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6.3. TESTING EULER EQUATIONS 179getµ Ã ! Ã !Ft − S t+1 PtCtln+ln = −γ ln +lnw t+1 .S t P t+1 C t+1ln(C t /C t+1 ) is correlated with ln w t+1 so you don’t get consistent estimateswith OLS—but you do get consistency with instrumental variablesand this is what GMM does. However, the regression analogytells us that the large estimate of γ and its large standard error canbe attributed to high variability in the excess return combined withlow variability in consumption growth. The difficulty that the Lucasmodel under CRRA utility to explain the data with small values of γ isnot conÞned to the foreign exchange market. The corresponding difficultyfor the model to simultaneously explain historical stock and bondreturns is what Mehra and Prescott [105] call the ‘equity premiumpuzzle’.Volatility BoundsHansen and Jagannathan [72] propose a framework to evaluate the extentto which the Euler equations from representative agent asset pricingmodels satisfy volatility restrictions on the intertemporal marginalrate of substitution.We will Þrst derive a lower bound on the volatility of the intertemporalmarginal rate of substitution predicted by the Euler equations ofthe intertemporal asset pricing model. Let r t+1 be an N-dimensionalvector of holding period returns from t to t+1 available to the agent,and µ t+1 = βu 0 (C t+1 )/u 0 (C t ) be the intertemporal marginal rate of substitution.We need to write the Euler equations in returns form. For equities,they take the form 1 = E t (µ t+1 rt+1) e wherert+1 e is the gross return. 6 Itreads—an asset with expected payoff E t (µ t+1 rt+1) e costsoneunitoftheconsumption good. An analogous returns form of the Euler equationholds for bonds. In the case of forward foreign exchange contracts, thereis no investment required in the current period so the returns form for(Fthe Euler equation is 0 = E t µ t−S t+1 )t+1 P t. Thus, the returns form of6 Take the equity Euler equation (4.12)) and divide both sides by e t u 1,t+1 . Letr e t+1 =(e t+1 + x t+1 )/e t to get the expression in the text.