Cointegration tests

in this video I want to talk about how we go about the testing whether YT and XT are Co integrated so in the last video I introduced the concept of Co integration and the idea here is that I can have two potentially non-stationary variables and if it so happens that I can multiply one of the variables buying some constant beta and then if I look at the difference between the original series YT and beta times X T so this top series here is YT and this bottom series here is beta times XT and if it looks like the difference between the two series is relatively constant or mathematically the way of stating that is that the difference between the two series is stationary then we might be able to conclude that we have some sort of meaningful relationship or some sort of economic relationship between y and X so the idea here is that if I take my original YT and I take a particular value of beta times XT and let's call that epsilon T the idea here is that if there is some meaningful relationship between YT and XT and that relationship holds for all time some sort of linear relationship that is then the idea is that ET here should be 0 in other words it's a stationary process which is weakly dependent so that's all fine and well when we actually know the parameter beta but most of the time we actually don't know the parameter beta we actually need to estimate it it might be the sole purpose of estimation to estimate the particular parameter beta so the idea here is that we will run a least squares regression which is YT is equal to alpha plus beta times XT plus UT and when we run least squares regression we get estimates for the parameters so we denote them by hat on these particular technically hat denotes an estimator but I'm using it to hear to mean the particular point value of that estimator so then what you might think would be a good thing to do here would be to take our estimated residuals or sorry our residuals which are estimated values of the error and they'd all just be equal to YT minus alpha hat minus beta hat times XT and if it is the case that YT and XT are Co integrated you might think that our residuals should themselves be stationary so they should be i0 much like we assumed that et or epsilon T up here should be i0 and that's essentially true essentially what we need to do is we do a Dickey fuller test on our residual so we apply a standard Dickey fuller test or residual so the idea here is that we are running a regression of the change in our residual on Delta 0 plus delta 1 times the lagged value of the residual and in principle we might want to add further lags to correct for any serial correlation and the idea here is then we would do some sort of and we would calculate a t statistic for this delta 1 or delta 1 hat technically and we would compare that with a standard Dickey fuller distribution and if the T stat was less than a particular critical value for that Dickey fuller distribution then we would reject the null and conclude that our error is i0 in other words at YT and X T or cointegrated but there is a problem here and it's because of the fact we don't actually know this true parameter beta we only estimate it and because of that carrying out this particular test which we've specified here by comparing the T stat on the estimated regression coefficient of on the lagged value of the residual is not quite right but it turns out there is actually quite a simple worker to this and essentially we keep everything the same we just we carry out this least-squares regression as we would before we carry out this particular test or the functional form of this test is exactly the same the only thing that needs to change is the particular comparison of the T with the Dickey fuller distribution it turns out that we can actually amend the Dickey fuller distribution to take into account the fact that we are estimating beta and it turns out that because of the fact we're estimating beta we compare it with a another Dickey fuller distribution which is slightly more stringent than the original Dickey fuller distribution in other words it's critical values are that much more negative than the original Dickey fuller distribution so it's that much less likely that we will reject the null hypothesis and there's some further intuition for this and it comes out the fact that essentially for this least spread regression up here the null hypothesis is the beta equals zero and it's only the alternative the beta does not equal zero so the null hypothesis here is essentially that we have run a spurious regression because if beta is zero there's no way that YT and BT can be Co integrated because we're saying that YT here is itself I one it's only under the case that we reject the null that we can start to think about there actually being some sort of Co integration between YT and XT and essentially that's part of the reason why we need to amend the Dickey fuller distribution is because of the fact that there is a significant chance that we might be running a spurious regression and because of the fact we might be running a spurious regression it might appear as if the variables are more cointegrated than they actually are and because of that we need to amend the original Dickey fuller distribution to take it that into account

Channel: Ben Lambert

Views: 114,014

Rating: 4.9582849 out of 5

Keywords: Econometrics (Field Of Study), time series, cointegration

Id: q5wbOSjbVW4

Length: 6min 28sec (388 seconds)

Published: Thu Sep 19 2013