CFA Level II Exam · Basics of Multiple Regression and Underlying Assumptions
Hypothesis Testing of Regression Coefficients for CFA Level II
Updated 7 October 2026 · Fact-checked
A regression coefficient test checks whether a slope differs from a hypothesized value, usually zero. Compute t = (estimated coefficient − hypothesized value) ÷ standard error, compare it with the critical t-value at n − k − 1 degrees of freedom, or compare the p-value with the significance level. Reject H0 if |t| exceeds the critical value.
Understand Hypothesis Testing of Regression Coefficients
A multiple regression gives you an estimated coefficient for each independent variable. These are estimates from a sample, so they carry error. A hypothesis test asks whether the true coefficient could plausibly equal some stated value, most often zero. If the true slope is zero, the variable adds nothing to explain the dependent variable, holding the other variables constant.
The standard error of a coefficient measures how much the estimate would vary from sample to sample. A small standard error relative to the coefficient means a precise estimate. The t-statistic converts the gap between the estimate and the hypothesized value into standard error units.
You compare the t-statistic with a critical value from the t-distribution. The degrees of freedom are n − k − 1, where n is the number of observations and k is the number of independent variables. For a two-tailed test at the 5% level with large degrees of freedom, the critical value is close to 1.96. Exact values will be given in the exhibit, so use what the vignette provides.
The p-value is the smallest significance level at which you could reject H0. If the p-value is less than your chosen significance level, reject H0. A p-value of 0.03 means you reject at 5% but not at 1%.
A confidence interval gives the range of hypothesized values you would not reject. If the interval excludes the hypothesized value, reject H0 at the matching significance level. This works for a two-tailed test.
Statistical significance is not the same as economic significance. A tiny coefficient can be statistically significant with a large sample. Also, these tests assume the regression assumptions hold. Under heteroskedasticity or serial correlation the standard errors, and so the t-statistics, are unreliable.
Key formulas to remember
- t-statistic for a coefficient
- t = (b̂ⱼ − bⱼ,H0) ÷ s(b̂ⱼ)
- b̂ⱼ is the estimated slope, bⱼ,H0 is the hypothesized value (often 0), s(b̂ⱼ) is its standard error.
- Degrees of freedom
- df = n − k − 1
- n observations, k independent variables. The +1 accounts for the intercept.
- Confidence interval for a coefficient
- b̂ⱼ ± t(critical) × s(b̂ⱼ)
- Use the two-tailed critical value for the chosen confidence level, with n − k − 1 degrees of freedom.
- Decision rule (t-test)
- Reject H0 if |t| > t(critical)
- For a one-tailed test, use the one-tailed critical value and check the sign of t.
- Decision rule (p-value)
- Reject H0 if p-value < α
- α is the significance level, such as 0.05.
How to solve Hypothesis Testing of Regression Coefficients questions
Use this method for any question on testing a single coefficient. Pull the numbers from the regression exhibit, then apply the rule.
- 1Identify the coefficient being tested and read its estimate and standard error from the exhibit.
- 2State H0 and Ha. Note the hypothesized value and whether the test is two-tailed or one-tailed.
- 3Compute degrees of freedom as n − k − 1 and find the critical t-value for the stated significance level.
- 4Compute t = (estimate − hypothesized value) ÷ standard error. Check whether the exhibit already gives t.
- 5Compare |t| with the critical value, or the p-value with α. State reject or fail to reject.
- 6For a confidence interval, compute estimate ± critical value × standard error, and see whether the hypothesized value lies inside.
- 7Translate the result into the answer in words: the variable is or is not significant, holding the others constant.
Quickest way: Rule-of-thumb significance check
When to use it: When the exhibit gives the estimate, standard error or t-statistic, and the question asks only whether a coefficient is significant.
- Divide the estimate by its standard error if t is not given.
- If the degrees of freedom are large, compare |t| with about 2 for a two-tailed 5% test, then confirm with the exhibit's critical value if the result is close.
- If a p-value is shown, just compare it with α and skip the t calculation.
- For a confidence interval question, check whether the interval contains the hypothesized value, usually zero.
- Watch for a hypothesized value other than zero. Then you must subtract it before dividing.
Common mistakes in Hypothesis Testing of Regression Coefficients
Using the wrong degrees of freedom, such as n − k instead of n − k − 1.
Students forget the intercept uses one degree of freedom.
Fix: Always write df = n − k − 1 and count k as the number of slope coefficients only.
Dividing the estimate by its standard error when H0 is not zero.
Zero is the usual hypothesized value, so students apply it automatically.
Fix: Subtract the hypothesized value from the estimate first, then divide.
Rejecting H0 when the p-value is larger than α.
Students confuse a large p-value with strong evidence.
Fix: A small p-value means strong evidence against H0. Reject only when p < α.
Using a two-tailed critical value for a one-tailed test.
Students overlook the direction in Ha.
Fix: Read Ha. If it says greater than or less than, use the one-tailed critical value and check that t has the right sign.
Interpreting a coefficient as the effect of the variable on its own.
Simple regression habits carry over.
Fix: In multiple regression, each slope is the change in the dependent variable per one-unit change in that variable, holding the other variables constant.
Treating a significant coefficient as proof the model is valid.
Students ignore assumption violations.
Fix: If the vignette mentions heteroskedasticity, serial correlation or multicollinearity, say the t-tests may be unreliable.
Worked examples
Example 1
An analyst regresses monthly excess returns of a fund on market excess return (MKT) and a size factor (SMB) using 62 observations. The estimated coefficient on SMB is 0.36 with a standard error of 0.15. Critical t-value for a two-tailed 5% test with 59 degrees of freedom is 2.00. (1) What are the degrees of freedom? (2) Is SMB significantly different from zero at 5%? (3) What is the 95% confidence interval?
Show the solution
- Degrees of freedom: n − k − 1 = 62 − 2 − 1 = 59, which matches the exhibit.
- t = (0.36 − 0) ÷ 0.15 = 2.40.
- Compare: 2.40 > 2.00, so reject H0.
- Confidence interval: 0.36 ± 2.00 × 0.15 = 0.36 ± 0.30.
- Lower limit = 0.06; upper limit = 0.66. Zero is outside the interval, which agrees with rejecting H0.
Answer: (1) 59. (2) Yes, t = 2.40 exceeds 2.00, so SMB is significant at 5%. (3) 0.06 to 0.66.
Example 2
A regression of a stock's return on 3 independent variables uses 43 observations. The coefficient on variable X2 is 0.80 with a standard error of 0.25. The analyst tests H0: coefficient = 0.50 against Ha: coefficient ≠ 0.50 at the 5% level. The two-tailed critical t-value with 39 degrees of freedom is 2.02. (1) Compute the test statistic. (2) State the conclusion. (3) Would a reported p-value of 0.24 be consistent with it?
Show the solution
- Degrees of freedom: 43 − 3 − 1 = 39, which matches the critical value given.
- t = (0.80 − 0.50) ÷ 0.25 = 0.30 ÷ 0.25 = 1.20.
- Compare: |1.20| < 2.02, so fail to reject H0.
- A p-value of 0.24 is greater than 0.05, so it also leads to failing to reject H0. A t of 1.20 with 39 degrees of freedom would give a two-tailed p-value of roughly 0.24, so it is consistent.
Answer: (1) t = 1.20. (2) Fail to reject H0; the data do not show the coefficient differs from 0.50 at 5%. (3) Yes, a p-value of 0.24 is above 0.05 and consistent with this result.
Exam tips
- Check whether the hypothesized value is zero. Questions often test a non-zero value to catch automatic division by the standard error.
- The exhibit usually prints t-statistics and p-values for each coefficient. Use them directly instead of recalculating.
- For answer choices that mention 'significant', confirm the significance level in the question, since a coefficient can be significant at 5% but not at 1%.
- Look for hints of assumption violations in the vignette. A question may ask whether the t-test conclusion is reliable.
- Confidence interval questions are really t-tests. If zero is inside the interval, the coefficient is not significant at that level.
Hypothesis Testing of Regression Coefficients in other exams
The same ground in other exams, if you are preparing for more than one or want another angle on it.
Hypothesis Testing of Regression Coefficients: frequently asked questions
How do I test if a slope coefficient is significant in multiple regression?
Compute t = estimate ÷ standard error for H0 that the slope is zero. Compare |t| with the critical value at n − k − 1 degrees of freedom. If |t| is larger, reject H0 and conclude the variable is significant, holding the others constant.
How do I interpret a p-value in multiple regression?
The p-value is the smallest significance level at which you can reject H0. If it is below your chosen α, such as 0.05, reject H0. A p-value of 0.03 rejects at 5% but not at 1%.
What is the formula for a confidence interval for a regression coefficient?
It is the estimated coefficient ± critical t-value × standard error of the coefficient. The critical value uses n − k − 1 degrees of freedom. If the interval excludes the hypothesized value, reject H0.
Does a significant t-test mean the variable matters economically?
No. Statistical significance only shows the coefficient is unlikely to be zero given the sampling error. With large samples, a very small effect can still be significant, so judge the size of the coefficient too.