In econometrics, the autoregressive conditional heteroskedasticity (ARCH) model is a statistical model for time series data that describes the variance of the current error term or innovation as a function of the actual sizes of the previous time periods' error terms;^[1] often the variance is related to the squares of the previous innovations. The ARCH model is appropriate when the error variance in a time series follows an autoregressive (AR) model; if an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.^[2]

ARCH models are commonly employed in modeling financial time series that exhibit time-varying volatility and 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.^[3]

Model specification

To model a time series using an ARCH process, let ${\displaystyle ~\epsilon _{t}~}$denote the error terms (return residuals, with respect to a mean process), i.e. the series terms. These ${\displaystyle ~\epsilon _{t}~}$ are split into a stochastic piece ${\displaystyle z_{t}}$ and a time-dependent standard deviation ${\displaystyle \sigma _{t}}$ characterizing the typical size of the terms so that

${\displaystyle ~\epsilon _{t}=\sigma _{t}z_{t}~}$

The random variable ${\displaystyle z_{t}}$ is a strong white noise process. The series ${\displaystyle \sigma _{t}^{2}}$ is modeled by

${\displaystyle \sigma _{t}^{2}=\alpha _{0}+\alpha _{1}\epsilon _{t-1}^{2}+\cdots +\alpha _{q}\epsilon _{t-q}^{2}=\alpha _{0}+\sum _{i=1}^{q}\alpha _{i}\epsilon _{t-i}^{2}}$,

where ${\displaystyle ~\alpha _{0}>0~}$ and ${\displaystyle \alpha _{i}\geq 0,~i>0}$.

An ARCH(q) model can be estimated using ordinary least squares. A method for testing whether the residuals ${\displaystyle \epsilon _{t}}$ exhibit time-varying heteroskedasticity using the Lagrange multiplier test was proposed by Engle (1982). This procedure is as follows:

1. Estimate the best fitting autoregressive model AR(q) ${\displaystyle y_{t}=a_{0}+a_{1}y_{t-1}+\cdots +a_{q}y_{t-q}+\epsilon _{t}=a_{0}+\sum _{i=1}^{q}a_{i}y_{t-i}+\epsilon _{t}}$.
2. Obtain the squares of the error ${\displaystyle {\hat {\epsilon }}^{2}}$ and regress them on a constant and q lagged values:

   ${\displaystyle {\hat {\epsilon }}_{t}^{2}=\alpha _{0}+\sum _{i=1}^{q}\alpha _{i}{\hat {\epsilon }}_{t-i}^{2}}$

   where q is the length of ARCH lags.

3. The null hypothesis is that, in the absence of ARCH components, we have ${\displaystyle \alpha _{i}=0}$ for all ${\displaystyle i=1,\cdots ,q}$. The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated ${\displaystyle \alpha _{i}}$ coefficients must be significant. In a sample of T residuals under the null hypothesis of no ARCH errors, the test statistic T'R² follows ${\displaystyle \chi ^{2}}$ distribution with q degrees of freedom, where ${\displaystyle T'}$ is the number of equations in the model which fits the residuals vs the lags (i.e. ${\displaystyle T'=T-q}$). If T'R² is greater than the Chi-square table value, we reject the null hypothesis and conclude there is an ARCH effect in the ARMA model. If T'R² is smaller than the Chi-square table value, we do not reject the null hypothesis.

GARCH

If an autoregressive moving average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.^[2]

In that case, the GARCH (p, q) model (where p is the order of the GARCH terms ${\displaystyle ~\sigma ^{2}}$ and q is the order of the ARCH terms ${\displaystyle ~\epsilon ^{2}}$ ), following the notation of the original paper, is given by

${\displaystyle y_{t}=x'_{t}b+\epsilon _{t}}$

${\displaystyle \epsilon _{t}|\psi _{t-1}\sim {\mathcal {N}}(0,\sigma _{t}^{2})}$

${\displaystyle {\mathrm{\sigma <!--\; \sigma \; -->}}_t^2=\mathrm{\omega <!--\; \omega \; -->}+{\mathrm{\alpha <!--\; \alpha \; -->}}_1{\mathrm{ϵ<!--\; ϵ\; -->}}_{t-<!--\; -\; -->1}^2+\cdots <!--\; \cdots \; -->+{\mathrm{\alpha <!--\; \alpha \; -->}}_q{\mathrm{ϵ<!--\; ϵ\; -->}}_t^{}}$Zdroj:https://en.wikipedia.org?pojem=GARCH
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