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6.1. DEVIATIONS FROM UIP 163Hansen and Hodrick’s Tests of UIPHansen and Hodrick [71] use generalized method of moments (GMM)to test uncovered interest parity. The GMM method is covered inchapter 2.2. The Hansen—Hodrick problem is that a moving-average serialcorrelation is induced into the regression error when the predictionhorizon exceeds the sampling interval of the data.The Hansen—Hodrick ProblemToseehowtheproblemarises,letf t,3 be the log 3-month forward exchangerate at time t, s t be the log spot rate, I t be the time t information ⇐(102)setavailabletomarketparticipants,andJ t be the time t informationset available to you, the econometrician. Even though you are workingwith 3-month forward rates, you will sample the data monthly. Youwant to test the hypothesisH 0 :E(s t+3 |I t )=f t,3 .In setting up the test, you note that I t is not observable but since J t isasubsetofI t and since f t,3 is contained in J t , you can use the law ofiterated expectations to testH 0 0 :E(s t+3|J t )=f t,3 ,which is implied by H 0 . You do this by taking a vector of economicvariables z t−3 in J t−3 , running the regressions t − f t−3,3 = z 0 t−3 β + ² t,3,and doing a joint test that the slope coefficients are zero.Under the null hypothesis, the forward rate is the market’s forecastof the spot rate 3 months ahead f t−3,3 =E(s t |J t−3 ). The observations,however, are collected every month. Let J t =(² t ,² t−1 ,...,z t ,z t−1 ,...).The regression error formed at time t − 3is² t = s t − E(s|J t−3 ). Att − 3, E(² t |J t−3 )=E(s t − E(s t |J t−3 )) = 0 so the error term is un- ⇐(103)predictable at time t − 3 when it is formed. But at time t − 2andt − 1 you get new information and you cannot say that E(² t |J t−1 )=E(s t |J t−1 )−E[E(s t |J t−3 )|J t−1 ] is zero. Using the law of iterated expectations,the Þrst autocovariance of the error E(² t ² t−1 )=E(² t−1 E(² t |J t−1 ))