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148 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESIt follows that the solution for the capital stock isˆk t+1 = ω 1ˆkt −"ρω 2 − ρ#[b 3 ρ + b 4 ]Ât. (5.26)To recover ŷ t , note that the Þrst-order expansion of the productionfunction gives y t = f(A, k) +f A Â t + f k kˆk t ,wheref A =1, andf k =(αy)/k. Rearrangement gives ŷ t = Ât + ˆk t .Torecoverî t , subtractthe steady state value γk = i +(1− δ)k from (5.8) and rearrange to getî t =(k/i)[γˆk t+1 −(1−δ)ˆk t ]. Finally, get ĉ t =ŷ t −î t from the adding-upconstraint (5.9). The log levels of the variables can be recovered byln(Y t ) = ŷ t +ln(X t )+ln(y),ln(C t ) = ĉ t +ln(X t )+ln(c),ln(I t ) = î t +ln(X t )+ln(i),ln(X t ) = ln(X 0 )+t ln(γ).Simulating the ModelWe’ll use the calibrated model to generate 96 time-series observationscorresponding to the number of observations in the data. From thesepseudo-observations, recover the implied log-levels and pass them throughthe Hodrick-Prescott Þlter. The steady state values arey =1.717, k =5.147, c =1.201, i/k =0.10.Plots of the Þltered log income, consumption, and investment observationsare given in Figure 5.3 and the associated descriptive statistics aregiven in Table 5.2. The autoregressive coefficient and the error varianceof the technology shock were selected to match the volatility of outputexactly. From the Þgure, you can see that both consumption and investmentexhibit high co-movements with output, and all three seriesdisplay persistence. However from Table 5.2 the implied investmentseries is seen to be more volatile than output but is less volatile thanthat found in the data. Consumption implied by the model is morevolatile than output, which is counterfactual.