Module 15: Advanced interactions
(The attached PDF file has better formatting.)
Practice problems for Interactions, dummy variables, F tests
(This posting covers Modules 14 and 15.)
*Question 15.1: Hypothesis testing
We use regression analysis to compare personal auto claim frequency in urban, suburban, and rural areas.
Claim frequency =
á + â1 D1 + â2 D2 + å, where
D1 = 1 for urban and 0 otherwise
D2 = 1 for sub-urban and 0 otherwise
How do we show that territory (urban vs sub-urban vs rural) is significant?
The t-values for both
â1 and â2 are greater than their critical values.
The t-value for either
â1 or â2 is greater than its critical value.
The t-values for both
â1 and â2 are less than their critical values.
The F-value for
â1 plus â2 is greater than its critical value.
The F-value for
â1 plus â2 is less than its critical value.
Answer 15.1: D
By "
â1 plus â2" we mean the F test examining the combination of these two coefficients. Fox uses qualitative explanatory variables in many examples, just as actuaries use age, sex, and other attributes of the insured.
*Question 15.2: Parameters
We regress personal auto claim frequency on (i) annual driving and (ii) urban, suburban, and rural areas.
Claim frequency =
á + â1 D1 + â2 D2 + â3 DD + å, where
D1 = 1 for urban and 0 otherwise
D2 = 1 for sub-urban and 0 otherwise
DD = annual driving distance in kilometers
What is the predicted difference in claim frequency for urban vs sub-urban insureds driving 30,000 kilometers a years?
â
1 + â2 + â3 × 30,000
â
1 – â2 + â3 × 30,000
â
1 + â2
â
1 – â2
â
1 D1 + â2 D2
Answer 15.2: D
*Question 15.3: Principle of marginality
An actuary regresses personal auto claim frequency on (i) amount of driving, (ii) income of driver, and (iii) territory. Driving and income are quantitative explanatory variables, and territory is a qualitative factor with three levels: urban, sub-urban, and rural.
Which of the following correctly reflects the principle of marginality?
The income by territory interaction is marginal to the income effect.
The amount of driving effect is marginal to the amount of driving by territory interaction.
We do not test the income by territory interaction until we test the income effect.
If we can rule out a main effect on theoretical grounds, we test the interaction effect.
None of A, B, C, or D is true.
Answer 15.3: B
(P135)
Fox emphasizes the principle of marginality. Some other textbooks do not discuss this topic. It is useful for actuarial analyses, which looks at the interactions of policyholder attributes.