OK, so welcome to lab, two of econometric and statistical methods.
As with the lab one video, this is intended for those of you who are not able to attend the live session for the lab.
So we are working through problems based on this problem from the main textbook and you need to have access to. Sorry about that. You need to have access to this file. This eviews workfile and you should be able to find it on canvas. So in the lab material.
Then there's the question, how do you get that onto your?
Your University M drive. In order to use within the E views on App Center and one solution to that is to access the version of Firefox available on the Apps Anywhere Server. So you go to apps anywhere, open Fire Fox and from Fire Fox. You can then just navigate to canvas and download the file.
And so since you're then using Firefox on our University application server, you have the option to download it to your M drive your University drive. So that's the method I would suggest. Just don't download it via the Apps Anywhere version of Firefox since you need to be on apps Anywhere, apps anywhere. Any way to access the views unless you have your own.
Copy of the views, in which case you just download it to your own computer locally.
OK, so once you've done that then.
Well, the questions are based around estimation of a Cobb Douglas production function commonly used within macroeconomics.
So you also financial economics sometimes, so you can. You can also model creation of credit and other things by these sorts of production functions.
Um, so here we have it. Capital Y National Income is now in terms of capital, important labor employee. Have these parameters. Beta 1 beta 2 beta 3 so we have the elasticities of output with respect to capital.
In beta two and the elasticity with respect to labor in beta three. So that's their interpretation. You can workout the elasticities that beta 1 beta three I mean beta 2 beta 3.
Um?
And if we have this special case where beta 2 plus beta three is equal to 1, then we have constant returns to scale. So you can see.
This here so homogeneity of degree one.
So if we multiply.
K&L by TI. Then we're multiplying the whole thing by.
Cheats the power one.
Um?
And the questions are to test well, first of all, to estimate this model, but then also to test various null hypothesis.
So to begin with, just testing the individual significance of these coefficients after linearizing so you take the log of both sides to get a linear model or model it's linear in the parameters and then use this method of just adding an error term and then you can estimate the coefficients here by least squares estimation of log Y on log K and log L.
And so the first issue is to estimate, But then to test for the individual significance.
And then to test for.
Well, that's just going to the question so.
To test beta 2 equal to beta three, in other words, to test whether the elasticities of output with respect to capital and labor are the same. So that's that's the first.
Well, not the first. The first is to look at the individual significance, but the second restriction for testing is this one, and then the final restriction we're testing is the constant returns to scale, so that's in Part 5 and what you should find is that in parts three and four where you're being asked to test in two different ways, using two different formulas for the F.
Test statistic that you get the same answer wears in parts five and six. The two different F test statistics should give you different answers. Very different answers. So then actually Part 5 is asking you to do both and so then Part Six is asking you why in Part 5 you get different answers for the two different methods.
So that's essentially what you have to do here. First of all, just confirm the estimation results or estimate the models using eviews and then test for individual significance.
Test this one here. The equal elasticity and then do that using two different methods, so the using the residual sum of squares and using the R-squared values.
So the idea is here is not to use the pointer click methods or the scripting methods in eviews. But to compute these quantities manually and then compare with the critical values of the F distribution that are available on canvas.
So to do it manually here, but then manually again using these two, these two different formulas for the F statistic. But for the restriction of constant returns to scale.
So.
Yeah, and then to think about.
Exit the final exercise. You know why? Why you get a different answer in the case where you have this restriction to the case.
Where you have this restriction? So why? Why in the one case are the results the same? And in this case are not?
So the solutions are available online. You can have a look at them straight away, but I would advise you just to spend 20 minutes having a go at this yourself and then have a look at the solutions for about 10 minutes.
And then there's the next part of the video, but I'm just going to briefly go through the solutions, specially focusing on the final question. So actually implementing the test statistics is just calculation, but this one requires some more thought.
OK, so here are the solutions or hints for us answering those questions. So there is a work file so you can have a look at the.
The workfile for the estimation outputs.
So to get to see whether the individual coefficients are significant, you can just look at the eviews regression outputs and the TV shows and corresponding P values. So they are given just after the estimation stage.
Sue
to compute the I mean to perform the F test for the.
For the elasticities being equal.
In part three, you're being asked to do it using the residual sum of squares, and so for that you need to go to lecture 2.
And look at the.
The statistic for the I mean the F statistic that uses the residual sum of squares. So the E dash E for the unrestricted model and the ER dash ER for the restricted model. So you get those values from the from the estimation of the unrestricted and restricted models. So you get the residual sum of squares available to you and.
Then need to just plug those into the formula.
And you should get this number.
That number for the the test statistic then.
Is to be compared with the relevant critical value of the F distribution under the null, and it's one restriction being made in the null hypothesis. So the.
The value of G that you use is 1 and the N -- K here is for the unrestricted model and it's 26 -- 3, so this is the distribution of the test statistic under the null hypothesis.
And at the 5% level, the critical value you should see is this. I think in the version that I uploaded for the FF distributions critical values, it wasn't to this many decimal places, but you should see it's essentially this and clearly the value for the test statistic is greater than the critical value, so you reject the null hypothesis that the elasticities are equal.
And.
Um?
Not sure why we're rejecting it again.
So Part 2 is about individual significance, wasn't it? So? Parts three and four were about.
This test for the elasticities, so I'm not sure. I think this might have been some kind of typo.
For Part 4, we're doing the same as in Part 3, so we actually get the same value for the statistic, but it's computed using the formula for the S statistic in terms of R ^2.
So again, if you go to lecture 2.
So you go to the slide in lecture 2 where the F statistic is defined in terms of the R-squared for the unrestricted model and for the restricted model and.
Again, when you estimate the models, you get the R-squared value, so you estimate the unrestricted model. The restricted model. In each case you get the value for R-squared in the regression output and you just plug into the formula for the F statistic and you should get exactly the same number. So I got slightly different, but that's just due to, you know, some something numerical happening, I think.
So I can't.
So I'm not sure why there's this difference. I think it's just something some small numerical glitch that's giving you the difference, but it's very, very small. Essentially the same number, and obviously it's the same test that's being done, so it's the same distribution for the null.
I mean under the null, and so again you compare this with the critical value and reject the null hypothesis.
So in Part 5, the task was to do this again, so use both methods for computing the F statistic and this time for testing the null hypothesis that beta 2 plus beta three was equal to 1, and that's the constant returns to scale null hypothesis and you should find this results here. So based on the sum of squared residuals.
So using the same methods as in Part 3, the statistic is very small.
Based on the method in Part 4, so using the R-squared values you should find an S statistic that's very large.
And in the one case.
You know we using? I mean it's it's one restriction for the null hypothesis, so G is equal to 1 again N -- K for the unrestricted model is again 23, so 26 -- 3. So we are using the same critical value as we were for the previous test and this value for the F statistic is clearly smaller than the critical value, so we cannot reject the null hypothesis.
Whereas if we use the formula in terms of R-squared, is the F statistic formula that uses the the R-squared from the unrestricted model and then for the restricted model we get the opposite results. So this is much larger than the critical value and so we do reject the null hypothesis and so.
Strange result in part six. We asked why there were two different methods.
And or.
Why is only one method valid?
And the answer is that only this method was valid.
If you go back to lecture two and the explanation of.
Of how we get to the formula for the F statistic in terms of R-squared, we needed to assume that the SST, the sum of the total sum of squares for the unrestricted and restricted models was the same and.
That's something we can do in part three and Part 4 where we're testing beta 2 equal to beta three. So in imposing that null hypothesis, we don't change the Y variable.
However, when we test beta 2 plus beta 3 equal to 1 and impose that null hypothesis in order to estimate the restricted model, we do end up changing the dependent variable in the model, so so the the left hand side of the regression equation for the restricted model is no longer just capital Y.
So.
When we compute the SST, the total sum of squares it's going to be different, so the total sum of squares for members, the sum of squared values of the Y variable minus it, sample mean. And that's going to be different if the Y variable is no longer the Y variable. That's why it's what is it? It's Y minus something else, so.
That's why that's why we get two different answers, because the formula in terms of R-squared is not valid. The SST is different now in the restricted model, so only the.
Version of the S statistic using the sum of squared residuals is valid for testing this null hypothesis.
So for more details about that, you can see the video for Lecture 2, but also have a look at Slide 36 of lecture two and see that the two methods are only the same in the special case where the SST's are the same.
OK, thank you. Let me know if there are any queries by email I can go into it more next time.