Autoregressive conditional heteroskedasticity

From Wikipedia, the free encyclopedia

"ARCH" redirects here. For other uses, see Arch (disambiguation).

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 error 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),^[1] 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 time-varying volatility clustering, i.e. periods of swings followed by periods of relative calm.

ARCH(q) model Specification

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

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

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

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

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

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

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

where q is the length of ARCH lags.
The null hypothesis is that, in the absence of ARCH components, we have $\alpha _{i}=0$ for all $i=1,\cdots ,q$ . The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated $\alpha _{i}$ coefficients must be significant. In a sample of T residuals under the null hypothesis of no ARCH errors, the test statistic TR² follows $\chi ^{2}$ distribution with q degrees of freedom. If TR² 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 TR² is smaller than the Chi-square 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 $~\sigma ^{2}$ and q is the order of the ARCH terms $~\epsilon ^{2}$ ) is given by

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

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).

EWMA is an alternative model in a separate class of exponential smoothing models. It can be an alternative to GARCH modelling as it has some attractive properties such as a greater weight upon more recent observations but also some drawbacks such as an arbitrary decay factor that introduce subjectivity into the estimation.

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

$y_{t}=a_{0}+a_{1}y_{{t-1}}+\cdots +a_{q}y_{{t-q}}+\epsilon _{t}=a_{0}+\sum _{{i=1}}^{q}a_{i}y_{{t-i}}+\epsilon _{t}$ .
Compute and plot the autocorrelations of $\epsilon ^{2}$ by

$\rho ={{\sum _{{t=i+1}}^{T}({\hat \epsilon }_{t}^{2}-{\hat \sigma }_{t}^{2})({\hat \epsilon }_{{t-1}}^{2}-{\hat \sigma }_{{t-1}}^{2})} \over {\sum _{{t=1}}^{T}({\hat \epsilon }_{t}^{2}-{\hat \sigma }_{t}^{2})^{2}}}$
The asymptotic, that is for large samples, standard deviation of $\rho (i)$ is $1/{\sqrt {T}}$ . Individual values that are larger than this indicate GARCH errors. To estimate the total number of lags, use the Ljung-Box test until the value of these are less than, say, 10% significant. The Ljung-Box Q-statistic follows $\chi ^{2}$ distribution with n degrees of freedom if the squared residuals $\epsilon _{t}^{2}$ 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 such errors exist 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.
$~\sigma _{{t}}^{2}=~\omega +~\alpha (~\epsilon _{{t-1}}-~\theta ~\sigma _{{t-1}})^{2}+~\beta ~\sigma _{{t-1}}^{2}$

$~\alpha ,~\beta \geq 0;~\omega >0$ .
For stock returns, parameter $~\theta$ 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.^[2]^[3]

This model shouldn't be confused with the NARCH model, together with the NGARCH extension, introduced by Higgins and Bera in 1992.

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

$\sum _{{i=1}}^{p}~\beta _{{i}}+\sum _{{i=1}}^{q}~\alpha _{{i}}=1$ .

EGARCH

The exponential general autoregressive conditional heteroskedastic (EGARCH) model by Nelson (1991) is another form of the GARCH model. Formally, an EGARCH(p,q):

$\log \sigma _{{t}}^{2}=\omega +\sum _{{k=1}}^{{q}}\beta _{{k}}g(Z_{{t-k}})+\sum _{{k=1}}^{{p}}\alpha _{{k}}\log \sigma _{{t-k}}^{{2}}$

where $g(Z_{{t}})=\theta Z_{{t}}+\lambda (|Z_{{t}}|-E(|Z_{{t}}|))$ , $\sigma _{{t}}^{{2}}$ is the conditional variance, $\omega$ , $\beta$ , $\alpha$ , $\theta$ and $\lambda$ 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.^[4]

Since $\log \sigma _{{t}}^{{2}}$ may be negative there are no (fewer) restrictions on the parameters.

GARCH-M

The GARCH-in-mean (GARCH-M) model adds a heteroskedasticity term into the mean equation. It has the specification:

$y_{t}=~\beta x_{t}+~\lambda ~\sigma _{t}+~\epsilon _{t}$

The residual $~\epsilon _{t}$ is defined as

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

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 $~\sigma _{t}$ is

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

where $z_{t}$ is i.i.d. and

$~\sigma _{t}^{2}=K+~\alpha ~\epsilon _{{t-1}}^{2}+~\beta ~\sigma _{{t-1}}^{2}+~\phi ~\epsilon _{{t-1}}$

GJR-GARCH

Similar to QGARCH, The Glosten-Chris Hughton-Runkle GARCH (GJR-GARCH) model by Glosten, Jagannathan and Runkle (1993) also models asymmetry in the ARCH process. The suggestion is to model $~\epsilon _{t}=~\sigma _{t}z_{t}$ where $z_{t}$ is i.i.d., and

$~\sigma _{t}^{2}=K+~\delta ~\sigma _{{t-1}}^{2}+~\alpha ~\epsilon _{{t-1}}^{2}+~\phi ~\epsilon _{{t-1}}^{2}I_{{t-1}}$

where $I_{{t-1}}=0$ if $~\epsilon _{{t-1}}\geq 0$ , and $I_{{t-1}}=1$ if $~\epsilon _{{t-1}}<0$ .

TGARCH model

The Threshold GARCH (TGARCH) model by Nikolai (2013) is similar to GJR GARCH, and the specification is one on conditional standard deviation instead of conditional variance:

$~\sigma _{t}=K+~\delta ~\sigma _{{t-1}}+~\alpha _{1}^{{+}}~\epsilon _{{t-1}}^{{+}}+~\alpha _{1}^{{-}}~\epsilon _{{t-1}}^{{-}}$

where $~\epsilon _{{t-1}}^{{+}}=~\epsilon _{{t-1}}$ if $~\epsilon _{{t-1}}>0$ , and $~\epsilon _{{t-1}}^{{+}}=0$ if $~\epsilon _{{t-1}}\leq 0$ . Likewise, $~\epsilon _{{t-1}}^{{-}}=~\epsilon _{{t-1}}$ if $~\epsilon _{{t-1}}\leq 0$ , and $~\epsilon _{{t-1}}^{{-}}=0$ if $~\epsilon _{{t-1}}>0$ .

fGARCH

Hentschel's fGARCH model,^[5] 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

↑ Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation Robert F. Engle, Econometrica , Vol. 50, No. 4 (Jul., 1982), pp. 987-1007. Published by: The Econometric Society Stable URL: http://www.jstor.org/stable/1912773
↑ Engle, R.F.; Ng, V.K. (1991). "Measuring and testing the impact of news on volatility". Journal of Finance 48 (5): 1749–1778. Cite uses deprecated parameters (help)
↑ 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.
↑ St. Pierre, Eilleen F. (1998). "Estimating EGARCH-M Models: Science or Art". The Quarterly Review of Economics and Finance 38 (2): 167–180. doi:10.1016/S1062-9769(99)80110-0.
↑ Hentschel, Ludger (1995). All in the family Nesting symmetric and asymmetric GARCH models, Journal of Financial Economics, Volume 39, Issue 1, Pages 71-104

Bollerslev, Tim (1986). "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics, 31:307-327
Bollerslev, Tim (2008). Glossary to ARCH (GARCH), working paper
Enders, W. (1995). Applied Econometrics Time Series, John-Wiley & Sons, 139-149, ISBN 0-471-11163-5
Engle, Robert F. (1982). "Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation", Econometrica 50:987-1008. (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):157-168. (a short, readable introduction) Preprint
Engle, R.F. (1995) ARCH: selected readings. Oxford University Press. ISBN 0-19-877432-X
Gujarati, D. N. (2003) Basic Econometrics, 856-862
Hacker, R. S. and Hatemi-J, 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: 347-370.

v t e Volatility

Modelling volatility	Implied volatility Volatility smile Volatility clustering Local volatility Stochastic volatility Jump-diffusion models ARCH and GARCH

Trading volatility	Volatility arbitrage Straddle Volatility swap IVX VIX

v t e Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itō diffusion Itō process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Markov random field Percolation Pitman–Yor process Point process Cox Poisson Potts model Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market SABR volatility Vašícek

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itō integral Itō's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prohorov theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

List of topics Category

Statistics

Descriptive statistics

Continuous data

Location	Mean (Arithmetic, Geometric, Harmonic) Median Mode

Dispersion	Range Standard deviation Coefficient of variation Percentile Interquartile range

Shape	Variance Skewness Kurtosis Moments L-moments

Count data

Index of dispersion

Summary tables

Dependence

Statistical graphics

Data collection

Designing studies	Effect size Standard error Statistical power Sample size determination

Survey methodology	Sampling Stratified sampling Cluster sampling Opinion poll Questionnaire

Controlled experiment	Design of experiments Randomized experiment Random assignment Replication Blocking Factorial experiment Optimal design

Uncontrolled studies	Natural experiment Quasi-experiment Observational study

Statistical inference

Statistical theory	Sampling distribution Order statistic Scan statistic Record value Sufficiency Completeness Exponential family Permutation test (Randomization test) Empirical distribution Bootstrap U statistic Efficiency Asymptotics Robustness

Frequentist inference	Unbiased estimator (Mean unbiased minimum variance, Median unbiased) Biased estimators (Maximum likelihood, Method of moments, Minimum distance, Density estimation) Confidence interval Testing hypotheses Power Parametric tests (Likelihood-ratio, Wald, Score)

Specific tests	Z (normal) Student's t-test F Goodness of fit (Chi-squared, G, Sample source, sample normality, Skewness & kurtosis Normality, Model comparison, Model quality) Signed-rank (1-sample, 2-sample, 1-way anova) Shapiro–Wilk Kolmogorov–Smirnov

Bayesian inference	Bayesian probability Prior Posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator

Correlation and regression analysis

Correlation	Pearson product–moment correlation Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models MARS

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) Binomial Poisson

Partition of variance	Analysis of variance (ANOVA) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical, multivariate, time-series, or survival analysis

Categorical data

Multivariate statistics

Time series analysis

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration

Specific tests	Granger causality Q-Statistic Durbin–Watson

Time domain	ACF PACF XCF ARMA model ARIMA model ARCH Vector autoregression

Frequency domain	Spectral density estimation Fourier analysis

Survival analysis

Applications

Biostatistics	Bioinformatics Clinical trials & studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process & Quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

Category
Portal
Outline
Index

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.