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Chapter 10. Binary choice models, tobit model and ML estimation

Chapter 1. Simple regression analysis | Chapter 2. Properties of regression coefficients and hypothesis testing | Chapter 3. Multiple regression analysis | Chapter 8. Stochastic regressors and measurement errors | Chapter 9 . Simultaneous equations estimation | Chapter 12. Autocorrelation | Chapter 13. Introduction to nonstationary time series |


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  7. B. In pairs, answer the following questions about model A.

1. Binary choice models

Question. What influences people’s choices between private and public transport?

- variables affecting choices (transportation costs, time of travel, availability of parking, etc.) Their index is defined by . For individual takes value .

General format of the model:

Special cases.

1) Linear model: , .

What is bad about this model (give two reasons)?

2) Probit model: - cumulative distribution function of the standard normal variable, - its density.

3) Logit model: - logistic function, - its density.

Marginal effect: In all cases the marginal effect of variable is

is hat-shaped. The marginal effect varies and is small when is large. Usually is calculated at mean values of observed independent variables.

Important: in probit and logit, the error is hidden in the model specification!

Significance testing: since the tests are asymptotical,

1) For one coefficient: stats instead of stats,

2) For all slopes (d. of f. = number of explanatory variables) instead of stats.

3) See also bottom of p. 298 (translate it to math and find a typo).

2. Censored regression (tobit model)

Provide motivation. Model with a lower bound : Using the observed values , define cut-off values and run the regression .

Sample selection bias model: more generally, instead of comparing the observed values with the lower or upper cutting value, one can compare to 0 a linear expression.

3. Maximum likelihood estimation

- density of .

Definition. Take as estimators those and which maximize the joint probability density.

Example. Suppose where .

Step 1. Write down the density of one error.

Step 2. Assuming independence, find the joint density , called a likelihood function.

Step 3. is maximized is maximized. Find the log-likelihood function .

Step 4. Write down and solve the FOCs for .

Result: - unbiased and efficient, - biased but consistent and more efficient than .

Goodness of fit and statistical tests

Significance testing: since the tests are asymptotical,

1) For one coefficient: stats instead of stats,

2) For all slopes instead of stats.

3) Pseudo- instead of . Here is the minimized log-likelihood when all parameters vary and is the minimized log-likelihood when all parameters vary, except the intercept.

Iterative methods

In general, may have many local minimums. An analytical solution may not be available. A numerical solution can be found using iterative methods. Such a method finds only one local minimum that depends on the initial approximation. The initial approximation is determined by the researcher.

Exercises. 10.2, 10.4, 10.5, 10.8, 10.9


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