Time Series for Quants: ARIMA and GARCH

1. An AR(1) process $r_t = c + \phi r_{t-1} + \varepsilon_t$ with $|\phi| < 1$ is covariance stationary. What is its unconditional variance, and what happens to the autocorrelation at lag $k$ as $k \to \infty$?

2. You apply the Augmented Dickey-Fuller test to the log-price series $\ln P_t$ of a liquid equity index and obtain ADF statistic $= -1.8$, p-value $= 0.38$. You then apply it to the log-return series $r_t = \Delta \ln P_t$ and obtain ADF statistic $= -28.4$, p-value $< 0.001$. What do you conclude, and what are the implications for model selection?

3. A GARCH(1,1) fitted to daily S&P 500 log-returns gives $\hat{\omega} = 2\times10^{-6}$, $\hat{\alpha} = 0.08$, $\hat{\beta} = 0.90$. Compute: (a) the unconditional daily variance, (b) the half-life of a variance shock in days, and (c) the 10-day-ahead conditional variance forecast if today's conditional variance $\sigma_T^2 = 4\times10^{-4}$.

4. In a GARCH(1,1) model with Gaussian innovations, if $\hat{\alpha} + \hat{\beta} = 1$ (the IGARCH case), the unconditional variance is infinite — the process is non-stationary in variance — even though conditional variance forecasts remain finite and the process can still be used for short-horizon volatility forecasting.

5. The GJR-GARCH model adds an asymmetric term to GARCH(1,1): $\sigma_t^2 = \omega + (\alpha + \gamma \mathbf{1}_{\varepsilon_{t-1}<0})\varepsilon_{t-1}^2 + \beta\sigma_{t-1}^2$. In equity markets, $\hat{\gamma} > 0$ is typically found. What economic mechanism explains this, and which GARCH specification would you use for VaR estimation on a long equity portfolio and why?

6. You fit an ARIMA model to daily equity returns and find BIC selects ARIMA(0,0,0) — a white noise process. A colleague insists on using ARIMA(3,0,2) because it has lower AIC. Who is right, and what does this reveal about equity market returns?