Most linear models and times series models estimated by OLS method assume homoscedastic disturbances. Heteroscedasticity is most expected in cross-sectional data, but also in financial time series. We present the Breusch-Pagan test valid for a general linear models and finally we show a specific LM test for testing the ARCH(1) model.
If we assume an usual linear regression model,
The test procedure is as follows:
It can be shown that this test procedure is equivalent to compute
where
is the squared of the determination coefficient
in a linear regression of
on
.
An inconvenient of this test, in practice, is the unacknowledge of the exogenous variables responsible of the heteroscedasticity. In that case, we present the following test.
The Lagrange multiplier test procedure is also adequate to test particular form of an ARCH(1) model.
Let it the hypotheses be
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(6.14) |
Under the null hypothesis, the test consists of deriving the score and the information matrix.
In this case, the score (1.6) can be written as
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(6.15) |
More generally, the partial derivation of the log-likelihood
function for a sample size is, under
,
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(6.16) |
The elements of the Hessian matrix can be calculated under the null hypothesis, and the information matrix is consistently estimated, taking into account expression (1.11), by
Applying previous Breusch-Pagan test and under the assumption that
the are normally distributed, is given by
Again, a statistic that is asymptotically equivalent to this one,
and which is also computationally convenient, can be obtained by
considering the square of the multiple correlation coefficient
in the regression of
on
. Given that adding a
constant and multiplying by a scalar will not change the
of a regression, this is also equivalent to the regression of
on
and a constant. The statistic will be
asymptotically distributed as chi square of one degree of freedom.
To carry out this regression, we save the residuals of the OLS
regression in the first stage. When we then regress the square
residuals on a constant and
, and test
as
.
In the ARCH() model the procedure is similar but taking in
account that the
vector is q-dimensional containing the
squared lagged perturbations and consequently the asymptotical
reference distribution is a
.
A direct application of the LM test to the Ibex35 data with the function archtest reveals a presence of heteroscedastic effects of order one in the conditional variance. If we write
; LM test for ARCH effects to Ibex35 return data arch= archtest(return,1,"LM")we obtain,
Contents of archt [1,] "Lag order Statistic 95\% Critical Value P-Value " [2,] "__________________________________________________" [3,] "" [4,] " 1 4.58716 3.84146 0.03221"