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.