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ECON 3210 (21)
Lecture

SolutionsChapter8.pdf

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School
York University
Department
Economics
Course
ECON 3210
Professor
Razvan Sufana
Semester
Fall

Description
CHAPTER 8 Exercise Solutions 271 Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 272 EXERCISE 8.1 2 2 When i N N N  x 2 22 2    xx  i1i1i i  i1 i  2 2 2  2 N   N 2 2 2 2   ix i i xx  xx   i  x  1 i1 i1 i1 Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 273 EXERCISE 8.2 1  2 (a) Multiplying the first normal equation by   i i  and the second one by  i yields  2  1 ˆ ˆ  1 2        i i  i 1 i i yx  i i i i   i i i i ix 2ˆi i i   * x x Subtracting the first of these two equations from the second yields 2      y 2 xy      * ˆ 1  1      i i i i ii ii iii 2 Thus, 2* x y x y 1   i i ii ii i    2  2   2    i i i i x   yx    y x i 22 2i i i  i     2   i ix  i i  2     i i   In this last expression, the second lin e is obtained from the first by making the   1   1 substitutions yi ii and xiii x , and by dividing numerator and denominator by 2 2 2 ˆ ˆ 1     ˆ   i  . Solving the first normal equationii i ii 1 2  x  y for  1   1   1 and making the substitutions yi ii and xiii x , yields 2 2   ˆ   i x i i  1    2   i i   2 2 2 2 2 2 2 2 (b) When i for all i, i ii ii yx ,  i iy i ,  i i i x , 2 2 ˆ and  i N . Making these substitutions into the expression fo2 yields   xii    yi i 2 22   yii  yx ˆ N NN    N 2 2  2 2 2    xi i   xi x 2  2  N N N    ˆ and that for 1 becomes  2 2y x   ˆ 2 1ˆ 2  i i      yx N N  2   These formulas are equal to those for the least squares estimatorsb and b . See pages 52 1 2 and 83-84 of the text. Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 274 Exercise 8.2 (continued) (c) The least squares estimator1and b2are functions of the following averages 1 1 1 1 2 x   xi yy  i  xii  xi N N N N For the generalized least squares estimator 1ornd 2, these unweighted averages are replaced by the weighted averages   2   2   2   22   i i   i i   i ii   i i  i2  i2  i2  i2         In these weighted averages each observaton is weighted by the inverse of the error variance. Reliable observations with small error variances are weighted more heavily than those with higher error variances that make them more unreliable. Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 275 EXERCISE 8.3 For the model e i ii 1 2  where var(ei i 22 , the transformed model that gives a constant error variance is yi i x1 i2  * * * where yi ii x , xi i x , and ei ii . This model can be estimated by least squares with the usual simple regression formulas, but with  and  reversed. Thus, the 1 2 generalized least squares estimators for 1 and 2 are ** * * ˆ  ˆNy i i i i and  y x * 1 1 2 * 2 * 2 Nx ()i i  Using observations on the transformed variables, we find y  7 , x  37 12 , xy*  47 8 , (x )  349 144  i  i  i i  i With N  5 , the generalized least squares estimates are ˆ 5(47 8) (37 12)(7) 1 2 2.984 5(349 144) (7 12) and ˆ ˆ * * (37 12) 2 1 x  (7 5) 2.984  0.44 5 Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 276 EXERCISE 8.4 (a) In the plot of the residuals against income the absolute value of the residuals increases as income increases, but the same effect is not apparent in the plot of the residuals against age. In this latter case there is no apparent relationship between the magnitude of the residuals and age. Thus, the graphs suggest that the error variance depends on income, but not age. (b) Since the residual plot shows that the erro r variance may increase when income increases, and this is a reasonable outcome since greater income implies greater flexibility in travel, 2 2 we set up the null and alternative hypotheses as the one tail test H 0 1 2 versus 2 2 2 2 H 1 1 2 , where  1 and 2 are artificial variance parameters for high and low income households. The value of the test statistic is 2 (2.9471 10 ) (100 4) F  2 7  2.8124 2 (1.0479 10 ) (100 4) The 5% critical value for (96, 96) degrees of freedom is F(0.95,96,96)401 . Thus, we reject H 0nd conclude that the error variance depends on income. Remark : An inspection of the file vacation.dat after the observations have been ordered according to INCOME reveals 7 middle observations with the same value for INCOME, namely 62. Thus, when the data are ordered only on the basis of INCOME, there is not one unique ordering, and the values for SSE and SSE will depend on the ordering chosen. 1 2 Those specified in the question were obtained by ordering first by INCOME and then by AGE. (c) (i) All three sets of estimates suggest that vacation miles travelled are directly related to household income and average age of all adu lts members but inversely related to the number of kids in the household. (ii) The White standard errors are slightly larger but very similar in magnitude to the conventional ones from least s quares. Thus, using White’s standard errors leads one to conclude estimation is less precise, but it does not have a big impact on assessment of the precision of estimation. (iii) The generalized least squares standard e rrors are less than the White standard errors for least squares, suggesting that genera lized least squares is a better estimation technique. Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 277 EXERCISE 8.5 (a) The table below displays the 95% confidence intervals obtained using the criticalt-value t(0.975,497)965 and both the least squares standard errors and the White’s standard errors. After recognizing heteroskedasticity and usin g White’s standard errors, the confidence intervals for CRIME, AGE and TAX are narrower while the confidence interval for ROOMS is wider. However, in terms of the magn itudes of the intervals, there is very little
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