Panel Data Models. James L. Powell Department of Economics University of California, Berkeley

Size: px
Start display at page:

Download "Panel Data Models. James L. Powell Department of Economics University of California, Berkeley"

Transcription

1 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 data models (or pooled cross section and time series models) are typically denoted using two (or more) subscripts, where the different subscripts indicate different characteristics of the variable usually indicating both individual and time, but possibly denoting location, group, etc. Some variants of the model ( fixed effects models) can be viewed as special cases of the classical linear regression model, while others ( random effects models) are special cases of the generalized regression model. The simplest model assumes that the dependent variable y it satisfies a linear model with an intercept that is specific to individual i, y it = x 0 itβ + α i + ε it, i =,...,N; t =,...,T. This is known as a balanced panel, since all individual observations are assumed observable for every time period (and vice versa); here, Kronecker product notation will useful when using matrix notation for the data set. In contrast, clustered or grouped data models typically assume the number of observations per group i can vary across groups, which in this notation would replace the common number of time periods T with a groups-specific numbert i of individuals. As for seemingly unrelated equations, the number of time periods T in most panel data applications is usually small relative to the number of individuals N. However, unlike SUR models, where it is more natural to stack observations by the second subscript first that is, across individuals for each equation, and then stack by equation for panel data models the usual convention is to stack observations in the opposite order of subscripts, that is, first collecting the observations across time for each individual as y i = X i β + α i ι T + ε i (T ) (T K) for i =,..., N, where y j and ε j are T -vectors and X i is a T K matrix, y i ε i y i = y i2 (T )..., ε i = ε i2 (T )..., X i = (T K) y it ε it x 0 i x 0 i2... x 0 it,

2 and ι T is a T -dimensional column vector of ones. Then, stacking the entire data set by individuals, y ε X y = y 2 (NT )..., ε = ε 2 (NT )..., X = X 2 (NT K)..., y N ε N X N and defining α (N ) = α α 2... α N, the data can be represented by the single (relatively simple) equation y = Xβ + Dα + ε, where D (NT N) I N ι T. In panel data (or clustered data) models, the individual intercept α i is meant to control for the effect of unobservable regressors that are specific to individual i, so that α i = w 0 iγ where the unobservable, individual-specific regressors w i might include ability, intelligence, family background, ambition, etc. Different assumptions on the relationship of the observable regressors x it to the intercept term α i, and thus to the unobservable regressors w i, yield different variations on the classical and generalized regression models. With this notation, it is understood that the matrix X does not include a column vector of ones, since otherwise a linear combination of the matrix D of individual-specific dummy variables would yield this vector of ones, D ι N = (I N ι T )(ι N ) = (I N ι N ι T ) = ι NT, so that the combined matrix [X, D] of regressors would not have full column rank. 2

3 Fixed Effect Model When the individual intercepts α i are treated as fixed constants, which can be arbitrarily related to the regression vectors x it, the resulting model, known as the fixed effect model, can be viewed as a special case of the classical linear model under the usual assumptions that X is nonrandom, [X, D] is of full column rank, E(ε) =0, and V(ε) =σ 2 I NT. By the usual residual regression formulae, the classical LS estimator of the subvector β of regression coefficients is ³ ˆβ FE = X 0 X X 0 ỹ, where ỹ ³I 0 NK D(D 0 D) D y are the residuals of a regression of y on D, with an analogous definition of X. (The subscript FE stands for fixed effects.) By the special structure of the matrix D (which has a lot of ones in it), it is straightforward to show that the subvector of ỹ corresponding to individual i canbeexpressedas ỹ i = (I T ι T (ι 0 T ι T ) ι 0 T )y i = y i y i ι T, where y i is the average value of y it over t, y i ι0 T y i ι 0 T ι T = T y it. This result can be interpreted as follows: from the original structural equation t= y it = x 0 itβ + α i + ε it it follows that y i = x 0 i β + α i + ε i, and subtracting the second equation from the first yields y it y i = (x it x i ) 0 β+(α i α i )+(ε it ε i ) = (x it x i ) 0 β +(ε it ε i ), so deviating y it and x it from their time-averages eliminates the fixed effect α i from the structural equation, much as taking deviations from means eliminates the intercept term in a classical regression model (with intercept). 3

4 An alternative interpretation of the fixed effect estimator ˆβ FE uses first-differences over time, rather than deviations from time-averages, to eliminate the fixed effect α i : y it y it y i,t = x 0 itβ + α i + ε it = x 0 itβ + ε it, for i =,...,N and t =2,..., T. For balanced panels, LS regression of y it on x it gives identical results to LS regression of y it y i on x it x i. Considering either interpretation, it is clear that the regression coefficients on any component of x it that is constant over time (x it x is ) will be unidentified, since that component of either x it or x it x i will be identically zero. It is only variation in the regressors across time for a given individual that allows the corresponding coefficient to be identified relative to the individual-specific, time-invariant intercept α i. If the error terms ε it are i.i.d. and normally distributed with zero mean and variance σ 2, then the fixed effect estimator ˆβ FE is also the maximum likelihood (ML) estimator of β, and the ML estimator of σ 2 is ˆσ 2 ML = NT i= t= ³ y it y i (x it x i ) 0 ˆβ FE 2. This estimator is biased, and, if N for fixed T (a reasonable approximation if N is large relative to T ), the ML estimator of σ 2 is also inconsistent, with ˆσ 2 ML p (T ) σ 2. T This inconsistency of the ML estimator of σ 2 is a classic example of the nefarious incidental parameters problem described by Neyman and Scott. Because the number N of intercept terms α i increases to infinity as N increases, and the corresponding ML estimator ˆα i y i x 0 i ˆβ FE p α i + ε i is inconsistent for α i, this inconsistency translates into inconsistency of the ML estimator ˆσ 2 ML. Fortunately, this doesn t cause inconsistency of ˆβ FE, and an unbiased and consistent estimator s 2 ML = N(T ) K i= t= ³y it y i (x it x i ) 0 ˆβ FE 2 4

5 of σ 2 is readily available. It is worth mentioning that the fixed effect label does not mean that the regressors x it or intercept terms α i must be viewed as nonrandom, or fixed in a statistical sense; they may be indeed be viewed as random and jointly distributed, provided the conditions on ε it are assumed to hold conditional on the realizations of X and ε. What characterizes a fixed effect model is that no structure on the relationship between α i and x it is imposed. Random Effect Model Adrawbackofthefixed-effect model is its failure to identify any components of β corresponding to regressors that are constant over time for a given individual; for such coefficients to be identified, stronger conditions on the relation of the individual-specific intercept α i to the regressors x it must be imposed. The random effects model uses a simple but very strong assumption to restrict this relationship: namely, that the intercept α i is a random variable which is not related to x it and ε it, in the sense that E(α i )=α, V ar(α i )=σ 2 α, and Cov(α i,ε it )=0, all assumed independent of x it. (These moments should be interpreted as conditional on X if the regressors are viewed as random.) Relabeling the variance of ε it as Var(ε it ) σ 2 ε, the original panel data model can be rewritten as y it = x 0 itβ + α + u it, where u it (α i α)+ε it has E(u it ) = 0, Var(u it ) = σ 2 α + σ 2 ε, Cov(u it,u is ) = σ 2 α, and Cov(u it,u js ) = 0 if i 6= j. This implies that the model can be written in matrix form as y = Xβ+αι NT + u, 5

6 where E(u) = 0, V(u) = σ 2 αdd 0 + σ 2 εi NT σ 2 εω, with Ω I NT + σ2 α σ 2 DD 0 ε = I NT + σ2 α IN σ 2 ι T ι 0 T. ε In the unlikely event that the ratio θ σ 2 α/σ 2 ε wereknown andthusω did not contain unknown parameters Aitken s GLS estimator of β and α would have the usual form µ ˆβGLS =(Z 0 Ω Z) Z 0 Ω y, ˆα GLS where Z [X, ι NT ]. Like the fixed effects estimator, there are a couple algebraically-equivalent representations of the GLS estimator ˆβ GLS of the slope coefficients. One interpretation is the coefficients of a LS regression of y it on x it, where y it y it y i + ω (y i y ), x it x it x i + ω (x i x ), for and y x ω NT NT s i= t= i= t= σ 2 ε Tσ 2 ε + σ 2 α r Tθ+, y it, x it, where as above θ σ 2 α/σ 2 ε. As the variability of the random effect σ 2 α declines to zero for σ 2 ε fixed, ω, and the GLS estimator reduces to the usual LS regression of the deviations y it y in the dependent 6

7 variable from its mean value on the corresponding deviations x it x in regressors. Using this result, we can obtain another interpretation of the GLS estimator as as a matrix-weighted average ˆβ GLS = Aˆβ FE +(I A)ˆβ B of the fixed effect estimator ˆβ FE defined above and the between estimator " N # X X N ˆβ B (x i x )(x i x ) 0 (x i x )(y i y ) i=t of β coming from the LS regression of the time-averages y i on x i and a constant. (The form of the matrix A = A(θ) is given in Ruud s textbook.) When the variances σ 2 α and σ 2 ε are unknown (as is always the case), an estimator of θ σ 2 α/σ 2 ε or ω ( + Tθ) /2 is needed to construct a Feasible GLS estimator. As noted above, the unbiased estimator s 2 FE of σ2 ε baseduponthefixed-effect estimator ˆβ FE will be consistent; for fixed T, the corresponding variance estimator for the between estimator s 2 B N K i=t ³(y i y ) (x i x ) 0 ˆβ B 2 will be unbiased and consistent for Var(ε i ) =σ 2 α + σ 2 ε/t. Hence i= ˆω s2 FE Ts 2 B p ω, and can be used in place of ω to construct a Feasible GLS estimator. Note that the corresponding estimator of σ 2 α, is not guaranteed to be positive. s 2 α s 2 B s 2 FE/T, Time Effects and Differences in Differences In addition to the assumption that the intercept term varies across individuals i, itmightalso be reasonable to assume that it varies across time t; defining η t to be the time-specific intercept term, a generalization of the basic panel data model would be y it = x 0 itβ + α i + η t + ε it, i =,...,N; t =,...,T, which, in matrix form, might be written as y = Xβ + Dα + Rη + ε, 7

8 for and η = (T ) η η 2... η T R (NT T ) ι N I T. Since Dι N = ι NT = Rι T, the columns of the matrix of regressors [X, D, R] would not be linearly independent, and a normalization on α or η would need to be imposed, e.g., η 0, which would imply that the first column of R could be dropped, along with the first component of η. Treating the α and η parameters as fixed effects, the corresponding fixed effect estimator of β is obtained by a regression of ỹ it on x it, where ỹ it y it y i y t + y, with and corresponding definitions for x it and x t. y t N i= y it If T is small (relative to N), the time-specific intercepts η are typically treated as fixed, which implies that the coefficients of any regressors that are time-specific (i.e., do not vary across individuals) would be unidentified. If such coefficients are of interest, the η coefficients can be treated as random effects, with corresponding variance component σ 2 η, along with the individual effects α. The corresponding GLS estimator of β would combine the fixed effects estimator with two different between estimators, one involving the regression of y i on x i and a constant and the other regressing y t on x t and a constant. The details are a bit messy, and can be found in many graduate texts. A special case of the fixed effects model with individual- and time-specific effects is the so-called differences in differences (or diffs indiffs ) framework, a name more descriptive of the estimation method than the model itself. The simplest version of this model has T =2and a single time-varying regressor x it which is binary. Specifically the N individual observations are classified into two groups, the controls (for i =,...,N c ), for which x it 0, and the treated (i = N c +,...,N), for which x i =0 8

9 and x i2 =. The scalar coefficient β is the treatment effect, i.e., the change in the average value of the dependent variable between the pre-treatment (t =)and post-treatment (t =2)periods. To repeat, 0 if i =,...,N c, x it = 0 if i = N c +,...,N, and t =; if i = N c +,...,N, and t =. Writing the structural equation y it = x it β + α i + η t + ε it, i =,...,N; t =, 2, and taking first differences yields y i2 = y i2 y i = x i2 β + η 2 + ε i2 = ½ η2 + ε i2 if i =,...,N c β + η 2 + ε i2 if i = N c +,...,N. A classical LS regression of y i2 on a constant and x i2 yields ˆβ FE = ȳ 2 ȳ, where ȳ XN c (y i2 y i ) and N c i= ȳ 2 N N c (y i2 y i ) i=n c + are the average changes in y it for the control and treatment groups, respectively. So the estimate of the treatment effect is the difference in the average change in the dependent variable across the two groups. As N c and N N c tend to infinity, it is easy to see that ȳ p η 2, ȳ 2 p β + η 2, so the diffs-in-diffs estimator of the treatment effect is consistent. Note that, with this fixed-effects approach, there is no need to assume the treatment assignment (or choice ) is independent of the individual effect α i or time effect η t. 9

10 Robust Covariance Estimation Like other GLS applications, statistical inference with panel data can be sensitive to heteroskedasticity or serial correlation of the error terms. Writing the panel data model as y it = z 0 itθ+u it, where z 0 it includes the row vector of regressors x0 it and the relevant row of the matrices D and R of dummy variables for the fixed effect, we may want to assume that the errors u it are uncorrelated across individuals i but have arbitrary variance-covariance patterns over time t for each individual i. That is, suppose E(u it ) = 0, Cov(u it,u is ) = σ i,ts, Cov(u it,u js ) = 0 if i 6= j. Stacking the observations in the usual matrix form y = Zθ + u, it follows that for V(u) Σ = Σ Σ Σ N Σ j [σ j,ts ]. (T T ) In the absence of a parametric form for Σ j, itwouldbereasonabletousetheclassicalleastsquares estimator ˆθ LS =(Z 0 Z) Z 0 y of the θ coefficients (i.e., the slope coefficients β and any individual or time fixed effects), which should be consistent and asymptotically normally distributed under fairly general conditions on u. As in other GLS applications, the trick is to find a consistent estimator for plim N V(ˆθ LS )=plim µ µ µ N Z0 Z N Z0 ΣZ N Z0 Z, 0

11 the asymptotic covariance matrix of the LS estimator. The middle matrix is the tricky one; since N Z0 ΣZ N i= t= s= σ i,ts z it z 0 is, the same reasoning that led to the Huber-Eicker-White robust covariance matrix estimator yields N Z0 ˆΣZ N i= t= s= û it û is z it z 0 is as a consistent estimator of the middle matrix of plim N V(ˆθ LS ), where û it are the LS residuals û it y it z 0 itˆθ LS. This estimator will be consistent as N under similar conditions as for consistency of the Huber- Eicker-White heteroskedasticity-robust covariance estimator. This robust covariance matrix estimator immediately extends to clustered data problems, where the number of time periods or group members T i depends upon the group index i; the formulae above are easily extended by changing T to T i throughout. Lagged Dependent Variables in Panel Data A more serious inference problem arises when the regressors include a lagged dependent variable in models with fixed effects. Consider the special case with T =2and y it = x 0 itβ+γy i,t + α i + ε it, where the {α i } are considered fixed effects and ε it satisfies the usual Gauss-Markov assumptions. Assuming y i0 is observable for all i and considered a nonrandom starting value, the fixed-effect estimator of β and γ is obtained by a LS regression on the differenced model y i2 = y i2 y i = x 0 i2β+γ y i + ε i2. But now Cov( y i, ε i2 ) = Cov ((y i y i0 ), (ε i2 ε i )) = Cov(y i,ε i ) = Var(ε i ) 6= 0,

12 so the LS regression of y i2 on x i2 and y i will yield biased and inconsistent estimators of β and γ in general. This problem is the same as the problem of inconsistency of LS with lagged dependent variables and serially-correlated errors; elimination of the fixed effect by differencing (or deviating from individual means) yields a differenced error term which is serially correlated, and thus related to the lagged dependent variable. A solution to the inconsistency of the LS estimator for this problem is the instrumental variables method to be discussed next. 2

Review 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 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 information

Econ 582 Fixed Effects Estimation of Panel Data

Econ 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 information

Zellner s Seemingly Unrelated Regressions Model. James L. Powell Department of Economics University of California, Berkeley

Zellner s Seemingly Unrelated Regressions Model. James L. Powell Department of Economics University of California, Berkeley Zellner s Seemingly Unrelated Regressions Model James L. Powell Department of Economics University of California, Berkeley Overview The seemingly unrelated regressions (SUR) model, proposed by Zellner,

More information

y it = α i + β 0 ix it + ε it (0.1) The panel data estimators for the linear model are all standard, either the application of OLS or GLS.

y it = α i + β 0 ix it + ε it (0.1) The panel data estimators for the linear model are all standard, either the application of OLS or GLS. 0.1. Panel Data. Suppose we have a panel of data for groups (e.g. people, countries or regions) i =1, 2,..., N over time periods t =1, 2,..., T on a dependent variable y it and a kx1 vector of independent

More information

Advanced Econometrics

Advanced 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 information

Multiple Equation GMM with Common Coefficients: Panel Data

Multiple Equation GMM with Common Coefficients: Panel Data Multiple Equation GMM with Common Coefficients: Panel Data Eric Zivot Winter 2013 Multi-equation GMM with common coefficients Example (panel wage equation) 69 = + 69 + + 69 + 1 80 = + 80 + + 80 + 2 Note:

More information

Repeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data

Repeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data Panel data Repeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data - possible to control for some unobserved heterogeneity - possible

More information

Recent Advances in the Field of Trade Theory and Policy Analysis Using Micro-Level Data

Recent Advances in the Field of Trade Theory and Policy Analysis Using Micro-Level Data Recent Advances in the Field of Trade Theory and Policy Analysis Using Micro-Level Data July 2012 Bangkok, Thailand Cosimo Beverelli (World Trade Organization) 1 Content a) Classical regression model b)

More information

Econometrics. Week 6. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague

Econometrics. Week 6. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Econometrics Week 6 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 21 Recommended Reading For the today Advanced Panel Data Methods. Chapter 14 (pp.

More information

Economics 582 Random Effects Estimation

Economics 582 Random Effects Estimation Economics 582 Random Effects Estimation Eric Zivot May 29, 2013 Random Effects Model Hence, the model can be re-written as = x 0 β + + [x ] = 0 (no endogeneity) [ x ] = = + x 0 β + + [x ] = 0 [ x ] = 0

More information

Panel Data Model (January 9, 2018)

Panel Data Model (January 9, 2018) Ch 11 Panel Data Model (January 9, 2018) 1 Introduction Data sets that combine time series and cross sections are common in econometrics For example, the published statistics of the OECD contain numerous

More information

Econometrics of Panel Data

Econometrics of Panel Data Econometrics of Panel Data Jakub Mućk Meeting # 6 Jakub Mućk Econometrics of Panel Data Meeting # 6 1 / 36 Outline 1 The First-Difference (FD) estimator 2 Dynamic panel data models 3 The Anderson and Hsiao

More information

1. The OLS Estimator. 1.1 Population model and notation

1. The OLS Estimator. 1.1 Population model and notation 1. The OLS Estimator OLS stands for Ordinary Least Squares. There are 6 assumptions ordinarily made, and the method of fitting a line through data is by least-squares. OLS is a common estimation methodology

More information

Non-linear panel data modeling

Non-linear panel data modeling Non-linear panel data modeling Laura Magazzini University of Verona laura.magazzini@univr.it http://dse.univr.it/magazzini May 2010 Laura Magazzini (@univr.it) Non-linear panel data modeling May 2010 1

More information

Econometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018

Econometrics 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 information

Econometrics of Panel Data

Econometrics of Panel Data Econometrics of Panel Data Jakub Mućk Meeting # 2 Jakub Mućk Econometrics of Panel Data Meeting # 2 1 / 26 Outline 1 Fixed effects model The Least Squares Dummy Variable Estimator The Fixed Effect (Within

More information

Topic 10: Panel Data Analysis

Topic 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 information

Non-Spherical Errors

Non-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 information

GLS and FGLS. Econ 671. Purdue University. Justin L. Tobias (Purdue) GLS and FGLS 1 / 22

GLS and FGLS. Econ 671. Purdue University. Justin L. Tobias (Purdue) GLS and FGLS 1 / 22 GLS and FGLS Econ 671 Purdue University Justin L. Tobias (Purdue) GLS and FGLS 1 / 22 In this lecture we continue to discuss properties associated with the GLS estimator. In addition we discuss the practical

More information

Econometrics of Panel Data

Econometrics of Panel Data Econometrics of Panel Data Jakub Mućk Meeting # 4 Jakub Mućk Econometrics of Panel Data Meeting # 4 1 / 30 Outline 1 Two-way Error Component Model Fixed effects model Random effects model 2 Non-spherical

More information

Econometrics Summary Algebraic and Statistical Preliminaries

Econometrics Summary Algebraic and Statistical Preliminaries Econometrics Summary Algebraic and Statistical Preliminaries Elasticity: The point elasticity of Y with respect to L is given by α = ( Y/ L)/(Y/L). The arc elasticity is given by ( Y/ L)/(Y/L), when L

More information

Outline. Overview of Issues. Spatial Regression. Luc Anselin

Outline. Overview of Issues. Spatial Regression. Luc Anselin Spatial Regression Luc Anselin University of Illinois, Urbana-Champaign http://www.spacestat.com Outline Overview of Issues Spatial Regression Specifications Space-Time Models Spatial Latent Variable Models

More information

Chapter 6. Panel Data. Joan Llull. Quantitative Statistical Methods II Barcelona GSE

Chapter 6. Panel Data. Joan Llull. Quantitative Statistical Methods II Barcelona GSE Chapter 6. Panel Data Joan Llull Quantitative Statistical Methods II Barcelona GSE Introduction Chapter 6. Panel Data 2 Panel data The term panel data refers to data sets with repeated observations over

More information

the error term could vary over the observations, in ways that are related

the error term could vary over the observations, in ways that are related Heteroskedasticity We now consider the implications of relaxing the assumption that the conditional variance Var(u i x i ) = σ 2 is common to all observations i = 1,..., n In many applications, we may

More information

Heteroskedasticity. We now consider the implications of relaxing the assumption that the conditional

Heteroskedasticity. We now consider the implications of relaxing the assumption that the conditional Heteroskedasticity We now consider the implications of relaxing the assumption that the conditional variance V (u i x i ) = σ 2 is common to all observations i = 1,..., In many applications, we may suspect

More information

10 Panel Data. Andrius Buteikis,

10 Panel Data. Andrius Buteikis, 10 Panel Data Andrius Buteikis, andrius.buteikis@mif.vu.lt http://web.vu.lt/mif/a.buteikis/ Introduction Panel data combines cross-sectional and time series data: the same individuals (persons, firms,

More information

Applied Econometrics (QEM)

Applied Econometrics (QEM) Applied Econometrics (QEM) The Simple Linear Regression Model based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #2 The Simple

More information

Lecture 10: Panel Data

Lecture 10: Panel Data Lecture 10: Instructor: Department of Economics Stanford University 2011 Random Effect Estimator: β R y it = x itβ + u it u it = α i + ɛ it i = 1,..., N, t = 1,..., T E (α i x i ) = E (ɛ it x i ) = 0.

More information

Panel Data Models. Chapter 5. Financial Econometrics. Michael Hauser WS17/18 1 / 63

Panel Data Models. Chapter 5. Financial Econometrics. Michael Hauser WS17/18 1 / 63 1 / 63 Panel Data Models Chapter 5 Financial Econometrics Michael Hauser WS17/18 2 / 63 Content Data structures: Times series, cross sectional, panel data, pooled data Static linear panel data models:

More information

11. Further Issues in Using OLS with TS Data

11. Further Issues in Using OLS with TS Data 11. Further Issues in Using OLS with TS Data With TS, including lags of the dependent variable often allow us to fit much better the variation in y Exact distribution theory is rarely available in TS applications,

More information

Introductory Econometrics

Introductory Econometrics Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 17, 2012 Outline Heteroskedasticity

More information

Econometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague

Econometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Econometrics Week 4 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 23 Recommended Reading For the today Serial correlation and heteroskedasticity in

More information

PANEL 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 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 information

Linear models. Linear models are computationally convenient and remain widely used in. applied econometric research

Linear models. Linear models are computationally convenient and remain widely used in. applied econometric research Linear models Linear models are computationally convenient and remain widely used in applied econometric research Our main focus in these lectures will be on single equation linear models of the form y

More information

1 Introduction to Generalized Least Squares

1 Introduction to Generalized Least Squares ECONOMICS 7344, Spring 2017 Bent E. Sørensen April 12, 2017 1 Introduction to Generalized Least Squares Consider the model Y = Xβ + ɛ, where the N K matrix of regressors X is fixed, independent of the

More information

Review of Econometrics

Review of Econometrics Review of Econometrics Zheng Tian June 5th, 2017 1 The Essence of the OLS Estimation Multiple regression model involves the models as follows Y i = β 0 + β 1 X 1i + β 2 X 2i + + β k X ki + u i, i = 1,...,

More information

Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16)

Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) 1 2 Model Consider a system of two regressions y 1 = β 1 y 2 + u 1 (1) y 2 = β 2 y 1 + u 2 (2) This is a simultaneous equation model

More information

Panel Data. March 2, () Applied Economoetrics: Topic 6 March 2, / 43

Panel Data. March 2, () Applied Economoetrics: Topic 6 March 2, / 43 Panel Data March 2, 212 () Applied Economoetrics: Topic March 2, 212 1 / 43 Overview Many economic applications involve panel data. Panel data has both cross-sectional and time series aspects. Regression

More information

Final Exam. Economics 835: Econometrics. Fall 2010

Final Exam. Economics 835: Econometrics. Fall 2010 Final Exam Economics 835: Econometrics Fall 2010 Please answer the question I ask - no more and no less - and remember that the correct answer is often short and simple. 1 Some short questions a) For each

More information

Econometrics of Panel Data

Econometrics of Panel Data Econometrics of Panel Data Jakub Mućk Meeting # 4 Jakub Mućk Econometrics of Panel Data Meeting # 4 1 / 26 Outline 1 Two-way Error Component Model Fixed effects model Random effects model 2 Hausman-Taylor

More information

Econometrics of Panel Data

Econometrics of Panel Data Econometrics of Panel Data Jakub Mućk Meeting # 1 Jakub Mućk Econometrics of Panel Data Meeting # 1 1 / 31 Outline 1 Course outline 2 Panel data Advantages of Panel Data Limitations of Panel Data 3 Pooled

More information

Introductory Econometrics

Introductory Econometrics Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 11, 2012 Outline Heteroskedasticity

More information

Econometrics. Week 8. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague

Econometrics. Week 8. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Econometrics Week 8 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 25 Recommended Reading For the today Instrumental Variables Estimation and Two Stage

More information

LECTURE 2 LINEAR REGRESSION MODEL AND OLS

LECTURE 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 information

Section 6: Heteroskedasticity and Serial Correlation

Section 6: Heteroskedasticity and Serial Correlation From the SelectedWorks of Econ 240B Section February, 2007 Section 6: Heteroskedasticity and Serial Correlation Jeffrey Greenbaum, University of California, Berkeley Available at: https://works.bepress.com/econ_240b_econometrics/14/

More information

EC327: Advanced Econometrics, Spring 2007

EC327: Advanced Econometrics, Spring 2007 EC327: Advanced Econometrics, Spring 2007 Wooldridge, Introductory Econometrics (3rd ed, 2006) Chapter 14: Advanced panel data methods Fixed effects estimators We discussed the first difference (FD) model

More information

Applied Economics. Panel Data. Department of Economics Universidad Carlos III de Madrid

Applied Economics. Panel Data. Department of Economics Universidad Carlos III de Madrid Applied Economics Panel Data Department of Economics Universidad Carlos III de Madrid See also Wooldridge (chapter 13), and Stock and Watson (chapter 10) 1 / 38 Panel Data vs Repeated Cross-sections In

More information

Time Series Models and Inference. James L. Powell Department of Economics University of California, Berkeley

Time Series Models and Inference. James L. Powell Department of Economics University of California, Berkeley Time Series Models and Inference James L. Powell Department of Economics University of California, Berkeley Overview In contrast to the classical linear regression model, in which the components of the

More information

Instrumental Variables, Simultaneous and Systems of Equations

Instrumental Variables, Simultaneous and Systems of Equations Chapter 6 Instrumental Variables, Simultaneous and Systems of Equations 61 Instrumental variables In the linear regression model y i = x iβ + ε i (61) we have been assuming that bf x i and ε i are uncorrelated

More information

Lesson 17: Vector AutoRegressive Models

Lesson 17: Vector AutoRegressive Models Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it Vector AutoRegressive models The extension of ARMA models into a multivariate framework

More information

Notes on Panel Data and Fixed Effects models

Notes on Panel Data and Fixed Effects models Notes on Panel Data and Fixed Effects models Michele Pellizzari IGIER-Bocconi, IZA and frdb These notes are based on a combination of the treatment of panel data in three books: (i) Arellano M 2003 Panel

More information

Econometric Analysis of Cross Section and Panel Data

Econometric Analysis of Cross Section and Panel Data Econometric Analysis of Cross Section and Panel Data Jeffrey M. Wooldridge / The MIT Press Cambridge, Massachusetts London, England Contents Preface Acknowledgments xvii xxiii I INTRODUCTION AND BACKGROUND

More information

Multiple Regression Analysis

Multiple Regression Analysis Multiple Regression Analysis y = 0 + 1 x 1 + x +... k x k + u 6. Heteroskedasticity What is Heteroskedasticity?! Recall the assumption of homoskedasticity implied that conditional on the explanatory variables,

More information

Introductory Econometrics

Introductory Econometrics Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna November 23, 2013 Outline Introduction

More information

Models, Testing, and Correction of Heteroskedasticity. James L. Powell Department of Economics University of California, Berkeley

Models, Testing, and Correction of Heteroskedasticity. James L. Powell Department of Economics University of California, Berkeley Models, Testing, and Correction of Heteroskedasticity James L. Powell Department of Economics University of California, Berkeley Aitken s GLS and Weighted LS The Generalized Classical Regression Model

More information

Applied Microeconometrics (L5): Panel Data-Basics

Applied Microeconometrics (L5): Panel Data-Basics Applied Microeconometrics (L5): Panel Data-Basics Nicholas Giannakopoulos University of Patras Department of Economics ngias@upatras.gr November 10, 2015 Nicholas Giannakopoulos (UPatras) MSc Applied Economics

More information

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)

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) 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

More information

Econometrics Master in Business and Quantitative Methods

Econometrics Master in Business and Quantitative Methods Econometrics Master in Business and Quantitative Methods Helena Veiga Universidad Carlos III de Madrid Models with discrete dependent variables and applications of panel data methods in all fields of economics

More information

A Course in Applied Econometrics Lecture 7: Cluster Sampling. Jeff Wooldridge IRP Lectures, UW Madison, August 2008

A Course in Applied Econometrics Lecture 7: Cluster Sampling. Jeff Wooldridge IRP Lectures, UW Madison, August 2008 A Course in Applied Econometrics Lecture 7: Cluster Sampling Jeff Wooldridge IRP Lectures, UW Madison, August 2008 1. The Linear Model with Cluster Effects 2. Estimation with a Small Number of roups and

More information

Dealing With Endogeneity

Dealing With Endogeneity Dealing With Endogeneity Junhui Qian December 22, 2014 Outline Introduction Instrumental Variable Instrumental Variable Estimation Two-Stage Least Square Estimation Panel Data Endogeneity in Econometrics

More information

Estimation of Time-invariant Effects in Static Panel Data Models

Estimation of Time-invariant Effects in Static Panel Data Models Estimation of Time-invariant Effects in Static Panel Data Models M. Hashem Pesaran University of Southern California, and Trinity College, Cambridge Qiankun Zhou University of Southern California September

More information

Panel Data Econometrics

Panel Data Econometrics Panel Data Econometrics RDB 24-28 September 2012 Contents 1 Advantages of Panel Longitudinal) Data 5 I Preliminaries 5 2 Some Common Estimators 5 21 Ordinary Least Squares OLS) 5 211 Small Sample Properties

More information

Lecture 9: Panel Data Model (Chapter 14, Wooldridge Textbook)

Lecture 9: Panel Data Model (Chapter 14, Wooldridge Textbook) Lecture 9: Panel Data Model (Chapter 14, Wooldridge Textbook) 1 2 Panel Data Panel data is obtained by observing the same person, firm, county, etc over several periods. Unlike the pooled cross sections,

More information

MS&E 226: Small Data. Lecture 11: Maximum likelihood (v2) Ramesh Johari

MS&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 information

Panel Threshold Regression Models with Endogenous Threshold Variables

Panel Threshold Regression Models with Endogenous Threshold Variables Panel Threshold Regression Models with Endogenous Threshold Variables Chien-Ho Wang National Taipei University Eric S. Lin National Tsing Hua University This Version: June 29, 2010 Abstract This paper

More information

Intermediate Econometrics

Intermediate Econometrics Intermediate Econometrics Heteroskedasticity Text: Wooldridge, 8 July 17, 2011 Heteroskedasticity Assumption of homoskedasticity, Var(u i x i1,..., x ik ) = E(u 2 i x i1,..., x ik ) = σ 2. That is, the

More information

Quantitative Analysis of Financial Markets. Summary of Part II. Key Concepts & Formulas. Christopher Ting. November 11, 2017

Quantitative 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 information

Nonstationary Panels

Nonstationary Panels Nonstationary Panels Based on chapters 12.4, 12.5, and 12.6 of Baltagi, B. (2005): Econometric Analysis of Panel Data, 3rd edition. Chichester, John Wiley & Sons. June 3, 2009 Agenda 1 Spurious Regressions

More information

CLUSTER EFFECTS AND SIMULTANEITY IN MULTILEVEL MODELS

CLUSTER EFFECTS AND SIMULTANEITY IN MULTILEVEL MODELS HEALTH ECONOMICS, VOL. 6: 439 443 (1997) HEALTH ECONOMICS LETTERS CLUSTER EFFECTS AND SIMULTANEITY IN MULTILEVEL MODELS RICHARD BLUNDELL 1 AND FRANK WINDMEIJER 2 * 1 Department of Economics, University

More information

α version (only brief introduction so far)

α version (only brief introduction so far) Econometrics I KS Module 8: Panel Data Econometrics Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: June 18, 2018 α version (only brief introduction so far) Alexander

More information

Lecture 4: Linear panel models

Lecture 4: Linear panel models Lecture 4: Linear panel models Luc Behaghel PSE February 2009 Luc Behaghel (PSE) Lecture 4 February 2009 1 / 47 Introduction Panel = repeated observations of the same individuals (e.g., rms, workers, countries)

More information

Sensitivity of GLS estimators in random effects models

Sensitivity of GLS estimators in random effects models of GLS estimators in random effects models Andrey L. Vasnev (University of Sydney) Tokyo, August 4, 2009 1 / 19 Plan Plan Simulation studies and estimators 2 / 19 Simulation studies Plan Simulation studies

More information

Munich Lecture Series 2 Non-linear panel data models: Binary response and ordered choice models and bias-corrected fixed effects models

Munich Lecture Series 2 Non-linear panel data models: Binary response and ordered choice models and bias-corrected fixed effects models Munich Lecture Series 2 Non-linear panel data models: Binary response and ordered choice models and bias-corrected fixed effects models Stefanie Schurer stefanie.schurer@rmit.edu.au RMIT University School

More information

What s New in Econometrics? Lecture 14 Quantile Methods

What s New in Econometrics? Lecture 14 Quantile Methods What s New in Econometrics? Lecture 14 Quantile Methods Jeff Wooldridge NBER Summer Institute, 2007 1. Reminders About Means, Medians, and Quantiles 2. Some Useful Asymptotic Results 3. Quantile Regression

More information

Short T Panels - Review

Short T Panels - Review Short T Panels - Review We have looked at methods for estimating parameters on time-varying explanatory variables consistently in panels with many cross-section observation units but a small number of

More information

The Simple Regression Model. Part II. The Simple Regression Model

The 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 information

Introduction to Econometrics Midterm Examination Fall 2005 Answer Key

Introduction to Econometrics Midterm Examination Fall 2005 Answer Key Introduction to Econometrics Midterm Examination Fall 2005 Answer Key Please answer all of the questions and show your work Clearly indicate your final answer to each question If you think a question is

More information

HOW IS GENERALIZED LEAST SQUARES RELATED TO WITHIN AND BETWEEN ESTIMATORS IN UNBALANCED PANEL DATA?

HOW IS GENERALIZED LEAST SQUARES RELATED TO WITHIN AND BETWEEN ESTIMATORS IN UNBALANCED PANEL DATA? HOW IS GENERALIZED LEAST SQUARES RELATED TO WITHIN AND BETWEEN ESTIMATORS IN UNBALANCED PANEL DATA? ERIK BIØRN Department of Economics University of Oslo P.O. Box 1095 Blindern 0317 Oslo Norway E-mail:

More information

1 Estimation of Persistent Dynamic Panel Data. Motivation

1 Estimation of Persistent Dynamic Panel Data. Motivation 1 Estimation of Persistent Dynamic Panel Data. Motivation Consider the following Dynamic Panel Data (DPD) model y it = y it 1 ρ + x it β + µ i + v it (1.1) with i = {1, 2,..., N} denoting the individual

More information

Econometrics of Panel Data

Econometrics of Panel Data Econometrics of Panel Data Jakub Mućk Meeting # 3 Jakub Mućk Econometrics of Panel Data Meeting # 3 1 / 21 Outline 1 Fixed or Random Hausman Test 2 Between Estimator 3 Coefficient of determination (R 2

More information

Problem Set #6: OLS. Economics 835: Econometrics. Fall 2012

Problem 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 information

Heteroskedasticity in Panel Data

Heteroskedasticity in Panel Data Essex Summer School in Social Science Data Analysis Panel Data Analysis for Comparative Research Heteroskedasticity in Panel Data Christopher Adolph Department of Political Science and Center for Statistics

More information

Panel Data: Linear Models

Panel Data: Linear Models Panel Data: Linear Models Laura Magazzini University of Verona laura.magazzini@univr.it http://dse.univr.it/magazzini Laura Magazzini (@univr.it) Panel Data: Linear Models 1 / 45 Introduction Outline What

More information

Seemingly Unrelated Regressions in Panel Models. Presented by Catherine Keppel, Michael Anreiter and Michael Greinecker

Seemingly Unrelated Regressions in Panel Models. Presented by Catherine Keppel, Michael Anreiter and Michael Greinecker Seemingly Unrelated Regressions in Panel Models Presented by Catherine Keppel, Michael Anreiter and Michael Greinecker Arnold Zellner An Efficient Method of Estimating Seemingly Unrelated Regressions and

More information

1 Outline. 1. Motivation. 2. SUR model. 3. Simultaneous equations. 4. Estimation

1 Outline. 1. Motivation. 2. SUR model. 3. Simultaneous equations. 4. Estimation 1 Outline. 1. Motivation 2. SUR model 3. Simultaneous equations 4. Estimation 2 Motivation. In this chapter, we will study simultaneous systems of econometric equations. Systems of simultaneous equations

More information

Simple Linear Regression: The Model

Simple 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 information

Homoskedasticity. Var (u X) = σ 2. (23)

Homoskedasticity. Var (u X) = σ 2. (23) Homoskedasticity How big is the difference between the OLS estimator and the true parameter? To answer this question, we make an additional assumption called homoskedasticity: Var (u X) = σ 2. (23) This

More information

Max. Likelihood Estimation. Outline. Econometrics II. Ricardo Mora. Notes. Notes

Max. Likelihood Estimation. Outline. Econometrics II. Ricardo Mora. Notes. Notes Maximum Likelihood Estimation Econometrics II Department of Economics Universidad Carlos III de Madrid Máster Universitario en Desarrollo y Crecimiento Económico Outline 1 3 4 General Approaches to Parameter

More information

Econ 510 B. Brown Spring 2014 Final Exam Answers

Econ 510 B. Brown Spring 2014 Final Exam Answers Econ 510 B. Brown Spring 2014 Final Exam Answers Answer five of the following questions. You must answer question 7. The question are weighted equally. You have 2.5 hours. You may use a calculator. Brevity

More information

Linear Regression. Junhui Qian. October 27, 2014

Linear 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 information

Lecture 8 Panel Data

Lecture 8 Panel Data Lecture 8 Panel Data Economics 8379 George Washington University Instructor: Prof. Ben Williams Introduction This lecture will discuss some common panel data methods and problems. Random effects vs. fixed

More information

Weighted Least Squares

Weighted 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 information

Business Economics BUSINESS ECONOMICS. PAPER No. : 8, FUNDAMENTALS OF ECONOMETRICS MODULE No. : 3, GAUSS MARKOV THEOREM

Business Economics BUSINESS ECONOMICS. PAPER No. : 8, FUNDAMENTALS OF ECONOMETRICS MODULE No. : 3, GAUSS MARKOV THEOREM Subject Business Economics Paper No and Title Module No and Title Module Tag 8, Fundamentals of Econometrics 3, The gauss Markov theorem BSE_P8_M3 1 TABLE OF CONTENTS 1. INTRODUCTION 2. ASSUMPTIONS OF

More information

Introduction to Estimation Methods for Time Series models. Lecture 1

Introduction 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 information

CRE METHODS FOR UNBALANCED PANELS Correlated Random Effects Panel Data Models IZA Summer School in Labor Economics May 13-19, 2013 Jeffrey M.

CRE METHODS FOR UNBALANCED PANELS Correlated Random Effects Panel Data Models IZA Summer School in Labor Economics May 13-19, 2013 Jeffrey M. CRE METHODS FOR UNBALANCED PANELS Correlated Random Effects Panel Data Models IZA Summer School in Labor Economics May 13-19, 2013 Jeffrey M. Wooldridge Michigan State University 1. Introduction 2. Linear

More information

An overview of applied econometrics

An overview of applied econometrics An overview of applied econometrics Jo Thori Lind September 4, 2011 1 Introduction This note is intended as a brief overview of what is necessary to read and understand journal articles with empirical

More information

Ma 3/103: Lecture 24 Linear Regression I: Estimation

Ma 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 information

Econometric Methods for Panel Data Part I

Econometric Methods for Panel Data Part I Econometric Methods for Panel Data Part I Robert M. Kunst University of Vienna February 011 1 Introduction The word panel is derived from Dutch and originally describes a rectangular board. In econometrics,

More information

STAT5044: Regression and Anova. Inyoung Kim

STAT5044: Regression and Anova. Inyoung Kim STAT5044: Regression and Anova Inyoung Kim 2 / 47 Outline 1 Regression 2 Simple Linear regression 3 Basic concepts in regression 4 How to estimate unknown parameters 5 Properties of Least Squares Estimators:

More information

Regression Models - Introduction

Regression Models - Introduction Regression Models - Introduction In regression models, two types of variables that are studied: A dependent variable, Y, also called response variable. It is modeled as random. An independent variable,

More information