1. Nguyen Ngoc Anh Nguyen Ha Trang
Applied Econometrics Maximum Likelihood Estimation and Discrete choice Modelling
Nguyen Ngoc Anh
Nguyen Ha Trang

2. Content
Basic introduction to principle of Maximum Likelihood Estimation
Binary choice
RUM
Extending the binary choice
Mutinomial
Ordinal

3. Maximum Likelihood Estimation
Back to square ONE
Population model: Y = α + βX + ε
Assume that the true slope is positive, so β > 0
Sample model: Y = a + bX + e
Least squares (LS) estimator of β:
bLS = (X′X)–1X′Y = Cov(X,Y) / Var(X)
Key assumptions
E(|x) = E( ) = 0  Cov(x, ) = E(x ) = 0
Adding: Error Normality Assumption – e is idd with normal distribution

4. Maximum Likelihood Estimation
joint estimation of all the unknown parameters of a statistical model.
 that the model in question be completely specified.
 Complete specification of the model includes specifying the specific form of the probability distribution of the model's random variables.
 joint estimation of the regression coefficient vector β and the scalar error variance σ2.

5. Maximum Likelihood Estimation
Step 1: Formulation of the sample likelihood function
Step 2: Maximization of the sample likelihood function with respect to the unknown parameters β and σ2 .

6. Maximum Likelihood Estimation
Step 1
Normal Distribution Function: if
Then we have the density function
From our assumption with e or u

7. Maximum Likelihood Estimation
Substitute for Y:
By random sampling, we have N independent observations, each with a pdf
joint pdf of all N sample values of Yi can be written as

8. Maximum Likelihood Estimation
Substitute for Y

9. Maximum Likelihood Estimation
the joint pdf f(y) is the sample likelihood function for the sample of N independent observations
The key difference between the joint pdf and the sample likelihood function is their interpretation, not their form.

10. Maximum Likelihood Estimation
The joint pdf is interpreted as a function of the observable random variables for given values of the parameters and
The sample likelihood function is interpreted as a function of the parameters β and σ2 for given values of the observable variables

11. Maximum Likelihood Estimation
STEP 2: Maximization of the Sample likelihood Function
Equivalence of maximizing the likelihood and log-likelihood functions : Because the natural logarithm is a positive monotonic transformation, the values of β and σthat maximize the likelihood function are the same as those that maximize the log-likelihood function
take the natural logarithm of the sample likelihood function to obtain the sample log-likelihood function.

12. Maximum Likelihood Estimation
Differentiation and prove that
MLE estimates is the same as OLS

13. Maximum Likelihood Estimation
Statistical Properties of the ML Parameter Estimators
Consistency
2. Asymptotic efficiency
3. Asymptotic normality
Shares the small sample properties of the OLS coefficient estimator

14. Binary Response Models: Linear Probability Model, Logit, and Probit
Many economic phenomena of interest, however, concern variables that are not continuous or perhaps not even quantitative
What characteristics (e.g. parental) affect the likelihood that an individual obtains a higher degree?
What determines labour force participation (employed vs not employed)?
What factors drive the incidence of civil war?

15. Binary Response Models
Consider the linear regression model
Quantity of interest

16. Binary Response Models
the change in the probability that Yi = 1 associated with a one-unit increase in Xj, holding constant the values of all other explanatory variables

17. Binary Response Models

18. Binary Response Models
Two Major limitation of OLS Estimation of BDV Models
Predictions outside the unit interval [0, 1]
The error terms ui are heteroskedastic – i.e., have nonconstant variances.

19. Binary Response Models: Logit - Probit
Link function approach

20. Binary Response Models
Latent variable approach
The problem is that we do not observe y*i. Instead, we observe the binary variable

21. Binary Response Models

22. Binary Response Models
Random utility model

23. Binary Response Models
Maximum Likelihood estimation
Measuring the Goodness of Fit

24. Binary Response Models
Interpreting the results: Marginal effects
In a binary outcome model, a given marginal effect is the ceteris paribus effect of changing one individual characteristic upon an individual’s probability of ‘success’.

25. STATA Example Logitprobit.dta Logitprobit description STATA command
Probit/logit inlf nwifeinc ed exp expsq age kidslt6 kidsge6
dprobit inlf nwifeinc ed exp expsq age kidslt6 kidsge6