Structural Approaches to Vector Autoregressions

Keating asks how structural restrictions can transform an estimated reduced-form VAR into a system of structural equations, presenting the paper as “an introduction to this developing literature” in structural VARs (Introduction, p. 37). He sets up a simultaneous-equations model Ax_t = C(L)x_{t-1} + Dz_t (Eq. 1), where A holds contemporaneous structural coefficients and D maps structural disturbances z_t into equations, and shows this model’s reduced form is exactly the estimable VAR x_t = β(L)x_{t-1} + e_t (Eq. 5), with reduced-form residuals related to structural shocks by e_t = A^{-1}Dε_t. The VAR “is shown to be a reduced-form for a linear simultaneous equations model,” making the identification problem one of recovering A and D from the data on e_t alone.

Identification requires restrictions on A, D, and the structural shock covariance matrix Σ_ε, since the reduced-form covariance matrix alone, Σ_e = A^{-1}DΣ_εD’A^{-1’} (Eq. 7), does not pin down these components uniquely. Keating normalizes the diagonal of Σ_ε to unity (independent, unit-variance structural shocks) and the diagonal of A to 1 (each equation normalized on one endogenous variable); these 2n normalizations still leave at least 3n(n-1)/2 additional exclusion or sign restrictions on A and D to be supplied by economic theory (pp. 39-40). The Choleski decomposition is described as a special case of this framework — “a recursive contemporaneous structure” — but the paper notes most economic theories do not actually imply a recursive ordering, and atheoretical Choleski shocks are linear combinations of the true structural disturbances, ν_t = R^{-1}A^{-1}Dε_t, which the paper (citing Cooley and LeRoy 1985) says makes such impulse responses difficult to interpret structurally because the ordering is chosen without theoretical justification among n! possibilities (pp. 42-43).

The illustrative contemporaneous model (Eqs. 8-11, pp. 40-41) uses the same 4-variable system — GNP deflator, real GNP, the 3-month T-bill rate, and M1, first-differenced, 1959:Q1-1991:Q3, 4 lags — and specifies: the price level as predetermined except for aggregate-supply shocks; a reduced-form IS equation for output as a function of price, interest rate, money, and an IS shock; a money-supply reaction function for the interest rate as a function of money supply; and a buffer-stock money-demand equation for nominal money as a function of output, the interest rate, lagged real money, and a money-demand shock. The paper reports this model’s estimated coefficients, impulse responses, and variance decomposition later in the article (Table 3, p. 51; Figures 5-8, pp. 52-55; Table 4, p. 56), covered below in Q8-Q10.

Long-run restrictions are imposed on the cumulative long-run multiplier matrix θ(1) = [I - β(1)]^{-1}A^{-1}D rather than on the contemporaneous impact matrix, which the paper states as an advantage because it “does not impose constraints on contemporaneous responses, allowing the data to determine short-run dynamics” (p. 42). Using [I - β(1)]^{-1}Σ_e[I - β(1)]^{-1’} = θ(1)Σ_εθ(1)’ (Eq. 17), Σ_ε and θ(1) can be recovered from the estimated reduced-form covariance matrix. In the illustrative long-run model (Eqs. 18-21, pp. 44-45), aggregate-supply shocks are restricted to be the sole source of permanent movements in output — the Blanchard and Quah (1989) restriction, generating three long-run zero restrictions — alongside a long-run IS equation (interest rate as a function of output), a long-run money-demand function, and a money-supply equation.

The estimated structural shock standard deviations are all statistically significant at the 5 percent level: aggregate supply 0.0144 (s.e. 0.0041), IS 0.0092 (s.e. 0.0028), money demand 0.0149 (s.e. 0.0031), and money supply 0.0172 (s.e. 0.0042) (Table 1, p. 45). Among the long-run structural coefficients, the money-demand coefficient on the interest rate (S3 = -2.276) and the money-supply coefficient on real money (S4 = -1.411) and on output (S6 = 0.9048) are significant at 5 percent, while the long-run IS coefficient on output (S1 = -0.1171) is not significantly different from zero. Keating notes the money-demand coefficients have the expected signs and that the money-supply parameters “can be interpreted as a Fed reaction function” in which the Fed reduces money when output rises but raises it when interest rates rise (pp. 45-46).

Impulse responses (with 90 percent Runkle 1987 Monte Carlo confidence bands) show aggregate-supply shocks raising output permanently and monotonically while lowering prices and slightly raising the interest rate (Figure 1, p. 46); IS shocks raising output only temporarily (by construction, output returns to zero in the long run) while raising prices and the interest rate permanently and lowering money (Figure 2, p. 47); money-demand shocks having virtually no effect on output, prices, or interest rates but a strong positive effect on nominal and real money, which the paper describes as “consistent with models where the Fed targets the interest rate rather than money” (Figure 3, p. 48); and money-supply shocks having a “gradual effect on output that peaks at 13 percent of the variance two years in the future,” which the paper flags as a “relatively small effect on output” that may be surprising, attributing it to the long-run model’s restriction that only supply shocks can explain permanent output movements (Figure 4, p. 49, and p. 46). Variance decompositions (Table 2, p. 50) show supply shocks dominating output variance at long horizons (90 percent at 48 quarters, versus 17 percent at horizon 1), IS shocks dominating interest-rate variance (95 percent at 48 quarters), money-demand shocks dominating real-money variance at short horizons (90 percent at horizon 1), and supply shocks dominating price-level variance at all reported horizons (73 percent at horizon 1, 54 percent at 48 quarters).

Keating notes in footnote 23 that the restriction identifying the long-run model — that aggregate-supply shocks are the sole source of permanent output movements — is an a priori identifying assumption, not a testable implication, and may be misspecified if interest-rate effects on capital accumulation create a second source of permanent output movements, in which case the estimated money-supply and money-demand shocks would actually be mixtures of the true structural disturbances. The paper also assumes stationarity is achieved by first-differencing (footnote 26), which the long-run model requires because it treats all shocks as having permanent effects. The paper’s own empirical comparison of this long-run model against the contemporaneous model (Q8-Q11 below) is where these caveats are put to the test.

Table 3 (p. 51) reports ten estimated structural coefficients for the contemporaneous model, and — in sharp contrast with the long-run model’s Table 1, where every shock standard deviation was significant at 5 percent — Keating states plainly that “each of these structural parameters is statistically insignificant” (p. 51). Several point estimates nonetheless carry theoretically sensible signs and magnitudes: in the money-demand equation the coefficient on nominal spending is “roughly one-half” and the interest-rate coefficient is “almost -1.0,” which Keating says makes “the parameter estimates in this structural model… consistent with economic theory” despite their individual insignificance; in the money-supply (interest-rate) equation the coefficient on money is positive, which the paper reads as supporting “the view that the central bank attempts to stabilize money growth by raising interest rates” (p. 51). By contrast, the IS equation’s coefficient on money is positive, which the paper flags as a sign “unexpected in a structural IS equation” (p. 51). The paper’s Concluding Remarks later summarize this contrast directly: “structural parameters in the long-run model are more precisely estimated than parameters in the contemporaneous model” (p. 51).

Because the aggregate-supply equation in the contemporaneous model is normalized on the price level rather than on output (unlike the long-run model), “an aggregate supply shock raises the price level and reduces output”; real money also decreases, with a “weak positive effect on money and the interest rate” (Figure 5, p. 52). An IS (“real spending”) shock “raises prices, output and the interest rate,” with real and nominal money initially increasing before subsequently falling (Figure 6, p. 53). Because the money-supply equation is normalized on the interest rate, a money-supply shock “raises the interest rate and causes a decline in nominal money, real money and the price level,” and — a result the paper calls “surprising” — “output rises briefly before it begins to decline” (Figure 7, p. 54). A money-demand shock “causes the interest rate, nominal and real money to increase while output falls,” with an accompanying rise in the price level that the paper calls “inconsistent with theory, although this effect is not statistically significant” (Figure 8, p. 55). Keating’s overall assessment is that “in contrast with the long-run model, there are a few unusual features in the impulse responses for the contemporaneous specification,” although “most of the dynamic patterns… are consistent with the structural model” (p. 51).

Table 4 (p. 56) reports variance decompositions for the contemporaneous model, and Keating explicitly compares them with the long-run model’s Table 2 results (Q6), finding both similarities and differences (p. 51). On the similarities: the aggregate-supply shock “gradually explains most of output’s variability, is the most important shock for the price level and is never an important source of interest rate movements,” while the IS shock “is the most important source of short-run output movement” and “explains most of the long-run variance of the interest rate” — echoing the long-run model’s pattern of supply-shock dominance for output and prices and IS-shock dominance for the interest rate. On the differences: the money-demand shock “has its greatest effect on output in the long run” and “explains a large amount of the short-run variance of the interest rate but virtually none of the long-run variance of real or nominal money balances,” while the money-supply shock “has essentially no effect on output, while accounting for a large amount of the variance in real money, even in the long run, and none of the variance of prices.” Keating states that “these results are inconsistent with most macroeconomic theories” (p. 51) — unlike the long-run model, where money-demand shocks dominated short-run real-money variance and money-supply shocks had the modest, delayed output effect described in Q6.

In its Concluding Remarks, Keating states directly that “wherever a significant discrepancy exists between the two models, the model with long-run restrictions yields sensible results, while the results from the contemporaneous model are inconsistent with standard economic theories” (p. 51), attributing this partly to the long-run model’s structural parameters being “more precisely estimated” (p. 51). He cautions that “these comparisons between contemporaneous and long-run specifications may not generalize to all structural VAR applications,” but suggests they indicate “long-run structural VARs may yield theoretically predicted results more frequently than VARs identified with short-run restrictions” (p. 57). He offers two candidate explanations: first, economic theories “may often have similar long-run properties but different short-run features” — e.g., output movements are driven solely by aggregate-supply shocks in a typical real-business-cycle model, whereas in Keynesian models supply shocks account for permanent output movements but “every shock may have some cyclical effect”; second, long-run structural VARs “typically do not impose contemporaneous exclusion restrictions,” which Keating (1990, Journal of Monetary Economics) argues can be inappropriate “in an environment with forward-looking agents who have rational expectations,” since any observable contemporaneous variable may carry information about future events (p. 57). He closes by calling for “further research” into other contemporaneous and long-run applications “to determine whether the superior performance of this paper’s long-run model is a special case or a more general result” (p. 57).