International macroe.. - Free

170CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCY(109)⇒Estimating p tWe have evidence that p t = f t − E(s t+1 |I t ) evolves as a random process,but what does it look like? You can get a quick estimate of p tby projecting the realized proÞt f t − s t+1 = p t − u t+1 on a vector ofobservations z t where u t+1 = s t+1 − E(s t+1 |I t ) is the rational predictionerror. Using the law of iterated expectations and the property thatE(u t+1 |z t )=0,youhaveE(f t − s t+1 |z t )=E(p t |z t ). If you run theregressionf t − s t+1 = z 0 tβ + u t+1 ,you can use the Þtted value of the regression as an estimate of the exante payoff, ˆp t = z 0 ˆβ. tA slightly more sophisticated estimate can be obtained from a vectorerror correction representation that incorporates the joint dynamics ofthe spot and forward rates. Here, the log spot and forward rates are assumedto be unit root processes and the forward premium is assumed tobe stationary. The spot and forward rates are cointegrated with cointegrationvector (1, −1). As shown in chapter 2.6, s t and f t have a vectorerror correction representation which can be represented equivalentlyas a bivariate vector autoregression in the forward premium (f t − s t )and the depreciation ∆s t .Let’s pursue the VAR option. Let y t=(f t − s t , ∆s t ) 0 follow thek−th order VARy t=kXj=1A j y t−j+ v t .Let e 2 =(0, 1) be a selection vector such that e 2 y t= ∆s t picks off thedepreciation, and H t =(y t,y t−1,...) be current and lagged values ofh Pkj=1 iy t.ThenE(∆s t+1 |H t )=e 2 E(y t+1|H t )=e 2 A j y t+1−j ,andyou