Theoretical Background¶
The models in arch.univariate divide an observed series \(r_t\), usually a
return, into a mean, a conditional volatility and a standardized shock
where \(\mu_t\) is the mean, \(\sigma^2_t\) is the conditional variance of
\(\epsilon_t\) and \(e_t\) is the standardized shock, which has mean 0 and
variance 1. An ARCH-type model specifies each of the three components: a mean model, a
volatility process and a distribution for \(e_t\). Model parameters are estimated
by maximum likelihood. The default estimator of the parameter covariance,
cov_type="robust", corresponds in the ARCH literature to the estimator of
Bollerslev and Wooldridge [BollerslevWooldridge1992]. The
introduction shows how to build and estimate a model, and
Forecasting and Model Results describe what can be done with it.
Mean Models¶
The mean model determines \(\mu_t\), see Mean Models.
ZeroMean, \(r_t = \epsilon_t\), andConstantMean, \(r_t = \mu + \epsilon_t\).LS, \(r_t = \mu + \gamma' x_t + \epsilon_t\), where \(x_t\) are exogenous regressors.ARXandHARX, which are an autoregression and a heterogeneous autoregression, both with optional exogenous regressors.MAMean, a moving average, \(r_t = \mu + \epsilon_t + \theta_1 \epsilon_{t-1} + \ldots + \theta_q \epsilon_{t-q}\).ARCHInMean, in which a function \(f\) of the conditional variance enters the mean [EngleLilienRobins1987], \(r_t = \mu + \kappa f(\sigma^2_t) + \sum_{i=1}^p \phi_{L_{i}} r_{t-L_{i}} + \gamma' x_t + \epsilon_t\).
Volatility Processes¶
The volatility process determines \(\sigma^2_t\), see Volatility Processes.
The GARCH class covers the ARCH, GARCH, GJR-GARCH, AVARCH,
AVGARCH and TARCH specifications. Its variance dynamics, using the power
\(\lambda\), are
The GARCH(1,1) of the introduction is the case \(p=q=1\), \(o=0\) and \(\lambda=2\).
Process |
Description |
|---|---|
\(\sigma_t^{2}=\omega+\sum_{i=1}^{p}\alpha_{i}\epsilon_{t-i}^{2}\) [Engle1982] |
|
ARCH, GARCH [Bollerslev1986], GJR-GARCH [GJR1993], AVARCH, AVGARCH and TARCH [Zakoian1994] specifications, using the equation above |
|
\(\ln\sigma_{t}^{2}\) is a function of the lagged absolute and signed standardized shocks \(e_{t-i}=\epsilon_{t-i}/\sigma_{t-i}\) and of lagged \(\ln\sigma_{t}^{2}\) [Nelson1991] |
|
Asymmetric power ARCH, which models \(\sigma_t^{\delta}\) using the terms \((|\epsilon_{t-i}|-\gamma_i\epsilon_{t-i})^{\delta}\) [DingGrangerEngle1993] |
|
Fractionally integrated GARCH, which has a fractional differencing parameter \(d\) and is estimated using its truncated ARCH(\(\infty\)) representation [BaillieBollerslevMikkelsen1996] |
|
Fractionally integrated asymmetric power ARCH, which extends the FIGARCH with the asymmetry and the power of the APARCH [Tse1998] |
|
Heterogeneous ARCH, in which the variance depends on averages of past squared shocks over several lag lengths |
|
Variance that depends on \(m\) lagged squared shocks, with weights \(\phi_i(\theta)\propto\Gamma(i+\theta)/(\Gamma(i+1)\Gamma(\theta))\) that sum to 1 [Foroni2013] [Sheppard2018] |
|
\(\sigma_t^{2}=\lambda\sigma_{t-1}^2 + (1-\lambda)\epsilon^2_{t-1}\), with \(\lambda=0.94\) by default [RiskMetrics1996] |
|
A weighted average of |
|
The same variance in all periods |
|
A variance that is provided by the user |
Distributions¶
The distribution is the distribution of \(e_t\), which is used to construct the likelihood, see Distributions. All of the distributions are standardized to have mean 0 and variance 1.
StudentsT, with one shape parameter \(\nu\) [Bollerslev1987].SkewStudent, a skewed Student’s t with the tail shape \(\eta\) and the skewness \(\lambda\) [Hansen1994]. When \(\lambda=0\) it is the standardized Student’s t.GeneralizedError, the generalized error distribution, with one shape parameter \(\nu\) [Nelson1991].
References¶
Baillie, R. T., Bollerslev, T., & Mikkelsen, H. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 74(1), 3-30.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3), 307-327.
Bollerslev, T. (1987). A conditionally heteroskedastic time series model for speculative prices and rates of return. The Review of Economics and Statistics, 69(3), 542-547.
Bollerslev, T., & Wooldridge, J. M. (1992). Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews, 11(2), 143-172.
Ding, Z., Granger, C. W. J., & Engle, R. F. (1993). A long memory property of stock market returns and a new model. Journal of Empirical Finance, 1(1), 83-106.
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4), 987-1007.
Engle, R. F., Lilien, D. M., & Robins, R. P. (1987). Estimating time varying risk premia in the term structure: The ARCH-M model. Econometrica, 55(2), 391-407.
Foroni, Claudia, and Massimiliano Marcellino. “A survey of Econometric Methods for Mixed-Frequency Data”. Norges Bank. (2013).
Glosten, L. R., Jagannathan, R., & Runkle, D. E. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. The Journal of Finance, 48(5), 1779-1801.
Hansen, B. E. (1994). Autoregressive conditional density estimation. International Economic Review, 35(3), 705-730.
Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica, 59(2), 347-370.
J.P. Morgan/Reuters (1996). RiskMetrics Technical Document, Fourth Edition. New York: J.P. Morgan/Reuters, December 17, 1996.
Sheppard, Kevin. “Direct volatility modeling”. Manuscript. (2018).
Tse, Y. K. (1998). The conditional heteroscedasticity of the yen-dollar exchange rate. Journal of Applied Econometrics, 13(1), 49-55.
Zakoian, J.-M. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and Control, 18(5), 931-955.