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180CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYthe Euler equations for asset pricing can generically be represented asv =E t (µ t+1 r t+1 ), (6.21)where v isavectorofconstantswhosei − th element v i =1ifassetiis a stock or bond, and v i =0ifasseti is a forward foreign exchangecontract.Taking the unconditional expectation on both sides of (6.21) andusing the law of iterated expectations givesv = E(µ t+1 r t+1 ). (6.22)(112)⇒Let θ µ ≡ E(µ t ), σµ 2 ≡ E(µ t − θ µ ) 2 , θ r ≡ E(r t ), andΣ r ≡ E(r t − θ r )(r t − θ r ) 0 .Project(µ t − θ µ )onto(r t − θ r )toobtain(µ t − θ µ )=(r t − θ r ) 0 β µ+ u t , (6.23)where β µis a vector of least squares projection coefficients, u t is theleast squares projection error andFurthermore, you know thatβ µ= Σ −1r E(r t − θ r )(µ t − θ µ ). (6.24)E(r t − θ r )(µ t − θ µ )=E(r t µ t ) −θ r θ µ , (6.25)| {z }vwhere E(r t µ t )=v comes from the returns form of the Euler equations.Upon substituting (6.25) into (6.24), we get, β µ= Σ −1r (v − θ r θ µ ).Computing the variance of the intertemporal marginal rate of substitutiongivesσ 2 µ = E(µ t − θ µ ) 2= E(µ t − θ µ ) 0 (µ t − θ µ )= E[(r t − θ r ) 0 β µ+ u t ] 0 [(r t − θ r ) 0 β µ+ u t ]= E[β 0 (r µ t − θ r )(r t − θ r ) 0 β µ]+σu 2 + β0 E(r µ t − θ r )u t + Eu t (r t − θ r ) 0 β µ| {z }(a)= E(µ t r t − θ µ θ r ) 0 Σ −1r Σ r Σ −1r E(µ t r t − θ µ θ r )+σu2= (v − θ µ θ r ) 0 Σ −1r (v − θ µθ r )+σu 2 . (6.26)