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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.

Section 3.6 Problem number 1 14 were doing a graphic exploration here. So what they give us is sort of fairly complicated. Here is a trick function, okay? And this trick function sort of a trick polynomial. If he were, there's a really good job of estimating the salt to function. And what they wanted to do is to do an exploration on that. So I've got assaulted function, which is this piece wise linear function. Okay, which you see, um, all graft there. Um, and what they want me to do is to say Okay, well, how well does this trick polynomial? Approximate that. And what you can see is pretty good. So if you look at it, um, you see, it has troubles right there at the turns, So it has a little bit of trouble every time there's a sharp corner, But otherwise, you can see this trick polynomial. You know, it's weaving a little bit, but it's sort of hard to tell. So that's a fairly good approximation when we look at what's going on here. So from, um, negative pie, too. Hi. And that should be a should be good. So that's a good approximation. So what they want us to do is to find the derivative of this function. So I went ahead and wrote it out. But really, I mean, it's how do you differentiate co sign you get a minus sign, and then you're gonna have to in front of it. So really, you see the pattern It's taking this expression that you see right here making it positive, putting a two in front, changing the co signed to sign, same thing putting the six in front, making it positive and changing the co sign to a science that this would be the derivative function that we see showing up. So now let's see what would be the derivatives. I want to take this function off. What would be the derivative of this function? Okay, if you look at this solitude function, um, it's going to be, you know, the slope is gonna be one on this first piece. Negative 11 and negative one. So what does the derivative look like? The derivative is just gonna oscillated between these values. So this is what the derivative of the salt to function looks like. OK, so it's gonna be, um oscillating between one and negative one and it's gonna be undefined these corners. Now I graphed the trigonometry trick in a metric polynomial. What would its derivative look like? And that's what you see here. So in this case, I'm looking at okay, When I graphed the derivative of this polynomial, I'm seeing that OK, things air sort of quite different. So my trick polynomial was pretty good in estimating the, um the function, but pretty bad when it comes to estimating the derivative will follow up on this in the next problem. So again, the overall consensus was I could use thes trick Modelo meals pretty well for estimating this salt to function. And it appeared to have troubles that it appeared to have troubles at the points that they're gonna be discontinuities in the derivatives. So it was a pretty good approximation, except for those points where that would happen. OK,

Okay, so I actually learned something new today. Apparently, there's a formula that kind of tells you when a fledgling bird is going to be able to fly on its own. And it's the ratio of two functions of time, one of which kind of gives an indication for how long their wings are and the other one is their body mass. Okay, so whenever these ratios kind of approach one, whenever FFT approaches one thing, the fledging fledgling is able to fly on its own. You didn't know all this question is asking us to do is to interpret the physical meaning behind those and described the units associated with them. Okay, so the 1st 2 shouldn't be too difficult, right? Because M prime of tear is the time derivative. It's the time derivative of em of tea. It's a rate of change. It's how fast this function changes with respect to time. And since we're given that the average body mass is measured in grams, this is gonna be essentially grams per per unit time and that the time is in weeks, by the way, since grams per weeks and just like I said, ah, the interpretation of physical meaning is how fast the body mass is changing with respect to time. It's the rate of change of body mats. Okay, for w prime of tea, that's again the time derivative of W T. Which is the length of the wings. And since they said that wing length was gonna be measured in millimeters, this rate of change is millimeters per week. And again, it's just how fast the length of the ones you're changing F prime of tea has to have special analysis because, um, f prime of TIA is the time derivative of f of tear. But f of tia is the ratio. Oh, the length of the wings and the average body mass. These air two functions that change. So we actually do have to use the quotient rule, which is lo de I minus high. Do you low swearing the bottom and we're gonna determine the units of this function As far as a physical meeting goes out, I said half of Tia as FFT approaches one, then the fledgling is gonna be more able to fly. This is a rate of change of that. So if this is a really, really positive number, then it's going to rapidly go towards, and it's gonna take a lot less time for to be able to fly. Maybe it's some sort of growth spurt. So all that remains to do is to find the units of this I'm of Tia. Ah was measured in grams w prime of tea we said was millimeters per week. Okay, W of tea waas um millimeters and M prime of tea is grams per week, divided by AM of T squared M of T is measured in grants. This is just grams squared. Okay, so we have a grams times of millimeters minus of grams, times of millimeters, all divided by a week. Okay, so this is gonna be some sort of of grams times millimeters. We don't know. Of course you know how exactly these variables are changing, so they're obviously not gonna be the same necessarily. So when you subtract two things that are like each other, you're gonna get something else that's like each but that's like those two things. So this is gonna be some grams millimeters divided by weeks, all divided by Graham Sward, which is gonna be grams times, millimeters times, weeks times one over. Graham squared. One of these grams cancels with one of those, and then we're left with millimeters Her grands week. Here we go.

All right. So really can't this trick in a metric polynomial, and I'm not going to write down the derivative function. Andi, I just did everything in a computer. Okay, so we do see a graph here that f gives a good approximation to the salt, whose function and this is actually really hopeful can lots of different things. Maybe I just won't even get into it. But you should believe me that this is a very useful thing. I mean, you might be thinking. I mean, you gotta be thinking I don't know what you're thinking. Maybe you're thinking, what's the point of this? But this is extremely important in image processing and a sound engineering being able to approximate functions like a salt tooth function in terms of colon quote nice functions like trick panem mules or other yeah, functions that are easy toe to sort of studies in calculus. OK, that's my That's my a few seconds of of advertising. Okay, so we see that this trigonometry polynomial is his nicely approximating on the salted function. But we're interested in the derivatives. Okay, So GFT is that sort of function, and so ji prom, if t so What is d prime? If thine it was just the slope. I mean, it's just a little piece. Wise regions of constant slope. So here's pie. Syria. Hi. And so Betweennegative pie and, uh, me. Okay. Surveillance plot. Cheap priority. So between negative pie and pie over too has a slope of wine in between pirate tunes. Zero as slip of Thank you, Dave. And then against one the slope of negative one. Okay. And I just think the derivative on the calculator and plotted death prime of tea. And so what's gonna happen is it's gonna actually start down here kind of wiggle. Down we go on, we go down, we go on back. Okay, So that's what the derivative that private he looks like. And so you'll notice that f is getting pretty is pretty good at approximate prime is a good thing to get approximately g prime. When g prime, it's just constant. Just kind of wiggles around, but then jump step. And so what? Thes in points negative by a negative IRA to zero higher. Too high where there's a jump in the function. It's really hard for f crime two. Approximate because it sort of stuck. Do I? Do I stay away. Appear one or jump down to negative one. So it has to jump down really quick in the actual value on this is sort of like a deep idea. Sort of ends up being exactly in between these these two sort of left hand limit and write him limit. Uh huh. Yes. Along. We're excited about that, but, uh, that's kind of the idea is going approximate well, except where there's jumps in the function, which is that negative pie and produce your power to comply.


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