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r total error is statistically more accurate. Clearly, a simple sum of the forecast errors wouldn’t work: positive and negative errors would cancel out. Alternatives include summing the squared forecast errors or the absolute forecast errors: ()Σ++=−=HTTtttyyHRMSE12ˆ1(RMSE)Error -Squared-Mean-RootΣ++=−=−−HTTtttyyHMAE1ˆ1(MAE)Error AbsoluteMean
Compare these statistics over different models for y. The model with the smallest values for RMSE/MAE provides the most accurate forecasts.
RMSE is based on a quadratic loss function (i.e., it squares the forecast errors). It therefore penalizes large forecast errors (outliers) more heavily than MAE. Σ++=−=−−−HTTttttyyyHMAPE1ˆ100(MAPE)Error PercentageAbsoluteMean
The MAE is often reported in % (scale-free) terms.
ΣΣ++=++=+=−−HTTttHTTttyHyHRMSETIC12121ˆ1(TIC)t CoefficienInequalityTheil
TIC lies between zero and one. A perfect forecast would imply TIC=0 (RMSE=0).
It’s also useful to see how well the forecast predicts:
i) The mean of the actual series
ii) The variance of the actual series
For example, the forecast may predict one of these well but not the other. In this context, Eviews reports a ‘bias proportion’ and ‘variance proportion’ which is informative about the accuracy of the forecasts of the mean and variance respectively of the series in the forecast period. Small bias and variance proportions indicate better forecasting of the mean and variance respectively. The covariance proportion captures the remaining unsystematic forecast error. This proportion will be large relative to the bias and variance proportions if the forecasts of the mean and variance are accurate (the bias, variance and covariance proportions sum to unity).
For the GMM CIP model:
• 56% of the total forecast error is due to inaccuracy in forecasting the mean of the forward premium (bias proportion).
• About 5% of the total forecast error is due to inaccuracy in forecasting the variance of the forward premium (variance proportion).
• The remaining 39% is due to unsystematic forecast error (covariance proportion).
This suggests that quite a large percentage of the total forecast error is due to inaccuracy in forecasting the mean of the forward premium. Can we build a model with more accurate forecasts of the forward premium? We will now go on to estimate and forecast an ARMA model for the forward premium.
2. Building an ARMA model for the forward premium
The three stages in the Box-Jenkins approach to ARMA modelling are:
1. Identification
2. Estimation
3. Testing
(see lecture 5 and Brooks 5.7)
Identification and estimation
Firstly we look at the ACF and PACF of the forward premium to help select an appropriate ARMA model.
Click on the Sample button (workfile toolbar) and re-set the sample to the in-sample estimation period: 5/09/2001 – 9/30/2004
Now open fp_3m and click View/Correlogram/Level: Autocorrelation Partial Correlation AC PAC Q-Stat Prob .|****** | .|****** | 10.7830.783545.220.000 .|****** | .|*** | 20.7870.4491096.60.000 .|****** | .|*** | 30.8060.3721675.60.000 .|****** | .|* | 40.7840.1952223.20.000 .|****** | .|** | 50.8000.2232794.10.000 .|****** | .|* | 60.7940.1593358.10.000 .|****** | .|* | 70.7890.1243915.30.000 .|****** | .|* | 80.7870.0914470.70.000 .|****** | .|* | 90.7930.1125034.30.000 .|****** | .|. | 100.7720.0135570.00.000
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