FRM Exam Part I · Regression with Multiple Explanatory Variables

Hypothesis Testing and Confidence Intervals for Regression Coefficients

Updated 11 October 2026 · Fact-checked

To test one coefficient in a multiple regression, compute t = (estimated coefficient − hypothesized value) ÷ standard error. Compare it with the critical t-value with n − k − 1 degrees of freedom, or compare the p-value with the significance level. A confidence interval is the estimate ± critical t × standard error.

Understand Hypothesis Testing and Confidence Intervals for Coefficients

A multiple regression gives you an estimated coefficient for each explanatory variable. These estimates come from a sample, so they carry sampling error. If you drew another sample, you would get slightly different numbers. The standard error of a coefficient measures how much that estimate would vary from sample to sample.

To ask whether a variable really matters, you test a hypothesis about its true coefficient. The usual null hypothesis is H0: β = 0, meaning that, holding the other variables fixed, the variable has no linear effect on the dependent variable. The alternative is H1: β ≠ 0 for a two-sided test. You convert the estimate into a t-statistic, which counts how many standard errors the estimate lies from the hypothesized value.

If the t-statistic is large in absolute value, the estimate is far from the null value relative to its noise. You then reject H0. The rejection rule uses the Student's t distribution with n − k − 1 degrees of freedom, where n is the number of observations and k is the number of explanatory variables (excluding the intercept). With large samples, the critical value for a 5% two-sided test is close to 1.96.

The p-value is the smallest significance level at which you could reject H0. It is the probability of seeing a t-statistic at least this extreme if H0 were true. It is not the probability that H0 is true. A confidence interval gives the range of hypothesized values you would not reject. If a 95% interval for β excludes 0, a two-sided 5% test rejects H0: β = 0.

In a multiple regression, each t-test is conditional on the other variables being in the model. A variable can look insignificant alone and significant in a different model, particularly when explanatory variables are highly correlated. To test several coefficients together, use the F-test, not a series of t-tests.

Key formulas to remember

t-statistic for a coefficient: t = (β̂ − β₀) ÷ SE(β̂); β₀ is the hypothesized value. It is 0 in the standard significance test.
Degrees of freedom: df = n − k − 1; k is the number of slope coefficients. The intercept uses one more degree of freedom.
Confidence interval for a coefficient: β̂ ± t_critical × SE(β̂); Use the two-sided critical value for the chosen confidence level, e.g. 5% total in both tails for 95%.
Decision rule (two-sided): Reject H0 if |t| > t_critical, or if p-value < α; The two rules always agree.
Approximate large-sample critical values: 90%: 1.645; 95%: 1.96; 99%: 2.576; These are normal values, a good approximation to t when df is large. For small df, use the t table given.

How to solve Hypothesis Testing and Confidence Intervals for Coefficients questions

Use the same sequence for any question on a single coefficient, whether it asks for a test, a p-value interpretation or an interval.

1Write the hypotheses: H0: β = β₀ (usually 0) and H1: β ≠ β₀, or a one-sided alternative if the question states a direction.
2Find the estimate β̂ and its standard error from the regression output.
3Compute t = (β̂ − β₀) ÷ SE. Keep the sign.
4Work out df = n − k − 1 and pick the critical t for the stated significance level. Halve α for each tail in a two-sided test.
5Decide: reject H0 if |t| exceeds the critical value (or the p-value is below α). Otherwise fail to reject.
6For an interval, compute β̂ ± t_critical × SE and check whether β₀ lies inside it.
7State the conclusion in words: the variable is or is not statistically significant, holding the other variables constant.

Quickest way: The rule-of-two shortcut

When to use it: Use when df is reasonably large (about 30 or more) and the question asks whether a coefficient is significant at 5%, or which option is correct.

Divide the coefficient by its standard error.
If |t| is clearly above about 2, reject H0 at 5%. If it is clearly below 1.96, do not reject.
For a 95% interval, go from β̂ − 2×SE to β̂ + 2×SE as a quick estimate, then match the nearest option.
If the value is close to the threshold or df is small, use the exact t critical value given in the question.
Check whether the interval contains 0 to confirm your decision.

Common mistakes in Hypothesis Testing and Confidence Intervals for Coefficients

Using n − 1 or n − 2 degrees of freedom in a multiple regression.
Students carry over the simple regression or sample mean formulas.
Fix: Use n − k − 1, where k counts all slope coefficients.
Reading the p-value as the probability that H0 is true.
The wording sounds like a probability about the hypothesis.
Fix: The p-value is the probability of a result at least this extreme if H0 were true. Reject H0 when p < α.
Forgetting to divide α by two for a two-sided test or confidence interval.
Students look up the 5% column instead of 2.5% per tail.
Fix: For a 95% interval, use 2.5% in each tail. Check whether the table is one-tailed or two-tailed.
Subtracting zero when the null value is not zero.
Habit from testing significance.
Fix: If the null is, for example, β = 1 (as for a market beta), compute t = (β̂ − 1) ÷ SE.
Concluding that a variable is unimportant because the t-test fails to reject.
Fail to reject is treated as proof of no effect.
Fix: It means the data do not provide enough evidence. High multicollinearity or a small sample can inflate standard errors.
Using several t-tests to test whether multiple coefficients are jointly zero.
It seems natural to test each variable.
Fix: Use the F-test for joint hypotheses. Individual t-tests do not control the overall error rate.

Worked examples

Example 1

A regression of a fund's monthly excess return on the market excess return and a size factor uses 64 observations. The market coefficient is 1.12 with a standard error of 0.05. Test H0: β = 1 against H1: β ≠ 1 at the 5% level. Use a critical t-value of 2.00 (df = 61).

Show the solution

Hypotheses: H0: β = 1; H1: β ≠ 1.
Degrees of freedom: n − k − 1 = 64 − 2 − 1 = 61.
t = (1.12 − 1) ÷ 0.05 = 0.12 ÷ 0.05 = 2.40.
Compare: |2.40| > 2.00, so reject H0.

Answer: Reject H0. The market beta is statistically different from 1 at the 5% level (t = 2.40).

Example 2

In a multiple regression with n = 43 observations and 3 explanatory variables, the estimated coefficient on variable X₂ is 0.80 with a standard error of 0.30. The two-sided 95% critical t-value for df = 39 is 2.02. Construct the 95% confidence interval for the coefficient and say whether X₂ is significant at 5%.

Show the solution

Degrees of freedom: 43 − 3 − 1 = 39, matching the given critical value.
Margin of error = 2.02 × 0.30 = 0.606.
Lower limit = 0.80 − 0.606 = 0.194.
Upper limit = 0.80 + 0.606 = 1.406.
Zero lies outside the interval (0.194 to 1.406), so reject H0: β = 0.
Cross-check: t = 0.80 ÷ 0.30 = 2.67, which exceeds 2.02.

Answer: The 95% confidence interval is 0.194 to 1.406. Because it excludes 0, X₂ is statistically significant at the 5% level.

Exam tips

Check what the null value is. A question about beta equal to 1 needs (β̂ − 1), not β̂ alone.
Read whether the question gives a one-sided or two-sided test, and whether the table value is for one or two tails.
Questions often give a p-value and ask for a conclusion. Compare it with α and answer in words without calculating anything.
Use the confidence interval as a check. If the interval excludes the null value, the two-sided test must reject.
When several answer options differ only in the decision, calculate t first. It usually eliminates two options quickly.

Practice questions from Regression with Multiple Explanatory Variables

Hypothesis Testing and Confidence Intervals for 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 and Confidence Intervals for Coefficients: frequently asked questions

How do I calculate a confidence interval for a regression coefficient?

Take the estimated coefficient and add and subtract the critical t-value times its standard error. The critical value uses n − k − 1 degrees of freedom and the two-sided significance level. For a 95% interval with a large sample, the critical value is about 1.96.

What does a p-value mean in a regression?

It is the probability of getting a t-statistic at least as extreme as the one observed if the null hypothesis were true. A small p-value, below your chosen significance level, leads you to reject the null. It does not give the probability that the null is true.

Why are degrees of freedom n − k − 1 in multiple regression?

You lose one degree of freedom for each estimated parameter. That is k slope coefficients plus the intercept. This leaves n − k − 1 for estimating the error variance.

When do I use a t-test and when an F-test in FRM Part I?

Use a t-test for one coefficient. Use an F-test when the null involves more than one restriction at once, such as several coefficients all equal to zero.