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Monetary Policy across Space and Time WP 18-14 Laura Liu Federal Reserve Board of Governors Christian Matthes Federal Reserve Bank of Richmond Katerina Petrova University of St. Andrews Monetary Policy across Space and Time Laura Liuy Christian Matthesz Katerina Petrovax August 13, 2018 Working Paper No. 18-14 Abstract In this paper we ask two questions: (i) is the conduct of monetary policy stable across time and similar across major economies, and (ii) do policy decisions of major central banks have international spillover e¤ects. To address these questions, we build on recent semi-parametric advances in time-varying parameter models that allow us to increase the VAR dimension and to jointly model three advanced economies (US, UK, and the Euro Area). In order to study policy spillovers, we jointly identify three economy-speci…c monetary policy shocks using a combination of sign and magnitude restrictions. JEL codes: C54, E30, E58 Keywords: Monetary policy spillovers, time-varying parameters, changing volatility We would like to thank Thomas Lubik as well as workshop participants at the EUI workshop on time-varying parameter models and the St. Andrews Workshop on time-varying uncertainty in macro. The views expressed in this paper are those of the authors and do not necessarily re‡ect those of the Federal Reserve Board of Governors, the Federal Reserve Bank of Richmond, or the Federal Reserve System. y Federal Reserve Board of Governors, Email: Laura.Liu@frb.gov z Federal Reserve Bank of Richmond, Email: Christian.Matthes@rich.frb.org x University of St. Andrews, Email: Katerina.Petrova@st-andrews.ac.uk 1 1 Introduction The past 50 years have seen three major economic events that have been shared across many industrialized economies: the Great In‡ation of the late 1970s and early 1980s, the Great Moderation starting in the mid-1980s, and the 2008 …nancial crisis and the subsequent recession. In this paper we use macroeconomic time series for three major world economies (the Euro Area, the US, and the UK) that feature these key events as a backdrop to ask what the conduct and e¤ects of monetary policy have been, how they have changed over time, and whether they have been similar across these economies. A question of particular importance for monetary policy concerns policy spillovers: do macroeconomic variables in major economies react to decisions of central banks in other major economies? Our choice of countries is motivated by the fact that the selected economies account for more than a third of total world GDP. Even though they all share the historic episodes mentioned before, the magnitude of these events has been dramatically di¤erent, as illustrated in Figure 1: the Great In‡ation was most severe in the UK, while the Bundesbank, which later joined the European Central Bank (ECB), is often credited with avoiding the large spikes in in‡ation we have seen in the other two countries. The Great Moderation period, characterized by low and stable in‡ation and unemployment, arrived later in the UK than in the US. Finally, the recession after the recent …nancial crisis and the associated recovery followed a very di¤erent path in the Euro Area (EA). In this paper, we estimate a time-varying parameter model with drifting volatility using a semiparametric approach to investigate monetary policy experiences across countries and during the three major events described above. We jointly identify three economy-speci…c monetary policy shocks and investigate the spillover e¤ects of policy across countries and time. To this end, we employ a combination of sign and magnitude restrictions, leaving most variables’responses unrestricted, and letting the data speak on their direction and size. The motivation behind our modeling choice is that vector autoregressive (VAR) models are well-suited for answering our research questions, as they allow for modeling interconnectedness and joint dynamics of macroeconomic time series. However, ignoring the structural changes and breaks in the past few decades that are both evident in the data and documented in the literature, can result in invalid inference; hence, allowing for time-variation in the parameters and volatility of the VAR model is essential. Because of the multi-country connections we want to explore, we also require a relatively large number of observables in the VAR model alongside parameter drift. 2 Figure 1: Unemployment, In‡ation and Interest Rates across countries Time-varying parameter (TVP) VAR models have been made popular by Cogley and Sargent (2005) and Primiceri (2005). One concern of the model speci…cation popularized in these papers is that it is generally not amenable to having more than a few variables in the VAR (the upper bound in the literature seems to be around …ve, as used, for example, in Amir-Ahmadi, Matthes and Wang (2016)) and few lags (the standard choice seems to be two lags for quarterly data, as in Cogley and Sargent (2005) and Primiceri (2005)). The reason for this concern is that state space methods used to …lter the drifts in the parameters are subject to the ‘curse of dimensionality’which is particularly severe when considering richly parameterized VAR models. This prevents researchers from using larger datasets that have been employed and found important in …xed-coe¢ cient VARs (see, for example, Christiano, Eichenbaum and Evans (1999)). To handle a larger number of drifting parameters, we use the quasi-Bayesian local likelihood approach to estimate time-varying parameter VARs introduced in Petrova (2018). The combination of the closed-form quasi-posterior expressions derived in Petrova (2018) with standard Minnesotatype priors used in the literature on …xed-coe¢ cient VARs speci…ed directly on the drifting parameters, allows the number of variables in the VAR to be large while also facilitating the parameter drift. Moreover, the quasi-Bayesian local likelihood approach of Petrova (2018) models the parameter time-variation nonparametrically, ensuring consistent estimation in a wide class of parameter processes and alleviating the risk of invalid inference due to misspecifying the state equations for the latent parameters in random coe¢ cient models. Finally, the availability of analytic expressions for the quasi-posterior density in the Gaussian VAR case eliminates the computational burden of 3 Markov chain Monte Carlo (MCMC) algorithms used for the estimation of TVP VARs by state space methods, making the proposed quasi-Bayesian procedure simple and computationally fast. Our main empirical results are organized in two parts. First, we compute various reduced-form quantities that help us understand the similarities and di¤erences between the three economies we study: we use our VAR to estimate standard quantities such as trends, gaps, volatilities, correlations, persistence measures, and a reduced-form view of the Phillips curve trade-o¤ in each economy. Furthermore, we build on recent advances in analyzing network structures using VAR estimates, such as Diebold and Yilmaz (2014), and show how the connectedness between variables has changed over time. Second, we identify the e¤ects of monetary policy shocks originating in each of the three economies using a combination of sign restrictions (as popularized by Faust (1998), Canova and Nicolo (2002), and Uhlig (2005)) and magnitude restrictions (De Graeve and Karas (2010)). These magnitude restrictions can substantially help with identi…cation, as recently emphasized by AmirAhmadi and Drautzburg (2017). Identifying a monetary policy shock serves two purposes in this paper: (i) we can compare the size of monetary policy surprise shocks and their domestic e¤ects over time and across countries, and (ii) we ask how an economy’s key variables react to unexpected policy changes by other central banks. The latter results can also be used to study the question coordination between central banks, a topic which we devote some attention to. There is a vast literature on monetary policy coordination and the associated welfare e¤ects, studied through a game theoretic framework (for example, Canzoneri and Gray (1985), Clarida, Gali and Gertler (2002), Corsetti and Pesenti (2005), Gali and Monacelli (2005), Benigno and Benigno (2006), and Coenen, Lombardo, Smets and Straub (2007)). This literature suggests that policy cooperation generally has some welfare gains as opposed to non-coordination or implementing an exchange rate peg, but it can sometimes generate incentives for central banks to ‘cheat’and deviate from commitments. Moreover, the results di¤er depending on the relative size of the economies considered (i.e., if the game is symmetric) as well as on whether players move simultaneously and whether the game is played repeatedly. There is also a long-standing literature on the international e¤ects of monetary policy in the absence of policy coordination. Most papers in the literature extend the classic static Mundell-Fleming-Dornbusch model to include dynamics, intertemporal preferences and various frictions. The conclusions in the literature di¤er considerably with some papers predicting negative ‘beggar-thy-neighbour’e¤ects on foreign variables by domestic monetary policy (Mundell (1963)), some …nding o¤-setting e¤ects (Obstfeld and Rogo¤ (1995)), while others 4 predicting positive ‘complementary’spillover e¤ects (Corsetti and Pesenti (2001)). The results in the literature are largely determined by the relative strength of the di¤erent transmission channels of monetary policy (exchange rate, terms of trade, or current account channels) imposed through the theoretical assumptions as well as the parameter calibrations of the models in these papers. Because there is a lack of clear consensus in the literature on the direction of policy spillover e¤ects and coordination of monetary policy, our approach is useful as it imposes fewer theoretical restrictions and lets the data speak, while also allowing for the possibility that these e¤ects might be changing over time. The empirical macroeconomic literature that has touched upon some of our research questions includes Lubik and Schorfheide (2006), Gerko and Rey (2017), Stephane, Pesaran, Smith and Smith (2013), to name a few. For example, Gerko and Rey (2017) compare the e¤ects of monetary policy shocks across the UK and US and …nd that the spillover e¤ects are stronger from the US to the UK than vice versa. Gerko and Rey (2017) are silent on how these e¤ects have varied over time, a focus of our paper. Papers that study how much co-movement there is across major economies include Canova, Ciccarelli and Ortega (2007) and Billio, Casarin, Ravazzolo and Van Dijk (2016). The latter studies a sample that includes the …nancial crisis and …nds, similar to our results, stronger co-movement in that period compared to earlier periods (focusing on the US and the EA only). Concerning monetary policy, some papers have analyzed di¤erences in the 1970s: DiCecio and Nelson (2009) emphasize similarities between the conduct of US and UK monetary policy in the 1970s, whereas Beyer, Gaspar, Gerberding and Issing (2008) emphasize large di¤erences in estimated policy rules between the US, UK, and Germany during that period. Most of the empirical literature either estimates a …xed-parameter VAR or DSGE model, typically considering a smaller subset of the sample to avoid estimation bias stemming from the structural change in the series, or estimates a time-varying model either through considering a small number of variables, possibly at the cost of omitted variable bias, or by imposing some additional (factor/ Markovswitching) structure on the parameter time-variation. The advantage of our approach is that we model jointly the three economies using a larger variable set and longer sample, while also allowing for time-variation in the dynamics of all variables. Our main reduced-form result is that once we allow for drifts in the model’s parameters, we …nd signi…cant time-variation in the cross-country interconnectedness, based on weighted directed networks constructed as in Diebold and Yilmaz (2014). Particularly, connectedness of economic and …nancial variables within and between countries is smaller during the Great Moderation and 5 increases considerably during the recent …nancial crisis, making spillover from …nancial to real variables as well as cross-country contagion much more severe. Our monetary policy shock analysis suggests several conclusions. First, monetary policy shocks are larger in magnitude and more persistent in the Great Moderation than in any subsequent periods in all economies. Second, we …nd positive spillover e¤ects of policy across countries in the 1980s (particularly from the EA to US and UK, as well as from US to UK and from UK to US) as well as evidence for policy coordination during that period, and smaller and sometimes negative ‘beggar-thy-neighbour’ spillover e¤ects and no coordination in the subsequent periods. Third, while we impose that the e¤ects of foreign monetary policy shocks are smaller on impact than domestic policy shocks, foreign spillovers can occasionally have more persistent e¤ects. The remainder of the paper is organized as follows. Section 2 outlines the Bayesian semiparametric methodology utilized in the paper and provides a brief comparison with alternative methods. Section 3.1 contains a detailed description of the model speci…cation, data, and priors. Sections 3.2 and 3.3 present the reduced-form empirical results and Section 3.4 contains the structural shock analysis of the paper. Finally, Section 4 concludes, and the supplementary Appendix contains additional results. 2 Methodology In this section, we outline the quasi-Bayesian local likelihood (QBLL) methodology developed for reduced-form VAR models in Petrova (2018). Before going into the technical details, we want to emphasize four major advantages of this approach. The …rst advantage is a remedy for the ‘curse of dimensionality’ problem. The standard approach to estimating time-varying parameter VARs with stochastic volatility involves casting these models in state space form (Cogley and Sargent (2002, 2005), Primiceri (2005)) and exploiting the MCMC algorithms for approximation of the posterior of the parameters (states). The most serious limitation to the practical use of this state space appoach to TVP VAR models is its inability to accommodate larger systems. The size and complexity of the state space increases with the VAR dimension, since an extra state equation is required for each parameter as well as an additional shock and additional coe¢ cients1 . As a result, state space methods are subject to the 1 In a state space setting, an M -dimensional TVP VAR (k) with stochastic volatility requires the addition of M (3=2+M (k +1=2)) state equations, so for instance, a simple …ve variable TVP VAR(4) requires 120 state equations. 6 ‘curse of dimensionality’ and their application to the estimation of TVP VAR models is limited to a model of four to …ve variables. This makes the use of the standard approach infeasible for our application. Additional estimation complexity of state space models arises from the use of MCMC algorithms. On the other hand, the QBLL methodology employed in this paper admits a closed-form quasi-posterior density, facilitating estimation of large VAR systems. For example, Petrova (2018) estimates an 80-variable VAR model with time-variation in the parameters and the covariance matrix in a little over a minute of computation time. Such a model in a state space setup would require 9,720 state equations just to allow for a single lag, which is clearly infeasible. An alternative to our approach would be to assume more structure on the VAR coe¢ cients or the volatilities; for example, a factor structure, as outlined by Canova and Ciccarelli (2009), Canova and Sala (2009), Amisano, Giannone and Lenza (2015), or a panel structure as in Koop and Korobilis (2018) or Canova et al. (2007). Our methodology does not require imposing such constraints a priori, which can be restrictive and even invalid if the model does not obey the assumed structure. Additionally, choosing prior distributions in models with a factor structure in the coe¢ cients can be burdensome. The second advantage of our methodology is that the standard state space approach is fully parametric and thus requires a parametric law of motion for the drifting parameters, with a random walk process being the most common assumption in the literature (see for example, Cogley and Sargent (2002, 2005), Primiceri (2005), Mumtaz and Surico (2009), Cogley, Primiceri and Sargent (2010) and Clark (2012)). While convenient, this assumption is restrictive and can provide invalid inference even asymptotically if the true law of motion is misspeci…ed. On the other hand, our methodology is nonparametric with respect to the parameter time-variation and as a result valid in a wide class of deterministic and stochastic processes (see Petrova (2018) for further discussion and Monte Carlo evidence using various data-generating processes). Third, to maintain symmetry and positive de…niteness of the drifting reduced-form covariance matrix, state space methods resort to diagonalization, e.g. Cogley and Sargent (2005), Primiceri (2005), Cogley et al. (2010) use Cholesky decomposition assuming that the diagonal elements follow a random walk in logarithms. This implies that the ordering of the variables in the VAR matters for inference2 , which can be undesirable particularly for reduced-form analysis. The QBLL approach permits direct estimation of the time-varying covariance matrix, which has a time-varying invertedWishart posterior density, remaining by construction symmetric and positive de…nite at each point 2 For a recent alternative parametric state space setup that does not share this problem, see Bognanni (2018). 7 in time. Finally, as will become evident below, our approach permits the use of exactly the same priors that have been well-designed and used by researchers for many years for …xed-coe¢ cient VARs. This makes prior elicitation substantially more straightforward than in the state space setup; for example, we do not have to rely on a training sample to obtain priors or starting values of the Kalman …lter, which is standard when using fully parametric TVP VAR models. Note that this does not mean that we are restricted to standard priors coming from …xed-coe¢ cient VARs. Our setup can accommodate ‡exible non-conjugate priors that can even facilitate time-varying prior beliefs. We now turn to a more formal description of our model and the estimation algorithm. Let an M 1 dimensional vector yt be generated by a stable time-varying parameter (TVP) heteroskedastic VAR model of lag order k: yt = B0t + Xk p=1 Bpt yt p 1=2 + " t ; "t = t; t t N ID(0; IM ) (1) where B0t is a vector of time-varying intercepts, Bpt are time-varying autoregressive matrices Pk p with all roots of the polynomial (z) = det IM outside the unit circle, and p=1 z Bpt 1 t a positive de…nite time-varying covariance matrix. Letting xt = (1; yt0 0 1 ; :::; yt k ) and Bt = (B0t ; B1t ; :::; Bkt ), the model (1) can be written as yt = (IM where t := vec(Bt0 ) is an M (M k + 1) xt ) t 1=2 + t t; (2) 1 vector for each t = 1; :::; T: Further, de…ne the matrices Y = (y1 ; :::; yT )0 , E = ("1 ; :::; "T )0 ; X = (x01 ; :::; x0T )0 ; and denote their vectorized forms by y = vec(Y ) and " = vec(E): In order to estimate the time-varying parameters t and t; we employ a quasi-Bayesian method- ology proposed by Petrova (2018), which builds on previous frequentist work by Giraitis, Kapetanios and Yates (2014). This class of semi-parametric estimators can handle both deterministic and stochastic time-variation and can provide valid inference for a wide class of models (see Giraitis et al. (2014) and Petrova (2018) for more details). For completeness, we include the conditions from these papers, su¢ cient for consistency and asymptotic normality of the time-varying parameter vector h i0 1 0 := ; vech below: t t t (i) t is a deterministic process t = f (t=T ) ; 8 (3) where f (:) is a piecewise di¤erentiable function or (ii) t is a stochastic process satisfying: sup j:jj tj h jj 2 j jj t = Op (h=t) for 1 h t as t ! 1: (4) Petrova (2018) proves that in the Bayesian setup the resulting quasi-posterior distributions are asymptotically valid for inference and con…dence interval construction in a general nonlinear likelihood setup. The intuition for this result is that the parameter vector t is assumed to vary slowly enough through (3) and (4) to permit consistent estimation. Petrova (2018) also veri…es that the required high-level assumptions are satis…ed for the special case of a time-varying (but otherwise linear) Gaussian model, for which a closed-form time-varying Normal-Wishart expression for the quasi-posterior density is provided. In particular, the method requires introducing a reweighting of the likelihoods of the observations (y1 ; :::; yT ) for the VAR(k) model (1). This weighting function gives greater weight to observations in the vicinity of the time period whose parameter values are of interest. The resulting local likelihood function at each point in time j is given by Lj (yj j; j ; X) = (2 ) M {T j =2 j 1 2 {T j =2 e jj PT t=1 #jt (yt (IM xt ) j) 0 j (yt (IM xt ) j) (5) where the weights #jt are computed using a kernel function and normalized in the following way #jt = {T j wjt = where {T j := PT t=1 2= wjt PT t=1 wjt ; PT t=1 wjt 2 wjt = K j t H for j; t 2 f1; :::; T g ; 1 . The kernel function K is assumed to be a non- negative, continuous, and bounded function with a bandwidth parameter H satisfying H ! 1 and H = o(T = log T ). The rate of convergence is given by {T j ; which behaves like H; implying a nonparametric rate; this is unsurprising, since we have an in…nite sequence of parameter vectors to estimate. For example, the widely used Normal kernel weights are given by p wjt = (1= 2 ) exp(( 1=2)((j t)=H)2 ) for j; t 2 f1; :::; T g ; while the rolling-window procedure results as a special case of the choice of a ‡at kernel weights: wjt = I( jt jj H) for j; t 2 f1; :::; T g : The weighted likelihood (5) can be written more compactly as Lj (yj j; j ; X) /j tr(Dj )=2 exp jj 1 (y 2 (IM 9 X) 0 j) ( j Dj )(y (IM X) j) (6) where Dj := diag(#j1 ; :::; #jT ) for j 2 f1:::; T g: The intuition behind what the kernel e¤ectively achieves when we estimate the parameters at time j is to give more weight to observations close to the speci…c point in time j and down-weigh distant observations. Next, we assume a Normal-Wishart prior distribution for jj j where 0j N 0j ; ( j is a vector of prior means, of the Wishart distribution, and 0j 0j 0j ) 1 ; j and W( j 0j ; is a positive de…nite matrix, for j 2 f1; :::; T g: j 0j ) 0j (7) is a scalar scale parameter is a positive de…nite matrix. Then, by Proposition 2 of Petrova (2018), combining this prior with the weighted likelihood Lj in (6) delivers a Normal-Wishart quasiposterior distribution for and j j j j j ; X; Y with posterior parameters: e = IM j ej = 0j ej 1 h for j = f1; :::; T g: N ej ; ( j ej ) X 0 Dj X) ^ j + (IM 0j ) XT e j = 0j + #jt ; ej = (IM + X 0 Dj X; where 1 ; 0j t=1 ^ = (IM j X 0 Dj X) 1 (IM j i 0j W(e j ; ej ); (8) ; (9) + Y 0 Dj Y + B0j X 0 Dj )y is the frequentist local likelihood estimator of Giraitis et al. (2014) for 0 0j B0j ej ej B e0 ; B j (10) j. Note that to generate a draw for the parameters at any point in time, we just need to draw from a conjugate Normal-Wishart posterior, which is very fast even for large systems. Thus, the main advantages of our approach over the standard fully parametric state space-based methods are its simplicity, computational e¢ ciency and robustness to misspeci…cation. 3 Empirical Application 3.1 Data and Priors We employ quarterly data starting in 1971Q1 until 2013Q4 on the unemployment rate, the shortterm nominal interest rate3 , the long-term (10-year) nominal interest rate on government bonds, year-on-year in‡ation (CPI-based for the US and EA, RPI-based for the UK), the annual growth 3 We consider the short-term interest rate as the main policy instrument. 10 rate of an exchange rate index for each country trade-weighted against a basket of currencies, and the annual growth rate of a stock price index4 for each country (S&P 500 for the US, DAX to proxy for the EA, and an all-share index for the UK from the Global Financial Database). For the pre-euro period of our sample, we follow the literature and use synthetic EA data constructed by Fagan, Henry and Mestre (2001) as a composite from individual countries’ data series. The price indices for the in‡ation calculations and the unemployment rates are seasonally adjusted. Finally, to account for movements in commodity prices, we also add a series on global commodity price in‡ation computed as the annual growth rate of the Moody’s commodity price index. For the estimation of our model, we use four lags and a Minnesota-style prior with overall shrinkage = 0:05. Since our VAR does not include variables with a clear stochastic trend, we follow standard practice (e.g., Ba´nbura, Giannone and Reichlin (2010) and Kilian and Luetkepohl (2017)) and center the coe¢ cient on the …rst lag of each variable at zero. We also impose that at each point in time, the companion form of our VAR has only eigenvalues less than one in absolute value. The prior for the Wishart parameters is set following Kadiyala and Karlsson (1997). We have performed a number of robustness checks and the main results in this section do not change. Some of these additional results can be found in the supplementary Appendix; the rest are available upon request. First, our results are robust to di¤erent values of the overall shrinkage parameter = 0.01, 0.3, 0.5, as well as lag orders of two, six, and eight quarters. In addition, our main results do not change when we replace EA data with German data (these additional results can be found in the supplementary Appendix). While the German data have the advantage that EA data aggregate various monetary policymakers into one arti…cial policymaker before the introduction of the euro, the EA data have the advantage over German data since the Bundesbank is not the sole decision-maker on German monetary policy after the introduction of the euro in 1999. These di¤erences are not important in practice since the EA and German series are highly correlated for most of our sample, and our robustness check in the Appendix con…rms this. Particularly, even before the advent of the euro, the Bundesbank had been a de facto leader in European policymaking decision (as emphasized by di Giovanni, McCrary and von Wachter (2009)). Conversely, the ECB has reacted, at least during its …rst years, strongly to German data (most likely to obtain a similar reputation to the Bundesbank), as highlighted in Alesina, Blanchard, Gali, Giavazzi and Uhlig (2001). 4 We will henceforth refer to this growth rate of stock indices as stock returns, but it useful to remember that we do not explicitly take into account dividends. 11 3.2 Reduced-Form Evidence We now present reduced-form evidence from our TVP VAR model. These following three subsections serve three purposes; we want to assess: (i) the di¤erences in the economic environments in the US, UK, and EA, (ii) whether these economies have become more similar (and more interconnected in a sense we will make precise below), and (iii) how similar the conduct of monetary policy is across these economies. Perhaps surprisingly, even without taking a stand on the identi…cation of a monetary policy, we can already glean some insights on both long-run di¤erences across countries and on the monetary policy stance (i.e. how tight monetary policy is in each country). Before we proceed to actual results, it is worth pointing out how our estimated parameters vary over time. In the supplementary Appendix, we show that the estimated parameter paths behave similarly to random walks with drifts, which is in line with the parametric assumption made in the bulk of the previous literature on TVP VAR models. Thus, our …ndings are not driven by an estimated parameter variation that is at odds with existing literature. To introduce the objects of our analysis, it will be useful for us to work with the companion form of the VAR model in equation (1): zt = 2 yt 6 6 6 yt 1 zt := 6 6 .. 6 . 4 yt p+1 3 7 7 7 7; 7 7 5 t + At zt 1 + t ; N (0; t ); t 2 3 2 B0t B1t B2t 6 7 6 6 7 6 6 0 7 6 IM 0 7 6 := 6 6 .. 7 ; At := 6 .. . .. 6 . 7 6 . 4 5 4 0 0 IM Because of the stability condition on the roots of the polynomial it follows that (At ) < 1; where (11) t Bpt 3 2 3 " 6 t 7 6 7 6 0 7 6 7: t := 6 7 6 0 7 4 5 0 Pk p (z) = det IM p=1 z Bpt , 7 7 0 7 7 .. 7 ; . 7 5 0 ( ) denotes the spectral radius. One additional advantage of our econometric approach over the state space approach of Cogley and Sargent (2005) to TVP VAR models is that given the stability condition above and the assumed time-variation of the parameters in conditions (3) and (4); Giraitis, Kapetanios and Yates (2018) show that the TVP VAR model in (11) can be approximated by an vector MA(1) process of the form zt = (IM At ) 1 t + 1 X Aht t j + op (1) : (12) h=0 For the reduced-form results in this section, we make use of this approximation5 to compute the 5 In the absence of such an approximation for state space models, the way the computation in (13) has been justi…ed in the literature is via an ‘anticipated utility approximation’, i.e. assuming that parameters will not change in the future. 12 Figure 2: Actual series and their long-run trends implied trends of the model’s variables: t The elements of t = (I At ) 1 t: (13) can be interpreted as long-run economic expectations or in…nite horizon forecasts implied by the model t = limh!1 E (zt+h jFt ) and have been used by Cogley and Sargent (2005) to study changes in ‘core’ or ‘natural’ rates. The approach is closely related to the notion of an in…nite horizon forecast as a trend embedded in the Beveridge-Nelson approach. We estimate the quasi-posterior of these trends using the quasi-posterior of the model’s parameters in (8) and, in Figure 2, we study how they have evolved over time. Figure 2 compares the actual series with the …tted long-run means, i.e. the posterior median of t, (in bold lines), computed using equation (13), and Figure 3 presents their implied 95% posterior posterior intervals. In terms of trends, it turns out that both for short-run and long-run nominal interest rates, the trends are very similar across economies at the end of the sample. If we interpret the short-term interest rate as the policy instrument of the central bank, this means that any major recent di¤erences across monetary policies in the three economies we study must be at higher frequencies. The trends in the short-term interest rates were also similar in the mid-1970s, with big di¤erences only emerging in the 1980s. Unsurprisingly, stock return trends are also very similar across economies. The most 13 Figure 3: Long-run trends striking di¤erence in terms of economic outcomes across countries is not in core in‡ation rates, but rather in natural unemployment rates, where the US and UK trends have converged around the year 2000, but the EA trend unemployment has remained considerably higher. Using this measure of long-run trends, we can derive a measure of the relationship between unemployment and in‡ation assessing whether there is a constant (Phillips curve) relationship in the data and whether this relationship is di¤erent across economies. We think of this as a useful …rst step to understand whether there are substantial di¤erences in the economic environment across the three economies (without taking a stand on the causes of potential di¤erences). Note that we also do not take a stand on whether this is an exploitable relationship. Rather, we only present a reduced-form measure, studying how deviations of in‡ation from its trend are related to deviations of unemployment from its trend6 . Figure 4 displays the estimated Phillips curve relationship in each economy. We …nd downward-sloping Phillips curves, but there is substantial heterogeneity in the estimated slopes. The absolute value of the slope is smallest in the US. Note that, given how we set up our Phillips curve regression, a smaller slope (in absolute value) means that any movement in in‡ation requires a larger movement in unemployment to stay on the curve. Interestingly, Phillips curve narratives are often featured in US monetary policy discussions. 6 We use the posterior mean of t as our measure of trend. 14 Figure 4: Phillips Curves across countries Using our estimated long-run trends (or natural rates), we can ask how deviations (or ‘gaps’) from these trends evolve over time. In particular, we look at a measure of the real interest rate gap. In many models of monetary policy (and in particular the new Keynesian model), the real e¤ects of monetary policy (due to sticky prices and sticky wages) appear because the central bank can in‡uence the real rate of interest. In these models, the long-run or steady-state level of the real rate is outside of the central bank’s control (and usually determined by technological progress). Our model naturally allows the implied long-run level of the real rate to vary over time. A natural measure of the monetary policy stance in our model is thus the deviation of the real rate (computed as the short-term nominal rate minus the rate of in‡ation) from its estimated long-run trend, which we plot in Figure 5. An advantage of this measure of the monetary policy stance is that we can compute it without having to identify a monetary policy rule or monetary policy shocks. To this end, the short-term interest rates need to be assumed as policy instruments of the central banks. We are not saying that this measure captures all aspects of monetary policy: during the Great Recession, for example, all three central banks in the economies in our sample used additional policy instruments such as quantitative easing and forward guidance. Nonetheless, by adopting this limited view of the monetary policy stance, we …nd policy di¤erences across economies. A …rst glance at Figure 5 reveals that the movements in our measure of the monetary policy stance are highly correlated across economies. The absolute levels (and the sign of the gap) can be substantially di¤erent across economies, though. The UK had a larger real rate gap in the 1970s, mainly driven by its higher in‡ation. Nonetheless, the timing of going to a negative gap and then back to a positive gap in the 1970s is broadly shared by all three of our economies (with the UK real rate gap lagging behind US and EA gaps). In the recent …nancial crisis, an interesting 15 Figure 5: Interest rate gaps di¤erence emerges: our estimates imply a tighter monetary policy in the EA for much longer during the Great Recession, which partly re‡ects the prolonged policy reactions to the European sovereign debt crisis. Next, we compute the measure for in‡ation persistence h steps ahead introduced by Cogley et al. (2010) and de…ned as 2 Rt;h =1 e e hP h 1 j j=0 At hP j 1 j=0 At i j0 0 A t t e i ; j0 0 t At e (14) 2 represents the proportion where the vector e selects in‡ation from the vector zt . The measure Rt;h of total variation explained by past shocks (or equivalently, one minus the proportion of total variation due to future shocks). The measure takes values between zero and one, with values close to unity implying that past shocks die out slowly, making in‡ation more persistent and hence predictable. 2 over time and horizon, computed using equation (14) Figure 6 presents the posterior mean of Rt;h for di¤erent draws of the parameters. One reason for a fall in in‡ation predictability is improvement in the ability of the central bank to handle in‡ation expectations through policy. Our results in Figure 6 for the US are similar to those presented in Cogley et al. (2010) and suggest that policy in the US started improving in the late 1970s and continued throughout the 1980s, making in‡ation harder to predict and analyze during this period. On the other hand, such policy advancements 16 Figure 6: In‡ation persistence across countries were already present in the EA earlier in the 1980s, and in‡ation persistence actually rose after the introduction of the euro in 1999. Finally, the UK in‡ation persistence is much higher throughout the sample relative to the US and EA, and the fall in UK persistence that we document occurs from 1990-2000, which is the period in which the Bank of England became an in‡ation-targeting central bank independent from the UK government. We now turn to the volatility of the series. We compute the unconditional variance of the series at each point in time using the companion form in (11) and the VMA(1) approximation in (12) as Ut = X1 j=0 Ajt t Ajt 0 : Figure 7 presents the unconditional volatility of the series in our model (given by the square roots of diagonal elements of the matrix Ut ) over time and across countries respectively. From the …gure, the fall in volatility during the Great Moderation period can be seen in most series and is consistent with previous …ndings in the literature (e.g., Sims (1980), Bernanke and Mihov (1998), Kim and Nelson (1999), McConnell and Perez Quiros (2000), Sims and Zha (2006) and Primiceri (2005)). Interestingly, we …nd that the Great Moderation period starts later in the UK. If well-designed policy was the cause of the fall in the macroeconomic volatility in the US, then one explanation for 17 Figure 7: Unconditional Volatility our result on volatility of British variables is that the UK underwent monetary policy improvements later. Particularly, the Bank of England adopted active in‡ation-targeting in 1992 and only became independent from the UK government in 1997. Unsurprisingly, we also discover an increase in the volatility of unemployment and stock market returns during the 2008 …nancial crisis. Similar trends are visible if we studied the conditional variances computed as var (zt jFt 1) = t: For the sake of brevity, we include these additional results in the supplementary Appendix. One novelty of the reduced-form results presented in this section is the use of a joint multicountry estimation of the time-varying parameters of the VAR as well the application of the novel quasi-Bayesian approach, which delivers estimates for the drifting volatilities without the need to take a stand on the ordering of the variables in the system. This joint estimation approach also allows us to compute the evolution of the unconditional correlations of the series across pairs of countries, which we present in Figure 8. The main result in Figure 8 is that the correlation between US and UK variables is predominantly positive and stronger than any other pair (USEU and UK-EU). One important …nding is the strong negative correlation between USD and EUR e¤ective exchange rates, weaker negative correlation between GBP and EUR rates, and the surprising periods of positive correlation between USD and GBP rates. These results are important for our structural shock analysis in Section 3.4, since the exchange rate channel is crucial for 18 Figure 8: Pair-wise Correlations international monetary policy transmission. We also see most pair-wise correlations increasing during the 2008 …nancial crisis, implying an increased interdependence during this period. For example, the estimated correlations between the stock market returns across countries reach values close to unity after 2008. More evidence on the system-wide connectedness of our model as well as the implied networks across variables and countries can be found in Section 3.3 below. 3.3 Time-varying connectedness In this section, we examine the interconnections between the variables/countries in our system and how they evolved over time using the framework of Diebold and Yilmaz (2014) and Demirer, Diebold, Liu and Yilmaz (2018). Our measure of connectedness is based on the variance decomposition of the VAR model and characterizes a weighted directed network. We adopt the generalized variance decomposition7 of Koop, Pesaran and Potter (1996) and Pesaran and Shin (1998) and make use of the VMA(1) approximation of our TVP VAR model in (12). The variance decomposition 7 If we had identi…ed as many structural shocks as observables in our structural VAR analysis in Section 3.4, we could have constructed the variance decomposition according to our identi…ed SVAR. Instead, in the next section we only identify three monetary policy shocks. The advantage of the results in this section is that they provide ‘reduced-form’evidence that does not rely on our speci…c identifying assumptions. 19 is given by 1 PH 1 h=0 t;ii Gij;t (H) = P H 1 h=0 where ei is an M k e0i Aht e0i Aht t t ej 2 0 Aht ei 1 selection vector with the i-th element unity and all the other elements zero. Roughly speaking, the (i; j)-th element of the variance decomposition matrix Gij;t (H) measures the fraction of variable i’s future uncertainty due to shocks to variable j at H horizons in the future. Here, we choose the forecasting horizon H to be four quarters, so we consider one-year-ahead uncertainty. Following Diebold and Yilmaz (2014) and Demirer et al. (2018), we further normalize the variance decomposition matrix so that the row sum equals to one, CiH j;t = Gij;t (H) PN . j=1 Gij;t (H) After constructing the connectedness table, we visualize it via network “spring graphs”. The color of each node represents which economy the variable is associated with: EU variables are red, UK variables are purple, US variables are blue, and the global commodity price in‡ation is green. Thickness of the edges of the graphs measures how strongly connected the two variables are, represented by the nodes at the ends of the edge. The strength of the edges is determined by the average pair-wise directional connectedness (i.e., (CiH H j;t + Cj i;t )=2). The size of the arrows at the end of each node, on the other hand, measures the pair-wise directional connectedness ‘to’ and ‘from’ (i.e., either CiH j;t or CjH i;t ). Figure 9 displays the network graphs for selected periods and reveals many interesting patterns across time. In general, we see a closer connection within each variable category. For example, the nodes for US, UK, and EA short-run interest rates are close to each other. This clustering feature reveals the synergy of economic environments and monetary policies across these three economies. We …nd that during the …rst oil shock in 1973, commodity price in‡ation, which includes the oil price, plays a central role in the system and explains a large part of the variation in exchange rates and in‡ation rates across countries, as implied by the thicker green arrows. We also see that the e¤ects on UK variables are larger in magnitude, which is consistent with the UK having experienced a much more severe oil crisis than the EA and the US. Similarly, we …nd an increased cross-variable and cross-country connectedness after the second oil shock of 1979, although commodity in‡ation does not play a central role anymore in 1982Q1, and instead we …nd that short- and long-run interest rates have become core in explaining variation across variables. During the Great Moderation period (1992Q1 and 1998Q1), we document an interesting separation in the system between …nancial and macroeconomic variables, with stock market returns and 20 Figure 9: Connectedness graphs for selected periods 1973Q3 1992Q1 2008Q3 1977Q1 1998Q1 2010Q3 1982Q1 2002Q1 2013Q3 Node name tags are abbreviated as follows. Unemployment: Unemp. In‡ation: Inf. Short-run interest rate: Int. Long-run interest rate: IntLong. Stock return: Stock. E¤ective exchange rate: X. Commodity price in‡ation: Comm. 21 exchange rates of our three economies forming a cluster, implying hardly any spillover e¤ects between …nancial markets and the macroeconomy. During these periods, we …nd that UK short- and long-run interest rates are strongly connected to UK in‡ation with large portions of the variation in in‡ation explained by the rates and vice versa, a possible consequence of the introduction of in‡ation targeting and its formal independence from the government in 1992 and 1997, respectively. In 2002Q1, after the short 2001 NBER recession, we continue to …nd some separation between …nancial and macro variables, although less clear than in the previous periods. During the 2008 crisis and the periods afterward, we uncover an increased global interdependence (implied by the thicker arrows) and importantly, we …nd that stock returns are now central in the estimated network, implying large contagion from …nancial markets to the macroeconomy. Particularly, stock returns are directly explaining large proportions of the variation in unemployment, in‡ation, and interest rates across countries. Our results are very similar in 2010Q3, with commodity price in‡ation now also playing a central role in the system. The graphs presented so far try to uncover relationships between di¤erent variables in our model. We can also use the information in those graphs to get a sense of how important unexpected movements in other variables are on average in our system over time. This gives us an estimate of how important interdependence or the ‘network structure’is in our data and, in particular, how it has evolved over time. To achieve this goal, we measure the interaction across variables net of the self e¤ect by averaging over the connectedness table excluding the diagonal elements, CtH = 1 XN CH i;j=1;i6=j i N j;t : Figure 10 presents the posterior median and the 64% estimated posterior sets of our system-wide connectedness measure, evolving over time, and the shaded grey top, middle, and bottom areas display recession dates in the US, UK, and EA, respectively8 . In general, Figure 10 suggests that periods characterized by economic downturns (such as the Great In‡ation in the early 1970s, the EA and UK recessions around the middle 1990s, as well as the …nancial crisis, the Great Recession, and the European sovereign debt crisis in the late 2000s) are associated with higher overall connectedness. Particularly, over 80% of the variance of our model is explained by past shocks to other variables during the recent crisis, compared to 60% in the Great Moderation. This result suggests that during economic distress, correlations and interdependencies between variables 8 For the US, we use NBER recession dates; for the UK and EA, we use an OECD monthly recession indicators from peak through trough and de…ne a recession in the UK and EA as …ve consecutive months of decline in each, respectively. 22 Figure 10: System-wise connectedness and across countries increase, which is consistent with …ndings in the …nancial GARCH literature (e.g., Engle, Ledoit and Wolf (2017)) as well as macroeconomic and …nancial network literature (e.g., Demirer et al. (2018)). Nevertheless, this …nding cannot be interpreted causally - these stronger network e¤ects could also be a consequence of the e¤ects of the Great Recession. 3.4 Structural analysis In this section, we turn to the very questions we asked at the beginning of this paper, namely how monetary policy and its associated domestic and international e¤ects di¤er across economies and over time. To address the further question of policy coordination, we focus on the behavior of foreign nominal interest rates to unexpected changes in the three policy shocks. This is a simplifying assumption we impose in our analysis, which might be unrealistic in practice, since we would expect that central banks also react to the systematic fraction of monetary policy of other major banks rather than only to unanticipated shocks. However, if this were indeed the case, the results presented in this section would only underestimate the true level of policy cooperation. Our identifying strategy for the economy-speci…c monetary policy shocks employs sign and 23 magnitude restrictions of the impulse responses, along the lines of Uhlig (2005), Canova and Nicolo (2002), and Faust (1998). Speci…cally, we impose the sign restrictions that in response to a monetary policy shock in country i, the short-term nominal rate in that country increases, the in‡ation rate decreases, and the unemployment rate increases. Moreover, to distinguish monetary policy shocks across countries, we also impose that a monetary policy shock in country i must have the largest e¤ect in magnitude on in‡ation, unemployment, and interest rate in country i. These magnitude restrictions can be powerful tools that help shrink the identi…ed set of impulse responses, as carefully detailed in Amir-Ahmadi and Drautzburg (2017). More formally, we use the VMA(1) approximation of the companion model in (12) and let Pt denote the Cholesky factor of t: To impose our identifying restrictions, we …rst draw from a family of orthogonal matrices Q 2 Q of size M , and the resulting impulse responses at every point in time, given by th = Ajt Pt Q0 ; span the space of all possible responses. Next, we employ rejection sampling, only retaining draws of th that satisfy all sign and magnitude restrictions. Namely, we check if a shock can be found for each economy i 2 f1; 2; 3g satisfying the following: an increase in domestic i > 0) reduces on impact (i.e., h = 0) domestic in‡ation ( interest rate (rt0 i t0 0), increases domestic unemployment (uit0 0), and, in addition, the shock has the highest magnitude e¤ect on the three domestic variables: i > rt0 j rt0 ; i t0 > j t0 and uit0 > ujt0 for all j 6= i: To implement the calculation of the impulse responses conditional on reduced-form estimates, we use the algorithm outlined in Rubio-Ramirez, Waggoner and Zha (2010), which allows us to e¢ ciently explore the space of candidate impulse responses. Notice that we leave unrestricted the direction in which a policy shock in a given country might a¤ect the remaining domestic variables (long-run interest rates, exchange rates, and stock returns) as well as all foreign variables. As a consequence, the results on the size and direction of the spillover e¤ects of shocks across countries are informed by the data only and not imposed as a maintained assumption. In the main text, we focus on the point-wise impulse responses (computed as the posterior mean) to a one standard deviation shock for selected time periods. The supplementary Appendix contains both the associated posterior error bands as well as 3D plots that display the entire evolution of the posterior mean responses. For the sake of clarity, throughout this section and the Appendix, we always use dark blue for the US policy shocks, lighter blue for UK shocks, and green for EA shocks. In Figure 11, we show the responses of all domestic variables to a monetary policy shock in that economy. First, the responses of the nominal interest rate in all economies are relatively short-lived and less persistent than, for example, the corresponding unemployment response. The persistence 24 Figure 11: Domestic e¤ects of the interest rate responses is in line with the literature (Christiano et al., 1999). The main di¤erence in the associated unemployment and in‡ation responses across economies seems to be in terms of persistence and magnitude - monetary policy shocks are much longer-lived in the 1980s and 1990s than in the subsequent periods. Moreover, EA monetary policy has smaller e¤ects on impact on EA in‡ation and unemployment, even though the impact on nominal interest rates across countries is of a similar magnitude. In general, we …nd a decreased responsiveness of in‡ation and unemployment over time for all economies, particularly during the Great Moderation. This fall in domestic responsiveness to monetary policy shocks after the 1980s is consistent with …ndings in Boivin and Giannoni (2006), who attribute it to policymakers becoming more e¢ cient in managing agents’expectations. In terms of stock return responses, the US is special in that for all selected time periods an unexpected increase in the policy instrument increases the stock market index, while for the UK and EA, we …nd that in periods characterized by …nancial distress and monetary stimulus (1980 and 2010) the responses are negative, suggesting asymmetry in the responses of …nancial markets to policy shocks. Finally, exchange rates mostly move in the expected direction: monetary tightening 25 causes an appreciation in the e¤ective exchange rate. One exception is the UK, as well as the EA in periods before the introduction of the euro. This is expected, as the UK and the aggregated individual European states prior to 1999 resemble more closely small open economies that have less control over the value of their currencies in international markets. Figure 12: Interest rate spillover e¤ects Next, we turn to cross-country spillover e¤ects9 . Figures 12-14 present the responses to monetary policy shocks of interest rates, in‡ation, and unemployment rates, respectively, in each economy (…rst row US shocks, second row UK shocks, and third row EA shocks) for various time periods (we thus repeat the domestic e¤ects in the diagonal of Figures 12-14 to ease comparison). Figure 12 helps us address the question of monetary policy coordination. Particularly, we …nd evidence of policy coordination only during the 1970s and early 1980s, and zero or negative cooperation afterwards. To put this result into historical context, the 1970s and 1980s were periods when 9 One caveat to the presentation of our results is that we relegate posterior bands to the Appendix. Given that our error bands feature both estimation and identi…cation uncertainty, some of the error bands contain zero. Nonetheless, we …nd it useful to characterize our model’s ‘best guess’(or point estimate) of what the spillovers are. 26 Figure 13: In‡ation spillover e¤ects central banks across the world faced large exogenous oil shocks, which suggests the possibility of policy cooperation as an optimal response, rather than a zero-sum game, in the face of these large exogenous shocks. The international e¤ects of policy shocks on in‡ation and unemployment display the same sign spillover e¤ects for most shocks in the early 1980s. Particularly, we …nd that during this period, these favorable spillover e¤ects are larger, more persistent, and often statistically signi…cant, which could be a consequence of the documented evidence of cooperation that we …nd between central banks in this period. After 1980, we …nd either negative ‘beggar-thy-neighbour’or insigni…cantly di¤erent from zero spillover e¤ects across economies. 4 Conclusion To answer the questions we posed in the beginning of this paper on how monetary policy spills across borders and whether it evolves over time, we adopt a novel quasi-Bayesian semi-parametric 27 Figure 14: Unemployment spillover e¤ects approach to estimate the parameter time-variation in a medium-sized VAR model. We …rst employ our estimated model to compute various reduced-form quantities in order to understand important di¤erences across countries in the long-run moments of the series implied by our model. We combine this novel estimation methodology with recent advances in the literature linking VARs and network analysis, in order to estimate the time-varying network structure of our model. An additional contribution of our paper is the design of an identi…cation strategy that allows us to jointly identify country-speci…c policy shocks through a combination of sign and magnitude restrictions, providing some useful insights on the international transmission of monetary policy. From our empirical results, several conclusions emerge. First, we …nd that the US and the UK share more similarities than the EA, which is natural since the ECB faces a considerably di¤erent environment as the policymaker of a currency union comprised of diverse member states. This is particularly evident in our measure of the monetary policy stance as well as in the domestic responses to monetary policy shocks. Our reduced-form analysis also points to some meaningful changes in the UK monetary policy after the Bank of England underwent structural changes in the 28 1990s. 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