CFA Level II Exam · Evaluating Regression Model Fit and Interpreting Model Results

Hypothesis Testing of Regression Coefficients in CFA Level II

Updated 7 October 2026 · Fact-checked

A regression coefficient test asks whether a slope differs from a hypothesized value, usually zero. Compute t = (estimated coefficient − hypothesized value) ÷ standard error, with n − k − 1 degrees of freedom. Reject the null 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 coefficient for each independent variable. These are sample estimates, so they carry error. Hypothesis testing asks whether the true coefficient in the population could plausibly be a specific value, most often zero.

The usual null hypothesis is H0: b = 0. If the true slope is zero, the variable adds nothing to explaining the dependent variable, holding the other variables constant. The alternative is H1: b ≠ 0 for a two-sided test. A one-sided test is used when the vignette states a directional view, such as a positive relationship.

The t-statistic measures how many standard errors the estimate sits from the hypothesized value. A large absolute t means the estimate is far from the null value relative to its noise. You compare it with a critical t-value from the t-distribution with n − k − 1 degrees of freedom, where n is the number of observations and k the number of slope coefficients.

The p-value is the smallest significance level at which you could reject the null. If the p-value is less than your chosen significance level (such as 5%), reject H0. It is not the probability that the null is true.

A confidence interval gives the range of hypothesized values you would not reject. It is the estimate plus or minus the critical value times the standard error. If a two-sided interval excludes zero, the coefficient is significant at the matching level. Exhibits usually give the coefficient, standard error, t-statistic and p-value, so most questions need only a small calculation or a correct reading.

Key formulas to remember

t-statistic for a coefficient: t = (b̂ − b₀) ÷ s(b̂); b̂ is the estimated coefficient, b₀ the hypothesized value (usually 0), s(b̂) the standard error. Exhibits often print t for b₀ = 0 only.
Degrees of freedom: df = n − k − 1; k is the number of independent variables, excluding the intercept.
Confidence interval: b̂ ± t(critical) × s(b̂); Use the two-sided critical value for the stated confidence level and df.
Decision rule (two-sided): Reject H0 if |t| > t(critical), or if p-value < α; For a one-sided test, use the one-tail critical value and check the sign of t.
Rearranged standard error: s(b̂) = b̂ ÷ t; Use when the exhibit gives the coefficient and its t-statistic for b₀ = 0 but not the standard error.

How to solve Hypothesis Testing of Regression Coefficients questions

Use the same sequence for any question on testing a coefficient. Pull the numbers from the exhibit first, then decide.

1Identify which coefficient is tested and write H0 and H1, including the hypothesized value and whether the test is one- or two-sided.
2Find the estimated coefficient and its standard error in the exhibit. If only t is given, check which null value it was computed for (normally 0).
3Compute df = n − k − 1 using the number of observations and slope variables in the vignette.
4Calculate t = (b̂ − b₀) ÷ s(b̂), or reuse the printed t if b₀ = 0.
5Get the critical t for the stated significance level and df, or compare the p-value with α.
6Decide: reject H0 if |t| exceeds critical value or p < α. State it in context.
7For an interval, compute b̂ ± t(critical) × s(b̂) and check whether the hypothesized value lies inside.

Quickest way: Rule of thumb check with p-value and interval

When to use it: Use when the exhibit prints p-values or the question asks only whether a coefficient is significant at a stated level.

Read the p-value for the coefficient and compare it with α. If p < α, reject H0 and stop.
If no p-value is printed, check |t| against the critical value given in the vignette. With large df, a two-sided 5% value is near 2.
For a hypothesized value other than zero, do not use the printed t. Recompute with (b̂ − b₀) ÷ s(b̂).
For an interval question, test whether the value falls between b̂ − t×s and b̂ + t×s.

Common mistakes in Hypothesis Testing of Regression Coefficients

Using the printed t-statistic to test a null value other than zero.
The exhibit's t is built for H0: b = 0, and it looks like the answer.
Fix: Recompute t = (b̂ − b₀) ÷ s(b̂) whenever the null is not zero.
Using n − 1 or n − k as degrees of freedom.
Students carry over the simple-mean rule from Level I.
Fix: Use n − k − 1. The extra 1 is for the intercept.
Reading the p-value as the probability that H0 is true.
The wording sounds like a probability of the hypothesis.
Fix: The p-value is the smallest significance level at which H0 is rejected. Just compare it with α.
Concluding a variable is unimportant when it is insignificant, or significant means large.
Significance is confused with economic size.
Fix: Significance means the estimate is distinguishable from the null given its standard error. Judge size by the coefficient itself.
Applying a one-sided critical value to a two-sided test, or ignoring the sign in a one-sided test.
Students look up the critical value before reading H1.
Fix: Write H1 first. For one-sided tests, the sign of t must match the alternative and exceed the one-tail critical value.
Interpreting a slope without 'holding other variables constant'.
Simple regression habits carry over.
Fix: In multiple regression, each slope is the effect of that variable with all others held constant.

Worked examples

Example 1

An analyst regresses a fund's monthly excess return on market excess return (MKT) and a size factor (SMB) using 60 observations. Estimates: intercept 0.20 (SE 0.15); MKT slope 1.10 (SE 0.08); SMB slope 0.25 (SE 0.12). The 5% two-sided critical t for 57 df is 2.00. (1) Is the SMB slope significant at 5%? (2) Test H0: MKT slope = 1 at 5%. (3) Is the intercept significant?

Show the solution

df = 60 − 2 − 1 = 57, so the critical value 2.00 applies.
(1) t for SMB = (0.25 − 0) ÷ 0.12 = 2.083. Since 2.083 > 2.00, reject H0.
(2) t for MKT against 1 = (1.10 − 1) ÷ 0.08 = 1.25. Since 1.25 < 2.00, do not reject H0.
(3) t for the intercept = 0.20 ÷ 0.15 = 1.333. This is below 2.00, so it is not significant.

Answer: (1) SMB slope is significant at 5%. (2) Cannot reject that the MKT slope equals 1. (3) Intercept is not significant.

Example 2

A regression of a company's quarterly sales growth on GDP growth uses 32 observations and three independent variables. The GDP growth slope is 0.80 with a standard error of 0.30. The 95% two-sided critical t for the relevant df is 2.048. (1) What are the df? (2) Construct the 95% confidence interval for the slope. (3) Can you reject H0: slope = 0 at 5%? Can you reject H0: slope = 1.5?

Show the solution

(1) df = 32 − 3 − 1 = 28.
(2) Margin = 2.048 × 0.30 = 0.6144. Interval = 0.80 ± 0.6144, which is 0.1856 to 1.4144.
(3) Zero is outside the interval, so reject H0: slope = 0. 1.5 is outside the interval (above 1.4144), so reject H0: slope = 1.5 too.
Check the second: t = (0.80 − 1.5) ÷ 0.30 = −2.333, and |−2.333| > 2.048, which agrees.

Answer: df = 28. The 95% interval is about 0.186 to 1.414. Reject both null hypotheses at the 5% level.

Exam tips

Check the null value first. If it is not zero, the printed t-statistic does not answer the question.
Compute df as n − k − 1 with k as slope count; vignettes often give n and the variables in different places.
If a p-value is printed, comparing it with α is the fastest route. Use it before any calculation.
A confidence interval question is also a hypothesis test: see whether the hypothesized value is inside the interval.
Eliminate options that call the p-value the probability that H0 is true or that equate significance with large economic effect.

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 coefficient is significant?

Set H0: b = 0, compute t = b̂ ÷ standard error, and compare |t| with the critical value for n − k − 1 degrees of freedom. If |t| is larger, reject H0 and call the coefficient significant. You can instead compare the p-value with your significance level.

What does the p-value mean in a regression output?

It is the smallest significance level at which you can reject the null hypothesis for that coefficient. A p-value of 0.03 means you reject at 5% but not at 1%. It is not the probability that the null is true.

How is a confidence interval linked to the t-test?

A two-sided confidence interval contains all hypothesized values you would not reject at the matching significance level. If zero is outside the 95% interval, the coefficient is significant at 5% in a two-sided test.

Why does the intercept also have a t-test?

The intercept is an estimated coefficient with its own standard error. You test it the same way, with H0: intercept = 0. Often it matters less in interpretation than the slopes, but the mechanics are identical.