nonstationary. Again, following Durlauf and Hall ( 1989c), if current and laggedand are used in the ht+1 projection, then the explosive part of the filtered noise will be recovered. However, this means that the ht+1 projection itself is nonzero. Therefore, the ht+1 projection will detect a deviation from the null model for any type of nonstationary noise except a rational bubble. Theorem 3 is therefore verified.
Theorem 2 is an implication of Theorem 1 and can be verified by a straightforward generalisation of Theorem 1.3 in Durlauf and Hall ( 1989c). As before, all projections are well defined so long as the histories of Pt and Mt are included in Lt(x).
To verify Theorem 4, observe that when prices obey, Δht+1 is an MA(1) process. The MA(1) coefficient is unrestricted as the model contains no implications for the variance/covariance of ζt, ξt, and ut. By a straightforward generalisation of Theorem 1, Δht+1 must be orthogonal to Lt-1(x).
Under the misspecification alternative, Δht+1 equals
Following the argument given in the proof of Theorem 1, the only case where the projection of Δht+1 onto Lt-1 (x) is zero while the projection ofonto Lt-1(x) is nonzero occurs when Nt is a rational bubble. This verifies Theorem 4.
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