## 16.5 Statistical Properties of SI Model Estimates

To determine the statistical properties of the plug-in principle/least squares/ML estimators $$\hat{\alpha}_{i}$$, $$\hat{\beta}_{i}$$ and $$\hat{\sigma}_{\epsilon,i}^{2}$$ in the SI model, we treat them as functions of the random variables $$\{(R_{i,t},R_{Mt})\}_{t=1}^{T}$$ where $$R_{t}$$ and $$R_{Mt}$$ are assumed to be generated by the SI model (16.1) - (16.5).

### 16.5.1 Bias

In the SI model, the estimators $$\hat{\alpha}_{i}$$, $$\hat{\beta}_{i}$$ and $$\hat{\sigma}_{\epsilon,i}^{2}$$ (with degrees-of-freedom adjustment) are unbiased: $\begin{eqnarray*} E[\hat{\alpha}_{i}] & = & \alpha_{i},\\ E[\hat{\beta}_{i}] & = & \beta_{i},\\ E[\hat{\sigma}_{\epsilon,i}^{2}] & = & \sigma_{\epsilon,i}^{2}. \end{eqnarray*}$ To shows that $$\hat{\alpha}_{i}$$ and $$\hat{\beta}_{i}$$ are unbiased, it is useful to consider the SI model for asset $$i$$ in matrix form for $$t=1,\ldots,T$$: $\mathbf{R}_{i}=\alpha_{i}\mathbf{1}+\beta_{i}\mathbf{R}_{M}+\epsilon_{i}=\mathbf{X}\gamma_{i}+\epsilon_{i},$ where $$\mathbf{R}_{i}=(R_{i1},\ldots,R_{iT})^{\prime},$$ $$\mathbf{R}_{M}=(R_{M1},\ldots,R_{MT})^{\prime}$$, $$\epsilon_{i}=(\epsilon_{i1},\ldots,\epsilon_{iT})^{\prime}$$, $$\mathbf{1}=(1,\ldots,1)^{\prime}$$, $$\mathbf{X}=(\begin{array}{cc} \mathbf{1} & \mathbf{R}_{M})\end{array},$$ and $$\gamma_{i}=(\alpha_{i},\beta_{i})^{\prime}.$$ The estimator for $$\gamma_{i}$$ is $\begin{eqnarray*} \hat{\gamma}_{i} & = & (\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\mathbf{R}_{i}. \end{eqnarray*}$ Pluging in $$\mathbf{R}_{i}=\mathbf{X}\gamma_{i}+\epsilon_{i}$$ gives $\begin{eqnarray*} \hat{\gamma}_{i} & = & (\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\left(\mathbf{X}\gamma_{i}+\epsilon_{i}\right)=(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\mathbf{X}\gamma_{i}+(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\epsilon_{i}\\ & = & \gamma_{i}+(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\epsilon_{i}. \end{eqnarray*}$ Then $\begin{eqnarray*} E[\hat{\gamma}_{i}] & = & \gamma_{i}+E\left[(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\epsilon_{i}\right]\\ & = & \gamma_{i}+E\left[(\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}\right]E[\epsilon_{i}]\,(\mathrm{because}\,\epsilon_{it}\,\mathrm{is}\,\mathrm{independent}\,\mathrm{of}\,R_{Mt})\\ & = & \gamma_{i}\,(\mathrm{because}\,E[\epsilon_{i}]=\mathbf{0}). \end{eqnarray*}$ The derivation of $$E[\hat{\sigma}_{\epsilon,i}^{2}]=\sigma_{\epsilon,i}^{2}$$ is beyond the scope of this book and can be found in graduate econometrics textbooks such as Hayashi (1980).

### 16.5.2 Precision

Under the assumptions of the SI model, analytic formulas for estimates of the standard errors for $$\hat{\alpha}_{i}$$ and $$\hat{\beta}_{i}$$ are given by: \begin{align} \widehat{\mathrm{se}}(\hat{\alpha}_{i}) & \approx\frac{\hat{\sigma}_{\varepsilon,i}}{\sqrt{T\cdot\hat{\sigma}_{M}^{2}}}\cdot\sqrt{\frac{1}{T}\sum_{t=1}^{T}r_{Mt}^{2}},\tag{16.36}\\ \widehat{\mathrm{se}}(\hat{\beta}_{i}) & \approx\frac{\hat{\sigma}_{\varepsilon,i}}{\sqrt{T\cdot\hat{\sigma}_{M}^{2}}},\tag{16.37} \end{align} where “$$\approx$$” denotes an approximation based on the CLT that gets more accurate the larger the sample size. Remarks:

• $$\widehat{\mathrm{se}}(\hat{\alpha}_{i})$$ and $$\widehat{\mathrm{se}}(\hat{\beta}_{i})$$ are smaller the smaller is $$\hat{\sigma}_{\varepsilon,i}$$. That is, the closer are returns to the fitted regression line the smaller are the estimation errors in $$\hat{\alpha}_{i}$$ and $$\hat{\beta}_{i}$$.
• $$\widehat{\mathrm{se}}(\hat{\beta}_{i})$$ is smaller the larger is $$\hat{\sigma}_{M}^{2}$$. That is, the greater the variability in the market return $$R_{Mt}$$ the smaller is the estimation error in the estimated slope coefficient $$\hat{\beta}_{i}$$. This is illustrated in Figure xxx. The left panel shows a data sample from the SI model with a small value of $$\hat{\sigma}_{M}^{2}$$ and the right panel shows a sample with a large value of $$\hat{\sigma}_{M}^{2}$$. The right panel shows that the high variability in $$R_{Mt}$$ makes it easier to identify the slope of the line.
• Both $$\widehat{\mathrm{se}}(\hat{\alpha}_{i})$$ and $$\widehat{\mathrm{se}}(\hat{\beta}_{i})$$ go to zero as the sample size, $$T$$, gets large. Since $$\hat{\alpha}_{i}$$ and $$\hat{\beta}_{i}$$ are unbiased estimators, this implies that they are also consistent estimators. That is, they converge to the true values $$\alpha_{i}$$ and $$\beta_{i}$$, respectively, as $$T\rightarrow\infty$$.
• In R, the standard error values (16.36) and (16.37) are computed using the summary() function on an “lm” object.

There are no easy formulas for the estimated standard errors for $$\hat{\sigma}_{\epsilon,i}^{2}$$, $$\hat{\sigma}_{\varepsilon,i}$$ and $$\hat{R}^{2}$$. Estimated standard errors for these estimators, however, can be easily computed using the bootstrap.

Example 2.10 (Computing estimated standard errors for $$\hat{\alpha}_{i}$$ and $$\hat{\beta}_{i}$$ in R)

to be completed

$$\blacksquare$$

### 16.5.3 Sampling distribution and confidence intervals.

Using arguments based on the CLT, it can be shown that for large enough $$T$$ the estimators $$\hat{\alpha}_{i}$$ and $$\hat{\beta}_{i}$$ are approximately normally distributed: $\begin{eqnarray*} \hat{\alpha}_{i} & \sim & N(\alpha_{i},\widehat{\mathrm{se}}(\hat{\alpha}_{i})^{2}),\\ \hat{\beta}_{i} & \sim & N(\beta_{i},\widehat{\mathrm{se}}(\hat{\beta}_{i})^{2}), \end{eqnarray*}$ where $$\widehat{\mathrm{se}}(\hat{\alpha}_{i})$$ and $$\widehat{\mathrm{se}}(\hat{\beta}_{i})$$ are given by (16.36) and (16.37), respectively.