Heteroskedasticity

1
Chapter 8
ECON 4550 Econometrics Memorial University of
Newfoundland

• Heteroskedasticity

Adapted from Vera Tabakovas 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

3
8.1 The Nature of Heteroskedasticity

(8.1)
(8.2)
(8.3)
4
8.1 The Nature of Heteroskedasticity

• Figure 8.1 Heteroskedastic Errors

5
8.1 The Nature of Heteroskedasticity

(8.4)
Food expenditure example
6
8.1 The Nature of Heteroskedasticity

• Figure 8.2 Least Squares Estimated Expenditure
  Function and Observed Data Points

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.

8
8.2 Using the Least Squares Estimator

(8.5)
(8.6)
(8.7)
9
8.2 Using the Least Squares Estimator

(8.8)
(8.9)
10
8.2 Using the Least Squares Estimator

We can use a robust estimator regress food_exp
income, robust
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 / TCRIT2.024 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
Slide 8-11
Principles of Econometrics, 3rd Edition
12
8.3 The Generalized Least Squares Estimator

(8.10)
13
8.3.1 Transforming the Model

(8.11)
(8.12)
(8.13)
14
8.3.1 Transforming the Model

(8.14)
(8.15)
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).

16
8.3.1 Transforming the Model

• The generalized least squares estimator is as a
  weighted least squares estimator. 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.

17
8.3.1 Transforming the Model

Food example again, where was the problem coming
from? regress food_exp income aweight
1/income
(8.16)
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 W1/INCOME OLS FOOD
INCOME / WEIGHTW 95 confidence interval, p.
205 CONFID INCOME / TCRIT2.024
(8.16)
Slide 8-18
Principles of Econometrics, 3rd Edition
19
Note that

• The residual statistics reported in a WLS
  regression (SIGMA2, 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

(8.17)
(8.18)
21
8.3.2 Estimating the Variance Function

(8.19)
(8.20)
22
8.3.2 Estimating the Variance Function

(8.21)
23
8.3.2 Estimating the Variance Function

(8.22)
(8.23)
(8.24)
24
8.3.2 Estimating the Variance Function

(8.25)
(8.26)
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

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 .

(8.27)
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(ehatehat) regress
lnehat2 z -------------------------------------
------- Feasible GLS -------------------------
------------------- predict sig2, xb gen wt
exp(sig2) regress food_exp income aweight 1/wt
Slide 8-27
Principles of Econometrics, 3rd Edition
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) / MODELMULT
Slide 8-28
Principles of Econometrics, 3rd Edition
29
8.3.3 A Heteroskedastic Partition

Using our wage data (cps2.dta)
(8.28)
(8.29a)
(8.29b)
???
30
8.3.3 A Heteroskedastic Partition

(8.30)
31
8.3.3 A Heteroskedastic Partition

(8.31a)
(8.31b)
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

(8.32)
33
8.3.3 A Heteroskedastic Partition

(8.33)
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
-------------------------------------------- regre
ss 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_r2rural)
(rmse_m2metro) regress wage educ exper metro
aweight 1/wt
STATA Commands
Slide 8-34
Principles of Econometrics, 3rd Edition
35
8.3.3 A Heteroskedastic Partition
Remark To implement the generalized least squares estimators described in this Section for three alternative heteroskedastic specifications, an assumption about the form of the heteroskedasticity is required. Using least squares with White standard errors avoids the need to make an assumption about the form of heteroskedasticity, but does not realize the potential efficiency gains from generalized least squares.
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.

37
8.4 Detecting Heteroskedasticity

• 8.4.2 The Goldfeld-Quandt Test

(8.34)
(8.35)
38
8.4 Detecting Heteroskedasticity

• 8.4.2 The Goldfeld-Quandt Test

--------------------------------------------
Goldfeld Quandt test ---------------------------
----------------- scalar GQ rmse_m2/rmse_r2 s
calar 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
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
40
8.4 Detecting Heteroskedasticity

• 8.4.2 The Goldfeld-Quandt Test

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 / CHOWONE20 HET
Principles of Econometrics, 3rd Edition
Slide 8-40
41
8.4 Detecting Heteroskedasticity

• 8.4.2 The Goldfeld-Quandt Test

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 / CHOWONE20 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
42
8.4 Detecting Heteroskedasticity

• 8.4.2 The Goldfeld-Quandt Test

SEQUENTIAL CHOW AND GOLDFELD-QUANDT TESTS     N1  
 N2    SSE1        SSE2       CHOW    PVALUE    G-
Q       DF1  DF2 PVALUE     20   20 0.23259E06  6
4346.     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
43
8.4 Detecting Heteroskedasticity

• 8.4.2 The Goldfeld-Quandt Test

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
44
8.4 Detecting Heteroskedasticity

• 8.4.3 Testing the Variance Function

(8.36)
(8.37)
45
8.4 Detecting Heteroskedasticity

• 8.4.3 Testing the Variance Function

(8.38)
(8.39)
46
8.4 Detecting Heteroskedasticity

• 8.4.3 Testing the Variance Function

(8.40)
(8.41)
(8.42)
(8.43)
47
8.4 Detecting Heteroskedasticity

• 8.4.3a The White Test

48
8.4 Detecting Heteroskedasticity

• 8.4.3b Testing the Food Expenditure Example

whitetst Or estat imtest, white
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   22
  480
•  DEPENDENT VARIABLE  FOOD            40 OBSERVATI
  ONS
•  REGRESSION COEFFICIENTS
•     10.2096426868       83.4160065402
•  HETEROSKEDASTICITY TESTS
•                              CHI-SQUARE     D.F.  
   P-VALUE
•                            TEST STATISTIC
•  E2 ON YHAT                      7.384     1   
   0.00658
•  E2 ON YHAT2                   7.549     1   
   0.00600
•  E2 ON LOG(YHAT2)              6.516     1   
   0.01069
•  E2 ON LAG(E2) ARCH TEST       0.089     1   
   0.76544
•  LOG(E2) ON X (HARVEY) TEST     10.654     1   
   0.00110
•  ABS(E) ON X (GLEJSER) TEST       11.466     1   
   0.00071
•  E2 ON X                 TEST
•            KOENKER(R2)             7.384     1   
   0.00658
•            B-P-G (SSR)             7.344     1   
   0.00673
•  E2 ON X X2    (WHITE) TEST
•            KOENKER(R2)             7.555     2   
   0.02288

Same in SLR
51
Keywords

• Breusch-Pagan test
• generalized least squares
• Goldfeld-Quandt test
• heteroskedastic partition
• heteroskedasticity
• heteroskedasticity-consistent standard errors
• homoskedasticity
• Lagrange multiplier test
• mean function
• residual plot
• transformed model
• variance function
• weighted least squares
• White test

52
Chapter 8 Appendices

• Appendix 8A Properties of the Least Squares
  Estimator
• Appendix 8B Variance Function Tests for
  Heteroskedasticity

53
Appendix 8A Properties of the Least Squares
Estimator

(8A.1)

54
Appendix 8A Properties of the Least Squares
Estimator

55
Appendix 8A Properties of the Least Squares
Estimator
(8A.2)
56
Appendix 8A Properties of the Least Squares
Estimator
(8A.3)
57
Appendix 8B Variance Function Tests for
Heteroskedasticity
(8B.1)
(8B.2)

58
Appendix 8B Variance Function Tests for
Heteroskedasticity
(8B.3)
(8B.4)
(8B.5)
59
Appendix 8B Variance Function Tests for
Heteroskedasticity
(8B.6)

(8B.7)
60
Appendix 8B Variance Function Tests for
Heteroskedasticity
(8B.8)