Estimating and Identifying Vector Autoregressions Under Diagonality and Block Exogeneity Restrictions

Transcription

1 Estimating and Identifying Vector Autoregressions Under Diagonality and Block Exogeneity Restrictions William D. Lastrapes Department of Economics Terry College of Business University of Georgia Athens, GA March 26, 2004 Abstract: I show how to estimate and identify a large-scale vector autoregression when the variables in a subset of the system are mutually independent after conditioning on a separate set of variables diagonality, and when the conditioning variables are independent of the former subset block exogeneity. Least squares estimation is efficient, and restrictions only on the set of common variables are sufficient to fully identify the economic structure. This approach will be most useful when using VARs to estimate the responses of a cross-section of variables, such as industry-level output or prices, to aggregate shocks. Keywords: VAR, impulse response functions, time-series JEL classification: C32

2 1. Introduction Vector autoregressions VARs are effective tools for analyzing the dynamics of a stochastic system, and for making economic inference. However, data constraints typically mean that large dimensional systems suffer from insufficient degrees of freedom and thus lack robustness. This note shows how to estimate and identify large-scale structural VARs when a the mutual correlation among a subset of the variables in the system is due solely to joint dependence on a separate subset variables, and b when the latter subset is independent of the former. The assumption of mutual independence conditional on a set of common factors imposes a diagonal structure on part of the VAR, while the second assumption implies that the common factors are block exogenous. I show that least squares methods are efficient, and that identification of the structural dynamic impulse responses of the model requires restrictions only on the subset of common variables. The approach will be most useful when it is desirable to use VAR models to examine the dynamic responses of a large cross-section of variables, such as industry-level prices or output, to aggregate shocks. 2. The model z1t Let z t = be an n-dimensional vector stochastic process, where z 1t is n 1 1, z 2t is n 2 1, and z 2t n = n 1 + n 2. Assume that this process is generated by the linear dynamic model: where u t = u1t u 2t A 0 z t = A 1 z t A p z t p + u t, 1 is a white noise vector process normalized so that Eu t u t = I, and A i, i = 0,..., p, is n n. The corresponding reduced form of this structural model is z t = A 1 0 A 1z t A 1 0 A pz t p + A 1 0 u t = B 1 z t B p z t p + ɛ t, Eɛ t ɛ t Ω. 2 The system in 2 is the VAR representation of the structural model in 1. The moving average representation of the structural model is z t = A 0 A 1 L... A p L p 1 u t 3a = D 0 + D 1 L + D 2 L u t 3b = DLu t. 3c Likewise, the reduced form moving average is z t = I B 1 L... B p L p 1 ɛ t 4a = I + C 1 L + C 2 L ɛ t 4b = CLɛ t. 4c 1

3 The objective is to identify the economic structure in 3 from the moving average in 4, which is directly determined by estimating the coefficients in 2. In particular, the parameters of interest are the structural dynamic multipliers or impulse response functions: z t+k u t = D k. The empirical strategy entails estimating BL and Ω from 2, then imposing restrictions on the structure to identify the parameters of interest from these estimates. 3. Estimation When n is large, estimating the unrestricted VAR in 2 may not be feasible for a reasonable lag structure, because of a lack of observations and degrees of freedom. Suppose for example that we are interested in estimating the dynamic effects of aggregate shocks on industry-level prices or output as in, for example, Loo and Lastrapes Consider then a VAR containing these variables z 1 and a standard set of macro variables z 2. A typical post-war sample of 40 years will contain 160 quarterly observations, so with, say, n 1 = 80 industries, estimating this VAR without restrictions will not be feasible for more than one common lag for each variable. However, consider two sets of over-identifying restrictions on the VAR: 1 the variables in z 1 are mutually independent after conditioning on z 2 ; and 2 z 2 is block exogenous with respect to z 1. The first set of restrictions imposes a diagonal structure on the relations among the variables in z 1, and fully accounts for the observed correlation across these variables through their joint dependence on the common factors, z 2. The plausibility of this assumption rests on the ability to obtain a sufficient set of common factors. The block exogeneity assumption means that z 2 is determined independently of z 1, which is surely plausible if z 2 contains aggregate variables and z 1 individual market or industry variables. Indeed, any strictly macro model necessarily makes such an assumption. I will show below that under these conditions, efficient estimation can be achieved by least squares applied equation-by-equation, and that the structural model is fully identified by restrictions only on z 2. In general, this restricted system may be feasible for large n 1 as long as n 2 is relatively small. 1 To clarify the restrictions and to note their effect on estimation, partition the coefficient matrices in equations 1 through 4 according to X h X h = 11 X12 h X21 h X22 h 5 for X = A, B, C, and D, where X h ij has dimension n i n j for all h and i, j = 1, 2. Partition the reduced 1 A typical strategy for estimating the effects of aggregate shocks on a large cross-section of individual variables, exemplified in Carlino and Defina 1998 who examine the response of US regions and states to monetary policy, is to estimate separate VARs for each element in the cross-section state, where each VAR contains specific cross-sectional variables and aggregate variables and the block-exogeneity assumption is not imposed. Clearly, this approach can be misleading since the estimation and identification of aggregate shocks under this strategy will vary across the separate VARs. 2

4 form covariance matrix Ω conformably: Ω = E ɛ1t ɛ 2t ɛ 1t ɛ 2t = Ω11 Ω 12 Ω 12 Ω The assumption of mutual independence restricts the n 1 n 1 matrix A h 11 to be diagonal, while block exogeneity restricts the n 2 n 1 matrix A h 21 to be 0, for h = 0, 1,..., p. Inverting A 0 implies A 1 0 = A A A 0 12A A so that both sets of restrictions carry over to the inverse. Furthermore, from 2,, 7 B i 11 = A A i 11, B i 21 = 0, i = 1,..., p; i = 1,..., p 8 the VAR coefficient matrices are similarly restricted. These restrictions imply that the VAR in 2 can be expressed as z1t z 2t = B i 11 B12 i z1t i i=1 0 B i 22 z 2t i + ɛ1t ɛ 2t, 9 where B i 11 is diagonal. In general, the conventional method of estimating each equation by least squares is inefficient for the restricted VAR in 9, since the right-hand-side variables differ across equations due both to the diagonality and block-exogeneity restrictions. However, taking advantage of block-exogeneity allows 9 to be re-parameterized and separated into independent parts Hamilton, 1994, pp : z 1t = z 2t = B11z i 1t i + G i z 2t i + v t i=1 i=0 B22z i 2t i + ɛ 2t, i=1 10a 10b where G 0 = Ω 12 Ω a G i = B i 12 G 0 B i 22, i = 1,..., p 11b Ev t v t H = Ω 11 Ω 12 Ω 1 22 Ω c Because the parameters in 10a are independent of those in 10b, estimating each sub-system separately is fully efficient. For the z 2 sub-system, this strategy simply entails estimating by least squares an unrestricted n 2 -dimensional VAR. Recall from 8 that B11 i is a diagonal matrix, so that the equation for each variable in 10a contains only its own lagged values as well as z 2 and its lags. It would thus appear that a full-information estimation 3

5 strategy is required for efficient estimation of 10a. However, given both sets of restrictions, the covariance matrix from the z 1 sub-system, H in 11c, is diagonal. To see this, note from 2 and the normalization of Eu t u t that Then 6 and 7 imply, ɛ t = A 1 0 u t, 12 Ω = A 1 0 A Ω 11 = A A A A 0 12A A A 0 12 A Ω 12 = A A 0 12A A Ω 22 = A A Finally, it can be shown by using 14 in 11c that H = A A which is diagonal. Therefore, there is no efficiency gain from generalized estimation methods; least squares applied to each equation in 10a individually is equivalent to a full-information approach and is thus efficient Theil, 1974, p In sum, I have shown that estimating the parameters in 10 by least squares for each equation is efficient. It is straightforward to use the mapping in 11 to determine the original VAR parameterization, BL and therefore CL and Ω. 4. Identification I now consider identifying the structure from the reduced form estimates obtained in the previous section. Using 12 in 4b, and comparing to 3b, it follows immediately that D 0 = A D i = C i D 0 for i = 1, 2, Partition D 0 as in 5, then use the partitioned expression of A 1 0 in 7 to obtain D 0 11 = A D 0 21 = 0; 18 thus, D 0 inherits the same diagonal upper-left partition and zero lower-left partition as A 0. Finally, using Hamilton 1994, p. 260, the solution to the inverted lag polynomial in 4a is C 0 = I C i = B 1 C i B p C i p i = 1, 2,

6 It is apparent that all C i will have the same restrictions as the VAR coefficient matrices, as will all D i from 16. Now, substitute 15 into 13 to get Ω = D 0 D 0, then partition the right-hand-side of this expression using 18: Ω11 Ω 12 Ω = 12 Ω 22 D 0 11 D D 0 12D 0 12 D 0 12D 0 22 D 0 22D 0 12 D 0 22D 0 22 Because of the block exogeneity of z 2, D 0 22 can be identified solely from the lower-right block of 20. For example, if D 0 22 is assumed to be lower triangular, then it is just-identified as the Cholesky factor of Ω 22, which is estimated directly from the z 2 sub-system and independently of z 1 ; see 6 and 10b. Once D 0 22 is identified, the upper-right matrix in 20 implies 2 20 Ω 12 = D 0 12D 0 22 D 0 12 = Ω 12 D Finally, from the upper-left matrix in 20: D 0 11D 0 11 = Ω 11 D 0 12D Ω 11 is directly estimated from 10 and 11, while D 0 12 is identified from 21. But since D 0 11 is diagonal from 17, each of its elements is just-identified as the positive square root of each of the elements of the known matrix on the right-hand-side of D 0 is now fully-identified; the entire set of structural impulse response functions can then be inferred from 16. I emphasize that no identifying restrictions on D 0 12 are necessary to identify how the system responds to the entire set of shocks. It is straightforward to show that infinite-horizon restrictions e.g. Blanchard and Quah 1989 from the z 2 sub-system are also sufficient to identify the dynamics of the full system. Suppose that z t = y t ; then the long-run multipliers of the levels are: y t+k lim = D1 = k u t i=0 Di 11 0 i=0 Di 12 i=0 Di 22 = D11 D12 0 D22, 23 where D 22 contains the long-run multipliers from the z 2 sub-system, and D 11 is clearly diagonal. From 16 and the mapping from D 0 to Ω: D1 = C1D 0 24 D1D1 = C1D 0 D 0C1 25 = C1ΩC D 0 22 must be fully identified, so the partial identification strategy based on the block recursive structure of Christiano, Eichenbaum and Evans 1999 is not sufficient for identification in this case. 3 If z 1 exhibits block diagonality e.g. z 1 contains industry-level prices and output, the implications for estimation are unchanged. However, additional identifying restrictions will be needed to sort out the structural parameters within each individual block. 5

7 Expanding 26 to express partitions, and noting the restrictions on C1, we have: D11 D 11 + D 12 D 12 D12 D 22 C11 C12 Ω11 Ω = 12 C 11 0 D 22 D 12 D22 D 22 0 C22 Ω 12 Ω 22 which implies C 12 C 22, 27 D 22 D 22 = C 22 Ω 22 C Thus, if we impose sufficient conditions on D 22, such as lower triangularity, it can be identified from the long-run covariance matrix of the reduced form in 28. Finally, once D 22 is known, from 28 and 20 Ω 22 = C 1 22 D 22 D 22 C 22 1 = D 0 22D Since Ω 22 is known from estimation, D 0 22 = C 1 22 D From this point, 21 and 22 identify D 0 12 and D 0 11 as before, and 16 yields the entire set of structural parameters. 5. Conclusion I have shown how to restrict a large-scale VAR to provide a consistent and feasible framework for estimating and identifying the dynamic effects of common e.g. aggregate shocks on a possibly large set of micro variables, such as individual markets, industries, states, or countries in a world-wide context. The key assumption is that a small set of common exogenous variables exist that can account for the correlations among the micro variables. Least squares is efficient for estimation, and identifying restrictions are minimal in that the structure of the micro variables need not be further constrained. References Blanchard, O. J., Quah, D., 1989, The Dynamic Effects of Aggregate Demand and Supply Disturbances. American Economic Review 79, Christiano, L. J., Eichenbaum, M., Evans, C.L., 1999, Monetary Policy Shocks: What Have We Learned and to What End? in: J. B. Taylor and M. Woodford, eds., Handbook of Macroeconomics, Vol. 1A North-Holland, Amsterdam Carlino, G., Defina, R., 1998, The Differential Regional Effects of Monetary Policy, Review of Economics and Statistics 80, Hamilton, J. D., 1994, Time Series Analysis Princeton University Press, Princeton. Loo, C. M., Lastrapes, W.D., 1998, Identifying the Effects of Money Supply Shocks on Industry-Level Output, Journal of Macroeconomics 20, Theil, H., 1971, Principles of Econometrics John Wiley & Sons, New York. 6