Lecture 13 Heteroskedasticity (Chapter 10.1–10.4, 10.7) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. Agenda for Today • Review Standard Errors (Chapter 5.2) • Heteroskedasticity (Chapter 10.1) • OLS and Heteroskedasticity (Chapter 10.2) • Tests for Heteroskedasticity (Chapter 10.3) • Generalized Least Squares (Chapter 10.4) • GLS: an Example (Chapter 10.5) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-2 Review of Standard Errors (Chapter 5.2) • Our DGP describes an underlying process we wish to study. • The model includes a stochastic component, the error term. • There are many possible realizations of the error term, but we get to observe only one. • The underlying process plus the particular realization of the error termour sample. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-3 Review of Standard Errors (cont.) • We observe only one sample. • Our estimator (OLS) provides a rule for going from one sample to one best guess of the underlying parameters. • If we could observe many samples, we could estimate many best guesses. • If we could observe many samples, we would have a distribution of estimates. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-4 Review of Standard Errors (cont.) • Using our model and our sample, we can estimate NOT ONLY the underlying parameters BUT ALSO the crosssample distribution of our estimates. • Estimating the cross-sample distribution lets us judge how likely our best guess is to be close to the real underlying parameters. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-5 Review of Standard Errors (cont.) Gauss–Markov DGP Yi 0 1 X 1i i E ( i ) 0 Var ( i ) 2 Cov( i , j ) 0, if i j X 's fixed across samples. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-6 Review of Standard Errors (cont.) Gauss–Markov DGP Yi 0 1 X 1i i E ( i ) 0 X 's fixed across samples. These assumptions suffice to derive the expectation of a linear estimator: E (wiYi ) 0 wi 1wi X i Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-7 Review of Standard Errors (cont.) • To derive the variance of a linear estimator, we need to add the two assumptions about the variance and covariances of the error term: Var( i ) 2 Cov( i , j ) 0, if i j Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-8 Review of Standard Errors (cont.) Variance of a Linear Estimator Var ( ˆ ) Var (wiYi ) n Var ( wiYi ) cov( wiYi , w jY j ) i 1 j i wi 2 Var (Yi ) wi 2 Var ( 0 1 X 1i i ) wi 2 2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-9 Review of Standard Errors (cont.) Variance of OLS with a Single Explanator Yi 0 1 X i i Var ( ˆ1 ) 2 w2ˆ 1i (Xi X ) wˆ i 1 ( X j X ) 2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-10 Review of Standard Errors (cont.) Variance of OLS with a Single Explanator (Xi X ) 2 2 2 ˆ Var ( 1 ) wi ( X j X ) 2 2 ( X j X ) 2 2 xi2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-11 Review of Standard Errors (cont.) • Problem: we do not know 2 • Solution: estimate 2 • We do not observe the ACTUAL error terms, i • We DO observe the residual, ei ei s n k 1 2 2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-12 Review of Standard Errors (cont.) • We can estimate the STANDARD ERROR of our OLS estimator (the cross-sample standard deviation). ei (n k 1) ˆ e.s.e.( ) 2 ( X i X ) 2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-13 Review of Standard Errors (cont.) • Estimated standard errors are used to calculate t-statistics, used for testing single hypotheses. * ˆ tˆ ˆ e.s.e.( ) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-14 Review of Standard Errors (cont.) • Estimated Standard Errors are also used to calculate Confidence Intervals. In % of samples, the C.I. will contain the true parameter value, . % C.I . ˆ t Copyright © 2006 Pearson Addison-Wesley. All rights reserved. (1 ) 2 n k 1 e.s.e.( ˆ ) 13-15 Review of Standard Errors (cont.) • Our formula for Estimated Standard Errors relied on ALL the Gauss–Markov DGP assumptions. • For this lecture, we will focus on the assumption of homoskedasticity. • What happens if we relax the 2 assumption that Var(i ) ? Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-16 Heteroskedasticity (Chapter 10.1) • HETEROSKEDASTICITY – The variance of i is NOT a constant 2. – The variance of i is greater for some observations than for others. Var(i ) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 2 i 13-17 Heteroskedasticity (cont.) • For example, consider a regression of housing expenditures on income. Renti 0 1 Incomei i • Consumers with low values of income have little scope for varying their rent expenditures. Var(i ) is low. • Wealthy consumers can choose to spend a lot of money on rent, or to spend less, depending on tastes. Var(i ) is high. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-18 Figure 10.1 Rents and Incomes for a Sample of New Yorkers Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-19 OLS and Heteroskedasticity (Chapter 10.2) • What are the implications of heteroskedasticity for OLS? • Under the Gauss–Markov assumptions (including homoskedasticity), OLS was the Best Linear Unbiased Estimator. • Under heteroskedasticity, is OLS still Unbiased? • Is OLS still Best? Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-20 OLS and Heteroskedasticity (cont.) • A DGP with Heteroskedasticity Yi 0 1 X 1i ... k X ki i E ( i ) 0 Var ( i ) 2 i Cov( i , j ) 0 for i j X ’s fixed across samples Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-21 OLS and Heteroskedasticity (cont.) • Is OLS still unbiased under this DGP? E(wiYi ) E(wiYi ) wi E(Yi ) wi ( 0 1 X1i .. k X ki ) E(wiYi ) 1 if wi 0, wi X1i 1, wi X 2i 0...wi X ki 0 • The unbiasedness conditions are the same as under the Gauss–Markov DGP. • OLS is still unbiased! Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-22 OLS and Heteroskedasticity (cont.) • To determine whether OLS is “Best” (i.e. the unbiased linear estimator with the lowest variance), we need to calculate the variance of a linear estimator under heteroskedasticity. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-23 OLS and Heteroskedasticity (cont.) • The variance of a linear estimator: n Var(wiYi ) Var(wiYi ) Cov(wiYi , w jY j ) i1 ji n Var(wiYi ) 0 wi 2Var(Yi ) wi w j Cov(Yi ,Y j ) i1 ji wi 2Var(Yi ) 0 wi 2 i 2 • How does this differ from the formula under homoskedasticity? Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-24 OLS and Heteroskedasticity • The variance of a linear estimator is wi i 2 2 • OLS minimizes 2wi 2 • OLS is no longer efficient! Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-25 OLS and Heteroskedasticity (cont.) • Under heteroskedasticity, OLS is unbiased but inefficient. • OLS does not have the smallest possible variance, but its variance may be acceptable. And the estimates are still unbiased. • However, we do have one very serious problem: our estimated standard error formulas are wrong! Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-26 OLS and Heteroskedasticity (cont.) • For the k = 1 case, the OLS weight to estimate 1 is wi xi x j 2 • The variance of the estimator is 2 2 x Var ( ˆ1 ) i 2 i 2 ( x j ) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-27 OLS and Heteroskedasticity (cont.) • For the k = 1 case, the variance of OLS is 2 2 x Var ( ˆ1 ) i 2 i 2 ( x j ) • However, our old e.s.e. formula is 2 2 e s i e.s.e.( ˆ1 ) x j 2 (n 2) x j 2 • This formula is no longer valid. Our C.I.’s and hypothesis tests will NOT be correct!!!!! Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-28 OLS and Heteroskedasticity (cont.) • Implications of Heteroskedasticity: – OLS is still unbiased. – OLS is no longer efficient; some other linear estimator will have a lower variance. – Estimated Standard Errors will be incorrect; C.I.’s and hypothesis tests (both t- and F- tests) will be incorrect. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-29 OLS and Heteroskedasticity (cont.) • Implications of Heteroskedasticity – OLS is no longer efficient; some other linear estimator will have a lower variance. • Can we use a better estimator? – Estimated Standard Errors will be incorrect; C.I.’s and hypothesis tests (both t- and F- tests) will be incorrect. • If we keep using OLS, can we calculate correct e.s.e.’s? Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-30 Tests for Heteroskedasticity (Chapter 10.3) • Before we turn to remedies for heteroskedasticity, let us first consider tests for the complication. • There are two types of tests: 1. Tests for continuous changes in variance: White and Breusch–Pagan tests 2. Tests for discrete (lumpy) changes in variance: the Goldfeld–Quandt test Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-31 The White Test • The White test for heteroskedasticity has a basic premise: if disturbances are homoskedastic, then squared errors are on average roughly constant. • Explanators should NOT be able to predict squared errors, or their proxy, squared residuals. • The White test is the most general test for heteroskedasticity. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-32 The White Test (cont.) • Five Steps of the White Test: 1. Regress Y against your various explanators using OLS 2. Compute the OLS residuals, e1...en 3. Regress ei2 against a constant, all of the explanators, the squares of the explanators, and all possible interactions between the explanators (p slopes total) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-33 The White Test (cont.) • Five Steps of the White Test (cont.) 4. Compute R2 from the “auxilliary equation” in step 3 5. Compare nR2 to the critical value from the Chi-squared distribution with p degrees of freedom. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-34 The White Test: Example (1) Estimate Wagei 0 1edi 2 expi 3 IQi i (2) Calculate e Wage ˆ ˆ ed ˆ exp ˆ IQ i i 0 1 i 2 i 3 i (3) Regress ei 2 0 1edi 2 edi 2 3 expi 4 exp 2 5 IQi 6 IQi 2 7 edi expi 8edi IQi 9 expi IQi vi (4) Compute nR 2 from (3) (5) Reject homoskedasticity if nR 2 Chi-Squared critical value with 9 degrees of freedom (16.92, if the significance level is 0.05) Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-35 The White Test • The White test is very general, and provides very explicit directions. The econometrician has no judgment calls to make. • The White test also burns through degrees of freedom very, very rapidly. • The White test is appropriate only for “large” sample sizes. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-36 The Breusch–Pagan Test • The Breusch–Pagan test is very similar to the White test. • The White test specifies exactly which explanators to include in the auxilliary equation. Because the test includes crossterms, the number of slopes (p) increases very quickly. • In the Breusch–Pagan test, the econometrician selects which explanators to include. Otherwise, the tests are the same. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-37 The Breusch–Pagan Test (cont.) • In the Breusch–Pagan test, the econometrician selects m explanators to include in the auxilliary equation. • Which explanators to include is a judgment call. • A good judgment call leads to a more powerful test than the White test. • A poor judgment call leads to a poor test. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-38 The Goldfeld–Quandt Test • Both the White test and the Breusch–Pagan test focus on smoothly changing variances for the disturbances. • The Goldfeld–Quandt test compares the variance of error terms across discrete subgroups. • Under homoskedasticity, all subgroups should have the same estimated variances. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-39 The Goldfeld–Quandt Test (cont.) • The Goldfeld–Quandt test compares the variance of error terms across discrete subgroups. • The econometrician must divide the data into h discrete subgroups. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-40 The Goldfeld–Quandt Test (cont.) • If the Goldfeld–Quandt test is appropriate, it will generally be clear which subgroups to use. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-41 The Goldfeld–Quandt Test (cont.) • For example, the econometrician might ask whether men and women’s incomes vary similarly around their predicted means, given education and experience. • To conduct a Goldfeld–Quandt test, divide the data into h = 2 groups, one for men and one for women. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-42 The Goldfeld–Quandt Test (cont.) (1) Divide the n observations into h groups, of sizes n1..nh (2) Choose two groups, say 1 and 2. H 0 : 12 2 2 against H a : 12 2 2 (3) Regress Y against the explanators for group 1. (4) Regress Y against the explanators for group 2. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-43 Goldfeld–Quandt Test (cont.) (5) Relabel the groups as L and S, such that SSRL SSRS nL k nS k SSRL nL k Compute G SSRS nS k (6) Compare G to the critical value for an F-statistic with (nL k) and (nS k) degrees of freedom. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-44 Goldfeld–Quandt Test: An Example • Do men and women’s incomes vary similarly about their respective means, given education and experience? • That is, do the error terms for an income equation have different variances for men and women? • We have a sample with 3,394 men and 3,146 women. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-45 Goldfeld–Quandt Test: An Example (cont.) (1) Divide the n observations into men and women, of sizes nm and nw . (2) We have only two groups, so choose both of them. H 0 : m 2 w2 against H a : m 2 w 2 (3) For the men, regress log(income)i 0 1edi 2 expi 3 expi 2 i (4) For the women, regress log(income)i 0 1edi 2 expi 3 expi 2 vi Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-46 Goldfeld–Quandt Test: An Example (cont.) (5) sm 2 sw 2 SSRm 1736.64 0.5123 n m k 3394 - 4 SSRw 1851.52 0.5893 n w k 3146 - 4 Compute G 0.5893 1.15 0.5123 (6) Compare G to the critical value for an F-statistic with 3142 and 3390 degrees of freedom, which is 0.99997 for the 5% significance level. We reject the null hypothesis at the 5% level. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-47 Generalized Least Squares (Chapter 10.4) • OLS is unbiased, but not efficient. • The OLS weights are not optimal. • Suppose we are estimating a straight line through the origin: Y X • Under homoskedasticity, observations with higher X values are relatively less distorted by the error term. • OLS places greater weight on observations with high X values. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-48 Figure 10.2 Homoskedastic Disturbances More Misleading at Smaller X ’s Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-49 Generalized Least Squares • Suppose observations with higher X values have error terms with much higher variances. • Under this DGP, observations with high X ’s (and high variances of ) may be more misleading than observations with low X ’s (and low variances of ). • In general, we want to put more weight on observations with smaller i2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-50 Figure 10.3 Heteroskedasticity with Smaller Disturbances at Smaller X ’s Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-51 Generalized Least Squares • To construct the BLUE Estimator for S, we follow the same steps as before, but with our new variance formula. The resulting estimator is “Generalized Least Squares.” Start with a linear estimator, wiYi Impose the unbiasedness conditions, wi X Ri 0 for R S, wi X Si 1 Find wi to minimize wi2 i2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-52 Generalized Least Squares (cont.) • For an example, consider the DGP Yi X i i E( i ) 0 Var( i ) di 2 2 Cov( i , j ) 0 for i j X i , di fixed across samples Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-53 Generalized Least Squares (cont.) min w 2 wi 2 di 2 i such that w d j j 1 Xi Solution: wi Copyright © 2006 Pearson Addison-Wesley. All rights reserved. di 2 X j d j 2 13-54 Generalized Least Squares (cont.) GLS ˆ Copyright © 2006 Pearson Addison-Wesley. All rights reserved. Yi X i d d i i Xj d j 2 13-55 Generalized Least Squares (cont.) • In practice, econometricians choose a different method for implementing GLS. • Historically, it was computationally difficult to program a new estimator (with its own weights) for every different dataset. • It was easier to re-weight the data first, and THEN apply the OLS estimator. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-56 Generalized Least Squares (cont.) • We want to transform the data so that it is homoskedastic. Then we can apply OLS. • It is convenient to rewrite the variance term of the heteroskedastic DGP as Var(i ) d 2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 2 i 13-57 Generalized Least Squares (cont.) • If we know the di factor for each observation, we can transform the data by dividing through by di. • Once we divide all variables by di, we obtain a new dataset that meets the Gauss–Markov conditions. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-58 GLS: DGP for Transformed Data X i i Yi 1 0 1 di di di di i E 0 di i Var di 1 1 2 2 2 d Var i i 2 2 d d i i i j 1 Cov ( i , j ) 0 Cov , di d j di d j Xi fixed across samples. di Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-59 Generalized Least Squares • This procedure, Generalized Least Squares, has two steps: 1. Divide all variables by di 2. Apply OLS to the transformed variables • This procedure optimally weights down observations with high di’s • GLS is unbiased and efficient Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-60 Generalized Least Squares (cont.) • Example: a straight line through the origin: 1. First, divide Yi , X i by di 2. Apply OLS to ˆ Yi X i , di di Yi X i d d i i Xj d j Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 2 13-61 Generalized Least Squares (cont.) • Note: we derive the same BLUE Estimator (Generalized Least Squares) whether we: 1. Find the optimal weights for heteroskedastic data, or 2. Transform the data to be homoskedastic, then use OLS weights Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-62 GLS: An Example (Chapter 10.5) • We can solve heteroskedasticity by dividing our variables through by di. • The DGP with the transformed data is Gauss–Markov. • The catch: we don’t observe di. How can we implement this strategy in practice? Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-63 GLS: An Example (cont.) • We want to estimate the relationship renti 0 1incomei i • We are concerned that higher income individuals are less constrained in how much income they spend in rent. Lower income individuals cram into what housing they can afford; higher income individuals find housing to suit their needs/tastes. • That is, Var(i ) may vary with income. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-64 GLS: An Example (cont.) • An initial guess: Var(i ) ·income 2 2 i • di = incomei • If we have modeled heteroskedasticity correctly, then the BLUE Estimator is: rent 1 0 1 vi income i incomei Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-65 TABLE 10.1 Rent and Income in New York Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-66 TABLE 10.5 Estimating a Transformed Rent–Income Relationship, var(i ) 2 X i2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-67 Checking Understanding • An initial guess: Var(i ) ·income 2 2 i • di = incomei rent 1 0 1 vi income i incomei • How can we test to see if we have correctly modeled the heteroskedasticity? Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-68 Checking Understanding • If we have the correct model of heteroskedasticity, then OLS with the transformed data should be homoskedastic. rent 1 0 1 vi income i incomei • We can apply either a White test or a Breusch–Pagan test for heteroskedasticity to the model with the transformed data. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-69 Checking Understanding (cont.) • To run the White test, we regress 1 1 ei 0 1 2 i 2 incomei incomei • nR2 = 7.17 • The critical value at the 0.05 significance level for a Chi-square statistic with 2 degrees of freedom is 5.99 • We reject the null hypothesis. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-70 GLS: An Example 2 2 Var( ) ·income • Our initial guess: i i • This guess didn’t do very well. Can we do better? • Instead of blindly guessing, let’s try looking at the data first. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-71 Figure 10.4 The Rent–Income Ratio Plotted Against the Inverse of Income Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-72 GLS: An Example • We seem to have overcorrected for heteroskedasticity. • Let’s try Var(i ) ·incomei 2 rent income i 0 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 1 income i 1 incomei vi 13-73 TABLE 10.6 Estimating a Second Transformed Rent–Income Relationship, var(i ) 2 Xi Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-74 GLS: An Example • Unthinking application of the White test procedures for the transformed data leads to 1 1 2 3 income incomei income i 1 4 incomei 5 income i income i ei 0 1 • The interaction term reduces to a constant, which we already have in the auxilliary equation, so we omit it and use only the first 4 explanators. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-75 GLS: An Example (cont.) • nR2 = 6.16 • The critical value at the 0.05 significance level for a Chi-squared statistic with 4 degrees of freedom is 9.49 • We fail to reject the null hypothesis that the transformed data are homoskedastic. • Warning: failing to reject a null hypothesis does NOT mean we can “accept” it. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-76 GLS: An Example (cont.) • Generalized Least Squares is not trivial to apply in practice. • Figuring out a reasonable di can be quite difficult. • Next time we will learn another approach to constructing di , Feasible Generalized Least Squares. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-77 Review • In this lecture, we began relaxing the Gauss– Markov assumptions, starting with the assumption of homoskedasticity. • Under heteroskedasticity, Var(i ) 2 di – OLS is still unbiased – OLS is no longer efficient – OLS e.s.e.’s are incorrect, so C.I., t-, and F- statistics are incorrect Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-78 Review (cont.) • Under heteroskedasticity, 2 2 2 ˆ Var ( ) w d i i • For a straight line through the origin, OLS 2 X i di ˆ Var ( ) 2 2 (X i ) 2 Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 2 13-79 Review (cont.) • We can use squared residuals to test for heteroskedasticity. • In the White test, we regress the squared residuals against all explanators, squares of explanators, and interactions of explanators. The nR2 of the auxilliary equation is distributed Chi-squared. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-80 Review (cont.) • The Breusch–Pagan test is similar, but the econometrician chooses the explanators for the auxilliary equation. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-81 Review (cont.) • In the Goldfeld–Quandt test, we first divide the data into distinct groups, and conduct our OLS regression on each group separately. • We then estimate s2 for each group. • The ratio of two s2 estimates is distributed as an F-statistic. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-82 Review (cont.) • Under heteroskedasticity, the BLUE Estimator is Generalized Least Squares • To implement GLS: 1. Divide all variables by di 2. Perform OLS on the transformed variables • If we have used the correct di , the transformed data are homoskedastic. We can test this property. Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 13-83