 Autoregressive conditional heteroskedasticity

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In econometrics, AutoRegressive Conditional Heteroskedasticity (ARCH) models are used to characterize and model observed time series. They are used whenever there is reason to believe that, at any point in a series, the terms will have a characteristic size, or variance. In particular ARCH models assume the variance of the current error term or innovation to be 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.
Such models are often called ARCH models (Engle, 1982), although a variety of other acronyms are applied to particular structures of model which have a similar basis. ARCH models are employed commonly in modeling financial time series that exhibit timevarying volatility clustering, i.e. periods of swings followed by periods of relative calm.
Contents
ARCH(q) model Specification
Suppose one wishes to model a time series using an ARCH process. Let denote the error terms (return residuals, with respect to a mean process) i.e. the series terms. These are split into a stochastic piece z_{t} and a timedependent standard deviation σ_{t} characterizing the typical size of the terms so that
where z_{t} is a random variable drawn from a Gaussian distribution centered at 0 with standard deviation equal to 1. (i.e. ) and where the series are modeled by
and where and .
An ARCH(q) model can be estimated using ordinary least squares. A methodology to test for the lag length of ARCH errors using the Lagrange multiplier test was proposed by Engle (1982). This procedure is as follows:
 Estimate the best fitting autoregressive model AR(q) .
 Obtain the squares of the error and regress them on a constant and q lagged values:
 where q is the length of ARCH lags.
 The null hypothesis is that, in the absence of ARCH components, we have α_{i} = 0 for all . The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated α_{i} coefficients must be significant. In a sample of T residuals under the null hypothesis of no ARCH errors, the test statistic TR² follows χ^{2} distribution with q degrees of freedom. If TR² is greater than the Chisquare table value, we reject the null hypothesis and conclude there is an ARCH effect in the ARMA model. If TR² is smaller than the Chisquare table value, we do not reject the null hypothesis.
GARCH
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
Generally, when testing for heteroskedasticity in econometric models, the best test is the White test. However, when dealing with time series data, this means to test for ARCH errors (as described above) and GARCH errors (below).
Prior to GARCH there was EWMA which has now been superseded by GARCH, although some people utilise both.
GARCH(p, q) model specification
The lag length p of a GARCH(p, q) process is established in three steps:
 Estimate the best fitting AR(q) model
 .
 Compute and plot the autocorrelations of ε^{2} by
 The asymptotic, that is for large samples, standard deviation of ρ(i) is . Individual values that are larger than this indicate GARCH errors. To estimate the total number of lags, use the LjungBox test until the value of these are less than, say, 10% significant. The LjungBox Qstatistic follows χ^{2} distribution with n degrees of freedom if the squared residuals are uncorrelated. It is recommended to consider up to T/4 values of n. The null hypothesis states that there are no ARCH or GARCH errors. Rejecting the null thus means that there are existing such errors in the conditional variance.
NGARCH
Nonlinear GARCH (NGARCH) also known as Nonlinear Asymmetric GARCH(1,1) (NAGARCH) was introduced by Engle and Ng in 1993.
.
For stock returns, parameter is usually estimated to be positive; in this case, it reflects the leverage effect, signifying that negative returns increase future volatility by a larger amount than positive returns of the same magnitude.^{[1]}^{[2]}This model shouldn't be confused with the NARCH model, together with the NGARCH extension, introduced by Higgins and Bera in 1992.^{[clarification needed]}
IGARCH
Integrated Generalized Autoregressive Conditional Heteroskedasticity IGARCH is a restricted version of the GARCH model, where the persistent parameters sum up to one, and therefore there is a unit root in the GARCH process. The condition for this is
.
EGARCH
The exponential general autoregressive conditional heteroskedastic (EGARCH) model by Nelson (1991) is another form of the GARCH model. Formally, an EGARCH(p,q):
where g(Z_{t}) = θZ_{t} + λ(  Z_{t}  − E(  Z_{t}  )), is the conditional variance, ω, β, α, θ and λ are coefficients, and Z_{t} may be a standard normal variable or come from a generalized error distribution. The formulation for g(Z_{t}) allows the sign and the magnitude of Z_{t} to have separate effects on the volatility. This is particularly useful in an asset pricing context.^{[3]}
Since may be negative there are no (fewer) restrictions on the parameters.
GARCHM
The GARCHinmean (GARCHM) model adds a heteroskedasticity term into the mean equation. It has the specification:
The residual is defined as
QGARCH
The Quadratic GARCH (QGARCH) model by Sentana (1995) is used to model symmetric effects of positive and negative shocks.
In the example of a GARCH(1,1) model, the residual process is
where z_{t} is i.i.d. and
GJRGARCH
Similar to QGARCH, The GlostenJagannathanRunkle GARCH (GJRGARCH) model by Glosten, Jagannathan and Runkle (1993) also models asymmetry in the ARCH process. The suggestion is to model where z_{t} is i.i.d., and
where I_{t − 1} = 0 if , and I_{t − 1} = 1 if .
TGARCH model
The Threshold GARCH (TGARCH) model by Zakoian (1994) is similar to GJR GARCH, and the specification is one on conditional standard deviation instead of conditional variance:
where if , and if . Likewise, if , and if .
fGARCH
Hentschel's fGARCH model,^{[4]} also known as Family GARCH, is an omnibus model that nests a variety of other popular symmetric and asymmetric GARCH models including APARCH, GJR, AVGARCH, NGARCH, etc.
References
 ^ Engle, R.F.; Ng, V.K. (1991). "Measuring and testing the impact of news on volatility". Journal of Finance 48 (5): 1749–1778. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=262096.
 ^ Posedel, Petra (2006). "Analysis Of The Exchange Rate And Pricing Foreign Currency Options On The Croatian Market: The Ngarch Model As An Alternative To The Black Scholes Model". Financial Theory and Practice 30 (4): 347–368. http://www.ijf.hr/eng/FTP/2006/4/posedel.pdf.
 ^ St. Pierre, Eilleen F (1998): Estimating EGARCHM Models: Science or Art, The Quarterly Review of Economics and Finance, Vol. 38, No. 2, pp. 167180 [1]
 ^ Hentschel, Ludger (1995). All in the family Nesting symmetric and asymmetric GARCH models, Journal of Financial Economics, Volume 39, Issue 1, Pages 71104
 Bollerslev, Tim (1986). "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics, 31:307327
 Bollerslev, Tim (2008). Glossary to ARCH (GARCH), working paper
 Enders, W. (1995). Applied Econometrics Time Series, JohnWiley & Sons, 139149, ISBN 0471111635
 Engle, Robert F. (1982). "Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation", Econometrica 50:9871008. (the paper which sparked the general interest in ARCH models)
 Engle, Robert F. (2001). "GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics", Journal of Economic Perspectives 15(4):157168. (a short, readable introduction) Preprint
 Engle, R.F. (1995) ARCH: selected readings. Oxford University Press. ISBN 019877432X
 Gujarati, D. N. (2003) Basic Econometrics, 856862
 Hacker, R. S. and HatemiJ, A. (2005). A Test for Multivariate ARCH Effects, Applied Economics Letters, 12(7), 411–417.
 Nelson, D. B. (1991). "Conditional heteroskedasticity in asset returns: A new approach", Econometrica 59: 347370.
Volatility Modelling volatility Implied volatility · Volatility smile · Volatility clustering · Local volatility · Stochastic volatility · Jumpdiffusion models · ARCH and GARCH
Trading volatility Volatility arbitrage · Straddle · Volatility swap · IVX · VIX
Categories: Time series analysis
 Nonlinear time series analysis
 Econometrics
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