Conditional Heteroskedasticity and GARCH

A return is typically serially uncorrelated yet dependent, the dependence carried by its conditional variance rather than its conditional mean. Large moves cluster, so the squared return holds the autocorrelation the return itself lacks. Writing $r_t=\mu_t+a_t$ with $a_t=\sigma_t\varepsilon_t$, where $\{\varepsilon_t\}$ is iid with mean zero and unit variance and $\sigma_t^2=\Var(a_t\gvn\mathcal F_{t-1})$ is the conditional variance, this post tests for that dependence and then models how the variance evolves [1], [2].

#Detecting ARCH effects

Volatility clustering shows up as autocorrelation in the squared returns even when the returns themselves are uncorrelated. The first task is to detect it, by testing whether past squared residuals help predict the current one.

::: theorem [thm

] In the auxiliary regression $\hat e_t^2=\alpha_0+\sum_{j=1}^q\alpha_j\hat e_{t-j}^2+v_t$, the score test of $H_0\!:\alpha_1=\dots=\alpha_q=0$ is $\mathrm{LM}=TR^2\cvD\chi^2_q$. :::

::: proof The Gaussian score for the ARCH$(q)$ parameters at $H_0$ is proportional to the sample covariance of $\hat e_t^2/\hat\sigma^2-1$ with the lagged $\hat e_{t-j}^2$. The Lagrange-multiplier statistic equals $T$ times the uncentred $R^2$ of the auxiliary regression, a quadratic form in $q$ asymptotically Gaussian scores, hence $\chi^2_q$. :::

ARCH-LM test for conditional heteroskedasticity
  input   residuals e_hat, lags q
  1  regress e_hat_t^2 on a constant and
     e_hat_{t-1}^2 .. e_hat_{t-q}^2
  2  LM <- T R^2 of the auxiliary regression
  3  reject the no-ARCH null when Prob(chi^2_q > LM) is small

#The ARCH and GARCH models

Detecting the effect is one thing, modelling its evolution another. The ARCH and GARCH families make the conditional variance an explicit function of past squared shocks and past variances.

::: definition [def

] The ARCH$(m)$ model sets $\sigma_t^2=\alpha_0+\sum_{i=1}^m\alpha_i a_{t-i}^2$ with $\alpha_0>0$ and $\alpha_i\ge0$. The GARCH$(m,s)$ model adds lagged variances, $\sigma_t^2=\alpha_0+\sum_{i=1}^m\alpha_i a_{t-i}^2+\sum_{j=1}^s\beta_j\sigma_{t-j}^2$, with $\beta_j\ge0$. :::

::: proposition [prop

] For the ARCH$(1)$ model with $0\le\alpha_1<1$ the unconditional variance is $\Var(a_t)=\alpha_0/(1-\alpha_1)$, and under Gaussian $\varepsilon_t$ with $3\alpha_1^2<1$ the unconditional kurtosis is $3(1-\alpha_1^2)/(1-3\alpha_1^2)>3$, so $a_t$ is heavier tailed than its driving noise. :::

::: proof The mean is zero since $\E[a_t]=\E[\sigma_t]\E[\varepsilon_t]=0$, the two factors being independent. By stationarity $\Var(a_t)=\E[\sigma_t^2]=\alpha_0+\alpha_1\E[a_{t-1}^2]=\alpha_0+\alpha_1\Var(a_t)$, which gives $\Var(a_t)=\alpha_0/(1-\alpha_1)$. Under normality $\E[a_t^4\gvn\mathcal F_{t-1}]=3\sigma_t^4$, so $m_4=\E[a_t^4]=3\E[(\alpha_0+\alpha_1 a_{t-1}^2)^2]$; solving the resulting stationary equation gives $m_4=3\alpha_0^2(1+\alpha_1)/[(1-\alpha_1)(1-3\alpha_1^2)]$, and $m_4/\Var(a_t)^2$ is the stated kurtosis, which exceeds $3$ for $\alpha_1>0$. :::

The same accounting extends to GARCH once the recursion is rewritten in the squared series, where the lagged variances fold into an autoregressive term.

::: proposition [prop

] Let $\eta_t=a_t^2-\sigma_t^2$. The GARCH$(m,s)$ recursion is equivalent to

where $\{\eta_t\}$ is a mean-zero martingale difference, so the squared process is an ARMA. When $\sum_i(\alpha_i+\beta_i)<1$ it is covariance stationary with $\Var(a_t)=\alpha_0/\bigl(1-\sum_i(\alpha_i+\beta_i)\bigr)$. :::

::: proof Substitute $\sigma_{t-j}^2=a_{t-j}^2-\eta_{t-j}$ into the GARCH recursion and collect the $a^2$ terms to reach Equation (1). The increment $\eta_t=\sigma_t^2(\varepsilon_t^2-1)$ has $\E[\eta_t\gvn\mathcal F_{t-1}]=0$, hence is a martingale difference and serially uncorrelated. Taking unconditional means of the ARMA equation, with $\E[\eta_t]=0$ and stationarity, leaves $\E[a_t^2]=\alpha_0+\sum_i(\alpha_i+\beta_i)\E[a_t^2]$, which rearranges to the stated variance. :::

The second moment settles whenever the persistence is below one, but the tails need the fourth moment, which exists only on a stricter region of the parameter space.

::: proposition [prop

] For the GARCH$(1,1)$ model with iid $\varepsilon_t$ of unit variance and finite fourth moment $\kappa=\E[\varepsilon_t^4]$, the fourth moment of $a_t$ is finite if and only if $(\alpha_1+\beta_1)^2+(\kappa-1)\alpha_1^2<1$, and then the kurtosis is

which exceeds $\kappa$ whenever $\alpha_1>0$, so the model adds tail weight beyond its innovation. :::

::: proof Write $\sigma_t^2=\alpha_0+\phi_{t-1}\sigma_{t-1}^2$ with $\phi_{t-1}=\alpha_1\varepsilon_{t-1}^2+\beta_1$, where $\phi_{t-1}$ is a function of $\varepsilon_{t-1}$ and so independent of $\sigma_{t-1}^2\in\mathcal F_{t-2}$. Then $\E[\phi]=\alpha_1+\beta_1$ and $\E[\phi^2]=\kappa\alpha_1^2+2\alpha_1\beta_1+\beta_1^2=(\alpha_1+\beta_1)^2+(\kappa-1)\alpha_1^2$. Squaring the recursion and taking expectations under stationarity gives $\E[\sigma_t^4]=\alpha_0^2+2\alpha_0\E[\phi]\E[\sigma_t^2]+\E[\phi^2]\E[\sigma_t^4]$, which has a positive solution exactly when $\E[\phi^2]<1$. Substituting $\E[\sigma_t^2]=\alpha_0/(1-\alpha_1-\beta_1)$ and $\E[a_t^4]=\kappa\E[\sigma_t^4]$ and dividing by $(\E[a_t^2])^2$ yields Equation (2), larger than $\kappa$ once $\alpha_1>0$. :::

#Estimation

Conditioning on the past turns the likelihood into a single forward pass, and the standardised residuals it leaves are what the diagnostics read.

::: proposition [prop

] Under $\varepsilon_t\sim\Nor(0,1)$ the parameter vector $\phi=(\alpha_0,\dots,\beta_s)$ maximises the prediction-error decomposition

each $\sigma_t^2(\phi)$ computed by the model recursion from the past, so the likelihood is evaluated in a single forward pass. :::

::: proof The joint density factors as $\prod_t p(a_t\gvn\mathcal F_{t-1})$ by the chain rule, and each conditional law is $\Nor(0,\sigma_t^2)$ with log density $-\tfrac12\ln(2\pi)-\tfrac12\ln\sigma_t^2-a_t^2/(2\sigma_t^2)$. Conditioning on $\mathcal F_{t-1}$ makes $\sigma_t^2$ measurable, so the recursion fixes it before $a_t$ is read. Dropping the constant and summing the remaining terms gives Equation (3). :::

Maximum-likelihood estimation of GARCH(1,1)
  input   return shocks a_t, initial variance sigma_0^2
  1  for a candidate (alpha_0, alpha_1, beta_1) run the recursion
     sigma_t^2 = alpha_0 + alpha_1 a_{t-1}^2 + beta_1 sigma_{t-1}^2
  2  ell <- -1/2 sum_t ( ln sigma_t^2 + a_t^2 / sigma_t^2 )
  3  maximise ell over alpha_0>0, alpha_1,beta_1>=0, alpha_1+beta_1<1
  4  return the fit and standardised residuals a_t / sigma_t

::: remark Maximising the Gaussian $\ell$ stays consistent and asymptotically normal when the true innovation law is not Gaussian, the Gaussian score remaining an unbiased estimating equation for the variance dynamics; this is the quasi-maximum-likelihood estimator. A fitted model is checked through the standardised residuals $\hat\varepsilon_t=a_t/\hat\sigma_t$, which should be close to iid. The Ljung-Box statistic on $\hat\varepsilon_t$ probes the mean specification and on $\hat\varepsilon_t^2$ the variance specification, ARCH effects surviving in $\hat\varepsilon_t^2$ signalling an order set too low. :::

#Persistence and forecasts

How long a shock to variance lasts is set by the persistence of the recursion, which also fixes how a multistep variance forecast decays back toward its level.

::: proposition [prop

] For the GARCH$(1,1)$ model the $\ell$-step variance forecast satisfies $\sigma_h^2(\ell)=\alpha_0+(\alpha_1+\beta_1)\sigma_h^2(\ell-1)$ for $\ell>1$, so when $\alpha_1+\beta_1<1$ it converges to the unconditional variance $\alpha_0/(1-\alpha_1-\beta_1)$ as $\ell\to\infty$. When $\alpha_1+\beta_1=1$ it follows the line $\sigma_h^2(\ell)=\sigma_h^2(1)+(\ell-1)\alpha_0$, the integrated GARCH case. :::

::: proof Write $\sigma_{t+1}^2=\alpha_0+(\alpha_1+\beta_1)\sigma_t^2+\alpha_1\sigma_t^2(\varepsilon_t^2-1)$ using $a_t^2=\sigma_t^2\varepsilon_t^2$. Since $\E[\varepsilon_t^2-1\gvn\mathcal F_{t-1}]=0$, the conditional expectation of the last term vanishes and the forecast obeys the affine recursion $\sigma_h^2(\ell)=\alpha_0+(\alpha_1+\beta_1)\sigma_h^2(\ell-1)$. A slope $\alpha_1+\beta_1<1$ contracts the map to its fixed point $\alpha_0/(1-\alpha_1-\beta_1)$, while at unit slope each step adds $\alpha_0$, giving the line. :::

::: proposition [prop

] In the IGARCH$(1,1)$ model $\alpha_1+\beta_1=1$ the recursion reads $\sigma_t^2=\alpha_0+(1-\beta_1)a_{t-1}^2+\beta_1\sigma_{t-1}^2$, and with $\alpha_0=0$ it is the exponentially weighted moving average $\sigma_t^2=(1-\beta_1)\sum_{i\ge0}\beta_1^i a_{t-1-i}^2$, whose multistep forecast is flat, $\sigma_h^2(\ell)=\sigma_h^2(1)$ for every $\ell$. :::

::: proof Setting $\alpha_1=1-\beta_1$ in the GARCH$(1,1)$ recursion gives the displayed form. With $\alpha_0=0$, unrolling $\sigma_t^2=(1-\beta_1)a_{t-1}^2+\beta_1\sigma_{t-1}^2$ backward produces geometric weights $(1-\beta_1)\beta_1^i$ that sum to one. The forecast recursion of ?? at unit slope with $\alpha_0=0$ holds every horizon at $\sigma_h^2(1)$. :::

#Asymmetry and the leverage effect

The GARCH variance enters past shocks only through their square, so it responds identically to a gain and a loss of equal size. Equity-like data instead carry a leverage effect, a larger variance after a drop, and an asymmetric law is needed to capture it.

::: definition [def

] The news impact curve is the function carrying the most recent shock $a_{t-1}$ to $\sigma_t^2$ with all earlier information held at its mean. For GARCH it is the parabola $\sigma_t^2=\text{const}+\alpha_1 a_{t-1}^2$, symmetric about the origin. :::

::: definition [def

] The threshold, or GJR, model adds a half-on quadratic term,

so a negative shock loads on $\alpha_1+\gamma$ and a positive one on $\alpha_1$ [3]. :::

::: proposition [prop

] With innovations symmetric about zero the threshold model is covariance stationary if and only if $\alpha_1+\gamma/2+\beta_1<1$, and then $\Var(a_t)=\alpha_0/(1-\alpha_1-\gamma/2-\beta_1)$. :::

::: proof Symmetry of $\varepsilon_t$ gives $\E[\one\{a_{t-1}<0\}a_{t-1}^2]=\tfrac12\E[a_{t-1}^2]$, the sign of $a_{t-1}$ being that of $\varepsilon_{t-1}$, independent of $\sigma_{t-1}$ and equally likely either way. Taking unconditional expectations of the recursion under stationarity gives $\Var(a_t)=\alpha_0+(\alpha_1+\gamma/2+\beta_1)\Var(a_t)$, which rearranges to the stated variance and is positive exactly when the coefficient sum is below one. :::

::: proposition [prop

] In the threshold model with symmetric innovations and $\gamma>0$, $\operatorname{Cov}(a_{t-1},\sigma_t^2)=\gamma\,\E[a_{t-1}^3\one\{a_{t-1}<0\}]<0$, so a fall raises next period's variance more than an equal rise. :::

::: proof Only the two shock terms of $\sigma_t^2$ correlate with $a_{t-1}$. Symmetry makes $\E[a_{t-1}^3]=0$, so $\operatorname{Cov}(a_{t-1},\alpha_1 a_{t-1}^2)=0$, leaving $\operatorname{Cov}(a_{t-1},\sigma_t^2)=\gamma\E[a_{t-1}^3\one\{a_{t-1}<0\}]$. The integrand is negative wherever it is nonzero, so for a nondegenerate shock with finite third moment the expectation is strictly negative when $\gamma>0$. :::

A second route to asymmetry models the log variance, which also frees the sign of the coefficients.

::: definition [def

] The exponential GARCH models the log variance as an autoregression driven by $g(\varepsilon_{t-1})=\theta\varepsilon_{t-1}+\gamma(|\varepsilon_{t-1}|-\E|\varepsilon_{t-1}|)$,

whose slope in $\varepsilon_{t-1}$ is $\theta+\gamma$ after a rise and $\theta-\gamma$ after a fall [4]. :::

::: proposition [prop

] The forcing $g(\varepsilon_{t-1})$ is iid with mean zero, so $\ln\sigma_t^2$ is a stationary autoregression whenever $|\beta|<1$, with $\E[\ln\sigma_t^2]=\alpha_0/(1-\beta)$, and modelling the log places no sign constraint on the parameters. :::

::: proof $\E[g(\varepsilon)]=\theta\E[\varepsilon]+\gamma(\E|\varepsilon|-\E|\varepsilon|)=0$, and $g(\varepsilon_{t-1})$ inherits the independence of $\varepsilon_{t-1}$. The recursion is then an autoregression with iid mean-zero forcing, stationary for $|\beta|<1$ with mean $\alpha_0/(1-\beta)$. Since $\sigma_t^2=\exp(\ln\sigma_t^2)>0$ for any real right-hand side, no nonnegativity constraint is needed. :::

GARCH is thus the variance-dynamics engine of financial time series. It turns the one fact a return's mean cannot show, that volatility clusters, into a parametric recursion whose persistence sets the half-life of a shock and whose asymmetric extensions carry the leverage effect. The clustering it models is exactly autocorrelation in the squared returns, and the conditional variance it produces is the discrete-time cousin of the stochastic-volatility diffusions that price options. A stochastic-volatility model, giving the log variance its own innovation rather than making it $\mathcal F_{t-1}$-measurable, is the natural next step when a latent variance is wanted at the cost of a closed-form likelihood.