Time Series Models of Volatility

Motivation

In many econometric applicaitons, model errors are heteroscedastic.

  • In addition to non-constant variance, the variance of the errors may be autocorrelated through time.
  • Financial time series have given rise to a rich literature in time series modeling of volatility.
  • However, non-financial time series may exhibit time-varying volatility.

Motivation

At most time scales, financial returns are known to exhibit:

  • Little or no serial correlation.
  • Dependence/serial correlation in the second moment.
  • Heavy tails, relative to a Gaussian distribution.

Motivation

library(quantmod)
getSymbols('SPY')
spyRets = dailyReturn(Cl(SPY))
png(filename='spyQQACF.png',,height=4,width=10,units='in',res=150)
par(mfrow=c(1,3))
qqnorm(spyRets)
qqline(spyRets)
acf(spyRets,lag.max=50)
acf(spyRets^2,lag.max=50)
dev.off()

Motivation

../_images/spyQQACF.png

Basic Model

Consider the following model

\[\begin{align} r_t & = \mu_t + \varepsilon_t, \,\,\, \varepsilon_t \sim N(0,\sigma_t^2). \end{align}\]
  • The mean, \(\smash{\mu_t}\), may follow a linear time series process
\[\begin{align} \mu_t & = \phi_0 + \sum_{i=1}^k \lambda_i x_{it} + \sum_{i=1}^p \phi_i r_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i}. \end{align}\]
  • The variables \(\smash{\{x_{it}\}}\) are exogenous explanatory variables (e.g. the market return, for the CAPM, or dummy variables for month/week/day effects).

Basic Model

Time series models of volatility augment the model of the mean with a process that governs conditional volatility, \(\smash{\sigma_t}\).

  • Some models specify a deterministic process for volatility (\(\smash{ARCH}\)/\(\smash{GARCH}\)).
  • Others specify a random process (stochastic volatility).

The \(\smash{ARCH}\) Model

Engle (1982) proposed the following model for volatility

\[\begin{align} \sigma_t^2 & = \alpha_0 + \sum_{i=1}^m \alpha_i \varepsilon_{t-i}^2. \end{align}\]

where \(\smash{\alpha_0 > 0}\) and \(\smash{\alpha_i \geq 0}\) for \(\smash{i=1,\ldots,m}\).

  • Recall, \(\smash{\varepsilon_t \sim N(0,\sigma_t^2)}\).
  • Alternatively, we could write \(\smash{\varepsilon_t = \sigma_t z_t}\), where \(\smash{z_t \sim N(0,1)}\).

\(\smash{ARCH}\) Order

Define \(\smash{\eta_t = \varepsilon_t^2 - \sigma_t^2}\).

  • It can be shown that \(\smash{\{\eta_t\}}\) is an uncorrelated, zero-mean series (not necessarily i.i.d.).
  • Thus the \(\smash{ARCH}\) model can be written as
\[\begin{align} \varepsilon_t^2 & = \alpha_0 + \sum_{i=1}^m \alpha_i \varepsilon_{t-i}^2 + \eta_t. \end{align}\]
  • As a result, the order can be diagnosed using the tools of \(\smash{AR}\) order determination (PACF).
  • This is done on the residuals, after estimating the mean equation.

Notes on \(\smash{ARCH}\)

  • It can be shown that the \(\smash{ARCH}\) model results in excess kurthosis (relative to a Normal).
  • The model parameters must be restricted in order to maintain finite unconditional variance and positive conditional variance.

\(\smash{ARCH}\) Estimation

Under the assumption of Normality, the conditional likelihood function is

\[\begin{gather} f(\varepsilon_{m+1},\ldots,\varepsilon_T|\alpha_0,\ldots,\alpha_m,\varepsilon_1,\ldots,\varepsilon_m) = \prod_{t=m+1}^T \frac{1}{\sqrt{2\pi \sigma_t^2}} \exp{\left\{ -\frac{\varepsilon_t^2}{2 \sigma_t^2}\right\}}. \end{gather}\]
  • MLE is conducted with an interative approach:
    • Compute a set of observed residuals \(\smash{\{\varepsilon_t\}}\).
    • For a candidate estimate of the parameters, iteratively compute \(\smash{\{\sigma_t^2\}_{m+1}^T}\) with the variance equation and evaluate the likelihood.
    • Update the parameters and repeat.
  • MLE is often conducted with a \(\smash{t}\) distribution or generalized error distribution.

The \(\smash{GARCH}\) Model

Bollerslev (1986) proposed the following extension to the \(\smash{ARCH}\) model:

\[\begin{align} \sigma_t^2 & = \alpha_0 + \sum_{i=1}^m \alpha_i \varepsilon_{t-i}^2 + \sum_{j=1}^s \beta_j \sigma_{t-j}^2. \end{align}\]

where \(\smash{\alpha_0 > 0}\), \(\smash{\alpha_i\geq 0}\), \(\smash{\beta_j\geq 0}\), and \(\smash{\sum_{i=1}^{\max(m,s)} (\alpha_i+\beta_i) < 1}\) for \(\smash{i=1,\ldots,m}\).

Notes on \(\smash{GARCH}\)

  • \(\smash{GARCH}\) is often preferred to \(\smash{ARCH}\) because it requires far fewer parameters.
  • MLE is conducted in a similar fashion, but requires an initial condition for \(\smash{\{\sigma_t\}_{m-s+1}^m}\).
  • As with \(\smash{ARCH}\) models, \(\smash{GARCH}\) models feature excess kurtosis.
  • Many refinements of the \(\smash{GARCH}\) model have been developed to account for asymmetry in volatility and other features.

Stochastic Volatility

Stochastic volatility models are an alternative to deterministic models of volatility (\(\smash{ARCH}\)/\(\smash{GARCH}\)).

  • Consider the simple model of Taylor (1986):
\[\begin{align} \log(\sigma_t^2) & = \eta_0 + \eta_1 \log(\sigma_{t-1}^2) + u_t, \,\,\, u_t \sim WN(0,\sigma_u^2). \end{align}\]
  • \(\smash{u_t}\) may be correlated with the return innovation \(\smash{\varepsilon_t}\).

Stochastic Volatility Estimation

The Kalman filter can be used to estimate the stochastic volatility model above.

  • Consider the state-space system of Nelson (1988):
\[\begin{split}\begin{align} \log(r_t^2) & = \log(\sigma_t^2) + \log(z_t^2), \,\,\, z_t \stackrel{i.i.d.}{\sim} N(0,1) \\ \log(\sigma_t^2) & = \eta_0 + \eta_1 \log(\sigma_{t-1}^2) + u_t, \,\,\, u_t \stackrel{i.i.d.}{\sim} N(0,\sigma_u^2). \end{align}\end{split}\]
  • The Kalman filter cannot be used for other, nonlinear stochastic volatility models.
  • Hamilton (1989) developed a nonlinear filtering method, similar to the Kalman filter, for such problems.