The Market Model

Abstract

The famed capital asset pricing model (CAPM) of Treynor (1962), Sharpe (1964), Lintner (1965), and Mossin (1966) is surprisingly elegant in design. To test the CAPM, the market model was invented in the form of a simple regression equation. Early evidence did not support the CAPM. Due to these findings, researchers later proposed new forms of the CAPM and alternative models extending the CAPM to multiple factors.

Appendices

Questions

• 1. Write the equation for the market model form of the CAPM. What is the intercept term in this model? Can the intercept be used to test the CAPM?

• 2. Why put a random error term in the market model? How would you estimate the market model with regression analysis?

• 3. Black, Jensen, and Scholes (BJS) (1972) tested the CAPM using the market model. Why did they use portfolios rather than individual stocks in their tests? How were the portfolios formed for their tests? Write the t-test that they performed to test whether the $\alpha _p$ for a portfolio was not equal to zero. What were the three main findings of their study? What did they conclude about the CAPM?

• 4. Fama and MacBeth (FM) (1973) tested the CAPM using cross-sectional regression analysis. What is the cross-sectional regression model that they tested?

• 5. What did Fama and MacBeth (1973) find in their cross-sectional tests of the CAPM?

• 6. Write a simple regression model with one explanatory variable. Also, write a multiple regression model with k explanatory variables $x_1, \ldots , x_k$. How would you test for a non-linear relationship between an independent variable and the dependent variable?

• 7. What are the statistical properties of the error term, or e, in a regression model? How should the regression coefficient $\beta$ be interpreted? How should the intercept $\alpha$ be interpreted from both a statistical perspective and an asset pricing perspective?

• 8. List the underlying classical assumptions in OLS regression analysis. Why are these assumptions important to OLS estimation of regression coefficients?

• 9. What does homoskedasticity of error terms mean in regression analysis? Are error terms correlated with one another? What is multicollinearity and can it cause problems in a multiple regression model? Lastly, what is the problem if error terms in the regression model are correlated with an independent variable?

• 10. If the CAPM does not hold, how does this affect the market model?

• 11. You want to test whether an estimated $\beta _j$ coefficient in an OLS regression model is significant. What is the null hypothesis? What is the statistical test of this hypothesis? What values of the t-statistic are needed to reach 5% and 1% significance levels? (Hint: this question is based on Appendix A.)

• 12. What is goodness-of-fit mean in regression analysis? How would you estimate it? Show a formula and explain its terms. Can it be adjusted for the degrees of freedom in the estimation of the regression equation? Which measure of goodness-of-fit should be used?

• 13. You want to test whether the market risk of a stock is the same as the market portfolio in CAPM. What is the null hypothesis? What is the alternative hypothesis?

x	y
4.6	10.6
11.1	22.6
19.5	29.7
3.2	2.6
9.5	23.4
10.8	9.0
14.2	26.8
8.6	15.0
21.9	38.9
9.2	17.5

Compute the OLS estimates for the intercept and the slope coefficient for the simple regression $y = \alpha + \beta x + e$ using Eqns. (4.11) and (4.12).

Problems

• 1. Update the data in Table 4.1 for Microsoft. Download monthly stock returns for MSFT and the S &P 500 index from Yahoo Finance. Download monthly Treasury bill rates from the Federal Reserve Bank of St. Louis (https://fred.stlouisfed.org/series/DGS1MO). Gather data for 36 months. What did you get for the intercept? Beta? Are these model parameters significant as determined by t-statistics? Also, draw a plot of the fitted regression line with excess MSFT returns on the Y-axis and excess S &P 500 index returns on the X-axis. Do the results support the CAPM?

• 2. In the text, it is proved that OLS estimation of the market model is the empirical counterpart of the CAPM under classical Gauss–Markov assumptions. Show that: (i) $E(u_{it}) = 0$, (ii) $Var \,(u_{it}) = \sigma _i^2(1 - \rho _{iM}^2) \equiv \sigma _{u_i}^2$ is constant, and (iii) $Cov \,(u_{it}, u_{is}) = 0$ for all $t \ne s$.

• 3. Consider the following observations for variables x and y:

Appendix A: Statistical Inference

Statistical inference gives us guidelines concerning what we can say about the population slope coefficients on the basis of the sample estimates. Confidence intervals and hypothesis testing are the major tools in this process. Statistical properties of the error term $e_{it}$ are critical. While the Gauss–Markov BLUE result needs only assumptions about the expected values, variances, and covariances, statistical inference needs more precise information about the whole distribution of the error terms.

The textbook assumption is that the error term is normally distributed, or $e_{i} \sim N(0, \sigma _{e}^2)$. While this assumption strengthens OLS estimator properties, as discussed in this chapter, it also determines the distribution (called the sampling distribution) of the estimator. An important property of the normal distribution is that any linear combinations of normal random variables are again normally distributed. Therefore, as an example considering $\hat{\beta }$ in Eq. (4.10), using fairly straightforward calculations, we can write

$$\begin{aligned} \hat{\beta }= \beta + \sum _{i = 1}^n w_i e_i, \end{aligned}$$

in which $w_{i} = (x_i - \bar{x}) / \sum _{i = 1}^n(x_i - \bar{x})^2$, such that $\hat{\beta }_i$ indeed is a linear combination of $e_i$. Thus, given $x_i$, $\hat{\beta }$ is normally distributed with expected value $\beta$ and variance

$$\begin{aligned} \sigma _{\hat{\beta }}^2 = \frac{\sigma _e^2}{\sum _{i = 1}^n(x_i - \bar{x})^2}. \end{aligned}$$

In the same fashion, $\hat{\alpha }$ is normal with expected value $\alpha$ and variance

$$\begin{aligned} \sigma _{\hat{\alpha }}^2 = \sigma _e^2\left( \frac{1}{n} + \frac{\bar{x}^2}{\sum _{i = 1}^n(x_i - \bar{x})^2}\right) . \end{aligned}$$

In the general case of Eq. (4.15), the variances of respective OLS coefficients are obtained from the diagonal of the matrix

$$\begin{aligned} Var \,(\hat{\boldsymbol{\beta }}) = \sigma _e^2(\mathbf{X'X})^{-1}. \end{aligned}$$

(A4.1)

If the error variance $\sigma _e^2$ were known, we could base our inferences directly on the normal distribution. In practice, however, variance $\sigma _e^2$ must be replaced by its regression estimate $s_e^2$ in Eq. (4.21) with the implication that inferences are based on the Student t-distribution. Thus, in a regression with k independent variables, the t-ratio for testing the null hypothesis

$$\begin{aligned} H_0 : \beta _j = \beta _j^* \end{aligned}$$

(A4.2)

becomes

$$\begin{aligned} t = \frac{\hat{\beta }_j - \beta _j^*}{s_{\hat{\beta }_j}}, \end{aligned}$$

(A4.3)

where $\beta _j^*$ is the hypothesized value of the population coefficient (i.e., a fixed value defined by the researcher), $s_{\hat{\beta }_j}$ is the estimated standard error of $\hat{\beta }_j$, equaling the square root of the jth diagonal element of Eq. (A4.1) with $s_e^2$ substituted for $\sigma _e^2$, $j = 0, 1, 2, \ldots , k$, and $\beta _0$ the intercept term (equaling $\alpha$ in the text). Under the null hypothesis, the t-ratio in Eq. (A4.3) is t-distributed with $n - k - 1$ degrees of freedom. Regarding the null hypothesis in Eq. (A4.2), the most common case is that we are interested in whether variable $x_j$ has any impact on y. In this case, $\beta _j^* = 0$ in (A4.2) so that the null hypothesis becomes

$$\begin{aligned} H_0: \beta _j = 0 \end{aligned}$$

and the t-statistic in (A4.3) reduces to

$$\begin{aligned} t = \frac{\hat{\beta }_j}{s_{\hat{\beta }_j}}, \end{aligned}$$

which is the t-ratio shown in standard computer outputs.

Confidence intervals are of the form $\hat{\beta }\pm t_ps_{\hat{\beta }}$ in which $t_p$ is an appropriate percentile from the t-distribution to match the desired confidence interval. Typical values are $p = .025$ for 95% and $p = .005$ for 99% interval. The $t_p$ value itself depends on the degrees of freedom. Typically in financial applications the number of observations is large. Therefore, as the t-distribution approaches the normal distribution when the degrees of freedom increases, $t_p \approx z_p$ such that for the 95% interval $t_{.025} = 1.96 \approx 2$ and for the 99% interval $t_{.005} = 2.58$. These t-statistics are interpreted as 5% and 1% significance levels, respectively. If t-statistics reach these levels or higher, they indicate that the estimated beta coefficient is statistically significant. Notably, even if the coefficient is statistically significant, it may not be economically significant. The magnitude of the beta coefficient needs to be large enough to suggest that a change in the independent variable will result in a economically meaningful change in the dependent variable.

If the normality of the error term does not hold, the Central Limit Theorem (CLT) provides us with so-called large sample results under fairly mild assumptions. The CLT implies that the above normal distribution results can be reliably used when the sample size is reasonably large. There is no exact definition of how large the sample should be. In general, with as small as 50 observations, the approximation is fairly close for practical purposes.

As discussed earlier in the chapter, if the error terms are heteroskedastic and/or autocorrelated, the major implication is bias in the standard errors leading in most cases to overconfidence about the precision of the estimates. For cross-sectional data, the White (1980) heteroskedastic robust variance estimators (HC) are routinely used to correct the bias. In time-series regression, the Newey and West  (1987) heteroskedastic autocorrelation robust measure (HAC) is used to correct standard errors. And, if data are clustered, Cameron, Gelbach, and Miller (2011) clustering robust standard errors can be used to take into account the correlation of error terms.

As an example, we apply HAC standard errors to our earlier example of Microsoft CAPM estimates in Sect. 4.3. For this purpose, we need more sophisticated regression software than provided by Excel. R (https://r-project.org) is a powerful open source software for general statistical analyses as well as a wide variety of regression estimation methods.

Below is R-code for the estimation in which we assume that the data is in an Excel file named msft.xlsx with the above headings on the first line.

We observe that the HAC standard is just slightly larger than those of OLS, which indicates that there is no material heteroskedasticity and/or autocorrelation in the OLS residuals. Therefore, the OLS results are equally reliable as those produced by the HAC approach, and the conclusions of Sect. 4.3 pass this robustness check.