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6.3. TESTING EULER EQUATIONS 181The term labeled (a) above is zero because u t is the least-squares projectionerror and is by construction orthogonal to r t .Sinceσ 2 u ≥ 0, thevolatility or standard deviation of the intertemporal marginal rate ofsubstitution must lie above σ r whereσ µ ≥ σ r ≡q(v − θ µ θ r ) 0 Σ −1r (v − θ µθ r ). (6.27)The right side of (6.27) is the lower bound on the volatility of theintertemporal marginal rate of substitution. If the assets are all equitiesor bonds v is a vector of ones and the volatility bound is a parabola in(θ µ , σ µ ) space. If the assets are all forward foreign exchange contracts,v is a vector of zeros and the lower volatility bound is a ray from theoriginh iσ r = θ µ θ0rΣ −1 1/2r θ r . (6.28)How does one construct and use the volatility bound in practice?First determine v and calculate θ r and Σ r from asset price data. Thenusing (6.28) you trace out σ r as a function of θ µ . Next, for a given functionalform of the utility function, use consumption data to calculatethe volatility of the intertemporal marginal rate of substitution, σ µ .Compare this estimate to the volatility bound and determine whetherthe bound is satisÞed.When we do this using quarterly US consumption and CPI dataand dollar exchange ratesqfor the pound, deutschemark, and yen from1973.1 to 1997.1, we get θ 0 rΣ −1 θ r =0.309. Now let the utility function ⇐(113)be CRRA with relative risk aversion coefficient γ. As we vary γ, wegenerate the entries in the following table.γ θ µ σ µ σ r2 0.982 0.015 0.3034 0.974 0.031 0.30110 0.953 0.078 0.29420 0.923 0.159 0.28530 0.901 0.248 0.27840 0.886 0.349 0.27350 0.879 0.469 0.27260 0.881 0.615 0.272