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138 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLES5.1 Calibrating the One-Sector Growth ModelWe begin simply enough, with the closed economy stochastic growthmodel with log utility and durable capital. Then we will constructan international real business cycle model by piecing two one-countrymodels together.MeasurementThe job of real business cycle models is to explain business cycles butthe data typically contains both trend and cyclical components. 1 Wewill think of a macroeconomic time series such as GDP, as being builtup of the two components, y t = y τt + y ct ,wherey τt is the long-runtrend component and y ct is the cyclical component. Since businesscycletheory is typically not well equipped to explain the trend, theÞrst thing that real business cycle theorists do is to remove the noncyclicalcomponents by Þltering the data.There are many ways to Þlter out the trend component. Two verycrude methods are either to work with Þrst-differenced data or to useleast-squares residuals from a linear or quadratic trend. Most real businesscycle theorists, however choose to work with Hodrick—Prescott [76]Þltered data. This technique, along with background information onthe spectral representation of time-series is covered in chapter 2.Our measurements are based on quarterly log real output, consumptionof nondurables plus services, and gross business Þxed investment inper capita terms for the US from 1973.1 to 1996.4. The output measureis GDP minus government expenditures. The raw data and Hodrick-Prescott trends are displayed in Figure 5.1. The Hodrick-Prescott cyclicalcomponents are displayed in Figure 5.2. Investment is the mostvolatile of the series and consumption is the smoothest but all threeare evidently highly correlated. That is, they display a high degree of‘co-movement.’Table 5.1 displays some descriptive statistics of the Þltered (cyclicalpart) data. Each series displays substantial persistence and a highdegree of co-movement with output.1 The data also contains seasonal and irregular components which we will ignore.