Suppose you believe that drunk driving fatalities are likely to occur at high altitude than at low altitude because higher elevations diminish the oxygen intake of the brain, increasing the impact of a given amount of alcohol. The following cross-sectional regression model is run across 48 states in the continental United States (note: the figures in the parentheses are t-scores in this question, they are not standard errors)
TFi = -3.36 - 0.002 BEERi + 0.17 HWSPi - 0.31 VSIPi + 0.011 (BEERi)(ALTi) (-0.08) (1.85) (-1.29) (4.05)
n = 48; Adjusted R2 = .499
where
TFi = traffic fatalities per motor vehicle mile driven in the ith state
BEERi = per capita beer consumption in the ith state
HWSPi = average highway speed limit in the ith state
VSIPi = a dummy variable equals to 1 if the ith state has a vehicle safety inspection
program and zero otherwise
ALTi = average altitude of metropolitan areas in the ith state
(a) Carefully explain the meaning of the coefficient of the interactive term (BEERi)(ALTi).
(b) Create and test hypothesis about the coefficient of the interactive term (BEERi)(ALTi) at significance level of 5% using critical t-score approach and confidence interval approach. Is it possible to find the p-value if we assume the sample size is very large? If yes, find it. If no, please explain why.
(c) In this model ALT is included in the interactive term but not as an independent regressor on its own. If a regression model includes an interactive variable, should both components of this interactive variable also be independent variable on its own in the model as a matter of course?