Autoregressive conditional heteroskedasticity

In econometrics, autoregressive conditional heteroscedasticity(ARCH) models are used to characterize and model time series. They are used at any point in a series where the error terms are thought to 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 that have a similar basis. ARCH models are commonly employed in modeling financial time series that exhibit time-varying 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.^[2]

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 $\alpha_i\ge 0,~i>0$.

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). This procedure is as follows:

1. Estimate the best fitting autoregressive model AR(q) $y_t = a_0 + a_1 y_{t-1} + \cdots + a_q y_{t-q} + \epsilon_t = a_0 + \sum_{i=1}^q a_i y_{t-i} + \epsilon_t$.
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 $\alpha_i = 0$ 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 T'R² follows $\chi^2$ distribution with q degrees of freedom, where $T'$ is the number of equations in the model which fits the residuals vs the lags (i.e. $T'=T-q$). 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 model (ARMA model) is assumed for the error variance, the model is a generalized autoregressive conditional heteroscedasticity(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 $~\epsilon^2$ ), following the notation of original paper is given by

$y_t=x'_t b +\epsilon_t$

$\epsilon_t| \psi_t \sim\mathcal{N}(0, \sigma^2_t)$

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

Generally, when testing for heteroscedasticity in econometric models, the best test is the White test. However, when dealing with time series data, this means to test for ARCH and GARCH errors.

Exponentially weighted moving average (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 drawbacks such as an arbitrary decay factor that introduces subjectivity into the estimation.

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 $\epsilon^2$ by

   $\rho = {{\sum^T_{t=i+1} (\hat \epsilon^2_t - \hat \sigma^2_t) (\hat \epsilon^2_{t-1} - \hat \sigma^2_{t-1})} \over {\sum^T_{t=1} (\hat \epsilon^2_t - \hat \sigma^2_t)^2}}$

3. 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 $\epsilon^2_t$ 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 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.^[3][4]

This model should not be confused with the NARCH model, together with the NGARCH extension, introduced by Higgins and Bera in 1992.

IGARCH

Integrated Generalized Autoregressive Conditional heteroscedasticity I GARCH is a restricted version of the GARCH model, where the persistent parameters sum up to one, and imports a unit root in the GARCH process. The condition for this is

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

EGARCH

The exponential generalized 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. $Z_{t}$ 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.^[5]

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 heteroscedasticity 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 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

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

GJR-GARCH

Similar to QGARCH, The Glosten-Jagannathan-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 $~\epsilon_{t-1} \ge 0$, and $I_{t-1} = 1$ if $~\epsilon_{t-1} < 0$.

TGARCH model

The Threshold GARCH (TGARCH) model by Zakoian (1994) is similar to GJR GARCH. 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 $~\epsilon_{t-1} > 0$.

fGARCH

Hentschel's fGARCH model,^[6] 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.

COGARCH

In 2004, Claudia Klüppelberg, Alexander Lindner and Ross Maller proposed a continuous-time generalization of the discrete-time GARCH(1,1) process. The idea is to start with the GARCH(1,1) model equations

$\epsilon_t = \sigma_t z_t,$

$\sigma_t^2 = \alpha_0 + \alpha_1 \epsilon^2_{t-1} + \beta_1 \sigma^2_{t-1} = \alpha_0 + \alpha_1 \sigma_{t-1}^2 z_{t-1}^2 + \beta_1 \sigma^2_{t-1},$

and then to replace the strong white noise process $z_t$ by the infinitesimal increments $\mathrm{d}L_t$ of a Lévy process $(L_t)_{t\geq0}$, and the squared noise process $z^2_t$ by the increments $\mathrm{d}[L,L]^\mathrm{d}_t$, where

$[L,L]^\mathrm{d}_t = \sum_{s\in[0,t]} (\Delta L_t)^2,\quad t\geq0,$

is the purely discontinuous part of the quadratic variation process of $L$. The result is the following system of stochastic differential equations:

$\mathrm{d}G_t = \sigma_{t-} \,\mathrm{d}L_t,$

$\mathrm{d}\sigma_t^2 = (\beta - \eta \sigma^2_t)\,\mathrm{d}t + \varphi \sigma_{t-}^2 \,\mathrm{d}[L,L]^\mathrm{d}_t,$

where the positive parameters $\beta$, $\eta$ and $\varphi$ are determined by $\alpha_0$, $\alpha_1$ and $\beta_1$. Now given some initial condition $(G_0,\sigma^2_0)$, the system above has a pathwise unique solution $(G_t,\sigma^2_t)_{t\geq0}$ which is then called the continuous-time GARCH (COGARCH) model.^[7]

References

1. Engle, Robert F. (1982). "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation". Econometrica 50 (4): 987–1007. JSTOR 1912773.
2. Brooks, Chris (2014). Introductory Econometrics for Finance (3rd ed.). Cambridge: Cambridge University Press. p. 461. ISBN 9781107661455.
3. Engle, R.F.; Ng, V.K. (1991). "Measuring and testing the impact of news on volatility". Journal of Finance 48 (5): 1749–1778. doi:10.1111/j.1540-6261.1993.tb05127.x.
4. 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" (PDF). Financial Theory and Practice 30 (4): 347–368.
5. 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.
6. Hentschel, Ludger (1995). "All in the family Nesting symmetric and asymmetric GARCH models". Journal of Financial Economics 39 (1): 71–104. doi:10.1016/0304-405X(94)00821-H.
7. Klüppelberg, C.; Lindner, A.; Maller, R. (2004). "A continuous-time GARCH process driven by a Lévy process: stationarity and second-order behaviour". Journal of Applied Probability 41 (3): 601–622. doi:10.1239/jap/1091543413.

Further reading

• Bollerslev, Tim (1986). "Generalized Autoregressive Conditional Heteroskedasticity". Journal of Econometrics 31 (3): 307–327. doi:10.1016/0304-4076(86)90063-1.
• Bollerslev, Tim (2008). "Glossary to ARCH (GARCH)" (PDF). working paper.
• Enders, W. (2004). "Modelling Volatility". Applied Econometrics Time Series (Second ed.). John-Wiley & Sons. pp. 108–155. ISBN 0-471-45173-8.
• Engle, Robert F. (1982). "Autoregressive Conditional Heteroscedasticity with Estimates of Variance of United Kingdom Inflation". Econometrica 50 (4): 987–1008. JSTOR 1912773. (the paper which sparked the general interest in ARCH models)
• Engle, Robert F. (1995). ARCH: selected readings. Oxford University Press. ISBN 0-19-877432-X.
• Engle, Robert F. (2001). "GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics". Journal of Economic Perspectives 15 (4): 157–168. doi:10.1257/jep.15.4.157. JSTOR 2696523. (a short, readable introduction)
• Gujarati, D. N. (2003). Basic Econometrics. pp. 856–862.
• Hacker, R. S.; Hatemi-J, A. (2005). "A Test for Multivariate ARCH Effects". Applied Economics Letters 12 (7): 411–417. doi:10.1080/13504850500092129.
• Nelson, D. B. (1991). "Conditional Heteroskedasticity in Asset Returns: A New Approach". Econometrica 59 (2): 347–370. JSTOR 2938260.