Monetary Policy with Uncertain Parameters Ulf S¨oderstr¨om∗ First version: March 1999 This version: February 2001 Forthcoming, Scandinavian Journal of Economics
Abstract This paper shows how uncertainty about the parameters in a dynamic macroeconomic model may lead to more aggressive monetary policy, in contrast to the accepted wisdom. In particular, when there is uncertainty about the persistence of inflation it may be optimal for the central bank to respond more aggressively to shocks to reduce uncertainty about the future development of inflation. Uncertainty about other parameters, on the other hand, acts to dampen the policy response. Keywords: Optimal monetary policy, parameter uncertainty, Brainard conservatism, interest rate smoothing. JEL Classification: E43, E52.
∗
Address: Research Department, Sveriges Riksbank, SE-103 37 Stockholm, Sweden;
[email protected]. I am grateful for helpful comments from Tore Ellingsen, Lars Ljungqvist, Marianne Ness´en, Anders Paalzow, Glenn Rudebusch, Anders Vredin, Carl Walsh, and two anonymous referees, as well as seminar participants at Sveriges Riksbank, Stockholm School of Economics, the ESEM99 conference in Santiago de Compostela and the ECB/CFS conference on “Monetary policy-making under uncertainty” in Frankfurt, December 1999. Financial support from the Tore Browaldh Foundation and the Jan Wallander and Tom Hedelius Foundation is also gratefully acknowledged. Any views expressed in this paper are those of the author, and do not necessarily reflect those of Sveriges Riksbank.
1
Introduction
After the seminal paper of Brainard (1967), policymakers have become comfortable with the notion that uncertainty about the parameters in the economy calls for a more cautious policy response to shocks than if acting under complete certainty (or certainty equivalence). The attractiveness of this result, named the “Brainard conservatism principle” by Blinder (1997, 1998), lies in both the simplicity of the original argument and in the underlying intuition. That the argument is well understood and used by central bankers in the practical policy process is made clear by, for example, Blinder (1998) and Goodhart (1999). Brainard was concerned with uncertainty about the transmission of policy to a target variable. It is less clear, however, whether his result also applies to uncertainty concerning other parameters in the economy. The purpose of this paper, therefore, is to analyze the effect of multiplicative parameter uncertainty in a dynamic macroeconomic model typically used for monetary policy analysis, developed by Svensson (1997, 1999). Recently, Svensson (1999) has shown that the Brainard result holds in a special case of that model: when there is uncertainty about some of the parameters, the optimal policy response to current inflation and output becomes less aggressive as the amount of uncertainty increases (that is, the coefficients in the policymaker’s optimal reaction function become smaller).1 Due to the complexity of the model with parameter uncertainty, however, Svensson chooses to analyze a special case, where only inflation (and no measure of output) enters the central bank’s objective function. In the present paper, Svensson’s analysis is extended to cover uncertainty about all parameters of the model, allowing the central bank to stabilize also output. Furthermore, both the optimal response coefficients and the time path of policy after a shock are examined. The results show that parameter uncertainty does not necessarily lead to more cautious policy responses, but may actually make policy more aggressive than under certainty equivalence. In particular, when the central bank puts some weight on stabilizing output in addition to inflation, uncertainty about the persistence of inflation increases the optimal reaction function coefficients. Uncertainty about other parameters, in contrast, always dampens the policy response. The reason is that when the dynamics of inflation are uncertain, the amount of uncertainty facing policymakers is greater the further away the inflation rate is from its target. Consequently, to 1
Similar results have been reached by, for example, Estrella and Mishkin (1999) and Wieland (2000). See also Schellekens (2000) for a discussion about caution and conservatism of monetary policy.
1
reduce the amount of uncertainty about the future path of inflation, optimal policy becomes more aggressive, pushing inflation closer to target.2 In contrast, the persistence of output is a crucial part of the transmission of policy to inflation, so uncertainty concerning the dynamics of output makes policy less aggressive. When parameter uncertainty does act to dampen the current policy response, it is optimal for the central bank to return to a neutral policy stance later than under certainty equivalence. This is due to the persistence of inflation and output: a smaller initial response leads to larger deviations of the goal variables from target in future periods, so policy needs to be away from neutral for a longer time to get the economy back on track. Thus, parameter uncertainty can lead to a smoother policy path in response to shocks, an issue analyzed in more detail by, for example, Sack (2000) and S¨oderstr¨om (2000). On the other hand, if the initial policy response is more aggressive under uncertainty, policy can return to a neutral stance earlier, leading to more a volatile policy response. The paper is organized as follows. In Section 2 the theoretical framework is presented and the optimal policy rule of the central bank is derived in a dynamic economy with stochastic parameters. Since analytical solutions of the model are unattainable, Section 3 presents numerical solutions for different configurations of uncertainty, to establish the effects of parameter uncertainty on optimal monetary policy. Finally, the results are discussed and some conclusions are drawn in Section 4.
2 2.1
The model Setup
The basic model used in the analysis is the dynamic aggregate supply-aggregate demand framework developed by Svensson (1997, 1999), which is similar to many other models used for monetary policy analysis, for example, by Ball (1999), Cecchetti (1998), Taylor (1994) and Orphanides and Wieland (2000). The model consists of two equations relating the output gap (the percentage deviation of output from its “natural” level) and the inflation rate to each other and to a monetary policy instrument, the short-term interest rate. Assuming a quadratic objective function for the central bank, the optimal policy rule is a linear function of current output and inflation, similar to a Taylor (1993) rule. 2 These results are closely related to those of Craine (1979), who shows that uncertainty about the impact effect of policy leads to less aggressive policy behavior, but uncertainty about the dynamics of the economy leads to more aggressive policy, albeit in a univariate model.
2
Important features of the model are the inclusion of control lags in the monetary transmission mechanism and the fact that monetary policy only affects the rate of inflation indirectly, via the output gap. Monetary policy is assumed to affect the output gap with a lag of one period, which in turn affects inflation in the subsequent period. Policymakers thus control the inflation rate with a lag of two periods. In the simplest version, including only one lag,3 the output gap in period t + 1, yt+1 , is related to the past output gap and the ex-post real interest rate in the previous period, it − πt , by the relationship yt+1 = αt+1 yt − βt+1 (it − πt ) + εyt+1 ,
(1)
where εyt+1 is an i.i.d. demand shock with mean zero and constant variance σy2 . The rate of inflation between periods t and t + 1, πt+1 , (or rather, its deviation from the long-run average inflation rate, given by the constant inflation target) depends on past inflation and the output gap in the previous period according to the Phillips curve relation πt+1 = δt+1 πt + γt+1 yt + επt+1 ,
(2)
where επt+1 is an i.i.d. supply shock with zero mean and variance σπ2 . Note that all variables are measured as deviations from their respective long-run averages. Thus, negative values of the interest rate are allowed. In the model presented here, there are two important modifications to the original Svensson framework: (1) the persistence parameter of the inflation process, δt+1 , is allowed to take values different from unity and (2) the parameters of the model are stochastic, and therefore time-varying. When the central bank sets its interest rate instrument at time t, it is assumed to know all realizations of the parameters up to and including period t, but it does not know their future realizations, and thus cannot be certain about the effect of policy on the economy.4 The parameters are assumed to be random variables with means E(αt+1 ) = α, E(βt+1 ) = β, E(γt+1 ) = γ 3
Rudebusch and Svensson (1999) and S¨ oderstr¨ om (2000) use a version of the model including four lags in each relationship, and estimate it on quarterly U.S. data. S¨ oderstr¨ om (2000) also formally tests the restrictions imposed by Svensson (1997, 1999). 4
That policymakers do not have complete information about the parameters in an economy is clearly not an unrealistic assumption. Holly and Hughes Hallett (1989) point to three reasons why a model’s parameters may be seen as stochastic: (1) they are genuinely random; (2) they are really fixed, but are impossible to estimate precisely, due to the sampling variability in a finite data set; and (3) they vary according to some well-defined but imperfectly known scheme, for example, because the model is a linearization around a trajectory of uncertain exogenous variables. Blinder (1997, 1998), Goodhart (1999) and Poole (1998) all stress the relevance of parameter uncertainty for practical monetary policy.
3
and E(δt+1 ) = δ, and variances σα2 , σβ2 , σγ2 and σδ2 . They are also assumed to be
independent of each other and of the structural shocks επt+1 and εyt+1 .5 Furthermore, the realizations of the parameters are drawn from the same distribution in each
period, so issues of learning and experimentation are disregarded.6 For simplicity, the model (1)–(2) does not include any forward-looking elements, a feature which could be seen as unrealistic.7 However, as shown by Estrella and Fuhrer (1998, 1999), purely forward-looking models of monetary policy are less successful in matching the data than backward-looking specifications, and they are not necessarily less sensitive to the Lucas critique. Also, hybrid models, including both forward- and backward-looking features, in many ways behave similarly to the purely backward-looking model used here. Furthermore, it is unclear how to solve models with forward-looking behavior and parameter uncertainty. 2.2
Optimal policy
To determine the optimal path for the interest rate, the central bank is assumed to minimize the expected discounted sum of future values of a loss function, which is quadratic in output and inflation deviations from target (here normalized to zero).8 Thus, the central bank solves the optimization problem min∞ Et
{it+τ }τ =0
∞ τ =0
φτ L(πt+τ , yt+τ ),
(3)
subject to (1)–(2), where in each period the loss function L(πt , yt ) is given by L(πt , yt ) = πt2 + λyt2 ,
(4)
5
The assumption of independence is convenient for the derivation of optimal policy, and may be realistic if the model equations (1) and (2) are interpreted as structural relationships. If, on the other hand, one interprets the model as reduced-form relations derived from microeconomic foundations, the parameters might well be correlated if they are derived from the same micro relations. Brainard (1967) shows how correlation between the parameters may overturn the conservatism result. Although such an extension would be interesting in a fully empirical analysis, it is not pursued here. 6 See Sack (1998) or Wieland (2000) for similar models of monetary policy including learning and experimentation; or Balvers and Cosimano (1994), Ba¸sar and Salmon (1990) and Bertocchi and Spagat (1993) for models in slightly different contexts. 7 Thus, any strategic interaction between the central bank and private agents is ruled out by construction. 8
The central bank is thus allowed to have explicit targets for both inflation and output. The output target is given by the natural level, so the central bank aims at a zero output gap (excluding the possibility of a systematic inflation bias). The inflation target pins down the long-run average inflation rate, so the target for π, the deviation of inflation from the average, is also zero.
4
and where φ is the central bank’s (constant) discount factor.9 The parameter λ ≥ 0 specifies the relative weight of output to inflation stabilization, and is assumed to be known and constant. In the simple case when parameters are non-stochastic, it is relatively straightforward to find an analytical solution for the optimization problem (3), as shown by Svensson (1997, 1999). When parameters are stochastic, however, finding an analytical solution is prohibitively difficult, so I shall here focus on numerical solutions.10 The inclusion of parameter uncertainty into this model will have an important effect on optimal policy. As is well known, optimal policy in a linear-quadratic framework with only additive uncertainty exhibits certainty equivalence. Consequently, the degree of uncertainty does not affect the optimal policy rule, which depends only on the expected value of the goal variables, so the central bank acts as in a non-stochastic economy. As will be clear below, when incorporating multiplicative parameter uncertainty into the model, certainty equivalence ceases to hold, and the variances of the state variables will affect the optimal policy rule. Thus, the amount of uncertainty facing policymakers has a decisive influence on their optimal behavior. To solve the central bank’s optimization problem it is convenient to rewrite the model (1)–(2) in state-space form as xt+1 = At+1 xt + Bt+1 it + εt+1 ,
(5)
where xt+1 = [ yt+1 πt+1 ] is a state vector and εt+1 = [ εyt+1 επt+1 ] is a vector of disturbances. The parameter matrices At+1 and Bt+1 are then stochastic with means α β −β , B = , A= (6) γ δ 0 9
The quadratic specification of the objective function is very common in the literature. Some authors, for example, Rudebusch and Svensson (1999) and Rudebusch (2000), include an interest rate smoothing objective in the loss function to capture the apparent preference of central banks for small persistent changes in the instrument. As shown by Sack (2000) and S¨ oderstr¨ om (2000), however, such a separate smoothing objective might not be necessary to mimic policy behavior in the U.S., at least not in an unrestricted VAR framework. Woodford (1999) demonstrates how a separate motive for interest rate smoothing can be beneficial in a forward-looking framework, since it brings the outcome of discretionary policy closer to the globally optimal outcome under commitment. 10
Svensson (1999) analytically solves a very simple case of parameter uncertainty, where δt+1 is non-stochastic and always equal to unity and where λ = 0. Since the most interesting results are obtained when λ > 0 and δt+1 is stochastic, I need a more general solution method.
5
and variance-covariance matrices
2 σα 0 0 σ2 β
ΣA =
0 0
0 0
0
0
σγ2
0
0
0
0
σδ2
,
ΣB =
σβ2
0
0
0
,
ΣAB =
0 −σβ2
0 0
.
0
0
0
0
(7)
Using the state-space formulation, the central bank’s optimization problem can be written as the control problem J(xt ) = min [xt Qxt + φEt J(xt+1 )] , it
(8)
subject to (5), where Q is a (2 × 2) preference matrix of the central bank, with λ and 1 on the diagonal and zeros elsewhere. The loss function will in this framework be quadratic, so J(xt+1 ) = xt+1 V xt+1 + w,
(9)
where the matrix V remains to be determined. To illustrate the effect of including multiplicative uncertainty into the model and show why certainty equivalence no longer holds, it is instructive to consider the expected value of the value function (9). In the general case, this expected value is Et J(xt+1 ) = (Et xt+1 ) V (Et xt+1 ) + tr(V Σt+1|t ) + w,
(10)
where Σt+1|t is the variance-covariance matrix of xt+1 , evaluated at time t, and the notation “tr” denotes the trace operator. The variance-covariance matrix is given by (11) Σt+1|t = Et [xt+1 − Et xt+1 ] [xt+1 − Et xt+1 ] , where xt+1 − Et xt+1 = (At+1 − A)xt + (Bt+1 − B)it + εt+1 .
(12)
When the parameters are non-stochastic, so At+1 = A and Bt+1 = B for all t, xt+1 − Et xt+1 = εt+1 , so the variance-covariance matrix Σt+1|t coincides with the variance-covariance matrix of the disturbance vector εt+1 , and thus is independent of the instrument it . Therefore, although the expected value of the objective function depends on the variances of the state variables, optimal policy cannot affect these variances. Consequently, the optimal policy rule is independent of the degree of uncertainty in the economy, so policy is certainty equivalent. In contrast, when the
6
parameters are uncertain, the variance-covariance matrix of the state variables also depends on the state of the economy (xt ), the instrument (it ) and the variances of the parameters, so it can be affected by monetary policy. Optimal policy will then minimize not only the future deviation of the expected state variables from target (via the term (Et xt+1 ) V (Et xt+1 )), but also their variance. Thus, certainty equivalence ceases to hold, and optimal policy depends crucially on the degree of uncertainty in the economy.11 Appendix A shows that the optimal decision rule for the central bank is to set the short-term interest rate as a linear function of the state vector in each period, that is, (13) it = f xt , where
f = − B (V + V ) B + 2v11 Σ11 B
−1
B (V + V ) A + 2v11 Σ11 AB .
(14)
Here Σij AB denotes the covariance matrix of the ith row of At+1 with the jth row of Bt+1 and vij denotes element (i, j) of the matrix V , which is given by iterating on the Ricatti equation V
= Q + φ(A + Bf ) V (A + Bf )
11 11 22 + φv11 Σ11 A + 2ΣAB f + f ΣB f + φv22 ΣA .
(15)
To obtain an analytical solution for this problem, one would need to solve equations (14)–(15) for the fixed-point value of V . For some simple configurations, for example, in the non-stochastic case, this is manageable (although tedious), since the system of equations obtained is relatively small. In this setup of multiplicative parameter uncertainty, however, the system of equations is highly non-linear and far too complicated to yield a usable solution. Therefore I proceed by numerical methods to analyze the optimal behavior of the central bank in this setting. 11
The same argument can be made by considering the expected future value of the loss function 2
2 + λyt+τ Et L(πt+τ , yt+τ ) = Et πt+τ 2 2 = (Et πt+τ ) + Vart (πt+τ ) + λ (Et yt+τ ) + Vart (yt+τ ) .
7
3
The effects of parameter uncertainty on optimal policy
Having derived the optimal policy rule (13) for the central bank, this section analyzes how optimal monetary policy is affected by the degree of parameter uncertainty in the economy. After choosing values for the parameter means (α, β, γ, δ) and the discount factor (φ), the analysis is conducted in two steps. The first step analyzes how the coefficients in the optimal rule (the elements in the vector f in equation (13)) are affected by parameter uncertainty, first considering uncertainty about one parameter at a time, then considering the combined effect of uncertainty about all parameters. The second step examines the optimal policy response over time to output and inflation shocks, when all parameters are uncertain. The values of the parameter means α, β and γ are taken from Orphanides and Wieland’s (2000) estimates for the Euro area, thus α = 0.77, β = 0.4, γ = 0.34.12 The parameter mean δ is set to unity, implying an accelerationist Phillips curve on average (which is also consistent with Orphanides and Wieland, 2000, who restrict δ to unity), and the discount factor φ is set to 0.99, implying a discount rate of 1% per period. When analyzing uncertainty about one parameter at a time, the relevant variance is set to 0.10, and the other variances to zero. (This fairly large value is chosen only to illustrate the effects. The qualitative effects do not depend on the size of the variances, or on the choice of the other parameter values.) When analyzing the combined effect of uncertainty about all parameters, on the other hand, the variances of αt+1 , βt+1 , and γt+1 are set to the squared estimated standard errors from Orphanides and Wieland (2000), so σα2 = 0.0121, σβ2 = 0.01, and σγ2 = 0.0169, whereas the variance of δt+1 (for which we have no empirical estimate) is allowed to take two different values, first a relatively small value (σδ2 = 0.01), then a relatively large value (σδ2 = 0.10).13 In each case, optimal policy is compared to the certainty equivalence case, when all parameters are constant and equal to their means. 12
Orphanides and Wieland (2000) estimate a very similar model (but without a lag in the transmission from the output gap to inflation) on annual data for the Euro area from 1976 to 1998. Although their use of the ex post real interest rate may risk overestimating the parameter β if the ex post real rate has been negative for part of the sample, but the ex ante real rate has been mainly positive, the close correspondence between their model and the one used here makes it natural to use their estimated parameter values. 13
The examples with positive variances for all parameters thus give an indication of the actual effects of parameter uncertainty in the Euro area. A complete empirical investigation would allow also for covariances between the parameters, which may have important additional effects. Since empirical estimates of parameter covariances are not easily obtained, however, the analysis abstracts from these issues.
8
Figure 1: Reaction function coefficients, impact parameters uncertain
Note: The variance of the uncertain parameter is set to 0.10; the other parameter variances to zero.
3.1
The initial policy response
As a first step, let us analyze the coefficients of the central bank’s optimal reaction function (that is, the initial response of policy to shocks) when there is uncertainty about each of the four parameters separately. Figures 1 and 2 show how the size of these coefficients vary as the preference parameter λ varies from 0, that is, “strict inflation targeting”, to 2, with a larger weight on stabilizing output than on stabilizing inflation. In each figure, panels (a) and (c) show the coefficient on output and panels (b) and (d ) the coefficient on inflation, with the solid line representing the certainty equivalent case and the dashed line representing the case with parameter uncertainty, with the variance of the uncertain parameter set to 0.10.14 First, Figure 1 shows the two cases of uncertainty about the impact parameters in the transmission mechanism, βt+1 and γt+1 . As is clear, the Brainard conservatism result is confirmed: when there is uncertainty concerning the impact parameters, the optimal response coefficients are smaller than under certainty equivalence, so policy is less aggressive. Increasing the variance of either parameter will weaken the response of the central bank and, in the limit, as the variances tend to infinity, 14
Note that the response coefficients on both output and inflation are decreasing in λ. This is because policy offsets shocks to both output and inflation by creating a recession. As the weight on output stabilization increases, optimal policy creates a smaller recession in response to a given shock (see Svensson, 1997).
9
Figure 2: Reaction function coefficients, persistence parameters uncertain
Note: The variance of the uncertain parameter is set to 0.10; the other parameter variances to zero.
the optimal response is to do nothing. This is true for all values of the preference parameter λ, although the effect of uncertainty is larger for small weights on output stabilization. Furthermore, uncertainty about βt+1 , the elasticity of the output gap with respect to the real interest rate, has a larger effect on policy than uncertainty concerning γt+1 , the parameter of transmission from output to inflation. Second, Figure 2 shows the response coefficients under uncertainty about the two persistence parameters, αt+1 and δt+1 . As seen in panels (a) and (b), uncertainty about the persistence of output affects policy in the same direction as uncertainty about the impact parameters (albeit to a smaller degree): the coefficients are smaller than under certainty equivalence. In contrast, uncertainty about the persistence of inflation in panels (c) and (d ) affects the optimal policy coefficients in the opposite direction. When λ > 0, optimal policy is more aggressive under uncertainty than under certainty equivalence, in contradiction to the Brainard intuition. When λ = 0, however, uncertainty about δt+1 has no effect on optimal policy.15 Since these results may be counterintuitive at first glance, they warrant some further consideration. As mentioned above, in this dynamic model the central bank wants to minimize the future deviation of expected inflation and output from target 15
Since writing the first version of this paper, I have discovered independent work by Srour (1999) and Shuetrim and Thompson (1999), who demonstrate versions of this result. Martin (1999) examines parameter uncertainty in a similar model, but only considers the case of strict inflation targeting, λ = 0, and thus finds no effect of uncertainty about the persistence of inflation.
10
as well as their variances. When parameters are non-stochastic, so there is only additive uncertainty in the model, the variances of inflation and output are constant, and thus independent of their distance from target. Under multiplicative parameter uncertainty, however, when the dynamics of the variables are uncertain, their variances increase with the distance from target, so when inflation and output are further away from target, the uncertainty about their future development is greater. Since the persistence of inflation only affects the dynamics of the economy, optimal policy reduces the amount of uncertainty about future inflation by acting more aggressively to push inflation closer to target.16 On the other hand, the persistence of the output process is a crucial part of the transmission mechanism from policy to inflation. This can be demonstrated by substituting the expression for output (1) into that for inflation (2), yielding17 πt+1 = δt+1 πt + γt+1 αt yt−1 − γt+1 βt (it−1 − πt−1 ) + γt+1 εyt + επt+1 .
(16)
While the parameter δt+1 determines only the dynamics of inflation, the parameter αt+1 determines not only the dynamics of output, but also the effect of policy on inflation (via the output gap). Therefore, uncertainty about the persistence of output has the traditional effect of making policy less aggressive. We also see from Figure 2 that the effect on policy of uncertainty concerning the persistence of inflation only operates when the central bank gives some weight to output in its objective function, so λ > 0. When the central bank cares only about stabilizing inflation (when λ = 0), it is always optimal to push inflation to target as quickly as possible (that is, after two periods). Then uncertainty about the persistence of inflation has no effect on optimal policy. When λ > 0, on the other hand, optimal policy closes only a fraction of the gap between expected inflation and target in each period, and with a larger λ, inflation is returned to target more slowly (see Svensson, 1997). As a consequence, uncertainty about the dynamics of inflation affects policy more strongly when λ increases, a pattern that is clear from panels (c) and (d ) of Figure 2.18 16
It should be stressed that the qualitative effect of parameter uncertainty do not depend on the actual parameter values chosen. Quantitatively, the effects become smaller as the persistence of inflation falls or the discount factor is decreased (since the future expected deviations of inflation from target drive the result). However, for any configuration of parameter values, increased uncertainty about the persistence of inflation always makes the policy response more aggressive. 17
I am grateful to a referee for suggesting this simple and intuitive explanation.
18
In practice, the case where λ = 0 is probably less realistic than that with a positive λ, since central banks typically want to avoid excessive real fluctuations. See, for example, Svensson (1998).
11
Figure 3: Reaction function coefficients, all parameters uncertain
Note: The variances of αt+1 , βt+1 , γt+1 are from Orphanides and Wieland (2000), and are 0.0121, 0.01, and 0.0169, respectively; the variance of δt+1 is 0.01 in panels (a) and (b) and 0.10 in panels (c) and (d ).
Finally, Figure 3 shows the case when there is uncertainty about all four parameters, with estimated variances for αt+1 , βt+1 and γt+1 , taken from Orphanides and Wieland (2000). Now we have two different possibilities: when the central bank puts little weight on output stabilization, so λ is small, optimal policy under uncertainty is more cautious than under certainty equivalence, since the uncertainty about αt+1 , βt+1 and γt+1 dampens the response coefficients, but the uncertainty about δt+1 has no or little effect. As λ increases, the uncertainty about δt+1 begins to have a positive effect on the coefficients, and eventually the response under uncertainty might get stronger than under certainty equivalence. For a given λ, whether the initial response coefficients are more or less aggressive under uncertainty then depends on the relative variances of αt+1 , βt+1 and γt+1 on the one hand and δt+1 on the other. When the degree of uncertainty about δt+1 is relatively small in panels (a) and (b) of Figure 3 (where σδ2 = 0.01), the response coefficient on inflation is larger under uncertainty for λ ≥ 17 (which falls outside the graph), whereas that on output is smaller under uncertainty for all λ. When uncertainty about δt+1 is relatively more important, however, in panels (c) and (d ) (where σδ2 = 0.10), policy is more likely to be more aggressive under uncertainty; the corresponding cutoff values are now λ ≥ 3.0 for output and λ ≥ 0.45 for inflation (as shown in panel (d )). Thus, the net effect on policy of parameter uncertainty depends not only on the relative
12
variances of the shocks, but also on the weight of output stabilization in the central bank’s objective function. In related work, Craine (1979) reaches a similar conclusion, using a dynamic model with one target variable: uncertainty about the impact of policy on the economy leads to less aggressive policy in response to shocks, but uncertainty about the dynamics of the economy leads to more aggressive policy.19 In that simple setup, it is straightforward to separate uncertainty about the transmission of policy from uncertainty about the dynamics of the economy. In the model used in this paper, such a separation is less clear-cut. Thus, the analysis above shows that Craine’s (1979) result is valid also in the current setup, but with one qualification: since policy affects inflation via output, the dynamics of the output process is an important part of the transmission of policy to inflation. Uncertainty about the dynamics of output therefore makes policy less aggressive. Similar results have also been reached within the literature on “robust control theory”: when the policymaker chooses policy to maximize welfare in the worst possible outcome under model uncertainty, particular configurations of uncertainty lead to more aggressive policy than under certainty equivalence (see, for example, Sargent, 1999; or Onatski and Stock, 2000).20 Intuitively, “cautious” policy can also mean that bad future outcomes are avoided by acting more aggressively today. 3.2
The policy response over time
The introduction of multiplicative parameter uncertainty also has interesting implications for the response of monetary policy over time after a shock. Figure 4 shows the response of monetary policy to inflation and output shocks over the first ten periods following a shock when λ = 1 (so the central bank gives equal weight to stabilizing output and inflation) and there is uncertainty about all parameters, as in Figure 3. Although in this parameter configuration the effects are typically small, it is clear that parameter uncertainty may make the policy response more or less gradualistic, again depending on the relative variances of the parameters and the preferences of the central bank. If the initial response is moderated by uncertainty (as in panels (a) and (b), where the variance of the persistence of inflation is small; σδ2 = 0.01), the policy response over time is more gradual, since the effects of the shock remain longer in the economy, so policy is away from “neutral” for a longer 19
See also Holly and Hughes Hallett (1989).
20
These analyses are performed within a backward-looking model. Using a forward-looking model, Hansen and Sargent (2000) show that the robust policy response under model uncertainty typically is less aggressive than under certainty.
13
Figure 4: Policy response over time, all parameters uncertain
Note: The variances of αt+1 , βt+1 , γt+1 are from Orphanides and Wieland (2000), and are 0.0121, 0.01, and 0.0169, respectively; the variance of δt+1 is 0.01 in panels (a) and (b) and 0.10 in panels (c) and (d ). In all panels, λ = 1.
period. On the other hand, if the initial response is more aggressive under uncertainty (as in panel (d ), where the variance of the persistence of inflation is large; σδ2 = 0.10), policy can return to neutral earlier, thus leading to larger swings in the interest rate response over time. Consequently, parameter uncertainty may lead to smoother paths of the interest rate than under certainty equivalence, without introducing an explicit smoothing objective into the central bank’s loss function. But this is not automatic; it depends on both the type of uncertainty and the preferences of the central bank. Casual observation suggests that central banks in practice tend to respond to shocks by first slowly moving the interest rate in one direction, and then gradually moving back to a more neutral stance. When parameters are certain, our model suggests a large initial move, and then a quick return to the original level, unless λ is very large. Under certain configurations of parameter uncertainty, the central bank behaves in a more gradual way: although the initial response is always the strongest, it is more modest under these cases of uncertainty, and the policy move is drawn out longer over time. In particular, the tendency of the bank to “whipsaw” the market by creating large swings in the interest rate is somewhat mitigated.21 In other configurations, 21
This issue of parameter uncertainty leading to more plausible paths of policy is examined more carefully in a companion paper, S¨ oderstr¨ om (2000). That analysis shows, however, that the
14
on the other hand, parameter uncertainty leads to even larger swings in the policy instrument.
4
Concluding remarks
This paper has demonstrated how uncertainty about parameters in a dynamic macroeconomic model can lead the central bank to pursue more aggressive monetary policy, providing a counterexample to the common wisdom following the results of Brainard (1967). When a policymaker is uncertain about the dynamics of the economy, it might be optimal to move more aggressively in response to shocks, so as to reduce uncertainty about the future path of the economy. Uncertainty about the impact effect of policy still leads to more cautious policy, in accordance with Brainard’s original analysis. It should be stressed that the model and the examples used are highly stylized and may not be entirely satisfactory from an empirical point of view, so any serious implications for policy are difficult to estimate. However, the qualitative results obtained from this simple model are also present in a more general empirical framework, similar to that of Rudebusch and Svensson (1999). It is possible that configurations of uncertainty in the real world are such that the Brainard result is typically valid, or to quote Blinder (1998, p. 12), “My intuition tells me that this finding is more general—or at least more wise—in the real world than the mathematics will support.” More complete empirical analyses of the effects of parameter uncertainty typically show that optimal policy is considerably more gradualistic under parameter uncertainty than under certainty equivalence,22 indicating that uncertainty about impact parameters is more important than about persistence parameters. Furthermore, Rudebusch (2000) argues that multiplicative parameter uncertainty has made Federal Reserve policy less aggressive, although it is not sufficient as an explanation for the Fed’s cautious behavior. Nevertheless, the main point in this paper is that the effects on optimal policy of parameter uncertainty may be less clear-cut than previously recognized. Determining the relevance of this result for actual policy should be an interesting topic for future research.
Svensson model always implies excessive volatility of the policy instrument, whereas optimal policy from an unrestricted VAR model is closer to the actual behavior of the Federal Reserve. 22
See Sack (2000) and S¨ oderstr¨ om (2000) for the U.S., Martin and Salmon (1999) for the U.K., and Shuetrim and Thompson (1999) for Australia.
15
A
Solving the control problem
First, the state vector xt+1 has expected value Et xt+1 = Axt + Bit , and covariance matrix
Σt+1|t =
(A1)
Σyt+1|t Σy,π t+1|t
,
π Σπ,y t+1|t Σt+1|t
(A2)
evaluated at t. Since all parameters are assumed independent, the off-diagonal elements of Σt+1|t are zero. The diagonal elements are Σyt+1|t = Vart [αt+1 yt − βt+1 (it − πt ) + εyt+1 ] 11 11 11 = xt Σ11 A xt + 2xt ΣAB it + it ΣB it + Σε ,
(A3)
and Σπt+1|t = Vart [δt+1 πt + γt+1 yt + επt+1 ] 22 = xt Σ22 A xt + Σε ,
(A4)
where Σij AB is the covariance matrix of the ith row of At+1 with the jth row of Bt+1 , that is, Σ11 A =
σα2
0
0
σβ2
,
Σ22 A =
2 11 Σ11 B = σβ , ΣAB =
σγ2
0
0
σδ2
0 −σβ2
,
(A5)
,
(A6)
and 2 22 2 Σ11 ε = σy , Σε = σπ .
(A7)
The trace term in equation (10) is then
11 11 11 tr(V Σt+1|t ) = v11 xt Σ11 A xt + 2xt ΣAB it + it ΣB it + Σε
22 + v22 xt Σ22 , A xt + Σε
where v11 and v22 are the diagonal elements of the matrix V .
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(A8)
Using equations (9), (10), and (A1) in the control problem (8), we can express the Bellman equation as xt V xt + w
= min xt Qxt + φ (Axt + Bit ) V (Axt + Bit ) + φtr(V Σt+1|t ) + φw , it
(A9)
which gives the necessary first-order condition as23
dtr(V Σt+1|t ) φ B (V + V )Axt + B (V + V )Bit + = 0, dit
(A10)
where, from (A8),
dtr(V Σt+1|t ) 11 = 2v11 Σ11 AB xt + ΣB it . dit
(A11)
Thus we get the optimal policy rule
it = − B (V + V ) B + 2v11 Σ11 B
−1
B (V + V ) A + 2v11 Σ11 AB xt
= f xt .
(A12)
Finally, using equation (A8) and the policy rule (A12) in the Bellman equation (A9) gives xt V xt + w = xt Qxt + φ [(Axt + Bf xt ) V (Axt + Bf xt ) + w]
11 11 11 + φv11 xt Σ11 A xt + 2xt ΣAB f xt + xt ft ΣB f xt + Σε
22 + φv22 xt Σ22 A xt + Σε
= xt
Q + φ(A + Bf ) V (A + Bf ) 11 11 22 +φv11 (Σ11 A + 2ΣAB f + ft ΣB f ) + φv22 ΣA
22 + φ w + v11 Σ11 ε + v22 Σε ,
xt
(A13)
so the matrix V is determined by V
= Q + φ(A + Bf ) V (A + Bf )
11 11 22 + φv11 Σ11 A + 2ΣAB f + f ΣB f + φv22 ΣA .
(A14)
See also Chow (1975). 23
Use the rules ∂x Ax/∂x = (A + A )x, ∂y Bz/∂y = Bz, and ∂y Bz/∂z = B y, see, e.g., Ljungqvist and Sargent (2000). Note also that V is not necessarily symmetric in this setup with multiplicative uncertainty.
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References Ba¸sar, Tamer and Mark Salmon, “Credibility and the value of information transmission in a model of monetary policy and inflation,” Journal of Economic Dynamics and Control 14 (1), 97–116, February 1990. Ball, Laurence, “Efficient rules for monetary policy,” International Finance 2 (1), 63–83, April 1999. Balvers, Ronald J. and Thomas F. Cosimano, “Inflation variability and gradualist monetary policy,” Review of Economic Studies 61 (4), 721–738, October 1994. Bertocchi, Graziella and Michael Spagat, “Learning, experimentation, and monetary policy,” Journal of Monetary Economics 32 (1), 169–183, August 1993. Blinder, Alan S., “What central bankers could learn from academics—and vice versa,” Journal of Economic Perspectives 11 (2), 3–19, Spring 1997. ———, Central Banking in Theory and Practice, The MIT Press, Cambridge, Mass., 1998. Brainard, William, “Uncertainty and the effectiveness of policy,” American Economic Review 57 (2), 411–425, May 1967. Cecchetti, Stephen G., “Policy rules and targets: Framing the central banker’s problem,” Federal Reserve Bank of New York Economic Policy Review 4 (2), 1–14, June 1998. Chow, Gregory C., Analysis and Control of Dynamic Economic Systems, John Wiley & Sons, New York, 1975. Craine, Roger, “Optimal monetary policy with uncertainty,” Journal of Economic Dynamics and Control 1 (1), 59–83, February 1979. Estrella, Arturo and Jeffrey C. Fuhrer, “Dynamic inconsistencies: Counterfactual implications of a class of rational expectations models,” Working Paper No. 98-5, Federal Reserve Bank of Boston, July 1998. Forthcoming, American Economic Review. ———, “Are ‘deep’ parameters stable? The Lucas critique as an empirical hypothesis,” Working Paper No. 99-4, Federal Reserve Bank of Boston, October 1999. Estrella, Arturo and Frederic S. Mishkin, “Rethinking the role of NAIRU in monetary policy: Implications of model formulation and uncertainty,” in Taylor, John B. (ed.), Monetary Policy Rules, Chicago University Press, 1999. Goodhart, Charles A. E., “Central bankers and uncertainty,” Bank of England Quarterly Bulletin 39 (1), 102–114, February 1999. Hansen, Lars Peter and Thomas J. Sargent, “Robust control and filtering of forwardlooking models,” manuscript, Stanford University, October 2000.
18
Holly, Sean and Andrew Hughes Hallett, Optimal Control, Expectations and Uncertainty, Cambridge University Press, 1989. Ljungqvist, Lars and Thomas J. Sargent, Recursive Macroeconomics, The MIT Press, Cambridge, Mass., 2000. Martin, Ben and Chris Salmon, “Should uncertain monetary policy-makers do less?” Working Paper No. 99, Bank of England, 1999. Martin, Ben, “Caution and gradualism in monetary policy under uncertainty,” Working Paper No. 105, Bank of England, December 1999. Onatski, Alexei and James H. Stock, “Robust monetary policy under model uncertainty in a small model of the U.S. economy,” Working Paper No. 7490, National Bureau of Economic Research, January 2000. Orphanides, Athanasios and Volker Wieland, “Inflation zone targeting,” European Economic Review 44 (7), 1351–1388, June 2000. Poole, William, “A policymaker confronts uncertainty,” Federal Reserve Bank of St. Louis Review 80 (5), 3–8, September/October 1998. Rudebusch, Glenn D. and Lars E. O. Svensson, “Policy rules for inflation targeting,” in Taylor, John B. (ed.), Monetary Policy Rules, Chicago University Press, 1999. Rudebusch, Glenn D., “Is the Fed too timid? Monetary policy in an uncertain world,” manuscript, Federal Reserve Bank of San Francisco, June 2000. Forthcoming, Review of Economics and Statistics. Sack, Brian, “Uncertainty, learning, and gradual monetary policy,” Finance and Economics Discussion Paper No. 1998-34, Board of Governors of the Federal Reserve System, August 1998. ———, “Does the Fed act gradually? A VAR analysis,” Journal of Monetary Economics 46 (1), 229–256, August 2000. Sargent, Thomas J., “Comment,” in Taylor, John B. (ed.), Monetary Policy Rules, Chicago University Press, 1999. Schellekens, Philip, “Caution and conservatism in the making of monetary policy,” Working Paper No. 25, European Central Bank, July 2000. Shuetrim, Geoffrey and Christopher Thompson, “The implications of uncertainty for monetary policy,” Research Discussion Paper 1999-10, Reserve Bank of Australia, November 1999. S¨oderstr¨om, Ulf, “Should central banks be more aggressive?” manuscript, Sveriges Riksbank, August 2000. Srour, Gabriel, “Inflation targeting under uncertainty,” Technical Report 85, Bank of Canada, April 1999.
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Svensson, Lars E. O., “Inflation forecast targeting: Implementing and monitoring inflation targets,” European Economic Review 41 (6), 1111–1146, June 1997. ———, “Inflation targeting in an open economy: Strict or flexible inflation targeting?” Victoria Economic Commentaries 15 (1), March 1998. ———, “Inflation targeting: Some extensions,” Scandinavian Journal of Economics 101 (3), 337–361, September 1999. Taylor, John B., “Discretion versus policy rules in practice,” Carnegie-Rochester Conference Series on Public Policy 39, 195–214, December 1993. ———, “The inflation/output variability trade-off revisited,” in Fuhrer, Jeffrey C. (ed.), Goals, Guidelines, and Constraints Facing Monetary Policymakers, Federal Reserve Bank of Boston, 1994. Wieland, Volker, “Monetary policy, parameter uncertainty, and optimal learning,” Journal of Monetary Economics 46 (1), 199–228, August 2000. Woodford, Michael, “Optimal monetary policy inertia,” Working Paper No. 7261, National Bureau of Economic Research, July 1999.
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