TimeSeriesBook.pdf

This took indeed place but first at a more moderate

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models predict a quick recovery. This took indeed place, but first at a more moderate pace. Starting in mid-2010 the unconventional monetary policy of quantitative easing, however, led to an unforeseen acceleration so that the forecasts turned out to be systematically too low for the later period. Interestingly, the smallest model fared significantly better than the other two. Finally, the results for the interest rates are very diverse. Whereas the VAR(2) model predicts a rise in the interest rate, the other models foresee a decline. The VAR(8) model even predicts a very drastic fall. However, all models miss the continuation of the low interest rate regime and forecasts an increase starting already in 2009. This error can again be attributed to the unforeseen low interest rate monetary policy which was implemented in conjunction with the quantitative easing. This misjudgement resulted in a relatively large bias proportion. Up to now, we have just been concerned with point forecasts . Point forecasts, however, describe only one possible outcome and do not reflect the inherent uncertainty surrounding the prediction problem. It is, thus, a question of scientific integrity to present in addition to the point forecasts also
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268 CHAPTER 14. FORECASTING WITH VAR MODELS 4.40 4.44 4.48 4.52 4.56 4.60 4.64 2008 2009 2010 2011 2012 2013 2014 actual VAR(2) VAR(5) VAR(8) (a) log Y t 5.32 5.36 5.40 5.44 5.48 5.52 5.56 5.60 2008 2009 2010 2011 2012 2013 2014 actual VAR(2) VAR(5) VAR(8) (b) log P t 2.5 2.6 2.7 2.8 2.9 3.0 2008 2009 2010 2011 2012 2013 2014 actual VAR(2) VAR(5) VAR(8) (c) log M t -2 -1 0 1 2 3 4 5 6 2008 2009 2010 2011 2012 2013 2014 actual VAR(2) VAR(5) VAR(8) (d) R t Figure 14.1: Forecast Comparison of Alternative Models
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14.4. EXAMPLE: VAR MODEL 269 Table 14.2: Forecast evaluation of alternative VAR models VAR(2) VAR(5) VAR(8) log Y t RMSE 8.387 6.406 5.678 bias proportion 0.960 0.961 0.951 variance proportion 0.020 0.010 0.001 covariance proportion 0.020 0.029 0.048 MAE 8.217 6.279 5.536 log P t RMSE 3.126 1.064 1.234 bias proportion 0.826 0.746 0.001 variance proportion 0.121 0.001 0.722 covariance proportion 0.053 0.253 0.278 MAE 2.853 0.934 0.928 log M t RMSE 5.616 6.780 9.299 bias proportion 0.036 0.011 0.002 variance proportion 0.499 0.622 0.352 covariance proportion 0.466 0.367 0.646 MAE 4.895 5.315 7.762 R t RMSE 2.195 2.204 2.845 bias proportion 0.367 0.606 0.404 variance proportion 0.042 0.337 0.539 covariance proportion 0.022 0.057 0.057 MAE 2.125 1.772 2.299 RMSE and MAE for log Y t , log P t , and log M t are multiplied by 100.
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270 CHAPTER 14. FORECASTING WITH VAR MODELS 4.40 4.44 4.48 4.52 4.56 4.60 4.64 2008 2009 2010 2011 2012 2013 2014 Actual forecast (a) log Y t 5.30 5.35 5.40 5.45 5.50 5.55 5.60 5.65 5.70 2008 2009 2010 2011 2012 2013 2014 Actual forecast (b) log P t 2.5 2.6 2.7 2.8 2.9 3.0 2008 2009 2010 2011 2012 2013 2014 Actual forecast (c) log M t -4 -2 0 2 4 6 8 10 2008 2009 2010 2011 2012 2013 2014 actual forecast (d) R t Figure 14.2: Forecast of VAR(8) model and 80 percent confidence intervals (red dotted lines) confidence intervals. One straightforward way to construct such intervals is by computing the matrix of mean-squared-errors MSE using equation (14.5). The diagonal elements of this matrix can be interpreted as a measure of the forecast error variances for each variable. Under the assumption that the innovations { Z t } are Gaussian, such confidence intervals can be easily computed. However, in practice this assumption is likely to be violated. This problem can be circumvented by using the empirical distribution function of
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