- CFA Exams
- 2023 Level I
- Topic 1. Quantitative Methods
- Learning Module 2. Multiple Regression
- Subject 2. Testing the Significance of a Regression Coefficient

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##### Subject 2. Testing the Significance of a Regression Coefficient PDF Download

Let βt = (b Reject H Reject H Reject H

t-statistic = (estimated regression coefficient - hypothesized population value of the regression coefficient) / standard error of the regression coefficient

= estimated regression coefficient / standard error of the regression coefficient

= column 1 / column 2.

The hypothesized value is β

_{k}* denote some hypothesized value of the regression coefficient β_{k}. If the basic assumptions hold, then the appropriate test statistic is

_{i}- β

_{k}*)/s

_{bi}

Case 1: Suppose we wish to test the null hypothesis H

_{0}: β_{k}= β_{k}* or H_{0}: β_{k}≤ β_{k}* against the one-sided alternative hypothesis H_{1}: β_{k}> β_{k}* using a level of significance α. Use the decision rule

_{0}in favor of H

_{1}if t > t

_{α, v}

where the degrees of freedom of the Student t distribution is v = (T - K - 1).

Case 2: To test the null hypothesis H

_{0}: β_{k}= β_{k}* or H_{0}: β_{k}≥ β_{k}* against the one-sided alternative hypothesis H_{1}: β_{k}< β_{k}*, use the decision rule

_{0}in favor of H

_{1}if t < t

_{α, v}

Case 3: To test the null hypothesis H

_{0}: β_{k}= β_{k}* against the two-sided alternative hypothesis H_{1}: β_{k}≠ β_{k}*, use the decision rule

_{0}in favor of H

_{1}if t < - t

_{α/2, v}or t > t

_{α/2, v}

Continue with the example in los a. It is reasonable to conjecture either that engine size has no effect on gasoline mileage or that the larger the engine, the lower the mileage. Thus, it is natural to test the null hypothesis H

_{0}: β_{1}= 0 against the one-sided alternative hypothesis H_{1}: β_{1}< 0. Test this hypothesis using a 5% level of significance.The data is shown below:

The t-statistic column reports the results of a t-test on the hypothesis that the population value of the regression intercept or slope coefficient equals 0.

t-statistic = (estimated regression coefficient - hypothesized population value of the regression coefficient) / standard error of the regression coefficient

= estimated regression coefficient / standard error of the regression coefficient

= column 1 / column 2.

We have α = 0.05, T = 10, and K = 2. The appropriate degrees of freedom is v = 10 - 2 - 1 = 7. The critical value of t having 7 degrees of freedom is -t0.05, 7 = -1.895.

The hypothesized value is β

_{1}* = 0. To test H_{0}, we calculate the t statistic t = (b_{1}- 0)/s_{b1}= -4.0129 / 1.737 = -2.31.Because this observed t statistic is less than the critical value, it falls in the rejection region. Thus, if we use α = 0.05, we reject H

_{0}in favor of H_{1}. However, if we had used the slightly different level of significance α = 0.025, then the critical value of the test would have been -t_{0.025}= -2.365. In this case, the observed t statistic would have fallen in the acceptance region and we would not have rejected H_{0}.In the model, the variable X

_{2}measures the weight of the car. We would expect heavy cars to get lower gasoline mileage than light cars, so it is natural to test H_{0}: β_{2}= 0 against H_{1}: β_{2}< 0. The appropriate t statistic is -4.48. This t statistic falls in the critical region if we use a 5% level of significance, so we reject the null hypothesis.We showed that if we performed a one-tailed test using a 2.5% level of significance, we would fail to reject the null hypothesis that β

_{1}= 0, but if we used a 5% level of significance we would reject the null hypothesis. Now the question is: Should we keep the variable X_{1}in the model or remove it and express gasoline mileage solely as a function of car weight? This is a difficult question to answer, and the decision depends partly on the opinions of the investigator. In los o when we discuss what is called the multicollinearity problem, we shall get more insight into why it is difficult to determine whether β_{1}= 0.

**Learning Outcome Statements**

d. interpret the results of hypothesis tests of regression coefficients;

CFA® 2023 Level I Curriculum, Volume 1, Module 2

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**User Contributed Comments**
4

User |
Comment |
---|---|

IIkDII |
what is r in this equation ? |

rt2007 |
correlation coeeficient |

Tony1234 |
Last paragraph - we should never accept the null hypothesis - always failed to reject it. |

Oksanata |
what a strange definition of H alternative for b2 parameter... |

I am using your study notes and I know of at least 5 other friends of mine who used it and passed the exam last Dec. Keep up your great work!