Chapter 7 Estimation of The GWN Model

Updated: May 6, 2021

Copyright © Eric Zivot 2015, 2016, 2017, 2020, 2021

The GWN model of asset returns presented in the previous chapter gives us a simple framework for interpreting the time series behavior of asset returns and prices. At the beginning of time \(t-1\), \(\mathbf{R}_{t}\) is an \(N\times1\) random vector representing the returns (simple or continuously compounded) on assets \(i=1,\ldots,N\) to be realized at time \(t\). The GWN model states that \(\mathbf{R}_{t}\sim iid~N(\mu,\Sigma)\). Our best guess for the return at \(t\) on asset \(i\) is \(E[R_{it}]=\mu_{i}\), our measure of uncertainty about our best guess is captured by \(\mathrm{SD}(R_{it})=\sigma_{i}\), and our measures of the direction and strength of linear association between \(R_{it}\) and \(R_{jt}\) are \(\sigma_{ij}=\mathrm{cov}(R_{it},R_{jt})\) and \(\rho_{ij}=\mathrm{cor}(R_{it},R_{jt})\), respectively. The GWN model assumes that the economic environment is constant over time (i.e., covariance stationary) so that the multivariate normal distribution characterizing returns is the same for all time periods \(t\).

Our life would be very easy if we knew the exact values of \(\mu_{i},\sigma_{i}^{2}\), \(\sigma_{ij}\) and \(\rho_{ij}\): the parameters of the GWN model. Then we could use the GWN model for risk and portfolio analysis. In actuality, however, we do not know these values with certainty. Therefore, a key task in financial econometrics is estimating these values from a history of observed return data. Given estimates of the GWN model parameters we can then apply the model to risk and portfolio analysis.

Suppose we observe returns on \(N\) different assets over the sample \(t=1,\ldots,T\). Denote this sample \(\{\mathbf{r}_{1},\ldots,\mathbf{r}_{T}\}=\{\mathbf{r}_{t}\}_{t=1}^{T}\), where \(\mathbf{r}_{t}=(\mathbf{r}_{1t},\ldots,\mathbf{r}_{Nt})^{\prime}\) is the \(N\times1\) vector of returns on \(N\) assets observed at time \(t\). It is assumed that the observed returns are realizations of the random variables \(\{\mathbf{R}_{t}\}_{t=1}^{T}\), where \(\mathbf{R}_{t}=(R_{1t},\ldots,R_{Nt})^{\prime}\) is a vector of \(N\) asset returns described by the GWN model:

\[\begin{equation} \mathbf{R}_{t}=\mu+\varepsilon_{t},~\varepsilon_{t}\sim \mathrm{GWN}(\mathbf{0},\Sigma).\tag{7.1} \end{equation}\]

Under these assumptions, we can use the observed returns \(\{\mathbf{r}_{t}\}_{t=1}^{T}\) to estimate the unknown parameters in \(\mu\) and \(\Sigma\) of the GWN model. However, before we describe the estimation of the GWN model in detail, it is necessary to review some fundamental concepts in the statistical theory of estimation. This is important because estimates of parameters have estimation error and statistical theory allows us to evaluate characteristics of estimation error and to develop quantitative measures of the typical magnitude of estimation error. As we shall see, some parameters of the GWN model are precisely estimated (have small estimation error) and some parameters are not (have large estimation error).

The chapter is outlined as follows. Section 7.1 reviews some of the statistical theory of estimation and discusses some important finite sample properties of estimators such as bias and precision. For many estimators it is difficult to derive exact finite sample properties which motivates the need to look at asymptotic properties of estimators. These are properties that hold exactly as the sample size goes to infinity but are used as approximations for a finite sample size. Estimators of the GWN model parameters are presented in section 7.4 and the statistical properties of these estimators are investigated in section 7.5. Section 7.6 illustrates how Monte Carlo simulation can be used to evaluate and understand the statistical properties of the GWN model estimators.

To be added:

  • Which asymptotic distribution formulas depend on normality assumption? Make clear which results depend on GWN model and which results do not.

In this chapter we use the R packages IntroCompFinR, mvtnorm, and PerformanceAnalytics. Make sure these packages are installed and loaded prior to running the R examples in the chapter.