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OutlineIntroduction

Park testGoldfeld-Quandt test

White testBreusch Pagan test

Tests of Heteroscedasticity

Prof. Rizzi Laura

January 20, 2009

Prof. Rizzi Laura Tests of Heteroscedasticity
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OutlineIntroduction



Brief Overview on:

Introduction

Park test Goldfeld-Quandt test

White test

Breusch Pagan test

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OutlineIntroduction



Introduction

Econometricians do not use always the same test to verify the presence of heteroscedasticdisturbances, because heteroscedasticity may assume different structures and sometimes it is noteasy to understand this structure.There is not a uniform approach in the choice of tests, but generally it is preferable to answer to

some questions before the application of whichever test:

are there specification errors in the chosen regression model?

are there possibilities of heteroscedastic errors in the phenomenon analized?

considering the graphical distribution of residuals versus each regressor, is there evidence ofheteroscedasticity? It is interesting to analyse the graphical distribution of residuals versussome regressors if they are thought to generate heteroscedasticity; if residuals appear related

(positively or negatively) with a regressor Z there may be heteroscedasticity. the assumption assumption of constant error variance can be checked throught the residual

plot. A residual plot is a scatterplot of the standardised residuals against the fitted values.


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OutlineIntroduction



Introduction

Residual plot - examples

The standardised residuals, si, are designed to overcome the problem of different variances of theraw residuals. The problem is solved by dividing each of the raw residuals by an appropriate term.Recall that the (standardised) residuals are the deviations of the observations away from the fittedvalues. If Assumptions of constant error variance is satisfied we would expect the residuals to varyrandomly around zero and we would expect the spread of the residuals to be about the same

throughout the plot.

Residual plot for the relationship between icecream consumption and temperature, icecream price, average annual family income,and the year.

The points in the plot seem to be fluctuatingrandomly around zero in an un-patternedfashion.The plot does not suggest violations of theassumption of constant variance of therandom errors.


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OutlineIntroduction



Introduction

If the residuals seem to increase or decrease in average magnitude with the fitted values, it is anindication that the variance of the residuals is not constant.If the points in the plot lie on a curve around zero, rather than fluctuating randomly, it is anindication that linearity assumption is broken.If a few points in the plot lie a long way from the rest of the points, they might be outliers, that is,data points for which the model is not appropriate.

Fig. a below shows a residual plot withno systematic patter.

In fig. b there is a clear curved pattern,lineaqrity assumption may be broken.

In fig. c the random variation of theresiduals increases as the fitted values

increase, then variance is not constant. Fig. d most of the residuals are

randomly scattered around 0, but oneobservation has produced a residualwhich is much larger than any of theother residuals. The point may be anoutlier.


O tline

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OutlineIntroduction



Park test

This test procedure requires 3 steps:

model OLS estimation to derive the OLS residuals, ei;

the derivation of the ln(e2i which are considered as dependent variable in the regressionwhere the only regressor is the log of the r.v. considered proportionality factor;

the estimation results of this model are used to verify the presence of heteroscedastic errors;

Then:1 - Let consider the regression model: yi = 0 + 1X1i + 2X2i + ui; the OLS estimation producesthe OLS residuals:

ei = yi (b0 + b1X1i + b2X2i)

2 - We derive the dependent variable ln(e2i ) for the regression:

ln(e2i ) = 0 + 1lnZi + vi

Where Z is a r.v. that may cause heteroscedasticity.

3 - We verify significance of the coefficient 1 using t test. If this coefficient is significant there is

heteroscedasticity explained by the r.v. Z.


Outline

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OutlineIntroduction



Park test

The Park test is not used frequentely because is not easy to chose the r.v. Z.When, in cross-section data, the observation units are regions, nations, provinces, etc. the r.v.(proportionality factor) to be chosen is a size variable which measures indirectly the observationalunits dimension.

Example

We use Park test to verify heteroscedasticity in the following data (n = 33):

Y is the number of customers of sampled restaurants;

C is the regressor measuring the number of competitive restaurants;

P is the regressor measuring the resident population;

I is the regressor measuring the average income of resident population.

Estimated equation is:

yi = 102, 2 + 9075 Ci+ 1288 Ii+ 0, 35 Pi(2053) (0, 54) (0, 073)4, 42 2, 37 4, 88


Outline

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Park test

Where R2 = 0, 58 and F = 15, 75. The estimated auxiliary regression is:

ln(e2i

) = 21, 05 + 0, 29 ln(Pi)(0, 63)0, 46

Given the sample value of the t statistic we accept the null hypothesis (1 = 0) then accept the

null hypothesis of omoscedasticity.


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Goldfeld-Quandt test

This test is frequentely used because it is easy to apply when one of the regressors (or another r.v.)is considered the proportionality factor of heteroscedasticity.The test has two limits: its difficulty to reject the null hypothesis of omoscedasticity and the factthat it do not allow to verify other forms of heteroscedasticity.This test is based on the hypothesis that the error variance is related to a regressor X.

The test procedure is the following:1 - the observations on Y and X are sorted following the ascending order of the regressor X

which is the proportionality factor;

2 - we divide the sample observations in three subsamples omitting the central one;

3 - we estimate throught OLS the regression models on the first and third subsample (then onnc

2 observations each; the number of observations considered has to be sufficiently large);

4 - we calculate the relative RSS, denoted as RSS1 and RSS2;5 - we derive the Goldfeld-Quandt test: GQ = R =

RSS2RSS1

;

6 - the test R under the null hypothesis has F distribution with degrees of freedom nc2k2both for numerator and denominator.


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IntroductionPark test

Goldfeld-Quandt testWhite test

Breusch Pagan test

Goldfeld-Quandt test

If the sample value of the test F is greater (in a.v.) than the critical value, at the chosen

significance level, we reject the null hypothesis of omoscedasticity.Idea: if R is large then RSS2 is greater that RSS1, which means that residuals increase with theregressor.The power of this test depends on the number of omitted observations (usually n3 observationshave to be omitted). If we exclude too much observations the RSS2 and RSS1 have too lowdegrees of freedom, if we exclude to few observations the test power is low because the comparisonbetween RSS2 and RSS1 becomes less effective.


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Breusch Pagan test

White test

Sometimes the researcher with to verify if more than one variable is proportionality factor in theheteroscedasticity process: in these situations it is preferable to consider the Breush Pagan test orthe White test.The White test has the advantage that it does not assume a specific form of heteroscedasticity.It is based on a auxiliary regression with suqred residuals as dependent variable and regressors

given by: the regressors of the initial model,, their squares and their cross-products.The White test procedure is as follows:

1 - we estimate the regression model throught OLS obtaining the OLS residuals, ei. Forinstance we estimate: yi = b0 + b1x1i + b2x2i, then ei = yi yi;

2 - we estimate an auxiliary regression model with e2i as dependent variable and initialregressors, their squares and cross-products as covariates. For instance, we estimate:e2i = 0 + 1x1i + 2x2i + 3x

21i + 4x

22i + 5x1ix2i.

3 - we verify the significance of the auxiliary regression throught the test nR2

, which, underthe null hypothesis (omoscedasticity) has 2(q), where the degrees of freedom q are equalto the number of regressors in the auxiliary model. In the example q = 5.

4 - if the sample value of the 2(q) is greater than the critical one we reject the nullhypothesis of omoscedasticity.


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Breusch Pagan test

White test

This test may have some problems when the number of regressors in the initial model is high.In these situation the cross-products of regressors may be omitted in the auxiliary model.

When in the initial model there are dummy variables their squares are not included in the auxiliary

regression to avoid multicollinearity problems.


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Breusch Pagan test

Breusch Pagan test

Given the model:yt = x

Tt + ut

With t = 1, 2, . . . , n and xTt = [1x2tx3t xkt].We assume that heteroscedasticity takes the form:

E(ut) = 0 for all t2t = E(u

2t) = h(z

Tt )

Where zTt = [1z2tz3t zpt] and = [12 p] is a vector of unknown coefficients and h() issome not specified function that must take only positive values.The null hypothesis (omoscedasticity) is then:

H0 : 2 = 3 = = p = 0

Under the null we have 2t = h(1) (constant).

The restricted model under the null is estimated throught OLS, assuming distubances normally

distributed.


OutlineI t d ti
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Breusch Pagan test

Breusch Pagan test

The test procedure is the following:

estimate the ooriginal model equation by OLS and obtain the OLS residuals, ei = yt xTtb,

and the estimated variance of disturbances, 2 =

e2t/n;

regress the variable e2t

2 on zt by OLS and compute the ESS of this regression;

under the null hypothesis, H0 we have that:12 ESS

a 2(p 1); omoscedasticity is rejected

if 12 ESS exceeds the relative critical value on the 2 distribution;

a simpler procedure requires the regression of e2t on zt; then the nR2 of this regression is

asymptotically distributed as a 2(p 1) under the null.

This test needs the knowledge of the regressors z but not the knowledge of the functional form

h(). Sometimes the regressors in z may be some regressors included in the original model, in such

case this test becomes an ad hoc version on the White test.

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