Slides 01

Advanced Econometrics I

Ingo Steinke, Anne Leucht, Enno Mammen

University of Mannheim

Fall 2014

Ingo Steinke (Uni Mannheim) Advanced Econometrics I Fall 2014 1

Organisation

Important dates

Start: 2013-10-07

End: 2013-12-05

Lectures:

Tuesday 10:15 - 11:45 in L 7, 3-5 - 001Thursday 10:15 - 11:45 in L 7, 3-5 - 001

Exercises:

Thursday 13:45 - 15:15 in L 9, 1-2 003Thursday 15:30 - 17:00 in L 9, 1-2 003teaching assistants: Maria Marchenko

Slides will be provided via Ilias, usually on Friday for the next week.


Exercise sheets

will be provided via Ilias and usually published on Tuesday (orWednesday).

hand in written solutions in lecture on Tuesday (you may work inpairs).

discussion of the solutions on Thursday.

There is 1 point per exercise (0.25,0.5,0.75,1).

You need 75% of the points of the Exercise sheet to get (at most)20% of the exam points.

There will be stared exercises which can be used to make up formissing points of one exercise sheet.


Exam

written exam, 180 min

Date: 2014-12-17

Contact

Office: L7, 3 - 5, room 142

Phone: 1940

E-Mail: [email protected]

Office hour: on appointment


Contents

Overview:

1 Probability theory

2 Asymptotic theory

3 Conditional expectations

4 Linear regression


Literature

Ash, R. B. and Doleans-Dade, C. (1999). Probability & MeasureTheory. Academic Press.

Billingsley, P. (1994). Probability and Measure. Wiley.

Hayashi, F. (2009). Econometrics. Princeton University Press.

Jacod, J. and Protter, P. (2000). Probability Essentials. Springer.

Van der Vaart, A. W. and Wellner, J. A. (2000). Weak Convergenceand Empirical Processes. With Applications to Statistics. New York:Springer.

Wooldridge, J. M. (2004). Introductory Econometrics: A ModernApproach. Thomson/Southwestern.


IntroductionMotivation

Application in Statistics ...

Study relationship between variables, e.g.

consumption and income How does raising income effect consumption behaviour?evaluation of effectiveness of job market training (treatment effects). . .

Econometrics (Wooldridge (2004)):

development of statistical methods for estimating economicrelationshipstesting economic theoriesevaluation of government and business policies


Classical model in econometrics: linear regression

Figure: http://en.wikipedia.org/wiki/File:Linear regression.svg

Y = 0 + 1X + u,

e.g. Y consumption, X wage, u error term typically data not generated byexperiments, error term collects all other effects on consumption besideswage variables somehow random How do we formalize randomness?


Aims of this course:

(1) Provide basic probabilistic framework and statistical tools foreconometric theory.

(2) Application of these tools to the classical multiple linear regressionmodel.

Application of these results to economic problems in AdvancedEconometrics II/III and follow-up elective courses.


Chapter 1: Probability theory

Chapter 1: Elementary probability theoryOverview

1 Probability measures

2 Probability measures on R3 Random variables

4 Expectation


Chapter 1: Probability theory 1.1 Probability measures

1.1 Probability measures

Aim: Formal description of probability measures

Setup:

The set 6= of the possible outcomes of an random experiment iscalled sample space, e.g. = N = {1, 2, }.A event, e.g. A = {2, 4, 6, 8, }outcome:

Want to assign a probability P(A) to event AConsider first the case that is a countable set, i.e.

= {1, 2, 3, }

(e.g. = N, = Z).



= { } denotes the empty set,P() = {A : A } the power set.

Defintion 1.1A probability measure P on a countable set is a set function thatmaps subsets of to [0, 1], i.e. P : P() [0, 1], and has the followingproperties:

(i) P() = 1.

(ii) It holds

P(i=1

Ai ) =i=1

P(Ai )

for any Ai , i N, that are pairwise disjoint, i.e Ai Aj = fori 6= j .



Recap: Index notation

Let I 6= some set and Ai for all i I . Thenx

iI

Ai j I : x Aj .

If I = {1, . . . , n} and J = N, theniI

Ai = A1 An =n

i=1

Ai ,

jJ

Aj = A1 An =j=1

Aj .

Especially, for I = A and Ax = {x},

A =xA{x}.



Recap: Series

A set is countable iff there is a set N N and a bijection(one-to-one-map) m : N A. Then A can be written A = {a1, a2, . . . , an}or A = {a1, a2, . . . , an, . . . }.A series is a infinite sum and defined by

s =k=1

ak = limn

nk=1

ai

if the limit exists. The series s is absolutely convergent if

k=1

|ak |


The series s is unconditionally well-defined if for any{k1, k2, k3, . . . } = N we have s =

i=1 aki , i.e. a rearrangement of its

members does not change the (infinite) sum which might be oder .Note:

If a series is absolutely convergent, then it is unconditionallyconvergent.

A series is unconditionally well-defined iff the (infinite) sums of all itspositive members or the sum of all negative members is finite.

Let I be countable and ai R for any i I . If I = {i1, i2, . . . }, theniI

ai :=j=1

aij ,

if the right-hand series is unconditionally convergent



Lemma 1.2For a countable sample space = {i}iI (with countable I ) and aprobability measure P on . Then for every A , it holds

P(A) =A

P({}).

Proof: Exercise.

Let . An event {} that only contains one element is also called anelementary event.



Arbitrary sample spaces

It is often impossible to define P appropriately for all subsets A such that Definition 1.1 holds true; see e.g. Billingsley (1994).

Definition 1.3 A family A of subsets of with(i) A,(ii) if A A, then AC = \A A,(iii) if A1,A2, A, then

i=1 Ai A,

is called a -field or -algebra.

For a -field A of holds: A P().



A -field A is called smallest--field, containing B P(), if for any-field C of holds: If B C, then A C.Notation: A = (B).

Example 1.4 Let 6= be a set.{, } is the smallest -field on and is called trivial -field.The power set P() is the largest -field on .If 6= B , the family {, ,B,BC} is the smallest -field on that contains B.

Suppose that A is a -field on a set . Then the tuple (,A) is called ameasurable space.



Definition 1.5A set function P : A [0,) is a measure on (,A) if for A1,A2, . . .pairwise disjoint A it holds that

P

( i=1

Ai

)=i=1

P(Ai ) ( additivity)

If, in addition, P() = 1, then it is called probability measure.

The triple (,A,P) is then called a probability space.Example 1.6 Let (,A) be a measurable space and 0 .Then (A) = |A|, A A, is the so-called counting measure.The Dirac measure 0 is then defined by

0(A) := 1A(0) A A.



Theorem 1.7 (Properties of probability measures)Suppose that (,A,P) is probability space. Let A,B,A1,A2, A.Then holds:

(i) P() = 0(ii) Finite additivity: A1, . . . ,An pairwise disjoint imply

P(n

i=1

Ai ) =n

i=1

P(Ai ).

(iii) P(AC ) = 1 P(A).(iv) P(A) 1 for all A A.(v) Subtractivity: A B implies P(B\A) = P(B) P(A).



(vi) Monotonicity: A B implies P(A) P(B).(vii) P(A B) = P(A) + P(B) P(A B).(viii) Continuity from below: An An+1 for all n N implies

P(An) nP(

k=1

Ak).

(ix) Continuity from above: An+1 An for all n N impliesP(An)

nP(k=1

Ak).

(x) Sub--additivity: P(n=1

An) n=1

P(An).

Proof: Exercise.


Chapter 1: Probability theory 1.2 Probability measures on R

1.2 Probability measures on R

Definition 1.8 The smallest -field B, that contains all open intervals(a, b) ( a b ), is called the Borel -field.A set A B is called a Borel set.

Theorem 1.9 Put

A1 = {(a, b] : a < b < +},A2 = {[a, b) : < a < b +},A3 = {[a, b] : < a b < +},A4 = {(, b] : < b < +}.

Then it follows for j = 1, . . . , 4: B = (Aj).

Proof: Exercise.



Definition 1.10A class A of subsets of is a field if

(i) A,(ii) A A, then AC A,(iii) A1,A2 A, then A1 A2 A.

Suppose that A is a field and define A as the smallest -field withA A (notion: A = (A)). Then a set function P : A [0,) s.t.for A1,A2, . . . pairwise disjoint A with

i=1 Ai A it holds that

P( i=1

Ai

)=i=1

P(Ai )

is called pre-measure.If, in addition, P() = 1 it is called probability pre-measure.



Theorem 1.11 (Caratheodory) Let A be a field, A = (A) and Pa probability pre-measure on A. Then there exists a uniqueprobability measure P on A with

P(A) = P(A) for A A.

For a proof see Ash and Doleans-Dade (1999), Theorem 1.3.10.

Definition 1.12 For a probability measure P on (R,B) the functionF : R [0, 1] given by

F (b) = P((, b]) b R

is called a (cumulative) distribution function (CDF).



Proposition 1.13 (Properties of the CDF) Suppose that F is thedistribution function of a probability measure P on (R,B). Then

(i) P((a, b]) = F (b) F (a) for a < b,(ii) F is non-decreasing (i.e. F (a) F (b) for a b),(iii) F is continuous from the right (i.e. F (bn) F (b) for bn b,

bn b (or for bn b)),(iv) limx F (x) = 0 and limx+ F (x) = 1.(v) F (b) := limn F (bn) = P((, b)) for any bn b.

(vi) P({b}) = F (b) F (b) for all b R.

Define P by F on A1: P((a, b]) = F (b) F (a), a < b.Can this function be uniquely extended to a set function on B?



Theorem 1.14 Consider a function F : R R satisfying (ii) to (iv) ofProposition 1.13. Then F is a distribution function (i.e. then thereexists a unique probability measure P on (R,B) withF (b) = P((, b]) for all b R).

Some ideas of the proof: First, define a set function P : A1 [0, 1] as

P((a, b]) = F (b) F (a).

Extend this function as follows: P : A [0, 1], where A consists of theempty set and all finite unions of sets of A1 and their complements, andfor disjoint intervals

P(

ni=1

(ai , bi ]

)=

ni=1

P((ai , bi ]) with notation (c,] = (c ,).



Discrete probability measures

The function f : R R is called probability mass function (pmf) if1 f (x) 0 for all x R and2 it holds

xSf f (x) = 1 with Sf = {x R : f (x) > 0}.

Note that Sf must be countable if 2. holds. Sf is called support of f .Define

P(A) =

xSf Af (x). (1)

Lemma 1.15 P, defined by (1), is a probability measure.

Then, the cdf is defined by F (x) =

aSf (,x] f (a).



Definition 1.16 A probability measure P on the measurable space (R,B)is discrete if there is an at most countable set A R such that P(A) = 1.By

SP = {a R : P({a}) > 0}

we denote the support of a discrete probability measure R.

Lemma 1.17 P is discrete iff f : R R, f (x) = P({x}), is a pmf.

Remark 1.18

1 SP A is countable and P(SP) = 1.2 If P is a discrete probability measure with support SP then F has

jumps at a SP with jump heights P({a}).



Example 1.19

1 Binomial distribution

P({i}) =(n

i

)pii (1 pi)(ni) for i = 0, 1, . . . , n,

P({i}) = 0 elsewhere. Parameter: 0 pi 1, n 1.2 Geometric distribution

P({i}) = (1 pi)i1pi for i = 1, 2, 3 . . .

P({i}) = 0 elsewhere. Parameter: 0 pi 1.3 Poisson distribution

P({i}) = i

i !e, i = 0, 1, 2, . . .

P({i}) = 0 elsewhere. Parameter: > 0.



Absolultely continuous probability measures

A (Riemann) integrable function f : R R is called probability densityfunction (pdf) if

1 f (x) 0 for all x and2 f (x)dx = 1.

In the following we assume that f is piecewise continuous, i.e. the is anat most countable index set I and pairwise disjoint open intervals Ai Rwith

iI Ai = R such that

f (x) is continuous on Ai for all i I .

Lemma 1.20 Let f : R R be a piecewise continuous pdf. Thenthere exists a unique probability measure on (R,B) such that

P((a, b]) =

ba

f (x)dx , for all a < b.



The corresponding distribution P is called absolutely continuous.Then the CDF is given by F (x) =

x f (t)dt.

Note that F is continuous and

F (x) = f (x),

if f is continuous at x .

A density is not unique but almost unique.

Lemma 1.21 Let f , g be piecewise continuous pdfs such that ba

f (x)dx =

ba

g(x)dx for all a < b.

Then {x : f (x) 6= g(x)} is countable.



Example 1.22

1 Normal distribution

f (x) =12pi

exp

(1

2

(x )22

).

Parameter: R, > 02 Uniform distribution

f (x) =1

b a1[a,b](x)

Parameters: < a < b 0



Extension to Rk

The Borel -field Bk is the -field generated by the open intervals(a1, b1) (ak , bk). As in the real-valued case, probability measures on(Rk ,Bk) are uniquely defined via the multivariate distribution function:

F (b1, . . . , bk) = P({(x1, . . . , xk) : x1 b1, . . . , xk bk}).

F is called absolutely continuous if:

F (b1, . . . , bk) =

b1 bk

f (x1, . . . , xk)dxk dx1

for all b1, ..., bk R. Here, if f continuous at (x1, . . . , xk), then

kF

x1 xk (x1, . . . , xk) = f (x1, . . . , xk).



The Borel -field Bk is the -field generated by the open intervals(a1, b1) (ak , bk).The function f : Rk R is called (multivariate) probability massfunction if

1 f (x) 0 for all x Rk and2 it holds

xSf f (x) = 1 with Sf = {x Rk : f (x) > 0}.

Define the discrete probability measure on (Rk ,Bk) by

P(A) =

xSf Af (x),

cf.(1), p.27.



A (Riemann) integrable function f : Rk R is called (multivariate)probability density function if

1 f (x) 0 for all x and2 it holds

Rkf (x)dx :=

f (x1, . . . , xk)dxk dx1 = 1.

Then by

P((a1, b1] (ak , bk ]) = b1a1

bkak

f (x1, . . . , xk)dxk dx1

a probability measure can be introduced on (Rk ,Bk) which is calledabsolutely continuous.


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