Question

Econometrics Question:
Discuss the intuition behind the maximum likelihood estimator. Discuss what, if any, desirable properties the maximum likelihood estimator posseses.

Answer 1

Maximum likelihood estimator :

Maximum likelihood estimation (MLE) is a method through which the values of the parameters of a model are determined . The values of the parameter values found are such that they lead to the maximisation of the likelihood i.e likehood of that the process described by the model produced the data that were actually observed.

For ex:

There is a collection of 20 data points for the time in length a student took to answer a MCQ question on an online test forum.

Student No	Time Taken
1	5
2	7
3	4
4	4
5	5
6	4
7	8
8	7
9	7
10	8
11	5
12	7
13	8
14	6
15	9
16	5
17	7
18	4
19	5
20	4

We have to first assume which model to use , we can assume that the data can be represented by a normal distribution ( as we can see intuitively that the numbers are clustered around a middle value ) . A normal distribution is described by its mean (μ) and its standard deviation (σ) . Using MLE which will help find values of (μ) & (σ) and that result in the curve that best fits the data.

The assumption here is that the distribution here is normal or gaussian and like that several intuitive assumptions can be made about the data at hand . The next proceeding steps are :

-> Formulating likelihood function of the data through the data generating function.

-> Find an estimator for the parameter using optimisation technique.

Desirable Properties of MLE are :

1 ) Unbiasedness : The Bias is = Difference between ( estimator - true parameter )

2) Efficiency : With minimum variance .

3) Consistency : When the number of observations reach till infinity and the estimator value converges to the true value , then this estimator is called a consistent estimator .

4) Minimum Mean Square Error : The expectation value of the square of the diff.( error) between estimator and the true value.