There are has c = 5 categories: dismissal, left job because of plant closures, “quit,” changed jobs for other reasons and no change in employment.
Table 11.1 shows that turnover declines as tenure increases.
To illustrate, consider a typical man in the 1992 sample where we have time = 16 and focus on dismissal probabilities.
For this value of time, the coefficient associated with tenure for dismissal is -0.221 + 16 (0.008) = -0.093 (due to the interaction term).
From this, we interpret an additional year of tenure to imply that the dismissal probability is exp(-0.093) = 91% of what it would be otherwise, representing a decline of 9%.
Scanner data - they are obtained from optical scanning of grocery purchases at check-out.
The subjects consist of n =100 households in Springfield, Missouri.
The response of interest is the type of yogurt purchased, consisting of four brands: Yoplait, Dannon, Weight Watchers and Hiland.
The households were monitored over a two-year period with the number of purchases ranging from 4 to 185; the total number of purchases is N =2,412.
The two marketing variables of interest are PRICE and FEATURES.
For the brand purchased, PRICE is recorded as price paid, that is, the shelf price net of the value of coupons redeemed. For other brands, PRICE is the shelf price.
FEATURES is a binary variable, defined to be one if there was a newspaper feature advertising the brand at time of purchase, and zero otherwise.
Note that the explanatory variables vary by alternative, suggesting the use of a multinomial (conditional) logit model.
A multinomial logit model was fit to the data, using
using Hiland as the omitted alternative.
Results:
Each parameter is statistically significantly different from zero.
For the FEATURES coefficient, a consumer is exp(0.4914) = 1.634 times more likely to purchase a product that is featured in a newspaper ad compared to one that is not.
For the PRICE coefficient, a one cent decrease in price suggests that a consumer is exp(0.3666) = 1.443 times more likely to purchase a brand of yogurt.
We now introduce a type of hierarchical model known as a nested logit model .
In the first stage, one chooses an alternative (say the first type) with probability
Conditional on not choosing the first alternative, the probability of choosing any one of the other alternatives follows a multinomial logit model with probabilities
The value of ρ = 1 reduces to the multinomial logit model.
Interpret Prob( y it = 1) to be a weighted average of values from the first choice and the others.
Conditional on not choosing the first category, the form of Prob( y it = j | y it 1) has the same form as the multinomial logit.
Random intercepts for Yoplait, Dannon and Weight Watchers were assumed to follow a multivariate normal distribution with an unstructured covariance matrix.
Here, we see that the coefficients for FEATURES and PRICE are qualitatively similar to the model without random effects.
Overall, the AIC statistic suggests that the model with random effects is the preferred model.
To simplify matters, assume that the initial state distribution, Prob( y i 1 ), is described by a different set of parameters than the transition distribution, f( y it | y i,t -1 ).
Thus, to estimate this latter set of parameters, one only needs to maximize the partial log-likelihood
Special case: specify separate components for each alternative
Idea: we can split up the data according to each (lagged) choice ( j )
determine MLEs for each alternative (at time t -1), in isolation of the others.
29.
Example 11.3 Choice of yogurt brands - Continued
Ignore the length of time between purchases
Sometimes referred to as “operational time.”
One’s most recent choice of a brand of yogurt has the same effect on today’s choice, regardless as to whether the prior choice was one day or one month ago.
Purchase probabilities for customers of Dannon, Weight Watchers and Hiland are more responsive to a newspaper ad than Yoplait customers.
Compared to the other three brands, Hiland customers are not price sensitive in that changes in PRICE have relatively impact on the purchase probability (it is not even statistically significant).
The total (minus two times the partial) log-likelihood is 2,379.8 + … + 281.5 = 6,850.3.
Without origin state, the corresponding value of 9,741.2.
The likelihood ratio test statistic is LRT = 2,890.9.
This corroborates the intuition that the most recent type of purchase has a strong influence in the current brand choice.
33.
Example 11.4 Income tax payments and tax preparers
Yogurt - the explanatory variables FEATURES and PRICE depend on the alternative.
The choice of a professional tax preparer - financial and demographic characteristics do not depend on the alternative.
97 tax filers never used a preparer in the five years under consideration and 89 always did, out of 258 subjects.
Table 11.8 shows the relationship between the current and most recent choice of preparer.
258 initial observations are not available for the transition analysis, reducing our sample size to 1,290 – 258 = 1,032.
Table 11.10 suggests that there are important differences in the transition probabilities for each lag 1 origin state ( y i,t -1 = Lag PREP) between levels of the lag two origin state ( y i,t -2 = Lag 2 PREP).
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