Abstract
Long-run asset pricing restrictions in a macro term structure model identify discretionary monetary policy separately from a policy rule. We find that policy discretion is an important contributor to aggregate risk. In addition, discretionary easing coincides with good news about the macroeconomy in the form of lower inflation, higher output growth, and lower risk premiums on short-term nominal bonds. However, it also coincides with bad news about long-term financial conditions in the form of higher risk premiums on long-term nominal bonds. Shocks to the rule correlate with changes in the yield curve’s level. Shocks to discretion correlate with changes in its slope.
When a central bank follows a monetary policy rule, it commits to a target interest rate that depends on the state of the macroeconomy. However, when policy is also subject to random departures from this rule, which we refer to as monetary policy discretion, the central bank also introduces a potential new source of risk.1 The interest rate we observe, which we refer to as the policy rate, therefore, is the sum of the target rate and discretion. Our goal is to add to the traditional analysis of how monetary policy affects the macroeconomy by exploring the relationship between policy discretion and financial market conditions.
Most studies to date explore the consequences of shocks to interest rate policy in one of two ways. The most common method imposes sufficient structure on the behavior of discretionary policy to identify an exogenous policy shock (e.g., Clarida, Gali, and Gertler 1998; Christiano, Eichenbaum, and Evans 1999). Alternatively, using futures data to measure unexpected shocks to the policy rate requires less structure, but these shocks will necessarily combine both target rate and discretionary shocks (e.g., Kuttner 2001; Piazzesi and Swanson 2008).
We use elements from these two approaches to explore different sources for monetary policy risk. We use financial market data to reveal the shock to the policy rate. We then impose just enough structure on the behavior of discretionary policy to separately identify its properties from those of the target rate. Since we don’t want to wed ourselves to any particular theory of discretion, we impose only long-run neutrality restrictions. These apply across a broad class of both new Keynesian and neoclassical models, and are likely to conform to most economists’ prior beliefs. The features of the data we uncover, therefore, can serve as a benchmark for assessing the relative merits of competing candidate theories.
We specify a macro term structure model that characterizes the dynamics of the pricing kernel as well as macro variables, such as inflation and output growth. This is a convenient setting in which to explore questions of monetary policy through interest rate targeting. In theory, this framework can characterize the dynamic responses of both real and nominal variables to unobservable shocks to policy discretion, but in practice, this model alone is not enough. In the words of Joslin, Le, and Singleton (2013): “several recent studies interpret the short-rate equation as a Taylor-style rule. However, without imposing additional economic structure the parameters are not meaningfully interpretable as the reaction coefficients of a central bank.” This echoes the concerns of Cochrane (2011), who writes “the crucial Taylor rule parameter is not identified in the new Keynesian model.”
The novelty in our solution to this identification problem is the addition of a long-run real asset pricing restriction to the more customary long-run real output restriction. We assume that the discretionary monetary policy shock does not have a permanent effect on either the level of real output or the level of real asset values. But since this does not restrict short-run responses, our empirical model is still consistent with most reduced-form econometric models as well as most structural models used in macroeconomics and macro-finance.
We should be clearer about our use of “shocks” in this informal discussion to avoid confusion later. Here we use it to describe unexpected changes from any and all sources. We use long-run restrictions on the impact of unexpected changes to identify structural parameters. We do not use these restrictions to create an orthogonal structure to the exogenous innovations of our dynamic system. As a result, we will not know which shock—or if any shock—in our model can be interpreted as an exogenous change in monetary policy that does not also cause a change in macro and financial variables. Therefore, we cannot conduct the impulse response exercises emblematic of most empirical work in this area, such as structural vector autoregressions. But since we do identify a process for discretionary policy—just not an exogenous innovation to that policy—we are still able to see how it relates to key macroeconomic and financial variables.
In that context, we find that most macroeconomic and financial market variables are related to a shock to the policy rate in much the same way as they are to a shock to just the target rate. What this masks, however, is that shocks to discretionary policy exhibit substantially different behavior, especially with respect to output growth and risk premiums. For example, both discretionary easing and target rate easing tend to coincide with good news about inflation. However, for output growth, discretionary easing tends to coincide with good news whereas easing through the target rate coincides with bad news. This evidence is broadly consistent with patterns in the empirical macro literature that find that discretionary policy exhibits a preference for interest rate smoothness or inertia (e.g., Clarida, Gali, and Gertler 2000; Rudebusch 2006), or slow adjustment of long-run inflation expectations (e.g., Gurkaynak, Sack, and Swanson 2005a).
With respect to financial markets, we find that discretionary policy is an important contributor to both the mean and variance of risk premiums. We can attribute as much as 20% of the average forward premium on a 10-year discount bond to discretionary policy, which contrasts with its negligible contribution for short-maturities. We can also attribute about 17% of the variance of forward premiums of all maturities to discretionary policy.
We find that easing through either the target rate or discretion tends to coincide with bad news about long-term financial market conditions in the form of an unexpected increase in the term premium on long-maturity nominal bonds. However, discretionary and target rate easing exhibit substantially different patterns in short-term nominal bond markets: discretionary easing tends to coincide with good news in the form of unexpected decreases in the risk premiums of short-term nominal bonds, whereas target rate easing coincides with increases in these same risk premiums.
Finally, since the patterns for target rate shocks are similar for both short- and long-term yields, movements in the target rate are closely related to movements in the level of the yield curve. However, since the discretionary shock is more closely related to movements in short-term yields, movements in discretionary policy are closely related to movements in the slope of the yield curve.
1. A Macro Term Structure Model
1.1 Abritrage-free pricing
Affine term structure models have become the standard framework for empirical term structure research. In the macro-finance branch of this literature, the state includes macroeconomic variables like inflation and output growth. Examples include Ang and Piazzesi (2003), Moench (2008), Rudebusch and Wu (2008), Chernov and Mueller (2012), Jardet, Monfort, and Pegoraro (2013), Hamilton and Wu (2012), Joslin, Le, and Singleton (2013), and Joslin, Priebsch, and Singleton (2014).
See Appendix A for the standard derivation of |$\mathcal{B}_0^{(h)}$| and |$\mathcal{B}^{(h)}$| as functions of the model’s parameters.
The risk-neutral dynamics of the state, |$x_t=A_0^*+A^*x_{t-1}+Bw_t$|, are governed by parameters |$A^* = A-B\lambda$| and |$A^*_0 = -B\lambda_0$| (see Appendix A), and |$\mathcal{A}^{*(h)}=(I-A^*)^{-1}(I-A^{*h})$|.
1.2 Term structure identification
Identification of this model, therefore, will be specific to the choice of the matrix |$T$| that defines the state variable. We adopt the canonical form used by Hamilton and Wu (2012), Joslin, Le, and Singleton (2013), and Joslin, Priebsch, and Singleton (2014), among many others, and choose a transformation that results in a diagonal matrix governing the dynamics under the risk-neutral distribution. We therefore set the columns of the matrix |$T^{-1}$| equal to the eigenvectors of the matrix |$A-B\lambda$|, which results in a diagonal matrix |$TA^*T^{-1}$|. It is important to note that this choice of a rotation affects only the interpretation of the state variable. It does not restrict the behavior of our model of the pricing kernel.
For notational simplicity, we drop the transformation given by the matrix |$T$|. From here on we will refer to the parameters we identify as |$A$|, |$B$|, and a diagonal matrix |$A^*$|. But the reader should be aware that by assuming a diagonal |$A^*$|, we have chosen to work with a specific rotation of the abstract state space.
Finally, note that the conditional variance of bond yields, |$\mathcal{B}^{(h)}BB^\top \mathcal{B}^{(h)\top}=a^\top (I-A^*)^{-1}(I-A^{*h})BB^\top(I-A^{*h})(I-A^*)^{-1}a$|, cannot distinguish |$a$| from |$B$|. We therefore set |$a$| equal to a vector of ones so that the factors, |$x_t$|, inherit the same scale as the short rate, |$i_t$|. Note that we will continue to use the same notation, but from hereon, |$a$| will denote an |$n$|-dimensional vector of ones.
1.3 Macro variables and the real pricing kernel
Our empirical exercise does not use any data on real asset prices; nonetheless, the real pricing kernel in Equation (5) will play an important role in the identification of discretionary monetary policy.
2. Monetary Policy
Nothing in the specification of our model would allow us to attach any particular economic interpretation to the state variables, |$x_t$|; hence, we are not yet at a point at which we can talk about policy shocks in a concrete way. To get to this point, we must first introduce a specific policy rule and explore how shocks to that rule evolve with the state variable, |$x_t$|.
2.1 A policy rule
The necessity for a policy discretion term is self-evident since data on the short rate will never perfectly align with the data on inflation and output growth that set the target rate. The fundamental determinants of this discretion, however, remain outside of our model. Conceptually, many possibilities exist that seem plausible. For example, it could reflect the short-term political pressure at odds with the long-term economic goals of a central bank that lacks complete independence. Or it could reflect the market microstructure of interactions between the central bank and its network of private brokers when implementing a particular policy rule. Or real-time measurement error in inflation and output. Or a central bank’s desire to keep the short rate positive. Or its response to a financial crisis. These all suggest that the central bank’s interest rate policy will depend on more than just |$\pi_t$| and |$g_t$| through the target rate, |$i^R_t$|. “Even strong proponents of simple policy rules generally advise that they be used only as guidelines, not as substitutes for more complete policy analyses," in the words of Chairman Bernanke (2005).
The presence of the policy discretion term, |$s_t$|, in the specification of monetary policy is intended to capture these possibilities. However, each of these examples, either individually or in some combination, is likely to result in a process for the policy discretion term that depends on the state of the economy in a different way. In the empirical exercise below, we won’t attempt to model |$s_t$| explicitly, rather we simply allow the data to determine the process for policy discretion. But as we show in the next section, since |$s_t$| is not directly observable, we must first place additional structure on the model to achieve identification.
2.2 Identification of monetary policy
Incorporating monetary policy into our term structure model introduces an additional identification problem beyond the term structure identification discussed above.4 To keep this discussion separate, assume that we already know the values of all of the parameters of the macro term structure model. Specifically, we know the factor loadings for inflation and output growth, that is, the parameter vectors |$b$| and |$c$|, as well as the intercept parameters |$a_0$|, |$b_0$|, and |$c_0$|. (Recall that the factor loadings for the interest rate have been scaled to equal a vector of ones.) The only parameters left to identify and estimate are those of the target rate, |$\tau_0$|, |$\tau_g$|, and |$\tau_\pi$|, and policy discretion, |$d$|.
2.2.1 The problem
Since monetary policy discretion is observed only indirectly as the difference between the policy rate and the target rate, a necessary first step must be to identify the parameters that determine the target rate separately from the parameters that determine policy discretion. For example, if we observe the policy rate increase when we see inflation and output increase, we could be tempted to conclude that this is a natural consequence of the policy rule. Likewise, if we see the policy rate increase when inflation and output are constant, we could be tempted to conclude that this must be the result of policy discretion. But since equilibrium inflation and output respond to both changes in the target rate and changes to policy discretion simultaneously, we can draw no such conclusions. Additional structure is needed to separately identify each of these effects.
In other words, monetary policy added |$n+1$| new restrictions to our system of equations. But it also added |$n+3$| new parameters, |$\tau_0$|, |$\tau_\pi$|, |$\tau_g$|, and |$d$|. Note also that there are no additional identifying restrictions provided by the equations for multiperiod bonds yields. Identification of the equations for bond yields did not depend on monetary policy, and likewise, monetary policy does not depend on those bond yields. Even if we had exact prior knowledge of all the parameters of the pricing kernel, they would be of no help identifying the parameters of the monetary policy rule.
Even if |$b_0$|, |$a_0$|, |$b$|, and |$A$| were known, we are still left with an underidentified system of equations: in this case |$n+1$| equations in |$n+2$| unknowns, |$\tau_0$|, |$\tau_\pi$| and |$d$|.
This example is simple but telling. Since |$s_t$| is unobservable, we cannot estimate |$d$| directly. And measuring |$s_t$| as a residual in the policy rate equation requires prior knowledge of the other policy parameters, |$\tau_0$|, |$\tau_\pi$|, and |$\tau_g$|, which we do not have. We need at least two additional restrictions on |$d$| to identify the Taylor rule in (6). Note that two restrictions will be enough to identify the two policy parameters |$\tau_\pi$| and |$\tau_g$|, so that |$\tau_0$| can be identified from the equation |$\tau_0=a_0-\tau_\pi b_0-\tau_t c_0$|, which will then identify the policy discretion term, |$s_t$|, as a residual so that we can estimate the remaining |$n$| independent parameters of |$d$|. Additional restrictions on |$d$| may add information, but are not strictly necessary for identification.
It is worth highlighting that this lack of identification has nothing to do with the reduced-form nature of our arbitrage-free macro term structure model and nothing to do with our definition of the policy rule. Placing additional economic structure on the pricing kernel, the macroeconomic variables, or the policy rule itself does not alter the monetary policy identification problem if that structure does not somehow restrict the policy parameters, |$\tau_0$|, |$\tau_\pi$|, |$\tau_g$|, or |$d$|. (See Appendix B for examples.)
2.2.2 The role of instrumental variables
Finally, identification via an instrumental variables estimator presupposes the existence of at least two valid instruments, |$z_{it}$|, such that |$E(z_{it} s_t)=0$|, for |$i=1,2$|. In the context of our state-space model, |$z_{it}$| is a function of the state variable, say |$z_{it}=\beta^\top_i x_t$|, for a vector of parameters |$\beta_i$|. Instrumental variables estimation, therefore, requires |$E(z_{it} s_t)=E(\beta_i^\top x_t x_t^\top d)=\beta_i^\top V_x d = 0$|. In other words, this identification is predicated on knowledge of at least two additional restrictions on the parameters of the policy rule, restrictions that are not part of the specification of the model. Without an economic theory of |$s_t$| that places additional structure on its behavior, the motivation and interpretation of any instrumental variables estimator is open to question. Unfortunately, macroeconomic theory is generally quite vague about the fundamental determinants of policy discretion even though a policy-discretion variable is a standard feature of most structural models of monetary policy.
The general feature of the identification problem highlighted in these simple examples extends to more complicated models: unless a model places explicit restrictions on the parameters of the policy rule it will be fundamentally underidentified. In the next section, we introduce and justify a set of such restrictions and integrate them into our macro term structure model.
3. The Long-Run Neutrality of Discretionary Shocks
If we had exact prior knowledge of the values of |$\tau_\pi$| and |$\tau_g$|, then identification of the discretionary shock would require nothing more beyond the identification of the parameters of the macro term structure model. To take a concrete example, if we were certain that |$\tau_\pi=1.5$| and |$\tau_g = 0.5$|, then given values for |$a$|, |$b$|, and |$c$|, the discretionary shock is simply |$s_t=(a-1.5b-0.5c)^\top x_t$|. Likewise, if we had exact prior knowledge of the values of |$d$|, then identification of the target rate parameters would require nothing more beyond the identification of the parameters of the macro term structure model. For example, if we were certain that the third factor in a three-factor macro term structure model was an exogenous AR(1) process for discretionary policy, that is, |$s_t=d_3 x_{3t}$|, then we also would be certain that |$d_1=0$| and |$d_2=0$|. The target rate parameters, |$\tau_\pi$| and |$\tau_g$|, would then simply solve the linear equations |$\tau_\pi b_1 + \tau_g c_1 = a_1$| and |$\tau_\pi b_2 + \tau_g c_2 = a_2$|. Unfortunately, we have neither a theoretical nor an empirical justification for assuming such exact prior knowledge either about the target rate parameters or about the discretionary policy parameters. And since identifying assumptions cannot be tested, relying on our personal intuition or sheer guesswork is unlikely to lead to convincing empirical conclusions.
Rather than assuming exact prior knowledge about specific parameter values governing monetary policy, perhaps we would have more confidence assuming exact prior knowledge of some properties of the joint distribution of variables in our model. For example, if we have exact prior knowledge that current and lagged values of the discretionary shock are uncorrelated with real output growth, that is, |$E[g_t s_t]=0$| and |$E[g_t s_{t-1}]=0$|, then parameters of the macro term structure model provide two linear equations, |$c^\top V_x (a-\tau_\pi b -\tau_g c)=0$| and |$c^\top A V_x (a-\tau_\pi b -\tau_g c)=0$|, that we can solve for |$\tau_\pi$| and |$\tau_g$|. Unfortunately, such assumptions are also unlikely to lead to convincing empirical conclusions since they are inconsistent with both new Keynesian sticky price models and neoclassical models with frictions. In essence, we would be assuming away one of the most interesting and long-standing questions in macroeconomics, ironically, for the sake of identifying monetary policy. That question being “Does monetary policy have real effects?” Did we simply make an unfortunate choice using |$g_t$|? Why not something like a long-term bond yield, |$y_t^{(n)}$|, instead? Couldn’t we just substitute |$\mathcal{B}^{(n)}$| for |$c$| in those two linear restrictions to identify the target rate parameters? In principle, yes of course. But if we want to allow for the possibility that real output growth affects the pricing kernel and, hence, bond valuations, while maintaining a channel for discretionary policy to affect real output growth, then we are right back in the same situation. We would have achieved identification by ruling out the most commonly used structural asset pricing models, and as a result, our empirical conclusions would rightly be viewed with skepticism.
Where does that leave us? We could go on and on describing potential identifying restrictions, then questioning their usefulness as the basis for our empirical exercise. Once again, without specifying a complete structural model, such an exercise would amount to little more than speculation and subjective personal intuition. But there is no free lunch! To proceed we must necessarily restrict some features of our model and our empirical conclusions must necessarily depend entirely on those restrictions.
We would ideally want an approach that is flexible enough to accommodate as large a set of structural models as possible yet still restrict the parameters of monetary policy. To that end, we will focus exclusively on the long-run consequences of temporary shocks to discretionary policy. One feature shared across a very broad class of structural models, and which is likely to conform to most economists’ prior beliefs, is that a temporary shock to discretionary monetary policy may affect the real economy, but that effect will not be permanent. We will now show how that seemingly weak and robust requirement can be used for identification, then how that identification translates to the empirical properties of our macro term structure model.
3.1 A long-run quantity restriction
In other words, even when |$n$| is arbitrarily large, the current shocks to the state of the economy, |$w_{t+1}$|, continue to have effects on |$\log Y_{t+n}$| directly through |$g_{t+1}$| and indirectly through the conditional means of |$g_{t+j}$|, |$j=2,\ldots,n$|.
This alternative derivation doesn’t add much value to the more intuitive direct method we originally used to derive Equation (10). In the next section, however, we will need to apply similar arguments to the nonlinear process for the real pricing kernel, |$m_{t+1}$|, in which case the analog to the Beveridge-Nelson decomposition given in Alvarez and Jermann (2005), Hansen and Scheinkman (2009), and Hansen (2012), will prove much more useful. Nonetheless, providing the permanent-transitory decomposition for |$g_{t+1}$| may help clarify the parallel interpretation of our two identifying restrictions. It is also helpful to see that a model with the restriction |$E_t[g_{t+1}s_{t+1}]=E_t[g_{t+1}^T s_{t+1}]$| still has ample (if not total) flexibility to match any short-run relationship between discretionary policy and real output evident in the data.
The restriction in Equation (10) is analogous to the long-run restriction on monetary policy shocks used in structural VAR models popularized by Blanchard and Quah (1989). In fact, if we could observe |$s_t$|, then it could be included in a VAR with |$g_t$|, and (10) would be the natural identifying restriction in the Blanchard-Quah methodology. In the context of our model, however, |$s_t$| is unobservable. Therefore, we use this restriction to identify structural parameters of the policy rule rather than to orthogonalize the shocks in a structural VAR. The result of this identification will be a process for discretionary monetary policy, |$s_t$|, with the desired long-run neutrality property.
It is also important to note that imposing comparable restrictions, implying that shocks to other variables in our model have no impact on the level of long-run real output, do not restrict the parameters of discretionary policy, |$d$|, or the target rate, |$\tau_0$|, |$\tau_\pi$|, or |$\tau_g$|. For example, assuming that the shock to inflation has no permanent effect on the level of real output implies the constraint |$c^\top (I-A)^{-1}BB^\top b = 0$|, or similarly the shock to nominal interest rates, |$c^\top (I-A)^{-1}BB^\top a = 0$|, or the shock to any long-term bond yield, |$c^\top (I-A)^{-1}BB^\top \mathcal{B}^{(n)} = 0$|. None of these constraints involves the parameters of discretionary policy, |$d$|, or the target rate, |$\tau_0$|, |$\tau_\pi$|, or |$\tau_g$|. Adding such restrictions will overidentify the parameters of the macro term structure model and may be of interest for other reasons, but they will not help identify the unknown policy parameters.
Alternatively, we could consider adding other observable real quantities to our macro term structure model, such as consumption and investment, and impose similar restrictions on the long-run responses of their levels to discretionary policy shocks as additional identifying restrictions. These would indeed involve |$d$| and, in principle, could help with identification. However, given the evidence that the levels of other real variables are cointegrated with the level of real output (e.g., Engle and Granger 1991), such restrictions are unlikely to add much new information beyond (10), and would at best provide a very weak identification. Instead, we make use of our pricing kernel model and explore real asset prices as the source for additional—and as-yet-unexplored—identifying restrictions.
3.2 A long-run asset pricing restriction
In other words, even when |$n$| is arbitrarily large, current shocks to the state of the economy, |$w_{t+1}$|, continue to have effects on real marginal valuations, |$\log \mathcal{M}_{t+n}$|, directly through |$m_{t+1}$| and indirectly through the conditional means of |$m_{t+j}$|, |$j=2,\ldots,n$|. Unlike our model of real quantities, however, the real pricing kernel has an additional channel for |$w_{t+1}$| to affect long-run real valuations: directly through the price of risk in |$m_{t+2}$| and indirectly through the conditional means of the prices of risk in |$m_{t+j}$|, |$j=3,\ldots,n$|.
Again, what concerns us isn’t how an arbitrary shock to the state of the economy affects long-run real valuations, but rather the shock specifically to discretionary policy, |$s_{t+1}=d^\top x_{t+1}$|. What we would like to rule out is that specific linear function of the state affecting the conditional forecast of |$\log \mathcal{M}_{t+n}$| when |$n$| is large.
That is, the real risk price of a shock to policy discretion can be measured with its covariance with the return on a real consol bond.
The first term in this expression is analogous to the permanent component of the Beveridge-Nelson decomposition we saw previously. But here it is applied to the process for the conditional mean of the risk-neutral distribution of the real pricing kernel. The second term recognizes the fact that shocks to the pricing kernel may also have a separate permanent effect through the price of risk. Therefore, we must restrict the combined correlation through both channels.5
Note that given the dependence of this restriction on the price-of-risk parameters, |$\lambda_0$| and |$\lambda$| (recall |$A^*=A-B\lambda$|), a well-specified pricing-kernel model is essential to the construction of this second identifying restriction.6
Intuitively, the value-added of this asset-pricing-based restriction relative to the quantity-based restriction in Equation (10), is in the way it restricts the long-run response of real risk prices to a discretionary monetary policy shock. To see this, consider the log real pricing kernel in a typical new Keynesian macro model like the one discussed in Section 2.2, |$m_{t+1}=\delta_0+\delta g_{t+1}$|, which is based on a representative agent with CRRA expected utility (we’re now interpreting |$g_{t+1}$| as real consumption growth as in an endowment economy). The real risk prices in this model are constant so that risk-neutral persistence is the same as physical persistence, that is, |$A^* = A$|. In addition, from Equation (B.1) we know that |$\lambda_0^\top B^{-1}=\delta c^\top$| and |$(b-a)^\top = \delta c^\top A$|. In this case, Equation (14) reduces to |$c^\top (I-A)^{-1}BB^\top d =0$|. This is identical to the restriction in Equation (10). A simpler way to see this, of course, is just to note that the moving-average coefficients in this linear time-series model of the log real pricing kernel are equal to |$\delta$| times the moving-average coefficients of |$g_{t+1}$|; hence, its permanent component is proportional to that of |$g_{t+1}$|. And obviously, any model that shares this feature with CRRA expected utility will likewise produce a redundant restriction. Since this is not the case for our model with state-dependent risk prices, Equation (14) will add an additional restriction that we can use to identify the parameters of monetary policy.
4. Empirical Results
Our empirical exercise proceeds sequentially. In the first step we estimate the parameters of the macro term structure model detailed in Section 1. Note that the restrictions in Section 3 that will ultimately identify monetary policy play no role in this step. By design, the identification assumptions of the factor model outlined in Section 1.2 do not restrict either the temporary or permanent components of the pricing kernel. And the same is true for completely unrestricted factor loadings for the macro variables in Equation (4). In other words, any equilibrium implications of monetary policy or the restrictions in Section 3 will be captured in the estimates of the macro term structure model through the observed behavior of interest rates, inflation, output growth, and multiperiod bond yields.
The second step is to substitute the parameter values from this macro term structure estimation into the two restrictions that identify the monetary policy parameters given in Equations (10) and (14), and solve for estimates of |$\tau_g$| and |$\tau_\pi$|. And those two values will then imply a value for |$d=a-\tau_\pi b - \tau_g c$|. In that sense, the parameters of our monetary policy model will be just identified even when the macro term structure model is overidentified.
4.1 Macro term structure estimation
We estimate the affine term structure model as outlined in Section 1 using quarterly U.S. data from 1980Q3 to 2019Q4 and generalized method of moments based on the conditional moment restrictions of our model. (See the Internet Appendix for details.) The sample period begins one year after Volcker’s ascendance as Fed chair to allow his monetary regime to establish credibility.7 For the quarterly interest rate, we use the Fama-Bliss data (available from CRSP), and for longer maturities we use continuously compounded default-free pure-discount bond yields as measured by Gurkaynak, Sack, and Wright (2007). All yields are measured on the last day of the quarter. Real gross domestic product (GDP) growth rates are from the National Income and Product Accounts, and core CPI inflation comes from the Bureau of Labor Statistics. Inflation is measured as the quarter-to-quarter change in the average monthly CPI. Growth rates are continuously compounded at annual rates. Figure 1 displays these standard data for our sample period.
U.S. GDP growth, CPI inflation, and yields
The time period is 1980Q3 to 2019Q4. Real GDP growth is from the NIPA and CPI inflation is from the BLS, both downloaded from FRED. The short rate is from Fama and Bliss (available from CRSP), and yields are from Gürkaynak, Sack, and Wright (2007). (Along with these three maturities, we also used a 12-quarter yield in our estimation, which is not plotted to avoid making the graph too dense.) All variables are reported as percentages, continuously compounded at annual rates.
Table 1 summarizes the results from the GMM estimation. The estimated parameter values have a number of noteworthy features. The risk-neutral dynamics encoded in the nonzero elements of |$A^*$|, are substantially more persistent than the actual dynamics of the state space in |$A$|. The absolute values of the four eigenvalues of |$A$|, which govern the persistence in the process for |$x_t$|, are |$0.9775$|, |$0.6705$|, |$0.6705$|, and |$0.3180$| (the middle pair correspond to complex conjugates). The diagonals of |$A^*$| are |$0.9961$|, |$0.8814$|, |$0.8348$|, and |$0.3638$|, and they are very precisely estimated—a consequence of the additional information in the cross-equation restrictions implied by the absence of arbitrage—and are all significantly different than zero.
GMM estimation
| A. State dynamics | ||||||||
|---|---|---|---|---|---|---|---|---|
| |$A$| | |$B$| | |||||||
| 0.9169 | 0.2825 | 0.2617 | 0.6601 | 0.0022 | 0 | 0 | 0 | |
| (0.0400) | (0.0459) | (0.0588) | (0.1311) | (0.0002) | ||||
| 0.4680 | 0.5501 | –0.0713 | –0.9401 | –0.0056 | 0.0110 | 0 | 0 | |
| (0.1885) | (0.0623) | (0.0354) | (0.4700) | (0.0030) | (0.0048) | |||
| –0.4394 | 0.1510 | 0.7531 | 0.6428 | 0.0042 | –0.0011 | 0.0022 | 0 | |
| (0.1676) | (0.0766) | (0.0686) | (0.4458) | (0.0031) | (0.0051) | (0.0002) | ||
| 0.0028 | –0.0668 | –0.0570 | 0.4147 | –0.0004 | 0.0010 | –0.0010 | 0.0014 | |
| (0.0036) | (0.0215) | (0.0287) | (0.0722) | (0.0003) | (0.0005) | (0.0003) | (0.0001) | |
| A. State dynamics | ||||||||
|---|---|---|---|---|---|---|---|---|
| |$A$| | |$B$| | |||||||
| 0.9169 | 0.2825 | 0.2617 | 0.6601 | 0.0022 | 0 | 0 | 0 | |
| (0.0400) | (0.0459) | (0.0588) | (0.1311) | (0.0002) | ||||
| 0.4680 | 0.5501 | –0.0713 | –0.9401 | –0.0056 | 0.0110 | 0 | 0 | |
| (0.1885) | (0.0623) | (0.0354) | (0.4700) | (0.0030) | (0.0048) | |||
| –0.4394 | 0.1510 | 0.7531 | 0.6428 | 0.0042 | –0.0011 | 0.0022 | 0 | |
| (0.1676) | (0.0766) | (0.0686) | (0.4458) | (0.0031) | (0.0051) | (0.0002) | ||
| 0.0028 | –0.0668 | –0.0570 | 0.4147 | –0.0004 | 0.0010 | –0.0010 | 0.0014 | |
| (0.0036) | (0.0215) | (0.0287) | (0.0722) | (0.0003) | (0.0005) | (0.0003) | (0.0001) | |
| |$A^*$| | |$\lambda_0$| | |$b$| | |$c$| | ||||
| 0.9961 | 0 | 0 | 0 | –0.0625 | 0.4947 | 0.0158 | |
| (0.0006) | (0.0053) | (0.0060) | (0.0071) | ||||
| 0 | 0.8814 | 0 | 0 | –0.1421 | 0.4243 | –0.1025 | |
| (0.0020) | (0.0191) | (0.0126) | (0.0379) | ||||
| 0 | 0 | 0.8348 | 0 | –0.0128 | 0.4751 | –0.2578 | |
| (0.0035) | (0.0355) | (0.0169) | (0.0438) | ||||
| 0 | 0 | 0 | 0.3638 | –0.5803 | 0.5927 | 0.0311 | |
| (0.0102) | (0.0777) | (0.0181) | (0.1221) | ||||
| |$A^*$| | |$\lambda_0$| | |$b$| | |$c$| | ||||
| 0.9961 | 0 | 0 | 0 | –0.0625 | 0.4947 | 0.0158 | |
| (0.0006) | (0.0053) | (0.0060) | (0.0071) | ||||
| 0 | 0.8814 | 0 | 0 | –0.1421 | 0.4243 | –0.1025 | |
| (0.0020) | (0.0191) | (0.0126) | (0.0379) | ||||
| 0 | 0 | 0.8348 | 0 | –0.0128 | 0.4751 | –0.2578 | |
| (0.0035) | (0.0355) | (0.0169) | (0.0438) | ||||
| 0 | 0 | 0 | 0.3638 | –0.5803 | 0.5927 | 0.0311 | |
| (0.0102) | (0.0777) | (0.0181) | (0.1221) | ||||
| |$\tau_\pi$| | |$\tau_g$| | |$d^\top$| | ||||
| 1.6230 | 0.6532 | 0.1868 | 0.3783 | 0.3973 | 0.0177 | |
| (0.3122) | (0.2164) | (0.1549) | (0.1471) | (0.1939) | (0.1732) | |
| |$\tau_\pi$| | |$\tau_g$| | |$d^\top$| | ||||
| 1.6230 | 0.6532 | 0.1868 | 0.3783 | 0.3973 | 0.0177 | |
| (0.3122) | (0.2164) | (0.1549) | (0.1471) | (0.1939) | (0.1732) | |
Based on moment restrictions (see Equations (3) and (4) in the Internet Appendix) for a sample period 1980Q3 to 2019Q4. Asymptotic standard errors are in parentheses. State dynamics: |$x_{t+1} = Ax_t + B w_{t+1}$|. Macro term-structure model: |$i_t = a_0+a^\top x_t$|, |$\pi_t = b_0+b^\top x_t$|, |$g_t = c_0+ c^\top x_t$|, |$h y_t^{(h)} = \mathcal{B}_0^{(h)} + \mathcal{B}^{(h)}x_t$|, and |$\mathcal{B}^{(h)} = a^\top (I-A^*)^{-1}(I-A^{*h})$|, |$\mathcal{B}_0^{(h)}=a_0+\mathcal{B}_0^{(h-1)}-\mathcal{B}^{(h-1)}B\lambda_0-\mathcal{B}^{(h-1)}BB^\top \mathcal{B}^{(h-1)\top}/2$|. Policy: |$i_t = \tau_0+\tau_\pi \pi_t+ \tau_g g_t + d^\top x_t$|. The state variable |$x_t$| is four-dimensional, |$i_t$| is the short interest rate (1 quarter), |$y_t^{(h)}$| is the yield on a discount bond of maturity |$h=4,\,12,\,20,\,40$| (quarters), |$\pi_t$| is the inflation rate, |$g_t$| is the growth rate of real GDP, and |$a^\top =[1 ;\ 1 ;\; 1 ;\; 1]$|. Values for intercepts are fixed at their sample means. The absolute value of the eigenvalues of |$A$| are 0|$.9775$|, |$0.6705$|, |$0.6705$|, and |$0.3180$|.
GMM estimation
| A. State dynamics | ||||||||
|---|---|---|---|---|---|---|---|---|
| |$A$| | |$B$| | |||||||
| 0.9169 | 0.2825 | 0.2617 | 0.6601 | 0.0022 | 0 | 0 | 0 | |
| (0.0400) | (0.0459) | (0.0588) | (0.1311) | (0.0002) | ||||
| 0.4680 | 0.5501 | –0.0713 | –0.9401 | –0.0056 | 0.0110 | 0 | 0 | |
| (0.1885) | (0.0623) | (0.0354) | (0.4700) | (0.0030) | (0.0048) | |||
| –0.4394 | 0.1510 | 0.7531 | 0.6428 | 0.0042 | –0.0011 | 0.0022 | 0 | |
| (0.1676) | (0.0766) | (0.0686) | (0.4458) | (0.0031) | (0.0051) | (0.0002) | ||
| 0.0028 | –0.0668 | –0.0570 | 0.4147 | –0.0004 | 0.0010 | –0.0010 | 0.0014 | |
| (0.0036) | (0.0215) | (0.0287) | (0.0722) | (0.0003) | (0.0005) | (0.0003) | (0.0001) | |
| A. State dynamics | ||||||||
|---|---|---|---|---|---|---|---|---|
| |$A$| | |$B$| | |||||||
| 0.9169 | 0.2825 | 0.2617 | 0.6601 | 0.0022 | 0 | 0 | 0 | |
| (0.0400) | (0.0459) | (0.0588) | (0.1311) | (0.0002) | ||||
| 0.4680 | 0.5501 | –0.0713 | –0.9401 | –0.0056 | 0.0110 | 0 | 0 | |
| (0.1885) | (0.0623) | (0.0354) | (0.4700) | (0.0030) | (0.0048) | |||
| –0.4394 | 0.1510 | 0.7531 | 0.6428 | 0.0042 | –0.0011 | 0.0022 | 0 | |
| (0.1676) | (0.0766) | (0.0686) | (0.4458) | (0.0031) | (0.0051) | (0.0002) | ||
| 0.0028 | –0.0668 | –0.0570 | 0.4147 | –0.0004 | 0.0010 | –0.0010 | 0.0014 | |
| (0.0036) | (0.0215) | (0.0287) | (0.0722) | (0.0003) | (0.0005) | (0.0003) | (0.0001) | |
| |$A^*$| | |$\lambda_0$| | |$b$| | |$c$| | ||||
| 0.9961 | 0 | 0 | 0 | –0.0625 | 0.4947 | 0.0158 | |
| (0.0006) | (0.0053) | (0.0060) | (0.0071) | ||||
| 0 | 0.8814 | 0 | 0 | –0.1421 | 0.4243 | –0.1025 | |
| (0.0020) | (0.0191) | (0.0126) | (0.0379) | ||||
| 0 | 0 | 0.8348 | 0 | –0.0128 | 0.4751 | –0.2578 | |
| (0.0035) | (0.0355) | (0.0169) | (0.0438) | ||||
| 0 | 0 | 0 | 0.3638 | –0.5803 | 0.5927 | 0.0311 | |
| (0.0102) | (0.0777) | (0.0181) | (0.1221) | ||||
| |$A^*$| | |$\lambda_0$| | |$b$| | |$c$| | ||||
| 0.9961 | 0 | 0 | 0 | –0.0625 | 0.4947 | 0.0158 | |
| (0.0006) | (0.0053) | (0.0060) | (0.0071) | ||||
| 0 | 0.8814 | 0 | 0 | –0.1421 | 0.4243 | –0.1025 | |
| (0.0020) | (0.0191) | (0.0126) | (0.0379) | ||||
| 0 | 0 | 0.8348 | 0 | –0.0128 | 0.4751 | –0.2578 | |
| (0.0035) | (0.0355) | (0.0169) | (0.0438) | ||||
| 0 | 0 | 0 | 0.3638 | –0.5803 | 0.5927 | 0.0311 | |
| (0.0102) | (0.0777) | (0.0181) | (0.1221) | ||||
| |$\tau_\pi$| | |$\tau_g$| | |$d^\top$| | ||||
| 1.6230 | 0.6532 | 0.1868 | 0.3783 | 0.3973 | 0.0177 | |
| (0.3122) | (0.2164) | (0.1549) | (0.1471) | (0.1939) | (0.1732) | |
| |$\tau_\pi$| | |$\tau_g$| | |$d^\top$| | ||||
| 1.6230 | 0.6532 | 0.1868 | 0.3783 | 0.3973 | 0.0177 | |
| (0.3122) | (0.2164) | (0.1549) | (0.1471) | (0.1939) | (0.1732) | |
Based on moment restrictions (see Equations (3) and (4) in the Internet Appendix) for a sample period 1980Q3 to 2019Q4. Asymptotic standard errors are in parentheses. State dynamics: |$x_{t+1} = Ax_t + B w_{t+1}$|. Macro term-structure model: |$i_t = a_0+a^\top x_t$|, |$\pi_t = b_0+b^\top x_t$|, |$g_t = c_0+ c^\top x_t$|, |$h y_t^{(h)} = \mathcal{B}_0^{(h)} + \mathcal{B}^{(h)}x_t$|, and |$\mathcal{B}^{(h)} = a^\top (I-A^*)^{-1}(I-A^{*h})$|, |$\mathcal{B}_0^{(h)}=a_0+\mathcal{B}_0^{(h-1)}-\mathcal{B}^{(h-1)}B\lambda_0-\mathcal{B}^{(h-1)}BB^\top \mathcal{B}^{(h-1)\top}/2$|. Policy: |$i_t = \tau_0+\tau_\pi \pi_t+ \tau_g g_t + d^\top x_t$|. The state variable |$x_t$| is four-dimensional, |$i_t$| is the short interest rate (1 quarter), |$y_t^{(h)}$| is the yield on a discount bond of maturity |$h=4,\,12,\,20,\,40$| (quarters), |$\pi_t$| is the inflation rate, |$g_t$| is the growth rate of real GDP, and |$a^\top =[1 ;\ 1 ;\; 1 ;\; 1]$|. Values for intercepts are fixed at their sample means. The absolute value of the eigenvalues of |$A$| are 0|$.9775$|, |$0.6705$|, |$0.6705$|, and |$0.3180$|.
Many of the off-diagonal elements of the matrix |$B$| are significantly different from zero, suggesting that our vector of state variables does not have orthogonal innovations. This will play an important role when we explore the model’s dynamics below.
The inflation rate has significant loadings (the values of |$b$|) for all four factors. On the other hand, real GDP growth has significant loadings (the values of |$c$|) for only the first three factors. This suggests that the model is capturing a purely nominal feature in the data with the fourth latent factor.
The average price of risk, |$\lambda_0$|, is negative for all four latent factors and appears to be different from zero for three of the four factors. The third factor—the third-most persistent under the risk-neutral distribution—is both small in absolute value and statistically insignificant. The average price of risk for the first factor—the most persistent factor under the risk-neutral distribution—is negative and close to zero, but it is statistically significant. The least persistent factor has an average prices of risk that is relatively larger and is statistically significant.
Finally, since we will be basing our monetary policy identification on the decomposition of the real pricing kernel into permanent and transitory components, it would be reassuring if those components were consistent with asset pricing restrictions beyond the term structure moments used in estimation. Alvarez and Jermann (2005) propose using an entropy bound for this purpose. (See Appendix C for closed-form expression for various measures of entropy for our model.) At our point estimates, the ratio of unconditional entropy of the permanent component of the real pricing kernel to the unconditional entropy of the real pricing kernel itself is |$1.0267$|, which is consistent with the estimates of lower bounds provided in Alvarez and Jermann (2005).
4.2 Monetary policy estimation
Given estimates of the parameters of the macro term structure model and the two additional restrictions in Equations (10) and (14), we can identify the target rate parameters, |$\tau_0$|, |$\tau_\pi$|, and |$\tau_g$|, as well as the factor loadings for policy discretion, |$d$|. Note that once we have identified |$\tau_\pi$| and |$\tau_g$|, identification of |$\tau_0$| and |$d$| follows from Equation (6): |$\tau_0 = a_0 - \tau_\pi b_0 - \tau_g c_0$| and |$d=a-\tau_\pi b - \tau_g c$|. The bottom panel of Table 1 presents the estimates for these parameters. Asymptotic standard errors are calculated using the delta method. (See the Internet Appendix for details.)
It is both reassuring and surprising that although we have adopted a novel asset pricing approach for identification, the estimates for the Taylor rule parameters are quite conventional. The coefficient for |$\pi_t$| is |$1.6230$|, which safely satisfies the Taylor-principle stability condition, |$\tau_\pi>1$|. It is larger than Taylor’s original specification of |$\tau_\pi=1.5$|, suggesting a somewhat more aggressive inflation policy. Our estimate of |$\tau_g$| is |$0.6532$|, which is quite close to Taylor’s value of |$\tau_g=0.5$|, however, the units are difficult to compare directly as we use output growth rather than deviations from a potential-output trend.
The estimates of the factor loadings for the policy discretion term, |$d$|, are significant for only two of the four factors. The loading on the first factor—the most persistent factor—is small and insignificant, which stands in contrast to real GDP growth which has a small but significant loading on that factor. But similar to GDP growth, the policy shock does not appear to depend on the fourth factor. If we interpret that factor as a purely nominal factor, then this suggests that policy discretion is not its source. On the other hand, |$s_t$| depends significantly on the second and third factors, but in the opposite direction to GDP growth, which suggests a smoothing role for discretion. This two-factor structure is consistent with the findings of Gurkaynak, Sack, and Swanson (2005b), who exploit high-frequency data around FOMC announcements to measure changes in |$s_t$| directly, which they then relate to high-frequency movements in asset pricing factors.
To gain a sense of the sign and the magnitude of these policy shocks, the upper panel of Figure 2 plots the Taylor rule with and without the discretionary shock |$s_{t}$|. Notice how much smoothness policy discretion imparts to the short rate relative to the target rate. In addition, we observe large and persistent discretionary tightening prior to the last three recessions, followed by large and persistent discretionary easing after those recessions. The zero lower bound shows up as large positive discretionary tightening in 2009–2010, followed by persistent discretionary easing over the subsequent post crisis years of our sample. How exactly does our identification strategy separate the observable effects of |$i^R_t$| and |$s_t$|? Our numerical estimates may help clarify this natural question. Consider two models, the first has no policy discretion so that |$i_t = \hat{\tau_0}+\hat{\tau}_\pi \pi_t + \hat{\tau_g} g_t$|. The implied values are |$\hat{\tau}_\pi=2.5590$| and |$\hat{\tau}_g=0.8371$| (using |$d_2$| and |$d_3$|), which are not unreasonable but imply a more aggressive policy than most central banker’s would likely admit to. The second model has discretion policy, but it looks just like the policy rule itself, |$s_t= d_\pi \pi_t + d_g g_t$|, so that |$i_t=\tau_0 + (\tau_\pi+d_\pi)\pi_t + (\tau_g + d_g) g_t$|. Obviously, without further structure, the two models are identical. To separate the effect of |$d_\pi$| and |$d_g$| from |$\tau_\pi$| and |$\tau_g$|, we also assume that the policy maker wants the nominal interest rate risk created by their discretion to be uncorrelated with the levels of the long-run real economy. That means we need to impose the two linear restrictions given by Equations (10) and (14) on this second model. This results in parameters for this theory of policy discretion of |$d_\pi = 0.9360$| and |$d_g=0.1838$|, which in turn imply our estimates for the policy rule, |$\tau_\pi=1.6230$| and |$\tau_g=0.6532$|. The policy maker’s discretion amplifies the response to inflation and output growth of a less aggressive policy rule, but in a limited way that ensures that the long-run risk for the levels of the real economy are unaffected.
Target rate and discretionary policy shocks
The top panel plots the policy rate including the discretionary shock, that is, the short rate, and excluding the shock, that is, the target rate, |$i^R_t=\tau_0+\tau_\pi \pi_t+\tau_g g_t$| using estimated parameter values for |$\tau_0$|, |$\tau_\pi$|, and |$\tau_g$| from Table 1. The difference is the value of the shock, |$s_{t}$|, plotted in the lower panel along with the 5-year forward spread, |$f^{(20)}_t-i_t$|, that is, a long forward rate minus the current interest rate.
Is this simple example a good structural model of policy discretion? Almost certainly not since it implies that the short rate is spanned by inflation and output growth, which is easy to reject. But more generally, we don’t take a stand on those kinds of questions. Building deeper structural models of policy discretion is beyond the scope of this paper, as is building a deeper structural model of our reduced-form parameters for the pricing kernel, output growth, and inflation. Nonetheless, by identifying the reduced-form parameters of policy discretion using restrictions that are consistent with a wide range of different theories, we have provided an important first step in this broader research program. We know that to be consistent with term structure evidence, any structural model of preferences must result in a marginal rate of intertemporal substitution that looks like our estimated affine pricing kernel. Now, we also know that to be consistent with the evidence, any structural model of policy discretion must have a reduced form that looks like our estimate of |$d$|.
5. Implications of Monetary Policy Shocks
Since we have identified the parameters of the target rate and the policy disturbance as part of a dynamic macro term structure model, we can use that model to get a better understanding of how policy shocks are related to the rest of the economy. We consider both unconditional moments and dynamic correlations. We also use our identification of the discretionary shock to shed light on some historical policy conundrums.
5.1 Unconditional moments
The first line in Equation (15) is the part of the average forward premium that is attributable to the target rate, the second line is attributable to policy discretion, and the third line captures their interaction through the “convexity” term.8
The blue line in Figure 3 represents the fraction of the expected value of the forward premium that we can attribute to policy discretion. You can see that it is very small for short-maturity bonds. Almost all of the average premium for short bonds seems to be associated with the target rate. This share rises steadily, however, before settling down at approximately 20% for long maturity bonds.
Forward premium share of discretionary policy
The solid line plots the fraction of the unconditional mean of the forward premium attributable to the discretionary policy shock. The dashed line plots the fraction of the unconditional variance of the forward premium attributable to the discretionary policy shock.
The red line in Figure 3 plots the share of the variance of the forward premium that we can attribute to policy discretion. It is at more than 15% for all maturities and rises to approximately 20% around 1-year maturities. In other words, a nontrivial fraction of the volatility we see in nominal risk premiums is associated with volatility in discretionary policy.
5.2 Dynamic properties
Our model has a VAR structure, so it is tempting to consider impulse response functions as a way to track marginal dynamic responses to specific shocks. However, our identifying assumptions do not create an orthogonal system of shocks. Rather, the identified policy shocks are potentially affected by the entire vector of innovations, |$w_t$|, as are the other variables in the model.
By substituting the appropriate factor loadings for output growth, inflation, bond yields, etc., in place of |$\beta$| in these formulas, we can get a sense of how the different sources of monetary policy shocks are related to different aspects of the economy. To control for scale, we report these covariances as correlation coefficients. And, again, we caution the reader not to interpret these as impulse responses. Even though they are similar in appearance, as bivariate correlations, they contain different information.
Figure 4 plots these dynamic correlations for the two macro variables in our empirical model, inflation and real output growth. The top panels display the combined effect of a policy shock. The lower panels decompose that shock into its two components, with the blue line representing a shock to the target rate and the red line representing a shock to policy discretion. (Dashed lines represent 2-standard deviation confidence bounds.)
Dynamic correlations: Macro variables
The top panels plot the dynamic correlation with a shock to the policy rate for inflation and real output growth. The lower panels decompose this into the two sources for a policy shock, with the solid line representing a shock to the target rate and the dashed line representing a shock to policy discretion. Dotted lines represent 2-standard-deviation confidence bounds.
We find that an unexpected shock to the policy rate is positively correlated with unexpected inflation. The correlation is large and persistent remaining significantly different from zero out to about 3 years. That is, inflation today is still correlated with policy shocks from 3 years ago. The two components of the policy shock have similar effects, but policy discretion is less correlated, and that correlation is less persistent. Our model fails to find much of a relationship between unexpected shocks to real output growth and unexpected shocks to the policy rate. However, that appears to be the result of offsetting effects of the two components of that policy shock. A shock to the target rate is positively correlated with shocks to output growth, whereas an unexpected shock to policy discretion is negatively correlated. Both of these are significant, but neither is very persistent becoming indistinguishable from zero after 2 or 3 quarters.
In summary, unexpected easing through the target rate tends to coincide with good macroeconomic news on inflation and bad news on growth. This is almost a necessary feature of the model given the structure of the Taylor rule. Discretionary easing also tends to coincide with good news on inflation, but in contrast to the target rate, it coincides with good news on growth. And as discussed in Section 3, although our identifying assumptions necessarily places a restriction across the parameters of the macro term structure model and the discretionary shock, they do not dictate either the size or the sign of these correlations. Rather these correlations are implications of the data.
In Figure 5, we repeat this exercise for the short rate, that is, the policy rate, and the 5-year bond yield. The response of the short rate to a policy shock is obviously perfectly correlated with itself, but it is also persistently correlated with future interest rates remaining significant beyond 3 years. And echoing the behavior we saw for inflation, the response of the short rate is similar for shocks coming from either source, with the correlation of policy discretion being both smaller and less persistent.
On the right side of Figure 5, one can see that an unanticipated shock to the policy rate is positively correlated with shocks to the long end of the yield curve measured here with the 5-year bond yield. And that correlation is also quite persistent. This is consistent with evidence that motivates the search for additional sources of monetary policy shocks as in Boyarchenko, Haddad and Plosser (2017), Gurkaynak, Sack, and Swanson (2005a), and Hanson and Stein (2015). What we see from the lower panel, however, tells a very different story depending on the source of the policy shock. Shocks to the long bond yield are not correlated with discretionary policy shocks. The relationship between a policy-rate shock and shocks to long bond yields is entirely attributable to the correlation with the shock to the target rate. Once again, this is not a feature of the model or the way we identify the policy discretion term. In principle our model has the freedom to take on virtually any correlation pattern with any variable. This outcome is strictly a feature of monetary policy as reflected in our data.
Dynamic correlations: Interest rates
The top panels plots the dynamic correlation with a shock to the policy rate for the short rate and the 5-year bond yield. The lower panels decompose this into the two sources for a policy shock, with the solid line representing a shock to the target rate and the dashed line representing a shock to policy discretion. Dotted lines represent 2-standard-deviation confidence bounds.
We can also think of these patterns in terms of the customary level and slope factors of empirical term structure models. A monetary policy easing originating in a shock to the target rate tends to coincide with an unexpected decrease in the level of the yield curve since its behavior is similar at both the long and short ends of the curve. Discretionary policy easing, however, tends to coincide with an unexpected increase in the slope of the yield curve since it is only related to movements at the short end of the curve. Figure 6 depicts these correlations.
Dynamic correlations: Level and slope factors
The top panels plots the dynamic correlation with a shock to the policy rate for the level and slope factors calculated from the first two principal components of the variance of a vector of forward rates for maturities 1 to 5 years. The lower panels decompose this into the two sources for a policy shock, with the solid line representing a shock to the target rate and the dashed line representing a shock to the policy disturbance. Dotted lines represent 2-standard-deviation confidence bounds.
Long-term bond yields embed forecasts of future interest rates as well as risk premiums. We have already observed that policy-rate shocks from both sources have persistent positive correlations with future interest rates. But what about risk premiums? Nominal forward premiums given in Equation (3) provide a clean decomposition of these two effects. Figure 7 plots dynamic correlations of forward premiums shocks with policy-rate shocks for short-maturity forward rates of 3 months and long-maturity forward rates of maturity 5 years. The top panels suggest that nominal risk premium shocks may have a small and short-lived negative correlation with policy-rate shocks. However, once again, this is masking two very different and offsetting effects for each of the sources of the policy-rate shock. The correlation of risk premiums at the very short end of the maturity structure with a shock to policy discretion is positive but short-lived. The correlation of risk premium on long-maturity bonds, on the other hand, are negative and more persistent (significant out to about 1 year). In contrast, the correlation with shocks to the target rate are negative for both long and short maturities, but the lagged correlation becomes significantly positive for long maturities. In other words, the risk characteristics of monetary policy shocks depends very much on the source of those shocks.
Dynamic correlations: Risk premiums
The top panels plot the dynamic correlation with a shock to the policy rate for the forward risk premiums at horizons of 3 months and 5 years. The lower panels decompose this into the two sources for a policy shock, with the solid line representing a shock to the target rate and the dashed line representing a shock to the policy disturbance. Dotted lines represent 2-standard-deviation confidence bounds.
In summary, an unanticipated easing originating with the target rate tends to be associated with bad news about financial conditions summarized by higher-than-expected risk premiums on nominal bonds of both long and short maturities. On the other hand, an unanticipated easing of discretionary policy tends to coincide with good news in the form of lower-than-expected risk premiums on short-term nominal bonds, but bad news in the form of higher-than-expected risk premiums on long-term nominal bonds.
5.3 Policy conundrums
Without a structural model of the fundamental source for policy discretion we can only speculate on the cause of the correlations we find. However, these results do cast some light on the so-called conundrum that has puzzled policy makers in the past: long nominal bond yields often move in ways that appear disconnected from discretionary policy as depicted in Figure 5. Alan Greenspan (1994) attributed the increase in long yields in early 1994 to expectations of increases in future values of |$g_t$| and |$\pi_t$|: “In early February, we thought long-term rates would move a little higher as we tightened. The sharp jump in [long] rates that occurred appeared to reflect the dramatic rise in market expectations of economic growth and associated concerns about possible inflation pressures.” What our results and Figure 2 suggest is that the policy tightening through the tightening of the target rate was larger than the overall increase in the policy rate and that discretionary easing continued until 1995. As we’ve seen, target rate shocks tend to coincide with increases in the level of long bond yields, whereas discretionary easing is unrelated to those yields. In other words, this particular combination of target rate and discretionary policies tends to coincide with both an increase in the level of the yield curve and and increase in its slope, which seems to be what Chairman Greenspan found puzzling.
A decade later Greenspan (2005) once again voiced puzzlement regarding the behavior of long yields: “Long-term interest rates have trended lower in recent months even as the Federal Reserve has raised the level of the target federal funds rate by 150 basis points. Historically, even distant forward rates have tended to rise in association with monetary policy tightening... For the moment, the broadly unanticipated behavior of world bond markets remains a conundrum.” What our findings and Figure 2 suggest is that much of the tightening of the target rate in the years prior to this episode was undone through the discretionary part of policy. Discretionary easing had averaged about 200 basis points in the previous 3 years. Discretionary tightening then averages more than 200 basis points in the subsequent 3 years. In this case, this particular combination of target rate and discretionary policies tends to coincide with not only an increase in the level of the yield curve as Chairman Greenspan expected but also a decrease in its slope.
To paint an image of these correlations, the lower panel of Figure 2 plots the discretionary shock, |$s_t$|, along with a standard measure of the slope of the yield curve, the 5-year forward spread, |$f^{(20)}_t-i_t$|. The negative correlation between shocks to discretionary monetary policy and the slope of the yield seems to be a fairly consistent pattern throughout this time period, not just during Greenspan’s conundrums.
6. Conclusion
We estimate a macro term structure model and use it, along with long-run restrictions that are consistent with a wide variety of both new Keynesian and classical monetary models, to arrive at a novel identification of shocks to discretionary monetary policy. We explore the properties of discretionary shocks through their dynamic correlations with shocks to macroeconomic and financial market conditions. A deeper understanding of the fundamental causes of discretionary shocks and their consequences for financial markets requires a structural model of monetary policy discretion capable of accounting for the empirical facts we identify.
The value we derive from integrating asset pricing models with time-varying risk premiums into the identification and estimation of monetary policy was foreshadowed by Backus and Wright (2007). They concluded that analyzing monetary policy through the narrow lens of constant risk premiums was problematic: “We follow a long line of work in suggesting that expectations-hypothesis intuition, based on constant term premiums, is likely to be misleading.” They saw a need for research that connected monetary policy to fundamental shocks and ultimately to risk premiums embedded in interest rates: “The next step, in our view, should be to develop models in which macroeconomic policy and behavior can be tied more directly to the properties of interest rates.” We believe the findings summarized in this paper demonstrate the value of this insight, and that future work in this area will continue to benefit from the integration of models of macroeconomic policy with models of asset pricing.
Appendix A. Equilibrium Term Structure
Appendix B. Lack of Identification in Macro Models
Cochrane’s example showing the lack of identification of Taylor rule parameters extends to models with more macroeconomic structure. Demonstrating this for a number of standard models is instructive, both for understanding the general identification problem and our particular identification strategy.
B.1 A Structural Real Pricing Kernel
B.2 A Model with a Phillips Curve
The restrictions in (B.3) are different in form than those in (9) or (B.2), but they share the same unavoidable underidentification: |$n+1$| equations with |$n+3$| unknowns. Unless additional restrictions that involve the policy parameters are imposed on the model, monetary policy remain unidentified. And this is the best-case scenario, in which there are no identification issues with any other parameters of the model.
B.3 A Forward-Looking Taylor Rule
B.4 A Taylor rule with a lagged interest rate
Popular applications of the Taylor rule often include a lagged interest rate term, presumably to capture the central bank’s desire to smooth interest rate changes. To see how this interest rate smoothing behavior is captured by the policy discretion term in our abstract state-space representation, it is instructive to work out a simple example.
Two things can be noted about this equilibrium. First, the parameter |$d_s$| clearly affects the dynamics of the model: when |$d_s=0$|, inflation and the log of the nominal pricing kernel exhibit the same AR(1) dynamics as the original state variable, but when |$d_s\ne 0$|, both of these processes become ARMA(1,1). In both cases, the nominal interest rate has the same AR(1) dynamics as the original state variable; hence, the dynamics of interest rates do not identify the value of |$d_s$|. Second, augmenting the Taylor rule to include an interest rate smoothing term did not change the basic identification problem. Even if |$d_s$| were known with certainty, |$\tau_\pi$| and |$\tau_g$| are still unidentified. And if |$d_s$| is also unknown, the lack of identification is even worse: we would need yet another restriction on the model to separately identify |$d_s$| beyond the restrictions that we need to identify |$\tau_\pi$| and |$\tau_g$|.
Appendix C. Entropy of the Real Pricing Kernel
Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.
Acknowledgement
Earlier versions of this work circulated with the title “Identifying Monetary Policy in Macro-Finance Models.” We are grateful to the anonymous referees and Andrew Karolyi and Itay Goldstein (the editors) for their thoughtful comments, which have helped us improve this work. We also thank Rich Clarida, John Cochrane, Patrick Feve, Mark Gertler, Valentin Haddad, Kei Kawai, Stacey Polloway, Guillaume Roussellet, Ken Singleton, Gregor Smith, Roman Šustek, Tommy Sveen, and Tao Zha for comments on earlier drafts and participants attending seminars at Aarhus University, ADEMU/CERGE-EI, Banque de France, BI Norwegian Business School, Bilkent University, the Euro Area Business Cycle Network, HEC Paris, INSEAD, New York University, Queen Mary University of London, Stony Brook University, the Swedish House of Finance, and the Toulouse School of Economics. David K. Backus passed away on June 12, 2016. Zviadadze gratefully acknowledges financial support from Investissements d’Avenir Labex [ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047]. Supplementary data can be found on The Review of Financial Studies web site.
Footnotes
Cochrane (2011) and Sims and Zha (2006), among many others, refer to the difference between the actual policy rate and the target rate as a policy disturbance. Taylor (1999) uses the value-loaded term of policy mistake.
The reduced-form empirical evidence for nonstationary inflation e.g., Stock and Watson (2007) is weaker for the sample period we will use below than for the entire postwar period, although such evidence can be difficult to interpret in finite samples. Moreover, a nonstationary inflation process in the context of our model would result in a Taylor rule incompatible with a stationary nominal short rate and would also imply a real yield curve that tends to |$-\infty$| as maturity increases, which is inconsistent with the evidence from Treasury inflation-protected securities (TIPS) markets in Gurkaynak, Sack, and Wright (2010).
Taylor’s original formulation of the rule used the deviation of output from a trend, that is, potential output. We simplify this by using output growth itself rather than the detrended level. This choice better aligns our model with the macro term structure literature, while still capturing the intent of Taylor’s original rule.
See Backus, Chernov, Zin (2015) for a more thorough discussion.
Following Hansen (2012), we can use |$m^P_{t+1}$| in our affine setup to form a “twisted” process for the state variable analogous to the risk-neutral distribution, |$x_{t+1}=A^P_0+A^P x_t+B w_{t+1}$|, with the property that the permanent component of the valuation of a random payoff |$\exp\{g^\top x_{t+1}\}$|, given by |$E_t\exp\{m^P_{t+1}+g^\top x_{t+1}\}$| is equal to the expected value of that payoff under the twisted process, |$E^P_t\exp\{g^\top x_{t+1}\}$|, where |$A^P=A^*$| and |$A_0^P=(b-a)^\top (I-A^*)^{-1}BB^\top + A_0^*$|. In that context, the restriction in (13) is equivalent to the restriction |$E^P_t s_{t+1}=E_t s_{t+1}$|. In other words, the discretionary shock does not contribute any long-run real risk.
The restriction in (14) still has content when the real asset pricing kernel corresponds to a literal risk-neutral model, |$-m_{t+1}=r_t$|. In that case, |$A^*=A$| and |$\lambda_0=b^\top B$|, so that |$\lambda_t-b^\top B=0$|. The real marginal utility of wealth becomes a simple linear process with innovations given by the real interest rate, so that its permanent component is given by the Beveridge-Nelson decomposition. The identifying restriction in (14) reduces to |$(a^\top-b^\top A) (I-A)^{-1}BB^\top d=0$|, which still says that the discretionary policy shock is uncorrelated with the permanent component in the real marginal utility of wealth.
Based on our understanding of the events of that time as summarized in Goofriend and King (2005), we believe it prudent to avoid the temporary disruption caused by the imposition of credit controls at the beginning of 1980, which the Fed was able to effectively work around by the summer of that year. Since our focus is on a stable Taylor rule, we avoid using pre-Volcker data. The end of our sample is dictated by the data available when we undertook the estimation.
Following Campbell and Ammer (1993), we apportion the conditional covariance that appears in the “convexity” term of the unconditional mean of the forward premium equally across the two variables. We do the same for the unconditional variance of the forward premium in Equation (16).
