 Binary models Logit and Probit  Binary models with correlation (multivariate)  Multinomial non ordered  Ordered models (rankings)  Count models (patents)

 Refer to Greene chapter (also uploaded in the site) and Montini document on fit measures.  Microeconometrics  Consumer choices (but not only)  Random utility framework (linked to Hicksian theory).. You observe what people choose, they choose what the like the best

Cdf= cumulative density function

 In deciding on the estimation technique, it is useful to derive the conditional mean and variance  E(y/x)= B0 + B1x1 + B2x2 +……Bkxk  Var (y/x)= XB(1- XB), where XB is B0 + B1x1 + B2x2 +……Bkxk.  OLS produces consistent and even unbiased estimates, BUT…  Heteroskedasticity is always an issue to be dealt by weighted least squares (het in stata)

 Always recall that HET affects s.e. not size of coefficients.. Correction should improve T ratios since it lower variances

 P(y=1/X)= B0 + B1x1 + B2x2 +……Bkxk  B1= dP(y=1)/dx1, assuming x1 is not related functionally to other covariates, B1 is the change in the probability of success given a one unit increase in x1. holding other Xj fixed  Unless x is restricted, the LPM cannot be a good description of the population response probability  There are values of Bx for which P is outside the unit 0-1 interval

 So what? We ve to find a model coherent with a probability framework  Here LOGIT and PROBIT enters

Used in MNL contexts

See Mancinelli, Mazzanti, Ponti and Rizza (2010), J of socio economics, also WP DEIT Non siamo in un contesto dove possiamo rappresentare B come elasticità, questo è vero anche in modelli lin-log, dove ad esempio la var dipendente (causa ‘0’ diffusi) non può essere rappresentata in log. The sign is given by the sign of B

 Linear model  Dy/dx = b; e=b*x/y  Log log  Dlny/dlnx= b*y/x  E=b  Lin log  Dy/dlnx = b*1/x  E= b*x/y= b/y

See various examples of papers in the site, mainly on innovation variables of adoption that take values 1/0

 Coefficient fo not represent marginal effects ◦ You can use dprobit in STATA for that  R2 is not a measure of fit, we have pseudo R2, es. McFadden R2 (see Montini document on that)  You should have good F test, reasonable R2 (0.2 excellent, but 0.05 fine as well), a set of *** coefficients.

Goodness of fit See Montini chapter

 Es. R&D, labor hours offered  First stage probit, then OLS  Get the inverse Mills ratio from first to inform the second and see whether the bias is there  Heckman vs Tobit models (different assumptions)

 Y1= X1B1 + u1  Y2= 1(x  2 + v2 >0)  Hp: x,y2 always observed, y1 only if y2=1, set to 0 if y2=0  E(y1/X, y2=1) = x1B1 +  (x  2)  OLS can produce biased inconsistent estimates of B1 if we do not account for the last term  OMITTED VAR problem

 We need an estimator of  2!  Obtain the probit estimates of  2 from first stage P(y2=1/x) =  (x  2) using all N units  Then estimate Inverse Mills ratio, = f(x  2)  Insert IMR in the OLS second equation and get B and  estimates  These estimators are consistent

 X1 covariates of OLS, X of probit  ** we dont need x1 to be a subset of X for identification. But X=X1 can introduce collinearity since can be approximated by a linear function of X  Example in Wooldridge, Econometric analysis of cross section and panel data, p.565 (wage equation for married women)  Estimates become imprecise when X=x1