Transcript:

Seeing as having stationary time series appears to be very, very important for linear regression. It seemed pertinent now to introduce one of the tests for whether a time series is stationary or specifically for the dickey-fuller test for a unit root. What we’re actually testing as our null hypothesis Is that it our time Series is actually non stationary. So the idea with the Dickey Fuller test is that we start off with an AR process, so we have the XT is equal to Alpha Plus. Rho Times XT, minus 1 plus some error E T and notice that I’ve included this alpha term here by default, so I’m not specifying whether Alpha Inc. Was not in which case I just have a random walk or we have random doesn’t. Oh, Alpha doesn’t equal to naught. Rather in which case we have a model with a stochastic time trend or we have a random walk with drift. It turns out that this particular test. It doesn’t matter, we don’t need to specify explicitly beforehand. What type of random what we’re talking about? So the null hypothesis Here is that we have. Rho is equal to 1 Because if Rho equals 1 we have a non stationary time series and the alternative hypothesis here is going to be that. Rho is less than 1 Because if Rho is less than 1 we’ve already proved that our particular time series or a our process rather is stationary. OK, so you might think that a way to go about testing. This would be just to test. Rho here and test whether Rho is different from 1 But the problem is under the null hypothesis, Both XT and XT minus 1 on on stationary and when we have time series, which are non stationary, the normal central limit theorem said apply. So it’s not like we can just readily test. Rho using a already sort of t-test. Actually, it seems like a better thing to do would be to. Let’s say we take. XT minus 1 from both sides. So if I take X T minus 1 from both sides, I have the X T minus XT. Minus 1 is equal to alpha plus open bracket row minus 1 times. X T minus 1 plus ET. Which, if a write is a little bit more neatly, the left hand side is just what we define is the change in X T and that’s equal to Alpha Plus Delta Times X T minus 1 plus ET and notice that under the multiples is here, that row equals 1 This particular term is. Delta term here would in fact vanish. So we wouldn’t actually have this XT on this side here. Whereas if we have, Rho is less than 1 we’re going to have that we do have this. XT on this right hand side here. But this XT minus 1 is going to be itself stationary, so it’s not a problem and just if I didn’t say when we were talking about row equals 1 if row equals 1 because this disappears this left hand side is stationary, and we don’t have any non stationary variables on a right hand side. So we’re better off than we were in this place, okay. So how do we actually go about testing? Whether we have a non stationary time series or whether we have a unit route because the unit route is when we have. Rho is equal to 1 in which case Delta is just equal to 0 Well, a way that you might think that we could do. This would be just to calculate an ordinary. T statistic on this Delta term here or specifically on the estimated value of Delta, which we call Delta hat and then if we compare that T statistic with a T distribution, perhaps then that would allow us to determine whether or not we had a stationary time series or a non stationary time series, but the problem is under the null hypothesis being true. XT minus 1 is itself non stationary, so the ordinary central limit theorem stones applying for when were thinking about the estimators for Delta or the least square’s estimating estimator of Delta, which we call Delta hat. So it’s not the case that under a large sample size or an asymptotic sample size that Delta has a given T distribution or a normal distribution, really. So what can we actually do here? Well, it turns out all is not lost because thanks to the fellows. Dickey and further who actually tabulated the asymptotic distribution of the least square’s estimator for Delta under the null hypothesis of it being a unit root. It turns out that we can actually just compare our ordinary. T statistic with the values of this Dickey for the distribution and it turns out that if the T is less than some critical value from the Dickey fuller distribution, then in only no circumstance, do we reject the null hypothesis and a note of caution here because Delta is going to be less than 1 We should suppose that. T itself is going to be negative and hence we’re comparing whether our negative value of T is less than some other negative value, so it’s not the case that the magnitude will necessarily be less than it is that the absolute actual true value of the T will be less than, so. Let me just say that again. It is that the T value itself will in real value terms, be less than the dictaphone a critical value.