How GARCH Separates Conditional and Unconditional Variance
Summary
The document explains why conditional heteroskedasticity does not conflict with a constant unconditional variance. A stationary return series may have finite, constant overall variance while its variance given past information changes over time. This changing conditional variance captures volatility clustering: periods of elevated volatility tend to persist, as do calmer periods.
For a GARCH(1,1) model, the conditional variance depends on the previous squared error and previous conditional variance. The response then relates squared errors to an ARMA representation and gives the resulting unconditional variance, which is finite when the sum of the model’s two persistence coefficients is below one. This derivation illustrates how time-varying conditional risk can coexist with a stable long-run variance. The explanation is limited to the basic model and its stated stationarity condition; it does not discuss estimation, model diagnostics, alternative error distributions, or forecasting performance.
Key ideas
- Conditional variance can change over time even when unconditional variance is constant.
- GARCH models represent volatility clustering through dependence on past errors and variance.
- In GARCH(1,1), the conditional variance is determined by the previous squared error and variance.
- The model’s unconditional variance is finite when the persistence coefficients sum to less than one.
- Conditional and unconditional moments describe different properties of a return series.
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# What implies "conditional heteroskedasticity" in (G)ARCH?
# What implies "conditional heteroskedasticity" in (G)ARCH?
I have trouble to understand what implies "conditional heteroskedasticity" term in (G)ARCH models. The residual $\epsilon$ is stationary, hence homoskedastic (unconditional variance is constant). Then, if we assume that residual has (G)ARCH structure, do we assume that it becomes heteroskedastic when it is conditioned w.r.t. its lags?
## Answer by Count (score 2)
https://quant.stackexchange.com/a/69056
Basically, you have to differ between conditional and unconditional moments. If you look at nearly every time series of returns, you observe 2 things:
- There is usually no doubt that the time series is stationary. Therefore, the unconditional variance $\text{Var}(r_t)$ must be constant and finite.
- You observe volatility clustering, i.e., periods of high volatility tend to be followed by periods of high volatility and vice versa. In other words, it seems like the conditional variance $\text{Var}(r_t \vert \Omega_{t-1})$ is not constant, but changes over time.
Note that even if $\text{Var}(r_t \vert \Omega_{t-1})$ changes over time, $\text{Var}(r_t)$ can be constant. There is no contradiction between those two things. GARCH models simply try to mimic those stylized facts.
Let's take a look at the GARCH(1,1) model. This model assumes that the returns can be modeled as: \begin{align} r_t&=\mu_t+\epsilon_t, \quad \epsilon_t\vert \Omega_{t-1}\sim WN(0,\sigma_t^2),\\ \epsilon_t&=\sigma_tu_t \quad u_t \overset{iid}{\sim}(0,1),\\ \sigma_t^2&=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2. \end{align}
Easy calculations show that in this model: $$ \text{Var}(r_t\vert\Omega_{t-1})=E((r_t-\mu_t)^2\vert \Omega_{t-1})=E(\epsilon_t^2\vert \Omega_{t-1})=\sigma_t^2 $$ Therefore: $$ \text{Var}(r_t \vert \Omega_{t-1})=E(\epsilon_t^2 \vert \Omega_{t-1})=\sigma_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2. $$ The conditional variance is not constant. On the other hand, you can express the squared error terms as an ARMA(1,1) process $$ \epsilon_{t}^2=\alpha_0+(\alpha_1+\beta_1)\epsilon_{t-1}^2+w_t-\beta_1w_{t-1} $$ from which you can derive the unconditional variance $$ \text{Var}(\epsilon_t)=E(\epsilon_t^2)=\frac{\alpha_0}{1-(\alpha_1+\beta_1)} $$ which is constant and finite if $\alpha_1+\beta_1<1$. This explains the name of the model "Generalized Conditional heteroskedasticity".Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.