS estimate, the Wald statistic associated with the IVX by Kostakis, Magdalinos, and Stamatogiannis (2015), the IVX-based bootstrap method by Demetrescu et al. (2022), and the RWB method in Equation (14). “*” means significance at a 10% level.

4 Conclusion

The popular IVX-based method by Kostakis, Magdalinos, and Stamatogiannis (2015) requires a zero intercept in predictors, and the empirical likelihood method by Zhu, Cai, and Peng (2014) excludes conditional heteroscedasticity in the error terms. It is open and practically essential to develop a unified predictability test regardless of the presence of non-zero intercept in a predictor, its degree of persistence, and conditional heteroscedastic errors. This article delivers a unified test by splitting data into two parts to construct score equations, using weighted inference to ensure either a normal limit or the product of a normal random variable and the sign of another random variable after random normalization, and adopting an RWB method to estimate the asymptotic variance in a unified way. Unlike the IVX-based method by Kostakis, Magdalinos, and Stamatogiannis (2015), our unified method is easy to implement and does not require constructing an instrument variable with a tuning parameter and correcting the covariance estimation. A sequence of simulation studies confirms the good finite sample performance of the new test and spots some issues with the famous IVX-based test in the literature. Specifically, the Wald test statistic in the IVX-based method could be negative, and the size is severely distorted in the case of a non-zero intercept in the predictor’s AR(1) model. Empirical analysis shows that, out of eleven widely used predictors, the new unified test identifies three that display predicting power on stock returns. At the same time, the IVX-based method fails to detect any valuable predictors. In future research, our method could be potentially extended to accommodate a predictive regression with multiple predictors of different persistent levels. Other limitations, such as the inability to conduct one-sided tests and the potential decrease in test power due to the data-splitting strategy, merit further investigation and could potentially be alleviated in future studies.

Supplemental Material

Supplemental material is available at Journal of Financial Econometrics online.

Funding

Yang’s research is partly supported by the National Natural Science Foundation of China (72173140, 71991474). Long’s research is partly supported by the 2022 SLA Faculty Research Award at Tulane University. Liu’s research is supported by the NSF of China (Grant No. 12471257), the NSF of the Science and Technology Department of Jiangxi Province (Grant No. 20243BCE51010), and the Jiangxi Province Key Laboratory of Data Science in Finance and Economics (Grant No. 2024SSY03201).

We thank the editor, Prof. Allan Timmermann, and two anonymous reviewers, for constructive suggestions. All remaining errors are our responsibility.

Footnotes

1

Usually when economists think about the persistence in xt, they have a “small, but highly persistent risk premium component” story in mind, as “high first-order autocorrelation implies persistence in expected returns” (Fama and French 1988). We would like to thank Prof. Timmermann, the handling editor, for suggesting this clarification.

2

We thank the editor for pointing this out.

3

See Zhang and Ling (2015) and Zhu and Ling (2015) for a detailed discussion on the general GARCH or G-GARCH process.

4

We also consider ρ=1−c/Td with d = 0.85, 0.9, and 0.95, respectively, and find quite similar results. For et, we also examined a more general case in which et follows ARMA(p, q) with p=q=1⁠, and found similar results. The optimal order of p and q is determined by the AIC. These results are in Supplementary Appendix Sections C and D. We thank an anonymous reviewer for suggesting this clarification.

5

In fact, our method allows for stochastic volatility models as the critical assumption in the generalized GARCH model is that volatility is independent of the current and future white noises. In Supplementary Appendix F, we assess the performance of our method in a stochastic volatility case and find it works well. We appreciate Prof. Timmermann, the handling editor, for suggesting this.

6

For RWB, we choose m=[T/2] throughout all simulations, so that the whole sample is evenly divided. The results are robust to other choices of m such as [T/3] and [2T/3]⁠, where the sample is not evenly divided. The simulation results are available upon request.

7

For Cases 1 and 2, we also consider θ∈{−0.6,0.6} so that the correlation coefficients become mild and approximately -0.5 and 0.5, and we find similar results. These tables are available upon request.

8

Data can be obtained from Professor Amit Goyal’s homepage: https://sites.google.com/view/agoyal145.

9

It is acknowledged that using the classical t-test is problematic when a predictor is highly persistent. We provide these heuristic results to offer initial insights, recognizing that developing a unified test for zero intercepts, irrespective of the properties of the underlying process, is beyond the scope of this article. In Supplementary Appendix E, we propose an alternative strategy to assess the significance of these estimated intercepts.

10

For these unit root tests, we adopt regression specifications with drifts.

11

We also examine the robustness of the empirical results by RWB to other choices of m such as [T/3] and [2T/3]⁠, where the sample is not evenly divided, and obtain similar findings. These results are available upon request. We sincerely thank one anonymous reviewer for suggesting this.

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