Testing for Unit Roots w Consider an AR 1 yt a ryt 1 et w Let H0 r 1 assume there is a unit root w Define q r 1 and subtract yt 1 from both sides to obtain Dyt a qyt 1 et w Unfortunately a simple t test is inappropriate since this is an I 1 process w A Dickey Fuller Test uses the t statistic but different critical values 1Economics 20 Prof Anderson Testing for Unit Roots cont w We can add p lags of Dyt to allow for more dynamics in the process w Still want to calculate the t statistic for q w Now it s called an augmented Dickey Fuller test but still the same critical values w The lags are intended to clear up any serial correlation if too few test won t be right 2Economics 20 Prof Anderson Testing for Unit Roots w Trends w If a series is clearly trending then we need to adjust for that or might mistake a trend stationary series for one with a unit root w Can just add a trend to the model w Still looking at the t statistic for q but the critical values for the Dickey Fuller test change 3Economics 20 Prof Anderson Spurious Regression w Consider running a simple regression of yt on xt where yt and xt are independent I 1 series w The usual OLS t statistic will often be statistically significant indicating a relationship where there is none w Called the spurious regression problem 4Economics 20 Prof Anderson Cointegration w Say for two I 1 processes yt and xt there is a b such that yt bxt is an I 0 process w If so we say that y and x are cointegrated and call b the cointegration parameter w If we know b testing for cointegration is straightforward if we define st yt bxt w Do Dickey Fuller test and if we reject a unit root then they are cointegrated 5Economics 20 Prof Anderson Cointegration continued w If b is unknown then we first have to estimate b which adds a complication w After estimating b we run a regression of D t on t 1 and compare t statistic on t 1 with the special critical values w If there are trends need to add it to the initial regression that estimates b and use different critical values for t statistic on t 1 6Economics 20 Prof Anderson Forecasting w Once we ve run a time series regression we can use it for forecasting into the future w Can calculate a point forecast and forecast interval in the same way we got a prediction and prediction interval with a cross section w Rather than use in sample criteria like adjusted R2 often want to use out of sample criteria to judge how good the forecast is 7Economics 20 Prof Anderson Out of Sample Criteria w Idea is to note use all of the data in estimating the equation but to save some for evaluating how well the model forecasts w Let total number of observations be n m and use n of them for estimating the model w Use the model to predict the next m observations and calculate the difference between your prediction and the truth 8Economics 20 Prof Anderson Out of Sample Criteria cont w Call this difference the forecast error which is n h 1 for h 0 1 m w Calculate the root mean square error RMSE 9Economics 20 Prof Anderson Out of Sample Criteria cont w Call this difference the forecast error which is n h 1 for h 0 1 m w Calculate the root mean square error and see which model has the smallest where 10Economics 20 Prof Anderson