Linear Probability

Подпись: (7.5)

Подпись: Suppose that УІ Suppose further that the Pr(yi =
Подпись: 1 if alternative is chosen 0 if alternative is not chosen 1) = ni. For a discrete variable

A linear probability model is a linear regression in which the dependent variable is an indicator variable. The model is estimated by least squares.

Подпись: (7.6)E[yi] =1 x Pr(yi = 1) + 0 x Pr(yi = 0) = пі

Thus, the mean of a binary random variable can be interpreted as a probability; it is the probability that y = 1. When the regression E[y^Xi2, Xi3,…, XiK] is linear then E[yi] = ві + в2Xi2 +.. .+вкXiK and the mean (probability) is modeled linearly.

E[yi|Xi2,Xi3, .. . , XiK] = Пі = ві + e2Xi2 + … + вкXiK (7.7)

The variance of a binanry random variable is

var[yi] = пі(1 – пі) (7.8)

which means that it will be different for each individual. Replacing the unobserved probability, E(yi), with the observed indicator variable requires adding an error to the model that we can estimate via least squares. In this following example we have 1140 observations from individuals who purchased Coke or Pepsi. The dependent variable takes the value of 1 if the person buys Coke and 0 if Pepsi. These depend on the ratio of the prices, pratio, and two indicator variables, disp_coke and disp_pepsi. These indicate whether the store selling the drinks had promotional displays of Coke or Pepsi at the time of purchase.

OLS, using observations 1-1140
Dependent variable: coke

Подпись: Sum squared resid 248.0043 R2 0.120059 F (3,1136) 56.55236 Подпись: S.E. of regression 0.467240 Adjusted R2 0.117736 P-value(F) 4.50e-34

Heteroskedasticity-robust standard errors, variant HC3

Coefficient

Std. Error

t-ratio

p-value

const

0.8902

0.0656

13.56

5.88e-039

pratio

-0.4009

0.0607

-6.60

6.26e-011

disp_coke

0.0772

0.0340

2.27

0.0235

disp_pepsi

-0.1657

0.0345

-4.81

1.74e-006

The model was estimated using a variance-covariance matrix estimator that is consistent when the error terms of the model have variances that depend on the observation. That is the case here. I’ll defer discussion of this issue until the next chapter when it will be discussed at some length.

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