[Gretl-users] Breusch-Pagan test for Heteroskedasticity (Gretl-users Digest, Vol 100, Issue 3)

Alecos Papadopoulos papadopalex at aueb.gr
Mon May 4 11:34:04 EDT 2015

I verified using your data set that Gretl calculates correctly the 
Breusch-Pagan LM test for heteroskedasticity, *as described in the 
original paper,**

A Simple Test for Heteroscedasticity and Random Coefficient Variation
Author(s): T. S. Breusch and A. R. Pagan
Source: Econometrica, Vol. 47, No. 5 (Sep., 1979), pp. 1287-1294
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/1911963

The auxiliary regression is performed using the *scaled* squared 
residuals as Dependent variable (as indicated in the Gretl's output), 
and the statistic is 0.5*R^2*(Total Sum of Squares of the Dependent 
Variable), the R^2 from this auxiliary regression, where I used the 
exact same regression matrix of the main regression (constant + the 
three regressors)

The scaling of the squared residual series is by the maximum likelihood 
estimator for the variance from the original model, ie. sum of squared 
residuals divided by nobs (not nobs-1).

Maddala's  Econometrics textbook (2001, 3d ed. pp 205-207) explains the 
relation of the above statistic with the approach you implemented 
(correctly) by hand, which is the "usual" way to obtain an LM statistic.
But these are only asymptotically equivalent, and in practice the 
estimated sigma^4 is involved, whose estimation is seriously biased in 
any case for samples much larger than yours.

So I do not think there is any bug or computational mistake involved, 
just an (educative) case of asymptotic equivalence not materializing at 
finite-sample level.
Personally I would go with the original Breusch-Pagan test statistic.

Alecos Papadopoulos
Athens University of Economics and Business, Greece
Department of Economics
fax: +30-210-8259763

On 4/5/2015 16:31, gretl-users-request at lists.wfu.edu wrote:
> Message: 1
> Date: Mon, 04 May 2015 11:07:53 +0100
> From: Graham Stark<graham.stark at virtual-worlds.biz>
> To:gretl-users at lists.wfu.edu
> Subject: [Gretl-users] Pagan  Heteroskedasticity Test
> Message-ID:<1430734073.12002.6.camel at virtual-worlds.biz>
> Content-Type: text/plain; charset="utf-8"
> Hi,
> I'm puzzled by the output of the Breusch-Pagan Heterokskedasticity test
> The example here:
> http://virtual-worlds-research.com/ou/econometrics/hetero_test.zip
> shows me attempting to replicate Gretl's Breusch-Pagan and White tests
> on a simple teaching dataset. The White Test is identical but the Pagan
> test is very different from how I always thought you did it. Apologies
> if I've misunderstood what you're doing.
> This is the packaged Gretl 1.9.91 from Debian Linux, but I'm pretty
> certain you get the same result on the latest Windows version (it was a
> student running Windows that asked me about this). As yet I haven't
> tried the latest Sourceforge versions, I'm afraid.
> While I'm here, I'd just like to thank everyone involved with this
> project for all their good work. I teach with Gretl at the UK Open
> University, several hundred students per year, and it's been a boon to
> us.
> regards,
> Graham
> -- Graham Stark, Virtual Worlds Research 
> http://www.virtual-worlds-research.com 136 Hainault Avenue, Milton 
> Keynes, MK14 5PG, UK t: (+044) 01908 618239 skype: graham_k_stark 
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