International macroe.. - Free

6.1. DEVIATIONS FROM UIP 167Hansen-Hodrick t-ratios t HH are given in square brackets and OLS t-ratios t OLS are given in double parentheses. The relative range is the2.5 to 97.5 percentile of the distribution with overlapping observationsdivided by the 2.5 to 97.5 percentile of the distribution with nonoverlappingobservations. 2 The empirical distribution of each statistic isbased on 2000 replications.You can see that there deÞnitely is an efficiency gain to using overlappingobservations. The range encompassing the 2.5 to 97.5 percentilesof the Monte Carlo distribution of the OLS estimator shrinksapproximately by half when going from nonoverlapping (quarterly) tooverlapping (monthly) observations. The tradeoff is that for very smallsamples, the distribution of the t-ratios under overlapping observationsare more fat-tailed and look less like the standard normal distributionthan the OLS t-ratios.Fama Decomposition RegressionsAlthough the preceding Monte Carlo experiment suggested that youcan achieve efficiency gains by using overlapping observations, in theinterests of simplicity, we will go back to working with the log oneperiodforward rate, f t = f t,1 to avoid inducing the moving averageerrors.DeÞne the expected excess nominal forward foreign exchange payoffto bep t ≡ f t − E t [s t+1 ], (6.1)where E t [s t+1 ]=E[s t+1 |I t ]. You already know from the Hansen—Hodrickregressions that p t is non zero and that it evolves overtime as a randomprocess. Adding and subtracting s t from both sides of (6.1) givesf t − s t =E t (s t+1 − s t )+p t . (6.2)Fama [48] shows how to deduce some properties of p t using the analysisof omitted variables bias in regression problems. First, considerthe regression of the ex post forward proÞt f t − s t+1 on the currentperiod forward premium f t − s t . Second, consider the regression of the2 Forexample,wegettherow1 relative range value 0.471 for the slope coefficientfrom (1.207-0.778)/(1.453-0.543).