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As Campbell and Shiller (1987) point out, if the discount factor is constant, the bivariate VAR includes all of the relevant information (because Liabilities/GDP is the expected present value of future Surplus/GDP). Of course, discount factors need not be constant so we also include the discount factor and consider a three-variable system. This allows us to determine whether the effect of an innovation to st on wt+1 is robust to conditioning on a, the discount factor, and to investigate the effect of innovations to st on future discount factors. Figure 4 shows impulse response functions from a VAR with a constant and two lags.

Once again the results are quite robust: adding a deterministic time trend, or using lag lengths of one, three or four, we get similar pictures. The VAR’s residuals are highly correlated; we show what would seem to be the two relevant orderings. In the top panel, Surplus/GDP comes first (as may make more sense in a FD regime), followed by Liabilities/GDP and a; in the bottom panel, Liabilities/GDP comes first (as may make more sense in a MD regime), followed by Surplus/GDP and a. The dashed lines represent the two standard deviation bands obtained by a Monte Carlo simulation with 500 repetitions.

Our basic results are robust to conditioning on Alpha. The response of Liabilities/GDP in periods 2 and 3 to an innovation to Surplus/GDP in period 1 is again negative and significant. In addition, the response of Surplus/GDP to that same innovation shows some persistence, but as before it is short lived. The new information in Figure 4 is the response of a to an innovation in Surplus/GDP. The initial impact period 1 is negative, and it appears to be significant. However, we are interested in the correlation between innovations in Surplus/GDP and future discount factors. In Figure 4, the response of future discount factors is quite small and statistically insignificant. Current innovations in Surplus/GDP appear to be uncorrelated with future discount factors.