Econometrics II. Seppo Pynnönen. Spring Department of Mathematics and Statistics, University of Vaasa, Finland

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1 Department of Mathematics and Statistics, University of Vaasa, Finland Spring 218

2 Part VI Vector Autoregression As of Feb 21, 218

3 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

4 Background 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

5 Background Example 1 Consider the following monthly values of FTAS (Financial Times All Share) and FTA dividend index values from Jan 1965 to Dec 25 (source: FT All Share Monthly Srock and Dividend Indexes div ret fta ftadiv Time

6 Background Potentially interesting questions: 1 Does one of the series (dividend) have a tendency to lead the other? 2 Are there feedbacks between the series? 3 How about contemporaneous movements? 4 How do impulses (shocks, innovations) transfer from one series to another? 5 How about common factors (disturbances, trend, yield component, risk)? Most of these questions can be empirically investigated using tools developed in multivariate time series analysis.

7 Background Some of these can be readily investigated by simple tools. For example lead-lag relations can be investigated by cross-autocorrelation function ˆρ xy (k) = ˆγ xy (k) ˆγx ()ˆγ y (), (1) where T k ˆγ xy (k) = (x t x)(y t+k ȳ) (2) t=1 is the cross-autocovariance function and ˆγ x () and ˆγ y () are the sample variances of x t and y t. Note that ˆρ xy (k) ˆρ xy ( k).

8 Background The following autocorrelation functions indicate that returns tend to lead dividends (negative cross-autocorrelation) Return Return & Dividend ACF Lag Lag Dividend & Return Dividend ACF Lag Lag

9 Background The following scatter diagram of (ret t 1, div t ) demonstrates the first order cross-correlation Returns and Dividends Ret t 1 (%) Div t (%) Vector autoregression (VAR) provides more sophisticated tools for deeper analyses.

10 The Model 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

11 The Model VAR(1) (i.e., first order vector autoregression) The univariate time series generalize straightforwardly to multivariate time series. Consider two time series x t and y t x t = φ 1 + φ (1) 11 x t 1 + φ (1) 12 y t 1 + ɛ 1t (3) y t = φ 2 + φ (1) 21 x t 1 + φ (1) 22 y t 1 + ɛ 2t (4) ɛ it and ɛ 2t are white noise error terms that can be contemporaneously correlated

12 The Model Generally: Suppose we have m time series y it, i = 1,..., m, and t = 1,..., T (common length of the time series). Then a vector autoregression model is defined as y 1t y 2t. y mt = µ 1 µ 2. µ m + + φ (1) 11 φ (1) 12 φ (1) 1m φ (1) 21 φ (1) 22 φ (1) 2m φ (1) m1 φ (1) m2 φ (1) mm φ (p) 11 φ (p) 12 φ (p) 1m φ (p) 21 φ (p) 22 φ (p) 2m φ (p) m1 φ (1) m2 φ mm (p) y 1,t p y 2,t p. y m,t p y 1,t 1 y 2,t 1. y m,t ɛ 1t ɛ 2t. ɛ mt (5).

13 The Model In matrix notations y t = µ + Φ 1 y t Φ p y t p + ɛ t, (6) which can be further simplified by adopting the matric form of a lag polynomial Φ(L) = I Φ 1 L... Φ p L p. (7) Using this in equation (6), it can be finally written shortly as Φ(L)y t = µ + ɛ t. (8) Note that in the above model each y it depends not only its own history but also on other series history (cross dependencies). This gives us several additional tools for analyzing causal as well as feedback effects as we shall see after a while.

14 The Model A basic assumption in the above model is that the residual vector follow a multivariate white noise, i.e. E[ɛ t ] = { E[ɛ t ɛ Σɛ if t = s s] = if t s (9) The coefficient matrices must satisfy certain constraints in order that the VAR-model is stationary. They are just analogies with the univariate case, but in matrix terms. It is required that roots of I Φ 1 z Φ 2 z 2 Φ p z p = (1) lie outside the unit circle. Estimation can be carried out by single equation least squares.

15 The Model Example 2 VAR(1) for the return/dividend series: VAR Estimation Results: ========================= Endogenous variables: ret, div Deterministic variables: const Sample size: 49 Log Likelihood: Roots of the characteristic polynomial: ret Estimate Std. Error t value Pr(> t ) ret.l div.l const * div ret.l < 2e-16 *** div.l const *** --- Signif. codes: ***.1 **.1 *.5..1 Correlation matrix of residuals: ret div ret div

16 Defining the order of a VAR-model 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

17 Defining the order of a VAR-model The number of estimated parameters grows rapidly very large. In the first step it is assumed that all the series in the VAR model have equal lag lengths. To determine the number of lags that should be included, criterion functions can be utilized the same manner as in the univariate case.

18 Defining the order of a VAR-model The underlying assumption is that the residuals follow a multivariate normal distribution, i.e. ɛ N m (, Σ ɛ ). (11) Akaike s criterion function (AIC) and Schwarz s criterion (BIC) have gained popularity in determination of the number of lags in VAR. In the original forms AIC and BIC are defined as AIC = 2 log L + 2s (12) BIC = 2 log L + s log T (13) where L stands for the Likelihood function, and s denotes the number of estimated parameters.

19 Defining the order of a VAR-model The best fitting model is the one that minimizes the criterion function. For example in a VAR(j) model with m equations there are s = m(1 + jm) + m(m + 1)/2 estimated parameters.

20 Defining the order of a VAR-model Under the normality assumption BIC can be simplified to jm 2 log T BIC(j) = log ˆΣ ɛ,j +, (14) T and AIC to 2jm 2 AIC(j) = log ˆΣ ɛ,j + T, (15) j =,..., p, where ˆΣ ɛ,j = 1 T T t=j+1 ˆɛ t,jˆɛ t,j = 1 T ÊjÊ j (16) with ˆɛ t,j the OLS residual vector of the VAR(j) model (i.e. VAR model estimated with j lags), and an m (T j) matrix. Ê j = [ˆɛ j+1,j, ˆɛ j+2,j,, ˆɛ T,j ] (17)

21 Defining the order of a VAR-model The likelihood ratio (LR) test can be also used in determining the order of a VAR. The test is generally of the form LR = T (log ˆΣ k log ˆΣ p ), (18) where ˆΣ k denotes the maximum likelihood estimate of the residual covariance matrix of VAR(k) and ˆΣ p the estimate of VAR(p) (p > k) residual covariance matrix. If VAR(k) (the shorter model) is the true one, i.e., the null hypothesis is true, then H : Φ k+1 = = Φ p = (19) LR χ 2 df, (2) where the degrees of freedom, df, equals the difference of in the number of estimated parameters between the two models.

22 Defining the order of a VAR-model In an m variate VAR(k)-model each series has p k lags less than those in VAR(p). Thus the difference in each equation is m(p k), so that in total df = m 2 (p k). Note that often, when T is small, a modified LR LR = (T mg)(log ˆΣ k log ˆΣ p ) is used to correct possible small sample bias, where g = p k. Yet another method is to use sequential LR LR(k) = (T m)(log ˆΣ k log ˆΣ k+1 ) which tests the null hypothesis H : Φ k+1 = given that Φ j = for j = k + 2, k + 3,..., p. EViews uses this method in View -> Lag Structure -> Lag Length Criteria....

23 Defining the order of a VAR-model Example 3 Consider series: FTAS, FTADIV, R91 (three months bond yield), R2 (2 year bond yield) Bond and Stock Market Series r2 rs fta ftadiv Time

24 Defining the order of a VAR-model Let p = 12 then in the equity-bond data different VAR models yield the following results. Below are EViews results. VAR Lag Order Selection Criteria Endogenous variables: DFTA DDIV DR2 DTBILL Exogenous variables: C Sample: 1965:1 1995:12 Included observations: 359 ========================================================= Lag LogL LR FPE AIC SC HQ NA * * * * * ========================================================= * indicates lag order selected by the criterion LR: sequential modified LR test statistic (each test at 5% level) FPE: Final prediction error AIC: Akaike information criterion SC: Schwarz information criterion HQ: Hannan-Quinn information criterion

25 Defining the order of a VAR-model Criterion function minima are all at VAR(1) (SC or BIC just borderline). LR-tests suggest VAR(8). Let us look next at the residual autocorrelations.

26 Defining the order of a VAR-model In order to investigate whether the VAR residuals are White Noise, the hypothesis to be tested is H : Υ 1 = = Υ h = (21) where Υ k = (ρ ij (k)) is the autocorrelation matrix (see later in the notes) of the residual series with ρ ij (k) the cross autocorrelation of order k of the residuals series i and j. A general purpose (portmanteau) test is the Q-statistic 8 Q h = T h tr( ˆΥ k k=1 ˆΥ 1 ˆΥ k ˆΥ 1 ), (22) where ˆΥ k = (ˆρ ij (k)) are the estimated (residual) autocorrelations, and ˆΥ the contemporaneous correlations of the residuals. 8 See e.g. Lütkepohl, Helmut (1993). Introduction to Multiple Time Series, 2nd Ed., Ch. 4.4

27 Defining the order of a VAR-model Alternatively (especially in small samples) a modified statistic is used Q h = T 2 h (T k) 1 tr( ˆΥ k k=1 ˆΥ 1 ˆΥ k ˆΥ 1 ). (23)

28 Defining the order of a VAR-model The asymptotic distribution is χ 2 with df = p 2 (h k). Note that in computer printouts h is running from 1, 2,... h with h specified by the user. VAR(1) Residual Portmanteau Tests for Autocorrelations H: no residual autocorrelations up to lag h Sample: 1966:2 1995:12 Included observations: 359 ================================================ Lags Q-Stat Prob. Adj Q-Stat Prob. df NA* NA* NA* ================================================ *The test is valid only for lags larger than the VAR lag order. df is degrees of freedom for (approximate) chi-square distribution There is still left some autocorrelation into the VAR(1) residuals. Let us next check the residuals of the VAR(2) model.

29 Defining the order of a VAR-model VAR Residual Portmanteau Tests for Autocorrelations H: no residual autocorrelations up to lag h Sample: 1965:1 1995:12 Included observations: 369 ============================================== Lags Q-Stat Prob. Adj Q-Stat Prob. df NA* NA* NA* NA* NA* NA* ============================================== *The test is valid only for lags larger than the VAR lag order. df is degrees of freedom for (approximate) chi-square distribution

30 Defining the order of a VAR-model Now the residuals pass the white noise test. On the basis of these residual analyses we can select VAR(2) as the specification for further analysis. Mills (1999) finds VAR(6) as the most appropriate one. Note that there ordinary differences (opposed to log-differences) are analyzed. Here, however, log transformations are preferred.

31 Vector ARMA (VARMA) 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

32 Vector ARMA (VARMA) Similarly as is done in the univariate case one can extend the VAR model to the vector ARMA model p q y t = µ + Φ i y t i + ɛ t Θ j ɛ t j (24) or i=1 j=1 Φ(L)y t = µ + Θ(L)ɛ t, (25) where y t, µ, and ɛ t are m 1 vectors, and Φ i s and Θ j s are m m matrices, and Φ(L) = I Φ 1 L... Φ p L p Θ(L) = I Θ 1 L... Θ q L q. Provided that Θ(L) is invertible, we always can write the VARMA(p, q)-model as a VAR( ) model with Π(L) = Θ 1 (L)Φ(L). The presence of a vector MA component, however, complicates the analysis to some extend. (26)

33 Autocorrelation and Autocovariance Matrices 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

34 Autocorrelation and Autocovariance Matrices The kth cross autocorrelation of the ith and jth time series, y it and y jt is defined as γ ij (k) = E(y it k µ i )(y jt µ j ). (27) Although for the usual autocovariance γ k = γ k, the same is not true for the cross autocovariance, but γ ij (k) γ ij ( k). The corresponding cross autocorrelations are ρ i,j (k) = The kth autocorrelation matrix is then γ ij (k) γi ()γ j (). (28) Υ k = ρ 1 (k) ρ 1,2 (k)... ρ 1,m (k) ρ 2,1 (k) ρ 2 (k)... ρ 2,m (k) ρ m,1 (k) ρ m,2 (k)... ρ m (k). (29)

35 Autocorrelation and Autocovariance Matrices Remark: Υ k = Υ k. The diagonal elements, ρ j (k), j = 1,..., m of Υ are the usual autocorrelations.

36 Autocorrelation and Autocovariance Matrices Example 4 Cross autocorrelations of the equity-bond data. ˆΥ 1 = Div t 1 Fta t 1 R2 t 1 Tbl t 1 Div t Fta t R2 t Tbl t Note: Correlations with absolute value exceeding 2 std err = 2 / are statistically significant at the 5% level (indicated by ).

37 Autocorrelation and Autocovariance Matrices Example 5 Consider individual security returns and portfolio returns. For example French and Roll i have found that daily returns of individual securities are slightly negatively correlated. The tables below, however suggest that daily returns of portfolios tend to be positively correlated! i French, K. and R. Roll (1986). Stock return variances: The arrival of information and reaction of traders. Journal of Financial Economics, 17, 5 26.

38 Autocorrelation and Autocovariance Matrices Source: Campbell, Lo, and MackKinlay (1997). Econometrics of Financial Markets.

39 Autocorrelation and Autocovariance Matrices One explanation could be that the cross autocorrelations are positive, which can be partially proved as follows: Let r pt = 1 m m r it = 1 m ι r t (3) i=1 denote the return of an equal weighted index, where ι = (1,..., 1) is a vector of ones, and r t = (r 1t,..., r mt ) is the vector of returns of the securities. Then [ ι ] r t 1 cov(r pt 1, r pt ) = cov m, ι r t = ι Γ 1 ι m m 2, (31) where Γ 1 is the first order autocovariance matrix.

40 Autocorrelation and Autocovariance Matrices Therefore m 2 cov(r pt 1, r pt ) = ι Γ 1 ι = ( ι Γ 1 ι tr(γ 1 ) ) + tr(γ 1 ), (32) where tr( ) is the trace operator which sums the diagonal elements of a square matrix. Consequently, because the right hand side tends to be positive and tr(γ 1 ) tends to be negative, ι Γ 1 ι tr(γ 1 ), which contains only cross autocovariances, must be positive, and larger than the absolute value of tr(γ 1 ), the autocovariances of individual stock returns.

41 Exogeneity and Causality 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

42 Exogeneity and Causality Consider the following extension of the VAR-model (multivariate dynamic regression model) y t = c + p A iy t i + i=1 p B ix t i + ɛ t, (33) where p + 1 t T, y t = (y 1t,..., y mt ), c is an m 1 vector of constants, A 1,..., A p are m m matrices of lag coefficients, x t = (x 1t,..., x kt ) is a k 1 vector of regressors, B, B 1,..., B p are k m coefficient matrices, and ɛ t is an m 1 vector of errors having the properties i= E[ɛ t ] = E[E[ɛ t Y t 1, x t ]] = (34) and E [ { ɛ t ɛ ] [ [ s = E E ɛt ɛ ]] Σɛ t = s s Y t 1, x t = t s, (35)

43 Exogeneity and Causality where Y t 1 = (y t 1, y t 2,..., y 1 ). (36) We can compile this into a matrix form where and Y = XB + U, (37) Y = (y p+1,..., y T ) X = (z p+1...., z T ) z t = (1, y t 1,..., y t p, x t,..., x t p) U = (ɛ p+1,..., ɛ T ), (38) B = (c, A 1,..., A p, B,..., B p). (39) The estimation theory for this model is basically the same as for the univariate linear regression.

44 Exogeneity and Causality For example, the LS and (approximate) ML estimator of B is ˆB = (X X) 1 X Y, (4) and the ML estimator of Σ ɛ is ˆΣ ɛ = 1 T Û Û, Û = Y XˆB. (41) We say that x t is weakly exogenous 2 if the stochastic structure of x contains no information that is relevant for estimation of the parameters of interest, B, and Σ ɛ. If the conditional distribution of x t given the past is independent of the history of y t then x t is said to be strongly exogenous. 2 For an in depth discussion of Exogeneity, see Engle, R.F., D.F. Hendry and J.F. Richard (1983). Exogeneity. Econometrica, 51:2,

45 Exogeneity and Causality 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

46 Exogeneity and Causality Granger-causality and measures of feedback One of the key questions that can be addressed to VAR-models is how useful some variables are for forecasting others. If the history of x does not help to predict the future values of y, we say that x does not Granger-cause y. 3 3 Granger, C.W. (1969). Econometrica 37, Sims, C.A. (1972). American Economic Review, 62,

47 Exogeneity and Causality Granger-causality and measures of feedback Usually the prediction ability is measured in terms of the MSE (Mean Square Error). Thus, x fails to Granger-cause y, if for all s > MSE(ŷ t+s y t, y t 1,...) = MSE(ŷ t+s y t, y t 1,..., x t, x t 1,...), (42) where (e.g.) MSE(ŷ t+s y t, y t 1,...) = E [ (y t+s ŷ t+s ) 2 y t, y t 1,... ]. (43)

48 Exogeneity and Causality Granger-causality and measures of feedback Remark: This is essentially the same that y is strongly exogenous to x. In terms of VAR models this can be expressed as follows: Consider the g = m + k dimensional vector z t = (y t, x t), which is assumed to follow a VAR(p) model z t = p Π i z t i + ν t, (44) i=1 where E[ν t ] = { E[ν t ν Σν, t = s s] =, t s. (45)

49 Exogeneity and Causality Granger-causality and measures of feedback Partition the VAR of z as y t = p i=1 C ix t i + p i=1 D iy t i + ν 1t x t = p i=1 E ix t i + (46) p i=1 F iy t i + ν 2t where ν t = (ν 1t, ν 2t ) and Π i and Σ ν are partitioned respectively, ( ) Ci D Π i = i, i = 1,..., p (47) E i F i and Σ ν = with E[ν kt ν lt ] = Σ kl, k, l = 1, 2. ( ) Σ11 Σ 21 Σ 21 Σ 22 (48)

50 Exogeneity and Causality Granger-causality and measures of feedback Now x does not Granger-cause y if and only if, C i = for all i = 1,..., p, or equivalently, if and only if, Σ 11 = Σ 1, where Σ 1 = E[η 1t η 1t ] with η 1t from the regression y t = p D i y t i + η 1t. (49) i=1 Changing the roles of the variables we get the necessary and sufficient condition of y not Granger-causing x.

51 Exogeneity and Causality Granger-causality and measures of feedback It is also said that x is block-exogenous with respect to y. Testing for the Granger-causality of x on y reduces to testing for the hypothesis H : C 1 = = C p =. (5)

52 Exogeneity and Causality This can be done with the likelihood ratio test by estimating with OLS the restricted 4 and non-restricted 4 regressions, and calculating the respective residual covariance matrices: Unrestricted: Restricted: ˆΣ 11 = 1 T p ˆΣ 1 = 1 T p T t=p+1 T t=p+1 ˆν 1t ˆν 1t. (51) ˆη 1t ˆη 1t. (52) 4 Perform OLS regressions of each of the elements in y on a constant, p lags of the elements of x and p lags of the elements of y. 4 Perform OLS regressions of each of the elements in y on a constant and p lags of the elements of y.

53 Exogeneity and Causality Granger-causality and measures of feedback The LR test is then if H is true. LR = (T p) ( ) log ˆΣ 1 log ˆΣ 11 χ 2 mkp, (53)

54 Exogeneity and Causality Example 6 Granger causality between pairwise equity-bond market series (by EViews) Pairwise Granger Causality Tests Sample: 1965:1 1995:12 Lags: 12 ================================================================ Null Hypothesis: Obs F-Statistic Probability ================================================================ DFTA does not Granger Cause DDIV DDIV does not Granger Cause DFTA DR2 does not Granger Cause DDIV DDIV does not Granger Cause DR DTBILL does not Granger Cause DDIV DDIV does not Granger Cause DTBILL DR2 does not Granger Cause DFTA DFTA does not Granger Cause DR DTBILL does not Granger Cause DFTA DFTA does not Granger Cause DTBILL DTBILL does not Granger Cause DR DR2 does not Granger Cause DTBILL ==============================================================

55 Exogeneity and Causality Granger-causality and measures of feedback The p-values indicate that FTA index returns Granger cause 2 year Gilts, and Gilts lead Treasury bill.

56 Exogeneity and Causality Granger-causality and measures of feedback Let us next examine the block exogeneity between the bond and equity markets (two lags). Test results are in the table below. =================================================================== Direction LoglU LoglR 2(LU-LR) df p-value (Tbill, R2) --> (FTA, Div) (FTA,Div) --> (Tbill, R2) =================================================================== The test results indicate that the equity markets are Granger-causing bond markets. That is, to some extend previous changes in stock markets can be used to predict bond markets.

57 Exogeneity and Causality 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

58 Exogeneity and Causality Geweke s measures of Linear Dependence Above we tested Granger-causality, but there are several other interesting relations that are worth investigating. Geweke 5 has suggested a measure for linear feedback from x to y based on the matrices Σ 1 and Σ 11 as F x y = log( Σ 1 / Σ 11 ), (54) so that the statement that x does not (Granger) cause y is equivalent to F x y =. Similarly the measure of linear feedback from y to x is defined by F y x = log( Σ 2 / Σ 22 ). (55) 5 Geweke (1982) Journal of the American Statistical Association, 79,

59 Exogeneity and Causality Geweke s measures of Linear Dependence It may also be interesting to investigate the instantaneous causality between the variables. For the purpose, define regression y t = p C i x t i + i= p D i y t i + ω 1t, (56) with C capturing the contemporaneous effect of x t on y t. Note: The coefficient matricese C i and D i are just generic notations, so they are logically different in different specifications of the models. i=1

60 Exogeneity and Causality Geweke s measures of Linear Dependence Similarly, the last k equations can be written as p p x t = E i x t i + F i y t i + ω 2t. (57) i=1 Denoting Σ ωi = E[ω it ω it ], i = 1, 2, there is instantaneous causality between y and x if and only if C and F or, equivalently, Σ 11 > Σ ω1 and Σ 22 > Σ ω2. Analogously to the linear feedback we can define instantaneous linear feedback F x y = log( Σ 11 / Σ ω1 ) = log( Σ 22 / Σ ω2 ). (58) Linear dependence, defined as i= F x,y = F x y + F y x + F x y. (59) measures overall (linear) dependence between x and y. Consequently, the linear dependence can be decomposed additively into three forms of feedback.

61 Exogeneity and Causality Geweke s measures of Linear Dependence Absence of a particular causal ordering is then equivalent to one of these feedback measures being zero. Using the method of least squares we get estimates for the various matrices above as ˆΣ i = 1 T ˆη it ˆη T p it, (6) for i = 1, 2. For example ˆΣ ii = 1 T p ˆΣ ωi = 1 T p t=p+1 T t=p+1 T t=p+1 ˆν it ˆν it, (61) ˆω it ˆω it, (62) ˆF x y = log( ˆΣ 1 / ˆΣ 11 ). (63)

62 Exogeneity and Causality Geweke s measures of Linear Dependence With these estimates one can test in particular: No Granger-causality: x y, H 1 : F x y = No Granger-causality: y x, H 2 : F y x = No instantaneous feedback: H 3 : F x y = No linear dependence: H 4 : F x,y = (T p) ˆF x y χ 2 mkp. (64) (T p) ˆF y x χ 2 mkp. (65) (T p) ˆF x y χ 2 mk. (66) (T p) ˆF x,y χ 2 mk(2p+1). (67) This last is due to the asymptotic independence of the measures F x y, F y x and F x y.

63 Exogeneity and Causality Geweke s measures of Linear Dependence Alternatively one can use asymptotically equivalent Wald and Lagrange Multiplier (LM) statistics for testing these hypotheses. Note that in each case (T p) ˆF is the LR-statistic.

64 Exogeneity and Causality Geweke s measures of Linear Dependence Example 7 The LR-statistics of the above measures and the associated χ 2 values for the equity-bond data are reported in the following table with p = 2. [y = ( log FTA t, log DIV t ) and x = ( log Tbill t, log r2 t )] ================================== LR DF P-VALUE x-->y y-->x x.y x,y ================================== Note. The results lead to the same inference as in Mills (1999), p. 251, although numerical values are different [in Mills VAR(6) is analyzed and here VAR(2)].

65 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

66 Consider a bivariate VAR(1) ( ) ( xt φ11 φ = 12 y t φ 21 φ 22 ) ( xt 1 y t 1 ) ( ɛ1t + ɛ 2t ). (68) The vector moving average (VMA) representation of is ( xt y t ) = ( ψ11 (i) ψ 12 (i) ) ( ɛ1,t i ) ψ 21 (i) ψ 22 (i) i= ɛ 2,t i (69) The coefficients ψ jk (i) can be used in one hand to trace the shocks (ɛ) on the future values of x and y or in the other hand on the variance of the prediction errors of x and y. The former is called impulse response analysis and the latter variance decomposition. Together these are called innovation accounting.

67 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

68 Impulse response analysis Consider the VAR(p) model where Φ(L)y t = ɛ t, (7) Φ(L) = I Φ 1 L Φ 2 L 2 Φ p L p (71) is the (matric) lag polynomial. Provided that the stationary conditions hold we have analogously to the univariate case the vector MA representation of y t as y t = Φ 1 (L)ɛ t = ɛ t + where Ψ i is an m m coefficient matrix. Ψ i ɛ t i, (72) i=1

69 Impulse response analysis The error terms ɛ t represent shocks in the system. Suppose we have a unit shock in ɛ t then its effect in y, s periods ahead is y t+s ɛ t = Ψ s. (73) Accordingly the interpretation of the Ψ matrices is that they represent marginal effects, or the model s response to a unit shock (or innovation) at time point t in each of the variables. Economists call such parameters are as dynamic multipliers.

70 Impulse response analysis For example, if we were told that the first element in ɛ t changes by δ 1, that the second element changes by δ 2,..., and the mth element changes by δ m, then the combined effect of these changes on the value of the vector y t+s would be given by y t+s = y t+s ɛ 1t δ y t+s ɛ mt δ m = Ψ s δ, where δ = (δ 1,..., δ m ). (74)

71 Impulse response analysis The response of y i to a unit shock in y j is given the sequence, known as the impulse response function, ψ ij,1, ψ ij,2, ψ ij,3,..., (75) where ψ ij,k is the ijth element of the matrix Ψ k (i, j = 1,..., m). Generally an impulse response function traces the effect of a one-time shock to one of the innovations on current and future values of the endogenous variables.

72 Impulse response analysis What about exogeneity (or Granger-causality)? Suppose we have a bivariate VAR system such that x t does not Granger cause y. Then we can write y t ( ) ( φ (1) ) = 11 yt 1 + x t + φ (1) 21 φ (1) 22 ( φ (p) 11 φ (p) 21 φ (p) 22 x t 1 ) ( yt p x t p ) ( ɛ1t + ɛ 2t ). (76)

73 Impulse response analysis Then under the stationarity condition where (I Φ(L)) 1 = I + Ψ i = ( Ψ i L i, (77) i=1 ψ (i) 11 ψ (i) 21 ψ (i) 22 ). (78) Hence, we see that variable y does not react to a shock of x.

74 Impulse response analysis Generally, if a variable, or a block of variables, are strictly exogenous, then the implied zero restrictions ensure that these variables do not react to a shock to any of the endogenous variables. Nevertheless it is advised to be careful when interpreting the possible (Granger) causalities in the philosophical sense of causality between variables. Remark: See also the critique of impulse response analysis in the end of this section.

75 Impulse response analysis Orthogonalized impulse response function Noting that E(ɛ t ɛ t) = Σ ɛ, we observe that the components of ɛ t are contemporaneously correlated, meaning that they have overlapping information to some extend.

76 Impulse response analysis Example 8 For example, in the equity-bond data the contemporaneous VAR(2)-residual correlations are ================================= FTA DIV R2 TBILL FTA 1 DIV R TBILL =================================

77 Impulse response analysis Many times, however, it is of interest to know how new information on y jt makes us revise our forecasts on y t+s. In order to single out the individual effects the residuals must be first orthogonalized, such that they become contemporaneously uncorrelated (they are already serially uncorrelated). Remark: If the error terms (ɛ t ) are already contemporaneously uncorrelated, naturally no orthogonalization is needed.

78 Impulse response analysis Unfortunately, orthogonalization is not unique in the sense that changing the order of variables in y chances the results. Nevertheless, there usually exist some guidelines (based on the economic theory) how the ordering could be done in a specific situation. Whatever the case, if we define a lower triangular matrix S such that SS = Σ ɛ, and ν t = S 1 ɛ t, (79) then I = S 1 Σ ɛ S 1, implying E[ν t ν t] = S 1 E[ɛ t ɛ t] S 1 = S 1 Σ ɛ S 1 = I. Consequently the new residuals are both uncorrelated over time as well as across equations. Furthermore, they have unit variances.

79 Impulse response analysis The new vector MA representation becomes y t = Ψ i ν t i, (8) i= where Ψ i = Ψ i S (m m matrices) so that Ψ = S. The impulse response function of y i to a unit shock in y j is then given by ψij,, ψij,1, ψij,2,... (81)

80 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

81 Variance decomposition The uncorrelatedness of ν t s allow the error variance of the s step-ahead forecast of y it to be decomposed into components accounted for by these shocks, or innovations (this is why this technique is usually called innovation accounting). Because the innovations have unit variances (besides the uncorrelatedness), the components of this error variance accounted for by innovations to y j is given by s k= ψ 2 ij,k. (82) Comparing this to the sum of innovation responses we get a relative measure how important variable js innovations are in explaining the variation in variable i at different step-ahead forecasts, i.e.,

82 Variance decomposition R 2 ij,s = 1 s k= ψ 2 ij,k m h=1 s k= ψ 2 ih,k. (83) Thus, while impulse response functions trace the effects of a shock to one endogenous variable on to the other variables in the VAR, variance decomposition separates the variation in an endogenous variable into the component shocks to the VAR. Letting s increase to infinity one gets the portion of the total variance of y j that is due to the disturbance term ɛ j of y j.

83 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

84 On the ordering of variables As was mentioned earlier, when there is contemporaneous correlation between the residuals, i.e., cov(ɛ t ) = Σ ɛ I the orthogonalized impulse response coefficients are not unique. There are no statistical methods to define the ordering. It must be done by the analyst!

85 On the ordering of variables It is recommended that various orderings should be tried to see whether the resulting interpretations are consistent. The principle is that the first variable should be selected such that it is the only one with potential immediate impact on all other variables. The second variable may have an immediate impact on the last m 2 components of y t, but not on y 1t, the first component, and so on. Of course this is usually a difficult task in practice.

86 On the ordering of variables Selection of the S matrix Selection of the S matrix, where SS = Σ ɛ, actually defines also the ordering of variables. Selecting it as a lower triangular matrix implies that the first variable is the one affecting (potentially) the all others, the second to the m 2 rest (besides itself) and so on.

87 On the ordering of variables One generally used method in choosing S is to use Cholesky decomposition which results to a lower triangular matrix with positive main diagonal elements. 6 6 For example, if the covariance matrix is ( ) σ 2 Σ = 1 σ 12 σ 21 σ2 2 then if Σ = SS, where We get ( ) σ 2 Σ = 1 σ 12 σ 21 σ2 2 = ( ) s11 S = s 21 s 22 ( ) ( ) ( s11 s11 s 21 s 2 = 11 s 11 s 21 s 21 s 22 s 22 s 21 s 11 s s2 22 Thus s11 2 = σ2 1, s 21s 11 = σ 21 = σ 12, and s s2 22 = σ2 2. Solving these yields s 11 = σ 1, s 21 = σ 21 /σ 1, and s 22 = σ2 2 σ2 21 /σ2 1. That is, ( ) σ1 S = σ 21 /σ 1 (σ2 2 σ2 21 /σ2 1 ) 1. 2 ).

88 On the ordering of variables Example 9 Let us choose in our example two orderings. One with stock market series first followed by bond market series [(I: FTA, DIV, R2, TBILL)], and an ordering with interest rate variables first followed by stock markets series [(II: TBILL, R2, DIV, FTA)]. In EViews the order is simply defined in the Cholesky ordering option. Below are results in graphs with I: FTA, DIV, R2, TBILL; II: R2, TBILL DIV, FTA, and III: General impulse response function.

89 On the ordering of variables Impulse responses: Order {FTA, DIV, R2, TBILL} Response to Cholesky One S.D. Innovations ± 2 S.E. Response of DFTA to DFTA Response of DFTA to DDIV Response of DFTA to DR2 Response of DFTA to DTBILL Response of DDIV to DFTA Response of DDIV to DDIV Response of DDIV to DR2 Response of DDIV to DTBILL Response of DR2 to DFTA Response of DR2 to DDIV Response of DR2 to DR2 Response of DR2 to DTBILL Response of DTBILL to DFTA Response of DTBILL to DDIV Response of DTBILL to DR2 Response of DTBILL to DTBILL

90 On the ordering of variables Order {TBILL, R2, DIV, FTA} Response to Cholesky One S.D. Innovations ± 2 S.E. Response of DFTA to DFTA Response of DFTA to DDIV Response of DFTA to DR2 Response of DFTA to DTBILL Response of DDIV to DFTA Response of DDIV to DDIV Response of DDIV to DR2 Response of DDIV to DTBILL Response of DR2 to DFTA Response of DR2 to DDIV Response of DR2 to DR2 Response of DR2 to DTBILL Response of DTBILL to DFTA Response of DTBILL to DDIV Response of DTBILL to DR2 Response of DTBILL to DTBILL

91 On the ordering of variables Variance decomposition graphs of the equity-bond data Variance Decomposition ± 2 S.E. Percent DFTA variance due to DFTA Percent DFTA variance due to DDIV Percent DFTA variance due to DR2 Percent DFTA variance due to DTBILL Percent DDIV variance due to DFTA Percent DDIV variance due to DDIV Percent DDIV variance due to DR2 Percent DDIV variance due to DTBILL Percent DR2 variance due to DFTA Percent DR2 variance due to DDIV Percent DR2 variance due to DR2 Percent DR2 variance due to DTBILL Percent DTBILL variance due to DFTA Percent DTBILL variance due to DDIV Percent DTBILL variance due to DR2 Percent DTBILL variance due to DTBILL

92 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

93 General impulse response function The general impulse response function are defined as 9 GI (j, δ i, F t 1 ) = E[y t+j ɛ it = δ i, F t 1 ] E[y t+j F t 1 ]. (84) That is difference of conditional expectation given an one time shock occurs in series j. These coincide with the orthogonalized impulse responses if the residual covariance matrix Σ is diagonal. 9 Pesaran, M. Hashem and Yongcheol Shin (1998). Impulse Response Analysis in Linear Multivariate Models, Economics Letters, 58,

94 General impulse response function Generalized impulse responses: Response to Generalized One S.D. Innovations ± 2 S.E. Response of DFTA to DFTA Response of DFTA to DDIV Response of DFTA to DR2 Response of DFTA to DTBILL Response of DDIV to DFTA Response of DDIV to DDIV Response of DDIV to DR2 Response of DDIV to DTBILL Response of DR2 to DFTA Response of DR2 to DDIV Response of DR2 to DR2 Response of DR2 to DTBILL Response of DTBILL to DFTA Response of DTBILL to DDIV Response of DTBILL to DR2 Response of DTBILL to DTBILL

95 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

96 Accumulated Responses Accumulated responses for s periods ahead of a unit shock in variable i on variable j may be determined by summing up the corresponding response coefficients. That is, ψ (s) ij = s ψ ij,k. (85) k= The total accumulated effect is obtained by ψ ( ) ij = ψ ij,k. (86) k= In economics this is called the total multiplier. Particularly these may be of interest if the variables are first differences, like the stock returns.

97 Accumulated Responses For the stock returns the impulse responses indicate the return effects while the accumulated responses indicate the price effects. All accumulated responses are obtained by summing the MA-matrices with Ψ ( ) = Ψ(1), where Ψ (s) = s Ψ k, (87) k= Ψ(L) = 1 + Ψ 1 L + Ψ 2 L 2 +. (88) is the lag polynomial of the MA-representation y t = Ψ(L)ɛ t. (89)

98 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

99 On estimation of the impulse response coefficients Consider the VAR(p) model y t = Φ 1 y t Φ p y t p + ɛ t (9) or Φ(L)y t = ɛ t. (91)

100 On estimation of the impulse response coefficients Then under stationarity the vector MA representation is y t = ɛ t + Ψ 1 ɛ t 1 + Ψ 2 ɛ t 2 + (92) When we have estimates of the AR-matrices Φ i denoted by ˆΦ i, i = 1,..., p the next problem is to construct estimates for MA matrices Ψ j. It can be shown that Ψ j = j Ψ j i Φ i (93) i=1 with Ψ = I, and Φ j = when i > p. Consequently, the estimates can be obtained by replacing Φ i s by the corresponding estimates.

101 On estimation of the impulse response coefficients Next we have to obtain the orthogonalized impulse response coefficients. This can be done easily, for letting S be the Cholesky decomposition of Σ ɛ such that Σ ɛ = SS, (94) we can write y t = i= Ψ iɛ t i = i= Ψ iss 1 ɛ t i = i= Ψ i ν t i, (95)

102 On estimation of the impulse response coefficients where and ν t = S 1 ɛ t. Then Ψ i = Ψ i S (96) cov[ν t ] = S 1 Σ ɛ S 1 = I. (97) The estimates for Ψ i are obtained by replacing Ψ t with its estimates and using Cholesky decomposition of ˆΣ ɛ.

103 1 Vector Autoregression (VAR) Background The Model Defining the order of a VAR-model Vector ARMA (VARMA) Autocorrelation and Autocovariance Matrices Exogeneity and Causality Granger-causality and measures of feedback Geweke s measures of Linear Dependence Impulse response analysis Variance decomposition On the ordering of variables General impulse response function Accumulated Responses On estimation of the impulse response coefficients Critique of impulse response analysis

104 Critique of impulse response analysis Critical quaestions: Ordering of variables is one problem. Interpretations related to Granger-causality from the ordered impulse response analysis may not be valid. If important variables are omitted from the system, their effects go to the residuals and hence may lead to major distortions in the impulse responses and the structural interpretations of the results.