The evolving transmission of uncertainty shocks in the United Kingdom

Mumtaz, Haroon Article The evolving transmission of uncertainty shocks in the United Kingdom Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Mumtaz, Haroon (2016) : The evolving transmission of uncertainty shocks in the United Kingdom, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 4, Iss. 1, pp. 1-18, https://doi.org/10.3390/econometrics4010016 This Version is available at: https://hdl.handle.net/10419/171869 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ econometrics Article The Evolving Transmission of Uncertainty Shocks in the United Kingdom Haroon Mumtaz School of Economics and Finance, Queen Mary College, London E1 4NS, UK; [email protected]; Tel.: +44-20-7882-8839 Academic Editors: Francesco Ravazzolo, Herman K. van Dijk, Nalan Basturk and Roberto Casarin Received: 4 September 2015; Accepted: 28 January 2016; Published: 14 March 2016 Abstract: This paper investigates if the impact of uncertainty shocks on the U.K. economy has changed over time. To this end, we propose an extended time-varying VAR model that simultaneously allows the estimation of a measure of uncertainty and its time-varying impact on key macroeconomic and financial variables. We find that the impact of uncertainty shocks on these variables has declined over time. The timing of the change coincides with the introduction of inflation targeting in the U.K. Keywords: TVP-VAR; stochastic volatility; uncertainty shocks JEL: C15; C32; E32 1. Introduction The recent financial crisis and ensuing recession have led to a renewed interest in the possible relationship between economic uncertainty and macroeconomic variables. A number of papers use VAR-based analyses to estimate the impact of uncertainty shocks for the U.S. and the U.K. (see, for example, [1] for the U.S. and [2] for the U.K.). In general, these studies report that uncertainty shocks have an adverse impact on the economy. For example, [2] find that uncertainty shocks depress GDP and industrial production. However, the estimates reported in these papers are typically based on data that span the last three or four decades and, thus, cover periods potentially characterised by changing dynamics, policy regimes and economic shocks. There has been limited focus on exploring whether the impact of uncertainty shocks has changed over time in the United Kingdom and identifying the factors that can possibly explain any temporal shifts.1 This paper attempts to fill this gap. We propose an extended TVP-VAR model that allows the estimation of a measure of uncertainty that encompasses volatility from the real and financial sectors of the economy and is a proxy for macroeconomic uncertainty. The proposed model incorporates time-varying parameters and simultaneously provides an estimate of the time-varying response of macroeconomic variables to shocks to this uncertainty measure, thus allowing the investigation of temporal shifts in a coherent manner. Our results suggest that the impact of uncertainty shocks on measures of real activity, inflation and interest rates has declined systematically over time, with the change coinciding with the introduction of inflation targeting in 1992. The impact of these shocks on stock returns has also declined, but the degree of the shift is smaller. 1Beetsma and Giuliodori [3] and Mumtaz and Theodoridis [4] investigate this question for the U.S. Econometrics 2016,4, 16; doi:10.3390/econometrics4010016 www.mdpi.com/journal/econometrics Econometrics 2016,4, 16 2 of 18 The analysis in the paper adds to the literature on uncertainty by systematically investigating how the impact of uncertainty has changed over time in the U.K. The empirical model proposed in the paper builds upon existing VAR models by simultaneously allowing the estimation of time-varying volatility and the time-varying impact of this volatility on the endogenous variables. Our results have important implications. Our empirical findings suggest that uncertainty is less of a concern for real activity and inflation than in the past, but continues to have an important impact on the stock market. This suggests that uncertainty shocks mainly affect the U.K. through a financial channel, and policies designed to ameliorate the impact of these shocks need to take this mechanism into account. The paper is organised as follows: Sections 2and 3introduce the empirical model and discuss the estimation method. The results from the empirical model are presented in Section 4.3. Section 5concludes. 2. Empirical Model The core of the empirical model is the following time-varying parameter vector autoregression (TVP VAR): Zt=ct+ P ∑ j=1 βtjZt−j+ J ∑ j=0 γtj ln λt−j+Ω1/2 tet(1) where Ztis a matrix of endogenous variables that we describe below. The law of motion for the VAR coefficients is given by: B=vec([c;β;γ]) (2) Bt=Bt−1+ηt,VAR (ηt)=QB As in [5], the covariance matrix of the residuals is defined as: Ωt=A−1 tHtA−10 t where Atis lower triangular. Each non-zero element of Atevolves as a random walk: at=at−1+gt,VAR(gt) = G(3) where Gis block diagonal, as in [5]. Following [6], the volatility of the shocks etis given by: Ht=λtS(4) S=diag(s1, .., sN) The overall volatility evolves as an AR(1) process: ln λt=α+Fln λt−1+¯ ηt,VAR(¯ ηt) = Qλ(5) and the diagonal elements of Sare scaling factors. The structure defined by Equation (4) suggests that the specification is characterised by the following feature. First, the model does not distinguish between the common and idiosyncratic component in volatility, and λtis a convolution of both components. In other words, Equation (5) implicitly imposes a factor structure on the volatilities where the loadings equal one and the idiosyncratic components are suppressed. With such a structure, λtis approximately the average volatility. While separating the unobserved components in Equation (4) may be interesting in its own right, it is not directly relevant for our application, where the key aim is to estimate a measure of the common volatility of the shocks, which, by definition, is a combination of the two components. Econometrics 2016,4, 16 3 of 18 As we show below, this simple scheme produces volatility estimates that are plausible from a historical perspective. The shock ¯ ηtrepresents the innovation to the volatility of the residuals etand is interpreted as an uncertainty shock. In the empirical analysis below, we examine the time-varying response of the endogenous variables to this shock. The formulation presented in Equations (4) and (5) is related to a number of recent empirical contributions. For example, the structure of the stochastic volatility model used above closely resembles the formulations used in time-varying VAR models (see [5,7]). Our model differs from these studies in that it allows a direct impact of the volatilities on the level of the endogenous variables. The model proposed above can be thought of as a multivariate extension of the stochastic volatility in the mean model proposed in [8] and applied in [9–11]. In addition, our model has similarities with the stochastic volatility models with leverage studied in [12] and the non-linear model proposed in Aruoba et al. [13]. Finally, the model is based on the VAR with stochastic volatility introduced in [14]. While [14] focus on the impact of volatility associated with the output shock, we focus on an overall measure of uncertainty that incorporates the variance of all shocks in the model. In addition, the model proposed above incorporates time variation, a feature missing from the studies that consider stochastic volatility in mean models.2 3. Estimation and Model Specification The model defined in Equations (1) and (5) is estimated using an MCMC algorithm. In this section, we summarise the key steps of the algorithm and provide the details of the prior distributions. 3.1. Priors and Starting Values 3.1.1. VAR Coefficients The initial conditions for the VAR coefficients B0are obtained via an OLS estimate of a fixed coefficient VAR using the first T0=180 observations of the sample period, which corresponds to the first 15 years of the monthly dataset described below. Let ˆ Bols and ˆ vols denote the OLS estimate of the VAR coefficients and the covariance matrix estimated on the pre-sample data described above. The prior for B0˜N(ˆ Bols,var(ˆ Bols)). The prior on QBis assumed to be inverse Wishart QB,0 ∼IW (¯ QB,0,TT0), where ¯ QB,0 is assumed to be T0×var(ˆ Bols)×k,T0is the length of the sample used for calibration and TT0equals 10 plus the columns of QB. Following [7], the scaling factor kis set to 3.5 ×10−4. 3.1.2. Elements of the AMatrix The prior for the off-diagonal elements Atis A0∼Nˆ aols,Vˆ aols, where ˆ aols are the off-diagonal elements of ˆ vols, with each row scaled by the corresponding element on the diagonal. Vˆ aolsis assumed to be diagonal with the elements set equal to 10-times the absolute value of the corresponding element of ˆ aols. The prior distribution for the blocks of Gis inverse Wishart: Gi,0 ∼IW(¯ Gi,Ki), where i=1..N−1 indexes the blocks of S. ¯ Giis calibrated using ˆ aols. Specifically, ¯ Giis a diagonal matrix with the relevant elements of ˆ aols multiplied by 10−3. This prior specification is used in previous studies, such as [15]. 2An exception is [4], who use an extended version of the proposed model to investigate the time-varying impact of uncertainty shocks in the U.S. The model in [4] incorporates a factor structure in the observation Equation (1) and, thus, incorporates more information. Econometrics 2016,4, 16 4 of 18 3.1.3. Elements of Sand the Parameters of the Stochastic Volatility Transition Equation The elements of Sare assumed to have an inverse Gamma prior: P(si)˜IG(S0,i,V0). The degrees of freedom V0are set equal to five. The prior scale parameters are set by estimating the following regression. ¯ λit =S0,i¯ λt+εt, where ¯ λtis the first principal component of the stochastic volatilities ¯ λit obtained using a univariate stochastic volatility model for the residuals of each equation of a VAR estimated via OLS using the endogenous variables Zt. We set a normal prior for the unconditional mean µ=α 1−F. This prior is N(µ0,Z0), where µ0=0 and Z0=10. The prior for Qλis IG Q0,VQ0, where Q0is the average of the variances of the transition equations of the initial univariate stochastic volatility estimates, and VQ0=5. The prior for Fis N(F0,L0), where F0=0.8 and L0=1. 3.1.4. Common Volatility λt The prior for the initial value of λtis defined as ln λ0∼N(ln µ0,I), where µ0is the initial value of ¯ λtdefined above. 3.2. MCMC Algorithm The Gibbs sampling algorithm is based on drawing from the following conditional posterior distributions: 1. G(Bt\Ξ). The distribution of the time-varying VAR coefficients Btconditional on all other parameters Ξis linear and Gaussian: Bt\Zt,Ξ∼NBT\T,PT\Tand Bt\Bt+1,Zt,Ξ∼ NBt\t+1,Bt+1,Pt\t+1,Bt+1, where t=T−1, ..1, Ξdenotes a vector that holds all of the other VAR parameters. As shown by [16] the simulation proceeds as follows. First, we use the Kalman filter to draw BT\Tand PT\Tand then proceed backwards in time using Bt|t+1=Bt|t+Pt|tP−1 t+1|t(Bt+1−Bt)and Bt|t+1=Bt|t−Pt|tP−1 t+1|tPt|t. 2. G(QB\Ξ). The conditional posterior for QBis inverse Wishart: IW (η0 tηt+¯ QB,0,T+T0),i.e., the posterior scale matrix is given by η0 tηt+¯ QB,0, and the degrees of freedom are T+T0. 3. G(At\Ξ). Given a draw for the VAR parameters, the model can be written as A0 t(vt)=et, where vtdenotes the VAR residuals. This is a system of linear equations with time-varying coefficients and a known form of heteroscedasticity. The j-th equation of this system is given as vjt =−ajtv−jt +ejt, where the subscript jdenotes the j-th column of v, while −jdenotes Columns 1 to j−1. Note that the variance of ejt is time-varying and given by λtsj. The time-varying coefficient follows the process ajt =ajt−1+gjt with the shocks to the j-th equation gjt uncorrelated with those from other equations. In other words, the covariance matrix var (g) is assumed to be block diagonal, as in [5]. With this assumption in place, the [16] algorithm can be applied to draw the time varying coefficients for each equation of this system separately. 4. G(S\Ξ). Given a draw for the VAR parameters, the model can be written as A0(vt)=et. The j-th equation of this system is given by vjt =−ajtv−jt +ejt, where the variance of ejt is time-varying and given by λtsj. Given a draw for λt, this equation can be re-written as ¯ vjt =−ajt ¯ v−jt +¯ ejt, where ¯ vjt =vjt λ1/2 t , and the variance of ¯ ejt is sj. The conditional posterior for this variance is inverse Gamma with scale parameter ¯ e0 jt ¯ ejt +S0,jand degrees of freedom V0+T. 5. G(λt\Ξ). Conditional on the VAR parameters, and the parameters of the transition equation, the model has a multivariate non-linear state-space representation. The work in [17] shows that the conditional distribution of the state variables in a general state-space model can be written as the product of three terms: ˜ ht\Zt,Ξ ∝ f˜ ht\˜ ht−1×f˜ ht+1\˜ ht×fZt\˜ ht,Ξ(6) Econometrics 2016,4, 16 5 of 18 where Ξdenotes all other parameters and ˜ ht=ln λt. In the context of stochastic volatility models, [18] show that this density is a product of log normal densities for λtand λt+1and a normal density for Zt. The work in [17] derives the general form of the mean and variance of the underlying normal density for f˜ ht\˜ ht−1,˜ ht+1,Ξ∝f˜ ht\˜ ht−1×f˜ ht+1\˜ htand shows that this is given as: f˜ ht\˜ ht−1,˜ ht+1,Ξ∼N(B2tb2t,B2t)(7) where B−1 2t=Q−1 λ+F0Q−1 λFand b2t=˜ ht−1F0Q−1 λ+˜ ht+1Q−1 λF. Note that due to the non-linearity of the observation equation of the model, an analytical expression for the complete conditional ˜ ht\Zt,Ξis unavailable, and a Metropolis step is required. Following [18], we draw from (6) using a date-by-date independence Metropolis step using the density in (7) as the candidate generating density. This choice implies that the acceptance probability is given by the ratio of the conditional likelihood fZt\˜ ht,Ξat the old and the new draw. To implement the algorithm, we begin with an initial estimate of ˜ h=ln ¯ λt. We set the matrix ˜ hold equal to the initial volatility estimate. Then, at each date, the following two steps are implemented: (a) Draw a candidate for the volatility ˜ hnew tusing the density 6, where b2t=˜ hnew t−1F0Q−1 λ+ ˜ hold t+1Q−1 λFand B−1 2t=Q−1 λ+F0Q−1 λF. (b) Update ˜ hold t=˜ hnew twith acceptance probability f(Zt\˜ hnew t,Ξ) f(Zt\˜ hold t,Ξ), where fZt\˜ ht,Ξis the likelihood of the VAR for observation tand defined as |Ωt|−0.5 −0.5 exp ˜ etΩ−1 t˜ e0 t, where ˜ et=Zt−ct+∑P j=1βtjZt−j+∑J j=0γtj ln λt−jand Ωt=A−1 texp(˜ ht)SA−10 t. Repeating these steps for the entire time series delivers a draw of the stochastic volatilities.3 6. G(α,F\Ξ). We re-write the transition equation in deviations from the mean: ˜ ht−µ=F˜ ht−1−µ+¯ ηt(8) where the elements of the mean vector µiare defined as αi 1−Fi. Conditional on a draw for ˜ ht and µ, the transition Equation (8) is simply a linear regression, and the standard normal and inverse Gamma conditional posteriors apply. Consider ˜ h∗ t=F˜ h∗ t−1+¯ ηt,VAR (¯ ηt)=Qλand ˜ h∗ t=˜ ht−µ,˜ h∗ t−1=˜ ht−1−µ. The conditional posterior of Fis N(θ∗,L∗), where: θ∗=L−1 0+1 Qλ ˜ h∗0 t−1˜ h∗ t−1−1L−1 0F0+1 Qλ ˜ h∗0 t−1˜ h∗ t L∗=L−1 0+1 Qλ ˜ h∗0 t−1˜ h∗ t−1−1 The conditional posterior of Qλis inverse Gamma with scale parameter ¯ η0 t¯ ηt+Q0and degrees of freedom T+VQ0. Given a draw for F, Equation (8) can be expressed as ¯ ∆˜ ht=Cµ+¯ ηt, where ¯ ∆˜ ht=˜ ht−F˜ ht−1 and C=1−F. The conditional posterior of µis N(µ∗,Z∗), where: µ∗=Z−1 0+1 Qλ C0C−1Z−1 0µ0+1 Qλ C0¯ ∆˜ ht Z∗=Z−1 0+1 Qλ C0C−1 3In order to take endpoints into account, the algorithm is modified slightly for the initial condition and the last observation. Details of these changes can be found in [18]. Econometrics 2016,4, 16 6 of 18 Note that αcan be recovered as µ(1−θ). 3.3. Estimation Using Artificial Data To test the algorithm, we conduct a small Monte Carlo experiment. Seven hundred twenty observations are generated from the following data-generating process with the number of variables N=2. The first 100 observations are discarded to remove the impact of initial conditions, and 120 observations of the remaining series are used as a training sample. Estimation is carried out using 500 observations. The length of the artificial sample broadly matches the monthly dataset used in the empirical analysis below. The DGP is defined as: Zt=βtZt−1+γtln λt+ct+Ω1/2 tet,et˜N(0, 1) Ωt=A−1 tHtA−10 t,Ht=λtS S= 1 0 0 2 ! λt=−0.1 +0.75λt−1+(0.5)1 2vt βt= β11,tβ12,t β21,tβ22,t!,γt= γ11,t γ21,t! where λtis generated once using vt˜N(0, 1)and fixed for all iterations of the experiment. Following [19], we assume that a one time shift defines the change in the VAR coefficients and the non-zero element of At. During the first 250 observations, these coefficients equal βt= 0.5 0.1 0.1 0.5 !, γt= −0.5 0.5 !and A=−1. During the next 250 observations, the coefficients change to βt= 0.5 0.1 0.1 0.5 !,γt= −1.5 1.5 !and A=0.1. The data is generate 1000 times. For each replication, the MCMC algorithm described above is run using 5000 iterations, and the last 1000 draws are used to compute the posterior mean of λt,Atand Bt. The figure below plots the median estimate and 84th percentile of λt,Atand Btacross Monte Carlo replications and compares these with the true underlying values. Figure 1shows that the estimated change in λ11 and λ21 closely matches the assumed shift in these coefficients. Note, however, that the model estimates the shift in these coefficients to be smoother than assumed in the DGP. This is not surprising, given the assumed random walk form for the transition of the VAR coefficients in the model, which contrasts with the one-time change in the DGP. The results do show that the model is able to pick up changes in the impact of uncertainty and is suited to the type of investigation undertaken by this paper. Figures 2and 3show that the Monte Carlo estimates of Atand ln λtare close to their true estimated values. Overall, the results provides some evidence that the MCMC algorithm delivers a satisfactory performance. Econometrics 2016,4, 16 7 of 18 Figure 1. Monte Carlo estimates of VAR coefficients. The black line is the true value of the parameter. The red line is the median estimate across 1000 replications, and the shaded area represents the 68% interval. Econometrics 2016,4, 16 8 of 18 Figure 2. Monte Carlo estimates of the non-zero and non-unity element of the Amatrix. The black line is the true value of the parameter. The red line is the median estimate across 1000 replications, and the shaded area represents the 68% interval. Econometrics 2016,4, 16 15 of 18 It is interesting to consider the possible factors that can explain the decline in the response to uncertainty shocks. A detailed DSGE-based analysis of this question is undertaken in [4] for the U.S. The simulations in that paper suggest that the decline in the response to uncertainty shocks may be consistent with an increase in weight placed on inflation in the policy rule employed by the central bank. When this coefficient rises and authorities react strongly to inflation, future inflation is expected to be on target. This reduces firms’ concerns about expected inflation and makes them less forward looking. In other words, the pricing bias decreases, and the link between inflation and marginal cost is renewed. In this case, authorities are able to cut the policy rate by more and for a longer period, which helps them to address the adverse effects from elevated uncertainty, thus ameliorating the decline in output and stock returns. Note that that the empirical results point to a change in the responses after the early 1990s when the Bank of England introduced inflation targeting. As documented in [23], there is strong evidence that the Bank placed a greater weight on inflation control after this date. This provides tentative evidence that the change in the response to uncertainty may be linked to a change in the practice of monetary policy. Of course, the U.K. economy was subject to other changes at the same time, and a more structural analysis is required to distinguish between different factors affecting the transmission mechanism of uncertainty shocks. 5. Conclusions This paper considers whether the impact of uncertainty shocks on the U.K. economy has changed over time. Using an extended TVP VAR model that allows the estimation of the time-varying impact of uncertainty shocks, we find that the responses of industrial production growth, CPI inflation, the short-term interest rate and stock market returns have declined over time. The main change in the response coincides with the adoption of inflation targeting in the U.K. and is consistent with simulations from a DSGE model that assumes an increase in the inflation coefficient in the monetary policy rule. In future work, it may be interesting to investigate more thoroughly the factors that may have led to the change in the response to uncertainty in the U.K. In addition, it may be useful to apply the model to a cross-section of countries that have had different historical experiences with regards to policy and structural changes. This would also allow the estimation of uncertainty indices for a larger range of countries. Acknowledgments: We thank the anonymous referees and the editor for useful comments. Conflicts of Interest: The author declares no conflict of interest. Econometrics 2016,4, 16 16 of 18 Appendix A. Convergence Figure A1. Recursive Means calculated every 100 Gibbs draws. Econometrics 2016,4, 16 17 of 18 B. Sensitivity Analysis Figure A2. Impulse responses from the model using a tighter prior. Econometrics 2016,4, 16 18 of 18 References 1. Bloom, N. The Impact of Uncertainty Shocks. Econometrica 2009,77, 623–685. 2. Denis, S.; Kannan, P. 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