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Econometrics II - uni- · PDF fileModule Statistics and Module Empirical Methods Descriptive statistics (Statistik I) How to process data? How to display data? Probability theory and

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  • Econometrics II

    Andrea Beccarini

    Winter2011/2012

    1

  • Introduction

    � Econometrics: application of statistical methods to empirical research in economics

    � Compare theory with facts (data)

    � Statistics: foundation of econometrics

    2

  • Module Statistics and Module Empirical Methods

    � Descriptive statistics (Statistik I) How to process data? How to display data?

    � Probability theory and statistical inference (Statistik II) Estimation of unknown parameters from random samples; hypothesis tests

    � Empirical research in economics (Empirische Wirtschaftsforschung) Applications of the linear model; statistical software

    3

  • Module Statistics/Econometrics/Empirical Economics I

    � Advanced Statistics Probability theory; multidimensional random variables; estimation and hypothesis testing

    � Econometrics I Simple and multiple linear regression model

    � Econometrics II Extensions of the multivariate linear regression model; simultaneous equation systems; dynamic models

    4

  • Module Statistics/Econometrics/Empirical Economics II

    � Time series analysis: Stochastic processes; stationarity; ergodicity; linear processes; unit root processes; cointegration; vector-autoregressive models

    � One further special course or seminar, e.g.

    Financial econometrics Panel data econometrics Introduction to R Poverty and inequality Statistical inference, bootstrap Wage and earnings dynamics

    5

  • Literature: Statistical basics

    � Karl Mosler and Friedrich Schmid, Wahrscheinlichkeitsrechnung and schließende Statistik, 2. Au., Springer, 2006.

    � Aris Spanos, Statistical Foundations of Econometric Modelling, Cambridge University Press, 1986.

    � Mood, A.M., Graybill, F.A. and D.C. Boes (1974). Introduction to the theory of statistics, 3rd ed., McGraw-Hill, Tokyo.

    6

  • Literature: Econometrics

    � Main book for this course: Ludwig von Auer, Ökonometrie: Eine Einführung, 4. Au., Springer, 2005.

    � Alternatively: William E. Gri¢ ths, R. Carter Hill and George G. Judge, Learning and Practicing Econometrics, John Wiley & Sons, 1993.

    � James Stock and Mark Watson, Introduction to Econometrics, Addison Wesley, 2003.

    � Russell Davidson and James MacKinnon, Econometric Theory and Methods, Oxford University Press, 2004.

    7

  • Class

    � Class teacher: Rainer Schüssler

    � Time and location: Tue, 14.00-16.00, CAWM1

    � A detailed schedule is available on the home page of this course http://www.wiwi.uni-muenster.de/statistik �! Studium �! Aktuelle Veranstaltungen �! Econometrics II

    8

  • Outline

    � Very brief revision of Econometrics I (chap. 8 to 14)

    � Violations of model assumptions (chap. 15 to 19, 21)

    � Stochastic exogenous variables (chap. 20)

    � Dynamic models (chap. 22)

    � Interdependent equation systems (chap. 23) 9

  • Multiple linear regression model (revision)

    Assumption A1: No relevant exogenous variable is omitted from the econometric model, and all exogenous variables in the model are relevant

    Assumption A2: The true functional dependence between X and y is linear

    Assumption A3: The parameters � are constant for all T observations (xt; yt)

    Assumptions B1 to B4:

    u � N � 0; �2IT

    � 10

  • Assumption C1: The exogenous variables x1t; : : : ; xKt are not stochastic, but can be controlled as in an experimental situation

    Assumption C2: No perfect multicollinearity:

    rank(X) = K + 1

    � Econometric model:

    y = X� + u

    � Point estimator (OLS):

    �̂ = � X0X

    ��1 X0y

    11

  • � Estimated model:

    ŷ = X�̂

    � Residuals:

    û = y � ŷ

    � Coe¢ cient of determination:

    R2 = Syy � Sûû Syy

    = Sŷŷ

    Syy =

    PK k=1

    b�kSky Syy

    12

  • � Unbiasedness:

    E(�̂) = �

    � Covariance matrix of �̂

    V(�̂) = �2 � X0X

    ��1

    � Gauss-Markov theorem: �̂ is BLUE

    13

  • � Distribution of y: y � N(X�; �2IT )

    � Distribution of �̂:

    �̂ � N � �; �2

    � X0X

    ��1�

    � Estimator of error term variance:

    �̂2 = Sûû

    T �K � 1

    � Unbiasedness: E(�̂2) = �2

    14

  • � Interval estimator of the component �k of �h �̂k � ta=2 � cse(�̂k) ; �̂k + ta=2 � cse(�̂k)i

    � t-test:

    H0 : r 0� = q

    H1 : r 0� 6= q

    where

    r = [r0; r1; : : : ; rK] 0

    � Test statistic:

    t = r0�̂ � qcse(r0�̂)

    15

  • � F -test:

    H0 : R� = q

    H0 : R� 6= q

    � Test statistic:

    F =

    � S0bubu � Sbubu�.L

    Sbubu/ (T �K � 1); or

    F =

    � R�̂ � q

    �0 h R � X0X

    ��1R0i�1 �R�̂ � q� =L û0û= (T �K � 1)

    where L is the number of restrictions in H0

    16

  • � Forecasting: Let x0 = [1; x10; x20; : : : ; xK0]0 be the vector of exogenous variables

    � Point forecast: ŷ0 = x00�̂

    � Variance of the forecast error:

    V ar (ŷ0 � y0) = �2 � 1 + x00

    � X0X

    ��1 x0

    � Violation of A1: Omitted or redundant variables

    � Violation of A2: Nonlinear functional forms 17

  • Qualitative exogenous variables

    � A3: The parameters � are constant for all T observations (xt; yt)

    � Example: Wage yt depends on both education x1t and age x2t yt = �+ �1x1t + �2x2t + ut

    � Suppose the parameters di¤er between men and women

    yt = �M + �M1x1t + �M2x2t + ut yt = �F + �F1x1t + �F2x2t + ut

    � What happens if the gender di¤erence is ignored? [dummy.R] 18

  • � Introduce a dummy variable

    Dt =

    ( 0 if male 1 if female

    � Extended model

    yt = �+Dt + �1x1t + �1Dtx1t + �2x2t + �2Dtx2t + ut

    � Submodels for men (Dt = 0) and women (Dt = 1)

    yt = � + �1 x1t + �2 x2t + ut yt = (�+ ) + (�1 + �1) x1t + (�2 + �2) x2t + ut

    � Interpretation of the coe¢ cients ; �1; �2 19

  • � Estimation of the model by OLS?

    � How does the matrix of exogenous variables X look like?

    � Apply t- or F -tests to check parameter constancy, e.g.

    H0 : = �1 = �2 = 0

    � Often, the models just include a level e¤ect, i.e.

    yt = �+ Dt + �1x1t + �2x2t + ut

    (use a t-test for )

    20

  • � If the qualitative exogenous variable has more than two values, we need more than one dummy variable

    � Example: Religion (protestant, catholic, other)

    D prot t =

    8>: 0 if other 1 if protestant 0 if catholic

    Dcatht =

    8>: 0 if other 0 if protestant 1 if catholic

    � Interpretation of the coe¤cients?

    21

  • � If there are two or more qualitative exogenous variables, interaction terms can be added

    � Example: Gender and citizenship

    D1t =

    ( 0 if male 1 if female

    D2t =

    ( 0 if German citizenship 1 else

    � Interpretation of the coe¢ cients 1; 2; � in the two models

    yt = �+ 1D1t + 2D2t + �xt + ut

    yt = �+ 1D1t + 2D2t + �D1tD2t + �xt + ut

    22

  • � What happens if there are two dummy variables

    Dfemalet =

    ( 0 if male 1 if female

    Dmalet =

    ( 1 if male 0 if female

    � What happens if the dummy variable is coded as

    Dt =

    ( 1 if male 2 if female

    23

  • � Compare the joint dummy variable model

    yt = �+Dt + �1x1t + �1Dtx1t + �2x2t + �2Dtx2t + ut

    with the two separated models

    yt = �M + �M1x1t + �M2x2t + ut for men yt = �F + �F1x1t + �F2x2t + ut for women

    [dummycomparison.R]

    � Questions: 1. Why are the point estimates identical? [1] 2. Why is the sum of squared residuals identical? [2] 3. Why are the standard errors di¤erent? [3]

    24

  • Heteroskedasticity

    � Assumption B2: V ar(ut) = �2 for t = 1; : : : ; T

    � Rent example: The rent yt depends on the distance xt from the city center

    t xt yt t xt yt 1 0,50 16,80 7 3,10 12,80 2 1,40 16,20 8 4,40 12,20 3 1,10 15,90 9 3,70 15,00 4 2,20 15,40 10 3,00 13,60 5 1,30 16,40 11 3,50 14,10 6 3,20 13,20 12 4,10 13,30

    25

  • � The scatterplot suggests that there might be heteroskedasticity:

    � What are the properties of �̂ if there is heteroskedasticity? [4]

    26

  • Transformation of the model

    � (Restrictive and arbitrary) assumption:

    �2t = � 2xt

    � Transformation of the model: ytp xt

    = � 1 p xt + �

    xtp xt +

    utp xt| {z }

    error term y�t = �z

    � t + �x

    � t + u

    � t

    � Properties of the new error term u�t [5] 27

  • � The transformed model satises all A-, B- and C-assump