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 $~\backslash epsilon\_t~$ denote the returns (or return residuals, net of a mean process) and assume that $~\backslash epsilon\_t=\backslash sigma\_t\; z\_t\; ~$, where $z\_t\backslash sim\; iid~\; (0,1)$ and where the series $\backslash sigma\_t^2$ are modeled by

$\backslash sigma\_t^2=\backslash alpha\_0+\backslash alpha\_1\; \backslash epsilon\_\{t-1\}^2+\backslash cdots+\backslash alpha\_p\; \backslash epsilon\_\{t-p\}^2$

and where $~\backslash alpha\_0>0~$ and $\backslash alpha\_i\backslash ge\; 0,~i>0$

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 is given by

$\backslash sigma\_t^2=\backslash alpha\_0+\backslash alpha\_1\; \backslash epsilon\_\{t-1\}^2+\backslash cdots+\backslash alpha\_p\; \backslash epsilon\_\{t-p\}^2\; +\backslash beta\_1\; \backslash sigma\_\{t-1\}^2+\backslash cdots+\backslash beta\_q\backslash sigma\_\{t-q\}^2$

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Prior to GARCH there was EWMA which has now been superseded by GARCH.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.

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)

External links

• ARCH and GARCH models for forecasting volatility, quantnotes.com

statistics econometrics | Time series analysis

ARCH-Modell | ARCH | ARCH模型