Autoregressive conditional heteroskedasticity

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In econometrics, a model featuring autoregressive conditional heteroskedasticity considers 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 is 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.

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[edit] ARCH(q) model Specification

Specifically, let ~\epsilon_t~ denote the error terms (return residuals, w.r.t. a mean process) and assume that ~\epsilon_t=\sigma_t z_t ~, where z_t\overset{\textrm{iid}}{\thicksim} 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_i\ge 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). The following steps show how to do it:

  1. 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 .
  2. 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.
  3. The null hypothesis is that, in the absence of ARCH components, we have αi = 0 for all i = 1, \cdots, q . The alternative hypothesis is that, in the presence of ARCH components, at least one of the estimated αi coefficients must be significant. In a sample of T residuals under the null hypothesis of no ARCH errors, the test statistic TR² follows χ2 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.

[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 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, the means[clarification needed] 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. Some people utilise both.

[edit] GARCH(p, q) model specification

The lag length p of a GARCH(p, q) process is established in three steps:

  1. 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 .
  2. Compute and plot the autocorrelations of ε2 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}}
  3. The asymptotic, that is for large samples, standard deviation of ρ(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 the these are less than, say, 10% significant. The Ljung-Box Q-statistic follows χ2 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 no ARCH or GARCH errors. Rejecting the null thus means that there are existing such errors in the conditional variance.

[edit] Nonlinear GARCH (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.[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]

[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

\sum^p_{i=1} ~\beta_{i} +\sum_{i=1}^q~\alpha_{i} = 1 .

[edit] 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_{t}+\sum_{k=1}^{p}\beta_{k}g(Z_{t-k})+\sum_{k=1}^{q}\alpha_{k}\log\sigma_{t-k}^{2}

where g(Zt) = θZt + λ( | Zt | − E( | Zt | )), \sigma_{t}^{2} is the conditional variance, ω, β, α, θ and λ are coefficients, and Zt is a standard normal variable.

Since \log\sigma_{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 + ~\lambda ~\sigma_t + ~\epsilon_t

The residual ~\epsilon_t is defined as

~\epsilon_t = ~\sigma_t ~\times 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 zt is i.i.d. and

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

[edit] GJR-GARCH

Similar to QGARCH, 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 zt 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 It − 1 = 0 if ~\epsilon_{t-1} \ge 0 , and It − 1 = 1 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] fGARCH

Hentschel's fGARCH model[3], 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.

[edit] References

  1. ^ Engle, R.F.; Ng, V.K.. "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. 
  2. ^ 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. 
  3. ^ Hentschel, Ludger (1995). All in the family Nesting symmetric and asymmetric GARCH models, Journal of Financial Economics, Volume 39, Issue 1, Pages 71-104
  • 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) [1]
  • Gujarati, D. N., Basic Econometrics, 856-862, 2003
  • Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach, Econometrica 59: 347-370.

[edit] External links

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Volatility
Modelling volatility
Trading volatility
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