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178CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYto you, the econometrician. Then (6.18) implies the following 3 × kestimable and testable equations 4E[w t+1 ⊗ z t ]=E[(µ m t+1 r t+1) ⊗ z t ]=0. (6.19)Now the question is what to choose for z t ? It is not a good idea touse too many variables because the estimation problem will becomeintractable and the small sample properties of the GMM estimator willsuffer. A good candidate is the forward premium since we know thatit is directly relevant to the problem at hand. Furthermore, it is notnecessary to use all the possible orthogonality conditions. To reducethe dimensionality of the estimation problem further, for each currencyi, letz it ="#1(F it −S it )S itbeavectorofinstrumentalvariablesconsistingoftheconstant1,andthe normalized forward premium. Estimating γ from the system of sixequations⎡⎤w 1t+1 z 1t⎢⎥E ⎣ w 2t+1 z 2t ⎦ , (6.20)w 3t+1 z 3tgives ˆγ =48.66 with asymptotic standard error of 79.36. The coef-Þcient of relative risk aversion is uncomfortably large and impreciselyestimated. However, the test of the Þve overidentifying restrictionsgives a chi-square statistic of 7.20 (p-value=0.206) does not reject atstandard levels of signiÞcance.Why does the data force ˆγ to be so big? We can get some intuitionby recasting the problem as a regression. Suppose you look at justone currency. If ( C t PC t+1, tP t+1, F t−S t+1S t) are jointly lognormally distributedthen w t+1 is also lognormal. 5 Taking logs, of both sides of (6.17), youµ 4 a11 a⊗ denotes the Kronecker product. Let A = 12and B be any n × ka 21 aµ 22a11 B amatrix. Then A ⊗ B =12 B.a 21 B a 22 B5 A random variable X is said to be lognormally distributed if ln(X) is normallydistributed.