This function contains all the information available in the sample about the population from which those observations were drawn. The strategy by which that information is used in estimation constitutes the estimator.
The maximum likelihood estimator [Fisher (1925)] is that function of the data which (as its name implies) maximizes the likelihood function (or, because it is usually more convenient, the log of the likelihood function). The motivation for this approach is most easily visualized in the setting of a discrete random variable. In this case, the likelihood function gives the joint probability for the observed sample observations, and the maximum likelihood estimator is the function of the sample information which makes the observed data most probable (at least by that criterion). Though the analogy is most intuitively appealing for a discrete variable, it carries over to continuous variables as well. Since this estimator is the subject of Chapter 17, which is quite lengthy, we will defer any formal discussion until then, and consider instead two applications to illustrate the techniques and underpinnings.
Least squares weighs negative and positive deviations equally and gives disproportionate weight to large deviations in the calculation. This property can be an advantage or a disadvantage, depending on the data-generating process. For normally distributed disturbances, this method is precisely the one needed to use the data most efficiently. If the data are generated by a normal distribution, then the log of the likelihood function is lnL = -2ln2n - 2lna2 - ^(y-Xp)'(y-Xp).
You can easily show that least squares is the estimator of choice for this model. Maximizing the function means minimizing the exponent, which is done by least squares for p and e'e/n for a2.
If the appropriate distribution is deemed to be something other than normal — perhaps on the basis of an observation that the tails of the disturbance distribution are too thick—see Example 5.1 and Section 17.6.3—then there are three ways one might proceed. First, as we have observed, the consistency of least squares is robust to this failure of the specification, so long as the conditional mean of the disturbances is still zero. Some correction to the standard errors is necessary for proper inferences. (See Section 10.3.) Second, one might want to proceed to an estimator with better finite sample properties. The least absolute deviations estimator discussed in Section 16.3.2 is a candidate. Finally, one might consider some other distribution which accommodates the observed discrepancy. For example, Ruud (2000) examines in some detail a linear regression model with disturbances distributed according to the t distribution with v degrees of freedom. As long as v is finite, this random variable will have a larger variance than the normal. Which way should one proceed? The third approach is the least appealing. Surely if the normal distribution is inappropriate, then it would be difficult to come up with a plausible mechanism whereby the t distribution would not be. The LAD estimator might well be preferable if the sample were small. If not, then least squares would probably remain the estimator of choice, with some allowance for the fact that standard inference tools would probably be misleading. Current practice is generally to adopt the first strategy.
The stochastic frontier model, discussed in detail in Section 17.6.3, is a regression-like model with a disturbance that is asymmetric and distinctly nonnormal. (See Figure 17.3.) The conditional density for the dependent variable in this model is f (y | x, p, a, X) =
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