Overheads ECON232 Heteroskedasticity Handout

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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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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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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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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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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Overheads ECON232 Heteroskedasticity Handout

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