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Introduction to Econometrics - Small and large sample properties of estimators

[Home] [Up] [Probability] [Axiom System] [Bayes Theorem] [Random Variables] [Matrix Algebra] [Distribution Theory] [Estimator Properties]

The property of unbiasedness (for an estimator of theta) is defined by

Introduction to Econometrics - Small and large sample properties of estimators


where the biasvector delta can be written as


and the precision vector as


which is a positive definite symmetric K by K matrix.

If two different estimators of the same parameter exist one can compute the difference between their precision vectors: if this vector is positive semi definite this means we know that the second estimator has a "smaller" covariance matrix and can therefore be called better than the first estimator.

An estimator is said to be efficient if it is unbiased and at the same the time no other estimator exists with a lower covariance matrix.

If Y is a random variable of independent observations with a probability distribution f then the joint distribution can be written as


The function of the unknown parameter, as a function of the values of the random variable, is called the likelihood function which has the same structure as the joint probability function but is dependent on the random variable in stead of the unknown parameter.

The information matrix is defined as the negative of the expected value of the Hessian matrix of the log likelihood function L



The Cramér-Rao lower bound is defined as the inverse of the information matrix


here denoted omega.

If an estimator is unbiased then


is a positive semi definite matrix. Expression (I.VI-6) is called the Cramér-Rao inequality.

Proof of this inequality can be easily obtained. If we con­sider only one parameter, by definition of the likelihood function we may write


which can be derived with respect to the parameter


Deriving a second time yields


This implies that E((D ln L)2) = - E(D2 ln L) which is e­quivalent to the information matrix.

If the estimator is unbiased then


It follows from (I.VI-10) that


On combining (I.VI-13) with (I.VI-12) and applying the Cauchy-Schwarz inequality we obtain


from which the Cramér-Rao inequality follows immediately.

Note that according to the Cramér-Rao lower bound


but not vice versa. This is because the Cramér-Rao lower bound is not always attainable (for unbiased estimators).

The property of sufficiency can be formulated as


while the property of consistency is defined as


where delta is a small scalar and epsilon is a vector containing elements with "small" values.

The large sample properties apply only when the number of observations converges towards infinity in the limit. Accordingly, we can define the large sample consistency as


where epsilon is "small".

By definition we can also use a shorter notation


were "plim" is the so-called "probability limit". In this case we say that the estimator for theta converges in probability to the population value of theta.

A short example will clarify the concept of large sample consistency. Let us take the sample mean as an estimator of the population mean. Then it is possible to prove large sample consistency on using eq. (I.III-47) applied to the sample mean:


The standard deviation of the sample mean is known to be


On combining (I.VI-20) and (I.VI-21) we obtain



Now it obvious that


where the RHS can be made arbitrarily close to 1 by increasing T (the number of sample observations). Now we may conclude


A sufficient, but not necessary, condition for large sample efficiency is


According to Slutsky's theorem the following holds



Other properties of plims are




(this is true even if both estimators are dependent on each other: this is not so with the mathematical expectation) and finally


where AT is a square parameter matrix.

Note the following definition of asymptotically distributed parameter vectors



The concept of asymptotic efficiency can be used to compare estimators. Formally this is written:





Finally we describe Cramér's theorem because it enables us to combine plims with convergence in distribution. Formally this theorem states that if




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Axiom System
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Random Variables
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