Research Method. Lecture 13 (Greene Ch 16) Maximum Likelihood Estimation (MLE). Basic idea. Maximum likelihood estimation (MLE) is a method to find the most likely density function that would have generated the data. Thus, MLE requires you to make a distributional assumption first.
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(Greene Ch 16)
Maximum Likelihood Estimation (MLE)
“Which distribution, A or B, is more likely to have generated the data?”
Answer to the question is A, because the data are clustered around the center of the distribution A, but not around the center of the distribution B.
Graphical illustration of the likelihood contribution
The likelihood contribution of the first observation
Then, you multiply the likelihood contributions of all the observations. This is called the likelihood function. We use the notation L.
This notation means you multiply from i=1 through n.
Then you find the values of μ and σ that maximize the likelihood function.
Fortunately, there are many optimization computer programs that can do this.
Suppose that you have this data, and you are interested in estimating the model: y=β0+β1x+u
The likelihood contribution of the 2nd observation
You choose the values of β0,β1, and σ that maximizes the likelihood function. These are the maximum likelihood estimators of of β0,β1, and σ .
If Y=1, it means that y*≥0
If Y=0, it means that y*<0
Then, what is the likelihood contribution of each observation? In this case, we only know if y* ≥0 or y*<0. We do not know the exact value of y* .
Take 2nd observation as an example. Since Y=0 for this observation, we know y*<0
Now, take 3nd observation as an example. Since Y=1 for this observation, we know y*≥0
You choose the values of β0 and β1 that maximizes the likelihood function. These are the maximum likelihood estimators of of β0 and β1 .
The score vector is the first derivative of the log likelihood function with respect to the parameters
Then, the standard errors of the parameters are given by the square root of the diagonal elements of the following matrix.