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 (SST) splits into the explained part (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 estimate in the next lesson, it has no units.

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). The analysis of variance (ANOVA) table in the next lesson builds it. It is a one-tailed, right-tailed test with 1 and $n-2$ degrees of freedom.

For the retailers, MSR = 62.727 and MSE = 2.7733 ÷ 4 = 0.6933, so $F = 62.727 \div 0.6933 = 90.47$. 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.

$$s_{b_1} = \frac{SEE}{\sqrt{\sum (X_i-\bar{X})^2}}$$

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.

$$t = \frac{b_1 - B_1}{s_{b_1}}, \qquad df = n-2$$

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 F-statistic and the slope’s t-statistic always agree in simple regression.

$$F = t_{b_1}^{\,2}$$

What about the intercept?

The test of the intercept works the same way, with $t = (b_0 - B_0) \div s_{b_0}$ and $n-2$ degrees of freedom. The standard error of the intercept is usually given in the output. For the retailers $s_{b_0} = 0.8215$ and $b_0 = 1.3867$, so $t = 1.3867 \div 0.8215 = 1.69$, which is below 2.776. The intercept is not significantly different from zero.

Is the regression any good?fit first, then tests

• Goodness of fit Memorize how much of Y the line explains
  • R² = SSR ÷ SST unit-free share, 0 to 1
  • F-test of the model H₀: slope is zero; one-tailed, df 1 and n − 2
• Tests of the coefficients Memorize t-tests with n − 2 degrees of freedom
  • Slope t = (b₁ − B₁) ÷ s_b₁
  • Intercept t = (b₀ − B₀) ÷ s_b₀
• 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

When X is an indicator variable

An indicator variable (or dummy) takes only the values 0 and 1. Suppose X = 1 on days when a central bank holds a press conference and 0 otherwise, and Y is the size of a currency pair’s daily move. The fitted line is Move = 0.35% + 0.40% × X.

• The intercept, 0.35%, is the average move on days without a press conference (X = 0).
• The slope, 0.40%, is the difference in averages, so press-conference days average 0.35% + 0.40% = 0.75%.
• The t-test of the slope is therefore a test of whether the two group means differ.

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

Go deeper Why F equals t squared only in simple regression

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.

Go deeper Testing a stock's beta against 1

The slope of a stock’s returns on the market’s is its beta. A portfolio manager often asks whether beta differs from 1, meaning the stock is not simply moving with the market. That is a slope test with $B_1 = 1$, exactly as in the retailer example above. A beta of 0.84 can look low 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.