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126 CHAPTER 4. THE LUCAS MODEL4.5 Calibrating the Lucas ModelMeasurement. The measurements that we ask the Lucas model tomatch are the volatility (standard deviation) and Þrst-order autocorrelationof the gross rate of depreciation, S t+1 /S t , the forward premiumF t /S t , the realized forward proÞt (F t − S t+1 )/S t ,andtheslopecoefficientfrom regressing the gross depreciation on the forward premium.Using quarterly data for the U.S. and Germany from 1973.1 to 1997.1,the measurements are given in the row labeled ‘data’ in Table 4.2.Table 4.2: Measured and Implied Moments, US-GermanyS t+1S tVolatilityF t (F t−S t+1 ) S t+1S t S t S tAutocorrelationSlopeData -0.293 0.060 0.008 0.061 0.007 0.888 0.026Model -1.444 0.014 0.006 0.029 0.105 0.006 0.628Note: Model values generated with γ = 10, θ =0.5.F t (F t−S t+1 )S t S tThe implied forward and spot exchange rates exhibit the so-calledforward premium puzzle–that the forward premium predicts the futuredepreciation, but with a negative sign. Recall that the uncoveredinterest parity condition implies that the forward premium predicts thefuture depreciation with a coefficient of 1. The depreciation and therealized proÞt exhibit volatility of similar magnitude which is muchlarger than the volatility of the forward premium. All three series exhibitsubstantial serial dependence.Calibration. Let random variables be denoted with a ‘tilde.’ The ‘technology’that underlies the model are the exogenous monetary growthrates ˜λ, ˜λ ∗ , and the exogenous output growth rates ˜g, ˜g ∗ . Let the statevector be ˜φ =(˜λ, ˜λ ∗ , ˜g, ˜g ∗ ). The process governing the state vector is aÞnite-state Markov chain with stationary probabilities (see the chapterCecchetti et.al. [24], Burnside [18], Gregory and Smith [67] show how calibrationmethods can be combined with classical statistical inference, but the practice hasnot caught on.