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6.3. TESTING EULER EQUATIONS 177This is why market efficiency does not mean that the exchange ratemust follow a random walk or that uncovered interest parity must hold.The Lucas model predicts that in equilibrium, it is the marginal utilityof the forward contract payoff that is unpredictable and that deviationsfrom UIP can emerge as compensation for risk bearing.6.3 Testing Euler EquationsUsing the methods of Hansen and Singleton [73], Mark [100] estimatedand tested the Euler equation restrictions using 1-month forward exchangerates. Modjtahedi [106] goes a step further and tests impliedEuler equation restrictions across the entire a term structure availablefor forward rates (1, 3, 6, and 12 months). The strategy is to estimatethe coefficient of relative risk aversion γ and test the orthogonalityconditions implied by the Euler equation (6.11) using GMM.Here, we use non-overlapping quarterly observations on dollar ratesof the pound, deutschemark, and yen from 1973.1 to 1997.1 and revisitMark’s analysis. To write the problem compactly, let r t+1 be the 3 × 1forward foreign exchange payoff vectorr t+1 =and let the 3 × 1vectorw t+1 be⎡⎢⎣(F 1t −S 1t+1 )(S 1t )(F 2t −S 2t+1 )(S 2t )(F 3t −S 3t+1 )(S 3t )⎤⎥⎦ ,w t+1 = µ m t+1r t+1 , (6.17)where µ m t+1 is the US representative investor’s intertemporal marginalrate of substitution of money under CRRA utility, u(C) =C 1−γ /(1−γ).Using the notation developed here to rewrite the Euler equations(6.11) you getE[w t+1 |I t ]=0. (6.18)Divide both sides by β sothatyouonlyneedtoestimateγ. (6.18)says that w t+1 is uncorrelated with any time-t information. Let z tbe a k−dimensional vector of time-t ‘instrumental variables,’ available