When we want to study the properties of the obtained estimators, it is convenient to distinguish between two categories of properties: i) the small (or finite) sample properties, which are valid whatever the sample size, and ii) the asymptotic properties, which are associated with large samples, i.e., when tends to .
Given that, as we obtained in the previous section, the OLS and ML estimates of lead to the same result, the following properties refer to both. In order to derive these properties, and on the basis of the classical assumptions, the vector of estimated coefficients can be written in the following alternative form:
The unbiasedness property of the estimators means that, if we have many samples for the random variable and we calculate the estimated value corresponding to each sample, the average of these estimated values approaches the unknown parameter. Nevertheless, we usually have only one sample (i.e, one realization of the random variable), so we can not assure anything about the distance between and . This fact leads us to employ the concept of variance, or the variance-covariance matrix if we have a vector of estimates. This concept measures the average distance between the estimated value obtained from the only sample we have and its expected value.
From the previous argument we can deduce that, although the unbiasedness property is not sufficient in itself, it is the minimum requirement to be satisfied by an estimator.
The consideration of allows us to define efficiency as a second finite sample property.
In order to study the efficiency property for the OLS and ML estimates of , we begin by defining , and the hessian matrix is expressed as a partitioned matrix of the form:
Following the Cramer-Rao inequality, constitutes the lower bound for the variance-covariance matrix of any unbiased estimator vector of the parameter vector , while is the corresponding bound for the variance of an unbiased estimator of .
According to (2.56), we can conclude that (or ), satisfies the efficiency property, given that their variance-covariance matrix coincides with .
With this aim, we define as a family of linear vectors of estimates of the parameter vector :
Taking into account (2.65) we have:
A general result matrix establishes that given any matrix P, then is a positive semidefinite matrix, so we can conclude that is positive semidefinite. This property means that the elements of its diagonal are non negative, so we deduce for every coefficient:
The set of results we have previously obtained, allows us to know the probability distribution for (or ). Given that these estimator vectors are linear with respect to the vector, and having a normal distribution, then:
According to expressions (2.34) and (2.53), the OLS and ML estimators of are different, despite both being constructed through . In order to obtain their properties, it is convenient to express as a function of the disturbance of the model. From the definition of in (2.26) we obtain:
Result (2.75), which means that is linear with respect to , can be extended in the following way:
From (2.76), and under the earlier mentioned properties of , the sum of squared residuals can be written as a quadratic form of the disturbance vector,
Note that from (2.75), it is also possible to write as a quadratic form of , yielding:
This expression for allows us to obtain a very simple way to calculate the OLS or ML estimator of . For example, for :
Having established these relations of interest, we now define the properties of and :
Nevertheless, given that is biased, this estimator can not be efficient, so we focus on the study of such a property for . With respect to the BLUE property, neither nor are linear, so they can not be BLUE.
Finite sample properties try to study the behavior of an estimator under the assumption of having many samples, and consequently many estimators of the parameter of interest. Thus, the average of these estimators should approach the parameter value (unbiasedness) or the average distance to the parameter value should be the smallest possible (efficiency). However, in practice we have only one sample, and the asymptotic properties are established by keeping this fact in mind but assuming that the sample is large enough.
Specifically, the asymptotic properties study the behavior of the estimators as increases; in this sense, an estimator which is calculated for different sample sizes can be understood as a sequence of random variables indexed by the sample sizes (for example, ). Two relevant aspects to analyze in this sequence are and .
A sequence of random variables is said to a constant or to another random variable , if
Result (2.91) implies that all the probability of the distribution becomes concentrated at points close to . Result (2.92) implies that the values that the variable may take that are not far from z become more probable as increases, and moreover, this probability tends to one.
A second form of convergence is convergence in distribution. If is a sequence of random variables with cumulative distribution function () , then the sequence to a variable with if
Having established these preliminary concepts, we now consider the following desirable asymptotic properties : asymptotic unbiasedness, consistency and asymptotic efficiency.
Note that the second part of (2.96) also means that the possible bias of disappears as increases, so we can deduce that an unbiased estimator is also an asymptotic unbiased estimator.
The second definition is based on the convergence in distribution of a sequence of random variables. According to this definition, an estimator is asymptotically unbiased if its asymptotic expectation, or expectation of its limit distribution, is the parameter . It is expressed as follows:
Since this second definition requires knowing the limit distribution of the sequence of random variables, and this is not always easy to know, the first definition is very often used.
In our case, since and are unbiased, it follows that they are asymptotically unbiased:
The simplest way of showing consistency consists of proving two sufficient conditions: i) the estimator must be asymptotically unbiased, and ii) its variance must converge to zero as n increases. These conditions are derived from the convergence in quadratic mean (or convergence in second moments), given that this concept of convergence implies convergence in probability (for a detailed study of the several modes of convergence and their relations, see Amemiya (1985), Spanos (1986) and White (1984)).
In our case, since the asymptotic unbiasedness of and has been shown earlier, we only have to prove the second condition. In this sense, we calculate:
where we have used the condition (2.6) included in assumption 1. Thus, result (2.101) proves the consistency of the OLS and ML estimators of the coefficient vector. As we mentioned before, this means that all the probability of the distribution of (or ) becomes concentrated at points close to , as increases.
Suppose we have applied a CLT, and we have:
The second definition of asymptotic variance, which does not require using any limit distribution, is obtained as:
If we consider the first approach of the asymptotic variance, the use of a CLT (see Judge, Carter, Griffiths, Lutkepohl and Lee (1988)) yields:
Finally, we should note that the finite sample efficiency implies asymptotic efficiency, and we could have used this fact to conclude the asymptotic efficiency of (or ), given the results of subsection about their finite sample properties.
With respect to the ML estimator of , which does not satisfy the finite sample unbiasedness (result (2.87)), we must calculate its asymptotic expectation. On the basis of the first definition of asymptotic unbiasedness, presented in (2.96), we have:
The second way to approach the asymptotic variance (see (2.104) ), leads to the following expressions:
As we have seen in the previous section, the quantlet gls allows us to estimate all the parameters of the MLRM. In addition, if we want to estimate the variance-covariance matrix of , which is given by , we can use the following quantlet