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8/12/2019 Overheads ECON232 Heteroskedasticity Handout
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Heteroskedasticity
Consequences of Heteroskedasticity
SR1 : The model is correctly specied.SR2 : E (e i ) = 0 for i = 1, ..., N SR3 : var (e i ) = 2 for i = 1, ..., N (homoskedasticity)SR4 : cov (e i , e j ) = 0 for i = 1, ..., N , j = 1, ..., N and i = j (no autocorrelation)SR5 : The variable x i is not random, and it must take at leasttwo different valuesSR6 : (optional) e i N (0, 2 )
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HeteroskedasticityDetecting Heteroskedasticity
Graphical Methods: Estimate the model by OLS, plot theresiduals against each of the variables, and look for evidence of the residual variance changing with the variable.
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Heteroskedasticity
Detecting Heteroskedasticity
Breusch-Pagan/Koenker (Lagrange Multiplier) Test:
Step 1: Estimate the main regression equation
y i = 1 + 2 x 2 i + 3 x 3 i + ... + k x ki + e i
by OLS and compute the residuals e i .
Step 2: Estimate the auxiliary regression by OLS, i.e.regress the squared residuals e 2i on the explanatoryvariables that are expected to have an effect on thevariance of the errors. e.g.
e 2i = 1 + 2 x 2 i + 3 x 3 i + ... + s x si + i
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HeteroskedasticityDetecting Heteroskedasticity
Breusch-Pagan/Koenker (Lagrange Multiplier) Test:
Step 3: Compute BP = N R 2 whereN = the number of observations;R 2 = the coefficient of determination
of the auxiliary regression.
Step 4:H 0 : 2 = 3 = ... = s = 0 (homoskedasticity)H 1 : j = 0 for some j = 1, ..., s (heteroskedasticity)
Step 5:If H 0 is true, then BP 2s 1 where s is the number of regression coefficients in the auxiliary regression.Therefore, reject H 0 if BP > 2
s 1
; .
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Heteroskedasticity
Detecting Heteroskedasticity
Whites Test:
Step 1: e.g. Consider the regression equation
y i = 1 + 2 x 2 i + 3 x 3 i + e i
Estimate by OLS and compute the residuals e i .
Step 2: Estimate the auxiliary regression by OLS, i.e.
regress the squared residuals e 2
i on all the explanatoryvariables, their squares and cross-products.
e 2i = 1 + 2 x 2 i + 3 x 3 i + 4 x 22 i + 5 x
23 i + 6 x 2 i x 3 i + i
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HeteroskedasticityDetecting Heteroskedasticity
Whites Test:
Step 3: Compute WH = N R 2 whereN = the number of observations;R 2 = the coefficient of determination
of the auxiliary regression.
Step 4:H 0 : 2 = 3 = ... = 6 = 0 (homoskedasticity)H 1 : j = 0 for some j = 1, ..., 6 (heteroskedasticity)
Step 5:If H 0 is true, then WH 2s 1 where s is the number of regression coefficients in the auxiliary regression.Therefore, reject H 0 if WH > 2
s 1 ; .
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HeteroskedasticityDealing with Heteroskedasticity
Whites Heteroskedasticity-Consistent VarianceEstimator:
e.g. for the two-variable regression model
var (b 2 ) =
N
i = 1(x i x )2 2i
N
i = 1(x i x )2
2
Whites estimator of this quantity is
var w (b 2 ) =N
i = 1(x i x )2 e 2i
N
i = 1(x i x )2
2
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Heteroskedasticity
Dealing with Heteroskedasticity
Whites Heteroskedasticity-Consistent VarianceEstimator:
var w (b j ) is a consistent estimator of var (b j ) The OLS estimator with Whites variance estimatorprovides an unbiased (but inefficient) coefficient estimatorwith valid t- and F-tests and valid condence intervals.
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HeteroskedasticityDealing with Heteroskedasticity
Whites Heteroskedasticity-Consistent Variance
Estimator: In Gretl, with cross-sectional data, choose the robust
standard errors option in the OLS dialogue box.
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HeteroskedasticityDealing with Heteroskedasticity
Generalised Least Squares (GLS):
e.g. consider the regression model
y i = 1 + 2 x 2 i + e i where var (e i ) = 2i (1)
Divide by i y i i
= 11 i
+ 2x 2 i i
+ e i i
i.e. y i = 1 x 1 i + 2 x 2 i + e i (2)
where y i = y i
i , x 1 i =
1 i
, x 2 i = x 2 i
i and e i =
e i i
.
Note that var (e i ) = var (e i i
) = 1 2i
var (e i ) = 2i
2i = 1
Regression (2) is homoskedastic.13/16
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Heteroskedasticity
Dealing with Heteroskedasticity
Generalised Least Squares (GLS):
y i = 1 x
1 i + 2 x
2 i + e
i (2)
Therefore, if the values of i , i = 1, ..., N were known,OLS could be used to estimate the parameters inEquation (2). This approach to estimating thecoefficients in Equation (1) is GLS.
Since i is unknown, the GLS estimator is infeasible.
Note: This GLS estimator may be derived by choosing theparameter values that minimise the sum of the squaredweighted errors, where the weights are 1
2i .
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HeteroskedasticityFeasible Generalised Least Squares (FGLS):
e.g. consider the regression model
y i = 1 + 2 x 2 i + e i where var (e i ) = 2i (1)Estimate Equation (1) by OLS. Compute the residuals e i .Use OLS to estimate the auxiliary regression
ln(e 2i ) = 1 + 2 x 2 i + u i (2)
Generate the tted values from the auxiliary regression
g i = 1 + 2 x 2 i , i = 1, ..., N (3)
Estimate 2i by s 2i = e
g i , i = 1, ..., N .
Create the transformed variables y +i =
y i s i
, x +2 i = x 2 i
s i , x +1 i =
1s i
Use OLS to estimate the regression
y +i
= 1 x +1i + 2 x +2
i + e +
i (4)
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Heteroskedasticity
Feasible Generalised Least Squares (FGLS):The FGLS estimator is biased, but is consistent and
asymptotically efficient. t- and F-tests are asymptoticallyvalid, and condence intervals have the correct probabilitycoverage.
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