In the bivariate regression model, the original parameterization is. Y i = β 1 + β 2 X2 + β 2 X2. + β 2 (X 2i X 2 ) + ε i (2)
|
|
- Dennis Little
- 5 years ago
- Views:
Transcription
1 RNy, econ460 autumn 04 Lecture note Orthogonalization and re-parameterization 5..3 and 7.. in HN Orthogonalization of variables, for example X i and X means that variables that are correlated are made perfectly uncorrelated (orthogonal) by a transformation. In the context of a model, e.g., a regression equation, variables can be orthogonalized, without changing the statistical properties of the models disturbance, however one or more of the original coefficients will be changed as a consequence of the transformation. We then have a re-parameterization of the model (with the aim of achieving orthogonalization) In the bivariate regression model, the original parameterization is Y i β + β X i + ε i () If the assumptions of the model holds, the disturbance has so called classical properties. Now consider adding and subtracting β X : Y i β + β X + β (X i X ) + ε i () δ This is a reparameterization of the model because one of the parameters, the constant term, has changed. Importantly, the disturbance is unaffected, hence the statistical properties of the model is unaffected by the re-parameterization. This means that the maximized value of the likelihood function for () must be the same value as for (). The OLS/ML estimators of δ and β are uncorrelated, although the estimators of β and β are correlated, as you will know from elementary econometrics. In 7.. in HN, the original parameterization of the model equation is like: We know that the reparameterization Y i β + β X i + β 3 X 3i + ε i (3) Y i β + β X + β X + β (X i X ) + β 3 (X i X 3 ) + ε i (4) δ secures that the two regressors are uncorrelated with the intercept. They are still correlated however. In 7... a parameterization is shown that gives uncorrelated regressors as well. The likelihood function can easily be maximized with respect to the parameters of the fully orthogonalized model (thus extending the step-wise maximization for the first model equation Y i β + ε i ) Structural interpretation in HN In economics, the label structural model is often given to theoretical relationships that are linearized and supplemented with an additive disturbance, and then estimated. In addition to theoretical interpretability, the book suggests that a structural model should have the features:
2 . Parameter constancy with respect to changes in sample (for example extensions). Parameter invariance to regime shifts 3. Invariance with respect to variable addition, for example from other studies. At this point in the exposition, the book only illustrates requirement 3. by the simplest example of omitted variable bias. Where the true model contains a second regressor: Y i b + b X i + b 3 X i + v i (5) From elementary econometrics you are able to derive the bias expression of the OLS/ML estimator of ˆβ given that (5) is the true model. Projection matrices Reference: Davidson and MacKinnon Ch. In particular page 57-8, or a suitable book chapter on matrix algebra! The matrix M I X(X X) X (6) is often called the residual maker. That nickname is easy to understand, since: My (I X(X X) X )y y X(X X) X y y X ˆβ ˆε M plays a central role in many derivations. The following properties are worth noting (and showing for yourself) : M M, symmetric matrix (7) M M, idempotent matrix (8) MX 0, regression of X on X gives perfect fit (9) Another way of interpreting MX 0 is that since M produces the OLS residuals, M is orthogonal to X. M does not affect the residuals: Since y Xβ + ε, we also have: Note that this gives: Mˆε M(y X ˆβ) My MX ˆβ ˆε (0) M ε M(y Xβ) ˆε. () ˆε ˆε ε Mε () which show that RSS is a quadratic form in the theoretical disturbances. A second important matrix in regression analysis is: which is called the prediction matrix, since P X(X X) X (3) ŷ X ˆβ X(X X) X y Py (4)
3 P is also symmetric and idempotent. In linear algebra, M and P are both known as projection matrices, Ch in DM, page 57, in particular gives the geometric interpretation. M and P are orthogonal: MP PM 0 (5) Since M and P are also complementary projections which gives and therefore: The scalar product y y then becomes: M I P, M + P I (6) (M + P)y Iy y ŷ + ˆε Py + My ŷ + ˆε (7) y y(y P + y M)(Py + My) ŷ ŷ + ˆε ˆε Written out, this is: Y i T SS Ŷ i ESS + ˆε i RSS You may be more used to write this famous decomposition as: (8) (Y i Ȳ ) } {{ } T SS (Ŷi Ŷ ) } {{ } ESS + ˆε i RSS (9) Question: Why is there no conflict between (8) and (9) as long as X contains a vector of ones (an intercept)? Alternative ways of writing RSS that sometimes come in handy: ˆε ˆε (y M )My y My y ˆε ˆε y (0) Partitioned regression ˆε ˆεy y y P Py y y y [ X(X X) X ] [ X(X X) X ] y () y y y X(X X) X y y y y X ˆβ y y ˆβ X y. We now let β [ ˆβ ˆβ ] where β has k parameters and β has k parameters. k + k k. [ The corresponding partitioning of X is X X n k X. The OLS estimated model with this notation: y X ˆβ +X ˆβ + ˆε, () We begin by writing out the normal equations: ˆε My. n k ] (X X) ˆβ X y 3
4 in terms of the partitioning: ( ) ( ) X X X X ˆβ X X X X ˆβ ( ) X y X. (3) y where we have used [ ] X X [X X ] X X see page 0 and Exercise.9 in DM. The vector of OLS estimators can therefore be expressed as: ( ) ( ) ˆβ X X X ( ) X X y ˆβ X X X X X. (4) y We need a result for the inverse of a partitioned matrix. There are several versions, but probably the most useful for us is: ( ) ( A A A A A A A ) A A A A A A (5) where With the aid of (5)-(7) it can be shown that A A A A A (6) A A A A A. (7) ( ) ˆβ ˆβ ( (X M X ) ) X M y (X M X ), (8) X M y where M and M are the two residual-makers: M I X (X X ) X (9) M I X (X X ) X. (30) As an application, start with: [ X ] [ X. X ι. X ] where ι X. n X... X k X... X k.. X n... X nk ι ι n so M I ι (ι ι) ι I ι (ι ι) ι M I ιι M ι 4
5 the so-called centering matrix. Show that it is: n M ι..... n n By using the general formula for partitioned regression above, we get ˆβ (X M X ) X M y ( [M ι X ] [M ι X ] ) [Mι X ] y [ (X X ) (X X ) ] (X X ) y (3) where X is the n (k ) matrix with the variable means in the k columns. In the exercise set to Seminar you are invited to show the same result for ˆβ more directly, without the use of the results from partitioned regression, by re-writing the regression model. Frisch-Waugh-Lovell (FWL) theorem The expression for ˆβ in (8) suggests that there is another simple method for finding ˆβ that involves M : We premultiply the model (), first with M and then with X : M y M X ˆβ + ˆε (3) X M y X M X ˆβ (33) where we have used M X 0 and M ˆε ˆε in (3) and X ˆε 0 in (3), since the influence of the k second variables has already been regressed out. We can find ˆβ from this normal equation: ˆβ It can be shown that the covariance matrix is ( ) Cov ˆβ ˆσ (X M X ). ( X M X ) X M y (34) where ˆσ denotes the unbiased estimator for the common variance of the disturbances: The corresponding expressions for ˆβ is: ˆσ ˆεˆε n k. ˆβ (X M X ) X M y (35) ( ) Cov ˆβ ˆσ (X M X ). Interpretation of (34):Use symmetry and idempotency of M : ˆβ ((M X ) (M X )) (M X ) M y Here, M y is the residual vector ˆε y X and M X are the matrix with k residuals obtained from regressing each variable in X on X, ˆε X X. But this means that ˆβ can be obtained by first running these two auxiliary regressions,saving the residuals in ˆε Y X and ˆε X X and then running a second regression with ˆε Y X as regressand, and with ˆε X X as the matrix with regressors: ˆβ (M X ) (M X ) ˆε X X ˆε X X (M X ) (M y ) (36) ˆε X X ˆε Y X 5
6 We have shown the Frisch-Waugh(-Lovell) theorem. [Followers of ECON 450 spring 03 of 04 will have seen a scalar version of this. The 450 lecture notes are still on those course pages]. In sum: If we want to estimate the partial effects on Y of changes in the variables in X (i.e., controlled for the influence of X ), we can do that in two ways. Either by estimating the full multivariate model y X ˆβ +X ˆβ + ˆε or, by following the above two-step procedure of obtaining residuals with X as regressor and then obtaining ˆβ from (36). Other applications: Trend correction Seasonal adjustment by seasonal dummy variables (DM p 69) Leverage (or influential observations), DM p
Reference: Davidson and MacKinnon Ch 2. In particular page
RNy, econ460 autumn 03 Lecture note Reference: Davidson and MacKinnon Ch. In particular page 57-8. Projection matrices The matrix M I X(X X) X () is often called the residual maker. That nickname is easy
More informationECON 4160, Autumn term Lecture 1
ECON 4160, Autumn term 2017. Lecture 1 a) Maximum Likelihood based inference. b) The bivariate normal model Ragnar Nymoen University of Oslo 24 August 2017 1 / 54 Principles of inference I Ordinary least
More informationLecture 6: Geometry of OLS Estimation of Linear Regession
Lecture 6: Geometry of OLS Estimation of Linear Regession Xuexin Wang WISE Oct 2013 1 / 22 Matrix Algebra An n m matrix A is a rectangular array that consists of nm elements arranged in n rows and m columns
More informationIntroduction to Estimation Methods for Time Series models. Lecture 1
Introduction to Estimation Methods for Time Series models Lecture 1 Fulvio Corsi SNS Pisa Fulvio Corsi Introduction to Estimation () Methods for Time Series models Lecture 1 SNS Pisa 1 / 19 Estimation
More informationFIRST MIDTERM EXAM ECON 7801 SPRING 2001
FIRST MIDTERM EXAM ECON 780 SPRING 200 ECONOMICS DEPARTMENT, UNIVERSITY OF UTAH Problem 2 points Let y be a n-vector (It may be a vector of observations of a random variable y, but it does not matter how
More informationThe Geometry of Linear Regression
Chapter 2 The Geometry of Linear Regression 21 Introduction In Chapter 1, we introduced regression models, both linear and nonlinear, and discussed how to estimate linear regression models by using the
More informationEconomics 240A, Section 3: Short and Long Regression (Ch. 17) and the Multivariate Normal Distribution (Ch. 18)
Economics 240A, Section 3: Short and Long Regression (Ch. 17) and the Multivariate Normal Distribution (Ch. 18) MichaelR.Roberts Department of Economics and Department of Statistics University of California
More informationECON 3150/4150 spring term 2014: Exercise set for the first seminar and DIY exercises for the first few weeks of the course
ECON 3150/4150 spring term 2014: Exercise set for the first seminar and DIY exercises for the first few weeks of the course Ragnar Nymoen 13 January 2014. Exercise set to seminar 1 (week 6, 3-7 Feb) This
More informationEcon 620. Matrix Differentiation. Let a and x are (k 1) vectors and A is an (k k) matrix. ) x. (a x) = a. x = a (x Ax) =(A + A (x Ax) x x =(A + A )
Econ 60 Matrix Differentiation Let a and x are k vectors and A is an k k matrix. a x a x = a = a x Ax =A + A x Ax x =A + A x Ax = xx A We don t want to prove the claim rigorously. But a x = k a i x i i=
More informationECON 3150/4150, Spring term Lecture 7
ECON 3150/4150, Spring term 2014. Lecture 7 The multivariate regression model (I) Ragnar Nymoen University of Oslo 4 February 2014 1 / 23 References to Lecture 7 and 8 SW Ch. 6 BN Kap 7.1-7.8 2 / 23 Omitted
More informationThe multiple regression model; Indicator variables as regressors
The multiple regression model; Indicator variables as regressors Ragnar Nymoen University of Oslo 28 February 2013 1 / 21 This lecture (#12): Based on the econometric model specification from Lecture 9
More informationMa 3/103: Lecture 24 Linear Regression I: Estimation
Ma 3/103: Lecture 24 Linear Regression I: Estimation March 3, 2017 KC Border Linear Regression I March 3, 2017 1 / 32 Regression analysis Regression analysis Estimate and test E(Y X) = f (X). f is the
More informationLinear Regression. In this problem sheet, we consider the problem of linear regression with p predictors and one intercept,
Linear Regression In this problem sheet, we consider the problem of linear regression with p predictors and one intercept, y = Xβ + ɛ, where y t = (y 1,..., y n ) is the column vector of target values,
More informationEcon 2120: Section 2
Econ 2120: Section 2 Part I - Linear Predictor Loose Ends Ashesh Rambachan Fall 2018 Outline Big Picture Matrix Version of the Linear Predictor and Least Squares Fit Linear Predictor Least Squares Omitted
More informationCh 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
More informationThe Multiple Regression Model Estimation
Lesson 5 The Multiple Regression Model Estimation Pilar González and Susan Orbe Dpt Applied Econometrics III (Econometrics and Statistics) Pilar González and Susan Orbe OCW 2014 Lesson 5 Regression model:
More information3. For a given dataset and linear model, what do you think is true about least squares estimates? Is Ŷ always unique? Yes. Is ˆβ always unique? No.
7. LEAST SQUARES ESTIMATION 1 EXERCISE: Least-Squares Estimation and Uniqueness of Estimates 1. For n real numbers a 1,...,a n, what value of a minimizes the sum of squared distances from a to each of
More informationEC3062 ECONOMETRICS. THE MULTIPLE REGRESSION MODEL Consider T realisations of the regression equation. (1) y = β 0 + β 1 x β k x k + ε,
THE MULTIPLE REGRESSION MODEL Consider T realisations of the regression equation (1) y = β 0 + β 1 x 1 + + β k x k + ε, which can be written in the following form: (2) y 1 y 2.. y T = 1 x 11... x 1k 1
More informationThe Linear Regression Model
The Linear Regression Model Carlo Favero Favero () The Linear Regression Model 1 / 67 OLS To illustrate how estimation can be performed to derive conditional expectations, consider the following general
More informationThe Simple Regression Model. Part II. The Simple Regression Model
Part II The Simple Regression Model As of Sep 22, 2015 Definition 1 The Simple Regression Model Definition Estimation of the model, OLS OLS Statistics Algebraic properties Goodness-of-Fit, the R-square
More informationL2: Two-variable regression model
L2: Two-variable regression model Feng Li feng.li@cufe.edu.cn School of Statistics and Mathematics Central University of Finance and Economics Revision: September 4, 2014 What we have learned last time...
More informationE 4101/5101 Lecture 9: Non-stationarity
E 4101/5101 Lecture 9: Non-stationarity Ragnar Nymoen 30 March 2011 Introduction I Main references: Hamilton Ch 15,16 and 17. Davidson and MacKinnon Ch 14.3 and 14.4 Also read Ch 2.4 and Ch 2.5 in Davidson
More informationEconometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
More informationLecture 5: Omitted Variables, Dummy Variables and Multicollinearity
Lecture 5: Omitted Variables, Dummy Variables and Multicollinearity R.G. Pierse 1 Omitted Variables Suppose that the true model is Y i β 1 + β X i + β 3 X 3i + u i, i 1,, n (1.1) where β 3 0 but that the
More informationLecture 4: Heteroskedasticity
Lecture 4: Heteroskedasticity Econometric Methods Warsaw School of Economics (4) Heteroskedasticity 1 / 24 Outline 1 What is heteroskedasticity? 2 Testing for heteroskedasticity White Goldfeld-Quandt Breusch-Pagan
More informationFENG CHIA UNIVERSITY ECONOMETRICS I: HOMEWORK 4. Prof. Mei-Yuan Chen Spring 2008
FENG CHIA UNIVERSITY ECONOMETRICS I: HOMEWORK 4 Prof. Mei-Yuan Chen Spring 008. Partition and rearrange the matrix X as [x i X i ]. That is, X i is the matrix X excluding the column x i. Let u i denote
More informationWeighted Least Squares
Weighted Least Squares The standard linear model assumes that Var(ε i ) = σ 2 for i = 1,..., n. As we have seen, however, there are instances where Var(Y X = x i ) = Var(ε i ) = σ2 w i. Here w 1,..., w
More informationMotivation for multiple regression
Motivation for multiple regression 1. Simple regression puts all factors other than X in u, and treats them as unobserved. Effectively the simple regression does not account for other factors. 2. The slope
More information3 Multiple Linear Regression
3 Multiple Linear Regression 3.1 The Model Essentially, all models are wrong, but some are useful. Quote by George E.P. Box. Models are supposed to be exact descriptions of the population, but that is
More informationLecture 4: Regression Analysis
Lecture 4: Regression Analysis 1 Regression Regression is a multivariate analysis, i.e., we are interested in relationship between several variables. For corporate audience, it is sufficient to show correlation.
More informationSimple Linear Regression: The Model
Simple Linear Regression: The Model task: quantifying the effect of change X in X on Y, with some constant β 1 : Y = β 1 X, linear relationship between X and Y, however, relationship subject to a random
More informationSo far our focus has been on estimation of the parameter vector β in the. y = Xβ + u
Interval estimation and hypothesis tests So far our focus has been on estimation of the parameter vector β in the linear model y i = β 1 x 1i + β 2 x 2i +... + β K x Ki + u i = x iβ + u i for i = 1, 2,...,
More informationLecture 9 SLR in Matrix Form
Lecture 9 SLR in Matrix Form STAT 51 Spring 011 Background Reading KNNL: Chapter 5 9-1 Topic Overview Matrix Equations for SLR Don t focus so much on the matrix arithmetic as on the form of the equations.
More informationLecture 3: Multiple Regression
Lecture 3: Multiple Regression R.G. Pierse 1 The General Linear Model Suppose that we have k explanatory variables Y i = β 1 + β X i + β 3 X 3i + + β k X ki + u i, i = 1,, n (1.1) or Y i = β j X ji + u
More informationProblem Set - Instrumental Variables
Problem Set - Instrumental Variables 1. Consider a simple model to estimate the effect of personal computer (PC) ownership on college grade point average for graduating seniors at a large public university:
More information14 Multiple Linear Regression
B.Sc./Cert./M.Sc. Qualif. - Statistics: Theory and Practice 14 Multiple Linear Regression 14.1 The multiple linear regression model In simple linear regression, the response variable y is expressed in
More informationEconometrics I. Professor William Greene Stern School of Business Department of Economics 3-1/29. Part 3: Least Squares Algebra
Econometrics I Professor William Greene Stern School of Business Department of Economics 3-1/29 Econometrics I Part 3 Least Squares Algebra 3-2/29 Vocabulary Some terms to be used in the discussion. Population
More informationNon-Spherical Errors
Non-Spherical Errors Krishna Pendakur February 15, 2016 1 Efficient OLS 1. Consider the model Y = Xβ + ε E [X ε = 0 K E [εε = Ω = σ 2 I N. 2. Consider the estimated OLS parameter vector ˆβ OLS = (X X)
More informationLECTURE 2 LINEAR REGRESSION MODEL AND OLS
SEPTEMBER 29, 2014 LECTURE 2 LINEAR REGRESSION MODEL AND OLS Definitions A common question in econometrics is to study the effect of one group of variables X i, usually called the regressors, on another
More informationInverse of a Square Matrix. For an N N square matrix A, the inverse of A, 1
Inverse of a Square Matrix For an N N square matrix A, the inverse of A, 1 A, exists if and only if A is of full rank, i.e., if and only if no column of A is a linear combination 1 of the others. A is
More informationLecture 15. Hypothesis testing in the linear model
14. Lecture 15. Hypothesis testing in the linear model Lecture 15. Hypothesis testing in the linear model 1 (1 1) Preliminary lemma 15. Hypothesis testing in the linear model 15.1. Preliminary lemma Lemma
More informationGreene, Econometric Analysis (7th ed, 2012)
EC771: Econometrics, Spring 2012 Greene, Econometric Analysis (7th ed, 2012) Chapters 2 3: Classical Linear Regression The classical linear regression model is the single most useful tool in econometrics.
More informationThe regression model with one fixed regressor cont d
The regression model with one fixed regressor cont d 3150/4150 Lecture 4 Ragnar Nymoen 27 January 2012 The model with transformed variables Regression with transformed variables I References HGL Ch 2.8
More informationINTRODUCTORY ECONOMETRICS
INTRODUCTORY ECONOMETRICS Lesson 2b Dr Javier Fernández etpfemaj@ehu.es Dpt. of Econometrics & Statistics UPV EHU c J Fernández (EA3-UPV/EHU), February 21, 2009 Introductory Econometrics - p. 1/192 GLRM:
More informationQuantitative Analysis of Financial Markets. Summary of Part II. Key Concepts & Formulas. Christopher Ting. November 11, 2017
Summary of Part II Key Concepts & Formulas Christopher Ting November 11, 2017 christopherting@smu.edu.sg http://www.mysmu.edu/faculty/christophert/ Christopher Ting 1 of 16 Why Regression Analysis? Understand
More informationThe regression model with one stochastic regressor.
The regression model with one stochastic regressor. 3150/4150 Lecture 6 Ragnar Nymoen 30 January 2012 We are now on Lecture topic 4 The main goal in this lecture is to extend the results of the regression
More informationApplied Statistics and Econometrics
Applied Statistics and Econometrics Lecture 6 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 53 Outline of Lecture 6 1 Omitted variable bias (SW 6.1) 2 Multiple
More informationPanel Data Models. James L. Powell Department of Economics University of California, Berkeley
Panel Data Models James L. Powell Department of Economics University of California, Berkeley Overview Like Zellner s seemingly unrelated regression models, the dependent and explanatory variables for panel
More informationPANEL DATA RANDOM AND FIXED EFFECTS MODEL. Professor Menelaos Karanasos. December Panel Data (Institute) PANEL DATA December / 1
PANEL DATA RANDOM AND FIXED EFFECTS MODEL Professor Menelaos Karanasos December 2011 PANEL DATA Notation y it is the value of the dependent variable for cross-section unit i at time t where i = 1,...,
More informationChapter 5 Matrix Approach to Simple Linear Regression
STAT 525 SPRING 2018 Chapter 5 Matrix Approach to Simple Linear Regression Professor Min Zhang Matrix Collection of elements arranged in rows and columns Elements will be numbers or symbols For example:
More informationLecture 19 - Decomposing a Time Series into its Trend and Cyclical Components
Lecture 19 - Decomposing a Time Series into its Trend and Cyclical Components It is often assumed that many macroeconomic time series are subject to two sorts of forces: those that influence the long-run
More informationSection I. Define or explain the following terms (3 points each) 1. centered vs. uncentered 2 R - 2. Frisch theorem -
First Exam: Economics 388, Econometrics Spring 006 in R. Butler s class YOUR NAME: Section I (30 points) Questions 1-10 (3 points each) Section II (40 points) Questions 11-15 (10 points each) Section III
More informationReview of Classical Least Squares. James L. Powell Department of Economics University of California, Berkeley
Review of Classical Least Squares James L. Powell Department of Economics University of California, Berkeley The Classical Linear Model The object of least squares regression methods is to model and estimate
More informationWeighted Least Squares
Weighted Least Squares The standard linear model assumes that Var(ε i ) = σ 2 for i = 1,..., n. As we have seen, however, there are instances where Var(Y X = x i ) = Var(ε i ) = σ2 w i. Here w 1,..., w
More informationBusiness Statistics. Tommaso Proietti. Linear Regression. DEF - Università di Roma 'Tor Vergata'
Business Statistics Tommaso Proietti DEF - Università di Roma 'Tor Vergata' Linear Regression Specication Let Y be a univariate quantitative response variable. We model Y as follows: Y = f(x) + ε where
More informationEmpirical Economic Research, Part II
Based on the text book by Ramanathan: Introductory Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 7, 2011 Outline Introduction
More informationVariable Selection and Model Building
LINEAR REGRESSION ANALYSIS MODULE XIII Lecture - 37 Variable Selection and Model Building Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur The complete regression
More informationRegression #2. Econ 671. Purdue University. Justin L. Tobias (Purdue) Regression #2 1 / 24
Regression #2 Econ 671 Purdue University Justin L. Tobias (Purdue) Regression #2 1 / 24 Estimation In this lecture, we address estimation of the linear regression model. There are many objective functions
More informationAndreas Steinhauer, University of Zurich Tobias Wuergler, University of Zurich. September 24, 2010
L C M E F -S IV R Andreas Steinhauer, University of Zurich Tobias Wuergler, University of Zurich September 24, 2010 Abstract This paper develops basic algebraic concepts for instrumental variables (IV)
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Midterm Review Week 7
MA 575 Linear Models: Cedric E. Ginestet, Boston University Midterm Review Week 7 1 Random Vectors Let a 0 and y be n 1 vectors, and let A be an n n matrix. Here, a 0 and A are non-random, whereas y is
More informationSimple Linear Regression
Simple Linear Regression September 24, 2008 Reading HH 8, GIll 4 Simple Linear Regression p.1/20 Problem Data: Observe pairs (Y i,x i ),i = 1,...n Response or dependent variable Y Predictor or independent
More informationProblem Set #6: OLS. Economics 835: Econometrics. Fall 2012
Problem Set #6: OLS Economics 835: Econometrics Fall 202 A preliminary result Suppose we have a random sample of size n on the scalar random variables (x, y) with finite means, variances, and covariance.
More informationAdvanced Econometrics I
Lecture Notes Autumn 2010 Dr. Getinet Haile, University of Mannheim 1. Introduction Introduction & CLRM, Autumn Term 2010 1 What is econometrics? Econometrics = economic statistics economic theory mathematics
More informationAdvanced Econometrics
Based on the textbook by Verbeek: A Guide to Modern Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna May 16, 2013 Outline Univariate
More informationSolutions for Econometrics I Homework No.1
Solutions for Econometrics I Homework No.1 due 2006-02-20 Feldkircher, Forstner, Ghoddusi, Grafenhofer, Pichler, Reiss, Yan, Zeugner Exercise 1.1 Structural form of the problem: 1. q d t = α 0 + α 1 p
More informationPeter Hoff Linear and multilinear models April 3, GLS for multivariate regression 5. 3 Covariance estimation for the GLM 8
Contents 1 Linear model 1 2 GLS for multivariate regression 5 3 Covariance estimation for the GLM 8 4 Testing the GLH 11 A reference for some of this material can be found somewhere. 1 Linear model Recall
More informationE 4160 Autumn term Lecture 9: Deterministic trends vs integrated series; Spurious regression; Dickey-Fuller distribution and test
E 4160 Autumn term 2016. Lecture 9: Deterministic trends vs integrated series; Spurious regression; Dickey-Fuller distribution and test Ragnar Nymoen Department of Economics, University of Oslo 24 October
More informationMS&E 226: Small Data. Lecture 11: Maximum likelihood (v2) Ramesh Johari
MS&E 226: Small Data Lecture 11: Maximum likelihood (v2) Ramesh Johari ramesh.johari@stanford.edu 1 / 18 The likelihood function 2 / 18 Estimating the parameter This lecture develops the methodology behind
More informationStatistics 910, #5 1. Regression Methods
Statistics 910, #5 1 Overview Regression Methods 1. Idea: effects of dependence 2. Examples of estimation (in R) 3. Review of regression 4. Comparisons and relative efficiencies Idea Decomposition Well-known
More informationMath 423/533: The Main Theoretical Topics
Math 423/533: The Main Theoretical Topics Notation sample size n, data index i number of predictors, p (p = 2 for simple linear regression) y i : response for individual i x i = (x i1,..., x ip ) (1 p)
More informationECON 3150/4150, Spring term Lecture 6
ECON 3150/4150, Spring term 2013. Lecture 6 Review of theoretical statistics for econometric modelling (II) Ragnar Nymoen University of Oslo 31 January 2013 1 / 25 References to Lecture 3 and 6 Lecture
More informationCh 3: Multiple Linear Regression
Ch 3: Multiple Linear Regression 1. Multiple Linear Regression Model Multiple regression model has more than one regressor. For example, we have one response variable and two regressor variables: 1. delivery
More information20.1. Balanced One-Way Classification Cell means parametrization: ε 1. ε I. + ˆɛ 2 ij =
20. ONE-WAY ANALYSIS OF VARIANCE 1 20.1. Balanced One-Way Classification Cell means parametrization: Y ij = µ i + ε ij, i = 1,..., I; j = 1,..., J, ε ij N(0, σ 2 ), In matrix form, Y = Xβ + ε, or 1 Y J
More information4 Multiple Linear Regression
4 Multiple Linear Regression 4. The Model Definition 4.. random variable Y fits a Multiple Linear Regression Model, iff there exist β, β,..., β k R so that for all (x, x 2,..., x k ) R k where ε N (, σ
More informationMAT2377. Rafa l Kulik. Version 2015/November/26. Rafa l Kulik
MAT2377 Rafa l Kulik Version 2015/November/26 Rafa l Kulik Bivariate data and scatterplot Data: Hydrocarbon level (x) and Oxygen level (y): x: 0.99, 1.02, 1.15, 1.29, 1.46, 1.36, 0.87, 1.23, 1.55, 1.40,
More informationLecture 14 Simple Linear Regression
Lecture 4 Simple Linear Regression Ordinary Least Squares (OLS) Consider the following simple linear regression model where, for each unit i, Y i is the dependent variable (response). X i is the independent
More informationMultiple Regression Model: I
Multiple Regression Model: I Suppose the data are generated according to y i 1 x i1 2 x i2 K x ik u i i 1...n Define y 1 x 11 x 1K 1 u 1 y y n X x n1 x nk K u u n So y n, X nxk, K, u n Rks: In many applications,
More informationEcon 582 Fixed Effects Estimation of Panel Data
Econ 582 Fixed Effects Estimation of Panel Data Eric Zivot May 28, 2012 Panel Data Framework = x 0 β + = 1 (individuals); =1 (time periods) y 1 = X β ( ) ( 1) + ε Main question: Is x uncorrelated with?
More informationThe regression model with one stochastic regressor (part II)
The regression model with one stochastic regressor (part II) 3150/4150 Lecture 7 Ragnar Nymoen 6 Feb 2012 We will finish Lecture topic 4: The regression model with stochastic regressor We will first look
More informationECON 4160, Spring term 2015 Lecture 7
ECON 4160, Spring term 2015 Lecture 7 Identification and estimation of SEMs (Part 1) Ragnar Nymoen Department of Economics 8 Oct 2015 1 / 55 HN Ch 15 References to Davidson and MacKinnon, Ch 8.1-8.5 Ch
More informationReliability of inference (1 of 2 lectures)
Reliability of inference (1 of 2 lectures) Ragnar Nymoen University of Oslo 5 March 2013 1 / 19 This lecture (#13 and 14): I The optimality of the OLS estimators and tests depend on the assumptions of
More informationRegression: Ordinary Least Squares
Regression: Ordinary Least Squares Mark Hendricks Autumn 2017 FINM Intro: Regression Outline Regression OLS Mathematics Linear Projection Hendricks, Autumn 2017 FINM Intro: Regression: Lecture 2/32 Regression
More informationAdvanced Quantitative Methods: ordinary least squares
Advanced Quantitative Methods: Ordinary Least Squares University College Dublin 31 January 2012 1 2 3 4 5 Terminology y is the dependent variable referred to also (by Greene) as a regressand X are the
More informationLecture 13: Simple Linear Regression in Matrix Format. 1 Expectations and Variances with Vectors and Matrices
Lecture 3: Simple Linear Regression in Matrix Format To move beyond simple regression we need to use matrix algebra We ll start by re-expressing simple linear regression in matrix form Linear algebra is
More informationINTRODUCING LINEAR REGRESSION MODELS Response or Dependent variable y
INTRODUCING LINEAR REGRESSION MODELS Response or Dependent variable y Predictor or Independent variable x Model with error: for i = 1,..., n, y i = α + βx i + ε i ε i : independent errors (sampling, measurement,
More informationLecture 19 Multiple (Linear) Regression
Lecture 19 Multiple (Linear) Regression Thais Paiva STA 111 - Summer 2013 Term II August 1, 2013 1 / 30 Thais Paiva STA 111 - Summer 2013 Term II Lecture 19, 08/01/2013 Lecture Plan 1 Multiple regression
More informationInterpreting Regression Results
Interpreting Regression Results Carlo Favero Favero () Interpreting Regression Results 1 / 42 Interpreting Regression Results Interpreting regression results is not a simple exercise. We propose to split
More informationMultivariate Regression Analysis
Matrices and vectors The model from the sample is: Y = Xβ +u with n individuals, l response variable, k regressors Y is a n 1 vector or a n l matrix with the notation Y T = (y 1,y 2,...,y n ) 1 x 11 x
More informationTopic 10: Panel Data Analysis
Topic 10: Panel Data Analysis Advanced Econometrics (I) Dong Chen School of Economics, Peking University 1 Introduction Panel data combine the features of cross section data time series. Usually a panel
More informationTime Invariant Variables and Panel Data Models : A Generalised Frisch- Waugh Theorem and its Implications
Time Invariant Variables and Panel Data Models : A Generalised Frisch- Waugh Theorem and its Implications Jaya Krishnakumar No 2004.01 Cahiers du département d économétrie Faculté des sciences économiques
More informationLecture 13: Simple Linear Regression in Matrix Format
See updates and corrections at http://www.stat.cmu.edu/~cshalizi/mreg/ Lecture 13: Simple Linear Regression in Matrix Format 36-401, Section B, Fall 2015 13 October 2015 Contents 1 Least Squares in Matrix
More informationRegression. ECO 312 Fall 2013 Chris Sims. January 12, 2014
ECO 312 Fall 2013 Chris Sims Regression January 12, 2014 c 2014 by Christopher A. Sims. This document is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License What
More informationRegression diagnostics
Regression diagnostics Kerby Shedden Department of Statistics, University of Michigan November 5, 018 1 / 6 Motivation When working with a linear model with design matrix X, the conventional linear model
More informationWe begin by thinking about population relationships.
Conditional Expectation Function (CEF) We begin by thinking about population relationships. CEF Decomposition Theorem: Given some outcome Y i and some covariates X i there is always a decomposition where
More informationLinear Regression. September 27, Chapter 3. Chapter 3 September 27, / 77
Linear Regression Chapter 3 September 27, 2016 Chapter 3 September 27, 2016 1 / 77 1 3.1. Simple linear regression 2 3.2 Multiple linear regression 3 3.3. The least squares estimation 4 3.4. The statistical
More informationLinear Regression. Junhui Qian. October 27, 2014
Linear Regression Junhui Qian October 27, 2014 Outline The Model Estimation Ordinary Least Square Method of Moments Maximum Likelihood Estimation Properties of OLS Estimator Unbiasedness Consistency Efficiency
More information9. Least squares data fitting
L. Vandenberghe EE133A (Spring 2017) 9. Least squares data fitting model fitting regression linear-in-parameters models time series examples validation least squares classification statistics interpretation
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON3150/ECON4150 Introductory Econometrics Date of exam: Wednesday, May 15, 013 Grades are given: June 6, 013 Time for exam: :30 p.m. 5:30 p.m. The problem
More informationSimple Linear Regression
Simple Linear Regression ST 430/514 Recall: A regression model describes how a dependent variable (or response) Y is affected, on average, by one or more independent variables (or factors, or covariates)
More informationREVIEW (MULTIVARIATE LINEAR REGRESSION) Explain/Obtain the LS estimator () of the vector of coe cients (b)
REVIEW (MULTIVARIATE LINEAR REGRESSION) Explain/Obtain the LS estimator () of the vector of coe cients (b) Explain/Obtain the variance-covariance matrix of Both in the bivariate case (two regressors) and
More information