Evaluating FAVAR with Time-Varying Parameters and Stochastic Volatility

Transcription

1 Evaluating FAVAR with Time-Varying Parameters and Stochastic Volatility Taiki Yamamura Queen Mary University of London September 217 Abstract This paper investigates the performance of FAVAR (Factor Augmented Vector Autoregression) with time-varying parameters and stochastic volatility (TVP/SV-FAVAR) in capturing time variation in monetary policy transmission, in comparison with that of small-scale TVP/SV-VAR. The analysis is conducted through Monte Carlo (MC)-based experiments using open-economy Dynamic Stochastic General Equilibrium (DSGE) and Smets-Wouters model (27) as the data-generating process. The experiments show that although TVP/SV- VAR does not adequately detect the time variation, TVP/SV-FAVAR does. This result is subsequently interpreted in terms of the information amounts of these two empirical models. Using the concept of informational deficiency and a technique to compute it, both of which were proposed in recent literature, I reveal that small-scale VAR easily suffers from two kinds of informational problems in identifying the time variation, whereas FAVAR can avoid them by containing a sufficient number of latent factors. Moreover, a reliable algorithm to estimate factors can give a further help to TVP/SV-FAVAR in terms of overcoming the informational problems, even if the model possesses only a limited number of factors. Keywords: Structural FAVAR, Structural VAR, Time-varying parameters, Stochastic volatility, Monetary transmission. JEL codes: C18, C32, E52. Preliminary version. I would like to thank Prof. Haroon Mumtaz for valuable supervision. I also benefited from useful comments by participants at the 217 North American Summer Meeting of the Econometric Society, the IAAE 217 Annual Conference, and the Econometric Study Group Conference 217. All remaining errors are those of the author alone. School of Economics and Finance, Queen Mary University of London, Mile End Road, London E1 4NS, UK. t.yamamura@qmul.ac.uk 1

2 1 Introduction In empirical macroeconomics, there are two explanations for the Great Moderation in the U.S. One is the good luck hypothesis supported by structural Vector Autoregression (SVAR) methods (see, for example, Stock and Watson (22), Primiceri (25), Sims and Zha (26), and Gambetti et al. (28)). It attributes the macroeconomic stability to exogenous cause, which is a decline in the volatility of exogenous shocks. On the contrary, researches using sticky-price Dynamic Stochastic General Equilibrium (DSGE) provide the good policy explanation, arguing that the volatility decline in macroeconomic variables was caused by a shift in the systematic components of the monetary policy rules (e.g., Lubik and Schorfheide (24), Boivin and Giannoni (26)). To discuss the inconsistency between these two explanations, Benati and Surico (29) investigate the nature of Vector Autoregression (VAR) results when there is a move from passive to active monetary policy regimes. Their finding suggests that impulse response functions (IRFs) to a monetary policy shock exhibit little change across the regimes. They claim that VAR provides such a misleading result because of a structural difference between DSGE and its SVAR representation 1. Against this background, the main focus of this paper is the performance of the Factor- Augmented Vector Autoregression (FAVAR) model when there is a structural difference between VAR and the underlying DSGE. The model to be examined is FAVAR with time-varying parameters and stochastic volatility (TVP/SV-FAVAR). Through Monte Carlo (MC)-based exercises, I investigate the performance of TVP/SV-FAVAR in detecting time variation in monetary policy transmission, in comparison with that of small-scale TVP/SV-VAR (3-variable TVP/SV-VAR). TVP/SV-FAVAR and TVP/SV-VAR have been used in a wide range of empirical analyses for macroeconomics. TVP/SV-VAR was originally proposed by Primiceri (25) 2. Using this model, previous studies have approached several important economic topics such as the Great Moderation (Primiceri (25), Gali and Gambetti (29), Bianchi et al. (29), Canova and Gambetti (21)); quantitative easing (Kapetanios et al. (212), Baumeister and Benati (213)); oil prices (Baumeister and Peersman (213)); and fiscal policy (Kilem et al. (215)). TVP/SV- FAVAR has been also developed in recent literature. Baumeister et al. (213), Ellis et al. (214), and Yamamura (217) used this model to analyze changes in monetary policy effects from the 197s to the 2s in the U.S., the U.K., and Japan, respectively. Although many studies analyze time variation in transmission of economic shocks by using TVP/SV-FAVAR or TVP/SV-VAR, only limited extant research discusses whether these empirical applications can identify such variation adequately. An example of such studies is Benati and Surico (29). As mentioned above, they point out the problem of the VAR methods. Graeve (216), on the contrary, argues that TVP/SV-VAR usually captures time variation in the dynamics of underlying DSGE models sufficiently, though he also admits that there are 1 They argue that changes in the interest rate equation (i.e. the monetary policy rule) of a structural VAR bear no clear-cut relationship with changes in the parameters of the monetary policy rule in the underlying DSGE model. 2 Cogley (25) and Cogley and Sargent (25) developed a similar model. However, as pointed out by Primiceri (25), their model assumes the simultaneous relation between volatilities to be time invariant. 2

3 occasional failures in this respect. As for FAVAR, no existing studies systematically analyze their credentials from a similar perspective. In this paper, TVP/SV-FAVAR and TVP/SV-VAR are examined through MC-based experiments. For this exercise, two DSGE models are used as the data-generating process (DGP): the open-economy DSGE and the Smets-Wouters model (27). Note that while the earlier works (Benati and Surico (29) and Graeve (216)) use small-scale DSGE models 3, I adopt mediumscale ones. By using medium-scale DSGE, a structural mismatch is more likely to occur between VAR and the DSGE. Using each of the DGPs, I conduct 1 pseudo-experiments where, in each experiment, TVP/SV-FAVAR and TVP/SV-VAR are estimated using one of the simulated samples (pseudo-data). Regardless of the DGPs, the experiments will reveal that TVP/SV-FAVAR adequately detects the time variation in monetary policy effects, whereas TVP/SV-VAR does not. In a sense that small-scale VAR does not display a good performance, my result is consistent with the finding of Benati and Surico (29). A novel finding of my experiments is that FAVAR can be reliable even when VAR does not behave in a good manner. As for why TVP/SV-FAVAR can display a good performance while TVP/SV-VAR does not, I interpret it in terms of the information amounts of the two models. In this discussion, I use a concept of informational deficiency and a technique to quantitatively evaluate it, both of which were recently proposed by Forni et al. (216). Informational deficiency represents how deficient (not sufficient) the information amount of an empirical model is in terms of estimating a certain structural shock. Using their approach, I evaluate the deficiency of the 3-variable VAR for the monetary policy shock. Moreover, the same exercise is applied to the VAR system comprising principal components and interest rate, by which the deficiency of FAVAR model is also discussed. Through these exercises, it will be shown that VAR easily suffers from two kinds of informational problems, whereas FAVAR can avoid them by containing a sufficient number of latent factors. Moreover, my experimental results suggest that a reliable algorithm to estimate factors can give a further help to TVP/SV-FAVAR in terms of overcoming the informational problems, even if the model possesses only a limited number of factors. These findings explain why TVP/SV-FAVAR is superior to TVP/SV-VAR in capturing time variation in monetary policy transmission. This paper is organized as follows. In Section 2, the empirical models (TVP/SV-FAVAR and TVP/SV-VAR) are reviewed. Section 3 presents the experimental results, which show that TVP/SV-FAVAR displays a good performance whereas it is not the case for TVP/SV-VAR. These results are interpreted in Section 4, and Section 5 provides a conclusion. 2 Empirical models This section gives an overview of TVP/SV-FAVAR and TVP/SV-VAR. 3 They use New-Keynesian and/or Real Business Cycle models. 3

4 2.1 TVP/SV-FAVAR The model The basic specification of TVP/SV-FAVAR follows a recommendation in the literature (see, for example, Bernanke et al. (25) and Baumeister et al. (213)). The equation system of the model is given by X i,t = Γ i Z t + e i,t (1) Z t = c t + B 1,t Z t B L,t Z t L + v t (2) where t = 1,..., T. Eq. (1) is the observation equation where X i,t (i = 1,..., N) represents a panel of N observed variables. The common factors Z t consist of latent factors (F t = (F 1 t,..., F K t )) and interest rate R t, where the latent factors F t summarize co-movements among the underlying series. e i,t are i.i.d. idiosyncratic components with E[e i,t ] = and E[e i,t e i,t ] = R, where R is assumed as diagonal. Z t and X i,t are related through the factor loadings Γ i. Eq. (2) is the transition equation. c t is an n 1 vector of time-varying coefficients (n K + 1), B k,t (k = 1,..., L) are n n matrices of time-varying coefficients, and v t is an n 1 vector of disturbances with variance covariance matrix Ω t. Without a loss of generality, Ω t is assumed to have the following form: Using these definitions, (2) can be rewritten as where the vector β t stacks all the VAR coefficients. Ω t = A 1 t Σ t Σ t(a t) 1 (3) Z t = Z tβ t + A 1 t Σ t u t, u t N(, I n ) Z t = I n [1, Z t 1,..., Z t L ] (4) To recover the unobserved structural shocks u t, restrictions are imposed to the residual terms A 1 t Σ t u t. Following common works in the literature, a Cholesky structure is imposed as follows: 1. a 21,t 1... A t = a n1,t a nn 1,t 1 σ 1,t. σ.. 2,t., Σ t = σ n,t where A t and Σ t are assumed to be lower-triangular and diagonal, respectively. Following Primiceri (25), time-varying processes of β t, A t, and Σ t are specified by (5) β t = β t 1 + η t α t = α t 1 + τ t (6) log σ t = log σ t 1 + ε t where the vector α t stacks non-zero and non-one elements of the matrix A t, and the vector σ t stacks diagonal components of the matrix Σ t. All the innovations in the model are assumed to 4

5 be jointly normally distributed, and their variance covariance matrix are specified by V = V ar u t η t τ t ε t I n = Q D G (7) where G is assumed to be diagonal (G = diag(g i )). Following Cogley and Sargent (25) and Primiceri (25), the lag length L in the transition equation (2) is set equal to two (L = 2). This choice is based on the fact that stability of VAR is adversely affected when the lag length is large Identification scheme The restrictions to identify monetary policy shock and latent factors are described below. Restrictions to identify monetary policy shock First, the common factors Z t are specified by Z t = {R t, F t } = {R t, F 1 t,..., F K t } Then, to identify the monetary policy shock, I use sign restrictions scheme. Following Canova and Nicol (22), Uhlig (25), and Ellis et al. (214), restrictions are placed on the contemporaneous response of the observed variables. As the restrictions, I assume that a tightening monetary policy shock increases the interest rate, and decreases the output and inflation on impact: (IRF ) Y t,h= < (IRF ) π t,h= < (8) (IRF ) R t,h= > where (IRF ) s t,h= represents the impulse response of the variable s (= Y, π, or R) to the monetary policy shock at time t for horizon h = (i.e. contemporaneous response). To impose the above restrictions, I take the procedure used by Ellis et al. (214). restrictions are introduced as follows. covariance matrix Ω t : The First, Cholesky decomposition is applied to the VAR Ω t = P t P t where P t corresponds to A 1 t Σ t in Eq. (3). Then, N N matrix ( J) is drawn from the N(, 1) distribution. Next, the QR decomposition is applied to J (i.e. J = QR with QQ = I), by which a candidate covariance matrix is obtained as Ω t = P t P t with P t = P t Q = A 1 t Σ t Q. Using this candidate matrix, I calculate the contemporaneous impulse responses of the observed variables to monetary policy shock. If these responses satisfy the restrictions (8), Pt is stored. This procedure is repeated until I have obtained 1 P t matrices. After storing 1 matrices, 5

6 the best P t is determined by choosing the P t matrix with elements closest to the median across the 1 estimates. Restrictions for unique identification of latent factors As mentioned in Bernanke et al. (25), it is important to impose restrictions to identify the latent factors and the associated loadings uniquely. Now, let me write the observation equation by X t = Λ f F t + Ψ R R t + e t By applying rotations to the factors (F t = AF t BR t ), this equation becomes X t = Λ f A 1 Ft + (Ψ R + Λ f A 1 B)R t + e t Therefore, for unique identification of the factors and the loadings, it needs to hold that Λ f A 1 = Λ f and Ψ R + Λ f A 1 B = Ψ R. According to Bernanke et al. (25), a sufficient condition for this is that the upper K K block of Λ f is identity matrix and the upper K 1 block of Ψ R is zero. As a common work in the literature, I adopt these restrictions Estimation methodology Estimation procedure The model is estimated by Bayesian approach described in Kim and Nelson (1999). The parameters to be estimated are: Γ (factor loadings); R = diag(r i ) (variance of the idiosyncratic components i (i = 1,..., N)); {β t } T t=1 (VAR coefficients); Q (covariance matrix for β t); {a ij,t } T t=1 (off-diagonal elements of A t ); D (covariance matrix for a ij,t ); {h i,t } T t=1 (diagonal elements of H t); g i (variance of ln(h i,t )); and {F t } T t=1 (factors). Priors and starting values for the parameters follow a standard recommendation in the literature 4. The estimation algorithm is made up of the following steps. To begin with, the factor loadings (Γ) are sampled along with the variance of the idiosyncratic components (R = diag(r i )), where Γ and R i are drawn from normal and inverse gamma distributions, respectively. The VAR coefficients (β t ), the off-diagonal elements of A t (a ij,t ), and their covariance matrices (Q and D) are then drawn through the method developed by Carter and Kohn (1994), while h i,t and g i are simulated by using the scheme described in Jacquier et al. (1994). Using the procedure described in 2.1.2, the variance matrix is obtained as Ω t = P t P t. The latent factors F t are sampled by relying on the algorithm of Carter and Kohn (1994) 5. The above steps are iterated 1, times, with the first 9, draws removed as a burn-in (M = 1, and M = 9, ). For more detail of the estimation procedure, see Appendix A. 4 For detail, see Appendix A.2. 5 The way to employ their algorithm is in line with Bernanke et al. (25) and Kim and Nelson (1999). 6

7 Computation of IRF (impulse response function) Impulse responses of the common factors Z t (F t and R t ) to monetary policy shock are computed at each point in time. Following Koop et al. (1996), the responses of the factors Z t at time t for horizon h are defined by (IRF ) Z t,h = E[Z t+h Ξ t, Z t 1, µ MP ] E[Z t+h Ξ t, Z t 1 ] where Ξ t represents all the parameters in the model at time t, and Z t 1 denotes the history of Z up to time (t 1). This equation indicates that IRFs are computed as difference between two conditional expectations. The first term is a forecast of Z t+h at the forecast origin t conditional on monetary policy shock µ MP, while the second term is the baseline forecast (i.e. the forecast conditional on a zero monetary shock). Note that when calculating the impulse responses, I do not take into account drift of the parameters over an impulse horizon (during t to t + h). Once the responses of Z t are obtained, it is straightforward to compute the responses of the observed variables X t through the observation equation: where Γ represents the factor loadings. (IRF ) X 1 t,h. (IRF ) X N t,h = Γ(IRF )Z t,h 2.2 TVP/SV-VAR As for TVP/SV-VAR, I use the specification of Primiceri (25). The equation system is given by X t = c t + B 1,t X t B L,t X t L + v t, V ar(v t ) = Ω t (9) where the vector X t consists of observed variables. Regarding time variation in the coefficients and shocks, the same assumptions are adopted as in the transition equation in TVP/SV-FAVAR, which are summarized by Eqs. (3)-(7). The lag length is set equal to two (L = 2). As in the case of TVP/SV-FAVAR, the model is estimated by using the MCMC algorithm of Kim and Nelson (1999), and the sign restrictions scheme is adopted to identify monetary policy shock. The parameters to be estimated are: {β t } T t=1 (VAR coefficients); Q (covariance matrix for β t ); {a ij,t } T t=1 (off-diagonal elements of A t); D (covariance matrix for a ij,t ); {h i,t } T t=1 (diagonal elements of H t ); and g i (variance of ln(h i,t )). Priors and starting values for these parameters are the same as in TVP/SV-FAVAR. Sampling procedure consists of the following steps: the VAR coefficients (β t ), the off-diagonal elements of A t (a ij,t ), and their covariance matrices (Q and D) are sampled through the method of Carter and Kohn (1994); while h i,t and g i are drawn by using the scheme of Jacquier et al. (1994). Using the procedure described in 2.1.2, the variance matrix is obtained as Ω t = P t P t. All the above steps are iterated 1, times, with the first 9, draws removed as a burn-in (M = 1, and M = 9, ). 7

8 Table 1: Macroeconomic variables in the open-economy DSGE model Notation Definition C t consumption Y t output π t inflation π H,t inflation for domestic goods π F,t inflation for imported goods S t terms of trade Ψ F,t price gap q t real exchange rate R t nominal interest rate Yt output in foreign economy π t inflation in foreign economy nominal interest rate in foreign economy R t 3 Monte Carlo (MC)-based experiments In this section, MC-based experiments are conducted to examine whether TVP/SV-FAVAR and TVP/SV-VAR correctly detect time variation in monetary policy effects. 3.1 Data generating process (DGP) As the baseline data-generating process (DGP), I use the open-economy DSGE developed by Justiniano and Preston (21). As alternative one, Smets-Wouters model (27) is also used. The experiments with the latter model will be performed as a robustness check (see 3.4.3). The open-economy DSGE model is a generalization of the model of Monacelli (25), where the generalization includes introduction of incomplete asset markets, habit formation, and indexation of prices to past inflation. The model consists of eight exogenous shocks and twelve macroeconomic endogenous variables. All the endogenous variables are listed in Table 1. The model is calibrated by using the estimates of Justiniano and Preston (21), but as described below, time variation is additionally imposed in the monetary policy rule. In the open-economy DSGE, the monetary policy rule is governed by r t = ρ i r t 1 + ψ π π t + ψ y y t + ψ y y t + ψ e e t + ε M,t (1) where r t denotes nominal interest rate, y t is output, π t and πt represent inflation in domestic and foreign economies, and ε M,t is monetary policy shock. Moreover, e t is a change in nominal exchange rate, which is defined by e t = (q t q t 1 ) + (π t πt ) where q t is real exchange rate. Note that all the variables in the above equation (r t, y t, π t etc.) represent log deviations from their respective steady-state level. As time variation in the policy rule, I impose the following 8

9 1 2 Output Regime1 Regime horizon.5 1 Inflation Regime1 Regime horizon 1.5 Interest rate Regime1 Regime2 5 1 horizon 15 Figure 1: Theoretical impulse response functions (IRFs) of output, inflation, and interest rate to monetary policy shock in the open-economy DSGE: Black and red lines represent the responses in Regimes 1 and 2, respectively. Monetary policy shock is defined by an increase of 1 basis points in the interest rate. structural break in ψ π : ψ π = { 1.8 (for 1 t 125; Regime 1).45 (for 126 t 25; Regime 2) (11) The value in Regime 1 (ψ π = 1.8) is the estimate of Justiniano and Preston (21), and in Regime 2, ψ π is set four times smaller than it (ψ π =.45). Note that the choice of four times is based on the observation of Fernández-Villaverde and Rubio-Ramírez (28) 6. Fig. 1 gives theoretical IRFs of three variables (output, inflation, and interest rate) to the monetary policy shock. It exhibits that the variation in the parameter ψ π significantly strengthens the impulse responses of output and inflation. The model is solved separately by regimes. Then, by simulating the models 1 times, 1 pseudo-data (simulation samples) are prepared. In each simulation, 5 observations are generated for each regime, by using the exogenous shocks drawn from normal distribution, and the first 375 observations are discarded to reduce impact of initial conditions. By this procedure, the sample length is set equal to Specifications of empirical models In TVP/SV-FAVAR, the observed vector X t and the common factors Z t are defined by X t = {Y t, π t, R t, C t, Y t, π t, π H,t, π F,t, S t, Ψ F,t, q t } (12) As for definition of each observed variable, see Table 1. defined by 6 They observed that γ π (equivalent parameter to ψ π) changes by four to five times through the Great Moderation. Z t = {R t, F t } = {R t, F 1 t,..., F K t } (13) In TVP/SV-VAR, the vector X t is X t = {R t, Y t, π t } (14) 9

10 The number of latent factors in TVP/SV-FAVAR When a large number of factors is added to TVP/SV-FAVAR, it adversely affects VAR s stability in the transition equation (due to the curse of dimensionality), so that the estimation algorithm does not work adequately. From this perspective, I consider only the models with K 3 throughout this paper, and under this situation, K is optimized. The optimization is conducted based on impulse responses of three key variables (output, inflation, and interest rate) to monetary policy shock. Note that I do not rely on data-driven method such as a use of the information criteria. This is due to the fact that the aim of the MC exercise is to evaluate the intrinsic ability of the empirical models, whereas performance of the optimization method for the model specification is not necessarily of interest. For this reason, the number of factors is optimized just by rough checks of the estimated impulse responses. Through such a procedure, I decide to adopt K = Experimental results Using the 1 simulation samples, I perform 1 pseudo-experiments where, in each experiment, TVP/SV-FAVAR and TVP/SV-VAR are estimated with one of the samples. The estimations are conducted by Bayesian approach. Fig. 2(a) displays the estimated median cumulative IRFs to monetary policy shock across the 1 experiments using TVP/SV-FAVAR. Note that the first 5 periods (t = 1 5) are not included in each plot, because those periods are used as a training sample. In the long-horizon region, it is observed that the responses of output and inflation strengthen around t = 125, which is consistent, in both timing and direction of variation, with the true change in the DSGE (see Fig. 1). Fig. 2(b) shows the corresponding plots obtained by TVP/SV-VAR. It suggests that the response of inflation strengthens around t = 125, whereas that of output decreases in an unexpected manner. The response of interest rate also exhibits a structural break at t = 125, and this should be also regarded as strange, because the theoretical IRF of interest rate is not much affected by the change in ψ π (see Fig. 1). As for the responses of output and inflation, I also check their time variation at a fixed impulse horizon of h = 24, which is shown in Fig. 3. In this figure, the estimated responses are compared with the true ones which are indicated by the red line. TVP/SV-FAVAR exhibits that the estimated response is consistent with the true one within estimation uncertainty, in both terms of output and inflation. However, in the case of TVP/SV-VAR, the estimated result for output deviates from the true response, where the gap between them can not be explained by estimation uncertainty at all. 3.4 Robustness checks The MC exercise in the previous subsection (I call it baseline exercise, henceforth) has observed that the performance of TVP/SV-FAVAR is reliable in terms of detecting time variation in monetary policy transmission, while that of TVP/SV-VAR is not. In this subsection, three types of alternative exercises are performed to check if the above observation is robust to changes in 1

11 Output Inflation Interest rate (a) TVP/SV-FAVAR Output Inflation Interest rate (b) TVP/SV-VAR Figure 2: Time-varying median cumulative impulse responses to monetary policy shock across the 1 pseudo-experiments: The estimated responses by (a) TVP/SV-FAVAR and (b) TVP/SV-VAR are depicted. Monetary policy shock is defined by an increase of one percent in the interest rate. 11

12 Output Inflation Time Time 2 25 (a) TVP/SV-FAVAR Output Inflation Time Time 2 25 (b) TVP/SV-VAR Figure 3: Time-varying cumulative impulse responses at an impulse horizon h = 24 obtained by the 1 pseudo-experiments: The estimated responses by (a) TVP/SV-FAVAR and (b) TVP/SV-VAR are depicted. Blue solid line, shaded area, and dashed lines represent the median, 16/84-percentiles, and 5/95-th percentiles of the estimated responses, respectively. Red solid line indicates the true response in the DSGE. 12

13 the experimental setup. The alternative setups are defined as follows: tight prior is used in the Bayesian estimation (in 3.4.1); smoothed time variation is imposed in the parameter ψ π (in 3.4.2); Smets-Wouters model (27) is used as another type of DGP (in 3.4.3) Tight prior In TVP/SV-FAVAR and TVP/SV-VAR, the time-varying coefficients β t play an important role in capturing time variation in monetary policy transmission. As described in Section 2, their volatility is controlled by the covariance matrix Q. In my estimation algorithm, the prior for Q is specified by the following inverse-wishart distribution 7 : Q iw (Q, T ) where Q = τ V ar[ β OLS ] T with τ = , and T is a length of the training sample 8. To perform a robustness check, the prior of Q is tightened to τ = Note that tighter prior of Q restricts the time-varying coefficients β t to be less volatile. Then, MC exercise is conducted, where the experimental setup is the same as in the baseline exercise except for the prior of Q. Fig. 4 shows the obtained cumulative responses (at an impulse horizon h = 24). In each plot, the median estimated response (blue solid line) is almost the same as that of the baseline exercise (red dashed line). This suggests that the experimental results are hardly affected by prior setup DGP with smoothed time variation In the baseline exercise, a structural break is imposed in the monetary policy rule (see Eq. (11)). As another option, let me consider the following smoothed time variation: ( ) 2π ψ π,t = A + B sin T t A = 1 2 (ψ max + ψ min ) B = 1 2 (ψ max ψ min ) (T = 25) where ψ max = 1.8 and ψ min =.45. Note that ψ π takes a different value for each t. Therefore, the solution form of the model is prepared for each t. Then, by simulating the model, 1 pseudodata are prepared. In each simulation, 75 observations are generated by using exogenous shocks drawn from normal distribution, and the first 5 observations are discarded to reduce impact of initial conditions. By this procedure, the sample length is set equal to 25. Using this DGP, I perform the same pseudo-experiments as in the baseline exercise, and Fig. 5 shows the obtained cumulative IRFs. In the case of TVP/SV-FAVAR, the responses of 7 For more detail, see A.2. As mentioned in 3.3, the periods t = 1-5 are used for the training. Moreover, βols represents an OLS estimate, which is obtained by OLS estimation over the training sample via a fixed-coefficient VAR model made up of the interest rate and the principal components (i.e. the VAR vector is defined as Z t = {R t, (P C) t}.). For more detail, see A.2. 13

14 Output Inflation Time Time 2 25 (a) TVP/SV-FAVAR Output Inflation Time Time 2 25 (b) TVP/SV-VAR Figure 4: Time-varying cumulative impulse responses at an impulse horizon h = 24 obtained by the 1 pseudo-experiments when prior for Q is tightened (τ = ): The estimated responses by (a) TVP/SV-FAVAR and (b) TVP/SV-VAR are depicted. Blue solid line, shaded area, and dashed lines represent the median, 16/84-percentiles, and 5/95-th percentiles of the estimated responses, respectively. Red dashed line indicates the estimated median response obtained in the baseline exercise in

15 Output Inflation Interest rate (a) TVP/SV-FAVAR Output Inflation Interest rate (b) TVP/SV-VAR Figure 5: Time-varying median cumulative impulse responses to monetary policy shock across the 1 experiments (when smoothed time variation is imposed in ψ π ): The estimated responses by (a) TVP/SV-FAVAR and (b) TVP/SV-VAR are depicted. Monetary policy shock is defined by an increase of one percent in the interest rate. output and inflation exhibit smoothed time variation but that of interest rate hardly varies. In the case of TVP/SV-VAR, the response of inflation varies smoothly over time but there is no such sign in the response of output, which is not a reasonable result. Furthermore, the response of interest rate is observed to vary over time, and this is also inconsistent with the truth. Fig. 6 illustrates the time variation in the cumulative responses of output and inflation at an impulse horizon h = 24. The estimated result with TVP/SV-FAVAR is in good agreement with the truth. However, TVP/SV-VAR shows that the estimated response of output is not consistent with the true response, whose inconsistency cannot be explained by estimation uncertainty. In summary, Figs. 5 and 6 give the same implication as in the baseline exercise: TVP/SV-FAVAR is superior to TVP/SV-VAR in terms of identifying time variation in monetary policy effects. 15

16 2 Output Inflation Time Time (a) TVP/SV-FAVAR Output Inflation Time Time 2 25 (b) TVP/SV-VAR Figure 6: Time-varying cumulative impulse responses at an impulse horizon h = 24 obtained by the 1 pseudo-experiments (when smoothed time variation is imposed in ψ π ): The estimated responses by (a) TVP/SV-FAVAR and (b) TVP/SV-VAR are depicted. Blue solid line, shaded area, and dashed lines represent the median, 16/84-percentiles, and 5/95-th percentiles of the estimated responses, respectively. Red solid line indicates the true response in the DSGE

17 Table 2: Endogenous variables in Smets-Wouters model (27) Notation Definition Y t output C t consumption L t labor (hours worked) R t nominal interest rate π t inflation W t real wage µ w t wage mark-up µ p t price mark-up I t investment q t value of capital stock K t capital installed Kt s capital used in production Rt k rental rate of capital z t capital utilization costs Use of another DSGE (Smets-Wouters model (27)) As another DGP, I use the medium-scale DSGE model of Smets and Wouters (27) (Smets- Wouters model (27)). As well as the open-economy DSGE, this model contains many shocks and frictions that affect households and firms. Its key features include nominal price and wage settings, habit information in consumption, and investment adjustment costs. On the basis of Smets and Wouters (23) and Christiano et al. (25), Smets and Wouters (27) extends monetary business cycle model so that it is consistent with a balanced steady-state growth path driven by deterministic technological progress. The model consists of seven exogenous shocks and fourteen endogenous variables. All the endogenous variables are listed in Table 2. In simulating the model, the sample length is set equal to 25. For calibration, I use the estimates of Smets and Wouters (27) with subsample for Regime 1 (1 t 125), and those with subsample for Regime 2 (126 t 25). The simulation procedure is the same as the one with the open-economy DSGE (see 3.1). First, the model is solved separately by regimes. Then, by simulating the models 1 times, 1 pseudo-data are prepared. In each simulation, 5 observations are generated for each regime, and the first 375 observations are discarded. Fig. 7 displays the theoretical IRFs of output, inflation, and interest rate. As key features, the response of output changes to a remarkable extent between the two regimes, whereas those of inflation and interest rate do not significantly. The observed vectors in TVP/SV-FAVAR and TVP/SV-VAR are defined by: X t = { z t, C t, K s t, W t, Y t, π t, L t, R k t } X t = {R t, Y t, π t } (for TVP/SV-FAVAR) (for TVP/SV-VAR) (15) For definition of the variables, see Table 2. Identification scheme and estimation procedure are basically the same as in the baseline exercise, but the exception is that the number of latent 17

18 2 4 Output Regime1 Regime horizon.5 Inflation Regime1 Regime horizon 1.5 Interest rate Regime1 Regime horizon Figure 7: Theoretical impulse response functions (IRFs) of output, inflation, and interest rate to monetary policy shock in the Smets-Wouters model (27): Black and red lines represent the responses in the cases of Regimes 1 and 2, respectively. Monetary policy shock is defined by an increase of 1 basis points in the interest rate. factors in TVP/SV-FAVAR is set equal to three (K = 3) 9. Figs. 8 and 9 display the estimated impulse responses to monetary policy shock, which are obtained from the pseudo-experiments. In the case of TVP/SV-FAVAR, the estimated response of output strengthens around t = 125, while those of inflation and interest rate do not clearly vary across the whole study period 1. In the case of TVP/SV-VAR, a clear time variation is not observed in any of the three variables. Moreover, it is especially strange that the response of interest rate is far below zero at a long horizon, which also indicates that TVP/SV-VAR fails to identify monetary policy shock correctly. To sum up, the exercise with Smets-Wouters model (27) has provided the same finding as in the baseline exercise and the other alternative exercises: TVP/SV-FAVAR shows a good performance whereas it is not the case for TVP/SV-VAR. 3.5 Further question on experimental results As mentioned in Section 1, Benati and Surico (29) investigate the nature of VAR results when there is a move from passive to active monetary policy regimes, and find that the estimated IRFs to monetary policy shock exhibit little change across the regimes. In a sense that small-scale VAR does not display a good performance, the experimental results in the previous subsections are consistent with the finding of Benati and Surico (29). As explained by Benati and Surico (29), VAR result is inadequate because of a structural difference between DSGE and its SVAR representation ( changes in the interest rate equation (i.e. the monetary policy rule) of a structural VAR bear no clear-cut relationship with changes in the parameters of the monetary policy rule in the underlying DSGE model ). A novel finding of my experiments is that FAVAR seems to overcome this problem. As for its reason, I need further discussion, which will be performed in Section 4. 9 For the optimization procedure for K, see In Fig. 8(a), it looks that the response of inflation strengthens around t = 125, but Fig. 9(a) shows this change is not statistically significant. 18

19 Output Inflation Interest rate (a) TVP/SV-FAVAR Output Inflation Interest rate (b) TVP/SV-VAR Figure 8: Time-varying median cumulative impulse responses to monetary policy shock across the 1 pseudo-experiments with the Smets-Wouters model (27): The estimated responses by (a) TVP/SV-FAVAR and (b) TVP/SV-VAR are depicted. Monetary policy shock is defined by an increase of one percent in the interest rate. 19

20 5 5 1 Output Inflation Time Time 2 25 (a) TVP/SV-FAVAR Output Inflation Time Time (b) TVP/SV-VAR Figure 9: Time-varying cumulative impulse responses at an impulse horizon h = 24 obtained by the 1 pseudo-experiments with the Smets-Wouters model (27): The estimated responses by (a) TVP/SV-FAVAR and (b) TVP/SV-VAR are depicted. Blue solid line, shaded area, and dashed lines represent the median, 16/84-percentiles, and 5/95-th percentiles of the estimated responses, respectively. Red solid line indicates the true response in the DSGE. 2

21 4 Why does TVP/SV-FAVAR adequately capture time variation in monetary policy transmission? In Section 3, TVP/SV-FAVAR displayed a good performance whereas TVP/SV-VAR did not, in terms of detecting time variation in monetary policy transmission. To interpret why such results were obtained, I use the approach of Forni et al. (216). In Section 4.1, I introduce the concept of informational deficiency and the technique used to compute it, both of which are developed by Forni et al. (216). Then, by using their technique, I evaluate the informational deficiency of the 3-variable VAR in terms of identifying monetary policy shock. The same exercise is also applied to a VAR system comprising principal components and interest rate, by which the deficiency of the FAVAR model is examined. Using the obtained results, a discussion will follow in Section 4.2, as to why TVP/SV-FAVAR can adequately detect time variation in monetary policy transmission even when TVP/SV-VAR cannot. Regarding evaluation of the informational deficiency (or sufficiency), the technique presented by Forni et al. (216) is not the only approach proposed in the literature. One popular method is the Granger causality test developed by Giannone and Reichlin (26) and Forni and Gambetti (214). Moreover, Canova (216) recently proposed a new method as a more robust approach. However, these methods are not designed to analyze the information amount of VAR for a certain specific shock 11. Because my interest is the informational sufficiency/deficiency for one specific shock (i.e. monetary policy shock), I adopt the approach of Forni et al. (216). 4.1 Informational deficiency Definition Let x t = [x 1,t,..., x n,t ] denote a vector of n variables used in VAR, and assume that this vector has moving average representation at the DGP level: x t = A x (l)u t l l= where u t = [u 1,t,..., u q,t ] are structural shocks (q 1 vector), and A x (l) represents the n q matrix of impulse response functions. Theoretically, VAR with lag L decomposes the vector x t to two orthogonal components: x t = P (x t H x t 1(L)) + ϵ (L) t where H x t (L) is a closed linear space defined by x t k with k =,..., (L 1); and P (a b) represents a projection function of a onto b. When u i,t (i-th component of u t ) is the shock of interest, it 11 Their methods are designed to check whether VAR system contains enough information to recover the vector of structural shocks (instead of one specific shock), which corresponds to a check of fundamentalness (Hansen and Sargent (1991), Lippi and Reichlin (1993), Lippi and Reichlin (1994), Chari et al. (28)), or informational sufficiency (Forni and Gambetti (214)). 21

22 can be written as u i,t = P (u i,t ϵ (L) t ) + e (L) i,t (16) In (16), e (L) i,t represents a discrepancy between the shock identified by VAR and the true shock, and this discrepancy is generated by a deficiency of the VAR information set. On this basis, Forni et al. (216) define informational deficiency of VAR for the shock u i,t by the fraction of unexplained variance in the projection function (16): δ i (L) = σ2 e i,(l) σ 2 u i. Note that δ i = represents a complete informational sufficiency, whereas δ i = 1 indicates that VAR contains no information on the shock u i,t. Formula Forni et al. (216) derive a simple formula to compute δ i (L) as follows. First, Proposition 2 in their paper gives P (u i,t ϵ (L) t ) = P (u i,t Ht x (L)) From this, δ i (L) = σ2 e i,(l) σu 2 = V ar[u i,t P (u i,t e (L) t )] i σu 2 i = V ar[u i,t P (u i,t Ht x (L))] σu 2 i (17) Using the definition y t [x t,..., x t L ], V ar[u i,t P (u i,t H x t (L))] = V ar[u i,t u i,t y t(y t y t) 1 y t ] = σu 2 i E[u i,t y t]σ 1 y E[u i,t y t ] = σu 2 i A (i) y () Σ 1 A y (i) () y (18) where A (i) y () represent the contemporaneous responses of y t to u i,t, and Σ y is the variance covariance matrix of y t. From (17) and (18), a formula to compute δ i is derived: δ i (L) = 1 A (i) y () Σ 1 y A (i) y ()/σ 2 u i (19) 22

23 Table 3: Informational deficiency of VARs for estimating a monetary policy shock in the openeconomy DSGE: x t represents the VAR vector. The lag of the VAR system is set equal to two (L = 2). (i) x t = [R t, Y t, π t ] Regime 1 Regime (ii) x t = [R t, (P C) 1,t,..., (P C) K,t ] Regime 1 Regime 2 K = K = K = K = K = K = Note that Σ y is obtained by Σ y = Γ Γ 1 Γ L Γ 1 Γ Γ L Γ L Γ Γ k = E[x t x t k ] = q σu 2 m m=1 l= ( ) A (m) x (l) A (m) x (l + k) where q is the number of structural shocks, and A (m) x (l) is the m-th column of the impulse response matrix A x (l). Computation I compute informational deficiency for the monetary policy shock in the open-economy DSGE and the Smets-Wouters model (27). As for the VAR vector x t, the following two cases are investigated: (i) x t = [R t, Y t, π t ] (ii) x t = [R t, (P C) 1,t,..., (P C) K,t ] where (P C) j,t is the j-th principal component obtained from all the endogenous variables used in TVP/SV-FAVAR estimation (see Eqs. (12) and (15)). The case (i) evaluates the deficiency of small-scale VAR, whereas the case (ii) pertains to the deficiency of FAVAR. Note also that the lag of VAR system is set equal to two (L = 2) in both cases. By Eq. (19), the informational deficiency is evaluated, and the obtained results are summarized in Tables 3 and 4. 23

24 Table 4: Informational deficiency of VARs for estimating a monetary policy shock in the Smets- Wouters model (27): x t represents the VAR vector. The lag of the VARs is set equal to two (L = 2). (i) x t = [R t, Y t, π t ] Regime 1 Regime (ii) x t = [R t, (P C) 1,t,..., (P C) K,t ] Regime 1 Regime 2 K = K = K = K = K = K = Discussion on performances of TVP/SV-FAVAR and TVP/SV-VAR Using the computation results for informational deficiency, I discuss the performances of TVP/SV- FAVAR and TVP/SV-VAR observed in Section 3 (TVP/SV-FAVAR displayed a good performance in terms of detecting time variation in monetary policy effects, whereas TVP/SV-VAR did not). In both of Tables 3 and 4, the information amount of the 3-variable VAR (x t = [R t, Y t, π t ] ) is found to be deficient in terms of identifying the monetary policy shock. On the contrary, FAVAR (x t = [R t, (P C) 1,t,..., (P C) K,t ] ) becomes informationally sufficient as the number of principal components increases. These findings imply that even if a small-scale VAR does not have sufficient information for capturing monetary policy shock, the information amount of FAVAR can be enough by virtue of sufficient number of latent factors. Another finding in Tables 3 and 4 is that the degree of informational deficiency can change across different regimes. This change is prominent, especially when the information amount of empirical models is not enough (see Table 4(i), or Tables 3(ii) and 4(ii) with small K). It should be emphasized that both informational deficiency and its variation diminish the ability of VAR to capture time variation in monetary policy transmission. The obtained implication is that small-scale VAR can easily suffer from both of these two problems, whereas FAVAR can overcome them by possessing a sufficient number of latent factors. As for the number of factors to achieve informational sufficiency, the implication of Tables 3 and 4 is not necessarily consistent with that of the MC experiments in Section 3. In Section 3, TVP/SV-FAVAR with K = 2 (when DGP is the open-economy DSGE) or K = 3 (when DGP is the Smets-Wouters model (27)) exhibited a good performance, whereas Tables 3 and 4 suggest that K should be four or even more than that. To discuss this inconsistency, it would be useful to recall that the exercise in Section 4.1 used principal components in examining the deficiency of FAVAR. As pointed out by Forni and Gambetti (214), an exercise using principal components may overestimate the number of latent factors needed to achieve informational sufficiency, because the principal components may be imperfect estimates of the latent factors. Regarding this point, Stock and Watson (21) describe that principal components estimation is 24

25 based on cross-sectional averaging 12. On the contrary, when I estimate TVP/SV-FAVAR in the MC experiments, the latent factors are estimated by using the algorithm of Carter and Kohn (1994), as explained in Section 2.1. Carter and Kohn s (1994) algorithm computes the factors using the information spanning across not only a series but also full sample period (through a Kalman filter and an additional backward recursive algorithm). Therefore, to the extent of relying on full period information, the obtained factors through their algorithm are expected to be better estimates than the principal components. Then, this can be regarded as the reason why TVP/SV-FAVAR could display a good performance in Section 3, even when it contained a limited number of factors (K 3) Conclusion This paper has investigated the performance of TVP/SV-FAVAR in capturing time variation in transmission of monetary policy shock, in comparison to that of small-scale TVP/SV-VAR. The analysis is conducted through MC-based experiments using the open-economy DSGE and Smets-Wouters model (27) as the data-generating process. The experiments showed that TVP/SV-VAR does not adequately detect time variation in monetary policy transmission, but TVP/SV-FAVAR does. Subsequently, the experimental results were interpreted in terms of the information amounts of these two empirical models. Using the technique of Forni et al. (216), it was quantitatively confirmed that VAR does not contain sufficient information to estimate the monetary policy shock. Furthermore, when the information amount of VAR is far from being enough, the extent of informational deficiency can vary significantly due to an effect of changes in DGP. It is generally expected that both informational deficiency and its variation diminish the ability of empirical models to capture time variation in structural shock transmission. Smallscale VAR easily faces both problems, but FAVAR can overcome them by virtue of a sufficient number of factors. Moreover, the experimental results also indicated that a reliable algorithm to estimate factors can give a further help to TVP/SV-FAVAR in terms of overcoming the informational problems, even if the model possesses a limited number of factors (K 3). These findings explain why TVP/SV-FAVAR is superior to TVP/SV-VAR in capturing time variation in monetary policy transmission. 12 This means that principal components at time t (= (P C) t) are estimated by using only the information at the contemporaneous period, instead of relying on the information spanning full sample period (i.e. 1 t T ). 13 The fact that TVP/SV-FAVAR with a small number of factors (i.e. K 3) behaves in a good manner is very important. This is because a large number of factors should not be added to TVP/SV-FAVAR in terms of the curse of dimensionality, as mentioned in Section 3.2. For the same reason, it is not favorable to use a medium- or large-scale TVP/SV-VAR. From such a perspective, the implication of the MC experiments can be expressed more carefully like the following: Even when a small-scale TVP/SV-VAR does not adequately detect time variation in monetary policy transmission, TVP/SV-FAVAR with a limited number of factors can possess a reliable ability to do that. 25

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