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172CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYcan estimate p t with⎡ˆp t =(f t − s t ) − e ⎣ 2kXj=1⎤Â j y ⎦ t+1−j. (6.9)(110)⇒Mark and Wu [102] used the VAR method to get quarterly estimatesof p t for the US dollar relative to the deutschemark, pound, and yen.Their estimates, shown in Figure 6.1, show that of E(∆s t+1 |H t )arepersistent for the pound and yen. Both ˆp t and Ê(∆s t+1|H t )alternatebetween positive and negative values but they change sign infrequently.The cross-sectional correlation across the three exchange rates is alsoevident. Each of the series spikes in early 1980 and 1981, the ˆp t saregenerally positive during the period of dollar strength from mid-1980 to1985 and are generally negative from 1990 to late 1993. You can alsosee in the Þgures the negative covariance between ˆp t and Êt(∆s t+1 )deduced by Fama’s regressions.Deviations from uncovered interest parity are a stylized fact of theforeign exchange market landscape. But whether the stochastic p t termßoating around is the byproduct of an inefficient market is an unresolvedissue. As per Fama’s deÞnition, we say that the foreign exchangemarket is efficient if the relevant prices are determined in accordancewith a model of market equilibrium. One possibility is that p t is a riskpremium. At this point, we revisit the Lucas model and use it to placesome structure on p t .6.2 Rational Risk PremiaHodrick [75] and Engel [44] show how to use the Lucas model to priceforward foreign exchange. We follow their use of Lucas model to understanddeviations from uncovered interest parity.Recall that forward foreign exchange contracts are like nominalbonds in the Lucas model in that they are not actually traded. Weare calculating shadow prices that keep them off the market. Let S t isthe nominal spot exchange rate expressed as the home currency priceofaunitofforeigncurrencyandF t be the price the foreign currencyfor one-period ahead delivery.