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Encyclopedia > Autoregressive Conditional Heteroskedasticity

In econometrics, an autoregressive conditional heteroscedasticity (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 clustering, i.e. periods of swings followed by periods of relative calm. Econometrics is concerned with the tasks of developing and applying quantitative or statistical methods to the study and elucidation of economic principles. ... Year 1982 (MCMLXXXII) was a common year starting on Friday (link displays the 1982 Gregorian calendar). ... This article is about mathematics. ... Volatility most frequently refers to the standard deviation of the change in value of a financial instrument with a specific time horizon. ...

Contents

1 Definition
- 1.1 How to specify an ARCH(q) model
2 GARCH
- 2.1 How to specify a GARCH(p, q) model
3 NGARCH
4 IGARCH
5 EGARCH
6 GARCH-M
7 QGARCH
8 GJR-GARCH
9 TGARCH model
10 APARCH
11 FIGARCH
12 FIEGARCH
13 FIAPARCH
14 HYGARCH
15 References
16 External links

[edit] Definition

Specifically, let $~epsilon_t~$ denote the returns (or return residuals, net of a mean process) and assume that $~epsilon_t=sigma_t z_t ~$ , where $z_tsim iid~ N(0,1)$ and where the series $sigma_t^2$ are modeled by

$sigma_t^2=alpha_0+alpha_1 epsilon_{t-1}^2+cdots+alpha_q epsilon_{t-q}^2 = alpha_0 + sum_{i=1}^q alpha_{i} epsilon_{t-i}^2$

and where $~alpha_0>0~$ and $alpha_ige 0,~i>0$ .

[edit] How to specify an ARCH(q) model

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). These steps show us how to do it: In regression analysis, least squares, also known as ordinary least squares analysis, is a method for linear regression that determines the values of unknown quantities in a statistical model by minimizing the sum of the residuals (the difference between the predicted and observed values) squared. ... Robert F. Engle (born 1942) received the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 2003, sharing the award with Clive Granger, for methods of analyzing economic time series with time-varying volatility (ARCH). He got his Ph. ...

Estimate the best fitting AR(q) model $y_t = a_0 + a_1 y_{t-1} + cdots + a_q y_{t-q} + epsilon_t = a_0 + sum_{i=1}^q a_i y_{t-i} + epsilon_t$ .
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 $α i = 0$ for all $i = 1, cdots, q$ . The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated $α i$ coefficients must be significant. In a sample of T residuals under the nullhypothesis of no ARCH errors, the test statistic TR2 follows $χ 2$ distribution with q degrees of freedom. If TR² is greater than the Chi-square table value, we reject the null hypothesis and find that there is no ARCH effect in the ARMA model. If TR² is smaller than the Chi-square table value, we accept the null hypothesis.

[edit] 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 statistics, autoregressive moving average (ARMA) models, sometimes called Box-Jenkins models after George Box and G. M. Jenkins, are typically applied to time series data. ...

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

$sigma_t^2=alpha_0 + alpha_1 epsilon_{t-1}^2 + cdots + alpha_q epsilon_{t-q}^2 + beta_1 sigma_{t-1}^2 + cdots + beta_psigma_{t-p}^2 = alpha_0 + sum_{i=1}^q alpha_i epsilon_{t-i}^2 + sum_{i=1}^p beta_i sigma_{t-i}^2$

Generally, when testing for heteroskedasticity in econometric models, the best test is the White test. However, when dealing with time series data, there means to test for ARCH errors (as described above) and GARCH errors (below). In statistics, a sequence or a vector of random variables is heteroskedastic if the random variables in the sequence or vector may have different variances. ... In statistics the White test is a test which establishes whether the residual variance of a variable in a regression model is constant (homoskedasticity). ... In statistics, signal processing, and econometrics, a time series is a sequence of data points, measured typically at successive times, spaced at (often uniform) time intervals. ...

Prior to GARCH there was EWMA which has now been superseded by GARCH. Some people utilise both. In statistics, a moving average is one of a family of similar techniques used to analyze time series data. ...

[edit] How to specify a GARCH(p, q) model

This methodology shows how to find the lag length p of a GARCH(p, q) process:

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 $ε 2$ by

$rho = {{sum^T_{t=i+1} (hat epsilon^2_t - hat sigma^2_t) (hat epsilon^2_{t-1} - hat sigma^2_{t-1})} over {sum^2_{t=1} (hat epsilon^2_t - hat sigma^2_t)^2}}$
The asymptotic, that is for large samples, standard deviation of $ρ(i)$ is T^{-½}. 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 the these are less than, say, 10% significant. The Ljung/Box Q-statistic follows $χ 2$ distribution with n degrees of freedom if the squared residuals $epsilon^2_t$ are uncorrelated. It is recommended to consider up to T/4 values of n. The null hypothesis states that there are ARCH or GARCH errors. Rejecting the null thus means that there are no such errors in the conditional variance.

In statistics, a portmanteau test (also called Ljung-Box test) tests whether the autocorrelations of a time series fulfill a certain criterion. ... In statistics, conditional variance is a special form of the variance. ...

[edit] NGARCH

Non Linear Generalized Autoregressive Conditional Heteroskedasticity.
$~sigma_(t+1)^2= ~omega + ~alpha * (~epsilon_t - ~theta * ~sigma_T)^2 + ~beta * ~sigma_T^2$
$~omega = V_L * ~gamma$
$~alpha , ~beta, ~theta < 1 ; ~omega > 0$
Theta is the leverage effect, signifying the inversely proportional relation between $R e t u r n s 2 a n d V a r i a n c e$

[edit] IGARCH

Integrated Generalized Autoregressive Conditional Heteroskedasticity IGARCH is a restricted version of the GARCH model, where the sum of the persistent parameters sum up to one, and therefore there is a unit root in the GARCH process. The condition for this is A unit root is a concept from autoregressive models in econometrics. ...

$sum^p_{i=1} ~beta_{i-t} ~sigma^2_t = 1$ .

[edit] EGARCH

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

$logsigma_{t}^{2}=omega_{t}+sum_{k=1}^{infty}beta_{k}g(Z_{t-k})+sum_{k=1}^{infty}alpha_{k}logsigma_{t-k}^{2}$

where $g (Z t) = θ Z t + λ( | Z t | - E (Z t))$ , $sigma_{t}^{2}$ is the conditional variance, $ω$ , $β$ , $α$ , $θ$ and $λ$ are coefficients, and $Z t$ is a standard normal variable. In statistics, conditional variance is a special form of the variance. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields, especially in physics and engineering. ...

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

[edit] GARCH-M

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

$y_t = ~beta x_t + ~sigma_t^{1/2} + ~epsilon_t$

The residual $~epsilon_t$ is defined as

$~epsilon_t = ~sigma_t^{1/2} z_t$

[edit] QGARCH

The Quadratic GARCH (QGARCH) model by Sentana (1995) is used to model asymmetric 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

$~sigma_t^2 = K + ~alpha ~epsilon_{t-1}^2 + ~beta ~sigma_{t-1}^2 + ~phi ~sigma_{t-1}$

[edit] GJR-GARCH

Similar to QGARCH, the The Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model by Glosten, Jagannathan and Runkle (1993) also models asymmetry in the GARCH 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 $I t - 1 = 1$ if $~epsilon_{t-1} ge 0$ , and $I t - 1 = 0$ if $~epsilon_{t-1} < 0$ .

[edit] TGARCH model

Finally, 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:

$~sigma_t = K + ~delta ~sigma_{t-1} + ~alpha_1^{+} ~epsilon_{t-1}^{+} + ~alpha_1^{-} ~epsilon_{t-1}^{-}$

where $~epsilon_{t-1}^{+} = ~epsilon_{t-1}$ if $~epsilon_{t-1} > 0$ , and $~epsilon_{t-1}^{+} = 0$ if $~epsilon_{t-1} le 0$ . Likewise, $~epsilon_{t-1}^{-} = ~epsilon_{t-1}$ if $~epsilon_{t-1} le 0$ , and $~epsilon_{t-1}^{-} = 0$ if $~epsilon_{t-1} > 0$ .

[edit] APARCH

[edit] FIGARCH

[edit] FIEGARCH

[edit] FIAPARCH

[edit] HYGARCH

[edit] References

Tim Bollerslev. "Generalized Autoregressive Conditional Heteroskedasticity", Journal of Econometrics, 31:307-327, 1986.

Enders, W., Applied Econometrics Time Series, John-Wiley & Sons, 139-149, 1995

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)

Gujarati, D. N., Basic Econometrics, 856-862, 2003

Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach, Econometrica 59: 347-370.

There are very few or no other articles that link to this one. ... Robert F. Engle (born 1942) received the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 2003, sharing the award with Clive Granger, for methods of analyzing economic time series with time-varying volatility (ARCH). He got his Ph. ... Econometrica is a prestigious academic journal of economics, publishing articles in not only econometrics but in many areas of economics. ...

[edit] External links

ARCH and GARCH models for forecasting volatility, quantnotes.com

v • d • e Volatility

Modelling volatility	Implied volatility · Volatility smile · Volatility clustering · Local volatility · Stochastic volatility · Jump-diffusion models · ARCH · GARCH Volatility most frequently refers to the standard deviation of the change in value of a financial instrument with a specific time horizon. ... In financial mathematics, the implied volatility of an option contract is the volatility implied by the market price of the option based on an option pricing model. ... Volatility Smile refers to the long-observed pattern in which at-the-money options tend to have lower implied volatilities than other options. ... In finance, volatility clustering refers to the observation, as noted by Mandelbrot, that large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes. ... Local volatility is a term used in quantitative finance to denote the set of diffusion coefficients, , that are consistent with the set of market prices for all option prices on a given underlier. ... Stochastic volatility models are used in the field of quantitative finance to evaluate derivative securities, such as options. ... A jump-diffusion model is an option valuation model in which a jump process is added in order to model the volatility smile. ... For other uses, see Arch (disambiguation). ... In econometrics, an autoregressive conditional heteroskedasticity (ARCH) model considers the variance of the current error term to be a function of the variances of the previous time periods error terms. ...

Trading volatility	Volatility arbitrage · Straddle · Volatility swap · Variance swap · VIX Volatility arbitrage, a. ... In finance, a straddle is an investment strategy involving the purchase or sale of particular derivatives. ... In finance, a volatility swap is a forward contract on the future realised volatility of a given underlying asset. ... A variance swap is a financial derivative whose payoff is the realised volatility squared of the underlier based on a prespecified set of sampling points. ... VIX Index from inception to Jan. ...

Categories: Time series analysis

Results from FactBites:

ARCH and GARCH Processes (498 words)
	Autoregressive conditional heteroskedastic (ARCH) processes are a form of stochastic process that are widely used in finance and economics for modeling conditional heteroskedasticity and volatility clustering.
	heteroskedasticity A condition where a stochastic process has non-constant second moments.
	Engle, Robert F. Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation, Econometrica, 50, 987-1008.

Autoregressive conditional heteroskedasticity - Wikipedia, the free encyclopedia (210 words)
	In econometrics, an autoregressive conditional heteroskedasticity (ARCH) model considers the variance of the current error term to be a function of the variances of the previous time period's error terms.
	If an autoregressive moving average model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model.
	Generally, when testing for heteroskedasticity in econometric models, the best test is the White test.

More results at FactBites »

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