Regression Fit, ANOVA, and Coefficient Tests

Updated for the 2026-2027 CFA® Level I curriculum.

Regression analysis involves more than estimating a line. You also need to assess how closely the line fits the data and whether the relationship is statistically significant.

For CFA Level I, these ideas come together through the sums of squares, , the standard error of estimate, the ANOVA table, the F-test, and coefficient t-tests. Once you understand how the sums of squares connect, the rest of the calculations become much easier to organize.

Key Takeaways About Regression Fit, ANOVA, and Coefficient Tests

• Total variation divides into explained and unexplained variation.

• The sums-of-squares relationship is .

• measures the proportion of total variation explained by the regression.

• The standard error of estimate measures the typical size of a residual in the units of .

• In simple linear regression, regression degrees of freedom equal 1.

• Error degrees of freedom equal .

• The ANOVA table converts sums of squares into mean squares.

• The F-statistic equals mean square regression divided by mean square error.

• A coefficient t-test compares an estimated coefficient with a hypothesized value.

• Statistical significance does not automatically mean the relationship is economically important.

• In a simple regression, when the slope is tested against zero using a two-tailed t-test.

• Hypothesis-test conclusions should use “reject” or “fail to reject” the null hypothesis.

What You Need to Know for CFA Level I

For CFA Level I, focus on:

• Calculating , , and .

• Using the identity .

• Calculating and interpreting the coefficient of determination.

• Calculating and interpreting the standard error of estimate.

• Completing a simple-regression ANOVA table.

• Calculating mean square regression and mean square error.

• Formulating and evaluating an F-test of overall regression significance.

• Formulating and evaluating a t-test for a regression coefficient.

• Using the correct degrees of freedom.

• Explaining the relationship between the slope t-test and the F-test.

The 2026 curriculum requires candidates to calculate and interpret regression fit measures, evaluate tests of fit and regression coefficients, interpret ANOVA results, and calculate the standard error of estimate.

How Is Variation Divided in Simple Linear Regression?

A regression model attempts to explain why the dependent variable differs across observations.

Some of that variation is explained by changes in the independent variable. The remaining variation appears in the residuals.

Three sums of squares describe this division.

Total Sum of Squares

The total sum of squares, , measures the total variation in the observed values of around their sample mean.

Regression Sum of Squares

The regression sum of squares, , measures the variation explained by the fitted regression model.

Error Sum of Squares

The error sum of squares, , measures the variation that remains unexplained.

Because the residual is:

The error sum of squares can also be written as:

For a regression that includes an intercept:

Where:

• = observed value of the dependent variable

• = fitted value of the dependent variable

• = sample mean of the dependent variable

• = residual for observation

• = number of observations

• = total sum of squares

• = regression sum of squares

• = error sum of squares

This identity gives you a useful arithmetic check. If two sums of squares are provided, you can calculate the third immediately.

What Is the Coefficient of Determination?

The coefficient of determination, written as , measures the proportion of total variation in explained by the regression.

Using , you can also calculate it as:

In a regression with an intercept, ranges from 0 to 1.

For example:

This means the regression explains 64% of the sample variation in the dependent variable. The remaining 36% appears in the residuals.

What R-Squared Does Not Tell You

A high does not confirm that:

• The independent variable causes changes in the dependent variable.

• The regression assumptions are satisfied.

• The chosen functional form is appropriate.

• The coefficient is economically meaningful.

• Predictions outside the sample will be accurate.

A model can have a high and still show curvature, changing variance, dependent errors, or influential observations.

What Is the Standard Error of Estimate?

The standard error of estimate measures the dispersion of the observed values around the fitted regression line.

For simple linear regression:

Because mean square error is:

The standard error of estimate can also be written as:

Where:

• = standard error of estimate

• = error sum of squares

• = number of observations

• = mean square error

The denominator is because the regression estimates two coefficients: the intercept and slope.

The standard error of estimate is expressed in the same units as the dependent variable. If is measured in percentage points, is also measured in percentage points.

Suppose:

You can interpret this as a typical regression error of approximately 1.58 units of . It summarizes the overall residual spread rather than guaranteeing that every residual will fall within 1.58 units of the line.

R-Squared vs Standard Error of Estimate

helps you judge the share of variation explained. The standard error of estimate helps you judge the practical size of the model’s errors.

How Does the ANOVA Table Work?

The regression ANOVA table organizes variation, degrees of freedom, and mean squares.

The degrees of freedom follow this relationship:

Regression Degrees of Freedom

Simple linear regression has one independent variable, so regression degrees of freedom equal 1.

Error Degrees of Freedom

Two coefficients are estimated from the sample, so error degrees of freedom equal .

Total Degrees of Freedom

Total degrees of freedom equal .

Mean Square Regression

Mean square regression is:

For simple linear regression:

Mean Square Error

Mean square error is:

For simple linear regression:

The ANOVA table is easier to remember as a calculation sequence:

1. Divide total variation into and .

2. Assign the correct degrees of freedom.

3. Divide each sum of squares by its degrees of freedom.

4. Divide by to calculate the F-statistic.

How Is the F-Test Used in Simple Linear Regression?

The F-test evaluates whether the regression model is statistically significant overall.

In simple linear regression, there is one slope coefficient. The hypotheses are:

The null hypothesis states that the independent variable has no linear relationship with the dependent variable in the population.

The F-statistic is:

Substituting the mean-square formulas gives:

The F-statistic has:

• 1 numerator degree of freedom

• denominator degrees of freedom

The F-test uses the right tail of the F-distribution. A large F-statistic means the explained variation is large relative to the unexplained variation.

F-Test Decision Rule

Using a critical value:

• Reject when the calculated F-statistic exceeds the critical F-value.

• Fail to reject when the calculated F-statistic does not exceed the critical value.

Using a p-value:

• Reject when the p-value is below the significance level.

• Fail to reject when the p-value is at or above the significance level.

Rejecting the null supports the conclusion that the model has a statistically significant linear relationship. It does not tell you whether the relationship is large enough to matter economically.

How Are Regression Coefficients Tested?

A coefficient t-test evaluates whether an individual population coefficient equals a hypothesized value.

The general hypotheses are:

The t-statistic is:

Where:

• = estimated regression coefficient

• = coefficient value stated under the null hypothesis

• = standard error of the estimated coefficient

• = calculated t-statistic

For a slope test against zero:

The statistic becomes:

The test uses degrees of freedom in simple linear regression.

The Hypothesized Value May Be Nonzero

Zero is the most common null value because it tests whether a linear relationship exists. Some questions use another value.

Suppose an analyst tests whether a stock’s population beta equals 1:

The t-statistic is:

Always use the hypothesized value stated in the question.

Coefficient-Test Decision Rule

For a two-tailed test:

• Reject when the absolute calculated t-statistic exceeds the critical t-value.

• Fail to reject when the absolute calculated t-statistic does not exceed the critical value.

Using a p-value:

• Reject when the p-value is below the significance level.

• Fail to reject when the p-value is at or above the significance level.

A test conclusion should be stated in context. For example:

The evidence supports the conclusion that the population slope differs from zero.

Avoid saying that the null hypothesis has been “accepted.” Failing to reject it means the evidence was insufficient to reject the stated value.

How Are the F-Test and Slope t-Test Related?

In simple linear regression:

This relationship applies when:

• The regression contains one independent variable.

• The F-test evaluates overall regression significance.

• The t-test evaluates the slope.

• The slope null hypothesis is .

• The t-test is two-tailed.

Under these conditions, the two tests evaluate the same population relationship and must reach the same decision.

For example:

Then:

The F-statistic is always nonnegative because it is based on squared quantities. The t-statistic may be positive or negative depending on the estimated slope.

The identity does not generally extend to a multiple regression with several independent variables. In that setting, the overall F-test evaluates the slopes jointly, while each coefficient t-test evaluates one coefficient.

Worked Example: Stock Return and Market Return

An analyst regresses a stock’s monthly excess return on the market’s monthly excess return using 38 observations.

The regression output reports:

• Estimated slope:

• Standard error of the slope:

Step 1: Calculate the Error Sum of Squares

Step 2: Calculate the Coefficient of Determination

The model explains 64% of the sample variation in the stock’s excess return.

The unexplained proportion is:

The remaining 36% of the variation appears in the residuals.

Step 3: Calculate the Standard Error of Estimate

The standard error of estimate is approximately 1.58 percentage points.

Step 4: Complete the ANOVA Table

Mean square regression is:

Mean square error is:

Step 5: Test Overall Regression Significance

At the 5% significance level, the critical F-value for 1 and 36 degrees of freedom is approximately 4.11.

Because:

Reject the null hypothesis. The market’s excess return has a statistically significant linear relationship with the stock’s excess return.

Step 6: Test the Slope Coefficient

The hypotheses are:

Calculate the t-statistic:

At the 5% significance level, the two-tailed critical t-value with 36 degrees of freedom is approximately 2.028.

Because:

Reject the null hypothesis. The population slope differs significantly from zero.

Step 7: Confirm the F and t Relationship

The two tests agree because the regression contains one independent variable and the slope is being tested against zero.

Common Exam Traps

• Reversing and . is explained variation. is unexplained variation.

• Using as . That ratio is the unexplained proportion, or .

• Interpreting as prediction accuracy. It measures explained variation, not the percentage of observations predicted correctly.

• Treating a high as proof of causation. It provides no causal evidence on its own.

• Using the wrong degrees of freedom. Simple regression uses 1 for regression, for error, and for total.

• Dividing . by . The standard error of estimate uses .

• Confusing with . The standard error of estimate is the square root of .

• Treating statistical significance as economic importance. A coefficient can be statistically significant but too small to influence an investment decision.

• Using the F-test as a two-tailed test. The rejection region lies in the right tail of the F-distribution.

• Forgetting the hypothesized coefficient value. Subtract the value stated under the null, which may be different from zero.

• Saying “accept the null.” Use “fail to reject the null hypothesis.”

• Applying too broadly. The relationship requires one independent variable and a two-tailed slope test against zero.

Practice Question

An analyst estimates a simple linear regression using 26 observations and reports the following partial ANOVA table:

The coefficient of determination is closest to:

1. 0.47

2. 0.53

3. 0.88

Solution

FAQs About Regression Fit, ANOVA, and Coefficient Tests