First — and Second-Order Conditions
The following conditions guarantee that the first - and second-order conditions for a maximum hold. Assumption 8.1: The parameter space © is convex and в0 is an interior point of …
Series Expansion of the Complex Logarithm
For the case x є К, |x | < 1, it follows from Taylor’s theorem that ln(1 + x) has the series representation TO ln(1 + x) = ^(-1)*-1 xk …
Conditional Expectations
3.1. Introduction Roll a die, and let the outcome be Y. Define the random variable X = 1 if Y is even, and X = 0 if Y is odd. …
Applications of the Uniform Weak Law of Large Numbers
6.4.2.1. Consistency of M-Estimators Chapter 5 introduced the concept of a parameter estimator and listed two desirable properties of estimators: unbiasedness and efficiency Another obviously See Appendix II. desirable property …
Gaussian Elimination of a Nonsquare Matrix
The Gaussian elimination of a nonsquare matrix is similar to the square case except that in the final result the upper-triangular matrix now becomes an echelon matrix: Definition I.10: Anm …
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 …
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 …
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 …
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 …
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 …
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 …
Properties of Conditional Expectations
As conjectured following (3.10), the condition E(| Y|) < to is also a sufficient condition for the existence of E(E[Y|&0]). The reason is twofold. First, I have already established in …
Generalized Slutsky’s Theorem
Another easy but useful corollary of Theorem 6.10 is the following generalization of Theorem 6.3: Theorem 6.12: (Generalized Slutsky’s theorem) Let Xn a sequence of random vectors in Kk converging …
Subspaces Spanned by the Columns and Rows of a Matrix
The result in Theorem I.9 also reads as follows: A = BU, where B = P-1L is a nonsingular matrix. Moreover, note that the size of U is the same …
The Normal Distribution
Several univariate continuous distributions that play a key role in statistical and econometric inference will be reviewed in this section, starting with the normal distribution. The standard normal distribution emerges …
Dependent Central Limit Theorems
7.1.4. Introduction As is true of the conditions for asymptotic normality of M-estimators in the 1.1. d. case (see Chapter 6), the crucial condition for asymptotic normality of the NLLS …
Supremum and Infimum
The supremum of a sequence of real numbers, or a real function, is akin to the notion of a maximum value. In the latter case the maximum value is taken …
Applications to Statistical Inference under Normality
5.6.1. Estimation Statistical inference is concerned with parameter estimation and parameter inference. The latter will be discussed next in this section. In a broad sense, an estimator of a parameter …
Asymptotic Efficiency of the ML Estimator
The ML estimation approach is a special case of the M-estimation approach discussed in Chapter 6. However, the position of the ML estimator among the M - estimators is a …
Conditional Probability Measures and Conditional Independence
The notion of a probability measure relative to a sub-a - algebra can be defined as in Definition 3.1 using the conditional expectation of an indicator function: Definition 3.2: Let …
Convergence in Distribution
Let Xn be a sequence of random variables (or vectors) with distribution functions Fn (x), and let X be a random variable (or conformable random vector) with distribution function F(x). …
Projections, Projection Matrices, and Idempotent Matrices
Consider the following problem: Which point on the line through the origin and point a in Figure I.3 is the closest to point b? The answer is point p in …
Distributions Related to the Standard Normal Distribution
The standard normal distribution generates, via various transformations, a few other distributions such as the chi-square, t, Cauchy, and F distributions. These distributions are fundamental in testing statistical hypotheses, as …
A Generic Central Limit Theorem
In this section I will explain McLeish’s (1974) central limit theorems for dependent random variables with an emphasis on stationary martingale difference processes. The following approximation of exp(i ■ x) …
Limsup and Liminf
Let an (n = 1, 2,...) be a sequence of real numbers, and define the sequence bn as bn = sup am m>n Then bn is a nonincreasing sequence: bn …
Confidence Intervals
Because estimators are approximations of unknown parameters, the question of how close they are arises. I will answer this question for the sample mean and the sample variance in the …
Testing Parameter Restrictions
8.5.1. The Pseudo t-Test and the Wald Test In view of Theorem 8.2 and Assumption 8.3, the matrix Й can be estimated consistently by the matrix Й in (8.53): If …
Conditioning on Increasing Sigma-Algebras
Consider a random variable Y defined on the probability space {^, &, P} satisfying E [|Y |] < to, and let &n be a nondecreasing sequence of sub-a-algebras of &: …
Convergence of Characteristic Functions
Recall that the characteristic function of a random vector X in Kk is defined as p(t) = E [exp(itTX)] = E [cos(tTX)] + i ■ E [sin(tTX)] for t e …
Inner Product, Orthogonal Bases, and Orthogonal Matrices
It follows from (I.10) that the cosine of the angle y between the vectors x in (I.2) and y in (I.5) is Figure I.5. Orthogonalization. Definition I.13: The quantity x …
The Student’s t Distribution
Let X ~ N(0, 1) and Yn ~ x2, where X and Yn are independent. Then the distribution of the random variable VYn/n is called the (Student’s2) t distribution with …
A.2. A Hilbert Space of Random Variables
Let U0 be the vector space of zero-mean random variables with finite second moments defined on a common probability space {&, .^, P} endowed with the innerproduct (X, Y) = …
Continuity of Concave and Convex Functions
A real function p on a subset of a Euclidean space is convex if, for each pair of points a, b and every X e [0, 1], p(Xa + (1 …
B. Extension of an Outer Measure to a Probability Measure
To use the outer measure as a probability measure for more general sets that those in F0, we have to extend the algebra F0 to a a-algebra F of events …
Testing Parameter Hypotheses
Suppose you consider starting a business to sell a new product in the United States such as a European car that is not yet being imported there. To determine whether …
