Autoregressive Conditional Heteroscedasticity
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An Autoregressive Conditional Heteroscedasticity is a serial correlation of the heteroskedasticity.
- AKA: ARCH, Autoregressive Conditional Heteroskedasticity.
- Context:
- It can be defined as the condition that one or more data points in a time series for which the variance of the current error term or innovation is a function of the actual sizes of the previous time periods' error terms
- It can be defined as a mean zero serially uncorrelated process with nonconstant variances conditional on the past, but constant unconditional variances.
- See: Stochastic process, Nonlinear Time Series, Time Series.
References
2016
- (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity 2016-08-13
- Autoregressive conditional heteroskedasticity (ARCH) is the condition that one or more data points in a series for which the variance of the current error term or innovation is a function of the actual sizes of the previous time periods' error terms: often the variance is related to the squares of the previous innovations. In econometrics, ARCH models are used to characterize and model time series. A variety of other acronyms are applied to particular structures that have a similar basis.
- ARCH models are commonly employed in modeling financial time series that exhibit time-varying volatility clustering, i.e. periods of swings interspersed with periods of relative calm. ARCH-type models are sometimes considered to be in the family of stochastic volatility models, although this is strictly incorrect since at time t the volatility is completely pre-determined (deterministic) given previous values.
- ARCH(q) model specification
- To model a time series using an ARCH process, let [math]\displaystyle{ ~\epsilon_t~ }[/math]denote the error terms (return residuals, with respect to a mean process), i.e. the series terms. These [math]\displaystyle{ ~\epsilon_t~ }[/math] are split into a stochastic piece [math]\displaystyle{ z_t }[/math] and a time-dependent standard deviation [math]\displaystyle{ \sigma_t }[/math] characterizing the typical size of the terms so that
- [math]\displaystyle{ ~\epsilon_t=\sigma_t z_t ~ }[/math]
- The random variable [math]\displaystyle{ z_t }[/math] is a strong white noise process. The series [math]\displaystyle{ \sigma_t^2 }[/math] is modelled by
- [math]\displaystyle{ \sigma_t^2=\alpha_0+\alpha_1 \epsilon_{t-1}^2+\cdots+\alpha_q \epsilon_{t-q}^2 = \alpha_0 + \sum_{i=1}^q \alpha_{i} \epsilon_{t-i}^2 }[/math]
- where [math]\displaystyle{ ~\alpha_0\gt 0~ }[/math] and [math]\displaystyle{ \alpha_i\ge 0,~i\gt 0 }[/math].
1982
- (Engle, 1982) ⇒ Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica: Journal of the Econometric Society, 987-1007. http://www.jstor.org/stable/1912773
- Traditional econometric models assume a constant one-period forecast variance. To generalize this implausible assumption, a new class of stochastic processes called autoregressive conditional heteroscedastic (ARCH) processes are introduced in this paper. These are mean zero, serially uncorrelated processes with nonconstant variances conditional on the past, but constant unconditional variances. For such processes, the recent past gives information about the one-period forecast variance. A regression model is then introduced with disturbances following an ARCH process. Maximum likelihood estimators are described and a simple scoring iteration formulated. Ordinary least squares maintains its optimality properties in this set-up, but maximum likelihood is more efficient. The relative efficiency is calculated and can be infinite. To test whether the disturbances follow an ARCH process, the Lagrange multiplier procedure is employed. The test is based simply on the autocorrelation of the squared OLS residuals. This model is used to estimate the means and variances of inflation in the U.K. The ARCH effect is found to be significant and the estimated variances increase substantially during the chaotic seventies.