LIMITED DEPENDENT VARIABLE REGRESSION MODELS CHAPTER 11 LIMITED DEPENDENT VARIABLE REGRESSION MODELS Damodar Gujarati Econometrics by Example, second edition

LIMITED DEPENDENT VARIABLE REGRESSION MODELS If we only look at individuals with an attribute, we have a censored sample, a sample in which information on the regressand is available only for some observations but not all, even though we may have information on the regressors, for all the units in the sample. The regressand can be left-censored (cannot take a value below a certain threshold) or right-censored (cannot take a value above a certain threshold), or it can be both left and right censored. A related model is the truncated sample model in which information on both the regressand and regressors is not available on some observations. Like the censored sample, the truncated sample can be left-censored, or right-censored or it can be both right- and left-censored. Damodar Gujarati Econometrics by Example, second edition

CENSORED REGRESSION MODELS Tobit model We have: (if 0 is the threshold), where Y* is a latent variable, the primary variable of interest. We do not actually observe this variable for all the observations. We only observe it for those observations with positive values because of censoring. The Tobit model uses the method of maximum likelihood (ML) In the Tobit model it is assumed that the error term follows the normal distribution with zero mean and constant variance (i.e., homoscedasticity) We cannot interpret the Tobit coefficient of a regressor as giving the marginal impact of that regressor on the mean value of the observed regressand. A unit change in the value of a regressor has two effects: (1) the effect on the mean value of the observed regressand, and (2) the effect on the probability that is actually observed. Damodar Gujarati Econometrics by Example, second edition

TRUNCATED SAMPLE REGRESSION MODELS In truncated samples if we do not have information on the regressand, we do not collect information on the regressors that may be associated with the regressand. We may then use the truncated normal distribution (using ML). As in the Tobit model, an individual regression coefficient measures the marginal effect of that variable on the mean value of the regressand for all observations, including the non-included observations. But if we consider only the observations in the (truncated) sample, then the relevant (partial) regression coefficient has to be multiplied by a factor which is smaller than 1. Hence, the within-sample marginal effect of a regressor is smaller (in absolute value) than the value of the coefficient of that variable, as in the case of the Tobit model. Damodar Gujarati Econometrics by Example, second edition

TOBIT VS. TRUNCATED REGRESSION MODEL Since the Tobit model uses more information than the truncated regression model, estimates obtained from Tobit are expected to be more efficient. This is the result of the fact that the Tobit likelihood function is the sum of the likelihood functions of truncated regression model and the probit likelihood function. Damodar Gujarati Econometrics by Example, second edition

APPENDIX: HECKMAN’S SELECTION MODEL An alternative to Tobit model is Jim Heckman’s selection model (“Heckit”), a two-step model. The first step is the selection equation, with Z, a binary variable, as the outcome. The first step determines whether or not Y is observed. The second step is the response equation, with Y as the outcome. The outcome in the second step, Y, is only observed when Z is equal to 1. Damodar Gujarati Econometrics by Example, second edition

APPENDIX: HECKMAN’S SELECTION MODEL In the probit function (first step), we include variables that are relevant to the decision (for example, to work or not work). Then, in the second step, we estimate the outcome Y (such as hours worked) with the relevant socio-economic variables as regressors and an additional variable, called the Inverse Mills Ratio (IMR) (also called the hazard rate), that is derived from the first-step probit function. Some or all the regressors in the two steps may be the same. Damodar Gujarati Econometrics by Example, second edition