6.5 ARCH(1) Regression Model

The ARCH process defined in the previous sections is used as a tool to capture the behaviour of the volatility when it is time-varying in a high-frequency data. However, in a wide variety of contexts, the information set could also be determinant in specifying a time-varying mean. In this section, we will define the information set in terms of the distribution of the errors of a dynamic linear regression model.

The ARCH regression model is obtained by assuming that the mean of the endogenous variable $ y_{t} $ is given by $ x_{t}^{\top }\beta$, a linear combination of lagged endogenous and exogenous variables included in the information set $ I_{t-1}$, with $ \beta $ a $ k\times1$ vector of unknown parameters.

That is to say:

$\displaystyle y_{t}\vert I_{t-1}$ $\displaystyle \sim$ $\displaystyle N(x_{t}^{\top }\beta,\sigma_{t}^{2})\cr
\sigma_{t}^{2}$ (6.17)

where $ u_t= y_t -x_{t}^{\top }\beta$.

Under these assumptions and considering that the regressors include no lagged endogenous variables, the unconditional mean and variance can be derived as:

$\displaystyle E[u_{t}]$ $\displaystyle =$ $\displaystyle 0,\ E[y_{t}]=x_{t}^{\top }\beta\cr V[y_{t}]$ (6.18)

It can be shown that the $ u_{t}$ are uncorrelated, so that we have:
$\displaystyle E[u]$ $\displaystyle =$ $\displaystyle 0\cr V[u]$ (6.19)

where

$\displaystyle \sigma^{2}=\frac{\alpha_{0}}{1-\alpha_{1}}$

Thus, the Gauss-Markov assumptions are satisfied and ordinary least squares is the best linear unbiased estimator for the model and the variance estimates are unbiased and consistent. However, OLS estimators do not achieve the Cramer-Rao bound.

By using maximum likelihood techniques, it is possible to find a nonlinear estimator that is asymptotically more efficient than OLS estimators.

The log-likelihood function for $ \alpha=(\alpha_{0},\alpha_{1})^{\top }$ and $ \beta $ can be written, ignoring a constant factor as

$\displaystyle L(\alpha,\beta)= \sum l_{t}$

and

$\displaystyle l_{t}=-\frac{1}{2}\log[\alpha_{0}+\alpha_{1}(y_{t}-x_{t-1}^{\top ...
...t-1}^{\top }\beta)^{2}}
{\alpha_{0}+\alpha_{1}(y_{t}-x_{t-1}^{\top }\beta)^{2}}$

The maximum likelihood estimator is found by solving the first order conditions. The derivative with respect to $ \beta $ is

$\displaystyle \frac{\partial l_t}{\partial \beta}= \frac{x_tu_t}{\sigma^2_t} + ...
...ac{\partial \sigma_t^2}{\partial \beta} \left(\frac{u^2_t}{\sigma_t^2}-1\right)$ (6.20)

For the particular case $ p=1$, we obtain:

$\displaystyle \frac{\partial l_t}{\partial \beta}= \frac{x_tu_t}{\sigma^2_t} - \frac{1}{\sigma^2_t} \left(\frac{u^2_t}{\sigma_t^2}-1\right)\alpha_1x_{t-1}u_{t-1}$ (6.21)

The Hessian matrix for $ \beta $ is given by

$\displaystyle \frac{\partial^2 l_t}{\partial \beta\partial \beta^\top } =
-\fra...
...tial \sigma_t^2}{\partial \beta^\top }
\left(\frac{u_t^2}{\sigma^2_t}\right) -
$

$\displaystyle \frac{2x_tu_t}{\sigma_t^4}\frac{\partial \sigma_t^2}{\partial \be...
...op } \left(\frac{1}{2\sigma^2_t}\frac{\partial\sigma_t^2}{\partial\beta}\right)$ (6.22)

Taking into account (6.3) and as the conditional perturbations are uncorrelated, the information matrix is given by

$\displaystyle I_{\beta\beta^\top } = \frac{1}{T}\sum_t E\left[\frac{x_tx_t^\top...
...al \beta} \frac{\partial \sigma^2_t}{\partial \beta^\top } \vert I_{t-1}\right]$ (6.23)

Simple calculus then reveals,

$\displaystyle \frac{\partial \sigma_t^2}{\partial\beta} = \alpha_1x_{t-1}u_{t-1}$ (6.24)

and, finally the Hessian matrix is consistently estimated by,

$\displaystyle {\hat I}_{\beta\beta^\top } = \frac{1}{T} \sum_t x_t x^\top _t \left[\frac{1}{\sigma_t^2}+ 2\frac{\alpha_1^2}{\sigma_t^4}u_{t-1}^2\right]$ (6.25)

The off-diagonal block of the information matrix is zero (see Engle (1982) for the conditions and the proof of this result). As a consequence, we can separately estimate vectors $ \alpha $ and $ \beta $.

The usual method used in the estimation is a two-stage procedure.

Initially, we find the OLS $ \beta $ estimate

$\displaystyle {\hat \beta} = (X^\top X)^{-1}X_t^\top y$ (6.26)

where $ X$ is the usual design matrix $ T \times k$ and $ y$ is the $ (T\times 1)$ vector of the endogenous variable. We calculate the residuals $ {\hat u}_t = y_t- x_t^\top \hat\beta$.

Secondly, given these residuals, we find an initial estimate of $ \alpha=(\alpha_0,\alpha_1)$, replacing $ y_t=u_t$ by $ {\hat u}_t$ in the maximum likelihood variance equations (6.6). In this way, we have an approximation of the parameters $ \alpha $ and $ \beta $.

The previous two steps are repeated until the convergence on $ \hat\alpha$ and $ \hat\beta$ is obtained.

Additionally, the Hessian matrix must be calculated and conditional expectations taken on it.

If an ARCH regression model is symmetric and regular, the off-diagonal blocks of the information matrix are zero [(see theorem 4, in Engle; 1982)].

Because of the block diagonality of the information matrix, the estimation of $ (\alpha_{0},\alpha_{1})$ and $ \beta $ can be considered separately without loss of asymptotic efficiency.

Alternatively, we can use an asymptotic estimator that is based on the scoring algorithm and which can be found using most least squares computer programs.

A homoscedastic test for this model is follows by a general LM test, where under the restricted model the conditional variance does not depend on the $ \alpha_1$. For a more detailed derivation of this test (see section 4.4, in Gouriéroux ; 1997).