Lecture 1: OLS derivations and inference
|
|
- Shannon Nash
- 6 years ago
- Views:
Transcription
1 Lecture 1: OLS derivations and inference Econometric Methods Warsaw School of Economics (1) OLS 1 / 43
2 Outline 1 Introduction Course information Econometrics: a reminder Preliminary data exploration 2 OLS: theoretical reminder Point estimation Measuring precision Model quality diagnostics under OLS 3 Multicollinearity (1) OLS 2 / 43
3 Outline 1 Introduction 2 OLS: theoretical reminder 3 Multicollinearity (1) OLS 3 / 43
4 Course information Course information lecturers: & Marcin Owczarczuk my website: (lecture slides, exercise les, literature, contact) nal grade: homework (50%) + written exam (50%); details: website (1) OLS 4 / 43
5 Course information Course information lecturers: & Marcin Owczarczuk my website: (lecture slides, exercise les, literature, contact) nal grade: homework (50%) + written exam (50%); details: website (1) OLS 4 / 43
6 Course information Course information lecturers: & Marcin Owczarczuk my website: (lecture slides, exercise les, literature, contact) nal grade: homework (50%) + written exam (50%); details: website (1) OLS 4 / 43
7 Econometrics: a reminder Why econometrics? investigation of relationships nding parameter values in economic models (eg elasticities) confronting economic theories with data forecasting simulating policy scenarios (1) OLS 5 / 43
8 Econometrics: a reminder Why econometrics? investigation of relationships nding parameter values in economic models (eg elasticities) confronting economic theories with data forecasting simulating policy scenarios (1) OLS 5 / 43
9 Econometrics: a reminder Why econometrics? investigation of relationships nding parameter values in economic models (eg elasticities) confronting economic theories with data forecasting simulating policy scenarios (1) OLS 5 / 43
10 Econometrics: a reminder Why econometrics? investigation of relationships nding parameter values in economic models (eg elasticities) confronting economic theories with data forecasting simulating policy scenarios (1) OLS 5 / 43
11 Econometrics: a reminder Why econometrics? investigation of relationships nding parameter values in economic models (eg elasticities) confronting economic theories with data forecasting simulating policy scenarios (1) OLS 5 / 43
12 Econometrics: a reminder Data structure 1 time series (industrial production, , monthly index) 2 cross section (exit polls with 1000 respondents) 3 longitudinal data (quarterly GDP dynamics in EU states, ) (1) OLS 6 / 43
13 Econometrics: a reminder Data structure 1 time series (industrial production, , monthly index) 2 cross section (exit polls with 1000 respondents) 3 longitudinal data (quarterly GDP dynamics in EU states, ) (1) OLS 6 / 43
14 Econometrics: a reminder Data structure 1 time series (industrial production, , monthly index) 2 cross section (exit polls with 1000 respondents) 3 longitudinal data (quarterly GDP dynamics in EU states, ) (1) OLS 6 / 43
15 Econometrics: a reminder Selection of explanatory variables theory, institutional setup, expert evaluation mechanical methods data mining correlation-matrix-based treatments from general to specic approach variable transformations (1) OLS 7 / 43
16 Econometrics: a reminder Selection of explanatory variables theory, institutional setup, expert evaluation mechanical methods data mining correlation-matrix-based treatments from general to specic approach variable transformations (1) OLS 7 / 43
17 Econometrics: a reminder Selection of explanatory variables theory, institutional setup, expert evaluation mechanical methods data mining correlation-matrix-based treatments from general to specic approach variable transformations (1) OLS 7 / 43
18 Econometrics: a reminder Selection of explanatory variables theory, institutional setup, expert evaluation mechanical methods data mining correlation-matrix-based treatments from general to specic approach variable transformations (1) OLS 7 / 43
19 Preliminary data exploration Preliminary exploration (1) distribution data errors, outliers? (1) OLS 8 / 43
20 Preliminary data exploration Preliminary exploration(2) time series graph trend? seasonality? volatility clustering? (1) OLS 9 / 43
21 Preliminary data exploration Preliminary exploration (3) scatterplot dependency, functional form, transformations? (1) OLS 10 / 43
22 Preliminary data exploration Preliminary exploration (4) correlations? (1) OLS 11 / 43
23 Preliminary data exploration Example (1/5) Student satisfaction survey Master students of Applied Econometrics at Warsaw School of Economics in Winter semester 2016/2017 were asked about their satisfaction from studying to be evaluated from 0 to 100 In addition, their average note from previous studies and their sex were registered 1 What kind of data is this? Cross-section, time series, panel? Frequency? Micro- or macroeconomic? 2 How can we quickly visualise a hypothesised causality from average note to satisfaction from studying? 3 Does such a relationship seem to be there? 4 How can sex of the respondent potentially aect the satisfaction from studies or the relationship in question? How can we visualise this? 5 Bottom line, what is the right specication of the linear regression model? (1) OLS 12 / 43
24 Outline 1 Introduction 2 OLS: theoretical reminder 3 Multicollinearity (1) OLS 13 / 43
25 Point estimation Linear regression model y i = β 0 + β 1 x 1,i + β 2 x 2,i + + β k x k,i + ε i = β 0 [ ] β 1 1 x1,i x 2,i x k,i β 2 + ε i = x i β + ε i β k Vector of parameters [ ] T β 0 β 1 β 2 β k is unknown Minimize the dispersion of ε i around zero, as measured eg by n ε 2 i t=1 (1) OLS 14 / 43
26 Point estimation Ordinary Least Squares (OLS) ε 2 i i=1 S = n = n i=1 (y i β 0 β 1 x 1,i β 2 x 2,i β k x k,i ) 2 min β 0,β 1, Denote: y = y 1 y 2 y n FOC: S β = 0 1 x 1,1 x 2,1 x k,1, X = 1 x 1,2 x 2,2 x k,2 1 x 1,n x 2,n x k,n and obtain:, β = β 0 β 1 β 2 β k β = ( X T X) 1 X T y (1) OLS 15 / 43
27 Point estimation Proof S = n ε 2 i = ε T ε = (y Xβ) T (y Xβ) = i=1 = y T y β T X T y y T Xβ + β T X T Xβ = = y T y 2y T Xβ + β T X T Xβ (2 and 3 component were transposed scalars, so they were equal) S β = 0 yt y β + βt X T Xβ β = 0 According to the rules of matrix calculus: 2y T X + β T ( 2X T ) X = 0 β T ( X T ) ( X = y T X X T ) X β = X T y β = ( X T X ) 1 X T y 2yT Xβ β (1) OLS 16 / 43
28 Point estimation Example (2/5) Student satisfaction survey 1 Run the regression model with an automated command in R 2 Write the equation and try to interpret the parameters Be careful it's tricky! (Why?) 3 Manually replicate the parameter estimates (1) OLS 17 / 43
29 Measuring precision Estimator as a random variable ˆβ is an estimator of the true parameter value β (function of the random sample choice) samples, and hence the values of ˆβ, can be dierent estimator as a (vector) random variable has its variance(-covariance matrix) ˆβ = ˆβ 0 ˆβ 1 ˆβ 2 ( ) Var ˆβ = ˆβ ( k ) var ˆβ 0 ( ) cov ˆβ 0, ˆβ 1 ( cov ˆβ0, ˆβ ) 2 ( ) cov ˆβ 0, ˆβ 1 ( ) var ˆβ 1 ( cov ˆβ1, ˆβ ) 2 ( ) cov ˆβ 0, ˆβ 2 ( ) cov ˆβ 1, ˆβ 2 ( ) var ˆβ2 ( ) var ˆβ k (1) OLS 18 / 43
30 Measuring precision Estimator as a random variable ˆβ is an estimator of the true parameter value β (function of the random sample choice) samples, and hence the values of ˆβ, can be dierent estimator as a (vector) random variable has its variance(-covariance matrix) ˆβ = ˆβ 0 ˆβ 1 ˆβ 2 ( ) Var ˆβ = ˆβ ( k ) var ˆβ 0 ( ) cov ˆβ 0, ˆβ 1 ( cov ˆβ0, ˆβ ) 2 ( ) cov ˆβ 0, ˆβ 1 ( ) var ˆβ 1 ( cov ˆβ1, ˆβ ) 2 ( ) cov ˆβ 0, ˆβ 2 ( ) cov ˆβ 1, ˆβ 2 ( ) var ˆβ2 ( ) var ˆβ k (1) OLS 18 / 43
31 Measuring precision Estimator as a random variable ˆβ is an estimator of the true parameter value β (function of the random sample choice) samples, and hence the values of ˆβ, can be dierent estimator as a (vector) random variable has its variance(-covariance matrix) ˆβ = ˆβ 0 ˆβ 1 ˆβ 2 ( ) Var ˆβ = ˆβ ( k ) var ˆβ 0 ( ) cov ˆβ 0, ˆβ 1 ( cov ˆβ0, ˆβ ) 2 ( ) cov ˆβ 0, ˆβ 1 ( ) var ˆβ 1 ( cov ˆβ1, ˆβ ) 2 ( ) cov ˆβ 0, ˆβ 2 ( ) cov ˆβ 1, ˆβ 2 ( ) var ˆβ2 ( ) var ˆβ k (1) OLS 18 / 43
32 Measuring precision Variance-covariance matrix of a random vector Denition: Var (β) = E {[β } E (β)] [β E (β)] T For a centered variable, ie E (ε) = 0, this denition simplies: Var (ε) = E ( εε T ) (1) OLS 19 / 43
33 Measuring precision OLS estimator: properties ˆβ = ( X T ) 1 X X T y is an estimator (function of the sample) of the true, unknown values β (population / data generating process) Under certain conditions (ia E(X T ε) = 0 E(εε T ) = σ 2 I ), the OLS estimator is: ) unbiased: E (ˆβ = β consistent: ˆβ converges to β with growing n ecient: least possible estimator variance (ie highest precision) (1) OLS 20 / 43
34 Measuring precision OLS estimator: properties ˆβ = ( X T ) 1 X X T y is an estimator (function of the sample) of the true, unknown values β (population / data generating process) Under certain conditions (ia E(X T ε) = 0 E(εε T ) = σ 2 I ), the OLS estimator is: ) unbiased: E (ˆβ = β consistent: ˆβ converges to β with growing n ecient: least possible estimator variance (ie highest precision) (1) OLS 20 / 43
35 Measuring precision OLS estimator: properties ˆβ = ( X T ) 1 X X T y is an estimator (function of the sample) of the true, unknown values β (population / data generating process) Under certain conditions (ia E(X T ε) = 0 E(εε T ) = σ 2 I ), the OLS estimator is: ) unbiased: E (ˆβ = β consistent: ˆβ converges to β with growing n ecient: least possible estimator variance (ie highest precision) (1) OLS 20 / 43
36 Measuring precision Variance of the error term (1) 1 Variance of the error term (scalar): ˆσ 2 = 1 n (k+1) Why such a formula if the general formula is n Var(X ) = 1 (x n 1 i x) 2? i=1 First of all note that ε = 0 (prove it on your own) n ε 2 i i=1 Second, we need to know why 1 turned into (k + 1) in the denominator (1) OLS 21 / 43
37 Measuring precision Variance of the error term (2) By Your intuition, what is the standard deviation in the following dataset of 3 observation? Without a correction in denominator: ] Var = [(3 2) 2 + (2 2) 2 + (1 2) 2 = (1) OLS 22 / 43
38 Measuring precision Variance of the error term (2) By Your intuition, what is the standard deviation in the following dataset of 3 observation? Without a correction in denominator: ] Var = [(3 2) 2 + (2 2) 2 + (1 2) 2 = (1) OLS 22 / 43
39 Measuring precision Variance of the error term (3) The intuition behind the standard deviation of 1 is build upon an implicit, graphical calibration of mean based on the data sample With an adequate correction for thatin denominator: ] Var = [(3 2) 2 + (2 2) 2 + (1 2) 2 = = 1 (1) OLS 23 / 43
40 Measuring precision Variance of the error term (3) The intuition behind the standard deviation of 1 is build upon an implicit, graphical calibration of mean based on the data sample With an adequate correction for thatin denominator: ] Var = [(3 2) 2 + (2 2) 2 + (1 2) 2 = = 1 (1) OLS 23 / 43
41 Measuring precision Variance of the error term (4) When X is directly observed, the terms like (x i x) consume one degree of freedom (there is one x estimated before) When ε is not observed, the terms ε i = y i ˆβ 0 ˆβ 1 x 1i ˆβ k x ki consume k + 1) degrees of freedom (there are k + 1 elements in vector ˆβ estimated before) (1) OLS 24 / 43
42 Measuring precision Variance-covariance matrix of the estimator ) [ ) ) ] T Var (ˆβ = E (ˆβ β (ˆβ β = { [( = E X X) T 1 ] [ T ( X y β X X) T 1 ] } T T X y β = { [( = E X X) T 1 ] [ ( T X (Xβ + ε) β X X) T 1 ] } T T X (Xβ + ε) β = [ ( ) = E X T 1 ( ) X T X ε ε T X X T 1 ] X = = ( X X) T 1 T X E ( εε ) ( T X X X) T 1 = = ( X X) T 1 ( T X σ 2 IX X X) T 1 = = σ ( ) 2 X T 1 ( T X X X X X) T 1 = = σ ( ) 2 X T 1 X ( ) Empirical counterpart: Var ˆβ = ˆσ ( 2 X X) T 1 [di,j ] (k+1) (k+1) (1) OLS 25 / 43
43 Measuring precision Standard errors of estimation Standard ( ) errors of estimation (vector for each parameter): s ˆβ 0 = ( ) d 1,1 s ˆβ 1 = ( ) d 2,2 s ˆβ 2 = d 3,3 Calculating SE 1 estimate parameters, 2 compute the empirical error terms, 3 estimate their variance, 4 compute the variance-covariance matrix of the OLS estimator, 5 compute the SE as a square root of its diagonal elements (1) OLS 26 / 43
44 Measuring precision Standard errors of estimation Standard ( ) errors of estimation (vector for each parameter): s ˆβ 0 = ( ) d 1,1 s ˆβ 1 = ( ) d 2,2 s ˆβ 2 = d 3,3 Calculating SE 1 estimate parameters, 2 compute the empirical error terms, 3 estimate their variance, 4 compute the variance-covariance matrix of the OLS estimator, 5 compute the SE as a square root of its diagonal elements (1) OLS 26 / 43
45 Measuring precision t-tests for variable signicance t-student test H 0 : β i = 0, ie i-th explanatory variable does not signicantly inuence y H 1 : β i 0, ie i-th explanatory variable does not signicantly inuence y Test statistic: t = ˆβ i s( ˆβ is distributed as t (n k 1) 1) p-value<α reject H 0 p-value>α do not reject H 0 (1) OLS 27 / 43
46 Measuring precision Example (3/5) Student satisfaction survey 1 Compute the tted values and the error terms 2 Use this result to estimate the variance of the error term 3 Estimate the variance-covariance matrix of the ˆβ estimates 4 Derive the standard errors from this matrix 5 Replicate and interpret the t statistics and the p-values (1) OLS 28 / 43
47 Model quality diagnostics under OLS R-squared (1) (1) OLS 29 / 43
48 Model quality diagnostics under OLS R-squared (2) (1) OLS 30 / 43
49 Model quality diagnostics under OLS R-squared (3) (1) OLS 31 / 43
50 Model quality diagnostics under OLS R-squared (4) (1) OLS 32 / 43
51 Model quality diagnostics under OLS R-squared (5) (1) OLS 33 / 43
52 Model quality diagnostics under OLS R-squared (6) (1) OLS 34 / 43
53 Model quality diagnostics under OLS R-squared (7) R 2 [0; 1] is a share of y t volatility explained by the model in total y t volatility: T (y t ȳ) 2 = T (ŷ t ȳ) 2 + T (y t ŷ t ) 2 R 2 = t=1 t=1 t=1 T (ŷ t ȳ) 2 t=1 T (y t ȳ) 2 t=1 Standard goodness-of-t measure in OLS regressions with a constant (1) OLS 35 / 43
54 Model quality diagnostics under OLS Wald test statistic Wald test H 0 : β 1 = β 2 = = β k = 0, ie no explanatory variable inuences y H 1 : i β i 0, at least 1 explanatory variable inuences y Test statistic: F = distributed as F (k, T k 1) R 2 /k (1 R 2 )/(T k 1) (1) OLS 36 / 43
55 Model quality diagnostics under OLS Adjusted R-squared R 2 = }{{} R 2 fit k ( ) 1 R 2 T (k + 1) }{{} penalty for overparametrization (1) OLS 37 / 43
56 Model quality diagnostics under OLS Example (4/5) Student satisfaction survey 1 Interpret the F-test result 2 Replicate the F statistic and its p-value manually 3 Interpret the R-squared 4 Replicate the R-squared and adjusted R-squared manually (1) OLS 38 / 43
57 Outline 1 Introduction 2 OLS: theoretical reminder 3 Multicollinearity (1) OLS 39 / 43
58 Multicollinearity Multicollinearity Regressors are not independent: some are a linear combination of others (extreme case), then X T X is a singular matrix and we cannot compute β = ( X T X) 1 X T y some are highly correlated (usual case), then X T X may not be singular, but its diagonal elements still close to zero then the diagonal elements of (and so the standard errors) ( X T X) 1 extremely high (1) OLS 40 / 43
59 Multicollinearity Multicollinearity diagnostics 1 correlation matrix only bilateral relationships no cut-o value above which the problem can be considered serious 2 ination variance factor (VIF) for regressor j VIF j = 1 1 R j 2 where Rj 2 is R-squared from the regression of variable j on the rest of the explanatory variables limit value: 10, above multicollinearity 3 condition index κ = λ max λ min where λ denotes eigenvalues of the matrix derived from X T X by division of every cell (i, j) by the product of diagonal elements (i, i) and (j, j) limit value: 20, above multicollinearity (1) OLS 41 / 43
60 Multicollinearity Multicollinearity solutions strengthen the precision of estimation by expanding sample size, removing a variable, imposing restrictions or calibrating the parameter manually increase the diagonal values in X T X (ridge regression) squeeze the common variance of the collinear variables into a lower number of new, independent ones (principal components) (1) OLS 42 / 43
61 Multicollinearity Multicollinearity solutions strengthen the precision of estimation by expanding sample size, removing a variable, imposing restrictions or calibrating the parameter manually increase the diagonal values in X T X (ridge regression) squeeze the common variance of the collinear variables into a lower number of new, independent ones (principal components) (1) OLS 42 / 43
62 Multicollinearity Multicollinearity solutions strengthen the precision of estimation by expanding sample size, removing a variable, imposing restrictions or calibrating the parameter manually increase the diagonal values in X T X (ridge regression) squeeze the common variance of the collinear variables into a lower number of new, independent ones (principal components) (1) OLS 42 / 43
63 Multicollinearity Example (5/5) Student satisfaction survey Investigate the issue of multicollinearity in the proposed model (1) OLS 43 / 43
Lecture 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 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 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 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 informationECO375 Tutorial 8 Instrumental Variables
ECO375 Tutorial 8 Instrumental Variables Matt Tudball University of Toronto Mississauga November 16, 2017 Matt Tudball (University of Toronto) ECO375H5 November 16, 2017 1 / 22 Review: Endogeneity Instrumental
More informationHomoskedasticity. 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 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 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 informationChapter 2: simple regression model
Chapter 2: simple regression model Goal: understand how to estimate and more importantly interpret the simple regression Reading: chapter 2 of the textbook Advice: this chapter is foundation of 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 informationProperties of the least squares estimates
Properties of the least squares estimates 2019-01-18 Warmup Let a and b be scalar constants, and X be a scalar random variable. Fill in the blanks E ax + b) = Var ax + b) = Goal Recall that the least squares
More informationRewrap ECON November 18, () Rewrap ECON 4135 November 18, / 35
Rewrap ECON 4135 November 18, 2011 () Rewrap ECON 4135 November 18, 2011 1 / 35 What should you now know? 1 What is econometrics? 2 Fundamental regression analysis 1 Bivariate regression 2 Multivariate
More informationAnalisi Statistica per le Imprese
, Analisi Statistica per le Imprese Dip. di Economia Politica e Statistica 4.3. 1 / 33 You should be able to:, Underst model building using multiple regression analysis Apply multiple regression analysis
More informationSpatial Econometrics
Spatial Econometrics Lecture 5: Single-source model of spatial regression. Combining GIS and regional analysis (5) Spatial Econometrics 1 / 47 Outline 1 Linear model vs SAR/SLM (Spatial Lag) Linear model
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 informationEconometrics 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 informationMultiple Linear Regression
Multiple Linear Regression Simple linear regression tries to fit a simple line between two variables Y and X. If X is linearly related to Y this explains some of the variability in Y. In most cases, there
More informationEconometrics I Lecture 3: The Simple Linear Regression Model
Econometrics I Lecture 3: The Simple Linear Regression Model Mohammad Vesal Graduate School of Management and Economics Sharif University of Technology 44716 Fall 1397 1 / 32 Outline Introduction Estimating
More informationECON Introductory Econometrics. Lecture 6: OLS with Multiple Regressors
ECON4150 - Introductory Econometrics Lecture 6: OLS with Multiple Regressors Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 6 Lecture outline 2 Violation of first Least Squares assumption
More informationECON The Simple Regression Model
ECON 351 - The Simple Regression Model Maggie Jones 1 / 41 The Simple Regression Model Our starting point will be the simple regression model where we look at the relationship between two variables In
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 informationMultiple Regression Analysis
Chapter 4 Multiple Regression Analysis The simple linear regression covered in Chapter 2 can be generalized to include more than one variable. Multiple regression analysis is an extension of the simple
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 informationFöreläsning /31
1/31 Föreläsning 10 090420 Chapter 13 Econometric Modeling: Model Speci cation and Diagnostic testing 2/31 Types of speci cation errors Consider the following models: Y i = β 1 + β 2 X i + β 3 X 2 i +
More informationIEOR 165 Lecture 7 1 Bias-Variance Tradeoff
IEOR 165 Lecture 7 Bias-Variance Tradeoff 1 Bias-Variance Tradeoff Consider the case of parametric regression with β R, and suppose we would like to analyze the error of the estimate ˆβ in comparison to
More informationEconometrics 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 informationIntroduction to Linear Regression Analysis
Introduction to Linear Regression Analysis Samuel Nocito Lecture 1 March 2nd, 2018 Econometrics: What is it? Interaction of economic theory, observed data and statistical methods. The science of testing
More informationEconometrics 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 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 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 informationIntermediate 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 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 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 informationEnvironmental Econometrics
Environmental Econometrics Syngjoo Choi Fall 2008 Environmental Econometrics (GR03) Fall 2008 1 / 37 Syllabus I This is an introductory econometrics course which assumes no prior knowledge on econometrics;
More informationIntroductory 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 informationL7: Multicollinearity
L7: Multicollinearity Feng Li feng.li@cufe.edu.cn School of Statistics and Mathematics Central University of Finance and Economics Introduction ï Example Whats wrong with it? Assume we have this data Y
More informationIntroduction to Linear Regression Analysis Interpretation of Results
Introduction to Linear Regression Analysis Interpretation of Results Samuel Nocito Lecture 2 March 8th, 2018 Lecture 1 Summary Why and how we use econometric tools in empirical research. Ordinary Least
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 informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postponed exam: ECON4160 Econometrics Modeling and systems estimation Date of exam: Wednesday, January 8, 2014 Time for exam: 09:00 a.m. 12:00 noon The problem
More informationEconometrics 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 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 informationMultiple Regression Analysis. Part III. Multiple Regression Analysis
Part III Multiple Regression Analysis As of Sep 26, 2017 1 Multiple Regression Analysis Estimation Matrix form Goodness-of-Fit R-square Adjusted R-square Expected values of the OLS estimators Irrelevant
More informationReview 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 informationAnswers to Problem Set #4
Answers to Problem Set #4 Problems. Suppose that, from a sample of 63 observations, the least squares estimates and the corresponding estimated variance covariance matrix are given by: bβ bβ 2 bβ 3 = 2
More informationApplied Econometrics - QEM Theme 1: Introduction to Econometrics Chapter 1 + Probability Primer + Appendix B in PoE
Applied Econometrics - QEM Theme 1: Introduction to Econometrics Chapter 1 + Probability Primer + Appendix B in PoE Warsaw School of Economics Outline 1. Introduction to econometrics 2. Denition of econometrics
More informationLesson 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 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 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 informationEconometrics I. Introduction to Regression Analysis (Outline of the course) Marijus Radavi ius
Econometrics I Introduction to Regression Analysis (Outline of the course) Marijus Radavi ius Vilnius 2012 Content 1 Introduction 2 1.1 Motivation................................. 2 1.2 Terminology................................
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 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 informationEconomics 471: Econometrics Department of Economics, Finance and Legal Studies University of Alabama
Economics 471: Econometrics Department of Economics, Finance and Legal Studies University of Alabama Course Packet The purpose of this packet is to show you one particular dataset and how it is used in
More informationEcon107 Applied Econometrics
Econ107 Applied Econometrics Topics 2-4: discussed under the classical Assumptions 1-6 (or 1-7 when normality is needed for finite-sample inference) Question: what if some of the classical assumptions
More informationECON Introductory Econometrics. Lecture 16: Instrumental variables
ECON4150 - Introductory Econometrics Lecture 16: Instrumental variables Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 12 Lecture outline 2 OLS assumptions and when they are violated Instrumental
More informationLeast Squares Estimation-Finite-Sample Properties
Least Squares Estimation-Finite-Sample Properties Ping Yu School of Economics and Finance The University of Hong Kong Ping Yu (HKU) Finite-Sample 1 / 29 Terminology and Assumptions 1 Terminology and Assumptions
More informationLinear Regression Models
Linear Regression Models November 13, 2018 1 / 89 1 Basic framework Model specification and assumptions Parameter estimation: least squares method Coefficient of determination R 2 Properties of the least
More informationLab 07 Introduction to Econometrics
Lab 07 Introduction to Econometrics Learning outcomes for this lab: Introduce the different typologies of data and the econometric models that can be used Understand the rationale behind econometrics Understand
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 informationFinite Sample Performance of A Minimum Distance Estimator Under Weak Instruments
Finite Sample Performance of A Minimum Distance Estimator Under Weak Instruments Tak Wai Chau February 20, 2014 Abstract This paper investigates the nite sample performance of a minimum distance estimator
More informationMaking sense of Econometrics: Basics
Making sense of Econometrics: Basics Lecture 7: Multicollinearity Egypt Scholars Economic Society November 22, 2014 Assignment & feedback Multicollinearity enter classroom at room name c28efb78 http://b.socrative.com/login/student/
More informationECON3150/4150 Spring 2015
ECON3150/4150 Spring 2015 Lecture 3&4 - The linear regression model Siv-Elisabeth Skjelbred University of Oslo January 29, 2015 1 / 67 Chapter 4 in S&W Section 17.1 in S&W (extended OLS assumptions) 2
More informationEconometrics 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 informationEstadística II Chapter 5. Regression analysis (second part)
Estadística II Chapter 5. Regression analysis (second part) Chapter 5. Regression analysis (second part) Contents Diagnostic: Residual analysis The ANOVA (ANalysis Of VAriance) decomposition Nonlinear
More informationThe Standard Linear Model: Hypothesis Testing
Department of Mathematics Ma 3/103 KC Border Introduction to Probability and Statistics Winter 2017 Lecture 25: The Standard Linear Model: Hypothesis Testing Relevant textbook passages: Larsen Marx [4]:
More informationApplied Econometrics (QEM)
Applied Econometrics (QEM) based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #3 1 / 42 Outline 1 2 3 t-test P-value Linear
More informationDay 4: Shrinkage Estimators
Day 4: Shrinkage Estimators Kenneth Benoit Data Mining and Statistical Learning March 9, 2015 n versus p (aka k) Classical regression framework: n > p. Without this inequality, the OLS coefficients have
More informationSimple Regression Model Setup Estimation Inference Prediction. Model Diagnostic. Multiple Regression. Model Setup and Estimation.
Statistical Computation Math 475 Jimin Ding Department of Mathematics Washington University in St. Louis www.math.wustl.edu/ jmding/math475/index.html October 10, 2013 Ridge Part IV October 10, 2013 1
More informationTwo-Variable Regression Model: The Problem of Estimation
Two-Variable Regression Model: The Problem of Estimation Introducing the Ordinary Least Squares Estimator Jamie Monogan University of Georgia Intermediate Political Methodology Jamie Monogan (UGA) Two-Variable
More informationLecture: 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 informationWISE International Masters
WISE International Masters ECONOMETRICS Instructor: Brett Graham INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This examination paper contains 32 questions. You are
More informationA Practitioner s Guide to Cluster-Robust Inference
A Practitioner s Guide to Cluster-Robust Inference A. C. Cameron and D. L. Miller presented by Federico Curci March 4, 2015 Cameron Miller Cluster Clinic II March 4, 2015 1 / 20 In the previous episode
More informationLectures 5 & 6: Hypothesis Testing
Lectures 5 & 6: Hypothesis Testing in which you learn to apply the concept of statistical significance to OLS estimates, learn the concept of t values, how to use them in regression work and come across
More informationStatistical Inference with Regression Analysis
Introductory Applied Econometrics EEP/IAS 118 Spring 2015 Steven Buck Lecture #13 Statistical Inference with Regression Analysis Next we turn to calculating confidence intervals and hypothesis testing
More informationThe Simple Linear Regression Model
The Simple Linear Regression Model Lesson 3 Ryan Safner 1 1 Department of Economics Hood College ECON 480 - Econometrics Fall 2017 Ryan Safner (Hood College) ECON 480 - Lesson 3 Fall 2017 1 / 77 Bivariate
More information4. Nonlinear regression functions
4. Nonlinear regression functions Up to now: Population regression function was assumed to be linear The slope(s) of the population regression function is (are) constant The effect on Y of a unit-change
More informationECON Introductory Econometrics. Lecture 4: Linear Regression with One Regressor
ECON4150 - Introductory Econometrics Lecture 4: Linear Regression with One Regressor Monique de Haan (moniqued@econ.uio.no) Stock and Watson Chapter 4 Lecture outline 2 The OLS estimators The effect of
More informationApplied 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 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 informationMa 3/103: Lecture 25 Linear Regression II: Hypothesis Testing and ANOVA
Ma 3/103: Lecture 25 Linear Regression II: Hypothesis Testing and ANOVA March 6, 2017 KC Border Linear Regression II March 6, 2017 1 / 44 1 OLS estimator 2 Restricted regression 3 Errors in variables 4
More informationManaging Uncertainty
Managing Uncertainty Bayesian Linear Regression and Kalman Filter December 4, 2017 Objectives The goal of this lab is multiple: 1. First it is a reminder of some central elementary notions of Bayesian
More informationLinear models and their mathematical foundations: Simple linear regression
Linear models and their mathematical foundations: Simple linear regression Steffen Unkel Department of Medical Statistics University Medical Center Göttingen, Germany Winter term 2018/19 1/21 Introduction
More informationApplied Quantitative Methods II
Applied Quantitative Methods II Lecture 4: OLS and Statistics revision Klára Kaĺıšková Klára Kaĺıšková AQM II - Lecture 4 VŠE, SS 2016/17 1 / 68 Outline 1 Econometric analysis Properties of an estimator
More informationLecture 3: Linear Models. Bruce Walsh lecture notes Uppsala EQG course version 28 Jan 2012
Lecture 3: Linear Models Bruce Walsh lecture notes Uppsala EQG course version 28 Jan 2012 1 Quick Review of the Major Points The general linear model can be written as y = X! + e y = vector of observed
More information2.1 Linear regression with matrices
21 Linear regression with matrices The values of the independent variables are united into the matrix X (design matrix), the values of the outcome and the coefficient are represented by the vectors Y and
More informationRecent 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 informationModelling the Electric Power Consumption in Germany
Modelling the Electric Power Consumption in Germany Cerasela Măgură Agricultural Food and Resource Economics (Master students) Rheinische Friedrich-Wilhelms-Universität Bonn cerasela.magura@gmail.com Codruța
More informationWISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, Academic Year Exam Version: A
WISE MA/PhD Programs Econometrics Instructor: Brett Graham Spring Semester, 2016-17 Academic Year Exam Version: A INSTRUCTIONS TO STUDENTS 1 The time allowed for this examination paper is 2 hours. 2 This
More informationECO220Y Simple Regression: Testing the Slope
ECO220Y Simple Regression: Testing the Slope Readings: Chapter 18 (Sections 18.3-18.5) Winter 2012 Lecture 19 (Winter 2012) Simple Regression Lecture 19 1 / 32 Simple Regression Model y i = β 0 + β 1 x
More informationCOMPREHENSIVE WRITTEN EXAMINATION, PAPER III FRIDAY AUGUST 26, 2005, 9:00 A.M. 1:00 P.M. STATISTICS 174 QUESTION
COMPREHENSIVE WRITTEN EXAMINATION, PAPER III FRIDAY AUGUST 26, 2005, 9:00 A.M. 1:00 P.M. STATISTICS 174 QUESTION Answer all parts. Closed book, calculators allowed. It is important to show all working,
More informationQuantitative Methods I: Regression diagnostics
Quantitative Methods I: Regression University College Dublin 10 December 2014 1 Assumptions and errors 2 3 4 Outline Assumptions and errors 1 Assumptions and errors 2 3 4 Assumptions: specification Linear
More informationØkonomisk Kandidateksamen 2004 (I) Econometrics 2. Rettevejledning
Økonomisk Kandidateksamen 2004 (I) Econometrics 2 Rettevejledning This is a closed-book exam (uden hjælpemidler). Answer all questions! The group of questions 1 to 4 have equal weight. Within each group,
More informationLinear Regression & Analysis of Variance
Linear Regression & Analysis of Variance with applications in R George Kapetanios Queen Mary, University of London Eurostat Short Course Outline Outline Outline Topics Topics In this course the following
More informationSection 2 NABE ASTEF 65
Section 2 NABE ASTEF 65 Econometric (Structural) Models 66 67 The Multiple Regression Model 68 69 Assumptions 70 Components of Model Endogenous variables -- Dependent variables, values of which are determined
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 informationSimple Linear Regression Analysis
LINEAR REGRESSION ANALYSIS MODULE II Lecture - 6 Simple Linear Regression Analysis Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Prediction of values of study
More informationCorrelation and Regression
Correlation and Regression October 25, 2017 STAT 151 Class 9 Slide 1 Outline of Topics 1 Associations 2 Scatter plot 3 Correlation 4 Regression 5 Testing and estimation 6 Goodness-of-fit STAT 151 Class
More informationEconometrics I KS. Module 1: Bivariate Linear Regression. Alexander Ahammer. This version: March 12, 2018
Econometrics I KS Module 1: Bivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: March 12, 2018 Alexander Ahammer (JKU) Module 1: Bivariate
More informationPractical Econometrics. for. Finance and Economics. (Econometrics 2)
Practical Econometrics for Finance and Economics (Econometrics 2) Seppo Pynnönen and Bernd Pape Department of Mathematics and Statistics, University of Vaasa 1. Introduction 1.1 Econometrics Econometrics
More informationFinal 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 informationViolation of OLS assumption- Multicollinearity
Violation of OLS assumption- Multicollinearity What, why and so what? Lars Forsberg Uppsala University, Department of Statistics October 17, 2014 Lars Forsberg (Uppsala University) 1110 - Multi - co -
More information