Time-Varying Vector Autoregressive Models with Structural Dynamic Factors Paolo Gorgi, Siem Jan Koopman, Julia Schaumburg http://sjkoopman.net Vrije Universiteit Amsterdam School of Business and Economics CREATES, Aarhus University ECB Workshop on Advances in short-term forecasting 29 September 2017
The VAR model Consider the vector autoregressive (VAR) model of order p, the VAR(p) model: y t = Φ [1] y t 1 +... + Φ [p] y t p + ε t, ε t NID(0, H), t = p + 1,..., T, where Φ [i] is coefficient matrix, i = 1,..., p, and H is variance matrix. For parameter estimation, forecasting, impulse response analysis, etc., we refer to Hamilton (1994) and Lutkepohl (2005), amongst many others. The VAR(p) model can be efficiently formulated as y t = ΦY t 1:p + ε t, Φ = [ Φ [1],..., Φ [p] ], Yt 1:p = ( y t 1,..., y t p We assume that the initial observation set {y 1,..., y p} is fixed and given. ) 2 / 32
Motivation for time-varying parameters in VAR model In macroeconometrics, vector-autoregressive (VAR) models are often extended with time-varying parameters, we have y t = Φ ty t 1:p + ε t, ε t NID(0, H t), t = p + 1,..., T. Earlier literature: Cogley/Sargent (2001 NBER): inflation-unemployment dynamics in changing monetary policy regimes Primiceri (2005 RES): role of monetary policy for macroeconomic performance in the 1970s and 1980s Canova/Ciccarelli (2004 JE, 2009 IER): Bayesian panel VAR models, multi-country analyses; Hubrich/Tetlow (2015 JME): amplification and feedback effects between financial sector shocks and economy in crises vs. normal times Prieto/Eickmeier/Marcellino (2016, JAE): role of financial shocks on macroeconomic variables during crisis 2008/09 3 / 32
Estimation for TV-VAR models The common approach to parameter estimation for VAR models with time-varying coefficients is based on Bayesian methods We develop an alternative methodology based on dynamic factors for VAR coefficient matrices and score-driven dynamics for the variance matrices: flexible modeling setup: it allows for wide variety of empirical specifications; simple, transparent and fast implementation: least squares methods and Kalman filter; relatively easy for estimation, impulse response, analysis and forecasting. Our approach is explored in more generality by Delle Monache et al. (2016): adaptive state space models 4 / 32
Outline Introduction Econometric Model Simulations Empirical application: Macro-financial linkages in the U.S. economy Conclusion 5 / 32
Time-varying autoregressive coefficient matrix TV-VAR model : y t = Φ ty t 1:p + ε t, ε t NID(0, H t), where we assume that sequence of variance matrices H p+1,..., H T is known and fixed, for the moment. The time-varying VAR coefficient is a matrix function, Φ t = Φ(f t) = Φ c + Φ f 1f 1,t + + Φ f r f r,t, where the unobserved r 1 vector f t has dynamic specification f t+1 = ϕf t + η t, η t N(0, Σ η), where ϕ is r r diagonal matrix of coefficients and Σ η = I r ϕϕ. 6 / 32
Linear Gaussian state space form We have y t = Φ ty t 1:p + ε t and Φ t = Φ(f t) = Φ c + Φ f 1 f 1,t + + Φ f r fr,t. Define ỹ t = y t Φ c Y t 1:p and consider the following equation equalities ] ỹ t = [Φ f 1 f t,1 +... + Φ f r ft,r Y t 1:p + ε t [ ] (ft = Φ f 1,, ) Φf r I Np Yt 1:p + ε t ( ]) = Y t 1:p [Φ f 1,, Φf r vec ( ) f t I Np + εt ( ]) = Y t 1:p [Φ f 1,, Φf r Q f t + ε t. We let ( ]) Z t = Y t 1:p [Φ f 1,, Φf r Q, to obtain the linear Gaussian state space form ỹ t = Z tf t + ε t, f t+1 = ϕf t + η t, where the properties of the disturbances ε t and η t are discussed above. 7 / 32
Kalman filter Prediction error is defined as v t = y t E(y t F t 1 ; ψ): F t 1 is set of all past information, including past observations; ψ is the parameter vector that collects all unknown coefficients in Φ c, Φ f 1,..., Φf r, H p+1,..., H T, ϕ; when model is correct, the sequence {v p+1,..., v T } is serially uncorrelated; variance matrix of the prediction error is F t = Var(v t F t 1 ; ψ) = Var(v t; ψ). For a given vector ψ, the Kalman filter is given by v t = ỹ t Z ta t, F t = Z tp tz t + Ht, K t = ϕp tz t F t 1, a t+1 = ϕa t + K tv t, P t+1 = ϕp t(ϕ K tz t) + Σ η, with a t = E(f t F t 1 ; ψ) and variance matrix P t = Var(f t a t F t 1 ; ψ), for t = p + 1,..., T. Loglikelihood function is T l(ψ) = l t(ψ), t=p+1 l t(ψ) = N 2 log 2π 1 2 log Ft 1 2 v t F 1 t v t. 8 / 32
Parameter estimation (1) For model y t = Φ ty t 1:p + ε t, Φ t = Φ(f t) = Φ c + Φ f 1 f 1,t + + Φ f r f r,t, f t+1 = ϕf t + η t, parameter estimation concentrates on ϕ, Φ c and Φ f i, for i = 1,..., r. MLE is maximisation of l(ψ) wrt ψ, heavy task. Our strategy : Step 1: Use economic information (if available) to restrict entries of Φ c and Φ f i. Step 2: Obtain estimates of Φ c via least squares method on static VAR. Step 3: Only place coefficients of ϕ and Φ f i in ψ. Step 4: Estimate this ψ by MLE using the Kalman filter. Least squares estimate of Φ from static VAR is consistent estimate of Φ c. Notice that E(f t) = 0. MLE is for a small dimension of ψ. 9 / 32
Time-varying variance matrix Each Kalman filter step at time t requires a value for H t. We use score-driven approach of Creal et al. (2013) to let variance matrix H t change recursively over time. We have N 1 vector ft σ = vech(h t) with N = N(N + 1)/2 and dynamic specification ft+1 σ = ω + B f t σ + A s t, where ω is constant vector, A and B are square coefficient matrices and s t is innovation vector. Distinguishing feature of score-driven model is definition of s t as the scaled score vector of l t l t(ψ) with respect to f σ t. We have s t = S t t where S t is scaling matrix and t is gradient vector. 10 / 32
Score-driven model for time-varying variance matrix The transpose of the gradient vector is given by t = lt ft σ = l t vec(f t) vec(f t) vech(h t) = l t vec(f t) vec(h t) vech(h t), last equality holds since F t = Z tp tz t + Ht and Zt does not depend on Ht and Pt is function of H p+1,..., H t 1, but not H t. l t vec(f t) = 1 2 [ vec(vt v t ) (vec(f t)) ] ( F 1 t ) Ft 1, vec(h t) vech(h t) = D N, where D N is the N 2 N duplication matrix, see Magnus and Neudecker (2007). It follows that t = 1 2 D N ( F 1 t ) Ft 1 (vec(vt v t ) vec(f ) t). 11 / 32
Score-driven model for time-varying variance matrix The inverse of the information matrix is taken as scaling matrix S t for gradient vector. Information matrix: I t = E[ t t F t 1] = 1 ( ) 4 D N Ft 1 Ft 1 Var [ vec(v t v t ) vec(ft) F ] ( t 1 Ft 1 = 1 ( ) 4 D N Ft 1 Ft 1 (I N 2 + C N )D N = 1 ( ) 2 D N Ft 1 Ft 1 D N, since Var[vec(v t v t ) vec(ft) F t 1] = (I N 2 + C N ) (F t F t) and (I N 2 + C N )D N = 2 D N, where C N is the N 2 N 2 commutation matrix. The inverse of the information matrix: It 1 = 2D n + (Ft Ft)D+ N, ) Ft 1 where D + N = (D N D N) 1 D N is the elimination matrix for symmetric matrices. 12 / 32
Score-driven model for time-varying variance matrix We set the scaling as S t = I 1 t. The scaled score s t = It 1 t becomes s t = D + N (Ft Ft)D+ N D N(Ft 1 Ft 1 )[vec(v t v t ) vec(f t)] = D + [ N vec(vt v t ) vec(f ] t) = vech(v t v t ) vech(ft). For the score-driven update of the variance factors in ft σ, we obtain for t = p + 1,..., T. f σ t+1 = ω + A [ vech(v t v t ) vech(ft)] + B f σ t, The score updating function can easily be incorporated in the Kalman filter: v t = ỹ t Z ta t, F t = Z tp tz t + Ht, K t = ϕp tz t F t 1, a t+1 = ϕa t + K tv t, P t+1 = ϕp t(ϕ K tz t) + Σ η, U t = v t v t Ft, ft+1 σ = ω + Avech(U t) + Bft σ, H t+1 = unvech(ft+1 σ 13 / 32
Direct updating for time-varying variance matrix We have vec(h t) = D N vech(h t) = D N f σ t and we obtain vec(h t+1 ) = D N ω + D N A D + N [vec(vt v t ) vec(f t)] + D N B D + N vec(ht), for t = p + 1,..., T. When we specify D N A D + N = A A and D N B D + N = B B, we have with vech(ω) = ω. H t+1 = Ω + A ( v t v t F t ) A + B H tb, In case A = a I N and B = b I N, the updating reduces simply to H t+1 = Ω + a ( v t v t Ft ) + b Ht. This time-varying variance matrix updating equation can even more conveniently be incorporated within the Kalman filter: v t = ỹ t Z ta t, F t = Z tp tz t + Ht, K t = ϕp tz t F t 1, a t+1 = ϕa t + K tv t, P t+1 = ϕp t(ϕ K tz t) + Σ η, H t+1 = Ω + A (v t v t Ft) A + B H tb. 14 / 32
Parameter estimation (2) For model y t = Φ ty t 1:p + ε t with ε t NID(0, H t) f σ t = vech(h t), f σ t+1 = ω + B f σ t + A s t, additional parameter estimation concentrates on ω, A and B. MLE is maximisation of l(ψ) wrt ψ, remains light task. Our strategy : Step 4: Notice that under stationarity, E(f σ t ) = (I B) 1 ω. Step 5: Hence least squares estimate of H in static VAR is consistent estimate of unvech[(i B) 1 ω]. Step 6: Only place coefficients of A and B in ψ. Step 7: Estimate slightly extended ψ by MLE using the Kalman filter with ft σ or direct H t updating. MLE is for a small dimension of ψ. 15 / 32
Outline Introduction Econometric Model Simulations Empirical application: Macro-financial linkages in the U.S. economy Conclusion 16 / 32
Simulation setup Zero-mean VAR(1) model with time-varying coefficient matrix and scalar factor f t: with y t = Φ ty t 1 + ε t, ε t N(0, H t), t = 1,..., T, Φ t = Φ c + Φ f f t, f t+1 = ϕf t + η t, η t N(0, 1 ϕ 2 ). Time-varying H t : step function or sine function: the sine function: the step function: ft σ f σ = 1 + 0.95 cos(2πt/150), t = 1.5 I (t > T /2). N = 5, 7; T = 250, 500 Parameter values: Σ ε = I N, Φ c ii = 0.5, Φ c i,j = 0.1, Φ f ii = 0.2, Φ f i,j = 0.1, ϕ = 0.6, for i, j = 1,..., N and i j. 17 / 32
True and filtered coefficient factor Series 1 0.6 0.2 0.2 0.6 1 0 1 2 0 100 200 300 400 500 0 100 200 300 400 500 Series 2 Series 3 2 1 0 1 2 2 1 0 1 2 0 100 200 300 400 500 0 100 200 300 400 500 Series 4 Series 5 1 0 1 2 2 1 0 1 2 0 100 200 300 400 500 0 100 200 300 400 500 18 / 32
True and filtered coefficient factor Series 1 0.6 0.2 0.2 0.6 2 1 0 1 2 0 100 200 300 400 500 0 100 200 300 400 500 Series 2 Series 3 2 1 0 1 2 3 2 1 0 1 2 3 0 100 200 300 400 500 0 100 200 300 400 500 Series 4 Series 5 2 1 0 1 2 2 1 0 1 2 3 0 100 200 300 400 500 0 100 200 300 400 500 19 / 32
Simulations: Mean Squared Errors variance pattern sine N=5 N=7 T=250 T=500 T=250 T=500 ϕ 0.01177 9e-04 0.00991 0.00048 Φ f 0.41609 0.15935 0.33568 0.18428 f t 0.06768 0.06491 0.06332 0.06485 variance pattern step N=5 N=7 T=250 T=500 T=250 T=500 ϕ 0.01122 0.00076 0.00828 0.00063 Φ f 0.24277 0.06938 0.20426 0.07465 f t 0.07205 0.06799 0.06396 0.06871 20 / 32
Simulations with sine function for variance Filtered f_t for T=500, N=7, sine variance 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0 100 200 300 400 500 Filtered variance H_1,1 for T=500, N=7 0.0 0.2 0.4 0.6 0.8 0 100 200 300 400 500 21 / 32
Simulations with step function for variance Filtered f_t for T=500, N=7, step variance 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0 100 200 300 400 500 Filtered variance H_1,1 for T=500, N=7 0.0 0.2 0.4 0.6 0.8 0 100 200 300 400 500 22 / 32
Outline 1. Introduction 2. Econometric Model 3. Simulations 4. Empirical application: Macro-financial linkages in the U.S. economy 5. Conclusion 23 / 32
Application: Macro-financial linkages Six-dimensional VAR with two lags, data from Prieto/Eickmeier/Marcellino (2016, JAE). Macroeconomic variables: Nominal GDP growth, inflation (GDP deflator). Financial variables: real house price inflation, corporate bond spread (Baa-Aaa), real stock price inflation, federal funds rate. Data transformations such that all time series are I (0). Sample: 1958Q1-2012Q2. 24 / 32
Transformed data set real house prices 8 6 4 2 0 2 4 fed funds rate 4 2 0 2 4 6 gdp deflator 0.0 0.5 1.0 1.5 2.0 2.5 3.0 real stocks prices 30 20 10 0 10 20 gdp growth 2 1 0 1 2 3 4 bond spread 1.0 0.5 0.0 0.5 1.0 1.5 1960 1970 1980 1990 2000 2010 1960 1970 1980 1990 2000 2010 25 / 32
Empirical specification VAR(2) model with two factors: where we assume that y t = Φ 1ty t 1 + Φ 2ty t 2 + ε t ε t N(0, H t) Φ jt = Φ c j + Φ f j,1f t,1 + Φ f j,2f t,2, j = 1, 2 Φ c 1 and Φ c 2 are full matrices, Φ f 1,1 and Φ f 2,1 are diagonal matrices and Φ f 1,2 and Φ f 2,2 have zero entries except for the four coefficients that measure the impact of the financial variables on GDP growth. Consequently, f t,1 captures the changing persistence in the six variables and f t,2 indicates how financial-macro spillovers vary over time. 26 / 32
Model specifications one lag two lags (1) (2) (3) (4) (5) (6) (7) (8) H H t f t,1 f t,2 Φ c #ψ 36 40 47 52 72 76 89 98 LogLik -1496.2-1437.2-1429.5-1414.5-1437.5-1393.7-1356.3-1348.5 AICc 3079.2 2973.1 2979.7 2966.6 3092.1 3022.9 3016.8 3057.5 27 / 32
Some estimation results ω 1 0.6964-0.3618 Φ f 1,11-1.1224 Φ f 2,13-0.0830 (1.0357) (0.2342) (1.2251) ω 2-0.0163-0.0467 Φ f 1,22-0.5944 Φ f 2,14-0.1355 (1.1812) (0.3513) (1.5007) A 0.3255-0.7284 Φ f 1,33 0.5313 Φ f 2,15 0.2935 (2.2364) (0.4356) (1.3403) B 0.9171 2.4032 Φ f 1,44 0.0149 Φ f 2,16 0.1391 (1.2238) (0.5280) (0.9997) ϕ 1 0.3554-0.5952 Φ f 1,55 0.0319 (0.2646) (0.4720) ϕ 2 0.9197 2.4380 Φ f 1,66 0.9570 (0.0970) (0.2649) 28 / 32
Filtered factors f t,1 and f t,2 Filtered f_1 0.4 0.0 0.2 0.4 1960 1970 1980 1990 2000 2010 Time Filtered f_2 1.5 0.5 0.5 1960 1970 1980 1990 2000 2010 Time 29 / 32
Time-varying variances variance gdp growth variance gdp deflator variance real house prices 0.5 1.0 1.5 2.0 0.3 0.4 0.5 0.6 0.7 0.5 1.0 1.5 2.0 2.5 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 variance bond spread variance real stock prices variance fed funds rate 1 2 3 4 5 6 0.5 1.0 1.5 2.0 2.5 3.0 1 2 3 4 5 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 30 / 32
Conclusion New frequentist estimation method for VAR models with time-varying coefficient matrices. Simple, fast and transparent implementation, but highly flexible for empirical specifications. Simulations: Good performance in filtering dynamic factors and estimation of constant parameters. Empirics: Evidence for time-variation in financial-macro spillover coefficients. Future steps: Impulse response functions. Forecasting Derivation of stability conditions, consistency and asymptotic theory. Extension of empirical analysis to include formal significant tests. 31 / 32
Thank you. 32 / 32
Creal, D. D., Koopman, S. J., and Lucas, A. (2013). Generalized autoregressive score models with applications. Journal of Applied Econometrics, 28:777 795. Delle Monache, D., Petrella, I., and Venditti, F. (2016). Adaptive state space models with applications to the business cycle and financial stress. Working Paper. Hamilton, J. (1994). Time Series Analysis. Princeton University Press, Princeton. Lutkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer-Verlag, Berlin. Magnus, J. and Neudecker, H. (2007). Matrix Differential Calculus with Applications in Statistics and Econometrics 3rd edition. Wiley Series in Probabiliy and Statistics. 32 / 32