TimeSeriesBook.pdf

144 example a simple var macroeconomic model for the

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14.4 Example: A Simple VAR Macroeconomic Model for the U.S. Economy In this section, we illustrate how to build and use VAR models for forecasting key macroeconomic variables. For this purpose, we consider the following four variables: GDP per capita ( { Y t } ), price level in terms of the consumer price index (CPI) ( { P t } ), real money stock M1 ( { M t } ), and the three month treasury bill rate ( { R t } ). All variables are for the U.S. and are, with the exception of the interest rate, in logged differences. 3 The components of X t are with the exception of the interest rate stationary. 4 Thus, we aim at modeling X t = (∆ log Y t , ∆ log P t , ∆ log M t , R t ) 0 . The sample runs from the first quarter 1959 to the first quarter 2012. We estimate our models, however, 3 Thus, ∆ log P t equals the inflation rate. 4 Although the unit root test indicate that R t is integrated of order one, we do not difference this variable. This specification will not affect the consistency of the estimates nor the choice of the lag-length (Sims et al., 1990), but has the advantage that each component of X t is expressed in percentage points which facilitates the interpretation.
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14.4. EXAMPLE: VAR MODEL 265 Table 14.1: Information Criteria for the VAR Models of Different Orders order AIC BIC HQ 0 -14.498 -14.429 -14.470 1 -17.956 -17.611 -17.817 2 -18.638 -18.016 -18.386 3 -18.741 -17.843 -18.377 4 -18.943 -17.768 -18.467 5 -19.081 -17.630 -18.493 6 -19.077 -17.349 -18.377 7 -19.076 -17.072 -18.264 8 -19.120 -16.839 -18.195 9 -18.988 -16.431 -17.952 10 -18.995 -16.162 -17.847 11 -18.900 -15.789 -17.639 12 -18.884 -15.497 -17.512 minimum in bold only up to the fourth quarter 2008 and reserve the last thirteen quarters, i.e. the period from the first quarter 2009 to first quarter of 2012, for an out-of- sample evaluation of the forecast performance. This forecast assessment has the advantage to account explicitly of the sampling variability in estimated parameter models. The first step in the modeling process is the determination of the lag- length. Allowing for a maximum of twelve lags, the different information criteria produce the values reported in Table 14.1. Unfortunately, the three criteria deliver different orders: AIC suggests 8 lags, HQ 5 lags, and BIC 2 lags. In such a situation it is wise to keep all three models and to perform additional diagnostic tests. 5 One such test is to run a horse-race between the three models in terms of their forecasting performance. Forecasts are evaluated according to the root-mean-squared-error (RMSE) 5 Such tests would include an analysis of the autocorrelation properties of the residuals and tests of structural breaks.
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266 CHAPTER 14. FORECASTING WITH VAR MODELS and the mean-absolute-error (MAE) 6 : RMSE : v u u t 1 h T + h X T +1 ( b X it - X it ) 2 (14.10) MAE : 1 h T + h X T +1 | b X it - X it | (14.11) where b X it and X it denote the forecast and the actual value of variable i in period t . Forecasts are computed for a horizon h starting in period T . We can gain further insights by decomposing the mean-squared-error additively into three components: 1 h T + h X T +1 ( b X it - X it ) 2 = 1 h T + h X T +1 b X it !
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