# Testing for ARCH Process

I have sales data that I think would be best modeled with an ARCH model.

I believe the following is true, but I'm not sure:

If the residuals are regressed against the sales data using the equation $e_t=b_0+b_1*X_t$, where $e$ represents the residual terms from the original regression and $X$ represents the sales data, if $b_1$ is statistically different from zero, then the regression model contains an ARCH process.

Is what I'm saying true?

No, this is not true. This is conditional heteroskedasticity, but not autoregressive conditional heteroskedasticity (ARCH). The latter is characterized by autocorrelation between squared residuals and their lags.

You could test for ARCH patterns up to order $m$ by checking the joint statistical significance of all slope coefficients in the model $$e^2_t=\alpha_0+\alpha_1 e^2_{t−1}+\dots+\alpha_m e^2_{t−m}+u$$ in the Lagrange multiplier form: take $R^2$ from the regression and multiply it by the number of observations $T$; this will be distributed as $\chi^2(m)$ under the null of no ARCH effects. This is known as ARCH-LM test. See more in Wikipedia.