Autoregressive conditional heteroskedasticity

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In econometrics, an autoregressive conditional heteroskedasticity (ARCH, Engle (1982)) model considers the variance of the current error term to be a function of the variances of the previous time period's error terms. ARCH relates the error variance to the square of a previous period's error. It is employed commonly in modeling financial time series that exhibit time-varying volatility.

Specifically, let $~\epsilon_t~$ denote the returns (or return residuals, net of a mean process) and assume that $~\epsilon_t=\sigma_t z_t ~$ , where $z_t\sim iid~ N(0,1)$ and where the series $\sigma_t^2$ are modeled by

$\sigma_t^2=\alpha_0+\alpha_1 \epsilon_{t-1}^2+\cdots+\alpha_q \epsilon_{t-q}^2$

and where $~\alpha_0>0~$ and $\alpha_i\ge 0,~i>0$

If an autoregressive moving average model (ARMA model) is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH, Bollerslev(1986)) model.

In that case, the GARCH(p,q) model (where p is the order of the GARCH terms and q is the order of the ARCH terms) is given by

$\sigma_t^2=\alpha_0+\alpha_1 \epsilon_{t-1}^2+\cdots+\alpha_q \epsilon_{t-q}^2 +\beta_1 \sigma_{t-1}^2+\cdots+\beta_p\sigma_{t-p}^2$

IGARCH or Integrated Generalized Autoregressive Conditional Heteroskedasticity is a restricted version of the GARCH model, where the sum of the persistent parameters sum up to one.

Generally, when testing for heteroskedasticity in econometric models, the best test is the White test. However, when dealing with time series data, the best test is Engle's ARCH test.

Prior to GARCH there was EWMA which has now been superseded by GARCH. Some people utilise both.

[edit] References

Tim Bollerslev. "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics, 31:307-327, 1986.

Robert F. Engle. "Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation", Econometrica 50:987-1008, 1982. (the paper which sparked the general interest in ARCH models)

Robert F. Engle. "GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics", Journal of Economic Perspectives 15(4):157-168, 2001. (a short, readable introduction)