Maximum likelihood estimation and specification tools

Maximum likelihood estimation and specification tools

1. 1. introduction

   • max. likelihood estimation technique
   • Assumption: distribution of y is known
   • ML: choose parameter estimates that most likely give the observed sample.

2. 2. example

   • large pool of red and yellow balls. our interest: fraction (p) of red balls.
     • N= random sample of i balls. Yi=1red balls. 0=otherwise
       • P{yi=1}=p and P{yi=0}=1-p
         • N1=red balls, N2=N-N1 yellow balls
           • P{N1 red balls, N-N1 yellow balls}= Pn1(1-p)N-N1
             • function of p or liklihood function
               • maximize with respect to p
                 • MAXIMUM LIKELIHOOD FUNCTION
                   • logL(p)_p gives first order condition
                     • • Solve for p and obtain

3. 4. Maximum likelihood: the intution

   • for distribution of data we determine
     • the likelihood of observing the sample
       • as a function of unknown parameter.
         • we choose values that gives highest likelihood.
           • consistent asymptotically normal and efficient esitmators

4. 3. the linear regression model with normal error terms

   • • the likelihood contribution of observation is given by density
   • Joint denstiy
     • • The likelihood function is identical to the joint density
         • but it is a function of unknown parameter ( )
           • • Log likelihood function is
               • • Maximizing this w.r.t. and reproduces the OLS estimator for

5. 5. general properties and notation

   • conditional distribution of y given x
     • the likelihood of the entire sample is
   • 1. consistent 2. asymptotically efficient 3. asypmtotic distribution

6. 6. specification test

   • 3 principles. we want to test
     • Wald test:
       • estimate by M, check =0, using cov matrix
     • Likelihood ratio test
       • estimate the model twice
         • with and without restrictions. compare the difference in logliklihood values
     • Lag range multiplier test
       • estimate the model with restriction imposed
         • check: the foc is sig. violated