Error correction model - part 1

in this video I want to talk about a particularly powerful type of dynamic model which is known as the error correction model so the idea is that if I have my YT and XT which are in general non stationary we spoke about one of the ways that we can get around that which is to regress first differences of YT on first differences next T so here I'm supposing the XT and YT lighten themselves bi1 which means that their first difference is stationary but if it so happens that YT and XT are Co integrated so that means that there is some sort of long-run or equilibrium value of y which is given by some linear combination of X then it turns out we can do something which is much more powerful than just regressing first differences on one another and because this relationship up here in the top right is essentially only a short-run relationship we'd like to try and include some sort of aspect of the long-run relationship if we possibly could so the idea here is that perhaps the Y which we observe YT might be different from the equilibrium value say there will be some sort of pendens not only on XT but there perhaps will be some dependence on XT minus 1 because the idea here being the YT takes some time to react to changes in XT and we're also suppose that there is some sort of dependent on the langt value of YT again this could represent some sort of time it takes for Y to adjust and the sort of new here might represent some degree of inertia and then finally I'm just including an error term VT ok so this is a sort of model which an experimenter empirical investigator might come across in reality but the problem with estimating this particular relationship is to follow one of them is that it doesn't really tell us anything about the dynamics of XT and YT it's just a whole range of lag so of XT and YT so there's no real economic content which is something on a daily which we'd like to have seeing as we're studying econometrics and we use that to study economic theory the second reason is a theoretical reason which is the fact that if Y and X are themselves non stationary we are very close to running into the problems of spurious regression so we know that if Y it is non stationary and X is non stationary there is a high probability that if I run this type of regression even if they're completely unrelated there will be some statistically significant value of Delta well which we obtain so we'd like to be able to combat both of these things we'd like to be able to estimate some sort of economic relationship and also we'd like to do away with this issue of spurious regression and an error correction model is a way of doing this and how we get to that particular model I'm going to explain now so the idea is that we start with this model here and then to begin with all we do is we take our YT and we take off y t minus 1 so the right hand side just stays the same so I have C plus Delta 1 times XT plus Delta 2 times XT minus 1 and the only difference here I'm gonna put a minus here 1 minus mute because of the fact that I've just taken YT minus 1 from both sides I did it for the left hand side so I have to do it to the right hand side okay so now this left-hand side here is just the change in Y T which is helpful because if Y is I 1 then the change in Y T we know is going to be AI 0 in other words it's going to be stationary ok so that's good but looking at this right hand side here we have still got these x and y which are themselves non stationary so we have got part of the way we haven't quite got all the way yet so the way in which we think about compound combating this is we would like to be dealing with first differences of X so we've gone up here at Delta 1 times X T what are we going to do down here is I'm gonna take away Delta 1 times X t minus 1 but I can't just do that arbitrarily I have to add in here Delta 1 times X t minus 1 and then I've still got my Delta 2 times XT minus 1 plus sorry 1 minus mu times y t minus 1 plus BT ok which if I write it a bit neater I have C plus Delta 1 times the change in X T and I'm actually gonna write this whole second half slightly differently I'm gonna write it as minus lambda which is a parameter I've just introduced I'm going to explain it in a minute why t minus 1 minus alpha minus beta times X t minus 1 plus some error VT and technically I should actually change the constant hit so I'm getting a C primed rather than just C so the idea here is that London is actually equal to 1 minus mu you can see that quite quickly because essentially the only coefficient I've got on YT minus 1 is minus lambda and minus lambda has got to be equal to minus 1 minus mu so that's easy enough the alpha and beta are slightly harder I'm not gonna show the alpha here but the beta is as it happens just equal to Delta 1 plus Delta 2 or dividing through by 1 minus mu and to see that you can just see here that in terms of the coefficients on the XT minus 1 I've got a delta 1 plus Delta 2 and then because I put this lumber outside I've essentially got to divide through by lambda and we know that lambda is just 1 minus mu so that's how we've got this particular relationship so why do we care about this particular relationship and is it useful to us so remember that the left-hand side here is the change in Y T and if Y is i1 we know that the change in Y T is i0 and similarly if XT is i1 the change in XT is itself zero so these are both stationary sir that's looking good but what about this thing in the parentheses here well you can probably get in the way in which I've written it in terms of defined in these parameters alpha and beta but essentially what I'm doing is I'm appealing to this long-run relationship between y and X and the idea is that if this long-run relationship between y and X exists then this particular term in the parenthesis here will be Co integrated in other words this term in the parenthesis here will be i0 so we've done away with our issue of spurious regression here if it so happens that Y and X are Co integrated and we know the parameters of cointegration alpha and beta frequently we don't know those parameters are from beta but I'm going to explain in the next video how we can actually estimate error correction models in circumstances where we don't know alpha and beta okay so that's the sort of theoretical reason what's the sort of economic reason for estimating these types of models well the idea here is that imagine that YT minus 1 is above alpha plus beta times XT minus 1 but we know from this model up here but that's essentially the same as saying that Y is above its equilibrium value so if Y is above its equilibrium value then we take off a little bit of Y so the change in Y T will be slightly negative hence we correct the error in the last period to adjust further towards the equilibrium value of y and this sort of error correction mechanism is why we call this an error correction so the idea is that this model allows for two types of things if I just try and write in if I try and get rid of this term here and have white to change in my T here well it allows for both short-run dynamics which is given by this sort of half of the expression here and it's short run because we're looking at first differences of YT regressed on first differences of XT but it also allows long run dynamics in to our model essentially it allows there to be some sort of long-run Co integrative relationship between YT and X T so the idea here is that the parameter lambda actually tells us the speed at which our bearable adjusts to any sort of dish equilibrium so the reason these models are highly favored it is because they allow for both interaction between short-run and long-run dynamic so it's a lot richer than just regressing first difference is here which is just short run or just regressing doing us a release and Lance operator so we need to get the long-run relationship

Channel: Ben Lambert

Views: 121,793

Rating: 4.9180326 out of 5

Keywords: time series, error correction model, ecm, Econometrics (Field Of Study)

Id: wYQ_v_0tk_c

Length: 10min 2sec (602 seconds)

Published: Tue Sep 24 2013