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Question

STATISTIER (20131.- Previous The Jclivuly; piedictlona [ oi Dunlosky cottelalion . computed Iuee couelton {eu [ne nrcde comolcie daba analysts ( malan Tctwich [Cam (rom thut Analyals / brlow Now Your involving CaM *coi cotutruling Ine Oa [fue 'cottelations Cl for each oftheEim SrixtExtn ? [Tot Fk 00] Jennon Cutuallo 50 (1-EandiPracbor RetiatPrareon Corrkon54 (I-lated)~Potntn Carellbo 50 (H-tatediRoRondn_schtoPeareon Coneuton503 (tud)CenHancanLOI audt-.taJiClaudia aircady started wotking on

STATISTIER (20131.- Previous The Jclivuly; piedictlona [ oi Dunlosky cottelalion . computed Iuee couelton {eu [ne nrcde comolcie daba analysts ( malan Tctwich [Cam (rom thut Analyals / brlow Now Your involving CaM *coi cotutruling Ine Oa [fue 'cottelations Cl for each ofthe Eim Srixt Extn ? [Tot Fk 00] Jennon Cutuallo 50 (1-Eandi Pracbor Retiat Prareon Corrkon 54 (I-lated) ~Potntn Carellbo 50 (H-tatedi RoRondn_schto Peareon Coneuton 503 (tud) Cen Hancan LOI audt-.taJi Claudia aircady started wotking on the 95% Ci for the practice rctricval and cxam score Sorrclation She used APpendix G t0 convert the sampie correlation of r 2264 Shc then plugged all the vilucs into the upper bound formula [0 a2 266l convered the upper bound z back and computed it_ Then she into an r by using Appendix G Her work below Upper boundary=(1,)-(rkjN5 2661+1.96 7oz - Upper boundary 2661+1.96 7o9 ~2661 - 96 ( 1005) 2661+.1970 = 44631. conversion: E of 4631 equal [0 an r of 43 (from So, alter Appendix G) using Appendix rclation In the population . rConvert the upper bound putations nd ( conversions approximately r Now: help Cack O an % the upper bound for the lower boundany Claudia by for the cOr - completing Lowct all of the com- boundary e(4,)-(za) Tou should have found [ (0 be .0701. Now = the lower E boundary 'compute the 95.% CIBC 'for the z to be.069 for the orher and the 'correlation E lower between- boundary spacing for the F and exam score 54na



Answers

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.

Part one. The test with strict exhaust Janet E gifts is row hat equal to minus 0.97 and the T value of minus 2.41 The regression that includes growth of minimum wage and growth of C P. I gives row hat equal to minus point 098 and it is statistic of minus 2.42 Roughly the same T values and roll had value. Therefore, we find evidence of some negative serial correlation and it does not matter which form of the tests we use. Yeah, this is the regression equation for part two and three with three types of standard. Erin's Yeah, note that the estimates are not changing. Only the standard errors change. The first line of standard error is the old LS the usual and probably incorrect type. The second line is the new E West standard payroll type, and the last line is hetero skate elasticity robots standard errors we have over 600 observations and the are square are the same for three types of standard herons, which is 30.293 So comparing the new e West standard error and the usual L s standard error for the variable growth of minimum wage. We find that the new E West than it Errol is much larger than the old l s one roughly four or five times larger. But for the variable growth of C P I, the new E West standard error is actually smaller than the old l s one that is the answer for part two. And for part three, we consider the hetero Scholastics city robots standard error with the newly West standard error, we don't find much difference. Yeah, So the difference between the two type of standard error is that, um, the last one only controls for different variants of the Errol terms. That's why it's called hetero Scholastic City. Robust, but new E was standard. Errol does more than that. The new E West Standard Erin's our robots to both hetero Scholastic City and serial Correlation, as we find little difference between the two, is probably because the negative serial correlation adjusting the standard error on, um, CP, I actually reduces it. Hetero Scholastic city does not have a major effect on the growth of CP I standard error. That is part three and part four. We run a BP test, We get F statistic equal to to 33.8. Yeah, which means the P value is almost zero. There is a very strong evidence of hetero ski elasticity. Part five, The usual F test is 4.53 with a P value of pointing 058 So, in the static model using the hetero Scholastic City Roberts T Statistic lead Teoh a less significant minimum wage effect. Okay, Actually, this one is there. This is for the usual F test and this is through the hetero stick elasticity. Robust test. All right, so the hetero see elasticity robots test for the legs show a very strong significance of the wage effect. Part six, the new we West version of theme F statistic is about 7.79 which show even mawr Statistic. Significance, then just the hetero Scholastic City. Robust statistic. So at just in the F start for hetero scholastic city or hetero ski elasticity and 12 order serial correlation leads to the conclusion that the LAX are very statistically significant. March 7 with 12 legs, the estimated long run propensity is about 0.198 and without the legs. The estimated L R P is just the coefficient on the growth of minimum wage. Yeah, which is 0.151 So when we include the LAX, L R P is about 30% larger using the new We West standard error the 95% confidence interval for the l r P. ISS from yeah, hauling 111 to you Point Thio 84 which easily contains the estimate from the aesthetic model.

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.

We're gonna be using Chevy Chefs rule from the Z score, we calculate uh that's the score is going to be Our X Value 205,550 -220258. So some big numbers with a pretty small difference, relatively A standard deviation of 5037 for the population gets us a Z score of negative 2.82 So we're gonna be plugging that into the cake for Chevy chefs of inequality. Here it is. Mhm. Well, I don't know why bother writing the negative. It's gonna be squared so positive by the way, And this is approximately equal to 87.4, Pretty high percentage. Uh we can therefore conclude that the other houses are above $205,000 and that this home is a good deal. And obviously we can use tv shows because it's um relatively high Z score with a good percentage


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