《计量经济学》课程教学资源（PPT课件讲稿，英文版）ch18 Testing for Unit roots

Testing for Unit roots 2 Consider an AR(1): y,=a+pvi-+er o Let Ho: p=1, (assume there is a unit root) ◆ Define 6=p I and subtract V/ rom both sides to obtain Ay,=a+ O-1+e, A Unfortunately, a simple: ot ls inappropriate, since this is an I(1)process 2 A Dickey-Fuller Test uses the t-statistic, but different critical values Economics 20- Prof anderson

Economics 20 - Prof. Anderson 1 Testing for Unit Roots Consider an AR(1): yt = a + ryt-1 + et Let H0 : r = 1, (assume there is a unit root) Define q = r – 1 and subtract yt-1 from both sides to obtain Dyt = a + qyt-1 + et Unfortunately, a simple t-test is inappropriate, since this is an I(1) process A Dickey-Fuller Test uses the t-statistic, but different critical values

Testing for Unit roots(cont) ◆ We can add p lags of△y, to allow for more dynamics In the process Still want to calculate the t-statistic for 0 e Now it's called an augmented Dickey Fuller test. but still the same critical values The lags are intended to clear up any serial correlation, if too few, test won't be right Economics 20- Prof anderson

Economics 20 - Prof. Anderson 2 Testing for Unit Roots (cont) We can add p lags of Dyt to allow for more dynamics in the process Still want to calculate the t-statistic for q Now it’s called an augmented Dickey￾Fuller test, but still the same critical values The lags are intended to clear up any serial correlation, if too few, test won’t be right

Testing for Unit roots w/Trends o 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 Can just add a trend to the model e Still looking at the t-statistic for 0, but the critical values for the Dickey-Fuller test change Economics 20- Prof anderson

Economics 20 - Prof. Anderson 3 Testing for Unit Roots w/ Trends 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 Can just add a trend to the model Still looking at the t-statistic for q, but the critical values for the Dickey-Fuller test change

Spurious regression e Consider running a simple regression of y on x, where y and x, are independent I(1) series The usual ols t-statistic will often be statistically significant, indicating a relationship where there is none o Called the spurious regression problem Economics 20- Prof anderson 4

Economics 20 - Prof. Anderson 4 Spurious Regression Consider running a simple regression of yt on xt where yt and xt are independent I(1) series The usual OLS t-statistic will often be statistically significant, indicating a relationship where there is none Called the spurious regression problem

Cointegration o Say for two I(1)processes, y, and x, there is a Such that y,Bx, is an I(0)process o If so, we say that y and x are cointegrated and ca ll B the cointegration parameter o If we know B, testing for cointegration is straightforward if we define st=y,-Bxr o Do Dickey-Fuller test and if we reject a unit root, then they are cointegrated Economics 20- Prof anderson 5

Economics 20 - Prof. Anderson 5 Cointegration Say for two I(1) processes, yt and xt , there is a b such that yt – bxt is an I(0) process If so, we say that y and x are cointegrated, and call b the cointegration parameter If we know b, testing for cointegration is straightforward if we define st = yt – bxt Do Dickey-Fuller test and if we reject a unit root, then they are cointegrated

Cointegration(continued) o If B is unknown, then we first have to estimate B, which adds a complication e After estimating B we run a regression of Au, on u,, and compare t-statistic ow/ with the special critical values o If there are trends. need to add it to the initial regression that estimates B and use different critical values for t-statistic oni./ Economics 20- Prof anderson 6

Economics 20 - Prof. Anderson 6 Cointegration (continued) If b is unknown, then we first have to estimate b , which adds a complication After estimating b we run a regression of Dût on ût-1 and compare t-statistic on ût-1 with the special critical values If there are trends, need to add it to the initial regression that estimates b and use different critical values for t-statistic on ût-1

Forecasting Once we've run a time-series regression we can use it for forecasting into the future o Can calculate a point forecast and forecast interval in the same way we got a prediction and prediction interval with a croSs-section o Rather than use in-sample criteria like adjusted R2, often want to use out-of-sample criteria to judge how good the forecast is Economics 20- Prof anderson 7

Economics 20 - Prof. Anderson 7 Forecasting Once we’ve run a time-series regression we can use it for forecasting into the future Can calculate a point forecast and forecast interval in the same way we got a prediction and prediction interval with a cross-section Rather than use in-sample criteria like adjusted R2 , often want to use out-of-sample criteria to judge how good the forecast is

Out-of-Sample criteria Idea is to note use all of the data in estimating the equation, but to save some for evaluating how well the model forecasts Let total number of observations be n +m and use n of them for estimating the model o Use the model to predict the next m observations. and calculate the difference between your prediction and the truth Economics 20- Prof anderson 8

Economics 20 - Prof. Anderson 8 Out-of-Sample Criteria Idea is to note use all of the data in estimating the equation, but to save some for evaluating how well the model forecasts Let total number of observations be n + m and use n of them for estimating the model Use the model to predict the next m observations, and calculate the difference between your prediction and the truth

Out-of-Sample Criteria(cont) Call this difference the forecast error which is en++ for h=0, 1,...,m Calculate the root mean square error (RMSE Economics 20- Prof anderson 9

Economics 20 - Prof. Anderson 9 Out-of-Sample Criteria (cont) Call this difference the forecast error, which is ên+h+1 for h = 0, 1, …, m Calculate the root mean square error (RMSE)

Out-of-Sample Criteria(cont) Call this difference the forecast error which is en+h+, for h=0, 1,...,m o Calculate the root mean square error and see which model has the smallest. where RMSE= m > e2 n+h+1 h=0 Economics 20- Prof anderson 10

Economics 20 - Prof. Anderson 10 Out-of-Sample Criteria (cont) Call this difference the forecast error, which is ên+h+1 for h = 0, 1, …, m Calculate the root mean square error and see which model has the smallest, where 1 2 1 0 2 1 1 ˆ       =  − = + + − m h n h RMSE m e