Structural Evolution of Postwar U.S. Economy

Transcription

1 Structural Evolution of Postwar U.S. Economy Yuelin Liu University of New South Wales James Morley University of New South Wales Preliminary draft, NOT under circulation This draft: March, 2013 Abstract We consider a time-varying parameter vector autoregressive model with stochastic volatility and mixture innovations to study the empirical relevance of the Lucas (1976) critique for the postwar U.S. economy. The model allows blocks of parameters to change at endogenously-estimated points of time. Contrary to the Lucas critique, there are large changes at certain points of time in the parameters describing monetary policy that do not correspond to changes in reduced-form parameters for inflation or the unemployment rate. However, the structure of the U.S. economy has evolved over the postwar period, with a reduction in the late 1980s in the impact of monetary policy shocks on inflation, but not on the unemployment rate. Somewhat related, we find changes in the Phillips Curve tradeoff between inflation and cyclical unemployment (measured as the deviation from the time-varying steady-state unemployment rate implied by the model) in the 1970s and especially since the mid-1990s. But, again, the changes do not appear to be due to changes in the systematic behaviour of policymakers. Keywords: Time-varying parameters, Mixture innovations, Lucas critique, Great Moderation, Natural Rate of Unemployment, Phillips Curve. 1 Introduction The U.S. economy has experienced large shifts in monetary policy regimes since World War II, as discussed by Lucas (1976) and Sargent (1999), amongst others. Therefore, an econometric model designed to study this phenomenon should allow for time-varying parameters. In addition, a substantial decline in the volatility of exogenous shocks, often referred to as Great Moderation, can be observed after mid-1980 s. These structural changes imply Corresponding author: Yuelin Liu. addresses: yuelin.liu@unsw.edu.au (Yuelin Liu); james.morley@unsw.edu.au (James Morley). 1

2 that a conventional time-invariant vector autoregressive (VAR) model is inadequate for the postwar U.S. data. In order to integrate time-varying parameters and stochastic volatility of exogenous shocks into an econometric framework, the literature has focused on two different approaches: Markov-switching (MS) models and time-varying parameter (TVP) models. Markov-switching VAR models presume that the economy switches between a few (possibly recurrent) regimes abruptly and the magnitude of variations among regimes can be large (see, for example, Sims and Zha, 2006). By contrast, time-varying parameter VAR models capture gradual changes (every period of time corresponds to a distinct regime) in conditional mean parameters and the variance-covariance matrix of exogenous shocks (see, among others, Cogley and Sargent (2001, 2005), Primiceri (2005), and Cogley et al. (2010). Koop et al. (2009) extend the stochastic volatility TVP model proposed by Primiceri (2005) to a framework with mixture innovations by introducing independent binary latent variables K t = (k 1t, k 2t, k 3t ), t = 1, 2,, n for each block of parameters (3 blocks: conditional mean parameters (k 1t ), variances (k 2t ) and covariances (k 3t ) of exogenous shocks) that allow data to determine the occurrence of break in each block independently in each period of time, such that P rob(k jt = 1) = p j and P rob(k jt = 0) = 1 p j, j = 1, 2, 3, t and corresponding hierarchical priors on p j, where p j Beta(λ 1j, λ 2j ). This framework forces all of the conditional mean parameters to change or stay at previous values all together by construction. 1 However, principal component analysis of the variance-covariance matrix governing the magnitude of shifts in the VAR parameters conducted by Cogley and Sargent (2005) and also suggested by Sargent (1999) argues that changes in the conditional mean parameters are highly structured due to cross-equation restrictions associated with the agent s optimization problem, although the way of imposing restrictions on reduced-form parameters is not clear-cut. This finding suggests that changes in the VAR parameters may occur in a highly structural way; that is to say, some coefficients may vary more frequently and more strongly, whilst others are approximately time-invariant. If this is the case, then a conventional TVP model will imply artificially high variation in some parameters and understate variation in others, which could distort our understanding of the evolution of the U.S. economy. In this paper, we extend Koop et al. (2009) to allow for structural changes at different times in subgroups of the VAR parameters, including different blocks of the conditional mean parameters. Because changes in the block j(j = 1, 2, 3, 4) of parameters is controlled by a Bernoulli distributed latent variable k jt with P rob(k jt = 1) = p j, the posterior density of p j 1 Note that this framework bridges the Markov-switching models and time-varying parameter models by imposing prior beliefs on p j. A pair of large λ 1j and small λ 2j a priori assumes frequent breaks in the model, on the contrary, small λ 1j and large λ 2j implies few breaks in the model. 2

3 reflects the frequency of occurrence of breaks in block j. If the true model is the stochastic volatility TVP model, as in Primiceri (2005), then the data will push the p j s to 1. However, if p j s are quite different from one another, then there is strong evidence of structural change in the VAR parameters. In fact, the evidence from our benchmark model supports the existence of structural change in the VAR parameters. Besides the extension of Koop et al. s (2009) modeling strategy, our contributions in this paper lie in the following three areas. First, because we explicitly divide the VAR parameters into policy and non-policy blocks, the frequency of changes in non-policy blocks relative to that of policy block can be used to test for the empirical relevance of Lucas critique, which states that a shift in systematic policy should introduce a change in the reduced-form parameters describing the correlations in the data. This is different than the simulation-based approaches to testing the Lucas critique in the literature; see, among others, Estrella and Fuhrer (2003), Linde (2001), Rudebusch (2005) and Lubik and Surico (2010). Our approach tells us whether the Lucas critique is empirically relevant for the VAR parameters, including the variance-covariance matrix of error terms. Second, based on standard short-run restrictions, we identify monetary policy shocks and study their effects on inflation and unemployment over time. Our findings can be compared with Primiceri (2005) and Koop et al. (2009), who find that there is no statistically significant shift in the effects of monetary policy after World War II, and Kuttner and Mosser (2002) and Boivin and Giannoni (2006), who find that the effects of monetary policy on the U.S. economy has weakened since 1980s. Based on our model, we find that the effects of monetary policy on inflation have only changed at the 3-9 quarter horizons and the effects of the unemployment rate have not changed. Third, we estimate the natural rate of unemployment as the timevarying steady-state of the unemployment rate implied by the model, as in Phelps (1994) and King and Morley (2007). Based on the estimated natural rate, we test for the existence of a Phillips curve trade-off between inflation and cyclical unemployment. We find strong evidence supporting the existence of this short-run trade-off. However, the short-run tradeoff has weakened since the late 1970s and even disappeared since the mid-1990s. The rest of this paper is organized as follows. Section 2 presents our model. Section 3 describes the data and elicitation of priors. Section 4 considers model comparisons and robustness analysis. Section 5 presents the test for the empirical relevance of Lucas critique. Section 6 reports the results on the evolution of impulse response functions for a monetary policy shock on inflation and the unemployment rate. Section 7 considers the natural rate of unemployment and the short-run trade-off between inflation and cyclical unemployment. Section 8 concludes. 3

4 2 The Model One of the contributions of this paper is to allow for structural variations in blocks of VAR parameters by linking them to latent variables K t s in the manner of Koop et al. (2009). As a byproduct, we propose a new approach to testing the empirical relevance of Lucas critique that does not rely on simulations from a DSGE model. We study two intuitive and plausible, although informal and atheoretic, ways of imposing structural changes in reduced-form VAR parameters: (1) by equations; (2) by variables. The details of the model structure are in the next subsections. 2.1 Stochastic Volatility Time-varying VAR and Identification of Monetary Policy Shock The time-varying reduced-form VAR with order p can be cast in the following form: y t = X tθ t + µ t, µ t i.i.d. N(0, Ω t ) X t = I n [ 1, y t 1,, y t p], where denotes the Kronecker product, y t is an n 1 vector including the current observations of endogenous variables, X t is an m n matrix including intercepts and lagged variables, θ t stacks time-varying reduced-form VAR coefficients and Ω t is the time-varying variance-covariance matrix of the error term µ t. In this paper, y t includes inflation, the unemployment rate and a short-term interest rate, so m = 21, n = 3 and we set p = 2 to reduce the dimension of parameter space and accord with the literature. 2 To identify the monetary policy shock, a structural VAR can be recovered from the triangular identification scheme i.e. we place endogenous variables in the order of y t = [π t u t i t ] in which π t, u t, i t are inflation, unemployment rate and short-term interest rate, respectively. This order of endogenous variables assumes that inflation and unemployment both respond to monetary policy shock with at least one-period lag. 3 Structural shocks ε t, t = 1, 2,, T, are identified through Cholesky decomposition of the variance-covariance matrix of the reduced-form error terms as follows: A t Ω t A t = Σ t Σ t, A 1 t Σ t ɛ t = µ t, ε t = Σ t ɛ t, ɛ t iid.n(0, I 3 ) 2 A trivariate model like ours that is also common in the literature, see among others, Rotemberg and Woodford (1997), Cogley and Sargent (2001, 2005), Primiceri (2005) and Koop et al. (2009). 3 It is well recognized that the order of variables in the recursive identification scheme matters. However, our results are robust to different orders of inflation and the unemployment rates. 4

5 1 0 0 A t = a 21,t 1 0 a 31,t a 32,t 1 3 3, Σ t = σ 11,t σ 22,t σ 33,t Note that ε t = [ε πt ε ut ε it ] where each element stands accordingly for fundamental shocks to inflation, unemployment and monetary policy shock. The reduced-form timevarying VAR can be rewritten as y t = X tθ t + A 1 t ε t, ε t i.i.d.n(0, Σ t Σ t) 3 3 X t = I 3 [ 1 y t 1 y t 2], (1) 2.2 Mixture Innovations of Time-varying Parameters and Variancecovariance Matrix The law of motion of time-varying parameters θ t obeys a driftless random walk 4 with mixture innovations: 5 θ t = θ t 1 + K t ξ t, ξ t i.i.d. N(0, Q), (2) where Q is positive definite and K t is a diagonal matrix whose diagonal elements are latent variables k it, i = 1, 2, 3, 4, taking 1 if a break occurs in the corresponding coefficients or 0 otherwise. We consider two types of restrictions on θ t, in both cases we force intercepts to vary together which reflects the moving targets for inflation, unemployment and nominal interest rate. The differences lie in the comovements of the slopes, in the first case slopes in the same equation are forced to move together and in another case slopes on the same variables are compelled to alter simultaneously. 6. The controlling matrix K t in the two cases is denoted by K (1) t = diag{k (1) 1t, K (1) 2t, K (1) 3t, K (1) 4t, K (1) 2t, K (1) 3t, K (1) 4t } (by equations) and 4 The assumption of a driftless random walk in time-varying parameters illustrates our belief, according with conventional wisdom, that the inflation and nominal interest rates are I(1) processes and the movement in the natural rate of unemployment (NRU) is persistent. Also, this helps to reduce the dimension of the parameter space. 5 Mixture innovations will allow for more flexible variations in VAR coefficients and potentially larger magnitude of innovations compared to conventional TVP model. 6 Cogley and Sargent (2005) conduct a simple principal components analysis and find that the parameters in inflation equation receive substantially higher loadings compared to the rest parameters i.e. the variations in reduced VAR parameters, if any, are highly structured which motivates our study. As demonstrated in Sargent (1999), the structured variation can be derived from the cross-equation restrictions associated with the agent s optimization problem. However, the cross-equation restrictions are not clear-cut. So, from the perspective of empirical study, we investigate another sensible way of restrictions according to variables instead of equations as well. 5

6 K (2) t = diag{k (2) 1t, K (2) 2t, K (2) 2t, K (2) 2t, K (2) 2t, K (2) 2t, K (2) 2t } (by variables), respectively, where K (1) 1t = K (2) 1t = k 1t k 1t 0, K (1) 2t = k 2t k 2t 0, K (1) 3t = k 3t k 3t 0, 0 0 k 1t 0 0 k 2t 0 0 k 3t K (1) 4t = k 4t k 4t 0, K (2) 2t = k 2t k 3t k 4t 0 0 k 4t Let α t be a vector collecting the non-zero and non-one elements in A t, σ t collect the diagonal elements in Σ t, then the evolutions of elements in α t and σ t are as follows: α t = α t 1 + k 5t η t, η t i.i.d. N(0, S), (3) ln σ t = ln σ t 1 + k 6t ζ t, ζ t i.i.d. N(0, W ), (4) where S, W are positive definite and S is block diagonal in which each block is corresponding to parameters in different equations and similarly, k jt = 1, j = 5, 6 if a break occurs and k jt = 0, j = 5, 6 otherwise. Moreover, we suppose that all of the innovations in the dynamic system are uncorrelated pairwisely at all lags and leads i.e., they are jointly normally distributed with the following variance-covariance matrix V as: V = V ar ɛ t ξ t η t ζ t I = 0 Q S W Estimation of parameters in this framework relies on Markov Chain Monte Carlo (MCMC) method. Particularly, under the assumption that k jt s are independent of one another, contemporaneously and at all lags and leads, K t, k 5t, k 6t can be drawn based on the reduced conditional sampling algorithm proposed in Gerlach et al. (2000) without conditioning on the state vector θ t that greatly improves the efficiency especially when K t and θ t are highly correlated. 7 Then, following Primiceri (2005), we can adapt Carter and Kohn (1994) and 7 Though it may be more desirable to introduce correlations between k jt s to fit the data better, the benefit will bring a huge cost in terms of increasing the computation burden substantially making the implementation infeasible. Another important reason is that, no matter we presume correlations between drifts in the policy block and non-policy blocks in any forms or not, data will choose the break timing in each parameter block under our model. We are able to find out the dependence of break timing among different parameter blocks and also the posterior probability of a break in other parameter blocks conditional on observing a drift in. 6

7 Kim et al. (1998) to draw state vectors θ t, α t and ln σ t from three Gaussian linear statespace systems separately. Nonetheless, we hold the same stance as Cogley and Sargent (2001, 2005) that in each period of time the state-space system associated with θ t must be stable, or inflation and unemployment rates will explode which is well recognized impossible in United States since well-designed policy is able to maintain macro aggregates in stable trajectories. Hence, we impose stationarity condition on θ t to exclude non-stable behaviours in each period of time. 8 See the detailed MCMC procedures in the technical appendix. 3 The Data and Priors 3.1 The Data As discussed above, we use a small three-variable VAR to study the evolution of U.S. monetary policy, measured by short-term nominal interest rate (federal funds rate, averaged from daily rates, series ID: FEDFUNDS), and its impacts on inflation (seasonally adjusted compounded annual rate of change of Personal Consumption Expenditures, series ID: PCECTPI) and the unemployment rate (seasonally adjusted civilian unemployment rate, all workers over 16, series ID: UNRATE). 9 The three time series are quarterly and run from 1954:Q3 to 2007:Q4. We execute 80, 000 replications of the Metropolis-within-Gibbs sampler in which the first 20, 000 draws, known as the burn-in, are discarded to allow for convergence to the ergodic distribution. Every 20-th draw is saved from the remaining 60,000 draws to economize the the policy block from the MCMC replications. It is this posterior conditional probability that measures the empirical relevance of the Lucas critique. 8 This procedure can be interpreted as a Metropolis-Hastings algorithm with acceptance probability α θt = T 1(θ t Θ S ) for proposal θ T = (θ1, θ2,, θt ), where Θ S denotes the collection of parameters making t=1 the state-space system stable and 1( ) is the indicator function. 9 All of the data are downloaded from FRED managed by Federal Reserve Bank of St. Louis at The results are robust to different measurements of inflation and short-term interest rates, for example, GDP deflator and 3-month treasury bill rate. Personal Consumption Expenditures (PCE) is used because it is more closely related to the cost of living at which the Fed might prefer to look. When 3-month treasury bill rate is employed instead of federal funds rate, an unemployment puzzle, a contraction monetary policy associated with a decline in unemployment rate, appears as in Primiceri (2005) and Koop et al. (2009). Therefore, we take the federal funds rate (FFR) which is directly under control of the Fed as the monetary policy instrument. Another potential concern is that the FFR might be less representative as a policy instrument from 1979 to 1982 during which the Fed is by law required to target various monetary aggregates. But Cook (1989) argues that even in that episode the FFR serves as a satisfactory policy indicator. Hence, it seems appropriate to treat the FFR as the policy instrument across the whole sample. 7

8 Figure 1: Sample 20th-order Autocorrelation Functions for All of the Parameters storage space and reduce the autocorrelations across draws. 10 Therefore, Bayesian inferences are carried out based on 3,000 draws from the posterior distribution. Convergence diagnostics is conducted by inspecting sample ACFs and recursive means of all of the parameters. As shown in Figure 1, sample 20th-order ACFs for all of the parameters (including hyperparameters) are well below 0.2 implying that the posterior draws mix well and the convergence check is satisfactory. 3.2 Prior Distributions Priors for state vectors and hyperparameters are calibrated following Primiceri (2005) and Koop et al. (2009) with only slight differences. Data of the first ten years (42 observations, 1954:Q3 1964:Q4) are employed to calibrate the priors. 11 Specifically, a time-invariant VAR is estimated using conditional MLE which produces the point estimate for VAR coefficients, ˆθ 0 and its corresponding variances V (ˆθ 0 ). Estimates of the variance-covariance matrix of VAR error terms, ˆΩ0, can be obtained as well and ˆα 0, ˆσ 0 can be derived from decomposing ˆΩ 0. The variance of ˆα 0, V (ˆα 0 ), is obtained by simulation from Wishart distribution with scatter matrix ˆΩ 0 and degree of freedom = 40, and we set the variance of 10 The acceptance rate in the Metropolis-Hastings procedure is approximately 50%. 11 The training sample of 10-year quarterly data is long enough to capture fundamental dynamics in a typical business cycle and short enough to make the priors as weak as possible. 8

9 ln(ˆσ 0 ) to 10I 3 which is huge in log-scale [ implying ] a small weight is put on the prior. As S 1 0 for the hyperparameters Q, S =, W, they are supposed to be governed by 0 S 2 inverse-wishart distributions. In order to put as least weight as possible on prior beliefs, the degree of freedom corresponding to each inverse-wishart distribution is set to the minimum plausible value dim(q) + 1 = 22, dim(s 1 ) + 1 = 2, dim(s 2 ) + 1 = 3, dim(w ) + 1 = 4, respectively. In summary, the priors are as follows: θ 0 N( ˆθ 0, 4V (ˆθ 0 ) ), α 0 N( ˆα 0, 4V (ˆα 0 ) ), ln σ 0 N( ln ˆσ 0, 10I 3 ), Q IW ( 40kQV 2 (ˆθ 0 ), 22 ), S 1 IW ( 2kSV 2 (ˆα 1,0 ), 2 ), S 2 IW ( 3kSV 2 (ˆα 2,0 ), 3 ), W IW ( 4kQI 2 3, 4 ), where k Q = k W = 0.01, k S = 0.1 and ˆα 1,0, ˆα 2,0 correspond to each block of ˆα To complete the model, hierarchical priors for K t, k 5t and k 6t need to be specified. We adopt a Bernoulli distribution Ber(p j ) with P rob(k jt = 1) = p j, j = 1, 2,, 6, (5) thus p j is the probability of a break occurring at time t for k jt. p j is supposed to be governed by Beta distribution Beta(λ 1j, λ 2j ), j = 1, 2,, 6 which form a conjugate prior with Bernoulli distributions. The values of λ 1j and λ 2j reflect our prior beliefs about the frequency of occurrence of breaks in the model. Small λ 1j and large λ 2j implies a few breaks (FB) model, say λ 1j = 0.01, λ 2j = 10 for all j, then E(p j ) = 0.001, se(p j ) = Large λ 1j and small λ 2j approaches the stochastic volatility TVP (SVTVP) model a la Primiceri (2005), say λ 1j = 1, λ 2j = 0.01 for all j, then E(p j ) = 0.99, se(p j ) = Priors for p j s are extremely tight in FB and SVTVP models which can be considered as approximations to time-invariant model and stochastic volatility TVP model, respectively. Our benchmark models set λ 1j = λ 2j = 1 for all j, then E(p j ) = 0.5, se(p j ) = 0.29 that means a priori we believe the occurrence of a break in each period of time is with 50% chance. We combine the benchmark priors for p j s with priors for state vectors θ t, α t, σ t and hyperparameters Q, S, W. The Bayesian inferences in the benchmark models with K (1) t or K (2) t rely on diffuse 12 See the reasons for choosing these values of k Q, k S, k W in Primiceri (2005). 9

10 priors which try to let data speak for themselves as much as possible. Comparisons of model fit are investigated in the next section. 4 Model Fit and Robustness Analysis We compare the fit of the benchmark models, the FB model and the SVTVP model by inspecting posterior means of p j s and the expected value of log-likelihood function as the empirical Bayesian metric described in Carlin and Louis (2000), instead of comparing Bayes factors that are more sensitive to prior information than posteriors especially in the case of a model with extremely high-dimensional parameter space. 13 As summarized in Table 1, two results stand out. First, our BEQ model with structural variations in intercepts and slopes in different equations receives strongest support from the data than BVA, FB and SVTVP models from the stance of expected log-likelihood E(log L) whose score is as high as Accord with Primiceri (2005), Cogley and Sargent (2005) and Koop et al. (2009), the BEQ model suggests that the volatility of error terms has evolved over time frequently with E(p 6 Y ) = (standard deviation ). However, the slopes in unemployment equation are relatively stable with E(p 3 Y ) = (standard deviation ) which is substantially smaller than the probabilities of occurrences of breaks in other blocks. It implies that on average a break occurs in the slopes in the unemployment equation every 7.5 quarters, whereas intercepts and slopes in inflation and interest rate equations are expected to be hit by a break every one or two quarters. This result provides strong evidence for the hypothesis that drift in reduced VAR parameters is highly structured. Second, we also find that even if extremely tight priors on p j s are employed in FB and SVTVP models, the data information is so strong that it pushes both models to the opposite side. This can be treated as one way of robustness analysis of our modeling strategy. For example, in the FBEQ model, the parameters of Beta(λ 1j, λ 2j ) priors on p j s are set to λ 1j = 0.01, λ 2j = 10, j = 1, 2,, 6, and E(p j ) = 0.001, se(p j ) = 0.01 which means that on average a break is expected to happen once every 1000 quarters a priori. Nevertheless, the posterior mean values of p j s show that E(p 1 Y ) = , E(p 2 Y ) = , E(p 3 Y ) = , E(p 4 Y ) = , E(p 5 Y ) = , E(p 6 Y ) = with standard deviations , , , , and , respectively. The posterior expected value 13 Hereafter, BEQ stands for the benchmark model varying with respect to equations, whereas BVA denotes the benchmark model varying with respect to variables. Likewise for FBEQ, FBVA, SVTVPEQ and SVTVPVA. 14 E(log L) is obtained by averaging log-likelihood from the state-space model (1) and (2) based on each draw of α T, σ T, Q, S, W, K T, k T 5, k T 6, λ which makes the state-space system stable, where (and hereafter) x T = (x 1, x 2,, x T ). 10

11 Table 1: Model comparison Models E(p 1 Y ) E(p 2 Y ) E(p 3 Y ) E(p 4 Y ) E(p 5 Y ) E(p 6 Y ) E(log L) BEQ (0.0502) (0.0590) (0.0376) (0.0622) (0.2356) (0.0585) BVA (0.0513) (0.0665) (0.0651) (0.0719) (0.2358) (0.0948) SVTVPEQ ( ) (0.0381) ( ) ( ) (0.0290) (0.0180) SVTVPVA (0.0342) (0.0632) (0.1875) (0.1480) (0.0250) (0.0189) FBEQ (0.0272) ( ) (0.0230) (0.0266) (0.0862) (0.0665) FBVA (0.0512) (0.0313) (0.0405) (0.0473) (0.0850) (0.0707) Standard deviations are listed in parentheses. of p 6 tells us volatilities of the reduced VAR error terms on average are changing approximately every 3 quarters which strongly rejects our prior belief that few breaks happen over time in the volatilities of error terms. Though the chances of observing breaks in other blocks are relatively smaller, it is still expected to observe a break in intercepts, slopes in inflation equation, slopes in interest rate equations or covariance matrix of error terms around every two years. On the other hand, in the SVTVPEQ model, λ 1j and λ 2j are set to 1 and 0.01, respectively. So a priori E(p j ) = 0.99 with standard deviation 0.08 favors a time-varying parameter with stochastic volatility model. However, the posterior mean values of p 1 up to p 6 are , , , , and , respectively. The probability of observing a break in the slopes in unemployment equation at every time period declines substantially from 0.99 to with standard deviation which significantly deviates from the prior belief. This suggests that the slopes in unemployment equation is more stable than the other blocks in the model, equivalently indicates a structured shift in reduced VAR parameters. In summary, our BEQ model receives the strongest support from the data. Hereafter, the empirical results about the evolution of US economy are derived from the benchmark i.e., the BEQ model. 11

12 5 Testing the Lucas Critique Ever since Lucas s seminal paper published in 1976, it is widely-recognized amongst macroeconomists that reduced-form econometric models will not be appropriate for policy analysis if there are changes in reduced form parameters due to shifts in policy. However, a relatively large literature see Fevero and Hendry (1992), Estrella and Fuhrer (2003), Linde (2001) and Rudebusch (2005), among others casts doubt on the empirical relevance of Lucas critique by considering Chow tests and superexogeneity tests. By contrast, Lubik and Surico (2010) find that by taking into account the stochastic volatility in the reduced-form error terms, one cannot reject the empirical relevance of Lucas critique. Specifically, a shift in policy rule has a great impact not only on reduced-form parameters, but also the variances of reducedform error terms. They criticized the Chow test and the superexogeneity test employed in Fevero and Hendry (1992), Estrella and Fuhrer (2003) and Rudebusch (2005), and others for implicitly assuming homoskedasticity of the reduced-form error terms which undermines the power of the Chow test and the superexogeneity test so as to lead to rejection of Lucas critique in the data. 15 Existing approaches for testing the Lucas critique typically rely on simulating data from some specified DSGE models that are treated as the true model of the world i.e. the true data generating process (DGP) of the macro variables of interest. The benefit of DSGE modeling is that the shift of parameters in different blocks, for example, policy block and non-policy block, is explicitly specified which makes it easy to distinguish changes in policy from changes derived from other sources. Nonetheless, nothing can be obtained at no cost. One may naturally doubt the assumption of treating DSGE models as true DGPs of the variables, furthermore, inferences based on simulated data from the hypothetic true DGPs. In contrast to Cogley and Sargent (2001, 2005), Primiceri (2005) and Koop et al. (2009), our approach allows the policy block and non-policy block to vary separately, which allows us to disentangle the sources of changes in the reduced-form VAR parameters. This implies it is possible, under our framework, to directly let actual data speak for themselves instead of simulating them from certain presumed true DGP of the variables of interest as conventional approach does. As the Fed can not obtain timely and reliable real-time data, as argued by McCallum and Nelson (1999), directly in the real world to formulate expectations of current and future economic conditions, hence, it is plausible to characterize a backward-looking feedback rule in the policy sector because the Fed has to formulate their 15 Another potential issue is the low power of the superexogeneity test in small samples, as discussed in Linde (2001) and Collard et al.(2001). 12

13 expectations of current and future endogenous macro aggregates x t+h (h = 0, 1, 2, ) by forecasting based on the information set available at time t 1. Generally, this feedback rule reads like: 16 p 1 p ( ) 2 p ( ) 3 i t = i t + ρ t,h Et 1 π t+h πt+h + φ t,h Et 1 u t+h u t+h + ϕ t,h (i t h i t h) + υ t h=0 h=0 h=1 where 1) i t is the target rate of short-term interest rate at time t when inflation and unemployment rates achieve their targets πt+h and u t+h (h 0), respectively; 2) The second and third terms specify a Taylor-type rule; 3) The last systematic term presents the inertia in setting short-term interest rate known as interest rate smoothing ; 4) υ t is a monetary policy shock. Since the expectations E t 1 π t+h and E t 1 u t+h are projections of current and future inflation and unemployment on information set available at time t 1, so it is sensible to characterize the monetary feedback rule in reality as the interest rate equation in the reduced-form VAR: 17 i t = c t + ρ 1,t π t 1 + φ 1,t u t 1 + ϕ 1,t i t 1 + ρ 2,t π t 2 + φ 2,t u t 2 + ϕ 2,t i t 2 + µ 3,t. First, it is interesting to investigate the timing of breaks across parameter blocks. As shown in Table 2, in the reduced-form VAR, intercepts (controlled by k 1 ), slopes in inflation equation (controlled by k 2 ), slopes in interest rate equation (controlled by k 4 ), covariances of VAR error terms (controlled by k 5 ) and standard deviations of fundamental shocks (controlled by k 6 ) pairwise comove frequently with a chance varying from 59% to 81% in terms of the fractions (posterior median values) of k it = k jt, i, j = 1, 2,, 6, i 5, j 5 over time. What is at odds with the above relatively strong pairwise dependence is the relationship between slopes in unemployment equation (controlled by k 3 ) and other parameter blocks which varies between 17% and 39%. 18 This relatively weak pairwise dependence between k 3 and k i, i 3 reinforces the existence of a structured shift in reduced form VAR parameters. In order to test the empirical relevance of the Lucas critique, we are interested in deter- 16 Clarida et al. (2000) and Rudebusch (2005) study similar empirical monetary policy rules. 17 In the paradigm of rational expectations, to drop the expectation operator, one may prefer to consider the last structural equation of interest rates in the structural VAR as the policy rule. Nonetheless, if we treat the structural parameters as the hidden Markov states as in the framework of TVP model proposed by Cogley and Sargent (2001, 2005), then the state-space form of the model is non-linear. Therefore, the likelihood can t be evaluated recursively based on the Kalman filter. That is to say, direct estimation of the structural parameters is technically infeasible. Also, note that when forecasting π t+h and u t+h (h 0) conditional on the information set available at time t 1, one must take into account the fundamental shocks to inflation and unemployment as well after dropping the symbol of expectation, because the forecast values are determined by the whole state-space system. 18 Posterior mean values deliver similar conclusions. 13

14 Table 2: Posterior median values of fractions of k it = k jt, i, j = 1, 2,, 6, i j k it = k jt, i j k 2 k 3 k 4 k 5 k 6 k (0.0474) (0.0393) (0.0383) (0.0832) (0.0402) k (0.0471) (0.0536) (0.1760) (0.0594) k (0.0485) (0.1785) (0.0511) k (0.1547) (0.0585) k (0.2080) Standard deviations are listed in parentheses. mining the probability of an occurrence of breaks in non-policy blocks conditional on a shift in the policy rule, as captured by the interest rate equation in the VAR. Different from the backward-looking policy reaction rules in Clarida et al. (2000) and Rudebusch (2005), we study a reduced-form VAR with intercepts that are combinations of targets of endogenous variables and reduce-form VAR parameters. Therefore, shift in intercepts may matter in the agent s decision making process. 19 As listed in Table 3, slopes in inflation equation and standard deviations of structural shocks Σ t vary with shift in policy closely, whereas slopes in unemployment equation and covariances of reduced-form error terms α t relate to policy evolution loosely. The posterior probabilities of occurrence of a break in slopes in unemployment equation and α t both significantly deviate from one, with medians 0.13 and 0.66, respectively. However, there is relatively strong evidence supporting empirical relevance of Lucas critique with respect to slopes in inflation equation and Σ t whose median posterior probabilities of observing a break conditional on shift in policy are 0.88 and 0.93, respectively. If we take the non-policy block as a whole, then it is unstable given shifts in the policy rule because of a high median posterior conditional probability of As for the variances of reduced-form error terms, in accordance with Lubik and Surico s (2010) finding, we find that there is strong evidence supporting shift in policy has an impact on the variances of reduced-form error terms whose median posterior conditional probability is as high as In words, a shift in the policy rule is very likely to induce not only drift in parameters in the non-policy block in the reduced VAR but also time-varying variances of error terms. But 19 The results are almost exactly the same when we do not treat shift in intercepts as a change in policy rule to which agents will respond. There are two potential reasons: 1). A shift in targets for endogenous variables is always accompanied with a shift in slopes of policy rule. 2). The variations in intercepts are essentially unavailable in the agent s information set, thus redundant for the agent to formulate expectations. 14

15 Table 3: Test of the Lucas critique: posterior probabilities of a shift in reduced VAR parameter blocks conditional on shift in policy rule Probability Standard Deviation 95% Credible Interval Inf a b [0.8582, ] Unem [0.1125, ] α t [0.5731, ] Σ t [0.9197, ] NP [0.8762, ] VarErr [0.9706, ] (a). Inf: slopes in inflation equation; Unem: slopes in unemployment equation; α t : covariances of reduced-form error terms; Σ t : standard deviations of structural shocks; NP: non-policy block; VarErr: variances of reduced-form error terms. (b). Probability: posterior medians. beyond that, the extent of empirical relevance of Lucas critique is not uniform for different parameter blocks which distinguishes our work from the existing literature. 6 Evolution of Impulse Responses It has been in a long-lasting debate that whether the monetary policy conducted since Paul Volker took office and succeeded by Alan Greenspan is responsible for stabilizing the US economy in which the volatilities of almost all of the macro aggregates have declined substantially since approximately 1985, a phenomenon known as the Great Moderation. To investigate the potential sources of this decline in volatility, one sensible way is to compare the response of macro aggregates (response) to monetary policy shock (impulse) over time. Primiceri (2005) and Koop et al. (2009) argue that there is no statistical significant evidence for a crucial role played by monetary policy, since the impulse responses of inflation and unemployment to a monetary policy shock do not significantly deviate from one another across time. Among others, however, Kuttner and Mosser (2002) and Boivin and Giannoni (2006) find that the impacts of monetary policy on output and inflation appear somewhat weaker recently compared to what they were before 1980s. Figures 2 - Figures 4 plot the evolutions of responses of inflation, unemployment and interest rate to 1% monetary policy shock at selected dates: 1975Q1, 1981Q3, 1996Q1 and 2006Q3. 20 It is clear that the magnitude of responses of inflation, unemployment and in- 20 Since Primiceri (2005) and Koop et al. (2009) report the impulse response functions for 1975Q1, 1981Q3, 1996Q1 and 2006Q3, to compare with their results, we study the impulse responses at the same dates. Though they take 3-month treasury bill rate as the proxy for the policy instrument and measure inflation by GDP deflator that both are different from our measures for inflation and short-term interest rate, the plots of impulse responses derived from our benchmark (BEQ) model are fundamentally the same for different 15

16 terest rates seems to be smaller since 1980s in median values. However, the evidence is not statistically significant except for the responses of inflation which shows that the impact of monetary policy shock on inflation at horizons of 3-9 quarters after 1980s presents significant downward deviation from what they were before 1980s, which are at odds with Primiceri (2005), Koop et al. (2009) and Boivin and Giannoni (2006). 21 Boivin and Giannoni (2006) study time-invariant VARs with respect to inflation, output and interest rates for two subsamples 1959:Q1-1979:Q3 and 1979:Q4-2002:Q2 and compare impulse responses evaluated from subsamples based on the recursive identification scheme. However, they only show point estimates of the impulse responses without conducting a rigorous statistical test of constant impulse responses across subsamples. As for Primiceri (2005) and Koop et al. (2009), since their modeling strategy are associated with ours, thus it is possible to figure out the source of difference at least to some extent. Given an identification scheme, every structural parameter is a mapping from the reduced VAR parameters. Furthermore, impulse responses are functions of structural parameters which implies that if the reduced VAR parameters somehow are inappropriately estimated, the impulse response functions will be contaminated as well. In this paper, we impose local-todate stationarity on the time-varying VAR parameters in each period of time a la Cogley and Sargent (2001, 2005) which is different from the VAR-based modeling strategy in Primiceri (2005) and Koop et al. (2009) in which explosive behaviors are not excluded. Nonetheless, the impulse responses plotted in Primiceri (2005) and Koop et al. (2009) are hump-shaped, i.e. no (at most, little) posterior probability is attached to explosive behaviors. Thus, it is sensible to expect that the assumption of local-to-date stationarity does not play a critical role in determining the shape of impulse responses which implies that instead it is the flexibly allowed structural variation in VAR parameters that make a difference in shaping impulse responses. As demonstrated in Table 1, our BEQ model is preferred to SVTVP and FB models in terms of model s fit, hence, we argue that it is the impulse responses derived from the BEQ model that are better estimates of the effects of monetary policy shock which leads to the finding of a weaker impulse response of inflation to monetary policy shock after 1980s. measurements of inflation and unemployment. What is more, instead of inspecting impulse response in single time period, the averages of impulse responses over periods deliver the same patterns. 21 An apparent price puzzle is evident in 1975Q1 and 1981Q3 which is common in a small monetary VAR like ours with the triangular identification scheme for pre-1980 US data. This might suggest evidence of mispecification of the model, e.g., ascribed to the reason that some informative variables engaged in the Fed and private sectors decision making process are missing from the model. As suggested by Sims (1992), one promising way to solve this problem is to include an index of commodity price. Nevertheless, for the sake of feasibility conditional on the already large dimension of parameter space, a trivariate model like ours that is also common in the literature is preferred. Also, because we are interested in the evolution of impulse responses instead of impulse responses per se, hence, the price puzzle should not matter for understanding the variations in impulse responses. 16

17 (a) (b) (c) (d) Figure 2: Responses of inflation to 1% monetary policy shock for 1975Q1, 1981Q3, 1996Q1 and 2006Q3: (a) medians of impulse responses; (b) response in 1981Q3 minuses response in 1975Q1 with 95% credible interval; (c) response in 1996Q1 minuses response in 1981Q3 with 95% credible interval; (d) response in 2006Q3 minuses response in 1996Q1 with 95% credible interval. 17

18 (a) (b) (c) (d) Figure 3: Responses of unemployment to 1% monetary policy shock for 1975Q1, 1981Q3, 1996Q1 and 2006Q3: (a) medians of impulse responses; (b) response in 1981Q3 minuses response in 1975Q1 with 95% credible interval; (c) response in 1996Q1 minuses response in 1981Q3 with 95% credible interval; (d) response in 2006Q3 minuses response in 1996Q1 with 95% credible interval. 18

19 (a) (b) (c) (d) Figure 4: Responses of interest rate to 1% monetary policy shock for 1975Q1, 1981Q3, 1996Q1 and 2006Q3: (a) medians of impulse responses; (b) response in 1981Q3 minuses response in 1975Q1 with 95% credible interval; (c) response in 1996Q1 minuses response in 1981Q3 with 95% credible interval; (d) response in 2006Q3 minuses response in 1996Q1 with 95% credible interval. 19

20 7 Natural Rate of Unemployment and Short Run Phillips Curve 7.1 Dynamics of Natural Rate of Unemployment Following Milton Friedman s (1968) presidential address to the American Economic Association, the natural rate of unemployment (NRU) and the related concept of the nonaccelarating inflation rate of unemployment (NAIRU) have been at the center of the macroeconomic theory. Traditional approaches to estimating the natural rate often impose some restrictions to make the natural rate constant (at most with discrete jumps at certain periods of time, see Papell et al., 2000), force the NRU a function of time using spline (see Staiger et al., 1997), or other techniques such as calibrated unobserved-components models (see Gorden, 1997), low-pass filtering (see Staiger et al., 2001) and the Hodrick-Prescott filter (see Ball and Mankiw, 2002). King and Morley (2007), hereafter KM, endogenize the NRU as the unique steady-state equilibrium derived from a structural vector autoregression (SVAR) in the spirit of Phelps (1994): In a useful shorthand one may characterize the theory here as endogenizing the natural unemployment rate defined now as the current equilibrium steady-state rate, given the current capital stock and any other state variables. Hence, the natural rate of unemployment is not necessarily a constant over time. They estimate the steady-state of unemployment following Beveridge and Nelson (1981) by treating the long-run forecast in levels y t = lim h E t y t+h conditional on the information set available at time t as the stochastic trend in y t. We follow this strategy and cast the VAR in the companion form: Y t = g t + F t Y t 1 + ε Y,t+1. We employ this companion form to conduct forecasting by assuming that VAR parameters remain constants at their current values as time goes forward, i.e. in each period of time, a local-to-date stationary time-invariant VAR is fit to the data relying on which long-run forecasting is carried out. 22 Therefore, it yields that Y t = (I F t ) 1 g t and u t = s u Y t, 22 This assumption is common in the literature on bounded rationality and learning, see anticipatedutility model in Kreps (1998). 20

21 where s u is a selector vector pointing to unemployment and u t is the time-varying NRU. 23 The point estimate (posterior median) of the natural rate of unemployment, its 68% credible interval and the actual unemployment rate are plotted in Figure 5. The point estimate of the NRU ranges from 3.69% to 8.13% which is less volatile compared to the range of 1.8% - 9.5% of the point estimate obtained by KM but is comparable to Phelps s (1994) estimate. The uncertainty of the point estimate has declined roughly after 1985, which may be due to the substantial decline in the volatility of exogenous shocks. Besides the difference in the range of the NRU, our estimate is smoother than that derived in KB. It is possible because our model allows any block of VAR coefficients to stay at their previous values so that the trend unemployment is not forced to drift as a random walk in each period of time and the magnitude of drift can be smaller if only parts of VAR coefficients drift. Another discrepancy between our estimate of the NRU and that presented in KM is that the timing of peaks of the NRU coincides with that of the actual unemployment rates in KM, however, the peaks of our estimate of the NRU will lead the peaks of the actual data. In other words, the path of the NRU well mimics the actual rates in KM, but ours do not. This result is likely derived from the time-invariant VAR coefficients, since with constant companion form parameters, the long-run forecast lim h E t y t+h is positively related to the actual data y t at time t. As shown in Figure 5, the NRU peaked around 1980 and slumped from then on which accords with an argument that it is the underestimation of the NRU that result in the Great Inflation in 1970s, see Orphanides (2002). If the NRU is somehow underestimated by policymakers so that a permanent shift in the NRU is mistakenly understood as a transitory shift in cyclical unemployment, policymakers will be tempted to conduct a more inflationary policy to bring down unemployment and believe this inflationary policy will not lead to a severe inflation pressure in the coming episodes. This inflationary policy does work if the increase in unemployment is transitory. But as depicted in Figure 5, the NRU is soaring during , thus a short-run trade-off, if any, between inflation and cyclical unemployment is unexplorable. 24 Thus, it finally turns out that the inflation is out of control and unemployment stay high as the economic conditions are becoming worse. 7.2 Test for Short-run Phillips Curve As we have estimated the natural rate of unemployment in the last section, we can construct cyclical unemployment and test for the existence of the short-run Phillips curve. Following 23 Note that though the NRU is allowed to vary with time in both KM and our paper, but KM fit the whole sample by a time-invariant VAR model, albeit allowing for a unit root in the unemployment rate. In contrast, we allow the VAR coefficients and variances of error terms to be time-varying. 24 We don t assume the existence of a short-run trade-off between inflation and cyclical unemployment in our paper. Actually, this potential short-run trade-off will be tested in the next section. 21

22 Figure 5: Natural rate of unemployment: posterior median and 68% credible interval Gordon (1997), we investigate a triangle model, however, the supply shocks are set to zero. Inflation is regressed on four lags of inflation and current cyclical unemployment u C t as follows: 25 π t = 4 δ i π t i + βu C t + ω t, E(ω t π t i ) = 0, E(ω t u C t ) = 0, t. (6) i=1 The regression results based on the full sample and subsamples are listed in Table Two features stand out: First, since we presume the existence of the NRU, so it would be strange to find some evidence against that. Along the lines of Solow-Tobin test, see Solow (1968) and Tobin (1968), for natural rate hypothesis (NRH), if δ significantly deviates downward from 1, then the NRH will be rejected. However, Sargent (1971) argued that only 25 To take into account the uncertainty in the NRU, we run an OLS regression with respect to the NRU derived from each replication in the MCMC simulation, then report the posterior median and 90% credible interval of the slope on contemporaneous cyclical unemployment. Results are robust to different specifications (including intercepts or not), say regressing inflation on its own four lags, contemporaneous cyclical unemployment and four lags of cyclical unemployment. Using this specification is mainly due to the purpose of comparison to evidence derived from impulse responses. 26 The break point in 1990:Q4 is chosen because in the existing literature, some authors discussed the short-run trade-offs between inflation and cyclical unemployment before and after 1990, see Atkeson and Ohanian (2001) and King and Morley (2007), among others. Actually, arbitrarily selected break points deliver similar conclusions. 22