Multivariate time series analysis is used when one wants to model and explain the interactions and comovements among a group of time series variables:

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1 Multivariate Time Series Consider n time series variables {y 1t },...,{y nt }. A multivariate time series is the (n 1) vector time series {Y t } where the i th row of {Y t } is {y it }.Thatis,for any time t, Y t =(y 1t,...,y nt ) 0. Multivariate time series analysis is used when one wants to model and explain the interactions and comovements among a group of time series variables: Consumption and income Stock prices and dividends Forward and spot exchange rates interest rates, money growth, income, inflation

2 Stock and Watson state that macroeconometricians do four things with multivariate time series 1. Describe and summarize macroeconomic data 2. Make macroeconomic forecasts 3. Quantify what we do or do not know about the true structure of the macroeconomy 4. Advise macroeconomic policymakers

3 Stationary and Ergodic Multivariate Time Series A multivariate time series Y t is covariance stationary and ergodic if all of its component time series are stationary and ergodic. E[Y t ]=μ =(μ 1,...,μ n ) 0 = var(y t )=Γ 0 = E[(Y t μ)(y t μ) 0 ] var(y 1t ) cov(y 1t,y 2t ) cov(y 1t,y nt ) cov(y 2t,y 1t ) var(y 2t ) cov(y 2t,y nt ) cov(y nt,y 1t ) cov(y nt,y 2t ) var(y nt ) The correlation matrix of Y t is the (n n) matrix corr(y t )=R 0 = D 1 Γ 0 D 1 where D is an (n n) diagonal matrix with j th diagonal element (γ 0 jj )1/2 =var(y jt ) 1/2.

4 The parameters μ, Γ 0 and R 0 are estimated from data (Y 1,...,Y T ) using the sample moments Ȳ = 1 T ˆΓ 0 = 1 T TX t=1 TX t=1 Y t p E[Yt ]=μ (Y t Ȳ)(Y t Ȳ) 0 p var(yt )=Γ 0 ˆR 0 = ˆD 1ˆΓ 0 ˆD 1 p corr(y t )=R 0 where ˆD is the (n n) diagonal matrix with the sample standard deviations of y jt along the diagonal. The Ergodic Theorem justifies convergence of the sample moments to their population counterparts.

5 Cross Covariance and Correlation Matrices With a multivariate time series Y t each component has autocovariances and autocorrelations but there are also cross lead-lag covariances and correlations between all possible pairs of components. The autocovariances and autocorrelations of y jt for j =1,...,n are defined as γ k jj = cov(y jt,y jt k ), ρ k jj = corr(y jt,y jt k )= γk jj γ 0 jj and these are symmetric in k: γ k jj = γ k jj, ρk jj = ρ k jj.

6 The cross lag covariances and cross lag correlations between y it and y jt are defined as γ k ij = cov(y it,y jt k ), ρ k ij = corr(y jt,y jt k )= γk ij q γ 0 ii γ 0 jj and they are not necessarily symmetric in k. In general, Defn: γ k ij = cov(y it,y jt k ) 6= cov(y it,y jt+k ) = cov(y jt,y it k )=γ k ij If γ k ij 6=0forsomek>0theny jt is said to lead y it. If γ k ij y jt. 6= 0forsomek>0theny it is said to lead

7 It is possible that y it leads y jt and vice-versa. In this case, there is said to be feedback between the two series.

8 All of the lag k cross covariances and correlations are summarizedinthe(n n) lagk cross covariance and lag k cross correlation matrices Γ k = E[(Y t μ)(y t k μ) 0 ]= cov(y 1t,y 1t k ) cov(y 1t,y 2t k ) cov(y 1t,y nt k ) cov(y 2t,y 1t k ) cov(y 2t,y 2t k ) cov(y 2t,y nt k ) cov(y nt,y 1t k ) cov(y nt,y 2t k ) cov(y nt,y nt k ) R k = D 1 Γ k D 1 The matrices Γ k and R k are not symmetric in k but it is easy to show that Γ k = Γ 0 k and R k = R 0 k. The matrices Γ k and R k are estimated from data (Y 1,...,Y T )using ˆΓ k = 1 T TX t=k+1 ˆR k = ˆD 1ˆΓ k ˆD 1 (Y t Ȳ)(Y t k Ȳ) 0

9 Multivariate Wold Representation Any (n 1) covariance stationary multivariate time series Y t has a Wold or linear process representation of the form Y t = μ + ε t + Ψ 1 ε t 1 + Ψ 2 ε t 2 + = μ + X k=0 Ψ k ε t k Ψ 0 = I n ε t WN(0, Σ) Ψ k is an (n n) matrixwith(i, j)th element ψ k ij.in lag operator notation, the Wold form is Y t = μ + Ψ(L)ε t Ψ(L) = X k=0 Ψ k L k elements of Ψ(L) are 1-summable The moments of Y t are given by E[Y t ]=μ, var(y t )= X k=0 Ψ k ΣΨ 0 k

10 Long Run Variance Let Y t be an (n 1) stationary and ergodic multivariatetimeserieswithe[y t ]=μ. Anderson s central limit theorem for stationary and ergodic process states or T (Ȳ μ) d N Ȳ A N μ, 1 T 0, X Γ j j= X Γ j j= Hence, the long-run variance of Y t is T times the asymptotic variance of Ȳ: LRV(Y t )=T avar(ȳ) = X Γ j j=

11 Since Γ j = Γ 0 j,lrv(y t) may be alternatively expressed as LRV(Y t )=Γ 0 + X j=1 (Γ j + Γ 0 j ) Using the Wold representation of Y t it can be shown that where Ψ(1) = P k=0 Ψ k. LRV(Y t )=Ψ(1)ΣΨ(1) 0

12 Non-parametric Estimate of the Long-Run Variance A consistent estimate of LRV(Y t ) may be computed using non-parametric methods. A popular estimator is the Newey-West weighted autocovariance estimator d LRV NW (Y t )=ˆΓ 0 + M T X j=1 w j,t ³ˆΓ j + ˆΓ 0 j where w j,t are weights which sum to unity and M T is a truncation lag parameter that satisfies M T = O(T 1/3 ). Usually, the Bartlett weights are used w Bartlett j,t =1 j M T +1

13 Vector Autoregression Models The vector autoregression (VAR) model is one of the most successful, flexible, and easy to use models for the analysis of multivariate time series. Made fameous in Chris Sims s paper Macroeconomics and Reality, ECTA It is a natural extension of the univariate autoregressive model to dynamic multivariate time series. Has proven to be especially useful for describing the dynamic behavior of economic and financial time series and for forecasting. It often provides superior forecasts to those from univariate time series models and elaborate theorybased simultaneous equations models.

14 Used for structural inference and policy analysis. In structural analysis, certain assumptions about the causal structure of the data under investigation are imposed, and the resulting causal impacts of unexpected shocks or innovations to specified variables on the variables in the model are summarized. Thesecausalimpactsareusuallysummarized with impulse response functions and forecast error variance

15 The Stationary Vector Autoregression Model Let Y t =(y 1t,y 2t,...,y nt ) 0 denote an (n 1) vector of time series variables. The basic p-lag vector autoregressive (VAR(p))modelhastheform Y t = c + Π 1 Y t 1 + Π 2 Y t Π p Y t p + ε t ε t WN (0, Σ) Example: Bivariate VAR(2) or à y1t y 2t! = à c1 c 2! + à π 2 11 π 2 12 π 2 21 π à π 1 11 π 1 12 π 1!à 21 π1! 22 y1t 2 y 2t 2!à y1t 1 + à ε1t ε 2t y 2t 1!! y 1t = c 1 + π 1 11 y 1t 1 + π 1 12 y 2t 1 +π 2 11 y 1t 2 + π 2 12 y 2t 2 + ε 1t y 2t = c 2 + π 1 21 y 1t 1 + π 1 22 y 2t 1 +π 2 21 y 1t 1 + π 2 22 y 2t 1 + ε 2t where cov(ε 1t,ε 2s )=σ 12 for t = s; 0otherwise.

16 Remarks: Each equation has the same regressors lagged values of y 1t and y 2t. Endogeneity is avoided by using lagged values of y 1t and y 2t. The VAR(p) model is just a seemingly unrelated regression (SUR) model with lagged variables and deterministic terms as common regressors.

17 In lag operator notation, the VAR(p) is written as Π(L)Y t = c + ε t Π(L) = I n Π 1 L... Π p L p The VAR(p) is stable if the roots of det (I n Π 1 z Π p z p )=0 lie outside the complex unit circle (have modulus greater than one), or, equivalently, if the eigenvalues of the companion matrix F = Π 1 Π 2 Π n I n I n 0 have modulus less than one. A stable VAR(p) process is stationary and ergodic with time invariant means, variances, and autocovariances.

18 Example: Stability of bivariate VAR(1) model à y1t y 2t Y t = ΠY t 1 + ε! à t π11 π = 12 π 21 π 22!à y1t 1 y 2t 1! + à ε1t ε 2t! Then det (I n Πz) = 0 becomes (1 π 11 z)(1 π 22 z) π 12 π 21 z 2 =0 Stability condition involves cross terms π 12 and π 21 If π 12 = π 21 = 0 (diagonal VAR) then bivariate stability condition reduces to univariate stability conditions for each equation.

19 If Y t is covariance stationary, then the unconditional mean is given by μ =(I n Π 1 Π p ) 1 c The mean-adjusted form of the VAR(p) is then Y t μ = Π 1 (Y t 1 μ)+π 2 (Y t 2 μ)+ +Π p (Y t p μ)+ε t The basic VAR(p) model may be too restrictive to represent sufficiently the main characteristics of the data. The general form of the VAR(p) model with deterministic terms and exogenous variables is given by Y t = Π 1 Y t 1 + Π 2 Y t Π p Y t p +ΦD t + GX t + ε t D t = deterministic terms X t = exogenous variables (E[X t ε t ]=0)

20 Wold Representation Consider the stationary VAR(p) model Π(L)Y t = c + ε t Π(L) = I n Π 1 L... Π p L p Since Y t is stationary, Π(L) 1 exists so that Note that Y t = Π(L) 1 c + Π(L) 1 ε t = μ + X k=0 Ψ 0 = I n Ψ k ε t k lim Ψ k =0 k Π(L) 1 = Ψ(L) = X k=0 Ψ k L k

21 The Wold coefficients Ψ k may be determined from the VAR coefficients Π k by solving which implies Ψ 1 = Π 1 Π(L)Ψ(L) =I n Ψ 2 = Π 1 Π 1 + Π 2. Ψ s = Π 1 Ψ s Π p Ψ s p Since Π(L) 1 = Ψ(L), the long-run variance for Y t has the form LRV VAR (Y t )=Ψ(1)ΣΨ(1) 0 = Π(1) 1 ΣΠ(1) 10 =(I n Π 1... Π p ) 1 Σ (I n Π 1... Π p ) 10

22 Estimation Assume that the VAR(p) model is covariance stationary, and there are no restrictions on the parameters of the model. In SUR notation, each equation in the VAR(p) maybewrittenas y i = Zπ i + e i, i =1,...,n y i is a (T 1) vector of observations on the i th equation Z is a (T k) matrixwitht th row given by Z 0 t = (1, Y 0 t 1,...,Y0 t p ) k = np +1 π i is a (k 1) vector of parameters and e i is a (T 1) error with covariance matrix σ 2 i I T.

23 Since the VAR(p) is in the form of a SUR model where each equation has the same explanatory variables, each equation may be estimated separately by ordinary least squares without losing efficiency relative to generalized least squares. Let ˆΠ =[ˆπ 1,...,ˆπ n ] denote the (k n) matrix of least squares coefficients for the n equations.let vec( ˆΠ) = ˆπ 1. ˆπ n Under standard assumptions regarding the behavior of stationary and ergodic VAR models (see Hamilton (1994) vec( ˆΠ) is consistent and asymptotically normally distributed with asymptotic covariance matrix where avar(vec( d ˆΠ)) = ˆΣ (Z 0 Z) 1 ˆΣ = 1 T k TX t=1 ˆε t = Y t ˆΠ 0 Z t ˆε tˆε 0 t

24 Lag Length Selection The lag length for the VAR(p) model may be determined using model selection criteria. The general approach is to fit VAR(p) models with orders p =0,...,p max and choose the value of p which minimizes some model selection criteria MSC(p) = ln Σ(p) + c T ϕ(n, p) Σ(p) = T 1 T X t=1 ˆε tˆε 0 t c T = function of sample size ϕ(n, p) = penalty function The three most common information criteria are the Akaike (AIC), Schwarz-Bayesian (BIC) and Hannan- Quinn (HQ): AIC(p) = ln Σ(p) + 2 T pn2 BIC(p) = ln Σ(p) + ln T T pn2 HQ(p) = ln Σ(p) + 2lnlnT T pn 2

25 Remarks: AIC criterion asymptotically overestimates the order with positive probability, BIC and HQ criteria estimate the order consistently under fairly general conditions if the true order p is less than or equal to p max.

26 Granger Causality One of the main uses of VAR models is forecasting. The following intuitive notion of a variable s forecasting ability is due to Granger (1969). If a variable, or group of variables, y 1 is found to be helpful for predicting another variable, or group of variables, y 2 then y 1 is said to Granger-cause y 2 ; otherwise it is said to fail to Granger-cause y 2. Formally, y 1 fails to Granger-cause y 2 if for all s>0 the MSE of a forecast of y 2,t+s based on (y 2,t,y 2,t 1,...) is the same as the MSE of a forecast of y 2,t+s basedon(y 2,t,y 2,t 1,...)and (y 1,t,y 1,t 1,...). The notion of Granger causality does not imply true causality. It only implies forecasting ability.

27 Example: Bivariate VAR Model In a bivariate VAR(p),y 2 fails to Granger-cause y 1 if all of the p VAR coefficient matrices Π 1,...,Π p are lower triangular: Ã! Ã! Ã!Ã! y1t c1 π 1 = y1t 1 + y 2t + c 2 Ã π p 11 0 π p 21 πp 22 π 1 21!Ã π1 22 y1t p y 2t p y 2t 1! + Ã ε1t ε 2t Similarly, y 1 fails to Granger-cause y 2 if all of the coefficientsonlaggedvaluesofy 1 are zero in the equation for y 2.Noticethatify 2 fails to Granger-cause y 1 and y 1 fails to Granger-cause y 2,thentheVARcoefficient matrices Π 1,...,Π p are diagonal.!

28 The p linear coefficient restrictions implied by Granger non-causality may be tested using the Wald statistic Wald = (R vec( ˆΠ) r) 0 n R h avar(vec( d ˆΠ)) i R 0o 1 (R vec( ˆΠ) r) Remark: In the bivariate model, testing H 0 : y 2 does not Granger-cause y 1 reduces to a testing H 0 : π 1 12 = = π p 12 = 0 from the linear regression y 1t = c 1 + π 1 11 y 1t π p 11 y 1t p +π 1 12 y 2t π p 12 y 2t p + ε 1t The test statistic is a simple F-statistic.

29 Example: Trivariate VAR Model In a trivariate VAR(p),y 2 and y 3 fail to Granger-cause y 1 if π j 12 = πj 13 =0forallj: y 1t y 2t y 3t = + c 1 c 2 c 3 + π p π p 21 πp 22 πp 32 π p 31 πp 32 πp 33 π π 1 21 π1 22 π1 23 π 1 31 π1 32 π1 33 y 1t p y 2t p y 3t p y 1t 1 y 2t 1 y 3t 1 + Note: One can also use a simple F-statistic from a linear regression in this situation: y 1t = c 1 + π 1 11 y 1t π p 11 y 1t p +π 1 12 y 2t π p 12 y 2t p +π 1 13 y 3t π p 13 y 3t p + ε 1t + ε 1t ε 2t ε 3t

30 Example: Trivariate VAR Model In a trivariate VAR(p), y 3 fails to Granger-cause y 1 and y 2 if all of the p VAR coefficient matrices Π 1,...,Π p are lower triangular: y 1t y 2t y 3t = + c 1 c 2 c 3 + π p π p 21 πp 22 0 π p 31 πp 32 πp 33 π π 1 21 π π 1 31 π1 32 π1 33 y 1t p y 2t p y 3t p y 1t 1 y 2t 1 y 3t 1 + Note: We cannot use a simple F-statistic in this case. We must use the general Wald statistic. + ε 1t ε 2t ε 3t

31 Forecasting Algorithms Forecasting from a VAR(p) is a straightforward extension of forecasting from an AR(p). The multivariate Wold form is Y t = μ + ε t + Ψ 1 ε t 1 + Ψ 2 ε t 2 + Y t+h = μ + ε t+h + Ψ 1 ε t+h 1 + +Ψ h 1 ε t+1 + Ψ h ε t + Note that E[Y t ] = μ ε t WN(0, Σ) var(y t ) = E[(Y t μ)(y t μ) 0 ] = E = X k=1 X k=1 Ψ k ΣΨ 0 k Ψ k ε t k X k=1 Ψ k ε t k 0

32 The minimum MSE linear forecast of Y t+h based on I t is Y t+h t = μ + Ψ h ε t + Ψ h+1 ε t 1 + The forecast error is ε t+h t = Y t+h Y t+h t = ε t+h + Ψ 1 ε t+h Ψ h 1 ε t+1 The forecast error MSE is MSE(ε t+h t ) = E[ε t+h t ε 0 t+h t ] = Σ + Ψ 1 ΣΨ Ψ h 1ΣΨ 0 h 1 = h 1 X s=1 Ψ s ΣΨ 0 s

33 Chain-rule of Forecasting The best linear predictor, in terms of minimum mean squared error (MSE), of Y t+1 or 1-step forecast based on information available at time T is Y T +1 T = c + Π 1 Y T + + Π p Y T p+1 Forecasts for longer horizons h (h-step forecasts) may be obtained using the chain-rule of forecasting as Y T +h T = c + Π 1 Y T +h 1 T + + Π p Y T +h p T Y T +j T = Y T +j for j 0. Note: Chain-rule may be derived from state-space framework (assume c = 0) Y t Y t 1. Y t p+1 = + Π 1 Π 2 Π p I n I n 0 ε t 0. 0 ξ t = Fξ t 1 + v t Y t Y t 1. Y t p+1

34 Then ξ t+1 t = Fξ t ξ t+2 t = Fξ t+1 t = F 2 ξ t. ξ t+h t = Fξ t+h 1 t = F h ξ t