Economics 421/521 - Econometrics - Winter 2011 - Lecture 5 (HD)

um, so where were we? We were talking about heteroskedasticity, how to test it today we are going to start talking about how to correct it and then I will do an example, they will also be examples done in a laboratory and you will get to do it live with your homework, but again I will do one equal to the problem of your homework, the what they'll do in a lab is a little further from the homework problem, it's exactly the same technique, so I hope at the end of the day Well, I know all the things we need to know about that, ah, what else do we need to talk about?

Let's keep going. I think of something we will do and then coordinate. This is how wide UC San Diego is in white text and this is what we call essentially a nonparametric test, so the first point about White's test is that it does not depend on a specific form of heteroscedasticity. So far we have looked at four models of heteroskedasticity and we needed a model to reduce the parameter space that we actually can. estimation, we looked at four heteroscedasticity models, one that said Sigma I squared is alpha x squared, something along those lines, one was the model, a Sigma squared is a must, ZT, log sigma, sit with those four models, so our tests above require us to model heteroscedasticity.

This doesn't mean you don't have to know the exact way now, it's better or worse, it depends if you are opposed to imposing a true constraint on a model, it improves efficiency because if the randomness takes you a little bit away from that constraint. and now if it's really true you want it to be true on the data so when you impose it you don't have randomness on the day to move it a little bit away from the constraint so when you impose a constraint on the data it makes things better by raising something true and making sure it shows up in your results and the data can't move you away from that truth even if you get a strange data set, so it's more efficient if you bother if you know the exact constraints, you know the exact shape of the heteroscedasticity, it's more efficient to make a monolith, so if you're pretty sure about scaling heteroscedasticity, writing a specific model is useful, it could be very useful, but if you put a false constraint on a model and it makes things much worse , so when you use a metric sentence model there is an advantage if you are right, it is much better.

There is a downside if you are wrong it is much worse so using parametric models depends critically on your confidence in your knowledge of the form of the underlying heteroskedasticity now theoretically sometimes you will know a lot about it sometimes you won't and it really depends , it's always a good idea to do White's test to verify this specification, you get the same answer on both. ways it's no big deal, you get different answers, white finds it, the other dozen is the other way around, so you have more to think about, you have to think about how correct your model is if you're using it, but in any case This It does not depend on the age-specific form of heteroscedasticity, we often don't know much about it, so it is a real advantage to have this type of test available; often you won't know much about it, so I'll do this in your projects.

I'll probably want you to do both, but this is just pressing a couple of buttons when you're done. I'll probably want you to try the other ones too just to show me that you can do it. and then comment on which one you think is best so that I know if you understand the kind of things I just talked about for the project, when we do them, I won't care much what your question is, what answers you find. What I will care about is that you show that you know and understand how to use the things we develop in class and this would be an example where you could do both.

I did this test. I did this test. I'm going to go with the white ones because I think I don't have a good idea of ​​the parametric structure of my air or I do or whatever and then I really know that you understand the underlying issues, which is what I want to see in the project and if you seems good or not. that is the first point about the white material, it is closely related to model a in the parametric test, but it is not model a, but it is like it is big, you know what this means, large sample LM test, if not , do not worry too much. this but this is another um anyway it's a great garage multiplier test let's not worry too much about that this is important it doesn't require normality of errors the other tests we did you had to have a parametric head to impose a particular distribution I can't do it right at the same time let me write a final table so they don't get it wrong um oh you're catching up let me talk yeah so um most of the tests we've done so far To get an F distribution or a distribution chi-square for any test distribution, I think we're using a hanging chi-square, you don't need Ali's norm of errors here, so it's robust to distributional assumptions for different distributions. yours, that's important because although maybe this is a good approximation, sometimes you're here on a normal wall, especially we have wrong specifications and this test will still work.

I mean, there are limits to how bad it would be, but for the most part it's okay and for these reasons. It's generally the recommended test, it's just stronger, parametric errors, stronger for distribution errors, law is like the first test anyway, so if you can do the first test, this one is probably just as good, maybe better in most cases, so let me give you an example. of how to do this later today I'll show you how to do it in a roundabout way, so let's say Y I beta 1 plus beta 2 model the slop itself at least you can think I care test YT there is beta 1 plus beta 2 X 2 I plus beta 3 X 3 I plus UI so let's assume now in this model what you remember with the variation of UI and Sigma al square, that is the problem, it is different for each observation it is from 1 to n, there was no eye there, it was the same Sigma square, you have a problem, so White's model assumes a particular structure, it is not a regular structure, It's something called flexible form.

So we're going to have a parametric form in a sense, but it's what we call pluggable flexible form and it adapts to almost anything, so there are certain types of flexible forms that you can use in

econometrics

that are very good. adjusting whatever the underlying truth is, they are very flexible in terms of how they can be adjusted, so what you do in this test is take all the variables as before, all the squares of the variables and all their cross products, it is always the same thing then our bearing model will be Sigma squared v is alpha 1 plus alpha 2 X 2i plus alpha 3 The Xs that are in your model now VP and that's why we used them before, they are not the same with all the over there.

It would be B 2 B 3 B 2 B 4 B 3 B 4 and so on all the inner products that are unique you would not have B 2 B 3 and B 3 B 2 because it is the same or X 2 so this is the model you don't actually end up having to estimate this then what do you actually do? I'm sorry. Wow, you don't press the buttons. You would do it yourself, so here are the steps involved with the tester and you anticipate. what will they be, let's estimate this model, what will we say, te squared, squared will be an estimate of this, I'll use te, the square of this first regression is an estimate of Sigma and squared, we do a UI squared regression. these variables, then we will look at n R squared as our test statistic and that will be a chi-square test statistic.

How many constraints do I need to make this variance concept? I need 2 alpha 3 4 5 and 6 all are 0 what is the constant variance so the null hypothesis we are aiming for is whether alpha 2 along the constant variance 6 or 0 vs. at least one is non-zero if at least one is non-zero in heteroscedasticity if we do not reject its zero homoscedasticity then estimate this model save the I squared estimate this model the T that statistic is M R squared and that is the chi square of the number of constraints in this case would be a chi square of 5 since 5 of the alpha have to be 0 and we will be specific smart about the Argentum hypothesis here incredulously they didn't actually let me write the hypothesis for the intercepts, so the value null that we want to test is alpha 2 is equal to alpha 3 is equal to L before with alpha 5 alpha 6 is equal to 0, that is 1 2 3 4 5 restrictions then the alternative is that at least one is non-zero, so a estimation by OLS this model here estimate this model by OLS now the next step the computer does it for you because it saves that variable called Raziel but I'm going to write it as a second step because not all packages do that and just to make the steps clear , the second step is to find your cat I equals y I minus theta 1 hat minus beta 2 hat X 2 I minus beta 3 hat the program, there is no need for this to happen, this just happens automatically in the background, anything with views, but technically it shouldn't, it should do it later, let's need to figure everything out for Becca, let's not confuse things, now there is a regression what you have, squared the constant X 2 X 3 X 2 squared X 3 squared X 2 and multiplied by R squared, so this is the goodness of fit, which is the number of observations, this is chi-square distributed with five degrees of freedom, so look up and look for chi-square five on the back from the book compared to n R squared very simple test, another reason for its popularity, so you simply compare an r-squared with the critical value for the chi-square test 5 and then again more sloppy, except it was more young man than that other thing or you reject as appropriate, okay, pretty son, so on to your task.

I have asked you to follow the white steps, as I said, there is a button you can press, the program performs the test automatically for the first time, at least I want you to do it explicitly yourself, run this regression, find the test statistic manually, etc It doesn't mean you can't check it with the program, but I want you to follow the steps because there is some intuition here behind the estimate that my packet is slow, so we did the test that we said we found heteroskedasticity. We know that all S is inefficient, what do we do right?

We use the efficient procedure called feasible generalized least squares, so we'll generalize the least squares procedure to handle heteroscedasticity along the way, we'll do some algebra that hopefully does that. Everything is clear, the first thing and it is simpler because we are not going to do anything theoretical with it is White's correction, so the first way to correct this is only White's, so White not only has a proof, but he has a correction, it is an intelligent correction. called white heteroscedastic consistent covariance creators, don't worry about that, so your audience is a HCC M procedure, so when you are looking for a lot of packages, it will simply tell you that you are using the HCC heteroscedastic consistent covariance matrix estimator and the procedures.

White direction, that's what it means, now we don't want to talk about maximum probability and information matrices yet, but this depends on some particular characteristics of something called an information matrix that arises from something called a map, some probability, which is something that will. week 10 on the last day of school, so use it, you've seen it, so let's not dig in and find out how this fix works. It's too much for us. It takes too much matrix algebra and bits of statistics. we haven't done it yet, that's unlikely, so this one is go to the computer, press the button and it works magically and there's a bunch of matrix algebra and stuff about statistics under the MIT insurers and it works fine, but this one is essentially a non-parametric test. it just reweighed things properly, we talk about heteroscedasticity, the problem with OLS is that too much weight is given to the variant variables, it has a high variance and too little weight to those with a small variance, essentially, this is a procedure weighing system that takes care of that. problem, so essentially everyone takes care of that problem by reweighing the schematic of some sort and this is no exception, so I'm debating whether to show you how to do this now that it's new or do it when I do the others. in the end, so I'm going to do all of this in one thing, in the end let me make the corrections for the other model first and then I'll go to therapy to show you how to do this kind of change of mind that's happening.

Well, the seventh way is called generalized or weighted least squares. Are there generalized least squares GLS estimators or something elsecalled WS wls weighted least squares estimators. a double a procedure, not all, not all, not all procedures live in both worlds, but this is probably most easily understood as a weighted least squares weighing scheme, but then we wait, we wait for the observations like I just described, but I. I will explain what the GLS is about; okay, let's start with the same model that I just deleted and then we'll use it as an example on how to do this.

Presumably you could generalize it to any number of variables, so I'm using a constant in the case of two x constant in two xIn this case, you should easily see how to generalize this to any number of and in a moment we'll have to talk about feasible generalized square sheets, so this is not feasible, this is how the model will do it. look, it's the same why I'm beta 1 plus beta 2 slowdown X 2i plus beta 3 It's up to us to take this data and re-weight it. Let me do it first as a weighing scheme to make it look simple.

Suppose we know Sigma squared. We don't know, but we can convince ourselves of it later, so for now let's assume we do. we don't know, that's what makes us unviable, but the guide is exactly a feasible procedure because we know how to estimate this, first we're going to pretend we know, then we'll figure out what to do and then we'll say, well, we don't know, how Can we get an estimate? Once you have it, we can do what we're doing here with the estimate, so this is setting this up for the way we're going. to do it soon let's take the data and wait a bit why I surpassed Sigma 1 over 2 Sigma I remember there is a 1 in front of here that is the variable actually there is a B 1 times 1 there is an X 1 Only all the ones so this is 1 over Sigma, I transform that variable X 2 I over Sigma I and taking this model and splitting it down both sides, for example, so my new model would be why I outperformed Sigma.

We will call it later why I will give it a new name, but why for Sigma I am beta 1. multiplied by 1 over Sigma I plus beta 2 X 2i over Sigma Phi plus theta 3 and X 3 I plus UI over C so if we knew Sigma we could do that so we'll call this regression and say okay that's why I point is beta 1 x 1 i star plus beta 2 x 2 i star plus theta 3 x 3 i star plus star UI so in our spreadsheet we would say you know it's equal to x 3 over sigma if you have data about sigma or data about x Form this new variable and run this regression on the transformation data with the stars.

This original model is not blue because we have heteroscedasticity. I claim and I will show in just a second that this model is blue will get the best unbiased linear estimate of the data is because we no longer have heteroskedasticity if I square this, what do I get? I understand you. I squared over Sigma. I squared if I take the expected value that a Sigma Y squared over Sigma Y squared, which is 1, so I don't want to have heteroscedasticity, I'll be mad, but that's what makes it work, so we take that this will be a trick that we will use over and over again whenever we have problems with these models, we will say how can I transform the data in a way that recovers the Gauss-Markov assumptions, is there a way to transform this data in a very simple way so that when finished, the model I run is still in beta and is there where I can move because I want to estimate them? things, that's the goal, that's what we're trying to figure out, since the most vain thing is a way to transform the data and get a blue s from the betas and in this case it turns out that they are, so let's make sure we see it .

Since we have solved the heteroscedasticity problem with that star model, then what is the variance of the UI star? So this just had a problem from all of us and we had that whole list of topics for Gauss-Markov there and evolving them into your first assignment. so whatever it was, they were fine, the only problem was we worked it out, so have we worked it out? So what's the variation of you? I emphasize that it is the variation of UI on Sigma. I have the same definition. I can find the UI star to be that, what is the variation of? a X X is a random variable and a is a constant a squared comes a variance of , why not? the concept changes the variants, all the constant does is change this distribution, there was an extension change, so if I add a to this, let's say this is X, this is X plus a, I get the same distribution in a new place Ctrl-c Ctrl-v exactly the same, so it doesn't change the spread of all the constant data when you multiply by a constant because you're taking the expected value of x minus x-bar squared, take an expected value ax minus ax bar squared. take an a squared and calculate the variance because variance involves squaring the expected value of F times the bar X squared when multiplied by a is a bar X minus X squared, which is a bar squared, so you get one square yards of X I guess this is all hat and look at me like that, that look means two things.

I have no idea how to talk about it. Keep going because I'm so bored. It is overlooked. I can't read you anyway. so this is 1 over Sigma hi multiplied by the square of the UI variance because this is just a number of cards, it's not a random variable number, we're an accident, it's not about what is the variance of your Sigma I squared, so this is 1 over Sigma I squared times Sigma, I squared, which is what 1 is, almost good asking directions, it's 1 everywhere, we don't have heteroskedasticity, so by going back to weighing the x and y we can solve the heteroskedasticity problem, if we know Sigma, we do not know Sigma. but we can use those models, we have model a B and C to estimate Sigma and make this procedure work.

Now look at what you're doing if this variation was really small, let's say it was point zero zero zero zero zero one sit on the very big what happens to the important by reducing them by dividing them by Sigma and if set, it reweights the observations we've been talking about, that makes this procedure optimal again now that we're giving each variable. the right weight exactly what you need to give you the best possible answer so this is weighted least squares yeah so what do you have you got it you have data so your spreadsheet with Netflix has Y I and a bunch of data, these are internal, we usually don't put it in the spreadsheet, this with the old ones, so you have of calculation that you simply extract. from someone and we don't know that the data works magically, we've given ourselves a perfect setup here, then you just calculate that divided by that, divided by that and that divided by that and that, but you might have new columns, those are your variables of star, so the next column will be Y star I and it will be just 1 1 divided by ABC II won one to two point five because this is squared.

I have to meet Sigma and this call would be X 1 1 over Sigma this call us X 2 over Sigma again the problem we do not know except in In very rare circumstances, what do we do now? We can estimate it and so we run the model, we give the estimate of this, we put it in our spreadsheet and then we use the estimate to correct and as long as the estimate is consistent and unbiased at least to a large extent. samples we will get the same answer because when n is large enough the estimate will go to the true Sigma squared, so for large n there should be a big difference if we use the actual value of the estimate as n gets smaller and smaller, of course, there is more variability in your estimates and the procedure doesn't work as well, but if there is more variability in the work even, that's where we're headed, it's good to debate with myself about something, okay, unless who has had linear algebra.

I can look for a second, but let me just for the people who can ever show you anything about the estimator. Here you can write a model using matrix algebra like this. This is a column. This is y 1 a y n. So this is it. a vector, this is a matrix, so the first column is all the units, this is X 2 1 bands that want. a beta K in this case beta 3 as I have it then you have all the errors e1 up to in I guess any of the common uses let me make it consistent and that is the model in the form of matrix algebra if you have had matrix algebra when you multiply rows by columns and you get y one is beta 1 plus beta 2 beta 3 X 3 2 plus u 2 and so on, this is a way to conveniently erase all data in a simple way.

When you do this, it turns out that the estimator here is square over the sum of the bar it's the sum of the x squared you take an x ​​squared essentially so there's x squared here's the and again, if you haven't had a linear algorithm, this will make some sense, we say the variance of U is Omega, so this is what we call the covariance of the variance. matrix, so what you can do with the errors is align them from u 1 to u n, which is u Prime and u 1 to u n, so we set them up this way to get a full matrix, this is n times 1 1 and n- by-n matrix of this, the first element is sympa, which is the square term u1, so this is the variance of the first observation, so this is Sigma 1 squared up to Sigma n squared, that's the stuff which we're worried about so when you multiply these together your 1 squared that's your Sigma 1 squared essentially we multiply them together you get 2 Sigma N squared now off the diagonal you get a bunch of UI UJ terms like you want u a u 1 u 3 u 1 or 8 we have an assumption that the errors are not correlated, so our assumption about this matrix whatever our substance is, there is no correlation between the errors, in the next chapter we will say: okay, what if that is not true?

What that means if you have a 0 here is to make everything but the diagonal variance covariance matrix zero? Heteroscedasticity says that everything has to be equal, each of these has to be equal, so the problems you may run into are double, or the elements on the diagonal of your variance covariance matrix under errors are not the same heteroskedasticity or there is a correlation between your errors, which is what we call autocorrelation. The next chapter will solve it. So what you can do about the problem is that your mistake is not your area. Your variance and covariance matrix is ​​not diagonal and the diagonal elements are not.

The problem is that people have non-zero off-diagonal elements in non-zero cami tuples and unequal elements along the diagonal. Now it turned out that the variance is a positive definite matrix. Maybe you know what that means. Maybe you don't know a positive definite. the matrix can always be decomposed q/v pitch transpose, it always exists into Q and V, so should I write this matrix this way? This is diagonal and you had matrix algebra. You know that the eigenvalues ​​are on the diagonal and these are the eigenvectors. it's actually used for dying eigenvectors when you put in the effort, okay, there's another way to do this.

I can make B equal to I without loss of generality, so I'm going to make this type identity matrix just all ones on the diagonal, so there have always been Q, so that's true, What I'm going to do is find a way totransform the model so that, instead of having this variance and covariance matrix, it has this one, so it sounds very difficult, what you do is estimate. during this time it will be just what you researched, you get this thing and you decompose it, there is a command that will decompose so that these two Vs are Q and Q transposed.

Do they take the variance covariance matrix that arises from the estimation and decompose it? so you just re-weight your data and say, okay, I'm going to take inverse Q and it's equal to inverse Q X beta plus inverse Q, this is what we did, our Q is 1 over Sigma, believe it or not, so we decompose variance and ours is really easy because it's our matrix, there are only zeros here and here our problem, these are just Sigma Squared, so QQ I just symbolize up Sigma I Q inverse is 1 over Sigma, so it's really simple in our case, so this inverse Q is just 1 over Sigma I for us is really simple, but in general this will always work because you look at the variance of this, you look at the expected value of Q, if you transpose Q, you transpose inverse Q, you get Oh, I get it backwards. you look at the headings of the inverse cube u u prime inverse Q we look at that variance covariance matrix I wrote this backwards this should have been on the other side sorry I asked you that way um you take that expectation you get inverse Q this is Omega Q transposed inverse so your head sorry get q inverse u u prime Q inverse transposed take that expectation you can q Omega Q inverse transposed so if you look here Q inverse Omega Q prime Q inverse transposed is the identity matrix so this is the identity matrix, so once you do this transformation, this will return the identity matrix.

Okay, now you can't say we never did any hard mapping, I know you didn't get any of that, it's for people in fifth, but there is a procedure when you use matrix algebra where you can generalize this. The point I wanted to make clear is that this is what it is called. here then it's called GLS estimator, so you can take this transformation data and you can use this X prime GLS estimator, so there is a way for this to be the general form of the OLS estimator. All you're doing is reading through the variance and covariance matrix, so you're somehow dividing both sides by the stereoscope variance matrix.

I'm sorry. It's not going as well as I expected. Oh, we kill the gun, so what we have we want to take the expected value of U 1 and G 1 X u u Prime, so this is the expected value of u 1 squared u 1 u 2 + 1 u 3. but like than 3 times 3, this will be u 2 u 1 u 2 squared u 2 u 3 u 3 u 1 u 3 u 2 and u 3 squared we assume we take the expected value we assume you wanted you to want u 3 to be uncorrelated , so all the errors are uncorrelated, so we take this expectation, you will get zeros here. this expectation is Sigma 1 squared 0 0, this is 0 Sigma 2 0 and 0, this is 0 0 Sigma 3 squared, so the variance The covariance matrices were concerned because we are simply stacking with the matrix, then when we decompose this, I can decompose it like this QQ prime, since it's going to be Sigma 1 Sigma 3 times Sigma 1 2 Sigma 3 2 off-diagonal zeros, so the QT r is really easy. because I multiply things I can Sigma 1 squared 2 Sigma 3 squared so it's just Sigma on the diagonal so Q inverse is just 1 over those when you say Sigma squared is safe so now we've proven that the variance and covariance matrix is ​​Sigma 1 squared under Sigma 3 squared 2 zeros on the diagonal which is equal to Sigma 1 Sigma 3 zeros off the diagonal times Sigma 1 2 Sigma 3 the zeros off the diagonal this is cute because it is diagonal this transpose Q then the inverse Q transpose is 1 over Sigma so when I take Q inverse Y just like Y over Sigma, so in this case it is very easy, but in general they will not be diagonal, but this trick will still work in 24, it's not sweat.

I know I messed up if we don't have zeros here I wasn't here that's not a religion yeah and in the next chapter what we're going to do, we're not actually going to let you download this again because that's how it's going to work, but in the next chapter everyone will be constants. They will all be the same, they will all be 11.3 and these will be non-zero and we will find a way to fix it. So we can imagine having both problems at once, this and this, in that case, this matrix procedure is really the way. go because when I decompose that cube it deals with both problems all at once instead of just one of the problems, so this GLS person just takes the variance and covariance matrix, splits it into two parts, takes that variance and covariance and creates a prime QQ. then just multiply by Q adversity, whether or not you have automatic correlation or heteroskedasticity, that inverse q will again weigh everything, both these and these, in a way that works, so if you have a legume matrix, this is simply just an orthogonal transformation, so we are just taking a linear transformation, we are taking a change of basis in a new space where the variance-covariance is diagonal, it is the same linear operator only with respect to a different basis set because you have the same operator linear operator on a different basis to obtain the same data.

I guess one of the points is that there is a way to do this with matrix algebra that makes it all very simple, so there is another algebra that takes all this complicated stuff and really does it. Pretty simple once you learn matrix algebra and I probably took the path to make me jump, well anyway back to our problem, we can't do any of this because I don't know what Omega is, I don't know what Sigma is. I don't know anything about that, so what do we do if we don't know Sigma I? So if our model is why is beta 1 plus beta 2 X 2 I plus beta 3 X 3 I plus UI, the variation of UI. this Sigma squared I and that's our problem, our solution is to reweigh 5 Sigma AND zero Sigma 1 over Sigma, so we weigh 1 over Sigma I, but I don't know Sigma what to do.

Shit it took waiting, we might not get to the example until Next time, one of the reasons I decided to do that was last time. I always thought this on Mondays and Wednesdays, so normally I had yesterday off, so we have an extra day, so I was killing time to keep us on schedule, let's take the simplest case verse, so the answer is use a variance model as a way to estimate, so we'll use Sigma I hat instead of Sigma and we'll get that Sigma I hat from a model and we'll start with one that's really easy to use because you don't actually have to do much, so this is the simplest case I can think of.

Suppose if the UI variance is Sigma squared, this is just a reason to call it alpha if you want. some number C T squared these are known data so we're giving ourselves some kind of variance because what we're saying is we know the variance up to a scale, we actually know the variance we just don't know if it's two or three or four so we know the degree of the scale factor, it's probably more information, but sometimes this is a decent model of the bearings. I use Tian's and it's probably all Dutch, so my next class is all keys and no eyes, so It arrives just as I have my eyes in my hand and I go to the next regiment, they say, with those eyes on, so the simplest case is when the variation follows this model.

This is almost like knowing the variation because you know everything with this book, the scale. with this number is and again, there's probably more information than we have, but this will work fine, let's say it looks, so what is the solution divided by this is Sigma I squared, so what is Sigma I, this model, Sigma Zi, this could be, you know, square feet? of housing or sales, the store or income, this is just a variable, you know, what we're going to do is just divide it by Zi, we would like to bite my signature, we don't know what it is, so we can ask ourselves, okay? what is the variation, then this is our star model.

We have star Y equals beta 1 X 1 in star plus beta 2 X 2 I star plus beta 3 in 3 I star plus star UI where they are this is X 3 star , so what is the variance? Do I stop? That's the variance UI Zi, which is 1 over Z, I squared the variance of U. I fly it again, so let me come here, have a sip of coffee when you catch up, sorry. Get excited, it's really fun, but when we say your variants, I was, this is one over Z, multiplied by Z squared, what is this? variance I just squared this, you started with that, which is equal to Sigma squared, almost okay asking now our variance covariance matrix has the same Sigma squared on the diagonal, so now we have a Barry non-animal, I don't understand how we are waiting. because if you are Zi Zi's group of constantly changing Zi, everyone will be divided by the same thing, different members were there to greet each X, so there are male relatives, it is this term if you want to focus on the majority, yeah , we.

We're, yeah, so what we're doing is taking it, we have something like this, the best intuitions about views rather than exes, you can do it another way, but really what we're doing, we're taking it. we have these big variants and we're weighting everything by dividing by a big number, so by dividing by that big number we make them less important, so Z being big actually reduces the weight of all of this when Z is small, so designate and you can rewrite the weighted here with the moving that does that so the new expectation is that's actually what Sal is doing another like we expect similar type of similar housing prices so maybe the size of the house yeah if you wanted a house you would have square footage and price. are related, you might see a lot more variation between square footage and price in high home prices and low home prices, say, so what you want to do is give more weight to homes in low price and less to high priced houses.

The way to do it is to simply take those prices and place them so that the errors are smaller because what we are never minimizing is the sum of the live squares and, therefore, warmings that are large have a lot of influence on the OLS estimator. . What we're doing is giving them less influence in the estimator procedure when I swear, it's big, don't pay so much attention to this, hey, less attention, okay, okay, and that's what you're doing. I think it's the best intuition. just think about this, I just want to wait down for the model to have a large variance and up, so when the variance is small now in the house price example, one assumption we're making is that we have the same linear relationship between Alice and square foot at the low ends we have end I, if it is not a linear model then we have other things to worry about, as long as that model is linear we can get more information about the low price now, for so you might think there is a different model. relationship then you have to put in a dummy variable or something to take care of that, but assuming that means there's no reason to look at the top price, which will just mislead you about what the relationship is, so we just want to tell the The procedure ignores them and splitting my IC is ignoring them because it has to do with this.

This is the amount of weight you give it in the estimator, so a low OLS outlier really raises the line a lot. You don't want these outliers to pull the line too much. a lot without shadows Estes because it's just because it aerates is big so you stopped the outlier to pull the anchor around the line and that makes it better now if you're not willing to adopt such a simple variation model and you stay on your feet , this won't work, so what if I don't know the variation and I'm not happy with that model? So I needed to generalize it a little more to have three other models that we can use for variation, so what? we can do this is called feasible generalized least squares, generally the same personal GLS, so we had three models of the variance Sigma I squared alpha 1 plus L 2 Z 2 plus Z P alpha C P I then we had a model B only Sigma I squared is equal to the same Sigma Squared, sorry, we have model C, which is the logarithm of Sigma.

I squared equals and now we are the same on the right side, so three models, so what we can imagine doing is exactly what you expect, you know, you can see it coming. correctly estimate this model, what do you get in Vermont and you get the I squared? So what would you do? Use this model to do this model to get an estimate of Sigma I squared. You will get Sigma I squared, so I can use this model to estimate Sigma I squared,then I can go back and divide by Y I over Sigma hi hat and then we will estimate the original model, save the estimate of this squared model and use our Sigma I estimate to correct the original model to have an electronic asbestos guide. a little sharper, you look at this side of everyone's life.

I was going to show you my article in the New York Times and we did what's going to get there, so if you want to read the opinion page, if you want to read something I wrote there, son's name is. the space for debate I wrote something about why there is a lag between the change in GDP and the changes in unemployment why it is longer than it used to be so there is a debate between four or five myths there they will call me P before class, all the adult PVs like a quote in the garbage registry who was some kind of Dylan Merkley is going to introduce a proposal to try to raise housing prices by forcing people to say they have problems with their mortgages, forcing financial companies to amortize their principal and rewrite their loans. so they wanted to know if that was a good procedure or not, so give me a joke here.

I am not happy. Let's review the clipping. Here are the steps. Estimate the regression of the original model Y on beta 1 beta 2 on a constant. X 2 XK KS 3 in our example you can also generate it and get beta hat ll s is equal to X prime I in that's why I minus beta 1 hat minus beta 2 hat X at least minus beta K hat XK so this is recent Sabre, okay? It's already saved for you, they have it depending on the model, so let's take three a, that's the first model of the variation, so for 3a you would do a regression in the UI application. squared the constant and Z 1 a Z P this is a constant let me look at the constant c2 a Z this is the model so what we're asking here is the UI that squared is alpha 0 plus alpha 1 alpha 1 plus alpha 2 Z 2 I plus alpha P Z P So I make this model for the UI scripts with the original model save the UI squared, estimate this which gives us alpha hats, so you will get alpha hats that are already estimated.

I need an estimate of the variance of the Sigma hat. square eye is cubit 1/2 plus alpha 2 1/2 z 2i plus alpha P hat Z P I this step will be a little confusing when we go to the data but you have UI square but those are not the estimates you have to use UI experts estimate this model and then take the estimated UI squared from this model. This is really UI, hat squared, hat, you pick an estimate, Molly, anyway. It's your estimate of Sigma I squared, you need to use the estimated model here to get the estimate of Sigma squared.

Now you know what to do, divide it by Sigma. I have an estimate and that model will be completely efficient. Now you will have a model. that satisfies the gauss-markov actually they are completely consistent asymptotically it is completely efficient when I say asymptotically that means the number of data points goes to infinity. I didn't understand, but this will be an efficient procedure, so the next step is to divide by Sigma. I packed something you passed on from the Alpha hats. I'll show you this is very simple when you enter the program. I'll show you how to do this so this is easy for you.

I'll have why I'm over Sigma, it's equal to beta 1 as 1 over Sigma, plus theta 2 times X 2i over Sigma plus plus beta k @ ki Sigma plus UI over Sigma, so just estimate what that model, please estimate that model , call this. and star X 1 star X 2 stars X K star estimates that model and is completely efficient, so here is the problem with this technique. The first is that there is no guarantee that the Sigma that I have squared is greater than zero when I make this estimate here when I get this. Sigma I squared it's possible for one of these to be negative, there's nothing to guarantee it won't be negative, it can't have a negative variance so it doesn't make much sense when that happens, generally what we use is the square root. of U of Sigma Y hat square so this is negative just take the absolute value so use the absolute value so the corrections are just to use the absolute value so one step you should probably do before you do the division right here you should probably form a new variable which is the absolute value of this variable most of the time you won't need to do that, but to be safe in case there is a negative variance just take the absolute value and you will be fine, what should I do next time?

It's come back here, go over this one more time slowly with models, start looking, do an example to make sure you understand it, and then let go of your task. I don't expect to have all of this in their heads yet. I'll get there, I'll get there, but I'll keep working, yeah.