Detecting Heteroscedasticity¶

Heteroscedasticity refers to the presence of changing variance within the residuals ($\hat{e}_{t}$) of a model, rather than the variance of the actual time series process itself. In a heteroscedastic model, the variance of the error term is time-dependent:
$$Var(e_{t}) = \sigma_{t}^{2}$$

1. Graphical Detection: ACF and PACF of Squared Residuals¶

A common way to detect non-constant variance is to examine the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots of the squared residuals ($e_{t}^{2}$).

Why squared residuals? Since we assume the mean of the errors is zero ($E(e_{t})=0$), the variance simplifies to the expectation of the squared errors: $V(e_{t}) = E(e_{t}^{2}) - [E(e_{t})]^{2} = E(e_{t}^{2})$.
Identification: If the variance is constant (homoscedastic), the ACF and PACF spikes of the squared residuals should remain within the 95% confidence limits, reflecting a completely random and stationary structure.

2. Statistical Tests for Heteroscedasticity¶

White's General Test¶

This test evaluates the null hypothesis ($H_{0}$) that the variance of the error term is constant ($\sigma_{e}^{2}$) given past information. It utilizes a regression of the squared residuals on the original lags, their squares, and their cross-products:
$$e_{t}^{2} = \alpha_{0} + \alpha_{1}Y_{t-1} + \alpha_{2} Y_{t-2} + \dots + \gamma_{1} Y_{t-1}^{2} + \gamma_{2}Y_{t-2}^{2} + \dots + \delta_{1}Y_{t-1}Y_{t-2} + \dots + u_{t}$$

Under Homoscedasticity: All coefficients except the intercept ($\alpha_{0}$) should be zero ($\alpha_{i} = \gamma_{i} = \delta_{i} = 0$).

Breusch-Pagan Test¶

This is a simpler version of White's test. It tests if the squared residuals are dependent on a linear combination of past terms ($Y_{t-1}, Y_{t-2}, \dots, Y_{t-m}$) without including the higher-degree squared or product terms. It essentially tests if all slope coefficients in the auxiliary regression are equal to zero.

3. Consequences and Solutions¶

Consequences: If heteroscedasticity is present, standard Ordinary Least Squares (OLS) estimates remain unbiased but are no longer efficient (they are not the Best Linear Unbiased Estimator). Additionally, the standard estimates of variance become biased, rendering OLS invalid for inference.
Required Action: To correct for this, Generalized Least Squares (GLS) or Weighted Least Squares (WLS) must be employed. Furthermore, you will need to model the volatility explicitly using ARCH (Autoregressive Conditional Heteroscedasticity) or GARCH models.