Modeling Volatility and GARCH Family › The ARCH Model

Robert Engle introduced the ARCH (Autoregressive Conditional Heteroskedasticity) model in 1982 to capture the observation that large shocks tend to be followed by large shocks. The name itself encodes the model's key features. "Autoregressive" indicates that the model uses its own past values to predict future values, much like the AR models we studied for returns themselves. "Conditional" emphasizes that we are modeling the variance conditional on past information, distinguishing it from the unconditional variance that averages across all market conditions. "Heteroskedasticity" simply means non-constant variance, the very phenomenon we seek to capture.

The key insight underlying ARCH is elegantly simple. Rather than treating variance as a fixed parameter to be estimated once from historical data, we model the conditional variance as a function of past squared residuals. This means that after observing a large price shock, our estimate of tomorrow's variance increases. Conversely, after a string of small price movements, we expect tomorrow's variance to be low. The model thus learns from recent market behavior and adapts its volatility estimate accordingly.

The ARCH(q) model specifies:

$$r_t = \mu + \epsilon_t$$

where:

• $r_t$: return at time $t$
• $\mu$: mean return
• $\epsilon_t$: innovation (shock) term

where the error term has time-varying variance:

$$\epsilon_t = \sigma_t z_t, \quad z_t \sim N(0,1)$$

where:

• $\sigma_t$: conditional standard deviation (volatility) at time $t$
• $z_t$: standardized residual, independent and identically distributed (i.i.d.)

This decomposition explains how ARCH works. The standardized residual $z_t$ captures the direction and magnitude of the shock in normalized terms, while $\sigma_t$ scales this shock to the appropriate level given current market conditions. When volatility is high, a typical standardized shock (say, $z_t = 1$) produces a large observed return shock $\epsilon_t$. The standardized residuals themselves are assumed to be i.i.d., meaning all the time-varying dynamics in returns come through the conditional variance.

The conditional variance follows:

$$\begin{aligned} \sigma_t^2 &= \omega + \alpha_1 \epsilon_{t-1}^2 + \alpha_2 \epsilon_{t-2}^2 + \cdots + \alpha_q \epsilon_{t-q}^2 \\ &= \omega + \sum_{i=1}^{q} \alpha_i \epsilon_{t-i}^2 \end{aligned}$$

where:

• $\sigma_t^2$: conditional variance at time $t$, given information at $t-1$
• $\omega$: constant term (intercept), must be positive
• $\alpha_i$: ARCH coefficients measuring the impact of past shocks
• $\epsilon_{t-i}^2$: squared residuals from $i$ periods ago (volatility shocks)
• $q$: number of lags in the ARCH model

This equation is the heart of the ARCH model. Reading it from right to left, today's conditional variance starts with a baseline level $\omega$ and then adds contributions from each of the past $q$ squared residuals. Each squared residual is weighted by its corresponding $\alpha_i$ coefficient, which measures how strongly that lag affects current variance. The summation structure creates the autoregressive nature: current variance depends on past squared shocks, just as AR models for returns make current returns depend on past returns.

The model captures a simple but powerful idea. If yesterday's return was unexpectedly large (positive or negative), today's conditional variance should be higher. Think of it as the market "remembering" recent turbulence. After a large price move, uncertainty increases, leading to more cautious or more reactive trading and larger subsequent price movements. The squaring of $\epsilon_{t-i}$ ensures that both positive and negative shocks increase volatility, consistent with the symmetric response we often observe in practice. A three percent gain creates as much "news" as a three percent loss in terms of its impact on expected future variance.

For the model to be well-defined and produce sensible variance forecasts, we need parameter constraints:

• $\omega > 0$ (positive baseline variance)
• $\alpha_i \geq 0$ for all $i$ (non-negative impact of shocks)
• $\sum_{i=1}^{q} \alpha_i < 1$ (ensures stationarity)

The first constraint guarantees that even in the complete absence of shocks, the model produces a positive variance. This makes economic sense: there is always some baseline uncertainty in financial markets. The second constraint ensures that past shocks cannot reduce variance, which would be counterintuitive. The third constraint, the stationarity condition, is particularly important because it guarantees that volatility doesn't explode over time. If the sum of the alpha coefficients exceeded one, a large shock would create even larger expected variance tomorrow, which would grow without bound.

When the stationarity condition holds, we find the unconditional (long-run) variance $\sigma^2$ by taking the expectation of the variance equation:

$$\begin{aligned} \sigma^2 &= \omega + \sum_{i=1}^{q} \alpha_i \sigma^2 \\ \sigma^2 \left(1 - \sum_{i=1}^{q} \alpha_i\right) &= \omega \end{aligned}$$

Solving for $\sigma^2$:

$$\sigma^2 = \frac{\omega}{1 - \sum_{i=1}^{q} \alpha_i}$$

where:

• $\sigma^2$: unconditional (long-run) variance
• $\omega$: constant baseline variance parameter
• $\alpha_i$: ARCH coefficients summing to less than 1

This formula reveals that the unconditional variance depends on both the baseline parameter $\omega$ and the persistence of the process captured by the sum of the alpha coefficients. When this sum is close to one, the denominator becomes small, making the unconditional variance large. Intuitively, high persistence means that shocks have lasting effects on variance, inflating the long-run average level.

While ARCH captures volatility clustering, it has practical limitations that became apparent as researchers applied it to financial data:

• Many parameters needed: Capturing persistence in volatility often requires high-order ARCH models (large q), meaning many parameters to estimate.
• Slow decay: Volatility shocks persist for months empirically.
• No leverage effect: The symmetric treatment of positive and negative shocks misses that negative returns often increase volatility more than positive returns of equal magnitude.

If volatility shocks truly persist for, say, 60 trading days, we would need an ARCH(60) model with 61 parameters in the variance equation alone. Estimating so many parameters reliably requires enormous datasets and provides little insight into the underlying dynamics. This parameter proliferation problem motivated the development of GARCH, which achieves the same modeling power with remarkable parsimony.