Testing the Zero-Process of Intraday Financial Returns for Non-Stationary Periodicity

Abstract

Recent studies show that the zero-process of observed intraday financial returns is frequently characterized by non-stationary periodicity. As liquidity varies across the trading day, not only does unconditional volatility change, but also the unconditional zero-probability. While scaling returns by the time-varying intraday volatility may stabilize volatility, it does not make the zero-process of scaled returns stationary. This invalidates standard methods of risk estimation, and recent studies document that the use of such invalid methods can have major effects on risk estimates. Formal tests for non-stationary periodicity in the zero-process can therefore be of great value in guiding the choice of a suitable risk estimation procedure. Despite this, little attention has been devoted to the derivation of such tests. Here, we help filling this gap by developing user-friendly yet flexible and powerful tests that hold under mild assumptions. Next, our empirical study reveals that intraday financial returns are widely characterized by non-stationary periodicity in the zero-process. This has important and potentially wide-ranging implications for future research.

It is well known that intraday financial returns are frequently zero. This can be due to low liquidity, discrete pricing, measurement issues, closed markets and other market microstructure issues. A number of contributions account for the observed zeros. In one strand of the literature, zeros occur when the true underlying price process (the “efficient price”) is not observed. Observed prices thus fail to update, that is, they are stale. Examples include Lesmond et al. (1999), Bandi et al. (2017), Bandi et al. (2020), Bandi et al. (2024), and Kolokolov and Reno (2024). In a second strand of the literature, zeros occur due to the discreteness of pricing. Examples include Hausman et al. (1992), Rydberg and Shephard (2003), Russell and Engle (2005), Liesenfeld et al. (2006), and Catania et al. (2022). In a third strand of the literature, price changes are decomposed multiplicatively into a zero-indicator and a continuous random variable. In effect, price changes are thus continuous except at zero. Examples include Hautsch et al. (2014), Kömm and Küsters (2015), and Harvey and Ito (2020). Finally, the Autoregressive Conditional Heteroscedasticity (ARCH) class of models may be considered a separate strand of the literature. There, the usual Quasi Maximum Likelihood Estimator (QMLE) is consistent subject to suitable assumptions on the standardized innovation, see, for example, Escanciano (2009), Francq and Thieu (2019), and the discussion in Sucarrat and Grønneberg (2022, section 1.5). In particular, under strict stationarity, QML estimation of the GARCH is generally consistent when the conditional zero-probability is time-varying and dependent on the past in unknown ways.

While the occurrence of zeros has received considerable attention under the assumption of strict stationarity of the zero-process, as in the majority of the works above, only recently scholars have discovered that the zero-process is in fact frequently non-stationary. That is, the unconditional zero-probability frequently changes over time. Kolokolov et al. (2020, section 6), for example, document that highly liquid stocks at the New York Stock Exchange (NYSE) exhibit an intraday non-stationary periodic pattern in the zero-probability. Specifically, they find that intraday periodicity is often characterized by an inverse U-pattern, with the unconditional zero-probability peaking in the middle of the day. Sucarrat and Grønneberg (2022, section 3) find that a non-stationary model of the zero-process provides the best fit for the zero-process of 178 daily stock returns (out of 1665) at the NYSE. Francq and Sucarrat (2023, section 5.2) show that the zero-process of intraday EUR/USD exchange rate returns exhibits a strong non-stationary periodic pattern.

A non-stationary zero-process has major effects on risk estimates, since standard risk estimation methods rely on strict stationarity of the transformed returns. If the zero-process is non-stationary, however, then common transformations will not lead to strict stationarity. To see this, let $ri$ denote the intraday financial return from the end of intraday period $i−1$ to the end of intraday period $i$⁠, and let $Ii$ be a binary variable equal to 0 if $ri$ is zero and 1 otherwise. It is widely believed that scaling observed return $ri$ by its unconditional volatility $σi$⁠, say, $E(ri2)1/2$⁠, produces a scaled process ${ri/σi}$ that is strictly stationary. Next, standard methods that rely on stationarity of the transformed series are used for risk estimation, see, for example, Andersen and Bollerslev (1997), Engle and Russell (1998, section 6.2), Mazur and Pipien (2012), Amado and Teräsvirta (2013), and Escribano and Sucarrat (2018). While the transformation may stabilize the unconditional volatility of scaled returns, the scaled process ${ri/σi}$ is not strictly stationary if the zero-process ${Ii}$ is not strictly stationary. The reason is that the probabilistic properties of the zero-process are unaffected by the scaling. Another approach to risk modeling is to model high-frequency volatility proxies directly, as in, for example, Bollerslev et al. (2016), Bollerslev et al. (2018), Buccheri and Corsi (2021), and Cipollini et al. (2021). However, none of these studies accommodate the fact that the intraday returns used to construct the proxies are characterized by non-stationary periodicity in the zero-process. Recent studies document that failing to accommodate the non-stationary zero-process can have major effects on risk estimates, see Kolokolov et al. (2020), Sucarrat and Grønneberg (2022), Francq and Sucarrat (2023), and Kolokolov and Reno (2024).

Formal tests for non-stationary periodicity in the zero-process can therefore be of great value in identifying a suitable risk estimation procedure. We derive three new tests for non-stationary periodicity in the zero-process: A Wald test, a Lagrange Multiplier (LM) test and a Likelihood Ratio (LR) test. These are the most common tests, and here they share the same theoretical basis. Accordingly, it is natural to consider all three of them. To the best of our knowledge, Kolokolov et al. (2020) is the only test currently available for non-stationary periodicity in the zero-process of intraday financial returns. The test assumes the length of each intraday interval tends to zero so that the number of intraday periods or “seasons” $S$ goes to infinity. In the three tests derived here, by contrast, $S$ is fixed (and chosen by the researcher) and can be as low as 2. In other words, in addition to situations where the number of intraday intervals is large, for example, at high frequencies, our tests can also be applied at low frequencies, for example, when the trading day is divided into, say, only two half-day intervals. The test of Kolokolov et al. (2020), by contrast, is only suitable for high-frequency data of highly liquid assets (their empirical illustration is on highly liquid NYSE stocks at the 30-second frequency). Another important difference concerns the assumed dependence. The assumptions of Kolokolov et al. (2020) implies strong restrictions on the within-day dependence of the zero-process. In our most versatile tests, by contrast, the Wald and LM tests, the within-day dependence is virtually unrestricted (we only need that the unconditional correlations of binary variables exist), and across days the two tests allow for substantial dependence (strong mixing) of unknown form. This is important, since the stylized facts in our empirical study show that the zero-process of intraday financial returns is often strongly dependent, both within the day and across days. Another attractive property of our tests is that they are intuitive and simple to implement, since they are made up of well-known and easy-to-compute ingredients, and since their asymptotic distribution is standard under the null. In deriving our tests, we establish exact and mild conditions under which the proportion of zeros across days in each intraday period is consistent for the unconditional zero-probabilities. Next, using the vector-of-seasons representation, we establish asymptotic normality and derive a consistent estimator of the asymptotic covariance matrix. With these ingredients, our three tests are derived and then compared in finite sample simulations. In the simulations, the LM test emerges as the preferred test, since its relative rejection frequency under the null is close to its nominal counterpart in finite samples of empirical relevance, and since the test is valid under mild assumptions of empirical relevance (i.e., under intraday and across day dependence). Next, the LM test is used in an empirical study of intraday financial returns. Our study reveals that intraday financial returns are comprehensively characterized by non-stationary periodicity in the zero-process at intraday frequencies. This has important and potentially wide-ranging implications for future research, since it invalidates standard methods of risk estimation (cf. our discussion above).

The remainder of the article is organized as follows. Section 1 provides the theoretical basis of our tests, derives the Wald, LM, and LR tests, and proves their consistency under the alternative. Section 2 studies the properties of the tests under the null in finite samples by simulation. In Section 3, our preferred test, the LM test, is used in a study of intraday financial returns. Section 4 contains our conclusions. The proofs of our theoretical results and additional supplementary material are contained in the Supplementary Material.

1 Theory

Let $ri=pi−pi−1$ denote the log-return at $i=1,2,…$⁠, where $pi$ is the observed intraday log-price of a financial asset at time $i$⁠. Typically, $pi$ will be the log-price at the end of an intraday period, say, an hourly interval, a 30-minute interval or a 10-minute interval. Let $s∈{1,…,S}$ denote the number of intraday periods, or “seasons”. For example, if returns are recorded 8 hours a day, then $S=8$ for hourly returns, $S=8×2$ for 30-minute returns, $S=8×6$ for 10-minute returns, and so on.

To develop tests under this null, we:

1. Derive a consistent estimator $θ^$ of the true parameter vector $θ0=(θ01,…,θ0S)′$⁠;

2. Establish that $T(θ^−θ0)→dN(0,Σ0)$ as $T→∞$⁠, and obtain an expression for the covariance matrix $Σ0$⁠;

3. Develop a consistent estimator $Σ^$ such that $Σ^→pΣ0$ as $T→∞$⁠.

Given these components, tests, such as Wald, LM, or LR, can be developed to see whether Equation (1.3) holds.

1.1 Consistency of $θ^$

To establish consistency, we rely on the following assumptions:

A1 For each $s∈{1,…,S}$⁠: $Θs$ is compact, $Θs⊊Θs*$ and $Θs*=(0,1)$⁠.

A2 Let $It=(I1t,…,ISt)′$ be a vector of a binary stochastic variables. ${It}$ is $ϕ$-mixing of size $−r/2(r−1)$⁠, or $α$-mixing of size $−r/(r−2),r>2$⁠, with $E(It)=θ0=(θ01,…,θ0S)′$ for all $t$⁠, and $θ0s∈Θs$⁠.

A3  $limT→∞1T∑t=1T∑l=1TE[(It−θ0)(Il−θ0)′]=Ω0$ is positive definite.

Assumption A1 is standard in the context of M-estimators, but means the boundary values 1 and 0 are not contained in $Θs$⁠. In empirical applications, this is not restrictive, since such series (e.g., a series without zeros) can simply be omitted from the test. A2 is a mild dependence assumption. In effect, it implies that the intraday series ${Ii}$ can be dependent both within the day and across days. It is convenient to introduce the dependence assumption on the seasonal vector representation $It$⁠, because the mixing properties are inherited element-wise, and so we can utilize the relevant theorems element-wise and rely on their multivariate extensions when needed. A3 is also standard, and it implies summable autocovariances of the mixing process ${It}$⁠, which is among the sufficient conditions needed to apply uniform weak law of large numbers (UWLLN). At the same time, it ensures that all the covariance matrices considered in our propositions and their proofs are positive definite (see, e.g., Supplementary Section 2). Ultimately, these assumptions are sufficient to demonstrate that $θ0s$ uniquely minimizes $limT→∞1T∑t=1TE(lt(Ist,θs))$ on $Θs$⁠.

Proposition 1 establishes the needed consistency result.

 

1.2 Asymptotic Normality

With $int(Θ)$⁠, we denote the interior of $Θ=Θ1×⋯×ΘS$⁠. For asymptotic normality, we rely on a single extra assumption, which is standard:

A4 The true parameter $θ0$ lies in $int(Θ)$⁠.

Proposition 2 establishes asymptotic normality of $θ^$⁠.

 

The covariance matrix in Proposition 2 possesses the usual “sandwich” form under serial correlation with $B0$ representing the long-run covariance matrix of the score. We dedicate the upcoming Section 1.3 to discuss its estimation methods. The full long-run covariance matrix can be significantly simplified when $B0=A0$⁠. This corresponds to the usual information matrix equality result. However, unlike in standard theory, it is not enough to have independence of the score over $t$ only. Additionally, diagonal $Ω0$ is necessary for the equality to hold as it ensures diagonal $B0$⁠. The intuition behind this unusual requirement is the following. Careful inspection of Equation (1.9) reveals that $∂2lt(It,θ)∂θu∂θv=0$ for every $u≠v$⁠, because the joint objective function is additive and each summand is evaluated at a single $θs$ for $s∈{1,…,S}$⁠. This leads to the diagonal $Ht(It,θ)$ for any $θ$⁠, and so $A0$ is diagonal, as well. Corollary 1 summarizes this discussion with the results on only independence over $t$ and the full independence.

 

While part b) of Corollary 1 produces information matrix equality, notice that we do not have a typical –1 in front of $B0$⁠. This is so, because we minimize the negative log-likelihood. In order to estimate $A0$⁠, we can simply use $A^=1T∑t=1THt(It,θ^)=1T∑t=1THt(It,θ0)+op(1)$⁠, which holds by the consistency of $θ^$ and the fact that ${Ht(It,θ)}$ obeys uniform weak law of large numbers (see the Supplement). Generally, $B0≠A0$⁠, therefore next we discuss construction of $B^$ and establish its consistency under different dependence assumptions.

1.3 Consistent Estimation of the Asymptotic Variance Matrix

A5 (i) For all $x∈R$⁠, we have $|k(x)|≤1$ and $k(x)=k(−x)$⁠; $k(0)=1$⁠; $k(x)$ is continuous at zero and almost everywhere on $R$⁠. Lastly, $∫R|k(x)|dx<∞$⁠.

(ii) Also, $|k(x)|≤l(x)$⁠, where $l(x)$ is a non-increasing function such that $∫R|x||l(x)|dx<∞$⁠.

A6  $ST→∞$ as $T→∞$ we have $ST=o(T1/2−1/c)$ for some $c∈(2,4]$⁠.

A7 For some $c∈(2,4]$ and some $p>c$⁠, we have $2(1/c−1/p)>(r−2)/r$ for $r>2$ when ${st(θ)}$ is $α$-mixing sequence.

Assumption A5 (i) is standard in the literature and it can be found in Hansen (1992), among others. It covers, non-exhaustively, the Bartlett kernel (set $k(x)=1−|x|$ for $x≤1$⁠), the Parzen kernel, and the Quadratic Spectral kernel. A5 (ii) and A6 are the requirements put forward by de Jong (2000) in his correction of Hansen’s proof. The latter condition regulates the expansion rate of the band-width $ST$⁠, while the former condition is needed in case of dependent processes in order to rigorously prove weak consistency of $B^$ first considered in Hansen (1992) (see Theorem 2 therein). Intuitively, it connects asymptotic behavior of $ST$ to the moments of $st(θ)$⁠. Finally, A7 makes sure that the mixing condition in A2 employed for asymptotic normality is also sufficient for the consistent estimation of $B0$⁠.

Further, based on the discussion of Corollary 1, let $A^=1T∑t=1THt(It,θ^)$⁠, which is an estimator of $A0$⁠, such that in total $Σ^=A^−1B^A^−1$ is the full estimator of the asymptotic variance. Proposition 3 establishes consistency of $Σ^$⁠.

 

As we can see, $Σ^$ is consistent under dependence over $t$ and across the intraday seasons in $It$⁠. Under full independence, Corollary 1 suggests that $Σ^=A^−1$⁠. That is, the matrix information equality is “reproduced” by the asymptotic variance estimator. However, in practice it likely that $Iut$ and $Ivt$ are still dependent for $u≠v$⁠, while serial correlation may be absent. Intuitively, in such case we expect that $1T∑t=1T−jst(θ^)st+j(θ^)′→p0S×S$ as $T→∞$⁠, and $1T∑t=1Tst(θ^)st(θ^)′$ should approximate $B0$⁠. Consequently, it should approximate $A0$ under independence across the seasons. Corollary 2 formalizes this discussion.

 

1.4 Wald Test

 

Part a) of the corollary says that the Wald statistic is distributed as a $χ2(S−1)$ in the limit, whereas part b) says it is consistent under the alternative. Note that under the conditions of Corollary 2 a), we have $(RΣ^R′)−1=(RA0−1B0A0−1R′)−1+op(1)$ with $B0=E(st(θ0)st(θ0)′)$⁠, whereas under b), $(RΣ^R′)−1=(RA0−1R′)−1+op(1)$⁠. Therefore, $WT→dχ2(S−1)$ as $T→∞$ under these respective independence conditions.

1.5 LM Test

 

Again, part a) in the corollary says the LM statistic is $χ2(S−1)$ under the null, and part b) says the test is consistent under the alternative. Similarly to the Wald statistic, $LMT$ is robustified against dependence over time $t$ and across the seasons due to normalization by $(RΣ˜R′)−1$⁠. Also, the chi-squared distribution remains under independence.

1.6 LR Test

 

Unlike Wald or LM tests, $LR$ test is more sensitive to dependence. Again, this occurs due to the information matrix equality. $LRT$ asymptotically attains nuisance-free chi-squared distribution if and only if $B0=A0$⁠, conditions of which are discussed in Corollary 1. The information matrix equality is necessary, because the time dynamics and other types of dependence should be fully specified under $H0$ for $LRT$ to have a nuisance-free asymptotic distribution (see White (1996) for a discussion on obtaining the distribution via simulation techniques). However, in our case, neither dependence over $t$⁠, nor across the seasons is explicitly modeled.

Independence over $t$ can be relaxed if $B0=ξ02A0$ for some $ξ02$⁠. Then we have $B0=diag[b0,11,…,b0,SS]=b0×IS$⁠, where $b0,uu=limT→∞1T∑t=1T∑l=1TE(sut(θ0)sul(θ0))=b0$ for $u∈{1,…,S}$ under $H0$⁠. This gives $ξ02=b0(1−θ0)θ0$⁠, as now $A0=1θ0(1−θ0)×IS$⁠. Then $LR˜T=ξ˜−2LRT→d χ2(S−1)$ as $T→∞$ under $H0$⁠, where we evaluate at $θ˜$⁠. Note that $ξ02=1$ under independence over $t$⁠, because $b0=1(1−θ0)θ0+limT→∞1T∑t=1T∑l≠tTE(sut(θ0)sul(θ0))=1(1−θ0)θ0$⁠.

2 Simulations

This section studies the finite sample properties of the tests. Here, our focus is exclusively on size properties under the null. The main reason for this is that, as we will see in the empirical study of Section 3, our preferred tests are clearly very powerful in actual real-world applications, since the null is comprehensively rejected at the usual significance levels. Moreover, as proved in Section 1, the tests are consistent under the alternative (i.e., they are asymptotically powerful). Simulations under the alternative are contained in the Supplement.

2.1 Dependency Schemes

2.2 Estimation of $Σ0$

Arguably, this is the most commonly used kernel in HAC estimation in empirical econometrics.

2.3 Results

We conduct an extensive number of simulations, only a subset is reported here. Table 1 contains the results of the Wald test. In the upper part of the table, $B0$ is estimated by $B^Ord$⁠, whereas in the lower part $B0$ is estimated by $B^HAC$⁠. In DGPs A and B, the Wald test performs well when the number of intraday periods $S$ is small, for example, $S=2$ or $S=8$⁠. As $S$ increases, however, the empirical size worsens considerably. This is the case in both the upper and lower parts of the table. Another characteristic present in both the upper and lower parts is that, when the sample $T$ increases, the relative rejection frequency gets closer and closer to its nominal counterpart $α$⁠. However, the value of $T$ required for this to happen is of little practical relevance. Only in rare situations will $T$ be sufficiently high in empirical applications. For DGPs C and D, the results are worse than for DGPs A and B. The relative rejection frequency is close to the nominal counterpart $α$ when $S=2$⁠. As $S$ increases, however, the empirical size worsens even more than in DGPs A and B. Again, this is the case for both estimators of $B0$⁠. As $T$ increases, the empirical size is better when $B^HAC$ is used instead of $B^Ord$⁠. However, the rate at which the empirical size tends to the nominal counterpart $α$ is again too low to be useful in practice. All in all, therefore, the simulations suggest the practical usefulness of the Wald test is limited.

Table 2 contains the results of the LM test. The results are considerably better than for the Wald test. For DGPs A and B, the test is fairly well sized across most sample sizes $T$⁠, intraday periods $S$ and both estimators of $B0$⁠. Moreover, as $T$ increases, the relative rejection frequency gets closer and closer—as it should—to its nominal counterpart $α$⁠. For DGPs C and D, this is also the case in the lower part of the table (where $B0$ is estimated by $B˜HAC$⁠), but not in the upper part (where $B0$ is estimated by $B˜Ord$⁠). Moreover, as $T$ increases in the upper part of the table, the empirical size diverges from its nominal counterpart $α$ for intermediate and large values of $S$⁠. This suggests the LM test is inherently flawed under dependence if the ordinary estimator $B˜Ord$ is used. In other words, the results underline the importance of a dependence robust estimator of $B0$ in empirical practice, since empirical observations are frequently dependent over time (see Section 3). An important characteristic of the results in the lower part of the table is that the relative rejection frequency is usually undersized when $S$ is large (essentially when $S>32$⁠). A practical remedy to this problem is to simply include every other, say, second or third, intraday period in the test. Our theoretical results are fully compatible with this. Illustrations of this are contained in Section 3. In summary, the simulations suggest the LM test performs fairly well in finite samples and that the dependence robust estimator $B˜HAC$ should be preferred to the ordinary estimator. Also, our results suggest very clearly that the LM test should be preferred to the Wald test.

Table 3 contains the results of the LR test. Under DGP A, the relative rejection frequency is very close to its nominal counterpart $α$ across values of $T$ and $S$⁠. In fact, the finite sample properties are slightly better than those of the LM test in the comparable part (i.e., DGP A in the upper part) of Table 2. Under DGP B, the relative rejection frequency of the LR test is also very close its nominal counterpart $α$⁠. It should be noted, however, that this is despite the asymptotic invalidity of the LR statistic under DGP B (i.e., contemporaneous cross-sectional dependence). See our discussion following Corollary 4. The reason the rejection frequency is approximately equal to $α$ in DGP B, even as $T$ increases, is that $Cov(Iut,Ivt)≈0$ for most pairs $u≠v$⁠. If, by contrast, the covariances were substantially different from 0 for $u≠v$ in the DGP, then the relative rejection frequency would also differ substantially from $α$ for large $T$⁠. Under DGPs C and D, the relative rejection frequency is not close to its nominal counterpart. Moreover, as $T$ increases it appears to diverge. This is as expected, since our asymptotic result is not compatible with dependence over $t$⁠. As we will see, the empirical results in Section 3 suggest the $Ist$’s can be substantially dependent over $t$⁠. All-in-all, therefore, our results suggest also here that the LM test should be preferred.

2.4 A Conservative Version of the LM Test

The term $LMT$ is the unadjusted LM statistic in Equation (2.15). The term $DF(S,T)$ is a non-random scalar that depends on $S$ and $T$ such that $DF(S,T)→0$ as $T→∞$ for fixed $S$⁠. Clearly, $LMT,Adj$ in Equation (2.7) has the same limiting distributions as $LMT$ in Equation (2.15). A simple example of the adjustment term is $DF(S,T)=S/T$⁠, in which $(1−DF(S,T))·T=T−S$⁠. This explains why we label the adjustment a Degrees-of-Freedom adjustment. Our exact specification of $DF(S,T)$ is nonlinear in both $S$ and $T$ and is derived from our simulations. The finite sample properties of the DF-adjusted LM statistic in Equation (2.7) are contained in Table 4. A comparison with the lower part of Table 2 confirms that our specification of $DF(S,T)$ makes the test more conservative for $S≤48$⁠, since the relative rejection frequency is now approximately equal to or lower than $α$ in almost all cases.

3 Is the Zero-Process Non-Stationary Periodic?

If the zero-processes of intraday returns are non-stationary periodic, then financial returns are not strictly stationary, even after scaling by unconditional volatility. This means a key assumption of most risk estimation methods is violated. Here, two empirical studies are conducted, one of intraday exchange rate returns, and one of intraday stock returns. We find that both are extensively characterized by a non-stationary periodic zero-process, in particular at high frequencies.

3.1 Intraday Exchange Rate Returns

Let $pi$ denote the log of an exchange rate at the end (close) of intraday period $i$⁠, and let $ri=pi−pi−1$ denote the log-return from one intraday period to the next. Table 5 contains the descriptive statistics of the non-zero process ${Ii}$ for the intraday returns of six exchange rates from 2 January 2017–31 December 2018 at the Forexite (https://www.forexite.com/) trading platform. The six exchange rates are USD/EUR, USD/JPY, GBP/USD, USD/CAD, AUD/USD, and USD/CHF. All exchange rates are quoted with four decimals. At the platform, trading takes place 24 hours on weekdays from 00:00 CET to 24:00 CET. There is no trading in the weekends or in holidays. At the hourly frequency, this corresponds to more than 12 000 observations for each exchange rate. At the 1-minute frequency, the highest frequency available in our data, this corresponds to about 746 000 observations for each exchange rate. As expected, the overall proportion of zero returns (⁠$π^0$⁠) increases with frequency. At the hourly frequency, the proportion ranges from 6.1% (GBP/USD) to 8.1% (USD/CHF), whereas at the 1-minute frequency the proportion ranges from 38.4% (GBP/USD) to 48.9% (AUD/USD and USD/CHF). In standard theory, intraday return is governed by $ri=σiηi$⁠, where $σi>0$ is the volatility and $ηi∈R$ is an iid innovation. This implies that also ${Ii}$ is an iid sequence. Empirically, however, the intraday first-order sample autocorrelations, denoted as $ρ^i(1)$ in Table 5, suggest ${Ii}$ is dependent, since the values range from 0.055 to 0.177 across exchange rates and frequencies. Moreover, the dependence appears to increase with frequency. Another notable characteristic is the dependence from one day to the next for an intraday period $s$ as measured by the maximum absolute first-order sample autocorrelation of $Ist$ and $Is,t−1$ (denoted as $maxs{|ρ^st(1)|}$ in the table). The values range from 0.097 to 0.191 across exchange rates and frequencies, which suggests there is slightly stronger first-order dependence across days for a period $s$ than intradaily, that is, from $i−1$ to $i$⁠. Interestingly, the dependence does not increase monotonously with frequency for all exchange rates. The dependence across days emphasizes the need for dependence robust testing procedures. Finally, Figure 1 contains the estimated unconditional intraday zero-probabilities at the 1-minute, 15-minute, and hourly frequencies. At the 1-minute frequency, the unconditional probability of a zero varies substantially within the day, and also within each of the three trading sessions: Asia (00 h CET—08h CET), Europe (08 h CET—17h CET) and the Americas (17 h CET—24h CET). At the 15-minute and hourly frequencies, the unconditional probabilities are lower throughout the day, and the variation within each of the three sessions is also lower.

The LM test emerged as the preferred test in the simulations of Section 2. Since the descriptive statistics suggest the $Ist$’s are dependent across days, we use the dependence robust estimator of the HAC-type described in Section 2 to estimate the asymptotic variance. Table 6 contains the results of two versions of the LM test. The first version of the LM test, denoted as $LMTHAC$⁠, is not subjected to the degrees of freedom adjustment described in Section 2.4. As is clear, the null of a constant unconditional intraday zero-probability is rejected at 1% for all exchange rates at all frequencies: Hourly, 30-minute, 15-minute, 5-minute(*), and 1-minute(*). The null is not rejected at the 5-minute frequency when all intraday periods are included in the test. However, as documented in the simulations, the test is undersized (and hence also under-powered) for large $S$⁠. When only a subset of the periods are considered, that is, the results labelled 5-minute(*), the tests also reject at 1%. Note that, for the 1-minute frequency, the tests corresponding to a maximum number of periods $S$ are not reported, since then $S$ is equal to or larger than $T$⁠.

The simulations in Section 2 showed that the $LMTHAC$ statistic can be slightly oversized for small $S$⁠, essentially when $S≤48$⁠, see the discussion in Section 2.4. For a more conservative test when $S$ is small, we derived the DF-adjusted LM statistic in Equation (3.7), which is more conservative in finite samples for small $S$⁠. Its statistic is denoted as $LMT,AdjHAC$ in Table 6. Qualitatively, the results are identical to the unadjusted LM test, since we again reject at 1% for all exchange rates at all frequencies. Also now is the null kept at the 5-minute frequency when all the intraday periods (⁠$S=288$⁠) are included. However, if we again only consider a subset of the periods to handle the problem of low power for large $S$⁠, that is, the results labelled 5-minute(*), then we again reject at 1% for all exchange rates.

Table 7 contains the results of the tests for the main trading period (i.e., when the European markets are open) from 08 h CET to 17 h CET. Compared to earlier, the results are more mixed. Still, if we focus on the adjusted test statistic $LMT,AdjHAC$⁠, our preferred test, then there is a clear pattern: Higher frequencies are more likely to be non-stationary. In particular, the null is always rejected at 1% at the 5-minute(*) and 1-minute(**) frequencies. At the 15-minute, 30-minute, and hourly frequencies, the null is sometimes kept, and the lower the frequency, the more often the null is kept.

In conclusion, our results suggest overwhelmingly that the zero-process of intraday exchange rate returns is characterized by non-stationary periodicity, even during the main trading hours, albeit not always at low frequencies.

3.2 Intraday Stock Price Returns

Let $pi$ denote the log of a stock price at the end (close) of intraday period $i$ so that $ri=pi−pi−1$ is the log-return from one intraday period to the next. We study the intraday zero-process of five stocks from 2 January 2019 to 31 December 2019. The five stocks are Amazon (AMZN), Facebook (FB), Microsoft (MSFT), Tesla (TSLA), and IBM (IBM). The datasource of the first four is First Rate Data (https://firstratedata.com/), whereas the datasource of the last is Kibot (http://www.kibot.com/). The stock prices are adjusted for splits and dividends. The price of FB is quoted with four decimals, the other four with two. Trading takes place from 04:00 EST to 20:00 EST on weekdays, but not in weekends or in holidays. The upper part of Figure 2 contains the estimated unconditional intraday zero-probabilities at the 1-minute frequency. For lower frequencies (middle and lower parts), the intraday evolution is similar but less erratic, and at lower probability levels. As is clear, the unconditional probability of a zero can vary greatly within the day, but the extent and exact pattern depends strongly on frequency. For the 1-minute frequency, for example, during the main trading hours from 09:30 to 16:00, the unconditional probability of a zero is low and usually below 5% for all five stocks (except IBM). Outside the main trading hours, by contrast, the unconditional probability of a zero is usually much higher, and much more erratic. As the frequency falls, the zero-probabilities fall and become less erratic, and the distinction between inside and outside main trading hours becomes less clear.

Table 8 contains the descriptive statistics of the main trading period. The overall proportion of zero returns (⁠$π^0$⁠) is much lower than those of the exchange rates, since they vary from 0.6% (Amazon at 60 minute) to 13.4% (IBM at 1 minute). However, just as for exchange rates, the overall proportion of zeros increases with frequency. The intraday first-order sample autocorrelations $ρ^i(1)$ range from 0.069 to 0.752 across stocks and frequencies. This means the intraday non-zero process ${Ii}$ can be strongly dependent over time intradaily, that is, from $i−1$ to $i$⁠. Indeed, the large maximum implies the dependence can be substantially stronger than for exchange rates. The dependence does not always increase with frequency, for example, the dependence decreases with frequency for Facebook, Microsoft and IBM. The dependence from one day to the next for an intraday period $s$ as measured by $maxs{|ρ^st(1)|}$ ranges from 0.016 to 0.496 across stocks and frequencies. Just as for the exchange rate returns, the dependence does not increase monotonously with frequency for all stocks. Contrary to the case of exchange rates, however, the first-order dependence across days for a period $s$ is almost always weaker than the intradaily dependence (as measured by $ρ^i(1)$⁠). The dependence across days emphasizes, again, the need for dependence robust testing procedures.

Table 9 contains the results of two tests applied to the main trading period. The two tests, $LMTHAC$ (unadjusted) and $LMT,AdjHAC$ (DF-adjusted), are the same as those used for exchange rates. Note that, in most cases, the number of intraday periods $S$ included in the test does not equal the maximum number available. The reason is that some of the across-day series ${Ist}t=1T$ are identical, and this leads to non-invertible matrices and hence incomputable test statistics. This primarily happens when liquidity is high and there are few zeros. To avoid non-invertible matrices, all but one of the identical series ${Ist}t=1T$ are therefore excluded so that the test-statistics can be computed. If we focus on the more conservative DF-adjusted test, then the null is rejected in at least one test at each frequency at 10% for Amazon, Facebook, and Tesla. In fact, in most of the tests for these stocks the $p$-value is less than 1%. For Microsoft and IBM, the results are more mixed, but overall there is extensive evidence of non-stationary periodicity also here at the usual significance levels (i.e., 10%, 5%, and 1%). For Microsoft, the $p$-value for the 30-minute frequency is 54%. However, for the other frequencies the $p$-value is always less than 5% for at least one test at each frequency. For IBM the $p$-values for the hourly and 15-minute frequencies are 67% and 10.2%, respectively, whereas for the other frequencies the $p$-values range from 0.1% to 2.7%.

Table 10 contains the results of the tests applied across all trading hours, that is, from 04:00 EST to 20:00 EST. As is clear, all the tests reject the null at all the usual significance levels at all frequencies. The only exception is the 5-minute test, where all periods are included. However, when only a subset of the periods are included, that is, 5-minute(*), then all tests reject at the usual significance levels.

All-in-all, our results suggest the zero-process of liquid intraday stock returns are comprehensively characterized by non-stationary periodicity. Not only over the whole trading day, but also in the main trading period. Additional exploration (see the Supplement) suggests the intraday periodicity in the main trading session may not be stable over time. An in-depth study of whether this is the case, and to what extent, is beyond the scope of this article. So we leave this for future research.

4 Conclusions

The zero-process of observed intraday financial returns can be periodic in non-stationary ways. When this is the case, standard risk estimation methods are invalid, since they rely on stationarity assumptions, possibly after scaling by unconditional volatility. Accordingly, tests for non-stationary periodicity in the zero-process of intraday financial returns can be of great value in identifying a suitable risk estimation method. We derive three tests for this purpose: a Wald test, a LM test, and a LR test. All three tests are user-friendly yet flexible and powerful. The tests are user-friendly in that they are intuitive and simple to implement, since they are made up of well-known and easy-to-compute ingredients, and since their asymptotic distributions under the null are standard. The tests are flexible, since they can be applied to both low and high-frequency data. The tests are powerful, since they are consistent under the alternative. The preferred test among the three is the LM test, since it is robust, and since the simulations under the null show that its relative rejection frequency is close to its nominal counterpart in finite samples of empirical relevance. The LM test is robust in the sense that knowledge of the exact specification that govern the zeros need not be known, and in the sense that the test holds under mild assumptions on the dependence of the zero-process, both intradaily and across days. Finally, our empirical study reveals that intraday financial returns of liquid exchange rates and liquid stocks are widely characterized by non-stationary periodicity in the zero-process.

Our findings have important and potentially wide-ranging implications for future research. Standard risk estimation methods are invalid when the zero-process is characterized by non-stationary periodicity. Our results, together with those of other recent studies, document that intraday financial returns are widely and comprehensively characterized by non-stationary periodicity in the zero-process. In our empirical study, the financial assets are very liquid. For illiquid assets, by contrast, which arguably constitute the greater portion of the world’s assets, the non-stationary characteristics are likely to be even more pronounced. Currently, very few risk estimation methods exist that accommodate non-stationary periodicity in the zero-process of intraday returns. Also, currently there exists very little research that explores the practical implications for actual decision-making. In consequence, studies that help fill these gaps will be of great value. A limitation of our tests is that they do not generate insight into the source or sources of the non-stationary periodicity, for example, external factors. This may be useful in the development of new methods for risk estimation.

Supplemental Material

Supplemental material is available at Journal of Financial Econometrics online.

We are grateful to the participants at the ISNPS 2024 (June, Braga), Zaragoza Workshop on Time Series Econometrics (April 2024), the CFE 2023 conference (December, Berlin), the QFFE 2023 workshop (June, Marseille), the internal economics seminar at BI Norwegian Business School (June 2023), and the econometrics seminar at the Lund University (May 2023) for their helpful comments, suggestions, and questions.

Footnotes

References