6.4 Testing Heteroscedasticity and ARCH(1) Disturbances

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.


6.4.1 The Breusch-Pagan Test

If we assume an usual linear regression model,

$\displaystyle y_t = x_t'\beta + u_t,
$

where $ x_t'\beta$ is a linear combination of exogenous variables included in the information set $ I_{t-1}$ whit $ \beta $ a $ k\times1$ vector of unknown parameters. It is assumed that heteroscedasticity takes the form,

$\displaystyle \sigma_t^2 = h(z_t'\alpha)
$

where $ z'_t=[1,z_{1t},\dots,z_{pt}]$ is a vector of known variables and $ \alpha'=[\alpha_0,\alpha_1,\dots,\alpha_p]$ is a vector of unknown coefficients and $ h(.)$ is an unspecified positive function. Let it the hypotheses be

$\displaystyle H_0:\alpha_1=\dots=\alpha_p=0
$

that is, $ \sigma^2=h(\alpha_0)$ is constant. Under the null hypothesis, and taken in account that this test is a LM test, the model is estimated simply applying OLS method.

The test procedure is as follows:

It can be shown that this test procedure is equivalent to compute $ TR^2$ where $ R^2$ is the squared of the determination coefficient in a linear regression of $ e_t$ on $ z_t$.

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.


6.4.2 ARCH(1) Disturbance Test

The Lagrange multiplier test procedure is also adequate to test particular form of an ARCH(1) model.

Let it the hypotheses be

$\displaystyle H_{0}$ $\displaystyle \equiv$ $\displaystyle \alpha_{1} = 0 \ (\sigma_{t}^{2} = \sigma^{2} =
\alpha_{0})\cr H_{a}$ (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

$\displaystyle \left.
\frac{\partial l_t}{\partial \alpha_0} \right\vert _{H_{0}}$ $\displaystyle =$ $\displaystyle \frac{1}{2\alpha_0 } \left(\frac{u_t^2}{\alpha_0}-1\right)\cr
\left.
\frac{\partial l_t}{\partial \alpha_1}\right\vert _{H_{0}}$ (6.15)

More generally, the partial derivation of the log-likelihood function for a $ T$ sample size is, under $ H_{0}$,

$\displaystyle \left.
\frac{\partial L}{\partial \alpha} \right\vert _{H_{0}} =
...
...}
\left(\frac{u_t^2}{\alpha_0}-1\right) = \frac{1}{2\alpha_0 }
z^{\top } w_{0}
$

where
$\displaystyle z_{t}^{\top }$ $\displaystyle =$ $\displaystyle (1,u_{t-1}^{2})\cr z^{\top }$ (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

$\displaystyle \left.\widehat{I}_{\alpha\alpha}\right\vert _{H_{0}} = \frac{1}{2\alpha_{0}^{4}}
z^{\top } z $

Applying previous Breusch-Pagan test and under the assumption that the $ u_{t}$ are normally distributed, is given by

$\displaystyle \frac{1}{2} w^{\top }_{0} z (z^{\top } z)^{-1} z^{\top }
w_{0}$

and, under $ H_{0}$, this statistic is asymptotically distributed as $ \chi_{1}^{2}$.

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 $ (R^{2})$ in the regression of $ w_{0}$ on $ z$. Given that adding a constant and multiplying by a scalar will not change the $ R^{2}$ of a regression, this is also equivalent to the regression of $ u_{t}^{2}$ on $ u_{t-1}^{2}$ 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 $ u_{t-1}^{2}$, and test $ TR^{2}$ as $ \chi_{1}^{2}$.

In the ARCH($ q$) model the procedure is similar but taking in account that the $ z_t$ vector is q-dimensional containing the squared lagged perturbations and consequently the asymptotical reference distribution is a $ \chi^2_{q}$.

6.4.2.0.1 Ibex35 data (continued)

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"
26777 XEGarch10.xpl