Category INTRODUCTION TO STATISTICS AND ECONOMETRICS

DETERMINANTS AND INVERSES

Throughout this section, all the matrices are square and n X n.

Before we give a formal definition of the determinant of a square matrix, let us give some examples. The determinant of a 1 X 1 matrix, or a scalar, is the scalar itself. Consider a 2 X 2 matrix

д __ an an

Cl 2i #22

Its determinant, denoted by |A| or det A, is defined by (11.3.1) |A| = <211^22 — <221«12-

The determinant of а З X 3 matrix

ап

а12

аЪ

а21

а 22

а23

а31

а32

аЪЪ

Подпись: (11.3.2) Подпись: |A| — an Подпись: a22 «23 «32 Язз Подпись: a2i Подпись: «12 e32 Подпись: «13 a33 Подпись: + «3! Подпись: «12 «13 «22 «23

is given by

= «цяггя’зз — «11«32«23 — «21«12«33 + «21 «32 «13 + «31 «12«23 — «31«22«13 •

Now we present a formal definition, given inductively on the assumption that the determinant of an (n – 1) X (n — 1) matrix has already been defined.

DEFINITION 11...

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CONFIDENCE INTERVALS

We shall assume that confidence is a number between 0 and 1 and use it in statements such as “a parameter 0 lies in the interval [a, b with 0.95 confidence,” or, equivalently, “a 0.95 confidence interval for 0 is [a, b].” A confidence interval is constructed using some estimator of the parameter in question. Although some textbooks define it in a more general way, we shall define a confidence interval mainly when the estimator used to construct it is either normal or asymptotically normal. This restriction is not a serious one, because most reasonable estimators are at least asymp­totically normal. (An exception occurs in Example 8.2.5, where a chi – square distribution is used to construct a confidence interval concerning a variance...

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Error Components Model

The error components model is useful when we wish to pool time-series and cross-section data. For example, we may want to estimate production functions using data collected on the annual inputs and outputs of many firms, of demand functions using data on the quantities and prices col­lected monthly from many consumers. By pooling time-series and cross – section data, we hope to be able to estimate the parameters of a relation­ship such as a production function or a demand function more efficiently than by using two sets of data separately. Still, we should not treat time – series and cross-section data homogeneously. At the least, we should try to account for the difference by introducing the specific effects of time and cross-section into the error term of the regression, as follows:

(1...

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Properties of a. and j3

First, we obtain the means and the variances of the least squares estimators a and (3. For this purpose it is convenient to use the formulae (10.2.12) and (10.2.16) rather than (10.2.4) and (10.2.5).

Inserting (10.1.1) into (10.2.12) and using (10.2.17) yields

(10.2.19) Подпись: by ut Пхї? P – (3 =

Since Eut = 0 and {x*’ are constants by our assumptions, we have from

(10.2.19) and Theorem 4.1.6,

(10.2.20) = p.

In other words, p is an unbiased estimator of p. Similarly, inserting

(10.1.1) into (10.2.16) and using (10.2.18) yields

Подпись:(10.2.21) d – a =

which implies

(10.2.22) Eol = a.

Using (10.2.19), the variance of P can be evaluated as follows:

(10.2.23) Up =——– ^ U(E x*ut) by Theorem 4.2.1

image522 Подпись: by Theorem 4.3.3

[Ц*?)2]2

S(xf)2

Similarly, we obtain from (10.2.21)

2

(10.2.24) Ud =————-

mn2

How good are the least squares est...

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Nonparametric Estimation

In parametric estimation we can use two methods.

(1) Distribution-specific method. In the distribution-specific method, the distribution is assumed to belong to a class of functions that are characterized by a fixed and finite number of parameters—for example, normal—and these parameters are estimated.

(2) Distribution-free method. In the distribution-free method, the distribution is not specified and the first few moments are estimated.

In nonparametric estimation we attempt to estimate the probability distribution itself. The estimation of a probability distribution is simple for a discrete random variable taking a few number of values but poses problems for a continuous random variable...

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SIMULTANEOUS LINEAR EQUATIONS

Throughout this section, A will denote an n X n square matrix and X a matrix that is not necessarily square. Generally, we shall assume that X is n X К with К ^ n.

Consider the following n linear equations:

anxi + a12x2 + ■ ■ ■ + alnxn = y

CL%X} T CL22X2 T ‘ ‘ ‘ T Cl2nXn y2

(11.4.1)

ani*i "f "f ■ ■ ■ T annxn yn

Define x = (xj, x2,. . . , x„)’ and у = (уь Уъ • • • > Уп)’ an(l let A be as in

(11.1.1) with n = m. Then (11.4.1) can be written in matrix notation as

(11.4.1) Ax = y.

A major goal of this section is to obtain a necessary and sufficient condi­tion on A such that (11.4.2) can be solved in terms of x for any y. Using the notation 3 (there exists), V (for any), and s. t. (such that), we can express the last clause of the previous sentence as

V у 3 x s. t...

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