12: Maximizing the Likelihood
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- 57762
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In the previous chapters, we defined "best" by how the sum of the squared residuals were minimized. We made bad things small. Another way of viewing the "best" line is to maximize good things. That is the idea behind maximum likelihood estimation. While the concepts seem a bit different from ordinary least squares, this method actually leads to the same estimators of \(\beta_0\) and \(\beta_1\). It also leads to a biased estimator of \(\sigma^2\). So, why do we look at it here? First, the bias is relatively minor and disappears as the sample size increases.
More importantly, the method is extremely flexible. OLS requires Normality in the residuals. MLE can be used with any distribution.
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In the previous chapters, we have progressed from our desire to minimize some function of the residuals. This led to several related techniques:
- ordinary least squares
- weighted least squares
- generalized least squares
- ordinary least absolutes
All of these techniques sought to make the "bad" things as small as possible, to produce a model that minimizes these residuals. However, our first definition of "best" from way back in Chapter 3 was based on making good things as large as possible, where that "good thing" is the likelihood of observing this particular data.
Technically, it is the "likelihood of the data given our parameter estimates."
The theory is that the estimate most likely to have produced the observed data is the "best" estimate. Note that this differs from previous estimation methods in both the objective function and the size we desire. Bigger is better... bigger in terms of the "likelihood."
The fundamental purpose of this chapter is to introduce you to the likelihood and the methods to maximize it. In trying to accomplish this, we will start with the simple and proceed to the less-simple.