L19 Diagnostic Checking 2

Detecting Heteroscedasticity

• changing variance for the residuals (not the actual time series process)

\[ Var(e_{t}) = \sigma_{t}^{2} \]

Using ACF-PACF plots (for Squared residuals)

• Why for squared residuals and not actual time series?
  • First, \(E(e_{t})=0\)
  • Second \(V(e_{t}) = E(e_{t}^{2})- E^{2}(e_{t})= E(e_{t}^{2})= e_{t}^{2}\)
• If variance of errors are constant
  • ACF and PACF should be within 95% limits
  • completely random and stationary structure in the \(e_{t}^{2}\) vs \(t\) plot

White's General Test

\[ H_{0}: Var(e_{t}) = E(e_{t}^{2}|Y_{t-1}, Y_{t-2},\dots) = \sigma_{e}^{2} \]

$$
\begin{align}

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}
\end{align}
$$

• These are all the terms on which \(e_{t}^{2}\) may be dependent if they are time dependent.

Thus, under homoscedastic case

\[ \alpha_{1} = \alpha_{2} = \dots = \gamma_{1} = \gamma_{2} = \dots = \delta_{1} = \delta_{2} = 0 \]

Breusch Pagan Test

Same as #White's General Test but, we base it on simple \(Y_{t-1}, Y_{t-2}\dots, Y_{t-m}\) terms instead of including second degree terms (squares and products).

Tests if all slope coefficients are equal to 0 or not.

Consequences of Heteroscedasticity

• If Heteroscedasticity
  • Parameter estimates are unbiased but not efficient
  • GLS or WLS has to be used (OLS invalid)
  • estimate of variance is also biased
• What to do?
  • We need to model volatility (ARCH, GARCH)