CFA Level I Exam · Applications of Simple Linear Regression in Finance

Hypothesis Testing of Regression Coefficients in CFA Level 1

Updated 7 October 2026 · Fact-checked

To test a regression coefficient, set H0 (often slope = 0), compute t = (estimated coefficient − hypothesized value) ÷ standard error of the coefficient, and compare it with the critical t-value at n − 2 degrees of freedom. Reject H0 if |t| exceeds the critical value or if the p-value is below the significance level.

Understand Hypothesis Testing of Regression Coefficients

A regression gives you an estimated slope and intercept from a sample. Because samples vary, the estimates are noisy. Hypothesis testing asks whether the true coefficient could plausibly be a specific value, most often zero.

The usual test is on the slope. If the true slope is zero, the independent variable has no linear relationship with the dependent variable. The null hypothesis is H0: B1 = 0, and the alternative is H1: B1 ≠ 0. You can also test a different value, such as B1 = 1 for a stock's beta.

The test statistic is a t-statistic. It measures how many standard errors the estimated slope lies from the hypothesized value. The standard error of the slope, s_b1, is given in the question or in the regression output. In simple linear regression the degrees of freedom are n − 2, because two parameters (slope and intercept) are estimated.

You can reach the same decision three ways. First, compare |t| with the critical value. Second, check whether the hypothesized value lies inside the confidence interval, estimate ± critical t × standard error. Third, compare the p-value with the significance level α. The p-value is the smallest significance level at which you could reject H0. Reject when p < α.

You can also test the correlation. The test of H0: ρ = 0 uses t = r√(n − 2) ÷ √(1 − r²) with n − 2 degrees of freedom. In simple regression this gives the same t-value as the test of the slope against zero. The same logic applies to the intercept, though the exam focuses on the slope.

Key formulas to remember

t-statistic for a coefficient: t = (b̂1 − B1) ÷ s_b1; B1 is the hypothesized slope, often 0. For the intercept use b̂0, B0 and s_b0. Degrees of freedom = n − 2.
Standard error of the slope: s_b1 = s_e ÷ √Σ(Xi − X̄)²; s_e is the standard error of the estimate. A larger s_e raises the slope's standard error. More spread in X lowers it.
Confidence interval for the slope: b̂1 ± t_c × s_b1; t_c is the critical value for n − 2 degrees of freedom. If the hypothesized value lies outside the interval, reject H0 at the matching significance level.
Decision rule (critical value): Reject H0 if |t| > t_c (two-tailed); 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; The p-value is the smallest significance level at which H0 can be rejected.
t-test for correlation: t = r√(n − 2) ÷ √(1 − r²); Tests H0: ρ = 0 with n − 2 degrees of freedom. It equals the slope t-statistic in simple regression.
Link to ANOVA: F = t² (slope, simple regression); The F-test of the slope and the two-tailed t-test give the same conclusion. Also R² = r².

How to solve Hypothesis Testing of Regression Coefficients questions

Use this sequence for any question that asks you to test or interpret a regression coefficient.

1State H0 and H1. Most often H0: B1 = 0 and H1: B1 ≠ 0. Note whether the test is one-tailed or two-tailed.
2Find the estimated coefficient, its standard error and the sample size n. Set the degrees of freedom to n − 2.
3Compute t = (estimate − hypothesized value) ÷ standard error.
4Find the critical t-value for the significance level and degrees of freedom. Use the right tail count (one or two tails).
5Compare. Reject H0 if |t| exceeds the critical value, or if the p-value is below α. For a one-tailed test, check the sign too.
6Or build the confidence interval, estimate ± t_c × standard error, and see whether the hypothesized value falls inside it.
7State the conclusion in words: the slope is or is not statistically different from the hypothesized value at that significance level.

Quickest way: Estimate ÷ standard error, then compare with the critical value

When to use it: Use this when the question gives the coefficient and its standard error, or when the answer options differ by the conclusion.

Divide the estimate (minus the hypothesized value) by its standard error. This is t.
Check n − 2 and look at the critical value given in the question. If none is given and df is large, a two-tailed 5% critical value is a little above 2.
If |t| is clearly bigger than the critical value, reject. If it is clearly smaller, do not reject.
If the question gives a p-value, skip the t-table. Reject only if p is below α.
If options include a confidence interval, check whether it contains the hypothesized value. Zero inside the interval means do not reject H0: B1 = 0.

Common mistakes in Hypothesis Testing of Regression Coefficients

Using n − 1 degrees of freedom instead of n − 2.
n − 1 is the familiar df for a single sample mean.
Fix: In simple linear regression two parameters are estimated, so df = n − 2 for coefficient and correlation tests.
Testing against zero when the question hypothesizes another value, such as a beta of 1.
Most practice questions use H0: B1 = 0, so the formula gets memorized without the hypothesized value.
Fix: Always subtract the hypothesized value from the estimate before dividing by the standard error.
Saying 'accept H0' when |t| is below the critical value.
Students treat a failure to reject as proof.
Fix: Say 'fail to reject H0'. The sample did not provide enough evidence of a difference.
Reading the p-value the wrong way round, for example rejecting H0 because the p-value is large.
Confusion between probability of the data under H0 and the size of the effect.
Fix: A small p-value is evidence against H0. Reject when p < α.
Using a two-tailed critical value for a one-tailed test, or the reverse.
The tail count is hidden in the wording, such as 'greater than' or 'different from'.
Fix: 'Different from' is two-tailed. 'Greater than' or 'less than' is one-tailed, with the whole α in one tail.
Confusing the standard error of the estimate (s_e) with the standard error of the slope (s_b1).
Both are called standard errors and appear in the same output.
Fix: The t-test for a coefficient always divides by the coefficient's own standard error, s_b1 (or s_b0 for the intercept).

Worked examples

Example 1

An analyst regresses a stock's excess returns on market excess returns using 27 monthly observations. The estimated slope (beta) is 1.35 and its standard error is 0.20. She tests H0: B1 = 1 against H1: B1 ≠ 1 at the 5% significance level. The critical t-value is 2.060 for the relevant degrees of freedom. Which conclusion is correct? A) The t-statistic is 1.75, so reject H0. B) The t-statistic is 1.75, so fail to reject H0. C) The t-statistic is 6.75, so reject H0.

Show the solution

Degrees of freedom = n − 2 = 27 − 2 = 25. The critical value 2.060 matches.
t = (1.35 − 1) ÷ 0.20 = 0.35 ÷ 0.20 = 1.75.
Compare: |1.75| < 2.060, so the statistic is inside the non-rejection region.
Cross-check with the confidence interval: 1.35 ± 2.060 × 0.20 = 1.35 ± 0.412, which is 0.938 to 1.762. The value 1 is inside it, so H0 is not rejected.
Option C gets 6.75 by dividing 1.35 by 0.20, which ignores the hypothesized value of 1.

Answer: B. t = 1.75 is below the critical value of 2.060, so you fail to reject H0. The beta is not statistically different from 1 at the 5% level.

Example 2

A sample of 20 paired observations of two return series has a sample correlation of 0.45. Test H0: ρ = 0 against H1: ρ ≠ 0 at the 5% level. The two-tailed critical t-value for 18 degrees of freedom is 2.101. What is the calculated t-statistic and the decision? A) t = 2.14, reject H0. B) t = 2.25, reject H0. C) t = 2.39, reject H0.

Show the solution

Degrees of freedom = n − 2 = 18.
Numerator: r√(n − 2) = 0.45 × √18 = 0.45 × 4.2426 = 1.909.
Denominator: √(1 − r²) = √(1 − 0.2025) = √0.7975 = 0.8930.
t = 1.909 ÷ 0.8930 = 2.14.
Compare: 2.14 > 2.101, so reject H0.
Option B comes from using n instead of n − 2 under the root. Option C comes from forgetting the square root in the denominator.

Answer: A. t ≈ 2.14 exceeds 2.101, so you reject H0. The correlation is statistically significantly different from zero at the 5% level.

Exam tips

Because options are listed from smallest to largest, work out t yourself first, then pick the matching value. Distractors often come from using n or n − 1 for df, or from skipping the hypothesized value.
Check the hypothesized value in the stem. If it is not zero, subtract it before dividing.
When a confidence interval is given, use it as a shortcut. If the hypothesized value is inside, you fail to reject.
Remember that in simple regression the slope t-test and the correlation t-test agree, and F = t². This can save a calculation.
Do not confuse statistical significance with economic significance. A tiny slope can be significant with a large sample, and the question may ask about practical meaning.

Practice questions from Applications of Simple Linear Regression in Finance

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 whether a regression slope is significant?

Set H0: B1 = 0, compute t = b̂1 ÷ s_b1, and compare |t| with the critical value at n − 2 degrees of freedom. If |t| is larger, reject H0 and conclude the slope is significantly different from zero.

What is the confidence interval for a slope coefficient?

It is b̂1 ± t_c × s_b1, where t_c is the critical t-value with n − 2 degrees of freedom. If the hypothesized value lies outside the interval, you reject H0 at the matching significance level.

Should I use the p-value or the critical value?

Both give the same decision. Use the p-value when the output provides it: reject if p is less than α. Use the critical value when you are given the t-table or a critical number.

Does the t-test of the slope relate to R-squared?

Yes. In simple linear regression, the slope t-statistic squared equals the F-statistic, and R² equals the squared correlation. A significant slope means the regression explains a significant part of the variation in the dependent variable.