# 27

# Understanding Regressions

## 27.1 KEY TAKEAWAYS

- A low
*R*-squared does not mean the model is flawed.
- Statistically significant observed relationships do not ensure causality.
- Interpreting regressions hinges on the nature of underlying observations.

There are a handful of key statistics used to describe linear regressions. Two of these define the line itself: The slope (beta) and *y*-intercept (alpha). Some statistics describe the “fit” or explanatory power of the model: *r* (correlation coefficient) and *R*-squared (the coefficient of determination). Other important statistics measure the precision or reliability of the model and its estimates under normal conditions, such as standard errors and t-statistics.

We use the calculation of a stock's beta, per the traditional CAPM approach, to estimating cost of equity, to frame our discussion of the key statistical measures related to linear regression analysis.

**Figure 27.1** Microsoft (MSFT) vs. S&P 500 Index returns: Jan '96–Dec '00

Figure 27.1 shows monthly Microsoft returns (on the vertical *y*-axis) plotted against monthly total returns of the S&P 500 Index (on the horizontal x-axis). It is common practice in regression analysis to present the dependent variable on the *y*-axis and the independent variable on the x-axis. Keep in mind that a regression equation only captures the degree of common variation among variables; correlation does not ...