Explanation

The regression equation $\varepsilon_t^2 = a_0 + a_1 \varepsilon_{t-1}^2 + u_t$ is testing for ARCH (Autoregressive Conditional Heteroskedasticity) effects.

Key points:

• When $a_1$ is positive and statistically significant, it indicates that the variance of the error term depends on past squared errors
• This is the definition of heteroskedastic errors in time series context
• ARCH models specifically capture this phenomenon where volatility clusters over time

Why other options are incorrect:

• Unit root (A): This relates to non-stationarity in the mean of the series, not the variance
• Mean-reverting (B): This refers to the series returning to its long-term mean, which is unrelated to variance behavior

The positive and significant $a_1$ coefficient directly indicates the presence of heteroskedastic errors following an ARCH(1) process.