The Lagrange Multiplier Test
The restricted ML estimator в can also be obtained from the first-order conditions of the Lagrange function <(в, /г) = ln(Ln (в)) - §2г, where д є Kr is a …
Conditional Expectations as the Best Forecast Schemes
I will now show that the conditional expectation of a random variable Y given a random variable or vectorXis the best forecasting scheme for Yin the sense that the mean-square …
The Central Limit Theorem
The prime example of the concept of convergence in distribution is the central limit theorem, which we have seen in action in Figures 6.4—6.6: Theorem 6.23: Let X1,Xn be i. …
Determinants of Block-Triangular Matrices
Consider a square matrix A partitioned as where A1,1 and A2 2 are submatrices of size k x k and m x m, respectively, A12 is a k x m …
The Standard Cauchy Distribution
The ti distribution is also known as the standard Cauchy distribution. Its density is Г(1) VnT(1/2)(1 + x2) 1 n (1 + x2)’ h1(x) (4.39) where …
A.5. Proof of the Wold Decomposition
Let Xt be a zero-mean covariance stationary process and E[X2] = a2. Then the Xt’s are members of the Hilbert space U0 defined in Section 7.A.2. Let S—TO be the …
Uniform Continuity
A function g on Кк is called uniformly continuous if for every є > 0 there exists a 8 > 0 such that |g(x) - g(y)| < є if ||x …
Borel Measurability, Integration, and Mathematical Expectations
2.1. Introduction Consider the following situation: You are sitting in a bar next to a guy who proposes to play the following game. He will roll dice and pay you …
Applications to Regression Analysis
5.1.1. The Linear Regression Model Consider a random sample Zj = (Yj, Xj)T, j = 1, 2,...,n from a ^-variate, nonsingular normal distribution, where Yj є К, Xj є R-1. …
Selecting a Test
The Wald, LR, and LM tests basically test the same null hypothesis against the same alternative, so which one should we use? The Wald test employs only the unrestricted ML …
Distributions and Transformations
This chapter reviews the most important univariate distributions and shows how to derive their expectations, variances, moment-generating functions (if they exist), and characteristic functions. Many distributions arise as transformations of …
Asymptotic Normality of M-Estimators
This section sets forth conditions for the asymptotic normality ofM-estimators in addition to the conditions for consistency An estimator в of a parameter в0 є Km is asymptotically normally distributed …
Inverse of a Matrix in Terms of Cofactors
Theorem I.31 now enables us to write the inverse of a matrix A in terms of cofactors and the determinant as follows. Define Definition I.20: The matrix is called the …
The Uniform Distribution and Its Relation to the Standard Normal Distribution
As we have seen before in Chapter 1, the uniform [0,1] distribution has density f (x) = 1 for 0 < x < 1, f (x) = 0 elsewhere. More …
Maximum Likelihood Theory
8.1. Introduction Consider a random sample Z ь..., Zn from a ^-variate distribution with density f (z0), where в0 є © c 1” is an unknown parameter vector with © …
The Mean Value Theorem
Consider a differentiable real function f (x) displayed as the curved line in Figure II.1. We can always find a point c in the interval [a, b] such that the …
Borel Measurability
Let g be a real function and let X be a random variable defined on the probability space {^, P}. For g(X) to be a random variable, we must have …
Least-Squares Estimation Observe that
E[(Y - X0)T(Y - X0)] = E[(U + X(00 - 0))T(U + X0 - 0))] = E[UTU] + 2(00 - 0)TE(XTE[U|X]) + (00 - 0 )T(E [XTX])(00 - 0) = …
Appendix I — Review of Linear Algebra
I.1. Vectors in a Euclidean Space A vector is a set of coordinates that locates a point in a Euclidean space. For example, in the two-dimensional Euclidean space K2 the …
The Binomial Distribution A random variable X has a binomial distribution if
P(X = k) =^n^jpk(1 - p)n-k for k = 0, 1, 2,..., n, P(X = k) = 0 elsewhere, (4.3) where 0 < p < 1. This distribution arises, for …
B.2. Slutsky’s Theorem
Theorem 6.B.1 can be used to prove Theorem 6.7. Theorem 6.3 was only proved for the special case that the probability limit Xis constant. However, the general result of Theorem …
Eigenvalues and Eigenvectors
I.15.1. Eigenvalues Eigenvalues and eigenvectors play a key role in modern econometrics - in particular in cointegration analysis. These econometric applications are confined to eigenvalues and eigenvectors of symmetric matrices, …
The Multivariate Normal Distribution and Its Application to Statistical Inference
5.1. Expectation and Variance of Random Vectors Multivariate distributions employ the concepts of the expectation vector and variance matrix. The expected “value” or, more precisely, the expectation vector (sometimes also …
Likelihood Functions
There are many cases in econometrics in which the distribution of the data is neither absolutely continuous nor discrete. The Tobit model discussed in Section 8.3 is such a case. …
Taylor’s Theorem
The mean value theorem implies that if, for two points a < b, f (a) = f (b),then there exists a point c є [a, b] such that f'(c) = …
Borel Sets
An important special case of Definition 1.4 is where ^ = К and © is the collection of all open intervals: © = {(a, b): Va < b, a, b …
Introduction to the Mathematical and Statistical Foundations of Econometrics
This book is intended for use in a rigorous introductory Ph. D.-level course in econometrics or in a field course in econometric theory. It is based on lecture notes that …
Outer Measure
Any subset A of [0,1] can always be completely covered by a finite or countably infinite union of sets in the algebra ^0: A c U°=j Aj, where Aj є …
Probability and Measure
1.1. The Texas Lotto 1.1.1. Introduction Texans used to play the lotto by selecting six different numbers between 1 and 50, which cost $1 for each combination.[1] Twice a week, …
Lebesgue Integral
The Lebesgue measure gives rise to a generalization of the Riemann integral. Recall that the Riemann integral of a nonnegative function f (x) over a finite interval (a, b] is …
Binomial Numbers
In general, the number of ways we can draw a set of k unordered objects out of a set of n objects without replacement is / n def. n! Ы …
Random Variables and Their Distributions
1.8.1. Random Variables and Vectors In broad terms, a random variable is a numerical translation of the outcomes of a statistical experiment. For example, flip a fair coin once. Then …
Sample Space
The Texas lotto is an example of a statistical experiment. The set of possible outcomes of this statistical experiment is called the sample space and is usually denoted by ^. …
Density Func tions
An important concept is that of a density function. Density functions are usually associated to differentiable distribution functions: Definition 1.13: The distribution of a random variable X is called absolutely …
Probability Measure
Let us return to the Texas lotto example. The odds, or probability, of winning are 1 /N for each valid combination rnj of six numbers; hence, if you play n …
