Category INTRODUCTION TO STATISTICS AND ECONOMETRICS

Neither of the two strategies discussed in Section 7.2.4 is the primary strategy of classical statisticians, although the second is less objectionable to them. Their primary strategy is that of defining a certain class of estimators within which we can find the best estimator in the sense of Definition 7.2.1. For example, in Example 7.2.1, if we eliminate W and Z from our consideration, T is the best estimator within the class consisting of only T and S. A certain degree of arbitrariness is unavoidable in this strategy. One of the classes most commonly considered is that of linear unbiased estimators. We first define

DEFINITION 7.2.4 0 is said to be an unbiasedestimator of 0 if £0 = 0 for

all 0 Є 0. We call £0-0 bias.

Among the three estimators in Example 7.2...

The t test is ideal when we have a single constraint, that is, q = 1. The F test, discussed in the next section, must be used if q > 1.

Since (J is normal as shown above, we have

(12.4.7) Q’0 ~ N[c, a2Q’ (X’X^Q] under the null hypothesis (that is, if Q’3 = c). Note that here Q’ is a row vector and c is a scalar. Therefore,

Q’0-r

(12.4.8) т z ~ N(0, 1).

<Wq'(x’x)-1q

The random variables defined in (12.4.2) and (12.4.8) are independent because of Theorem 12.4.2. Hence, by Definition 2 of the Appendix, we have

QP ~ c

ctVQ'(X’X)_1Q

Student’s t with T — К degrees of freedom, where a is the square root of

л

the unbiased estimator of a defined in equation (12.2.29). Note that the denominator in (12.4.9) is an estimate of the standard deviation of the numerator...

We have so far considered only situations in which both the null and the alternative hypotheses are simple in the sense of Definition 9.1.1. Now we shall turn to the case where the null hypothesis is simple and the alterna­tive hypothesis is composite.

We can mathematically express the present case as testing H0: 0 = 0O against Hi: 0 Є 01; where 0X is a subset of the parameter space. If 0j consists of a single element, it is reduced to the simple hypothesis consid­ered in the previous sections. Definition 9.2.4 defined the concept of the most powerful test of size a in the case of testing a simple against a simple

(i +e,)2

hypothesis...

The qualitative response model or discrete variables model is the statistical model that specifies the probability distribution of one or more discrete depend­ent variables as a function of independent variables. It is analogous to a regression model in that it characterizes a relationship between two sets
of variables, but differs from a regression model in that not all of the information of the model is fully captured by specifying conditional means and variances of the dependent variables, given the independent variables. The same remark holds for the models of the subsequent two sections.

The qualitative response model originated in the biometric field, where it was used to analyze phenomena such as whether a patient was cured by a medical treatment, or whether insects died after the a...

The need to predict a value of the dependent variable outside the sample (a future value if we are dealing with time series) when the corresponding value of the independent variable is known arises frequently in practice. We add the following “prediction period” equation to the model (10.1.1):

(10.2.82) yp = a + $xp + up,

where jp and up are both unobservable, xp is a known constant, and up is independent of {ut}, t = 1, 2, . . . , T, with Eup = 0 and Vup = a2. Note that the parameters a, (3, and cr2 are the same as in the model (10.1.1). Consider the class of predictors of yp which can be written in the form

(10.2.83) yp = a + $xp,

where a and (3 are arbitrary unbiased estimators of a and (3, which are linear in {yt}, t = 1, 2, . . . , T...

Thus far we have discussed only the finite sample properties of estimators. It is frequently difficult, however, to obtain the exact moments, let alone the exact distribution, of estimators. In such cases we must obtain an approximation of the distribution or the moments. Asymptotic approxima­tion is obtained by considering the limit of the sample size going to infinity. In Chapter 6 we studied the techniques necessary for this most useful approximation.

One of the most important asymptotic properties of an estimator is consistency.

DEFINITION 7.2.5 We say 0 is a consistent estimator of 0 if plim„^oo 0 = 0. (See Definition 6.1.2.)

In Examples 6.4.1 and 6.4...