Conditional Variance

The conditional variance of a return series measures the time-varying volatility given past information.

Definition

For a return series $\{r_t\}$, the conditional variance at time $t$ given information set $F_{t-1}$ is:

$${\sigma }_t^2=\text{Var}(r_t|F_{t-1})=E(r_t^2|r_{t-1},r_{t-2},...)$$

The last equality holds when $E(r_t|F_{t-1})=0$ (zero conditional mean), which is typical for financial returns.

Why Conditional Variance?

Interpretation

1. An asset is risky if its return $r_t$ is volatile (changes a lot over time)
2. In statistics, we use variance to measure volatility (dispersion), and thus risk
3. We care about conditional variance because we want to use past history to forecast future variance

Characteristics of Volatility

1. Volatility Clustering: High volatility periods cluster together; low volatility periods cluster together
2. Continuity: Volatility evolves continuously—jumps are rare
3. Stationarity: Volatility varies within a fixed range (mean-reverting)
4. Leverage Effect: Volatility responds differently to price increases vs decreases

In ARCH/GARCH Models

ARCH/GARCH models specifically model this conditional variance:

• ARMA models the conditional mean: $E(r_t|F_{t-1})$
• ARCH/GARCH models the conditional variance: $\text{Var}(r_t|F_{t-1})={\sigma }_t^2$

Together, they provide a complete description of the conditional distribution of returns.

Related

• Volatility Clustering
• ARCH(m) Model
• GARCH(m,s) Model