International macroe.. - Free

168CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYone-period ahead depreciation s t+1 − s t on the current period forwardpremium. The regressions aref t − s t+1 = α 1 + β 1 (f t − s t )+ε 1t+1 , (6.3)s t+1 − s t = α 2 + β 2 (f t − s t )+ε 2t+1 . (6.4)(6.3) and (6.4) are not independent because when you add them togetheryou getα 1 + α 2 = 0,β 1 + β 2 = 1, (6.5)ε 1t+1 + ε 2t+1 = 0. (6.6)In addition, these regressions have no structural interpretation. Sowhy was Fama interested in running them? Because it allowed him toestimate moments and functions of moments that characterize the jointdistribution of p t and E t (s t+1 − s t ).The population value of the slope coefficient in the Þrst regression(6.3) is β 1 =Cov[(f t − s t+1 ), (f t − s t )]/Var[f t − s t ]. Using the deÞnitionof p t , it follows that the forward premium can be expressed as,f t − s t = p t +E(∆s t+1 |I t ) whose variance is Var(f t − s t )=Var(p t )+Var[E(∆s t+1 |I t )]+2Cov[p t , E(∆s t+1 |I t )]. Now add and subtract E(s t+1 |I t )to the realized proÞt togetf t − s t+1 = p t − u t+1 where u t+1 = s t+1 −E(s t+1 |I t )=∆s t+1 − E(∆s t+1 |I t ) is the unexpected depreciation. Nowyou have, Cov[(f t −s t+1 ), (f t −s t )] = Cov[(p t −u t+1 ), (p t +E(∆s t+1 |I t ))]=Var(p t )+Cov[p t , E(∆s t+1 |I t ))]. With the aid of these calculations,the slope coefficient from the Þrst regression can be expressed asβ 1 =Var(p t )+Cov[p t , E t (∆s t+1 )]Var(p t )+Var[E t (∆s t+1 )] + 2Cov[p t , E t (∆s t+1 )] . (6.7)In the second regression (6.4), the population value of the slope coefficientis β 2 =Cov[(∆s t+1 ), (f t −s t )]/Var(f t −s t ). Making the analogoussubstitutions yieldsβ 2 =Var[E t (∆s t+1 )] + Cov[p t ,E t (∆s t+1 )]Var(p t )+Var[E t (∆s t+1 )] + 2Cov[p t , E t (∆s t+1 )] . (6.8)