Goodness of fit and testing the regression coefficients

How well does the line fit?

The coefficient of determination , $R^2$, is the share of the variation in Y that the model explains. The total variation, the sum of squares total (SST), splits into the explained part, the sum of squares regression (SSR), and the unexplained part, the sum of squares error (SSE), so SST = SSR + SSE.

$$R^2 = \frac{SSR}{SST} = 1 - \frac{SSE}{SST}$$

In simple regression $R^2$ is also the square of the correlation between X and Y. Unlike the standard error of the estimate in the next lesson, it has no units. Three numbers summarize fit: $R^2$, the F-statistic and the standard error of the estimate ($s_e$, also called SEE; lesson 4). $R^2$ describes the fit; it is not a test of significance.

Going the other way, the correlation is the square root of $R^2$, and it takes the sign of the slope.

$$r = \pm\sqrt{R^2} \quad (\text{same sign as } \hat{b}_1)$$

Is the model significant? The F-test

The F-test of the model asks whether the regression explains anything at all. In simple regression the null is that the slope is zero. The statistic is the explained variation per slope divided by the unexplained variation per remaining degree of freedom: the mean square regression (MSR) divided by the mean square error (MSE). It is a one-tailed, right-tailed test with 1 and $n-2$ degrees of freedom.

Read F as “big is bad for the null”: a large F rejects the claim that the model explains nothing, and an F near zero means it explains little. Lesson 4 builds F from the analysis of variance (ANOVA) table.

For the retailers, MSR = 62.727 and MSE = 2.7733 ÷ 4 = 0.69333, so $F = 62.727 \div 0.69333 = 90.47$. The 5% critical value of F with 1 and 4 degrees of freedom is 7.71, so reject: the model is significant.

How do you test the slope?

Each coefficient has an estimate and a standard error. The standard error of the slope coefficient falls when the residuals are small and when X is spread out widely.

$$s_{\hat{b}_1} = \frac{s_e}{\sqrt{\sum (X_i-\bar{X})^2}}$$

The t-test of the slope then asks how many standard errors the estimate sits from the hypothesized slope $B_1$. The test has $n-2$ degrees of freedom because two coefficients were estimated.

$$t = \frac{\hat{b}_1 - B_1}{s_{\hat{b}_1}}, \qquad df = n-2$$

If $|t|$ exceeds the critical value, reject $H_0$. Software also reports a p-value : the smallest significance level at which you would reject. Reject when the p-value is below your chosen $\alpha$.

The alternative can be two-sided (the slope differs from $B_1$) or one-sided (it is above, or below, $B_1$); a one-sided test puts all of $\alpha$ in one tail. Lowering $\alpha$ cuts the risk of a Type I error (rejecting a true null) but raises the risk of a Type II error (missing a false one).

With one independent variable, the F-test and the slope’s t-test against zero reach the same conclusion. With several independent variables, only F tests the slopes together.

What about the intercept?

The test of the intercept works the same way, with $t = (\hat{b}_0 - B_0) \div s_{\hat{b}_0}$ and $n-2$ degrees of freedom. Output tables give the standard error of the intercept ; you can also build it:

$$s_{\hat{b}_0} = s_e\sqrt{\frac{1}{n} + \frac{\bar{X}^2}{\sum (X_i-\bar{X})^2}}$$

Is the regression any good?fit first, then tests

• Goodness of fit Memorize how much of Y the line explains
  • R² = SSR ÷ SST explained share, 0 to 1; describes, does not test
  • F-test of the model H₀: slope is zero; a large F rejects
  • s_e typical miss in the units of Y (lesson 4)
• Tests of the coefficients Memorize t-tests with n − 2 degrees of freedom
  • Slope t = (b̂₁ − B₁) ÷ s_b̂₁
  • Intercept t = (b̂₀ − B₀) ÷ s_b̂₀; unbiased forecasts: intercept 0, slope 1
• Reading the result Use it compare with the critical value or p-value
  • |t| above critical value, or p below α reject H₀: the coefficient differs from the hypothesized value
  • |t| at or below critical value fail to reject H₀

Are the forecasts unbiased?

Regress actual outcomes (Y) on forecasts (X). Unbiased forecasts mean an intercept of 0 and a slope of 1, so test each against its own hypothesized value. This is the test of unbiased forecasts .

Testing whether two group means differ

Lesson 1 showed that with an indicator variable as X, the slope is the difference between the two group means. Its t-test is therefore a test of whether the means differ.

Exam tip When a question asks whether beta “differs from 1”, write the subtraction down before you divide.

Go deeper Why the F-test and the t-test agree here

The F-test checks whether all slope coefficients are jointly zero. With a single slope, that is the same claim the slope’s t-test makes, and the square of a t-statistic with $n-2$ degrees of freedom follows the F-distribution with 1 and $n-2$. With several independent variables, F tests all slopes at once and no single t-statistic equals it. You can check it on the retailers: with t to four decimals, $9.5116^2 \approx 90.47$. Treat it as a check, not a formula to learn.

Go deeper Testing a stock's beta against 1

Regress a stock’s excess returns on the market’s excess returns and the slope is its beta (lesson 7). Asking whether beta differs from 1 is a slope test with $B_1 = 1$, exactly as above. A beta of 1.22 can look high but still be statistically indistinguishable from 1 when the standard error is large.

Go deeper Statistically significant versus economically meaningful

With a large sample, a tiny slope can have a huge t-statistic. Statistical significance tells you the slope is unlikely to be zero, not that it is large enough to matter for a decision. Judge the size of the coefficient in its units alongside its t-statistic.