An example：LPM and Logit model

Like credit cards,debit cards are now used extensively by consumers. Vendor prefer them because when you use a debit card, the amount of your purchase is automatically deducted from your checking or other designated account. To found out what factors determine the use of the debit card, we obtained data on 60 customers and considered the following model:
$Y_i=\beta_1+\beta_2X_{2i}+\beta_3X_{3i}+\beta_4 X_{4i}+\mu_i$Yi​=β1​+β2​X2i​+β3​X3i​+β4​X4i​+μi​
Where Y=1 for debit card holder, 0 otherwise;X₂=account balance in dollars; X₃=number of ATM transactions;X₄=1 if interest is received on the account, 0 otherwise.
The following table summarizes the estimated linear probability model (LPM), Logit models (standard errores are in parentheses below the estimated coefficients).

1. In LPM model, explain whether the signs of explanatory variables (except intercept)are consistent to your expectation. Are all the variables statistically significant?
   Answer: Consistent to the expectation:the higher the balance in the account, more likely to hold debit card; more transactions in ATM means more transaction fees which leads less likely to hold debit card (this variable is not significant though); if interests is received in the account, less likely to hold debit card. “Balance” is statistically significant at 5% level.

2. The second customer in the data having $500 balance in the account, and number of ATM transactions is 3; the account does receive intersts. Based on LPM and Logit estimates, what are the porbability respectively that this customer will hold the debit card?
   Answer:
   LPM: 0.3631+0.00028Balance-0.0269ATM-0,3019Interest=0.121
   Logit: L=-0.57490+0.001248Balance-0.120225ATM-1.352086Interest=-1.6637
   for Logit distribution:
   $F(t)=\frac{1}{1+e^{-t}}=\frac{e^t}{1+e^t}$F(t)=1+e−t1​=1+etet​
   then,$P=F(L)=0.15927$P=F(L)=0.15927

3. Based on the output table, calculate respectively for LPM and Logit mode, the marginal effect of a one extra dollar deposited in the account on the probability of the customer described in 2. holding the debit card.
   Answer:
   LPM: 0.028%
   Logit: P*(1-P)*$\beta_{Balance}$βBalance​=0.000167. So the probability increase 0.0167%

Microeconometrics course, University of LIVERPOOL