Engle's ARCH LM Test

Worked Solution

How to Think About It: You have a return series and you want to check whether its volatility is time-varying. Under the null, the squared residuals are just white noise -- they do not predict their own future values. The LM test checks this by running a simple regression of $\hat{\varepsilon}_t^2$ on its own lags and testing whether the regression has explanatory power. If the $R^2$ of this auxiliary regression is high, there is evidence of ARCH effects. This is appealing because you never actually need to estimate the ARCH model -- you only estimate the (simpler) null model.

Key Insight: The LM test exploits the fact that, under $H_0$ (no ARCH), $\varepsilon_t^2$ has constant conditional expectation $\omega$. Any predictability of $\varepsilon_t^2$ from its own lags is evidence against $H_0$. The test statistic is just $N \cdot R^2$ from an auxiliary OLS regression.

The Method:

Step 1 -- Fit the null model. Under $H_0$, the conditional variance is constant ($\sigma_t^2 = \omega$). Estimate the conditional mean model (e.g., $r_t = \mu + \varepsilon_t$ or an AR model) by OLS. Obtain the residuals $\hat{\varepsilon}_t$.

Step 2 -- Run the auxiliary regression. Regress the squared residuals on their $p$ lags:

$\hat{\varepsilon}_t^2 = c_0 + c_1 \hat{\varepsilon}_{t-1}^2 + c_2 \hat{\varepsilon}_{t-2}^2 + \cdots + c_p \hat{\varepsilon}_{t-p}^2 + u_t$

Compute the $R^2$ of this regression using observations $t = p+1, \ldots, N$.

Step 3 -- Compute the test statistic. The LM statistic is:

$T_{LM} = (N - p) \cdot R^2$

where $N - p$ is the effective sample size of the auxiliary regression.

Step 4 -- Derivation of the asymptotic distribution.

Under $H_0\colon \alpha_1 = \cdots = \alpha_p = 0$, the residuals $\varepsilon_t$ are i.i.d. with mean zero and constant variance $\omega$. Therefore $\varepsilon_t^2$ is also i.i.d. with mean $\omega$ and finite variance $\text{Var}(\varepsilon_t^2) = E[\varepsilon_t^4] - \omega^2$.

The auxiliary regression of $\hat{\varepsilon}_t^2$ on its lags is an OLS regression of i.i.d. variables on their own lags. Under $H_0$, the true coefficients $c_1 = \cdots = c_p = 0$, and standard OLS asymptotic theory gives:

$T_{LM} = (N-p) \cdot R^2 \xrightarrow{d} \chi^2(p)$

This follows from the general LM test principle: under $H_0$, the score vector (proportional to the regression coefficients) is asymptotically normal, and the quadratic form in the score gives a chi-squared distribution with degrees of freedom equal to the number of restrictions being tested.

Reject $H_0$ at level $\alpha$ if $T_{LM} > \chi^2_\alpha(p)$. Large values indicate ARCH effects.

Step 5 -- Controlling for conditional mean misspecification.

If the conditional mean model is wrong (e.g., you fit a constant mean when the true model is AR(1)), then $\hat{\varepsilon}_t = r_t - \hat{\mu}$ contains mean-model errors that induce spurious autocorrelation in $\hat{\varepsilon}_t^2$, inflating the LM statistic.

To guard against this:

1. Over-fit the mean model: Include enough AR/MA lags to absorb any mean dynamics before computing residuals. Use AIC/BIC to select the lag order.

1. Pre-whiten the residuals: Apply a Ljung-Box test to $\hat{\varepsilon}_t$ (not $\hat{\varepsilon}_t^2$) first. If there is serial correlation in the levels, the mean model is misspecified and should be corrected before running the ARCH LM test.

1. Use robust variants: Wooldridge (1991) proposed a regression-based test that is robust to certain forms of mean misspecification by including additional regressors that capture mean-model error.

1. Include mean regressors in the auxiliary regression: Add the original mean-model regressors (e.g., lagged returns) to the auxiliary regression of $\hat{\varepsilon}_t^2$ on its lags. This "partials out" the effect of mean misspecification.

Practical Considerations:

• The choice of $p$ matters. Too small and you miss higher-order ARCH effects; too large and you lose power. Common defaults: $p = 5$ for daily data, $p = 12$ for monthly.
• The test assumes finite fourth moments. For heavy-tailed returns (e.g., $t$-distributed with low degrees of freedom), the chi-squared approximation can be poor. Bootstrap the test statistic in such cases.
• The ARCH LM test detects ARCH but not GARCH directly. However, since GARCH($p$,$q$) implies ARCH($\infty$), the LM test with moderate $p$ typically has good power against GARCH alternatives.

Answer: The ARCH LM test statistic is $T_{LM} = (N-p) \cdot R^2$ from the regression of $\hat{\varepsilon}_t^2$ on $p$ lags of itself. Under $H_0$ (no ARCH), $T_{LM} \xrightarrow{d} \chi^2(p)$. To control for mean misspecification, over-fit the conditional mean model before extracting residuals, and verify that $\hat{\varepsilon}_t$ itself shows no serial correlation.