paper.md
In stochvolTMB: Likelihood Estimation of Stochastic Volatility Models

title: 'stochvolTMB: An R-package for likelihood estimation of stochastic volatility models' tags: - R - maximum likelihood - laplace approximation - volatility - finance date: "8 December 2020" output: rticles::joss_article authors: - name: Jens Christian Wahl orcid: 0000-0002-3812-5111 affiliation: 1 - name: Norwegian Computing Center index: 1 year: 2020 formatted_doi: XX.XXXXX/joss.XXXXX bibliography: paper.bib citation_author: Wahl journal: JOSS

Summary

Stochastic volatility (SV) models are often used to model financial returns that exhibit time-varying and autocorrelated variance. The first SV model was introduced by @Taylor1982 and models the logarithm of the variance as a latent autoregressive process of order one. Parameter estimation of stochastic volatility models can be challenging and a variety of methods have been proposed, such as simulated likelihood [@Liesenfeld2006], quasi-maximum likelihood [@Harvey1994] and Markov Chain Monte Carlo methods (MCMC) [@Shepard1998; @Kastner2016]. stochvolTMB takes a frequentist approach and estimates the parameters using maximum likelihood, similar to @Skaug2014. The latent variables are integrated out using the Laplace approximation. The models are implemented in C++ using the R-package [@rCore] TMB [@TMB2016] for fast and efficient estimation. TMB utilizes the Eigen [@Eigen2010] library for numerical linear algebra and CppAD [@CppAD2005] for automatic differentiation of the negative log-likelihood. This can lead to substantial speed-up compared to MCMC methods.

Implementation

stochvolTMB implements stochastic volatility models of the form

\begin{equation} \begin{aligned} y_t &= \sigma_y e^{h_t/2} \epsilon_t, \quad t = 1, \dots, T, \ h_{t+1} &= \phi h_{t} + \sigma_h \eta_t, \quad t = 1, \dots, T-1, \ \eta_t &\stackrel{\text{iid}}{\sim} \mathcal{N}(0,1), \ \epsilon_t &\stackrel{\text{iid}} {\sim} F, \ h_1 &\sim \mathcal{N} \bigg (0, \frac{\sigma_h}{\sqrt{(1 - \phi^2)}} \bigg ) \end{aligned} \end{equation} where $y_t$ is the observed log return for day $t$, $h_t$ is the logarithm of the conditional variance of day $t$ and $\boldsymbol{\theta} = (\phi, \sigma_y, \sigma_h)$ are the fixed parameters. Four distributions are implemented for $\epsilon_t$: (1) The standard normal distribution; (2) The t-distribution with $\nu$ degrees of freedom; (3) The skew-normal distribution with skewness parameter $\alpha$; and (4) The leverage model where $(\epsilon_t, \eta_t)$ are multivariate normal with zero mean and correlation coefficient $\rho$. The last three distributions add an additional fixed parameter to $\boldsymbol{\theta}$. stochvolTMB also supports generic functions such as plot, summary, predict and AIC. The plotting is implemented using ggplot2 (@ggplot2) and data processing utilizes the R-package data.table [@datatableRpackage].

The parameter estimation is done in an iterative two-step procedure: (1) Optimize the joint negative log-likelihood with respect to the latent log-volatility $\boldsymbol{h} = (h_1, \ldots, h_T)$ holding $\boldsymbol{\theta}$ fixed, and (2) Optimizing the Laplace approximation of the joint negative log-likelihood w.r.t $\boldsymbol{\theta}$. This procedure is iterated until convergence. Standard deviations for the log-volatility and the fixed parameters are obtained by the delta-method [@TMB2016].

stochvolTMB different from R-package stochvol [@Kastner2016] as stochvol performs Bayesian inference using MCMC. By using optimization instead of simulation we are able to obtain a 5-10 times speed up, dependent on the data, model and number of observations.