In previous lecture, we highlighted 3 shortcomings of the LPM. The most serious one is the unboundedness problem, i.e., the LPM may make the nonsense predictions that an event will occur with probability greater than 1 or less than 0. In this lecture, we will deal with this problem using the logit model. 1 Adapted from “Introduction to Econometrics” by Christopher Dougherty

The usual way of avoiding this problem is to hypothesize that the probability is a sigmoid (S-shaped) function of Z, F(Z), where Z is a function of the explanatory variables. The "input" is Z and the "output" is F(Z). BINARY CHOICE MODELS: LOGIT ANALYSIS 2

Several mathematical functions are sigmoid in character. One is the logistic function. The logistic function is useful because it can take as an input any value from negative infinity to positive infinity, whereas the output is confined to values between 0 and 1. The variable Z represents the exposure to some set of independent variables, in this case, Z =  1 +  2 X. Z is also known as logit. F(Z) represents the probability of an event occurring (probability of success, p), given that set of explanatory variables. 3

As Z goes to + , e –Z goes to 0 and p goes to 1 (but cannot exceed 1). As Z goes to - , e –Z goes to  and p goes to 0 (but cannot be below 0). 4

For a nonlinear model of this kind, maximum likelihood estimation is much superior to the use of OLS for estimating the parameters. ILLUSTRATION 1 Why do some people graduate from high school while others drop out? Our LPM model is in the form (see previous lecture): GRAD =  1 +  2 ASVABC +  i Our logit model is in the form: Z =  1 +  2 ASVABC i 5

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. logit GRAD ASVABC Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = Logit estimates Number of obs = 540 LR chi2(1) = Prob > chi2 = Log likelihood = Pseudo R2 = GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval] ASVABC | _cons |

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What is the probability that an individual with a score of 30 will graduate? When ASVABC = 30, An individual with a score of 30 has a 67 percent probability of graduating. 9

The coefficients of the Z function do not have any direct intuitive interpretation. However, we can use them to quantify the marginal effect of a change in ASVABC on the probability of graduating. We will do this theoretically for the general case where Z is a function of several explanatory variables. Since p is a function of Z, and Z is a function of the X variables, the marginal effect of X i on p can be written as: 10

(1) 11

(2) (3) Hence we obtain an expression for the marginal effect of X i on p. The marginal effect is not constant because it depends on the value of Z, which in turn depends on the values of the explanatory variables. A common procedure is to evaluate it for the sample means of the explanatory variables. 12

The sample mean of ASVABC in this sample is sum GRAD ASVABC Variable | Obs Mean Std. Dev. Min Max GRAD | ASVABC | When evaluated at the mean, Z is equal to

The marginal effect, evaluated at the mean, is therefore This implies that a one point increase in ASVABC would increase the probability of graduating from high school by 0.4%. 14

As noted earlier, the marginal effect does vary because it depends on the value of Z, which in turn depends on the values of X. How much is the marginal effect when ASVABC equal to 30? A one point increase in ASVABC then increases the probability of graduating by 2.9% 15

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. logit GRAD ASVABC SM SF MALE Iteration 0: log likelihood = Iteration 1: log likelihood = Iteration 2: log likelihood = Iteration 3: log likelihood = Iteration 4: log likelihood = Iteration 5: log likelihood = Here we consider a multivariate model with a somewhat better specification. ILLUSTRATION 2 Why do some people graduate from high school while others drop out? 17

Logit estimates Number of obs = 540 LR chi2(4) = Prob > chi2 = Log likelihood = Pseudo R2 = GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval] ASVABC | SM | SF | MALE | _cons | sum GRAD ASVABC SM SF MALE Variable | Obs Mean Std. Dev. Min Max GRAD | ASVABC | SM | SF | MALE |

ASVABC SM11.58–0.023–0.269 SF MALE Constant1.00–3.252–3.252 Total3.514 We will estimate the marginal effects, putting all the explanatory variables equal to their sample means. Step 1: Calculate Z, when the X variables are equal to their sample means. 19

Step 2: Calculate Step 3: Calculate Note that: 20

We see that the effect of ASVABC is about the same as before. Mother's schooling (SM) has negligible effect and father's schooling (SF) has no discernible effect at all. Males have 0.4 percent higher probability of graduating than females. ASVABC SM11.58–0.023– –0.001 SF MALE

Suppose that we want to examine the effect of a teaching method known as PSI on the performance of students in GD The question was whether students exposed to the method scored higher on exams in the class. Data were collected from 32 students in two classes, one in which PSI was used and another in which a traditional teaching method was employed. 1) GPA — Grade point average before taking the class. 2) TUCE — the score on an exam given at the beginning of the term to test entering knowledge of the material. 3) PSI — a dummy variable indicating the teaching method used (1 = used Psi, 0 = other method). 4) GRADE — coded 1 if the final grade was an A, 0 if the final grade was a B or C. GRADE was the dependent variable, and of particular interest was whether PSI had a significant effect on GRADE. TUCE and GPA are included as control variables. ILLUSTRATION 3 22

. logit GRADE GPA TUCE PSI Logit estimates Number of obs = 32 LR chi2(4) = Prob > chi2 = Log likelihood = Pseudo R2 = GRADE | Coef. Std. Err. z P>|z| [95% Conf. Interval] GPA | TUCE | PSI | _cons |

GPA TUCE PSI Constant1.00–13.021– Total What is the probability of getting an A for a student who has a GPA of 3.0, is taught by traditional methods, and has a score of 20 on TUCE? This student has a 6.66% chance of getting an A. 24

GPA TUCE PSI Constant1.00–13.021– Total What is the probability of getting an A for the same student if he is placed in the PSI class? Hence, getting into the PSI class would substantially increase the chances of getting an A, i.e., the student would have about a 37% better chance. 25