7.1 Estimators and Estimates

Let $R_{t}$ be the return on a single asset (simple or continuously compounded) described by the GWN model and let $\theta$ denote some characteristic (parameter) of the GWN model we are interested in estimating. For simplicity, assume that $\theta\in\mathbb{R}$ is a single parameter. For example, if we are interested in the expected return on the asset, then $\theta=\mu$; if we are interested in the variance of the asset returns, then $\theta=\sigma^{2}$; if we are interested in the first lag autocorrelation of returns then $\theta=\rho_{1}$. The goal is to estimate $\theta$ based on a sample of size $T$ of the observed data when we believe the data is generated from the GWN return model.

Definition 7.1 Let $\{R_{1},\ldots,R_{T}\}$ denote a collection of $T$ random returns from the GWN model, and let $\theta$ denote some characteristic of the model. An estimator of $\theta$, denoted $\hat{\theta}$, is a rule or algorithm for estimating $\theta$ as a function of the random variables $\{R_{1},\ldots,R_{T}\}$. Here, $\hat{\theta}$ is a random variable.

Definition 2.2 Let $\{r_{1},\ldots,r_{T}\}$ denote an observed sample of size $T$ from the GWN model, and let $\theta$ denote some characteristic of the model. An estimate of $\theta$, denoted $\hat{\theta}$, is simply the value of the estimator for $\theta$ based on the observed sample $\{r_{1},\ldots,r_{T}\}$. Here, $\hat{\theta}$ is a number.

Example 4.2 (The sample average as an estimator and an estimate)

Let $R_{t}$ be the return on a single asset described by the GWN model, and suppose we are interested in estimating $\theta=\mu=E[R_{t}]$ from the sample of observed returns $\{r_{t}\}_{t=1}^{T}$. The sample average is an algorithm for computing an estimate of the expected return $\mu$. Before the sample is observed, we can think of $\hat{\mu}=\frac{1}{T}\sum_{t=1}^{T}R_{t}$ as a simple linear function of the random variables $\{R_{t}\}_{t=1}^{T}$ and so is itself a random variable. We can study the properties of $\hat{\mu}$ using the tools of probability theory presented in chapter 2.

After the sample is observed, the sample average can be evaluated using the observed data $\{r_{t}\}_{t=1}^{T}$ giving $\hat{\mu}=\frac{1}{T}\sum_{t=1}^{T}r_{t}$, which produces the numerical estimate of $\mu$. For example, suppose $T=5$ and the realized values of the returns are $r_{1}=0.1,r_{2}=0.05,r_{3}=0.025,r_{4}=-0.1,r_{5}=-0.05$. Then the estimate of $\mu$ using the sample average is: $$\hat{\mu}=\frac{1}{5}(0.1+0.05+0.025+-0.1+-0.05)=0.005.$$

$\blacksquare$

The example above illustrates the distinction between an estimator and an estimate of a parameter $\theta$. However, typically in the statistics literature we use the same symbol, $\hat{\theta}$, to denote both an estimator and an estimate. When $\hat{\theta}$ is treated as a function of the random returns it denotes an estimator and is a random variable. When $\hat{\theta}$ is evaluated using the observed data it denotes an estimate and is simply a number. The context in which we discuss $\hat{\theta}$ will determine how to interpret it.