- CFA Exams
- 2025 Level II
- Topic 1. Quantitative Methods
- Learning Module 2. Evaluating Regression Model Fit and Interpreting Model Results
- Subject 4. Confidence Intervals for Regression Coefficients in a Multiple Regression

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##### Subject 4. Confidence Intervals for Regression Coefficients in a Multiple Regression PDF Download

A 100 (1 - α)% confidence interval for a coefficient β

_{k}takes the form

_{k}- t

_{α/2, v}s

_{bk}, b

_{k}+ t

_{α/2, v}s

_{bk})

where s

_{bk}is the estimated standard deviation of bk, and t

_{α/2, v}is the critical value of the t distribution that has v = (T - K - 1) degrees of freedom such that P(t > t

_{α/2, v}) = α/2.

Let's construct 95% confidence intervals for each of the regression coefficients for the sample regression equation in los a.

We have T = 10 observations and K = 2 explanatory variables in the model, so the appropriate degrees of freedom is v = 10 - 2 - 1 = 7. To construct a 95% confidence interval, we need the critical values from the t distribution having 7 degrees of freedom. We obtain t

_{0.025, 7}= 2.365.

The desired confidence intervals are:

β

_{0}: 54.318 ± 2.365(2.485) or (48.441, 60.195)

β

_{1}: -4.013 ± 2.365(1.737) or (-8.121, 0.095)

β

_{2}: -15.981 ± 2.365(3.565) or (-24.412, -7.55)

Recall that the variable X

_{1}is engine size, so the coefficient b1 measures the estimated influence of engine size on gasoline mileage. We would expect this influence to be negative; that is, the larger the engine, the lower the gasoline mileage. However, observe that the 95% confidence interval for β1 contains the value 0. This implies that if we performed a two sided test using α = 0.05, we would not reject H

_{0}: β

_{1}= 0.

The confidence interval for β

_{2}is fairly wide, which implies that our sample of data has not provided a very precise estimate of the effect of car weight on gas mileage. Nevertheless, the confidence interval for β

_{2}does not contain the value 0, so we can feel quite confident that increasing a car's weight does have a negative effect on gas mileage.

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