5. Testing the volatility restrictions (Cecchetti et. al. [25]). This exercise develops the volatility bounds analysis so that we can do classical statistical hypothesis tests to compare the implied volatility of the intertemporal marginal rate of substitution and the lower volatility bound. Begin defining φ as a vector of parameters that characterize the utility function, and ψ as a vector of parameters associated with the stochastic process governing consumption growth. Stack the parameters that must be estimated from the data into the vector θ θ =   µr vec(Σr) ψ   , where vec(Σr) is the vector obtained by stacking all of the unique elements of the symmetric matrix, Σr. Let θ0 be the true value of θ,and let àθ be a consistent estimator of θ0 such that √ T(àθ − θ0) D → N(0, Σθ). Assume that consistent estimators of both θ0 and Σθ are available. Now make explicit the fact that the moments of the intertemporal marginal rate of substitution and the volatility bound depend on sample information. The estimated mean and standard deviation of predicted by the model are, ൵ = µµ(φ;ψà) and àσµ = σµ(φ; ψà), while the estimated volatility bound is σàr = σr(φ; àθ) = r³ µàq − µµ(φ;ψà)àµr ¥0 Σà −1 r ³ µàq − µµ(φ; ψà)àµr ¥ . Let ∆(φ; àθ) = σM(φ;ψà) − σr(φ; àθ), be the difference between the estimated volatility bound and the estimated volatility of the intertemporal marginal rate of substitution. Using the ëdelta method,í (a first-order Taylor expansion about the true parameter vector), show that √ T(∆(φ; àθ) − ∆(φ; θ0)) D → N(0, σ2 ∆), where σ2 ∆ = µ∂∆ ∂θ0 ¶ θ0 (àθ − θ0)(àθ − θ0) 0 µ∂∆ ∂θ ¶ θ0 . How can this result be used to conduct a statistical test of whether a particular model attains the volatility restrictions?