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Reconsider the sugar cane supply response model that was discussed in lecture: Consider the following specification of the model:Ln (AREA:) = B +Bz (PRICE): + â‚...

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Reconsider the sugar cane supply response model that was discussed in lecture: Consider the following specification of the model:Ln (AREA:) = B +Bz (PRICE): + €tYou are given three sets of output: (1) OLS output which utilizes the OLS standard errors: (2) OLS output which utilizes the OLS output with HAC standard errors; and output under the assumption that the error is an AR (1) process. You are also given correlogram for the OLS output: (35 points)Given the correlogram the OLS output; determ

Reconsider the sugar cane supply response model that was discussed in lecture: Consider the following specification of the model: Ln (AREA:) = B +Bz (PRICE): + €t You are given three sets of output: (1) OLS output which utilizes the OLS standard errors: (2) OLS output which utilizes the OLS output with HAC standard errors; and output under the assumption that the error is an AR (1) process. You are also given correlogram for the OLS output: (35 points) Given the correlogram the OLS output; determine what autocorrelalions are significantly different from zero_ Conduct form test (e.g calculate significant bounds and compare correlogram) Interpret results: [5 points] Conduct formal test for autocorrelalion using OLS results utilizing Durbin Watson Statistic at a=.05. [15 points] Set up null hypothesis and alternative hypothesis for conducting Durbin-Watson Test If results indicate evidence of significant auto correlation, examine the output for the AR (1) model to examine if the model has been corrected for autocorrelation: Conduct second Durbin Watson test utilizing output from AR(L) model Find 95% confidence intervals for Lhe elasticity of supply for the (1) OLS model, (2) OLS model using HAC standard errors, and (3) the ARI model Compare and contrast the ARI interval estimates with the other 2. [10 points] Would ignoring autocorrelation (if present) lead unreliable estimates and tests of statistical inference Explain thoroughly. points) [5 points]



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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.

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.

Part one. The coefficient on Bs is about minus 0.52 Okay, it's usual. Standard error is 0.110 Please give a T statistic of minus 4.7. Remember, the teen statistic is calculated by taking beta hat divided by its standard error. So you will take minus 0.52 divided by 0.110 Yeah. Even these t statistic, right? For the case of usual standard error, we can conclude that beta hat of BS is highly significant. This T value is much larger than the critical value of 2.33 So you may see in your, um, statistical program statistical software. This variable should be significant at the 1% level just for the usual standard error. Okay, When we estimate the robust than that error, we get a much larger number. We get point to a three, and that reduces the T statistic. The new T statistic is about minus 1.8 and even the critical value the smallest, smallest critical value of one. That is, for the 10% level of significance. We can conclude that the coefficient of be as variable is marginally significant. If you use a statistical software. You may see that this variable is significant at the 10% level. So because the standard error has changed by a lot, when we move to the option of robust, we may conclude that there is serious hetero scad elasticity in the data. This is the symbol for exist. The estimate of this beta is not small. Yeah, by economic sense. Although it is not close to the hypothesized value of one. Yeah, should be minus one. Okay, let's move. Thio Part three So I switched part two and three by Mystic. We will get to part to you. After this we will include four dummies in the regression. Okay? And we can see that after we include the dumbest. Yeah, the estimates are still the same. Okay, Before dummies do not change our result. And looking at the coefficients of the dummies, we see that all the observation 15 08 only this observation has a significant coefficient. It is. Statistic is very large. Minus 6.1. You may see that none of the other three dummies is significant even at the 10% level. Okay, Back to part two. We will drop four observations with variable B s greater than 40.5. Then we will re estimate the initial regression equation. And we look for the change in the beta hat on BS. The beta head we got is minus one point. Sorry. Minus pulling 86 And the robust T statistic is minus 1.27 So the practical significance of the coefficient is much lower. And given this T statistic, this coefficient is not significant at while the 5% level, you should check that again. So this T value is still larger than one in absolute value, but it is very close to one. Would you like to hear from you? That's why I guess it is not statistically significant at the 5% level, but maybe at the 10% level. But we use the 5% level as the standard. Okay, let's move to part four. And in this part, we win. Estimate four regression in each regression we win. Drop one. Observation One influential observation from the sample Recall that initially before dropping any observations the beta had of bs is minus 0.52 Now, if we drop observation 15 08 only we will get beta head off Bs as minus 0.20 If we drop observation 68 only beta head is minus 680.5. Next observation, 11 27. Beta head is minus 0.54 And the last observation 16 07 We get minus point 53 So the only big change happens when we drop observation. 15 08 We can conclude that the estimate is Onley sensitive to the inclusion of observation 15 08 part five. Yeah, The results we got so far reveal that if an observation is extreme enough, it can really influence the O. L s estimate we have a large sample in this problem are simple is about 1800 observations. However, the old L s estimates has shown earlier are not immune to the influential observations. Let me write that down. Extreme observation. Yeah, can influence all L s estimates substantially even when the sample is large. In this problem, the influential observation 15 08 happens to be the observation with the lowest average salary and the highest Bs ratio. So including this observation clearly ships the estimate toward minus one part six In Part six, we will adopt a new estimation method. We will use L A D. Which represents least absolute deviation. This is a substitute method for our l s in the case, we have influential observations. We don't need to drop these observations from our simple when we use thes method. Unlike O l s, uh, s maximize. Sorry. Minimize the sum of square of errors L a de minimis the sum of absolute values of the residuals. The L A D. Estimate of beta head on B s with the foreign. Simple is minus 0.109 And the T statistic is minus 1.1 When we drop observation 15 08 from the sample. An estimate The regression equation with L I d. Again, we get beta hat on B s as minus 0.97 and the T statistic is minus point 81 Looking at these numbers, you can see that the change in the estimate is modest and based on the value of the T statistic, very close to one. Beta hat on bs is statistically insignificant. In both cases,


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