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This question asks you to study the so-called Beveridge Curve from the perspective of cointegration analysis. The U.S. monthly data from December 2000 through February 2012 are in BEVERIDGE.RAW.
(i) Test for a unit root in urate using the usual Dickey-Fuller test (with a constant) and the augmented DF with two lags of curate. What do you conclude? Are the lags of curate in the augmented DF test statistically significant? Does it matter to the outcome of the unit root test?
(ii) Repeat part (i) but with the vacancy rate, vrate.
(iii) Assuming that urate and vrate are both I(1), the Beveridge curve,
$$u r a t e_{t}=\alpha+\beta vrate +u_{t}$$
only makes sense if urate and vrate are cointegrated (with cointegrating parameter $\beta<0 )$ . Test for cointegration using the Engle-Granger test with no lags. Are urate and vrate cointegrated at
the 10$\%$ significance level? What about at the 5$\%$ level?
(iv) Obtain the leads and lags estimator with $cvrate_{t}$, $cvrate_{t-1}$ and $cvrate_{t+1}$ as the I(O) explanatory variables added to the equation in part (ii). Obtain the Newey- West standard error for $\hat{\beta}$ using four lags $(\mathrm{so} g=4$ in the notation of Section 12.5$) .$ What is the resulting 95$\%$ confidence interval for $\beta$ How does it compare with the confidence interval that is not robust to serial correlation (or heteroskedasticity)?
(v) Redo the Engle-Granger test but with two lags in the augmented DF regression. What happens? What do you conclude about the robustness of the claim that urate and vrate are cointegrated?

First one. The test for a unit root in series you rate unemployment rate using the usual dickey fuller test with a constant yeah. And the augmented dickey fuller with two legs of change of unemployment rate. I find that seven both times we are unable to reject the now hypothesis that unemployment rate series is a unit fruit. The legs are not significant. However, the significance of the legs matters. So the outcome of the unit root test, we will repeat what we have done in part one two series vacancy rate and report the result in part two. I guess similar result. So the rate is a unit root. Well part one and two. I use package the R. Package A. T. S. A. And the function is a D. F. Dot test. R. Three. We assuming that unemployment rate and vacation re rate are both integrated of level one. We test for co integration using the angle grandeur test with no legs. So the step the steps are as follow. We first regress, you read on the rate then we yet the residual and we run the key fuller has on the residual to see whether the residuals our unit root. I find that you're right and we rate Arco integrated at the 5% level. Yeah Heart Forest. I get the leads and lacks estimator of the change in vacancy rate and I did note that uh CB rates up minus one. This is for the lack and plus one is for the lead. This is a regression result. So the usual centered errors are in green and in round brackets, the robots that Iran's are in blue and in square brackets you can see that the main estimate on vacancy rate is highly significant. This one is not correct. So the centered errol the usual one for the estimate of the first lack of change in vacancy rate is 164 In all cases except for the estimate of the lead of C. V. Right. The robust standard Iran's are larger than the usual standard errors. This is usually the case it happens but rare that the robot standard errors are smaller than the usual standard errors. The r square of this regression is 0.77 So for the rate, because the robot standard error is larger than the usual standard error. So we will get a wider confidence interval if we use a robot standard error and for confidence interval you will run this function in our count in and you impose the name of the regression. It was spits all the 95% confidence intervals for all explanatory variables. The default version is the 95% interval. But because the standard barrel of this estimate is are very close, two versions are very close to each other so the confidence intervals should be roughly equal. Yeah. Last part. What you could say about real business of the claim that you rate and the rate are co integrated. Yeah. When I run the test and good grandeur, the results are not consistent across alternative types of process. In one case I can reject the notion that the residuals are united and for all the cases I cannot reject. So I conclude that the claim that you rate and be rate our co integrated is not robust.

Yeah. All right guys, the first thing we need to do here is we do identify on are null and alternative hypothesis are no hypothesis. Excuse me being that the variations are going to be the same. And the turn of prosthesis is that they're going to have different variations as you can see right here. Okay, so the next thing I did to solve the problem was I opened a google she and I am putting a lot of the data. You'll notice all the data is and put it right there. Okay. And then, so what I'm gonna do from there, what I'm gonna do from there is this I'm gonna go ahead and calculate the variances for each. The first variance I'm gonna calculate using google sheets. So I'm gonna type equals var open parentheses and then just highlight everything I want. I know have an extra box in there. That's not a big deal. And there is my first variant. Second variants will be found the exact same way it looks like it already knows what I want, which is great. And there is our second variance. Alright, so from there I need to find my f statistic. Okay, so my F statistic is just going to be the greater variance divided by the smaller variance and there we go, there's my f statistic. Once I found my F statistic, I'm ready to find my P value. Actually. Sorry, scratch that. Once I find my F statistic, I'm almost ready to find my P value. But there's one more step in order to find a P value. Using google sheets with an F statistic. First need to calculate degrees of freedom. Okay, very easy to do in google sheets. So degrees of freedom is just all the data, the amount of data you have minus one. So how do you find that? Google sheets is equals count. Just highlight everything. I have noticed how I do have an extra box there. That's not a big deal to subtract one. And that I have 20 in this case 24 degrees of freedom. And in our second case again equal count Open parentheses, highlight everything and I do need to subtract one. And the other case have 15 15 degrees of freedom. Okay, so now I'm ready to find my P. Value. So my P value uh through here is going to be something called an F. Distribution or F. Dist. And then open parentheses. I need to put three values. Number one, my F statistic. The second value is going to be the number of entries from S one squared. I'm sorry the degrees of freedom from s one square in this case is 24. And my last and final value is the degrees of freedom from my S two squared. So that's right here. I close all that up and there is my P value. However, I'm not quite finished yet. Okay, that is a P value for a one tailed distribution. What do I mean by that? That's a good P value. If that was a greater than or less than sign, this is a not equal to sign. Okay. That's not equal to sign. Which means um uh that which means that I have actually had the wrong P value right now. Since this is for a one tailed distribution and I need a two tailed distribution because they're not equal sign, I need to take this value multiplied by two. So equals I click on the value multiply symbol in google sheets. Asterisk hit A two and I'm done. That is my P value. Notice that P value is just a touch above 0.05, meaning I'm going to fail to reject the no hypothesis. I'm gonna fail to reject. And all hypothesis

The following is a nova test based on the mean salaries for different metropolitan areas. So the alternative or the null hypothesis is that all the means are the same. So there are six metropolitan areas, I think it goes Chicago, Dallas Miami, Denver san Diego and Seattle. Uh So the null hypothesis is that all the means are the same. And then the alternative is that at least one of them is different. The second step is to find the critical value and you can do that using either software or a table, But they're essentially three things you need. The first thing is your alpha value, your significance level and that's usually given to you the problem and that's .05. Then you need the degrees of freedom for the numerator and the degrees of freedom for the denominator. And the way you find that Is the degrees of freedom for the numerator is the number of categories -1. So there were six cities that we looked at our metropolitan areas, so 6 -1° of freedom would be five for the numerator. And then for the denominators, the total number of data values minus the number of categories. So there were 36 data values minus the six metropolitan areas. So 30 is your degrees of freedom for the denominator. So that should be enough to use a table. But I use a calculator and I wrote a program in here called inverse. F. I'm not going to show you how to how to write the program. You can youtube it if you wish. Um But this is what I do. So um I put in my area which is my alpha value, my degrees of freedom is five and then my degrees of freedom for the denominator is 30 and That gives me my critical value. About 2.534 2534 is my critical value. I call f. star. So 2.534. Okay so anything greater than 2.534. We reject the annual hypothesis that all the means are the same And anything less than 2.534. We failed to reject meaning the h not is true. Okay so the second step is to find the F statistic and there's a formula but it's a bit of a mess. I always use software you know technology is a great thing. So if you go to stat and you can type in your data values. So these are the mean salaries um So again L1 I think was Chicago and then this is the mean salary for Dallas Miami Denver San Diego and Seattle. So there are six categories. And if you go to stat tests and then we're gonna go to the Unova test and then you just type in your columns separated by commas remember there were six columns, six data columns that we used and we need to make sure that all of them are in there and last one and then also you know make sure you separate those by commons, otherwise it's going to read it wrong. So then um that gives us everything we need. So the F. Is the F statistic, that's the third step. So we're looking at this it's about 2.281 as our F. Value. So two point 281 is our f statistic Which is actually barely in the non rejection region 2.281. So that means we fail to reject. Okay and also we can verify that with this p value here. So the p values 0.7 which is a pretty small p value, but it's still in this case greater than the alpha value. So the alpha value remembers point oh five, so it's barely greater than the alpha value. And whenever it's greater than the alpha value, uh we failed to reject, I should probably put H not there, so we failed to reject H not whenever the P values greater than the alpha. Okay. So then the last step is to summarize everything with actual words. So what does this all mean? It just means that there is not sufficient evidence, there is not sufficient. I guess you could say statistical evidence to suggest that the mean salaries from the different metropolitan areas are different. Okay. And that's the five step process for an Innova one way and over test

Uh, h note in that new one is mother than or equal to me too. And each one is that new one is bigger than mutant. So the degree of freedom, which is the minimum off anyone Where? This one in two minus one. So is 16 and 13. So the minimum 13 and the critical value in the row Corresponding toe degree. Freedom is off. Ableto appoint one on table five. So the critical value is 1.35 So the distances is Is that x one bar minus X tow bar over. Squared off S one squared over N one plus is two squared over into. She is equal to 3.429 So at 3.429 is it's a bigger than 135 So we reject this offer is won t This value is from table five


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