# Category INTRODUCTION TO STATISTICS AND ECONOMETRICS

## EXAMPLES OF HYPOTHESIS TESTS

In the preceding sections we have studied the theory of hypothesis testing. In this section we shall apply it to various practical problems.

EXAMPLE 9.6.1 (mean of binomial) It is expected that a particular coin is biased in such a way that a head is more probable than a tail. We toss this coin ten times and a head comes up eight times. Should we conclude that the coin is biased at the 5% significance level (more precisely, size)? What if the significance level is 10%?

From the wording of the question we know we must put (9.6.1) H0: p = У2 and Яр p > У2.

From Example 9.4.2, we know that we should use X ~ B(10,p), the number of heads in ten tosses, as the test statistic, and the critical region should be of the form

(9.6.2) R = {c, c + 1, . . . , 10},

where c (the critical value) should...

## Multinomial Model

We illustrate the multinomial model by considering the case of three alternatives, which for convenience we associate with three integers 1, 2, and 3. One example of the three-response model is the commuter’s choice of mode of transportation, where the three alternatives are private car, bus, and train. Another example is the worker’s choice of three types of employment: being fully employed, partially employed, and self-employed.

We extend (13.5.2) to the case of three alternatives as (13.5.8) Uu = x’u 0 + uu

Ub = X2; P + щі

Ubi = Хзі (З + uSi,

where (щі, Щі, иы)are i. i.d. It is assumed that the individual chooses the alternative with the largest utility. Therefore, if we represent the ith per­son’s discrete choice by the variable yv our model is defined by

Р(Уі = 1) =...

## Tests for Structural Change

Suppose we have two regression regimes

 (10.3.7) and Уь = a + 3i*T + ut, t= 1, 2, . . ■ • ,Ti (10.3.8) Tit = a + 32*21 + u2t, t = 1, 2, . . • , T2,

where each equation satisfies the assumptions of the model (10.1.1). We

9 9

denote Vun = crj and Vu% = a2 . In addition, we assume that (щ() and {u2t)
are normally distributed and independent of each other. This two-regres­sion model is useful to analyze the possible occurrence of a structural change from one period to another. For example, (10.3.7) may represent a relationship between у and x in the prewar period and (10.3.8) in the postwar period.

First, we study the test of the null hypothesis H0: (3j = (32, assuming

9 9

aj = ct2 under either the null or the alternative hypothesis...

## Continuous Sample

For the continuous case, the principle of the maximum likelihood estima­tor is essentially the same as for the discrete case, and we need to modify Definition 7.3.1 only slightly.

DEFINITION 7.S.2 Let (Xj, X%, . . . , X„) be a random sample on a con­tinuous population with a density function /(• 10), where 0 = Ob 02, ■ ■ • , 0jf), and let хг be the observed value of X,. Then we call L = IIf= f{xt | 0) the likelihood function of 0 given (x, x%,. . • , xn) and the value of 0 that maximizes L, the maximum likelihood estimator.

example 7.3.3 Let {X,-}, і = 1, 2, . . . , n, be a random sample on

2   N{p, a ) and let {x,| be their observed values. Then the likelihood func­tion is given by

7h 1 11

(7.3.14) log L = ~ log(2ir) – ^ log a2 – —- X (*« “ P)2- 2 2 2cr2 i=

E...

## GENERALIZED LEAST SQUARES

In this section we consider the regression model (13.1.1) у = X0 + u,

where we assume that X is a full-rank T X К matrix of known constants and u is a Г-dimensional vector of random variables such that £u = 0 and

(13.1.2) £uu’ = X.

We assume only that X is a positive definite matrix. This model differs from the classical regression model only in its general specification of the variance-covariance matrix given in (13.1.2).

## TESTING ABOUT A VECTOR PARAMETER

Those who are not familiar with matrix analysis should study Chapter 11 before reading this section. The results of this chapter will not be needed to understand Chapter 10. Insofar as possible, we shall illustrate our results in the two-dimensional case.

We consider the problem of testing H0: 0 = 00 against Ну 0 Ф 0O, where 0 is a A-dimensional vector of parameters. We are to use the test statistic 0 ~ N(Q, X), where X is а К X К variance-covariance matrix: that is, X = £(0 — 0)(0 — 0)’. (Throughout this section a matrix is denoted by a boldface capital letter and a vector by a boldface lower-case letter.) In

0,

FIGURE 9.9 Critical region for testing about two parameters

Section 9.7.1 we consider the case where X is completely known, and in Section 9.7...