Category Introduction to the Mathematical and Statistical Foundations of Econometrics

The Nonlinear Case

If we denote G (x) = Ax + b, G-i(y) = A-i( y – b), then the result of Theo­rem 4.3 reads h(y) = f (G-i(y))|det(9G-i(y)/9y)|. This suggests that Theo­rem 4.3 can be generalized as follows:

Theorem 4.4: Let X be k-variate, absolutely continuously distributed with joint density f(x),x = (xi,…,xk)T, and let Y = G(X), where G(x) = (gi(x),…, gk(x))T is a one-to-one mapping with inverse mapping x = G-i(y) = (gj(y),…,gl(y))T whose components are differentiable in the components ofy = (yi;yk)T. Let J(y) = dx/dy = dG-i(y)/dy, that is, J(y) is the matrix with i, j’s element dgf(y)/dyj, which is called the Jacobian...

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Uniform Weak Laws of Large Numbers

7.1.1. Random Functions Depending on Finite-Dimensional Random Vectors

On the basis of Theorem 7.7, all the convergence in probability results in Chapter 6 for i. i.d. random variables or vectors carry over to strictly stationary time series processes with an a-mixing base. In particular, the uniform weak law of large numbers can now be restated as follows:

Подпись: plimn image545 Подпись: = 0.

Theorem 7.8(a): (UWLLN) Let Xt be a strictly stationary k-variate time se­ries process with an a-mixing base, and let в e © be nonrandom vectors in a compact subset © c Km. Moreover, letg(x, в) be a Borel-measurablefunction on Kk x © such that for each x, g(x, в) is a continuous function on ©. Finally, assume that E[supee©|g(X,,в)|] < to. Then

Theorem 7...

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Appendix II – Miscellaneous Mathematics

This appendix reviews various mathematical concepts, topics, and related re­sults that are used throughout the main text.

11.1. Sets and Set Operations

11.1.1. General Set Operations

The union A U B of two sets A and B is the set of elements that belong to either A or B or to both. Thus, if we denote “belongs to” or “is an element of” by the symbol e, x e A U B implies that x є A or x є B, or in both, and vice versa. A finite union Un=1 Aj of sets Ai,…, An is the set having the property that for each x e Un=1 Aj there exists an index i, 1 < i < n, for which x e Ai, and vice versa: Ifx e Ai for some index i, 1 < i < n, thenx e Un=1 Aj...

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Distributions of Quadratic Forms of Multivariate Normal Random Variables

As we will see in Section 5.6, quadratic forms of multivariate normal random variables play a key role in statistical testing theory. The two most important results are stated in Theorems 5.9 and 5.10:

Theorem 5.9: Let X be distributed Nn(0, Y), where Y is nonsingular. Then XT£-1 X is distributed as x„.

Proof: Denote Y = (Y1,…, Yn)T = Y- /2X. Then Yis n-variate, standard normally distributed; hence, Y1,…,Yn are independent identically distributed (i. i.d.) N(0, 1), and thus, XT Y-1 X = YTY = Ynj=1 Y2 – x2. Q. E.D.

The next theorem employs the concept of an idempotent matrix. Recall from Appendix I that a square matrix M is idempotent if M2 = M...

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Generic Conditions for Consistency and Asymptotic Normality

The ML estimator is a special case of an M-estimator. In Chapter 6, the generic conditions for consistency and asymptotic normality of M-estimators, which in most cases apply to ML estimators as well, were derived. The case (8.11) is one of the exceptions, though. In particular, if

Assumption 8.2: plimn^msupee©ln(L„(в)/Ln(в0)) _ E[ln(L„(в)/Ln(0o))]

| = 0 and limn^m supeЄ©E[ln(Ln(в)/Ln(в0))] — і(вв0) = 0, where і(вв0) is a continuous function in в0 such that, for arbitrarily small 8 > 0,

supeЄ©:\в—в0\>8^(вв0) < 0,

then the ML estimator is consistent.

Theorem 8.3: Under Assumption 8.2, plim^^O = в0.

The conditions in Assumption 8.2 need to be verified on a case-by-case basis...

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Appendix IV – Tables of Critical Values

Table IV1: Critical values of the two-sided tk test at the 5% and 10% significance levels

k

5%

10%

k

5%

10%

k

5%

10%

і

12.704

6.313

11

2.201

1.796

21

2.080

1.721

2

4.303

2.920

12

2.179

1.782

22

2.074

1.717

3

3.183

2.353

13

2.160

1.771

23

2.069

1.714

4

2.776

2.132

14

2.145

1.761

24

2.064

1.711

5

2.571

2.015

15

2.131

1.753

25

2.059

1.708

6

2.447

1.943

16

2.120

1.746

26

2.056

1.706

7

2.365

1.895

17

2.110

1.740

27

2.052

1.703

8

2.306

1.859

18

2.101

1.734

28

2.048

1.701

9

2.262

1.833

19

2.093

1.729

29

2.045

1.699

10

2.228

1.813

20

2.086

1.725

30

2.042

1.697

Table IV2: Cr...

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