Brownian Bridge™

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 $k \to \infty$ ?

Unconditional variance

= \sigma_\varepsilon^2/(1-\phi^2)

; autocorrelation at lag

k

\phi^k \to 0

Unconditional variance

= \sigma_\varepsilon^2/(1-\phi)

; autocorrelation at lag

k

\phi^k \to 0

Unconditional variance

= \sigma_\varepsilon^2

; autocorrelation at lag

k

\phi^k \to 0

Unconditional variance

= \sigma_\varepsilon^2/(1-\phi^2)

; autocorrelation at lag

k

k\phi^k \to 0

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?

Log-prices are stationary; log-returns are not. Use ARIMA(p,1,q) on log-pricesLog-prices have a unit root (non-stationary); log-returns are stationary. Use ARIMA(p,0,q) directly on log-returns (i.e., ARMA on returns, which is

d=0

)Both series are non-stationary. Use GARCH on log-pricesBoth series are stationary. Use ARIMA(0,1,0) on log-prices

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 $\sigma_T^2 = 4\times10^{-4}$ .

(a)

1\times10^{-4}

, (b) 34 days, (c)

2\times10^{-4}

(a)

1\times10^{-4}

, (b) 34 days, (c)

1.74\times10^{-4}

(a)

2\times10^{-5}

, (b) 17 days, (c)

1.74\times10^{-4}

(a)

1\times10^{-4}

, (b) 69 days, (c)

3\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.

truefalse

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?

\hat{\gamma} > 0

reflects that positive news increases vol more than negative news; use standard GARCH for VaR since tail risk is symmetric

\hat{\gamma} > 0

reflects the leverage effect: falling prices increase financial leverage, raising equity risk; for a long portfolio, use GJR-GARCH with Student-t innovations since it captures both the asymmetric vol response and the fat tails that drive VaR exceedances

\hat{\gamma} > 0

reflects that volatility is mean-reverting faster after negative shocks; use EGARCH for VaR since it guarantees positivity

\hat{\gamma} > 0

is a statistical artefact; use standard GARCH with Gaussian innovations for simplicity and regulatory acceptability

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?

The colleague is right: lower AIC always indicates a better model regardless of the criterion usedYou are right: BIC penalises complexity more heavily (penalty

k\ln T

2k

). For

T > e^2 \approx 7

, BIC is stricter than AIC. ARIMA(0,0,0) being selected by BIC but not AIC suggests the additional AR/MA terms improve in-sample fit marginally but are not parsimonious — consistent with the near-efficiency of daily equity returns (ACF near zero)You are right: AIC is always wrong for financial time series because returns are non-normalThe colleague is right: more parameters always improve out-of-sample forecasting by capturing latent structure

Quiz: Time Series for Quants: ARIMA and GARCH

Quick Quiz