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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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| Identification | | | Chapter 11. Models using time-series data |