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6.6. NOISE-TRADERS 205and let ˆθ be a consistent estimator of θ 0 such that√T (ˆθ − θ0 ) D → N(0, Σ θ ).Assume that consistent estimators of both θ 0 and Σ θ are available.Now make explicit the fact that the moments of the intertemporalmarginal rate of substitution and the volatility bound depend on sampleinformation. The estimated mean and standard deviation of predictedby the model are, ˆµ µ = µ µ (φ; ˆψ) andˆσ µ = σ µ (φ; ˆψ), while theestimated volatility bound isrˆσ r = σ r (φ; ˆθ) =³ˆµq− µ µ (φ; ˆψ)ˆµ r´0 ˆΣ−1 r³ˆµ q− µ µ (φ; r´ ˆψ)ˆµ .Let∆(φ; ˆθ) =σ M (φ; ˆψ) − σ r (φ; ˆθ),be the difference between the estimated volatility bound and the estimatedvolatility of the intertemporal marginal rate of substitution.Using the ‘delta method,’ (a Þrst-order Taylor expansion about thetrue parameter vector), show that√T (∆(φ; ˆθ) − ∆(φ; θ0 )) → D N(0, σ∆),2whereµ µ ∂∆∂∆σ∆ 2 =∂θ 0 (ˆθ − θ 0 )(ˆθ − θ 0 ) 0 .θ 0∂θ θ 0How can this result be used to conduct a statistical test of whether aparticular model attains the volatility restrictions?6. (Peso problem). Let the fundamentals, f t = m t −m ∗ t −λ(y t −yt ∗ ) followthe random walk with drift, f t+1 = δ 0 + f t + v t+1 ,wherev t ∼ iid withE(v t )=0andE(vt 2 )=σv. 2 Agents know the fundamentals processwith certainty. Forward iteration on (6.33) yields the present valueformula∞Xs t = γ E t (f t+j ).j=1Verify the solution (6.38) by direct substitution of E t (f t+j ).Now let agents believe that the drift may have increased to δ = δ 1 .Show that E t (f t+j )=f t + j(δ 0 − δ 1 )p 0t + jδ 1 . Use direct substitutionof this forecasting formula in the present value formula to verify thesolution (6.43) in the text.