90
3. Decision Theoretic Approach
1. Introduction
Loss Function
We dene the loss function (t, ), which assigns disutility to each
set of an estimate t and parameter value . Examples are
(a) (t, ) = (t )2 , squared error loss
(b) (t, ) = |t |, absolute err

17
2. Random Variables, Distributions and Densities
1. Random Variables and Distributions
Let a probability space (, F , P) be given. We dene
Denition 1 A random variable X is a measurable function from to R.
The random variable X has two dening propertie

Advanced Probability and Statistics
for
Economists
Yoosoon Chang and Joon Y. Park Department of Economics
Indiana University
August 2011
c 2011 by Yoosoon Chang & Joon Y. Park All rights reserved.
Part I
Probability
2
1. Introduction to Probability
1. Pro

80
2. Estimation
1. Sample Analogue Estimation
Let x = (x1 , . . . , xn ) be an observation from X = (X1 , . . . , Xn ) . Assume that the
random variables X1 , . . . , Xn are independent and has common underlying distribution, which we parameterize as
P =

33
3. Expectations
1. Expectation
We dene the expectation E(X ) of a random variable X by
E(X ) = X dP.
More generally, let X be an n-dimensional random vector dened on a probability
space (, F , P) and f : Rn R be measurable, and dene
Ef (X ) = f (X ) dP

61
5. Multivariate Normal Distribution
1. Introduction
An n-dimensional random vector X is said to have (multivariate) normal distribution
with parameters and , and denoted by X =d N(, ) (or Nn (, ) to emphasize
the dimension of X ), if it has probability

Part II
Statistics
72
1. Preliminaries
1. Introduction to Statistical Inference
In this section, we introduce the basic framework for statistical inference.
General Setting Suppose that we are given n-numbers x1 , . . . , xn , which are believed to be gen

48
4. Families of Distributions and Transformations
1. Common Families of Distributions
In this section, we introduce some commonly used families of distributions. The index
(or indices) used to denote a member of a family is called the parameter.
Uniform

E571: ECONOMETRICS I - Statistical Foundations
FALL 2010
Class meets: 11:15 a.m. 12:30 p.m. MW in Wylie 101
Instructor: Yoon-Jin Lee, Wylie Hall 201, e-mail: lee243@indiana.edu
Economists are interested in economic inference, hypothesis testing, and forec

Joon Y. Park
Fall 2012
ECON E571
Indiana University
ECONOMETRICS I: STATISTICAL FOUNDATIONS
The lectures will follow closely the lecture notes, prepared by myself and Professor Yoosoon
Chang. Other references for additional reading are provided below.
Gra

1
Basic Matrix Theory
1. Matrix as a Linear Transformation
Let A be an n m matrix of real numbers. Instead of dening a matrix
as a rectangular array of real numbers, we view it as a linear function from
Rm to Rn .
First, A is interpreted as a function so

126
3. Asymptotics for Maximum Likelihood Estimation
In this section, we show consistency and asymptotic normality of maximum likelihood
estimator (MLE). The tests based on MLE, such as likelihood ratio (LR), Wald (W)
and Lagrange multiplier (LM) tests, a

121
2. Law of Large Numbers and Central Limit Theorem
1. Law of Large Numbers (LLN)
Let cfw_i be a sequence of random variables such that E i = 0. Then under general
regularity conditions
1 a .s . or p
i 0,
n i=1
n
which is called law of large numbers (L

Joon Y. Park
Fall 2011
ECON E571
Indiana University
ECONOMETRICS I: STATISTICAL FOUNDATIONS
The lectures will follow closely the lecture notes, prepared by myself and Professor Yoosoon
Chang. Other references for additional reading are provided below.
Gra

Part III
Asymptotic Theory
108
1. Introduction
1. Modes of Convergence
In this section, we will study various modes of convergence for a sequence cfw_Xn of
random variables.
Denition 1 (a.s. convergence) Let cfw_Xn be dened on a common probability space

97
4. Hypothesis Testing
1. Introduction
Let X = (X1 , . . . , Xn ) be a random sample, and suppose the distribution of X is
given by a parametric family P = cfw_P | . We partition the parameter set as
= 0 1 .
The null hypothesis is given by
H0 : 0 ,
whi