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Discrete models

Discrete models. Types of discrete models. Binary models Logit and Probit Binary models with correlation (multivariate) Multinomial non ordered Ordered models (rankings) Count models (patents). We focus on binary models.

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Discrete models

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  1. Discrete models

  2. Types of discrete models • Binary models Logit and Probit • Binary models with correlation (multivariate) • Multinomial non ordered • Ordered models (rankings) • Count models (patents)

  3. We focus on binarymodels • 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

  4. Cdf= cumulative density function

  5. Mean and variance for Bernoulli random var • 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)

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

  7. Linear probability model for binarymodels • 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

  8. LPM thennotusable • So what? We ve to find a model coherent with a probability framework • Here LOGIT and PROBIT enters

  9. Used in MNL contexts

  10. The sign is given by the sign of B 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.

  11. Elasticity • 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

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

  13. Practical issues to interpretestimates • 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.

  14. Goodness of fit See Montini chapter

  15. Two stage Heckman • 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)

  16. 2 stage Heckman or heckit model: sample selection • 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

  17. 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

  18. 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

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