Testing for ARCH Effects

Procedure

Two main tests for detecting autoregressive conditional heteroscedasticity in time series residuals.

Test 1: Ljung-Box Test on Squared Residuals

Tests whether the squared residuals $\{a_t^2\}$ exhibit autocorrelation.

Hypotheses

• $H_0:{\rho }_1={\rho }_2=...={\rho }_m=0$ (no ARCH effects)
• $H_1:$ At least one ${\rho }_i\ne 0$ (ARCH effects present)

Test Statistic

$$Q(m)=T(T+2)\sum _{i=1}^m\frac{{\hat{\rho }}_i^2}{T-i}\sim {\chi }^2(m)$$

where ${\hat{\rho }}_i$ is the sample autocorrelation of $a_t^2$ at lag $i$.

Decision Rule

Reject $H_0$ if $Q(m)>{\chi }_{\alpha }^2(m)$

Interpretation: Rejection implies the presence of ARCH effects that need to be modeled.

Test 2: Lagrange Multiplier (LM) Test

A regression-based test proposed by Engle (1982) that uses an auxiliary regression.

Auxiliary Regression

Regress squared residuals on their own lags:

$$a_t^2={\alpha }_0+{\alpha }_1{a}_{t-1}^2+...+{\alpha }_m{a}_{t-m}^2+e_t$$

Hypotheses

• $H_0:{\alpha }_1={\alpha }_2=...={\alpha }_m=0$ (no ARCH effects)
• $H_1:$ At least one ${\alpha }_i>0$ (ARCH effects present)

Test Statistics

LM Statistic:

$$LM=T\cdot R^2\sim {\chi }^2(m)$$

where $T$ is the sample size and $R^2$ is the coefficient of determination from the auxiliary regression.

Alternative F-Statistic:

$$F=\frac{(SS{R}_0-SS{R}_1)/m}{SS{R}_1/(T-2m-1)}\sim F(m,T-2m-1)$$

where:

• $SS{R}_0={\sum }_{t=m+1}^T(a_t^2-\bar{\omega }{)}^2$ (restricted, $\bar{\omega }=\frac{1}{T}\sum a_t^2$)
• $SS{R}_1={\sum }_{t=m+1}^T{\hat{e}}_t^2$ (unrestricted)

Decision Rule

Reject $H_0$ if $LM>{\chi }_{\alpha }^2(m)$ or if $p$-value $<\alpha$.

Practical Example

For Intel stock returns:

• LM test: $F=53.62$, $p\approx 0$
• Ljung-Box on $a_t^2$: $Q(12)=89.85$, $p\approx 0$

Both strongly reject $H_0$, confirming significant ARCH effects.

Related

• ARCH(m) Model
• Building a Volatility Model