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94 CHAPTER 3. THE MONETARY MODELSubstitute (3.23) into (3.21) to getζ t = e 2 Y t =∞Xj=1= e 1⎛⎝ψ j e 1 B j Y t∞Xj=1ψ j B j ⎞⎠ Y t (3.24)= e 1 ψB(I − ψB) −1 Y t .Equating coefficients on elements of Y t yields a set of nontrivial restrictionspredicted by the theory which can be subjected to statisticalhypothesis testse 2 (I − ψB) =e 1 ψB. (3.25)Estimating and Testing the Present-Value ModelWe use quarterly US and German observations on the exchange rate,money supplies and industrial production indices from the InternationalFinancial Statistics CD-ROM from 1973.1 to 1997.4, to re-estimate theMacDonald and Taylor formulation and test the restrictions (3.25). Weview the US as the home country. The bivariate VAR is run on (∆f t , ζ t )with observations demeaned prior to estimation. The fundamentals aregiven by f t =(m t −m ∗ t )−(y t −y ∗ t ) where the income elasticity of moneydemand is Þxed at φ =1.The BIC (chapter 2.1) tells us that a VAR(4) is the appropriate.Estimation proceeds by letting x 0 t =(∆f t−1 ,...,∆f t−4 , ζ t−1 ,...,ζ t−4 )and running least squares on∆f t = x 0 t β + ² t,ζ t = x 0 t δ + v t.Expanding (3.25) and making the correspondence between the coefficientsin the matrix B and the regressions, we write out the testablerestrictions explicitly asβ 1 + δ 1 =0, β 5 + δ 5 =1/ψ,β 2 + δ 2 =0, β 6 + δ 6 =0,β 3 + δ 3 =0, β 7 + δ 7 =0,β 4 + δ 4 =0, β 8 + δ 8 =0.