PREDICTION FROM THE DYNAMIC SIMULTANEOUS EQUATION MODEL WITH VECTOR AUTOREGRESSIVE ERRORS

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1 Econometrica, Vol. 49, No. 5 (September, 1981) PREDICTION FROM THE DYNAMIC SIMULTANEOUS EQUATION MODEL WITH VECTOR AUTOREGRESSIVE ERRORS BY RICHARD T. BAILLIE' The asymptotic distribution of prediction is derived for the general simultaneous equation model with lagged endogenous variables and vector autoregressive errors. The results turn out to be particularly simple when no lagged endogenous variables are present. 1. INTRODUCTION IN THIS PAPER we derive the asymptotic distribution of multistep prediction from the general dynamic simultaneous equation model. The approach taken is a generalization of that of Schmidt [4] who considered a model with uncorrelated errors. It seems worthwhile to extend Schmidt's results, since the estimation of dynamic simultaneous equation models with autoregressive errors is now fairly commonplace in the literature. Also, as noted by Schmidt [5], the effects of parameter estimation can produce a significant effect on forecast confidence intervals. In fact, the results we obtain are computationally feasible making it possible to calculate the term due to parameter estimation in a forecast confidence interval. For the seemingly unrelated regression model with vector autoregressive errors, but with no lagged dependent variables, the results turn out to be particularly straightforward generalizations of those obtained by Baillie [1], who considered the single equation regression model with autoregressive errors. 2. PREDICTION FROM THE GENERAL MODEL The dynamic simultaneous equation model with maximum lags of r on the endogenous variables and p on the autoregressive error process is given by () C(L)y, = Bx, + u1, A(L)u, =, C(L) = > r =C L', A(L) = oajalj, A = Co = I; C), Aj, and B are respectively g x g, g x g and g x k matrices of coefficients; y1 and x1 are vectors of the tth observation on the g endogenous and k exogenous variables; L is the lag operator and the vector of disturbances e is defined such that E(1) = O and E(=t) = E, r t, O, r7t 'The author is currently Visiting Associate Professor, Wayne State University, Detroit, MI 4822, USA. Many of the results in this paper are extensions of ideas contained in my 1978 Ph.D thesis at the University of London. I gratefully acknowledge the very helpful comments of my thesis supervisors J. Durbin and K. F. Wallis. All errors are my own. 1331

2 1332 RICHARD T. BAILLIE It is further assumed that all the roots of det C(L) and deta (L) lie outside the unit circle. For subsequent analysis, it is convenient to express (1) in companion form as (2) Y= FY, -1 + DX, +, Y = [yfyt, ***x =.. [x'xt,.i, x>p] and t = [E,..., ] are vectors of dimension g(p + r), k(p + 1), and g(p + r) respectively. The square matrix F is of dimension g(p + r), and is given by G1 G2... Gp+r F= I O *.. : O I... *.. I Gj = j-='.uaj C, A = for i > p, and Ci = for i > r. The matrix D is of dimension g(p + r) x k(p + 1) and consists entirely of zeros except for the first g rows which are given by [ B,A IB,A2B,..., ApB]. Hence, the first g rows of (2) give the model (1), and all the other rows are merely identities. At time n + 1, equation (2) can be written as Yn+, = E F"ln+,-j+ E FJDXn+,-j+ F'Yn j=o j=o and the predictor of yn+ in model (1), made at period n, / steps ahead, is thus given by Xn+l Yn= [N'D, N'FD,..., N'F'- 1D, NF'] Xn+ I yn N' = [I ], is of dimension g X g(p + r). On use of the row stacking operator, R, the predictor can be expressed as (3) Yn= W1D1, (4) Wl =(I Xn+1) * Y Xn+ 1), g) Yn')) and U = {R(N'D ), R (N'FD),..., R(N'F'-D ), R(N'F')}.

3 SIMULTANEOUS EQUATION MODEL 1333 With known parameters, (3) is the minimum mean squared error predictor with 1 step prediction mean squared error given by N' F'N N'F'"N. j=o In order to construct a predictor in practice, we have to obtain maximum likelihood estimates of the parameters 9, 9' = (a'3'-y') and a' = { R(Ai)",. R.,R(Ap)'}, ' = R(B)' and -y' = {R(Cl)',..., R(CQ)'}. The corresponding maximum likelihood estimates 6 are assumed to be such that n (9-9) has a limiting normal distribution with mean vector zero and covariance matrix Q. The parameter estimates 9 can be used to form the estimated predictor weights i,, so that the practical predictor becomes Yn I= ld By using the fact that A,j =,j + H,( - ) A it can be shown that conditional on W1, Vn(Ynd - of the form (S) N(O, W,H,Q2Hj Wj ) In,) has a limiting distribution A parametric expression for H,, obtained by using standard results on matrix differentiation due to Neudecker [3], and a lemma due to Schmidt [4] and Yamamoto [6], is given in the Appendix to this paper. By using the fact that (9-9) and W, are asymptotically independent to order O(n 1), the asymptotic mean squared error of 1 step prediction is then given by (6) N' F'NEN'F'"N+ - { W, H,2HWl }. 3. PREDICTION FROM THE SEEMINGLY UNRELATED REGRESSION MODEL WITH VECTOR AUTOREGRESSIVE ERRORS When r =, so that C(L) = I, the general model (1) becomes (7) y= Bx, + u,; A(L)u, = Ec. Rather than consider the multivariate regression model with the same exogenous variables in each equation, we can take (7) to be the seemingly unrelated

4 1334 RICHARD T. BAILLIE regression model with X/~~~~~~~~~~~~~~~~/j X/ ~ / xt=.. and B= so that the model can be expressed as X y1=x1,/+ u1, A(L)u1=c, f /' = [ #1J32,. * * ' g] In this case, (,, the matrix of predictor weights, takes a particularly simple form and we can express the predictor as Yn, = xn+l,3 + N'F'(Yn -Xnf), Al A2... Ap I. F= I.... I O which is a straightforward multivariate generalization of equation (3.8) in Baillie [1], which is the predictor for the single equation regression model with autoregressive errors. The practical predictor is then A = x,+fi + N'F(l Y-Xn X/) - xn+i + { I ( Yn -Xn3)'}R(N'F')'. But as shown in Baillie [2], R(N'Fl) = R(N'Fl) + (&- a)'g, G = MI F"I Fl-'-'M i=o so that it is a special case of (A3).

5 SIMULTANEOUS EQUATION MODEL 1335 Then Yn,l= Xn+1/ +( -/3)} + { I (Yn-Xn/Y)}{ R(N'Fl)'+ G (&-a)} and Yn,l n,l,, = -xn+g D (/3-/3) (-{I C( -Xn/3)'}G/(&-a) + N'F Xn (,-B) On generalizing result (3.9) in Baillie [1], it can be shown that V{(a--,/)(&- a)}has a multivariate normal distribution with mean vector zero, and covariance matrix (Y2B- L Y2(Z O or-,) E(U,Ut') = r and U,' = (ut,u1, U_p + Hence, the asymptotic distribution of An' will be multivariate normal with mean vector yn + and covariance matrix given to order O(n -1) by (8) N' E F'N N'F'jN j= +?- -(xn+-n'f'xn)b -'(xn+, - N'F'Xn)' , has (i, j)th element Gl( Er-) )G1K and K is a square matrix of dimension g2p, containing square submatrices, all of which are of dimension gp and are null matrices, except the (i, j)th submatrix which is r. The expression (8) is thus a direct simultaneous equation generalization of (3.12) in Baillie [1]. The first term in braces in equation (8) is purely due to the estimation of the regression parameters, while the last term is due to estimating the autoregressive parameters. 4. CONCLUDING REMARKS The results presented in this paper will hopefully be of use in the construction of forecast confidence intervals and testing model stability from post sample dynamic simulations. For models with substantial dynamics, both in the error process and in the endogenous variables, the form of H, in the mean squared error formula (6) can be complicated. The efficient computation of H, would require exploiting some of the recursions that exist in its matrix elements. University of Birmingham Manuscript received Febrmary, 198; revision received July, 198.

6 1336 RICHARD T. BAILLIE APPENDIX By using standard matrix differentiation theorems, e.g., Neudecker [3], we can find a parametric expression for the matrix (Al) H ar(n'd) ar(n'fd) ar ('F''D) ar(n'f') } On considering a typical submatrix of HI, we find that (A2) ar(n'fjd) a)(i X D) dr(n'fj a ) ((I D) The first matrix derivative in (A2) can be written as ar(n'fj) a{r(n'f)}' ar(n'fi) a9 a9 a{r(n'f)}' and as noted by Baillie [2], by modifying the result due to Schmidt [4] and Yamamoto [6], ar(n'fi ) 8R(N'F)' j-1 (A3) a M' E F'?FJI-'-M, i=o M' = [I ] and is of dimension g(p + r) X g2(p + r)2. It can also be shown that ar(nf) {K,FK2. = Km} Kj = EJ - Pi and Pi = {(I C,)(A>1 I)}. The three null matrices in Pi span g2(] _ i- 1), g{k + g(p -j + 2i - 1)} and g2(r - i) columns respectively. The second matrix derivative in (A2) is found to be a(i xd). JS., ao = {?Q1, QPJsl,.,spo) the first null matrix is g(p + r) x g2p and Qi is composed of p submatrices, all of which are null matrices of order g(p + r) x g2; except the ith submatrix which is b1 b2 bg bjis the]jth row of B.

7 SIMULTANEOUS EQUATION MODEL 1337 The matrix J is defined as J= i is a 1 x g row vector composed entirely of ones and S. = 1 a (i)l a (i)2 a(i)p a ijis the jth column vector of A,. Finally, the last null matrix is of order g(p + r) X g2r. REFERENCES [1] BAILLIE, R. T.: "The Asymptotic Mean Squared Error of Multistep Prediction from the Regression Model with Autoregressive Errors," Journal of the American Statistical Association, 74(1979), [2] "Asymptotic Prediction Mean Squared Error for Vector Autoregressive Models," Biometrika, 66(1979), [3] NEUDECKER, H.: "Some Theorems on Matrix Differentiation with Special Reference to Kronecker Matrix Products," Journal of the American Statistical Association, 64(1969), [4] SCHMIDT, P.: "The Asymptotic Distribution of Forecasts in the Dynamic Simulation of an Econometric Model," Econometrica, 42(1974), [5] "Some Small Sample Evidence on the Distribution of Dynamic Simulation Forecasts," Econometrica, 45(1977), [6] YAMAMOTO, T.: "Asymptotic Mean Square Prediction Error for an Autoregressive Model with Estimated Coefficients," Journal of the Royal Statistical Society, C, 25(1976),