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Chapter 8

ECON 4550 Econometrics Memorial University of Newfoundland. Heteroskedasticity. Chapter 8. Adapted from Vera Tabakova’s notes . Chapter 8: Heteroskedasticity. 8.1 The Nature of Heteroskedasticity 8.2 Using the Least Squares Estimator 8.3 The Generalized Least Squares Estimator

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Chapter 8

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  1. ECON 4550 Econometrics Memorial University of Newfoundland Heteroskedasticity Chapter 8 Adapted from Vera Tabakova’s notes

  2. Chapter 8: Heteroskedasticity • 8.1 The Nature of Heteroskedasticity • 8.2 Using the Least Squares Estimator • 8.3 The Generalized Least Squares Estimator • 8.4 Detecting Heteroskedasticity Principles of Econometrics, 3rd Edition

  3. 8.1 The Nature of Heteroskedasticity Principles of Econometrics, 3rd Edition

  4. 8.1 The Nature of Heteroskedasticity Figure 8.1 Heteroskedastic Errors Principles of Econometrics, 3rd Edition

  5. 8.1 The Nature of Heteroskedasticity Food expenditure example: Principles of Econometrics, 3rd Edition

  6. 8.1 The Nature of Heteroskedasticity Figure 8.2 Least Squares Estimated Expenditure Function and Observed Data Points Principles of Econometrics, 3rd Edition

  7. 8.2 Using the Least Squares Estimator The existence of heteroskedasticity implies: • The least squares estimator is still a linear and unbiased estimator, but it is no longer best. There is another estimator with a smaller variance. • The standard errors usually computed for the least squares estimator are incorrect. Confidence intervals and hypothesis tests that use these standard errors may be misleading. Principles of Econometrics, 3rd Edition

  8. 8.2 Using the Least Squares Estimator Principles of Econometrics, 3rd Edition

  9. 8.2 Using the Least Squares Estimator Principles of Econometrics, 3rd Edition

  10. 8.2 Using the Least Squares Estimator We can use a robust estimator: regress food_exp income, robust Principles of Econometrics, 3rd Edition

  11. 8.2 Using the Least Squares Estimator In SHAZAM the HETCOV option on the OLS command reports the White's heteroskedasticity-consistent standard errors OLS FOOD INCOME / HETCOV Confidence interval estimate using the White standard errors CONFID INCOME / TCRIT=2.024 But the White standard errors reported by SHAZAM have numeric differences compared to our textbook results. There is a correction we could apply, but we will not worry about it for now Principles of Econometrics, 3rd Edition Slide 8-11

  12. 8.3 The Generalized Least Squares Estimator Principles of Econometrics, 3rd Edition

  13. 8.3.1 Transforming the Model Principles of Econometrics, 3rd Edition

  14. 8.3.1 Transforming the Model Principles of Econometrics, 3rd Edition

  15. 8.3.1 Transforming the Model To obtain the best linear unbiased estimator for a model with heteroskedasticity of the type specified in equation (8.11): • Calculate the transformed variables given in (8.13). • Use least squares to estimate the transformed model given in (8.14). Principles of Econometrics, 3rd Edition

  16. 8.3.1 Transforming the Model The generalized least squares estimator is as a weighted least squaresestimator. Minimizing the sum of squared transformed errors that is given by: When is small, the data contain more information about the regression function and the observations are weighted heavily. When is large, the data contain less information and the observations are weighted lightly. Principles of Econometrics, 3rd Edition

  17. 8.3.1 Transforming the Model Food example again, where was the problem coming from? regress food_exp income [aweight = 1/income] Principles of Econometrics, 3rd Edition

  18. 8.3.1 Transforming the Model Food example again, where was the problem coming from? Specify a weight variable (SHAZAM works with the inverse) GENR W=1/INCOME OLS FOOD INCOME / WEIGHT=W 95% confidence interval, p. 205 CONFID INCOME / TCRIT=2.024 Principles of Econometrics, 3rd Edition Slide 8-18

  19. Note that The residual statistics reported in a WLS regression (SIGMA**2, STANDARD ERROR OF THE ESTIMATE-SIGMA, and SUM OF SQUARED ERRORS-SSE) are all based on the transformed (weighted) residuals You should remember when making model comparisons, that the high-variance observations are systematically underweighted by this procedure This may be a good thing, if you want to avoid having these observations dominate the model selection comparisons. But if you want model selection to be based on how well the alternative models fit the original (untransformed) data, you must base the model selection tests on the untransformed residuals.

  20. 8.3.2 Estimating the Variance Function Principles of Econometrics, 3rd Edition

  21. 8.3.2 Estimating the Variance Function Principles of Econometrics, 3rd Edition

  22. 8.3.2 Estimating the Variance Function Principles of Econometrics, 3rd Edition

  23. 8.3.2 Estimating the Variance Function Principles of Econometrics, 3rd Edition

  24. 8.3.2 Estimating the Variance Function Principles of Econometrics, 3rd Edition

  25. 8.3.2 Estimating the Variance Function The steps for obtaining a feasible generalized least squares estimator for are: • 1. Estimate (8.25) by least squares and compute the squares of the least squares residuals . • 2. Estimate by applying least squares to the equation Principles of Econometrics, 3rd Edition

  26. 8.3.2 Estimating the Variance Function 3. Compute variance estimates . 4. Compute the transformed observations defined by (8.23), including if . 5. Apply least squares to (8.24), or to an extended version of (8.24) if . Principles of Econometrics, 3rd Edition

  27. 8.3.2 Estimating the Variance Function For our food expenditure example: gen z = log(income) regress food_exp income predict ehat, residual gen lnehat2 = log(ehat*ehat) regress lnehat2 z * -------------------------------------------- * Feasible GLS * -------------------------------------------- predict sig2, xb gen wt = exp(sig2) regress food_exp income [aweight = 1/wt] Principles of Econometrics, 3rd Edition Slide 8-27

  28. 8.3.2 Estimating the Variance Function The HET command can be used for Maximum Likelihood Estimation of the model given in Equations (8.25) and (8.26), p. 207. This method is an alternative estimation method to the GLS method discussed in the text (so the results will also be different): HET FOOD INCOME (INCOME) / MODEL=MULT Principles of Econometrics, 3rd Edition Slide 8-28

  29. 8.3.3 A Heteroskedastic Partition Using our wage data (cps2.dta): ??? Principles of Econometrics, 3rd Edition

  30. 8.3.3 A Heteroskedastic Partition Principles of Econometrics, 3rd Edition

  31. 8.3.3 A Heteroskedastic Partition Principles of Econometrics, 3rd Edition

  32. 8.3.3 A Heteroskedastic Partition Feasible generalized least squares: 1. Obtain estimated and by applying least squares separately to the metropolitan and rural observations. 2. 3. Apply least squares to the transformed model Principles of Econometrics, 3rd Edition

  33. 8.3.3 A Heteroskedastic Partition Principles of Econometrics, 3rd Edition

  34. 8.3.3 A Heteroskedastic Partition * -------------------------------------------- * Rural subsample regression * -------------------------------------------- regress wage educ exper if metro == 0 scalar rmse_r = e(rmse) scalar df_r = e(df_r) * -------------------------------------------- * Urban subsample regression * -------------------------------------------- regress wage educ exper if metro == 1 scalar rmse_m = e(rmse) scalar df_m = e(df_r) * -------------------------------------------- * Groupwise heteroskedastic regression using FGLS * -------------------------------------------- gen rural = 1 - metro gen wt=(rmse_r^2*rural) + (rmse_m^2*metro) regress wage educ exper metro [aweight = 1/wt] STATA Commands: Principles of Econometrics, 3rd Edition Slide 8-34

  35. 8.3.3 A Heteroskedastic Partition Principles of Econometrics, 3rd Edition

  36. 8.4 Detecting Heteroskedasticity 8.4.1 Residual Plots • Estimate the model using least squares and plot the least squares residuals. • With more than one explanatory variable, plot the least squares residuals against each explanatory variable, or against , to see if those residuals vary in a systematic way relative to the specified variable. Principles of Econometrics, 3rd Edition

  37. 8.4 Detecting Heteroskedasticity 8.4.2 The Goldfeld-Quandt Test Principles of Econometrics, 3rd Edition

  38. 8.4 Detecting Heteroskedasticity * -------------------------------------------- * Goldfeld Quandt test * -------------------------------------------- scalar GQ = rmse_m^2/rmse_r^2 scalar crit = invFtail(df_m,df_r,.05) scalar pvalue = Ftail(df_m,df_r,GQ) scalar list GQ pvalue crit Principles of Econometrics, 3rd Edition Slide 8-38 8.4.2 The Goldfeld-Quandt Test

  39. 8.4 Detecting Heteroskedasticity 8.4.2 The Goldfeld-Quandt Test For the food expenditure data You should now be able to obtain this test statistic And check whether it exceeds the critical value Principles of Econometrics, 3rd Edition

  40. 8.4 Detecting Heteroskedasticity Sort the data by income: SORT INCOME FOOD / DESC OLS FOOD INCOME On the DIAGNOS command the CHOWONE= option reports the Goldfeld-Quandt test for heteroskedasticity (bottom of page 212) with a p-value for a one-sided test. The HET option reports the tests for heteroskedasticity reported on page 215. DIAGNOS / CHOWONE=20 HET Principles of Econometrics, 3rd Edition Slide 8-40 8.4.2 The Goldfeld-Quandt Test

  41. 8.4 Detecting Heteroskedasticity SORT INCOME FOOD / DESC OLS FOOD INCOME On the DIAGNOS command the CHOWONE= option reports the Goldfeld-Quandt test for heteroskedasticity (bottom of page 212) with a p-value for a one-sided test. The HET option reports the tests for heteroskedasticity reported on page 215. DIAGNOS / CHOWONE=20 HET Of course, this option also computes the Chow test statistic for structural change (that is, tests the null of parameter stability in the two subsamples). Principles of Econometrics, 3rd Edition Slide 8-41 8.4.2 The Goldfeld-Quandt Test

  42. 8.4 Detecting Heteroskedasticity SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS     N1   N2    SSE1        SSE2       CHOW    PVALUE    G-Q       DF1  DF2 PVALUE     20   20 0.23259E+06  64346.     0.45855     0.636  3.615       18   18 0.005              CHOW TEST - F DISTRIBUTION WITH DF1=   2 AND DF2=  36 Principles of Econometrics, 3rd Edition Slide 8-42 8.4.2 The Goldfeld-Quandt Test

  43. 8.4 Detecting Heteroskedasticity SHAZAM considers that the alternative hypothesis is smaller error variance in the second subset relative to the first subset. Some authors present the alternative as larger variance in the second subset. Goldfeld and Quandt recommend ordering the observations by the values of one of the explanatory variables. This can be done with the SORT command in SHAZAM. The DESC option on the SORT command should be used if it is assumed that the variance is positively related to the value of the sort variable. Principles of Econometrics, 3rd Edition Slide 8-43 8.4.2 The Goldfeld-Quandt Test

  44. 8.4 Detecting Heteroskedasticity 8.4.3 Testing the Variance Function Principles of Econometrics, 3rd Edition

  45. 8.4 Detecting Heteroskedasticity 8.4.3 Testing the Variance Function Principles of Econometrics, 3rd Edition

  46. 8.4 Detecting Heteroskedasticity 8.4.3 Testing the Variance Function Principles of Econometrics, 3rd Edition

  47. 8.4 Detecting Heteroskedasticity 8.4.3a The White Test Principles of Econometrics, 3rd Edition

  48. 8.4 Detecting Heteroskedasticity 8.4.3b Testing the Food Expenditure Example whitetst Or estat imtest, white Principles of Econometrics, 3rd Edition

  49. Further testing in SHAZAM DIAGNOS / HET Will yield a battery of heteroskedasticity tests using different specifications

  50. SHAZAM for food example DIAGNOS\HET  REQUIRED MEMORY IS PAR=       7 CURRENT PAR=   22480  DEPENDENT VARIABLE = FOOD            40 OBSERVATIONS  REGRESSION COEFFICIENTS     10.2096426868       83.4160065402  HETEROSKEDASTICITY TESTS                              CHI-SQUARE     D.F.   P-VALUE                            TEST STATISTIC  E**2 ON YHAT:                      7.384     1    0.00658  E**2 ON YHAT**2:                   7.549     1    0.00600  E**2 ON LOG(YHAT**2):              6.516     1    0.01069  E**2 ON LAG(E**2) ARCH TEST:       0.089     1    0.76544  LOG(E**2) ON X (HARVEY) TEST:     10.654     1    0.00110  ABS(E) ON X (GLEJSER) TEST:       11.466     1    0.00071  E**2 ON X                 TEST:            KOENKER(R2):             7.384     1    0.00658            B-P-G (SSR) :            7.344     1    0.00673  E**2 ON X X**2    (WHITE) TEST:            KOENKER(R2):             7.555     2    0.02288            B-P-G (SSR) :            7.514     2    0.02336 Same in SLR

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