HETEROSCEDASTICITY
Regression of lnsalary on years of experience for professors- USE DATA 3-11
Original Equation-with Uncorrected HSK
. reg lnsalary years yearssq
Source
SS
df
MS
Model
Residual
10.8438849
9.38050436
2 5.42194244
219 .042833353
Total
20.2243892
221 .091513074
lnsalary
Coef.
years
yearssq
_cons
.0438528
-.0006273
3.809365
Std. Err.
.0048287
.0001209
.0413383
t
9.08
-5.19
92.15
Number of obs
F( 2, 219)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.000
0.000
=
=
=
=
=
=
222
126.58
0.0000
0.5362
0.5319
.20696
[95% Conf. Interval]
.0343361
-.0008655
3.727894
.0533696
-.0003891
3.890837
Formal Tests of HSK
To carry out the White’s Test: under equation above, estat imtest, white
To carry out a Breusch-Pagan test: estat hettest, normal or iid or fsstat
To carry out a specification test for non-linearity: estat ovtest
(This is Ramsey’s specification error test)
Breusch-Pagan / Cook-Weisberg test for heteroskedasticity
.estat hettest, normal
Ho: Constant variance
Variables: fitted values of lnsalary
chi2(1) = 16.20
Prob > chi2 = 0.0001
White Heteroskedasticity Test:
.estat imtest, white
White's test for Ho: homoskedasticity
against Ha: unrestricted heteroskedasticity
chi2(4) = 20.00
Prob > chi2 = 0.0005
.5
0
-1
-.5
Residuals
3.8
4
4.2
Fitted values
4.4
4.6
An important command to remember is rvfplot. This command generates the above graph
when used after regression (reg) command and provides a visual check on the existence
of heteroscedasticity. There seems to be evidence for heteroscedasticity given the
changing variance of the residuals across observations.
Methods for Correcting for Heteroscedasticity
The easiest way to correct for HSK is to use the command:
.reg Y X1 X2, vce(robust)
This yields HSK corrected robust standard errors (if the structure of HSK is
unknown). This method generates Eicker-Huber-White HSK corrected standard
errors..
Below is the result of such a correction using the robust option.
. rvfplot
. reg
lnsalary years yearssq, vce(robust)
Linear regression
Number of obs
F( 2,
219)
Prob > F
R-squared
Root MSE
lnsalary
Coef.
years
yearssq
_cons
.0438528
-.0006273
3.809365
Robust
Std. Err.
.0043609
.0001179
.026119
t
10.06
-5.32
145.85
P>|t|
0.000
0.000
0.000
=
=
=
=
=
222
216.87
0.0000
0.5362
.20696
[95% Conf. Interval]
.0352582
-.0008597
3.757889
.0524475
-.000395
3.860842
. display e(r2_a)
.5319428
Since this option does not yield adj. Rsq, we need to use an additional command to get it:
.display e(r2_a)
This command yields the adj. Rsq=0.5319428
However, using the vce(robust) option is not the same as using the Generalized Least
Squares (Weighted Least Squares Method) which must be preferred to correct fully the
problem of HSK. Here are the steps to be taken to carry out a Feasible Generalized Least
Squares Method (when HSK structure is unknown).
1)
2)
3)
4)
5)
run the regression, predict the uhat (the estimated residuals), call it u!
generate squared u
run the regression of usq on the independent variables
predict the fitted values from this regression, call it v!
rerun the regression by weighing each variable with 1/v (see class notes for
generating this method)
Correcting for HSK: FGLS Method: General Method
. reg
lnsalary years yearssq
Source
SS
df
MS
Model
Residual
10.8438849
9.38050436
2
219
5.42194244
.042833353
Total
20.2243892
221
.091513074
lnsalary
Coef.
years
yearssq
_cons
.0438528
-.0006273
3.809365
Std. Err.
.0048287
.0001209
.0413383
t
9.08
-5.19
92.15
Number of obs
F( 2,
219)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.000
0.000
=
=
=
=
=
=
222
126.58
0.0000
0.5362
0.5319
.20696
[95% Conf. Interval]
.0343361
-.0008655
3.727894
.0533696
-.0003891
3.890837
. predict u, residual
. gen usq=u^2
. reg usq
years yearssq
Source
SS
df
MS
Model
Residual
.080454658
.996377813
2
219
.040227329
.00454967
Total
1.07683247
221
.004872545
usq
Coef.
years
yearssq
_cons
.0060837
-.0001288
-.011086
Std. Err.
.0015737
.0000394
.0134726
t
3.87
-3.27
-0.82
Number of obs
F( 2,
219)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.001
0.411
=
=
=
=
=
=
222
8.84
0.0002
0.0747
0.0663
.06745
[95% Conf. Interval]
.0029821
-.0002065
-.0376386
.0091853
-.0000512
.0154665
. predict v
(option xb assumed; fitted values)
. reg lnsalary years yearssq [aweight=1/v]
(sum of wgt is
1.9809e+04)
Source
SS
df
MS
Model
Residual
12.4573794
2.58113177
2
216
6.22868971
.011949684
Total
15.0385112
218
.068983996
lnsalary
Coef.
years
yearssq
_cons
.0312545
-.0002426
3.867963
Std. Err.
.0025661
.000063
.0114291
t
12.18
-3.85
338.43
Number of obs
F( 2,
216)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.000
0.000
Alternatively, one can use the command vwls to correct for HSK.
=
=
=
=
=
=
219
521.24
0.0000
0.8284
0.8268
.10931
[95% Conf. Interval]
.0261968
-.0003667
3.845436
.0363122
-.0001185
3.89049
. gen sqrtusq=sqrt(usq)
. vwls
lnsalary years yearssq, sd(sqrtusq)
Variance-weighted least-squares regression
Goodness-of-fit chi2(219) = 218.59
Prob > chi2
= 0.4951
lnsalary
Coef.
years
yearssq
_cons
.0436123
-.0006114
3.807325
Std. Err.
.0006536
.0000188
.0037622
Number of obs
Model chi2(2)
Prob > chi2
z
66.73
-32.44
1011.99
P>|z|
0.000
0.000
0.000
=
222
= 15451.64
= 0.0000
[95% Conf. Interval]
.0423313
-.0006483
3.799951
.0448933
-.0005744
3.814699
Notice that there are some slight differences in the coefficient estimates between these
two approaches, and the vwls command does not generate adj.Rsq.
I would prefer to use the previous method (FGLS) in steps.
Estimation by FGLS under HSK disturbances
a) Breusch-Pagan Specification
1. Regress the original model, calculate uhat and uhatsq, then regress uhatsq on
known factors causing HSK. This is the auxiliary regression.
2. Then type: “predict yhat” to get the estimated (fitted) values of uhatsq.
3. Weight (w)used in FGLS estimation is the inverse of the squared root of the yhat
(the fitted uhatsq)
Problem: No guarantee that the yhat will be positive, may not take the square root. If
this situation arises, treat negative values as positive (by taking the absolute value)
and then take the square root.
4. Multiply each variable with the weights (w), including the constant.
5. Regress: reg Yw w X1w X2w, no constant (suppressing the constant term)
where Yw is the product of the dependent variable and the inverse of the squared
root of the yhat (the fitted uhatsq)
b) Glesjer Specification
1. Regress the original model, calculate uhat and absuhat (by gen
absuhat=abs(uhat)), then regress absuhat on known factors causing HSK. This is
the auxiliary regression.
2. Then type: “predict yhat2” to get the estimated (fitted) values of absuhat.
3. Weight (w)used in FGLS estimation is the inverse of the yhat2 (the fitted absuhat)
Problem: No guarantee that the yhat will be positive, may not take the square root. If
this situation arises, treat negative values as positive (by taking the absolute value).
4. Multiply each variable with the weights (w), including the constant.
5. Regress: reg Yw w X1w X2w, no constant (suppressing the constant term) where
Yw is the product of the dependent variable and the inverse of the squared root of
the yhat (the fitted absuhat)
c) Harvey-Godfrey Specification
1. Regress the original model, calculate uhat and lnuhat (by gen lnhat=log(uhat)),
then regress lnuhat on known factors causing HSK. This is the auxiliary
regression.
2. Then type: “predict yhat3” to get the estimated (fitted) values of lnuhat.
3. Weight (w)used in FGLS estimation is the inverse of the squared root of the
antilog of yhat3 (the fitted lnuhat). Taking anti-log means to “exponentiate.”
There is no problem of negative values because exponentiation generates only
positive values.
4. Multiply each variable with the weights (w), including the constant.
5. Regress: reg Yw w X1w X2w, no constant (suppressing the constant term) where
Yw is the product of the dependent variable and the inverse of the squared root of
the yhat (the fitted lnuhat)
All these methods are alternative ways of correcting for HSK.