Time Series for Quants: ARIMA and GARCH

Setup

Why Time Series Models Matter in Quant Finance

Time series methods appear in two distinct roles on a quant desk:

1. Return prediction: Can past prices or returns predict future ones? ARIMA provides a rigorous framework for modelling serial dependence in levels and returns — and usually confirms that equity daily returns have almost none.
2. Volatility forecasting: Volatility is serially correlated even when returns are not. GARCH models capture volatility clustering — the empirical observation that large moves tend to cluster in time — and are essential for option pricing under stochastic vol, VaR estimation, and risk-adjusted sizing.

Conventions throughout. All rates of return are continuously compounded: $r_t = \ln(P_t / P_{t-1})$. Volatility $\sigma_t$ is annualised unless stated otherwise. Daily returns assumed to have 252 trading days per year. All series assumed to be observed at equally spaced intervals.

Theory

1. Stationarity

A time series $(r_t)_{t \in \mathbb{Z}}$ is weakly stationary (covariance stationary) if:

1. $\mathbb{E}[r_t] = \mu < \infty$ for all $t$,
2. $\text{Var}(r_t) = \sigma^2 < \infty$ for all $t$,
3. $\text{Cov}(r_t, r_{t-k}) = \gamma(k)$ depends only on lag $k$, not on $t$.

Why it matters. Statistical inference on time series requires stationarity: parameter estimates have no asymptotic meaning for non-stationary series. Log-prices $\ln P_t$ are typically non-stationary (unit root); log-returns $r_t = \Delta \ln P_t$ are typically stationary.

Unit root test. The Augmented Dickey-Fuller (ADF) test tests $H_0$: unit root present (non-stationary) against $H_A$: stationary. Reject $H_0$ at the 5% level if the ADF test statistic is below the critical value (approximately $-2.86$ for no constant, $-3.43$ for constant and trend).

2. ARMA Models

AR($p$) — Autoregressive. The series depends linearly on its own past $p$ values:

$r_t = c + \phi_1 r_{t-1} + \cdots + \phi_p r_{t-p} + \varepsilon_t, \qquad \varepsilon_t \overset{\text{iid}}{\sim} (0, \sigma^2).$

Using the lag operator $L$ ($L^k r_t = r_{t-k}$), write as $\Phi(L) r_t = c + \varepsilon_t$ where $\Phi(z) = 1 - \phi_1 z - \cdots - \phi_p z^p$. The process is stationary iff all roots of $\Phi(z) = 0$ lie outside the unit circle.

MA($q$) — Moving Average. The series is a linear combination of current and past shocks:

$r_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \cdots + \theta_q \varepsilon_{t-q} = \mu + \Theta(L)\varepsilon_t.$

An MA($q$) process is always stationary. It is invertible (representable as an infinite AR) iff all roots of $\Theta(z) = 0$ lie outside the unit circle.

ARMA($p, q$): $\Phi(L) r_t = c + \Theta(L) \varepsilon_t.$

ARIMA($p, d, q$): Apply differencing $d$ times before fitting ARMA($p, q$). For $d=1$: model applies to $\Delta r_t = r_t - r_{t-1}$. For daily equity returns, $d=0$ is appropriate (returns are already stationary). For log-prices, $d=1$ produces returns.

Model selection. Use information criteria:

• AIC: $-2\ell(\hat{\theta}) + 2k$, where $\ell$ is log-likelihood and $k$ is number of parameters.
• BIC: $-2\ell(\hat{\theta}) + k\ln T$.

BIC penalises complexity more heavily and is preferred when the true model is parsimonious. Inspect ACF (autocorrelation function) and PACF (partial ACF) to guide $p$ and $q$ choices.

3. GARCH: Generalised ARCH

Motivation. Equity returns $r_t$ are approximately serially uncorrelated (ACF of $r_t$ near zero), but $r_t^2$ has significant positive autocorrelation at many lags. This is volatility clustering — Mandelbrot (1963) observed that "large changes tend to be followed by large changes, of either sign." ARCH/GARCH models this explicitly.

GARCH($p, q$) — Bollerslev (1986). Decompose the return as:

$r_t = \mu + \varepsilon_t, \qquad \varepsilon_t = \sigma_t z_t, \qquad z_t \overset{\text{iid}}{\sim} (0,1),$

where the conditional variance $\sigma_t^2$ follows:

$\sigma_t^2 = \omega + \sum_{i=1}^q \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^p \beta_j \sigma_{t-j}^2.$

Parameters and constraints.

• $\omega > 0$, $\alpha_i \geq 0$, $\beta_j \geq 0$ — ensures $\sigma_t^2 > 0$ a.s.
• Stationarity: $\sum_{i=1}^q \alpha_i + \sum_{j=1}^p \beta_j < 1$ ensures the variance process is covariance stationary.
• Unconditional variance: $\mathbb{E}[\sigma_t^2] = \bar{\sigma}^2 = \omega / (1 - \sum \alpha_i - \sum \beta_j)$, which exists only when the stationarity condition holds.

GARCH(1,1) is the workhorse: $\sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2.$

Typical estimated values for daily equity returns: $\alpha \approx 0.08$, $\beta \approx 0.90$, giving $\alpha + \beta \approx 0.98$ — high persistence. The half-life of a variance shock is $\ln(0.5) / \ln(\alpha + \beta) \approx 34$ days for these parameters.

Volatility mean-reversion. Write $\sigma_t^2 = \bar{\sigma}^2 + (\alpha + \beta)(\sigma_{t-1}^2 - \bar{\sigma}^2) + \alpha(\varepsilon_{t-1}^2 - \sigma_{t-1}^2)$. The term $(\alpha + \beta)^h$ governs mean-reversion speed; for $\alpha + \beta < 1$ it decays geometrically.

Maximum likelihood estimation. Assume $z_t \sim \mathcal{N}(0,1)$. The log-likelihood is:

$\ell(\theta) = -\frac{1}{2}\sum_{t=1}^T \left[\ln(2\pi) + \ln \sigma_t^2 + \frac{\varepsilon_t^2}{\sigma_t^2}\right].$