金融统计与计量课程讲义4
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(2013-14 1st Semester) Financial Statistics and Econometrics August 17, 2013 9 / 48
The White test
Basic premise: if disturbances are homoscedastic, then squared errors are on average roughly constant, and explanators should NOT be able to predict squared errors, or their proxy, squared residuals. The White test uses all explanators, squared explanators, and cross products of explanators to predict squared residuals. The performance of above predicatory (auxiliary) regression can be used to test hypotheses: H0 : Homoscedasticity against Ha : Heteroscedasticity
(2013-14 1st Semester) Financial Statistics and Econometrics August 17, 2013 11 / 48
Step 3 — Compute test statistic nR2 , where R2 is the coefficient of determination from the auxiliary equation in step 2. Step 4 — Make decision: Compare nR2 to the critical value χ2 α (p), the upper α-quantile of the Chi-squared distribution with p degrees of freedom. If nR2 > χ2 α (p), then reject the null hypothesis of homoscedasticity at the α level of significance.
(2013-14 1st Semester)
Financial Statistics and Econometrics
August 17, 2013
4 / 48
Example 4.1: Housing expenditures
Consider a regression of housing expenditures on income: renti = β0 + β1 (income)i + εi . Consumers with low values of income have little scope for varying their rent expenditures, and hence Var(εi ) is low. On the other hand, wealthy consumers can choose to spend a lot of money on rent, or to spend less, depending on tastes, as a result, Var(εi ) is high. This heteroscedasticity can be clearly seen from the scatter plot, see Figure 4.1.
(2013-14 1st Semester)
Financial Statistics and Econometrics
August 17, 2013
10 / 48
Steps of the White Test:
Step 1 — Generate residuals: Regress Y against your various explanators Xi ’s using OLS, compute the OLS residuals e1 , · · · , en . Step 2 — Conduct auxiliary regression: Regress e2 i against a constant (intercept), all of the explanators Xi ’s, the squares of the explanators Xi2 ’s, and all possible interactions between the explanators (p slopes total).
(2013-14 1st Semester) Financial Statistics and Econometrics August 17, 2013 8 / 48
§4.2
Tests for Heteroscedasticity
There are two types of tests: Tests for continuous changes in variance: White and Breusch-Pagan tests.
Financial Statistics and Econometrics
Chapter 4 Heteroscedastic Disturbances
Master of Finance Graduate School at Shenzhen Tsinghua University
(2013-14 1st Semester)
7 / 48
百度文库
Our strategy
Test the existence of heteroscedasticity. If heteroscedasticity exists, find estimates more efficient than OLS estimates. The variances of OLS estimates may be acceptable, and the estimates are still unbiased. However, we do have one very serious problem: our estimated varince/standard error formulas are wrong! Since in this case, we cannot simply
2 is a continuous Continuous changing variances means σi
function (e.g., linear) of a (some) continuous explanator(s).
Tests for discrete (lumpy) changes in variance: the Goldfeld-Quandt test.
+α7 (X1 · X2 )i + α8 (X1 · X3 )i + α9 (X2 · X3 )i + νi . (4.4) Here we have total p = 9 slopes. Again, OLS method is used to estimate αi ’s in this model.
Financial Statistics and Econometrics
August 17, 2013
1 / 48
Contents
Heteroscedastic disturbances Tests for heteroscedasticity Generalized Lease Squares (GLS) Feasible Generalized Lease Squares (FGLS) White’s Robust Standard Errors
n n i=1 wij Yi ,
0 ≤ j ≤ k.
Then, under heteroscedasticity, the variances are Var(βj ) =
i=1 2 2 wij σj ,
(4.3)
instead of Var(βj ) = this case.
(2013-14 1st Semester)
Discrete (lumpy) changing variances means variances are different between some sub-groups of observations, while within each sub-group, disturbances are homoscedastic.
2 by the residual variance s2 . Consequently, replace all σj
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?
For example, if we have three explanators Xi , i = 1, 2, 3, in step 1, then the auxiliary regression is e2 i
2 2 2 = α0 + α1 X1i + α2 X2i + α3 X3i + α4 X1 i + α5 X2i + α6 X3i
2 Var(εi ) = σi ,
i = 1, 2, · · · , n.
(4.2)
Besides, we still assume that εi ’s are independent of each other and each has a normal distribution with common mean 0.
Financial Statistics and Econometrics
August 17, 2013
6 / 48
Implications of heteroscedasticity
Under heteroscedasticity, the OLS estimate b is unbiased but inefficient since heteroscedasticity does affect the variance of b but does not affect its expectation. Denote the OLS estimates by βj =
(2013-14 1st Semester)
Financial Statistics and Econometrics
August 17, 2013
3 / 48
Heteroscedasticity
Heteroscedasticity means the variance of εi , and hence of Yi , is NOT a constant σ 2 . In general, we have
(2013-14 1st Semester)
Financial Statistics and Econometrics
August 17, 2013
2 / 48
§4.1
Heteroscedastic Disturbances
We consider the following multiple linear regression model in this section: Yi = β0 + β1 X1i + β2 X2i + · · · + βk Xki + εi , i = 1, 2, · · · , n. Or, in matrix form, Y = Xb + e. (4.1)
(2013-14 1st Semester)
Financial Statistics and Econometrics
August 17, 2013
5 / 48
Figure 4.1 Rents and Incomes for a Sample of New Yorkers
(2013-14 1st Semester)
n 2 2 i=1 wij σ
under homoscedasticity. As
a result, the OLS weights {wj } fail to minimize Var(βj ) in
Financial Statistics and Econometrics
August 17, 2013
The White test
Basic premise: if disturbances are homoscedastic, then squared errors are on average roughly constant, and explanators should NOT be able to predict squared errors, or their proxy, squared residuals. The White test uses all explanators, squared explanators, and cross products of explanators to predict squared residuals. The performance of above predicatory (auxiliary) regression can be used to test hypotheses: H0 : Homoscedasticity against Ha : Heteroscedasticity
(2013-14 1st Semester) Financial Statistics and Econometrics August 17, 2013 11 / 48
Step 3 — Compute test statistic nR2 , where R2 is the coefficient of determination from the auxiliary equation in step 2. Step 4 — Make decision: Compare nR2 to the critical value χ2 α (p), the upper α-quantile of the Chi-squared distribution with p degrees of freedom. If nR2 > χ2 α (p), then reject the null hypothesis of homoscedasticity at the α level of significance.
(2013-14 1st Semester)
Financial Statistics and Econometrics
August 17, 2013
4 / 48
Example 4.1: Housing expenditures
Consider a regression of housing expenditures on income: renti = β0 + β1 (income)i + εi . Consumers with low values of income have little scope for varying their rent expenditures, and hence Var(εi ) is low. On the other hand, wealthy consumers can choose to spend a lot of money on rent, or to spend less, depending on tastes, as a result, Var(εi ) is high. This heteroscedasticity can be clearly seen from the scatter plot, see Figure 4.1.
(2013-14 1st Semester)
Financial Statistics and Econometrics
August 17, 2013
10 / 48
Steps of the White Test:
Step 1 — Generate residuals: Regress Y against your various explanators Xi ’s using OLS, compute the OLS residuals e1 , · · · , en . Step 2 — Conduct auxiliary regression: Regress e2 i against a constant (intercept), all of the explanators Xi ’s, the squares of the explanators Xi2 ’s, and all possible interactions between the explanators (p slopes total).
(2013-14 1st Semester) Financial Statistics and Econometrics August 17, 2013 8 / 48
§4.2
Tests for Heteroscedasticity
There are two types of tests: Tests for continuous changes in variance: White and Breusch-Pagan tests.
Financial Statistics and Econometrics
Chapter 4 Heteroscedastic Disturbances
Master of Finance Graduate School at Shenzhen Tsinghua University
(2013-14 1st Semester)
7 / 48
百度文库
Our strategy
Test the existence of heteroscedasticity. If heteroscedasticity exists, find estimates more efficient than OLS estimates. The variances of OLS estimates may be acceptable, and the estimates are still unbiased. However, we do have one very serious problem: our estimated varince/standard error formulas are wrong! Since in this case, we cannot simply
2 is a continuous Continuous changing variances means σi
function (e.g., linear) of a (some) continuous explanator(s).
Tests for discrete (lumpy) changes in variance: the Goldfeld-Quandt test.
+α7 (X1 · X2 )i + α8 (X1 · X3 )i + α9 (X2 · X3 )i + νi . (4.4) Here we have total p = 9 slopes. Again, OLS method is used to estimate αi ’s in this model.
Financial Statistics and Econometrics
August 17, 2013
1 / 48
Contents
Heteroscedastic disturbances Tests for heteroscedasticity Generalized Lease Squares (GLS) Feasible Generalized Lease Squares (FGLS) White’s Robust Standard Errors
n n i=1 wij Yi ,
0 ≤ j ≤ k.
Then, under heteroscedasticity, the variances are Var(βj ) =
i=1 2 2 wij σj ,
(4.3)
instead of Var(βj ) = this case.
(2013-14 1st Semester)
Discrete (lumpy) changing variances means variances are different between some sub-groups of observations, while within each sub-group, disturbances are homoscedastic.
2 by the residual variance s2 . Consequently, replace all σj
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?
For example, if we have three explanators Xi , i = 1, 2, 3, in step 1, then the auxiliary regression is e2 i
2 2 2 = α0 + α1 X1i + α2 X2i + α3 X3i + α4 X1 i + α5 X2i + α6 X3i
2 Var(εi ) = σi ,
i = 1, 2, · · · , n.
(4.2)
Besides, we still assume that εi ’s are independent of each other and each has a normal distribution with common mean 0.
Financial Statistics and Econometrics
August 17, 2013
6 / 48
Implications of heteroscedasticity
Under heteroscedasticity, the OLS estimate b is unbiased but inefficient since heteroscedasticity does affect the variance of b but does not affect its expectation. Denote the OLS estimates by βj =
(2013-14 1st Semester)
Financial Statistics and Econometrics
August 17, 2013
3 / 48
Heteroscedasticity
Heteroscedasticity means the variance of εi , and hence of Yi , is NOT a constant σ 2 . In general, we have
(2013-14 1st Semester)
Financial Statistics and Econometrics
August 17, 2013
2 / 48
§4.1
Heteroscedastic Disturbances
We consider the following multiple linear regression model in this section: Yi = β0 + β1 X1i + β2 X2i + · · · + βk Xki + εi , i = 1, 2, · · · , n. Or, in matrix form, Y = Xb + e. (4.1)
(2013-14 1st Semester)
Financial Statistics and Econometrics
August 17, 2013
5 / 48
Figure 4.1 Rents and Incomes for a Sample of New Yorkers
(2013-14 1st Semester)
n 2 2 i=1 wij σ
under homoscedasticity. As
a result, the OLS weights {wj } fail to minimize Var(βj ) in
Financial Statistics and Econometrics
August 17, 2013