Lecture 9: Logit/ProbitProf. Sharyn O’Halloran Sustainable Development U9611Econometrics IIReview of Linear EstimationSo far, we know how to handle linearestimation models of the type:Y = β0+ β1*X1+ β2*X2+ … + ε ≡ Xβ+ εSometimes we had to transform or add variables to get the equation to be linear: Taking logs of Y and/or the X’s Adding squared terms Adding interactionsThen we can run our estimation, do model checking, visualize results, etc.Nonlinear EstimationIn all these models Y, the dependent variable, was continuous.  Independent variables could be dichotomous (dummy variables), but not the dependent var.This week we’ll start our exploration of non-linear estimation with dichotomous Y vars.These arise in many social science problems Legislator Votes: Aye/Nay Regime Type: Autocratic/Democratic Involved in an Armed Conflict: Yes/NoLink FunctionsBefore plunging in, let’s introduce the concept of a link function This is a function linking the actual Y to the estimated Y in an econometric modelWe have one example of this already: logs Start with Y = Xβ+ ε Then change to log(Y) ≡ Y′ = Xβ+ ε Run this like a regular OLS equation Then you have to “back out” the resultsLink FunctionsBefore plunging in, let’s introduce the concept of a link function This is a function linking the actual Y to the estimated Y in an econometric modelWe have one example of this already: logs Start with Y = Xβ+ ε Then change to log(Y) ≡ Y′ = Xβ + ε Run this like a regular OLS equation Then you have to “back out” the resultsDifferentβ’s hereLink FunctionsIf the coefficient on some particular X is β, then a 1 unit ∆X Æ β⋅∆(Y′) = β⋅∆[log(Y))]= eβ⋅∆(Y)  Since for small values of β, eβ≈ 1+β , this is almost the same as saying a β% increase in Y (This is why you should use natural log transformations rather than base-10 logs)In general, a link function is some F(⋅) s.t. F(Y) = Xβ+ εIn our example, F(Y) = log(Y)Dichotomous Independent Vars.How does this apply to situations with dichotomous dependent variables? I.e., assume that Yiœ {0,1}First, let’s look at what would happen if we tried to run this as a linear regressionAs a specific example, take the election of minorities to the Georgia state legislature Y = 0: Non-minority elected Y = 1: Minority electedDichotomous Independent Vars.0 1Black Representative Elected0 .2 .4 .6 .8 1Black Voting Age PopulationThe data look like this.The only values Y can have are 0 and 1Dichotomous Independent Vars.And here’s a linear fit of the dataNote that the line goes below 0 andabove 10 10 .2 .4 .6 .8 1Black Voting Age PopulationBlack Representative Elected Fitted valuesDichotomous Independent Vars.The line doesn’t fit the data very well.And if we take values of Y between 0 and 1 to be probabilities, this doesn’t make sense0 10 .2 .4 .6 .8 1Black Voting Age PopulationBlack Representative Elected Fitted valuesRedefining the Dependent Var.How to solve this problem?We need to transform the dichotomous Y into a continuous variable Y′ œ (-∞,∞)So we need a link function F(Y) that takes a dichotomous Y and gives us a continuous, real-valued Y′Then we can runF(Y) = Y′ = Xβ+ εRedefining the Dependent Var.01OriginalYRedefining the Dependent Var.01OriginalYY as aProbability01Redefining the Dependent Var.01OriginalYY as aProbability01Y′ œ (- ∞,∞)?- ∞∞Redefining the Dependent Var.What function F(Y) goes from the [0,1] interval to the real line?Well, we know at least one function that goes the other way around. That is, given any real value it produces a number (probability) between 0 and 1.This is the…Redefining the Dependent Var.What function F(Y) goes from the [0,1] interval to the real line?Well, we know at least one function that goes the other way around. That is, given any real value it produces a number (probability) between 0 and 1.This is the cumulative normal distribution Φ That is, given any Z-score, Φ(Z) œ [0,1]Redefining the Dependent Var.So we would say thatY = Φ(Xβ+ ε)Φ−1(Y) = Xβ+ εY′ = Xβ+ εThen our link function is F(Y) = Φ−1(Y) This link function is known as the Probit link This term was coined in the 1930’s by biologists studying the dosage-cure rate link It is short for “probability unit”Probit EstimationAfter estimation, you can back out probabilities using the standard normal dist.0 .1 .2 .3 .4-4 -2 0 2 4Probit EstimationSay that for a given observation, Xβ = -10 .1 .2 .3 .4-4 -2 0 2 40 .1 .2 .3 .4-4 -2 0 2 4Probit EstimationSay that for a given observation, Xβ = -10 .1 .2 .3 .4-4 -2 0 2 4Probit EstimationSay that for a given observation, Xβ = -10 .1 .2 .3 .4-4 -2 0 2 4Probit EstimationSay that for a given observation, Xβ = -1Prob(Y=1)0 .1 .2 .3 .4-4 -2 0 2 4Probit EstimationSay that for a given observation, Xβ = 20 .1 .2 .3 .4-4 -2 0 2 4Probit EstimationSay that for a given observation, Xβ = 20 .1 .2 .3 .4-4 -2 0 2 4Probit EstimationSay that for a given observation, Xβ = 2Prob(Y=1)Probit EstimationIn a probit model, the value of Xβ is taken to be the z-value of a normal distribution Higher values of Xβ mean that the event is more likely to happen Have to be careful about the interpretation of estimation results here A one unit change in Xileads to a βichange in the z-scoreof Y (more on this later…)The estimated curve is an S-shaped cumulative normal distributionProbit Estimation• This fits the data much better than the linear estimation• Always lies between 0 and 10 .2 .4 .6 .8 1Probability of Electing a Black Rep.0 .2 .4 .6 .8 1Black Voting Age PopulationProbit Estimation• Can estimate, for instance, the BVAP at which Pr(Y=1) = 50%• This is the “point of equal opportunity”0 .2 .4 .6 .8 1Probability of Electing a Black Rep.0 .2 .4 .6 .8 1Black Voting Age PopulationProbit Estimation• Can estimate, for instance, the BVAP at which Pr(Y=1) = 50%• This is the “point of equal opportunity”0 .2 .4 .6 .8 1Probability of Electing a Black Rep.0 .2 .4 .6 .8 1Black Voting Age PopulationProbit Estimation• Can estimate, for instance, the BVAP at which Pr(Y=1) = 50%• This is the “point of equal opportunity”0 .2 .4 .6 .8 1Probability of Electing a Black Rep.0 .2 .4 .6 .8 1Black Voting Age PopulationProbit Estimation• This