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Use the data in FERTIL3 for this exercise.(i) Add $p e_{t-3}$ and $p e_{t-4}$ to equation $(10.19) .$ Test for joint significance of these lags.(ii) Find the estimated long-run propensity and its standard error in the model from part (i). Compare these with those obtained from equation $(10.19) .$(iii) Estimate the polynomial distributed lag model from Problem $6 .$ Find the estimated LRP and compare this with what is obtained from the unrestricted model.

(i) See video. Lags are jointly insignificant (ii) LRP = .129, se = .03 (iii) LRP =.135, restrictions are not rejected by the data

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Chapter 10

Basic Regression Analysis with Time Series Data

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hard one. This is Theis, estimated equation. We have several explanatory variables. P E Sebti PSF team minus one up to T minus four, and the coefficients are in blue ink, and their standard errors are in green inside the bracket. Besides P E and it's lack, we also have another two explanatory variables. Double W two T and pill sub d. This is a regression equation with an intercept. The dependent variable is G f R Sebti, and we have 68 observations with the Art square and adjusted our square as follows. Yeah, Given this estimation results, we can conduct an F test where we test the joint significance of P E E sub t minus three N p e sub T minus four. So the 3rd and 4th leg of P E the P value off the F statistic is about 0.94 Yeah, which means that yeah, we cannot reject the null hypothesis. The evidence against the joint significance of these too lax is very weak. Let's move to Part two. We have a, um we need to estimate the long run propensity, and it's standard error. It is actually the coefficient and standard error on P E Sebti in this regression, so we will regret G r f sub t on p e sub d. And the difference between every lack of p e with, um, the difference between every lack of peace empty and peace empty itself. Also, we have two other explanatory variables. W w two and pill. The result. We get the beta of P E. Sebti. No, wait ahead. Actually, it's the long run propensity. It's equal 0.1 to 9 with a centered error of 0.3 So it is significant. This estimate is oh, a little bit above the estimated long run propensity with Ali two legs, which is 0.101 or three. We will estimate the polynomial distributed lag model with to additional variables W W two and Pulte Pierce Empty. Yeah, to estimate gamma gamma, not gamma one and gamma two. We define three variables. Z not Z one and Z two z Not is the sum of P E Sebti and its legs up to four. That and the one is the sum of P E sub T minus one up to B E 17 minus four. Same to Z two. The only difference is in the, um the weight. As you can see here, Z one and Z two are different. Linear combination of P 70 minus one up to pee 70 minus four. That is the first step. And in the second step, we will use these newly defined variables. We win regress G R F t on the three variables and the two additional variables. We will get gamma not as a point. 069 gamma one minus 0.57 gamma 2.0 12 And based on these Gammas values, we can calculate This should be ah, Delta. We can calculate the value of delta not to Delta two and Delta, not with equal gamma notes. So it is 0.69 Again. My one No, sorry. Delta one is gamma, not plus gamma, one plus gamut. You and plug in the numbers. You wouldn't get 0.24 Next we have Delta Three, Delta to Delta three and Delta four. Mhm. I forgot you make space for Delta three and four. It's okay. So Delta two is gamma, not plus to gamma one plus four Gamma too. And you can get 0.3 Delta three is Gamma, not plus three gamma one plus nine gamma two. And you will get point 006 for Delta four. Delta four is the sum of gamma, not four gamma one and 16. Gamma too. The value of the Delta four is 40.33 The long run propensity is the sum of all the data. Yeah, from Delta one, Delta zero to Delta four. You wouldn't get the value of the long run propensity as okay, 0.135 This is a little bit above their point. 1 to 9 we get from the unrestricted model in part two. The last part of part three, we will run a an F test. We will test the restrictions imposed by the polynomial distributed lack model. Recall the formula of the F test. We we need the ah square of the unrestricted and the restricted models. Que Here is the number of restrictions. Kay is the number of independent variables and n is the number of observation. Let me write down. We have okay. The R square of the unrestricted model as 0.537 Our square of the restricted model is slightly smaller. 0.536 we have to restrictions and and minus K minus one is 60. We will get the value of the F statistic as zero 0.0 65 This is a very, very small value. It is insignificant. We cannot reject the null hypothesis. Okay. In other words, there restrictions are not rejected by the data.

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