Method of Moments and Generalised Method of Moments Estimation - part 1

in this video I want to provide an introduction to method of moments and generalized method of moments estimator by means of an example so in method of moments and generalized method of moment what we do is we first of all think about there being some underlying population process which is generating our data so I'm going to define a population and within that population data is being generated by a normal process so X here is thought to be distributed normally in our population with a mean of mu and a variance of Sigma squared and because of the fact we're dealing with a normal distribution we know certain moments of that distribution we know certain properties of it one of them we know is that the expected value of X is equal to MU we've just defined it to be the case okay so that's easy enough the second condition we know is that the variance of our random variable X is equal to Sigma squared okay so those are two things that we know about our population process in both method of moments and generalized method Romans what we do is we recreate those two moment conditions but within our sample so suppose that we've got some sort of sample from a population what we're going to do is we're going to recreate each of these moment conditions but with sample analogs so this first moment condition is that the expectation of X is equal to MU so what we actually have to do is we have to recreate what is the sample equivalent of an expectation well of course that's just the sample mean so the sample mean here is just 1 over N times the sum from I equals 1 to N of X I and within our sample we set that equal to our estimator of the population parameter mu which I'm just going to call Mew hat so within method of moments estimator mu hat is just given by the sample mean okay so that's the first moment condition then we have to do this for the second one here so the second moment condition here could be reformulated in terms of an expectation it's just the same thing if I write it out a little bit above is the same thing as the expected value of X minus mu all squared and setting that equal to Sigma squared so what we're going to do again is we're going to replace this expectation by the sample equivalent which is just the sample mean but now it's not the sample mean of X it's the sample mean of ie the sum from I equals 1 to n of X I minus mu all squared and what I do is I replace mu by our estimator which I medical Mew hat and then that defines in turn our estimator for Sigma or Sigma squared in the population which we only call Sigma hat squared so notice in this case we had two unknowns and we had two equations which corresponding to the two moment conditions which we have within our population so what that meant was that there were exact and unique solutions so as it turns out this is what the finds method of moments estimator so if I destroy sort of dotted line on here everything above the dotted line corresponds exactly to method of moments estimators oh we've got the number of moment conditions equals exactly the number of parameters we're trying to estimate so that in turn defines exact solutions for our estimators of the parameters but of course for a normal distribution there are higher order moment conditions which we could have included as well so one of them is that we know for a normal distribution that the skewness is 0 so that's equivalent to saying that the expectation of X minus mu all to the power 3 is equal to 0 within our population we also know that the ketosis or talking about the ketosis we can actually define what the fourth order moment condition is for a normal random variable which is that the expectation of X minus mu all to the power 4 is equal to 3 Sigma to the power so what we could do then is we could just carry on as we did before and we could formulate the sample equivalents of each of these population moment conditions so let's go ahead and do that so this first one defines 1 over N times the sum of I equals 1 to N of X I now replacing mu with new hat mu hat or cubed is equal to 0 so that's in the first moment condition after the dotted line and then if we have the sort of last moment condition we are going to replace this expectation by a sample mean so it just becomes the sum from I equals 1 to N of X I minus mu hat all to the power 4 is equal to 3 Sigma hat to the powerful but notice now that we've got four equations but we've only got two branches we're trying to estimate so we know from sort of rudimentary algebra that there aren't going to be exact solutions to this in fact it's going to be very unlikely for the vast majority of cases that we come across that we're going to be able to satisfy each of these for moment conditions within our sample exactly and that defines the issue of GMM and it contrasts it with method of moments so in GMM the number of moment conditions which I'm going to call n exceeds the number of parameters which you're trying to estimate which I'm going to call K here so in that circumstance it's not likely going to be the case that we're going to get exact solutions so there aren't going to be exact solutions to all of the moment conditions or the sample equivalent of the population moment conditions which are going to mean that there's not going to be a particular value of MU hats and a particular value of Sigma hat squared which mean that all of these four conditions are satisfied so when we think about generalized method of moments we need to reformulate the way in which we approach the problem to deal with this particular issue and the way in which we actually go about it is by formulating what we call cost functions so I'm going to define for the first moment condition a cost function which I'm gonna call g1 which is actually how far away from this sort of theoretical equality are we actually when we estimate our parameters so g1 is going to be defined as the difference between the sum from I equals 1 to n of X I minus mu hat ok so note that this is the deviation of this above equation from the value it should theoretically take on let's do the same for the second condition here so for the second moment condition here we're going to reformulate it as a cost function so we're going to then have that g2 is just defined as the sum across I equals 1 to n but now we've got a slightly different thing which is the sum from I equals on to n of X I minus mu hat or squared minus Sigma hat squared and notice that if this was being or if the second moment condition was being upheld exactly this thing would actually have on a value of 0 okay so that's the first two memory conditions then if we do the same for the third and the fourth for the third moment condition here within our sample what we have is we have a cost which we call g3 which is 1 over n times the sum from I equals 1 to N of X I minus mu hat cubed and then in theory which take up zero but taking off zero is equivalent to not doing anything to the right hand side so notice again that if this was being upheld exactly by our value of mu hat which we estimate then this would actually have on a value of 0 as well then finally if we do the same with the fourth and final moment condition which we've created here we get g4 which is the difference or the deviation of this moment condition from its theoretical value which is just the sum from I equals 1 to N of X I minus mu hat all to the power 4 minus 3 Sigma hat to the power 4 so what we then need to do is we need to think about essentially our estimate mu hat and Sigma hat squared as being the estimates of the parameters which minimize some sort of sum of functions of these particular costs here and we're going to talk about how we actually formulate that's um in a particular appropriate way in the next video

Channel: Ox educ

Views: 159,218

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Keywords: Method Of Moments, Estimation, Econometrics (Field Of Study), Generalized Method Of Moments

Id: pIIEmUEnjhY

Length: 8min 59sec (539 seconds)

Published: Fri Aug 08 2014