Exam 16: Additional Topics in Time Series Regression

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There are various methods available for forecasting the inflation rate and the unemployment rate.One basic distinction is whether or not the two variables are forecasted separately,or jointly as a system of two equations.Another distinction involves one period ahead forecasts vs.multiperiod forecasts.Finally,if multiperiod forecasts are used,then there is a multiperiod forecasting regression method vs.an interated forecast method.
Univariate Regression Methods: Here either the change in the inflation rate or the unemployment rate is modeled as an AR(p)and estimated by OLS.Observed values for the regressors are then substituted to produce a one period ahead forecast.(The one period ahead forecast for the inflation rate can then also be derived. )Statistical methods,such as the BIC or AIC can be used for choosing the number of lags.There are two important properties of the forecasts: the best forecast of either the change of the inflation rate or the unemployment rate depends only on the most recent p past values,and the errors are serially uncorrelated.These follow from the OLS assumptions.The multiperiod regression method for making an h-period ahead forecast of the change in inflation or unemployment rate using the AR(p)involves regressing these variables on its p lags,starting from (t-h),i.e. ,Yt = δ0 + δ1Yt-h + ...+ δpYt-p-h+1 + ut.Since the error term is serially correlated for the multiperiod regression,HAC standard errors must be used to have a reliable basis for inference.The iterated AR forecast method for the AR(p)is achieved by forecasting one period ahead initially,then using the forecasted value for the two period ahead forecast,and so on.More formally,the two-period ahead forecast is = 0 + 1 + 2Yt-2 + 3Yt-3 + ...+ pYt-p,while the three-period ahead forecast is = 0 + 1 + 2Yt-2 3 + 3Yt-3 + ...+ pYt-p,etc.
Multiple Predictors: If economic theory suggests that other variables could help forecast either the change in the inflation rate or the unemployment rate,then lags of these variables can be included.The Granger-causality test can be used to determine whether or not these additional variables belong in the regression.The same methods that were used for the AR(p)model can be applied for the ADL(p,q)model.For example,in the multiperiod forecasting using multivariate forecasts,all regressors must be lagged h periods to produce the h-period ahead forecast.To forecast both the change in the inflation and unemployment rate,regressions for each of the two dependent variables have to be estimated first,i.e. ,for both variables the following regression is estimated by OLS: Yt = δ0 + δ1Yt-h + ...+ δpYt-p-h + 1 + δp + 1Xt-h + ...+ δ2pXt-p-h + 1 + ut.Then the estimated coefficients are used to make the h-period ahead forecast.The interated forecast method now involves making one-period ahead forecasts using the estimated VAR specification,and using these forecasted values for both variables in subsequent forecasts.The two period ahead forecast,for example,for variables would be calculated as follows: = 10 + 11 + 12 Yt-2 + 13Yt-3 + ...+ 1pYt-p
+ 11 t-1 + Xt-2 + 13Xt-3 + ...+ 1pXt-p.
The decision on which method to use depends on the quality of the specification.If the AR(p)or the VAR is a good approximation to the underlying relationship,then the iterated forecast method is better.Note that if multiple predictors are involved,the ADL is not an alternative,since the additional predictors have to be forecasted themselves.However,even if one of the VAR equations is not a good representation of the underlying process,then the multiperiod regression forecasts are more accurate on average.Since the difference between the two methods is typically small,the textbook suggests to use the one "which is most conveniently implemented in your software."