ADVANCED ECONOMETRICS I Theory (1/3) Instructor: Joaquim J. S. - - PowerPoint PPT Presentation

Theory (1/3)

ADVANCED ECONOMETRICS I

Instructor: Joaquim J. S. Ramalho E.mail: jjsro@iscte-iul.pt Personal Website: http://home.iscte-iul.pt/~jjsro Office: D5.10 Course Website: https://jjsramalho.wixsite.com/advecoi Fénix: https://fenix.iscte-iul.pt/disciplinas/03089

Joaquim J.S. Ramalho

This course provides an introduction to the modern econometric techniques used in the analysis of cross-sectional and panel data in the area of microeconometrics:

▪ The interaction between theory and empirical econometric analysis is emphasized ▪ Students will be trained in formulating and testing economic models using real data

Pre-requisites (recommended):

▪ Introductory Econometrics

Grading:

▪ Two problem sets (50%) + Final (open book) exam (50%)

– Weighted mean of at least 9,5/20 – Minimum grade at the exam of 7,5/20

▪ No re-sit examinations

Course Description

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• i. Introduction
• 1. Linear Regression Analysis
• 2. Nonlinear Regression Analysis
• 3. Discrete Choice Models
• 4. Models for Continuous Limited Dependent Variable Models

Contents - Theory

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Recommended:

▪ Cameron, A. and P.K. Trivedi (2005), Microeconometrics: Methods and Applications, Cambridge University Press

Others:

▪ Baltagi, B. (2013), Econometric Analysis of Panel Data, John Wiley and Sons (5th Edition) ▪ Davidson, R. and J.G. MacKinnon (2003), Econometric Theory and Methods, Oxford University Press ▪ Greene, W. (2011), Econometric Analysis, Pearson (7th Edition) ▪ Verbeek, M. (2017), A Guide to Modern Econometrics, Wiley (5th Edition) ▪ Wooldridge, J.M. (2010), Econometric Analysis of Cross Section and Panel Data, MIT Press (2nd Edition) ▪ Wooldridge, J.M. (2015), Introductory Econometrics: A Modern Approach, South Western (6th Edition).

Textbooks

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• 1. Determinants of Firm Debt
• 2. Estimating the Returns to Schooling
• 3. Explaining Individual Wages
• 4. Explaining Capital Structure
• 5. Modelling the Choice Between Two Brands
• 6. Health Care Expenses and Consultations
• 7. Explaining Firm’s Credit Ratings
• 8. Travel Mode Choice
• 9. Health Care Expenses and Consultations (revisited)
• 10. Determinants of Firm Debt (revisited)

Contents - Illustrations

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Recommended:

▪ Stata: http://www.stata.com ▪ R: https://cran.r-project.org

Others:

▪ Gauss: http://www.aptech.com/products/gauss-mathematical-and- statistical-system ▪ Matlab: https://www.mathworks.com/products/matlab

Software

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i.1. Econometric Methodology i.2. The Structure of Economic and Financial Data i.3. Dependent Variables and Econometric Models i.4. Types of Explanatory Variables

• i. Introduction

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Econometrics:

Definition:

▪ Application of statistical techniques to the analysis of economic, financial, social... data aiming at estimating relationships between a dependent variable and a set of explanatory variables

Ultimate purpose:

▪ Theory validation ▪ Prediction / Forecasting ▪ Policy recommendation

• i. Introduction

i.1. Econometric Methodology

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• i. Introduction

i.1. Econometric Methodology

Model specification Data collection Model estimation Model evaluation Suitable model Unsuitable model Result interpretation Theory validation Policy recommendation Prediction / Forecasting

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Cross-sectional data:

N cross-sectional units (individuals / firms / cities / ...) 1 time observation per unit

Time series:

1 unit T time observations per unit

Panel data

N units T time observations per unit

• i. Introduction

i.2. The Structure of Economic and Financial Data

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Main types of econometric models:

Regression Model:

▪ Aim: explaining 𝐹 𝑍|𝑌

Probabilistic model:

▪ Aim: explaining 𝑄𝑠 𝑍|𝑌 ▪ Usually incorporates also a regression model for 𝐹 𝑍|𝑌

Each type of econometric model has many variants

• i. Introduction

i.3. Dependent Variables and Econometric Models

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𝑍: dependent variable 𝑌: explanatory variables 𝐹 𝑍|𝑌 : expected value for 𝑍 given 𝑌 𝑄𝑠 𝑍|𝑌 : probability of 𝑍 being equal to a specific value given 𝑌

Joaquim J.S. Ramalho

• i. Introduction

i.3. Dependent Variables and Econometric Models 𝑍 Type of outcome Main model ] − ∞, +∞[ Unbounded data Linear [0, +∞[ Nonnegative data Exponential [0,1] Fractional data Fractional Logit,... {0,1} Binary choices Logit,... {0,1,2, … , 𝐾 − 1} Multinomial choices Multinomial logit,… {0,1,2, … , 𝐾 − 1} Ordered choices Ordered logit,… {0,1,2, … } Count data Poisson,…

The numeric characteristics of the dependent variable restricts the variants that may be applied in each case:

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Model Transformations and Adaptations:

Bounded continuous outcomes may often be transformed in such a way that they give rise to unbounded outcomes which may be modelled using a linear model Any econometric model may require adaptations:

▪ Data structure: cross-section, time series, panel ▪ Non-random samples: stratified, censored, truncated ▪ Measurement error ▪ Endogenous explanatory variables ▪ Corner solutions

• i. Introduction

i.3. Dependent Variables and Econometric Models

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Explanatory variables:

Their characteristics are not relevant for the choice of econometric model, but affect the interpretation of the results Quantitative variables (examples):

▪ Levels (Euro, kilograms, meters,…) ▪ Levels and squares ▪ Logs ▪ Growth rates ▪ Per capita values

Qualitative variables

▪ Binary (dummy) variables:

𝑌 = 0,1

▪ Interaction variables:

𝑌 = 𝐸𝑣𝑛𝑛𝑧 𝑤𝑏𝑠. ∗ 𝑅𝑣𝑏𝑜𝑢𝑗𝑢𝑏𝑢𝑗𝑤𝑓 𝑝𝑠 𝑒𝑣𝑛𝑛𝑧 𝑤𝑏𝑠.

• i. Introduction

i.4. Quantitative and Qualitative Explanatory Variables

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1.1. The Linear Regression Model with Cross-Sectional Data

1.1.1. Exogenous Explanatory Variables Specification Estimation Interpretation Inference Model Evaluation

RESET Test Tests for Heteroskedascity Chow Test

• 1. Linear Regression Analysis

1.1. The Linear Regression Model with Cross-Sectional Data

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Model Specification:

𝑍

𝑗 = 𝛾0 + 𝛾1𝑌𝑗1 + ⋯ + 𝛾𝑙𝑌𝑗𝑙 + 𝑣𝑗

𝑗 = 1, ⋯ , 𝑂

• r

𝑧 = 𝑌𝛾 + 𝑣

𝑧 = 𝑍

1

𝑍

2

⋮ 𝑍

𝑂

𝑌 = 1 𝑌11 𝑌12 ⋯ 𝑌1𝑙 1 𝑌21 𝑌22 ⋯ 𝑌2𝑙 ⋮ ⋮ ⋮ ⋱ ⋮ 1 𝑌𝑂1 𝑌𝑂2 ⋯ 𝑌𝑂𝑙 𝛾 = 𝛾0 𝛾1 ⋮ 𝛾𝑙 𝑣 = 𝑣1 𝑣2 ⋮ 𝑣𝑂

1.1.1. Exogenous Explanatory Variables Specification

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𝑣: error term 𝛾: parameters 𝑙: n. explanatory variables 𝑞: n. parameters (= 𝑙 + 1) 𝑂: n. observations

Joaquim J.S. Ramalho

Model estimation:

𝛾 is unknown and needs to be estimated Most popular estimation method - Ordinary Least Squares (OLS)

min ෍

𝑗=1 𝑂

ො 𝑣𝑗

2 ,

ො 𝑣𝑗 = 𝑍

𝑗 − ෠

𝑍

𝑗,

෠ 𝑍

𝑗 = መ

𝛾0 + መ 𝛾1𝑌𝑗1 + ⋯ + መ 𝛾𝑙𝑌𝑗𝑙

• r

min ො 𝑣′ ො 𝑣 , ො 𝑣 = 𝑧 − ො 𝑧, ො 𝑧 = 𝑌 መ 𝛾

1.1.1. Exogenous Explanatory Variables Estimation

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ො 𝑣: residuals ෠ 𝑍: fitted values of 𝑍 ෠ 𝛾: estimator for 𝛾

Joaquim J.S. Ramalho

OLS estimators:

𝜖ෝ 𝑣′ෝ 𝑣 𝜖෡ 𝛾 = −2𝑌′ 𝑧 − 𝑌′ መ

𝛾 = 0 (Note: implies 𝑌′ ො 𝑣 = 0) 𝑌′𝑧 = 𝑌′𝑌 መ 𝛾 መ 𝛾 = 𝑌′𝑌 −1𝑌′𝑧 1.1.1. Exogenous Explanatory Variables Estimation

Stata regress Y 𝑌1 ⋯ 𝑌𝑙

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Model assumptions:

• 1. Linearity in parameters
• 2. Random sampling
• 3. 𝑭 𝒗|𝒀

= 𝟏

• 4. No perfect colinearity
• 5. Homoskedasticity: 𝑊𝑏𝑠 𝑣|𝑌 = 𝜏2𝐽
• 6. Normality: 𝑣~𝒪 0, 𝜏2𝐽

1.1.1. Exogenous Explanatory Variables Estimation

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𝜏2: error variance

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Estimator properties:

Finite samples:

▪ Assumptions 1-4: Unbiasedness ▪ Assumptions 1-5: Unbiasedness and efficiency ▪ Assumptions 1-6: Unbiasedness, efficiency and normality

Asymptotically:

▪ Assumptions 1-4: Consistency ▪ Assumptions 1-5: Consistency, efficiency and normality

1.1.1. Exogenous Explanatory Variables Estimation

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Unbiasedness: 𝐹 ෠ 𝛾 = 𝛾 Efficiency: in the group of linear unbiased estimators, OLS displays the smallest variance [𝜏෡

𝛾 2 or 𝑊𝑏𝑠 ෠

𝛾 ] Normality: ෠ 𝛾 ∼ 𝒪 𝛾, 𝜏෡

𝛾 2

Consistency: lim

𝑂→∞ 𝐹 ෠

𝛾 = 𝛾

Joaquim J.S. Ramalho

Goodness-of-fit:

Sums of squares – measures of the sample variations in 𝑧, ො 𝑧 and ො 𝑣:

▪ Total Sum of Squares: 𝑇𝑇𝑈 = σ𝑗=1

𝑂

𝑍

𝑗 − ത

𝑍 2 ▪ Explained Sum of Squares: 𝑇𝑇𝐹 = σ𝑗=1

𝑂

෠ 𝑍

𝑗 − ത

෠ 𝑍

2

▪ Residual Sum of Squares: 𝑇𝑇𝑆 = σ𝑗=1

𝑂

ො 𝑣𝑗

2

The total variation in 𝑧 can be expressed as the sum of the explained and unexplained variation: 𝑇𝑇𝑈 = 𝑇𝑇𝐹 + 𝑇𝑇𝑆 Coefficient of determination (𝑆2):

▪ Proportion of the variation of the dependent variable that is explained by the explanatory variables: 𝑆2 = 𝑇𝑇𝐹 𝑇𝑇𝑈 = 𝑠

𝑧, ො 𝑧 2

1.1.1. Exogenous Explanatory Variables Estimation

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𝑠: coefficient of correlation

Joaquim J.S. Ramalho

Effects from unitary changes in a given explanatory variable: ∆𝑌

𝑘= 1 ⇒ ∆𝑍 = 𝛾𝑘

Needs adaptation for:

▪ Transformed dependent variables ▪ Transformed quantitative explanatory variables ▪ Qualitative explanatory variables

Aims:

▪ Most of the time: testing whether the effect is null or significantly different from zero → it is equivalent to test whether a parameter or a set of a parameters or a linear combination of parameters are significantly different from zero ▪ Most of the time: analyze the sign of the effect (null, positive, negative) ▪ Sometimes: calculate and analyze the magnitude of the effect

1.1.1. Exogenous Explanatory Variables Interpretation

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Partial effect for quantitative variables:

(Ceteris paribus) effect over the dependent variable originated by a unitary change in a given explanatory variable:

∆𝑌𝑗𝑘 = 1 unit ⇒ ∆𝑍

𝑗 = ?

It is approximated by differentiating the dependent variable in

• rder to the explanatory variable:

▪ Model in levels - 𝑍

𝑗 = ⋯ + 𝛾𝑘𝑌𝑗𝑘 + ⋯ + 𝑣𝑗: 𝜖𝑍𝑗 𝜖𝑌𝑗𝑘 = 𝛾𝑘, which implies that ∆𝑌𝑗𝑘 = 1 unit ⇒ ∆𝑍 𝑗 = 𝛾𝑘 units

▪ Quadratic model - 𝑍

𝑗 = ⋯ + 𝛾𝑘𝑌𝑗𝑘 + 𝛾𝑛𝑌𝑗𝑘 2 + ⋯ + 𝑣𝑗: 𝜖𝑍𝑗 𝜖𝑌𝑗𝑘 = 𝛾𝑘 + 2𝛾𝑛𝑌𝑗𝑘, which implies that

∆𝑌𝑗𝑘 = 1 unit ⇒ ∆𝑍

𝑗 = 𝛾𝑘 + 2𝛾𝑛𝑌𝑗𝑘 units

1.1.1. Exogenous Explanatory Variables Interpretation

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▪ Model in logs - ln 𝑍

𝑗 = ⋯ + 𝛾𝑘ln 𝑌𝑗𝑘 + ⋯ + 𝑣𝑗:

– Re-transformed model: 𝑍

𝑗 = 𝑓⋯+𝛾𝑘ln 𝑌𝑗𝑘 +⋯+𝑣𝑗

– Derivative:

𝜖𝑍𝑗 𝜖𝑌𝑗𝑘 = 𝛾𝑘 𝑌𝑗𝑘 𝑓⋯+𝛾𝑘ln 𝑌𝑗𝑘 +⋯+𝑣𝑗

≈ ∆𝑍𝑗

∆𝑌𝑗𝑘 = 𝛾𝑘 𝑌𝑗𝑘 𝑍 𝑗

⟺ ∆𝑍

𝑗

𝑍

𝑗

× 100 = 𝛾𝑘 ∆𝑌𝑗𝑘 𝑌𝑗𝑘 × 100 ⟺ %∆𝑍

𝑗 = 𝛾𝑘%∆𝑌𝑗𝑘

– Effect: %∆𝑌𝑗𝑘 = 1% ⇒ %∆𝑍

𝑗 = 𝛾𝑘%

» 𝛾𝑘 represents an elasticity

1.1.1. Exogenous Explanatory Variables Interpretation

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▪ Log-linear model - ln 𝑍

𝑗 = ⋯ + 𝛾𝑘𝑌𝑗𝑘 + ⋯ + 𝑣𝑗:

– Re-transformed model: 𝑍

𝑗 = 𝑓⋯+𝛾𝑘𝑌𝑗𝑘+⋯+𝑣𝑗

– Derivative: 𝜖𝑍

𝑗

𝜖𝑌𝑗𝑘 = 𝛾𝑘𝑓⋯+𝛾𝑘𝑌𝑗𝑘+⋯+𝑣𝑗 ≈ ∆𝑍

𝑗

∆𝑌𝑗𝑘 = 𝛾𝑘𝑍

𝑗

⟺ ∆𝑍

𝑗

𝑍

𝑗

× 100 = 100 × 𝛾𝑘∆𝑌𝑗𝑘 ⟺ %∆𝑍

𝑗 = 100𝛾𝑘∆𝑌𝑗𝑘

– Effect: ∆𝑌𝑗𝑘 = 1 unit ⇒ %∆𝑍

𝑗 = 100𝛾𝑘%

1.1.1. Exogenous Explanatory Variables Interpretation

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▪ Semi-logarithmic model - 𝑍

𝑗 = ⋯ + 𝛾𝑘ln 𝑌𝑗𝑘 + ⋯ + 𝑣𝑗:

– Derivative: 𝜖𝑍

𝑗

𝜖𝑌𝑗𝑘 = 𝛾𝑘 𝑌𝑗𝑘 ≈ ∆𝑍

𝑗 × 100 = 𝛾𝑘

∆𝑌𝑗𝑘 𝑌𝑗𝑘 × 100 ⟺ ∆𝑍

𝑗 = 𝛾𝑘

100 %∆𝑌𝑗𝑘 – Effect: %∆𝑌𝑗𝑘 = 1% ⇒ ∆𝑍

𝑗 = 𝛾𝑘

100 units

1.1.1. Exogenous Explanatory Variables Interpretation

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Partial effect for dummy variables:

(Ceteris paribus) difference on the value of the dependent variable between two groups The effect is given by the parameter associated to the dummy variable:

▪ Definition of a dummy variable: 𝐸𝑗 = ቊ1 if the individual belongs to group 𝐻𝐵 0 if the individual belongs to group 𝐻𝐶 ▪ Example:

– Base model: 𝑍

𝑗 = 𝛾0 + 𝛾1𝑌𝑗1 + ⋯ + 𝛾𝑙𝑌𝑗𝑙 + 𝛾𝑒𝐸𝑗 + 𝑣𝑗

– If 𝐸𝑗 = 1, then 𝑍

𝑗 = 𝛾0 + 𝛾1𝑌𝑗1 + ⋯ + 𝛾𝑙𝑌𝑗𝑙 + 𝛾𝑒 + 𝑣𝑗

– If 𝐸𝑗 = 0, then 𝑍

𝑗 = 𝛾0 + 𝛾1𝑌𝑗1 + ⋯ + 𝛾𝑙𝑌𝑗𝑙 + 𝑣𝑗

– Difference (effect of belonging to group 𝐻𝐵): 𝛾𝑒

1.1.1. Exogenous Explanatory Variables Interpretation

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Partial effect for interaction variables:

(Ceteris paribus) difference between two groups on the effect

• ver the value of the dependent variable of a unitary change in a

given explanatory variable The effect is given by the parameter associated to the interaction variable:

▪ Example:

– Base model: 𝑍

𝑗 = 𝛾0 + 𝛾1𝑌𝑗1 + ⋯ + 𝛾𝑙𝑌𝑗𝑙 + 𝛾𝑛𝑌𝑗𝑛 + 𝛾𝑒𝑛 𝐸𝑗 ∗ 𝑌𝑗𝑛 + 𝑣𝑗

– If 𝐸𝑗 = 1, then 𝑍

𝑗 = 𝛾0 + 𝛾1𝑌𝑗1 + ⋯ + 𝛾𝑙𝑌𝑗𝑙 + 𝛾𝑛 + 𝛾𝑒𝑛 𝑌𝑗𝑛 + 𝑣𝑗

– If 𝐸𝑗 = 0, then 𝑍

𝑗 = 𝛾0 + 𝛾1𝑌𝑗1 + ⋯ + 𝛾𝑙𝑌𝑗𝑙 + 𝛾𝑛𝑌𝑗𝑛 + 𝑣𝑗

– Partial effect of 𝑌𝑗𝑛 for those in group 𝐻𝐵: 𝛾𝑛 + 𝛾𝑒𝑛 – Partial effect of 𝑌𝑗𝑛 for those in group 𝐻𝐶: 𝛾𝑛 – Difference: 𝛾𝑒𝑛

1.1.1. Exogenous Explanatory Variables Interpretation

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Estimators for the variance of the parameter estimators:

Standard – assumes homoskedasticity: 𝑊𝑏𝑠 መ

𝛾 = ො 𝜏2 𝑌′𝑌 −1

Robust – allows for heteroskedasticity: 𝑊𝑏𝑠 መ 𝛾 = 𝑌′𝑌 −1𝑌′ ෡ Φ𝑌 𝑌′𝑌 −1, ෡ Φ = ො 𝑣1

2

⋯ ො 𝑣2

2

⋮ ⋱ ⋮ ⋯ ො 𝑣𝑂

2

Cluster-robust – specific for panel data Bootstrap – simulated variance 1.1.1. Exogenous Explanatory Variables Inference

Stata regress Y 𝑌1 … 𝑌𝑙 regress Y 𝑌1 … 𝑌𝑙, vce(robust) regress Y 𝑌1 … 𝑌𝑙, vce(cluster clustvar) regress Y 𝑌1 … 𝑌𝑙, vce(bootstrap)

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Hypothesis tests:

Require specification of the:

▪ Null and alternative hypothesis; typically: – 𝐼0: Partial effect = 0 – 𝐼1: Partial effect ≠ 0 (> 0 or < 0 also possible) ▪ Significance level (𝛽):

– Probability of rejecting the null hypothesis when it is true – Typically, 𝛽 = 0.01, 0.05 or 0.10

p-value:

▪ The p-value of the result of a test is the probability of obtaining a value at least as extreme when the null hypothesis is true; therefore:

– 𝑞 < 𝛽 ⟹ Reject 𝐼0 – 𝑞 > 𝛽 ⟹ Do not reject 𝐼0

1.1.1. Exogenous Explanatory Variables Inference

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Main tests:

▪ Test for the individual significance of a parameter: t test ▪ Test for the joint significance of a set of parameters: F test

t test:

𝐼0: 𝛾𝑘 = 0 𝐼1: 𝛾𝑘 ≠ 0 𝑢 = መ 𝛾𝑘 ො 𝜏෡

𝛾𝑘

~𝑢𝑂−𝑞

Τ 𝛽 2

𝑢 < 𝑢𝑂−𝑞

Τ 𝛽 2 ⟹ Do not reject 𝐼0

𝑢 > 𝑢𝑂−𝑞

Τ 𝛽 2 ⟹ Reject 𝐼0

1.1.1. Exogenous Explanatory Variables Inference

Stata regress Y 𝑌1 … 𝑌𝑙 regress Y 𝑌1 … 𝑌𝑙, vce(robust) regress Y 𝑌1 … 𝑌𝑙, vce(cluster clustvar) regress Y 𝑌1 … 𝑌𝑙, vce(bootstrap)

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F test:

Model: 𝑍 = 𝛾0 + 𝛾1𝑌1 + ⋯ +𝛾𝑕 𝑌𝑕 +𝛾𝑕+1 𝑌𝑕+1 + ⋯ + 𝛾𝑙𝑌𝑙 + 𝑤 Hypotheses: 𝐼0: 𝛾𝑕+1 = ⋯ = 𝛾𝑙 = 0 𝐼1: No 𝐼0 Test:

𝐺 =

𝑆2−𝑆∗

2

1−𝑆2 𝑂−𝑞 𝑟 ~𝐺 𝑂−𝑞 𝑟

→ valid only under homoskedasticity 𝐺 = 𝑂 መ 𝛾∗

′ Var መ

𝛾∗

−1 መ

𝛾∗~𝜓𝑟

2 → general formula

▪ Decision:

𝐺 < 𝐺

𝑂−𝑞 𝑟

⟹ Do not reject 𝐼0 𝐺 > 𝐺

𝑂−𝑞 𝑟

⟹ Reject 𝐼0

1.1.1. Exogenous Explanatory Variables Inference

Stata regress Y 𝑌1 ⋯ 𝑌𝑕 𝑌𝑕+1 ⋯ 𝑌𝑙,… test 𝑌𝑕+1 ⋯ 𝑌𝑙

32 Advanced Econometrics I 2020/2021

𝑆2: coefficient of determination of the full model 𝑆∗

2: coefficient of determination of the restricted model

𝑟: n. of parameters to be tested

Joaquim J.S. Ramalho

Specification tests:

Model functional form: RESET test Heteroskedasticity: Breusch-Pagan (BP) test Structural breaks: Chow tests 1.1.1. Exogenous Explanatory Variables Model Evaluation

33 Advanced Econometrics I 2020/2021

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RESET test:

Intuition:

▪ Any model of the type 𝐹 𝑍|𝑌 = 𝑇 𝑌𝛾 may be approximated by 𝐹 𝑍|𝑌 = 𝑀 𝑌𝛾 + σ𝑘=1

∞

γj 𝑌 መ 𝛾

𝑘+1

▪ Assume a linear form for 𝑀 ∙ and check if γj = 0

Implementation:

▪ Estimate the original model and get መ 𝛾 ▪ Generate the variables 𝑌 መ 𝛾

2, 𝑌 መ

𝛾

3, 𝑌 መ

𝛾

4, …

▪ Add the generated variables to the original model and estimate the following auxiliary model: 𝑍 = 𝛾0 + 𝛾1𝑌1 + ⋯ + 𝛾𝑙𝑌𝑙 + γ1 𝑌 መ 𝛾

2 + γ2 𝑌 መ

𝛾

3 + γ3 𝑌 መ

𝛾

4 + ⋯ + 𝑤

▪ Apply an F test for the significance of the added variables: 𝐼0: γ1 = γ2 = γ3 = ⋯ = 0 (suitable model functional form) 𝐼1: No 𝐼0 (unsuitable model functional form)

1.1.1. Exogenous Explanatory Variables Model Selection - RESET Test

Stata

(only test version based on three fitted powers)

regress Y 𝑌1 … 𝑌𝑙 estat ovtest

34 Advanced Econometrics I 2020/2021

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BP Test:

Intuition:

▪ Because the mean of the error term is zero, the variance of the error term is given by the sum of the squared error terms ▪ The squared residuals are an estimate of the squared error terms ▪ Check if the squared residuals and the explanatory variables are correlated

Implementation:

▪ Estimate the original model and generate the variable ො 𝑣2 ▪ Replace 𝑍 by ො 𝑣2 in the original model and estimate the following auxiliary model: ො 𝑣2 = γ0 + γ1𝑌1 + ⋯ + γ𝑙𝑌𝑙 + 𝑤 ▪ Apply an F test for the joint significance of the right-hand side variables

• f the previous auxiliary model:

𝐼0: γ1 = ⋯ = γ𝑙 = 0 (homoskedasticity) 𝐼1: Não 𝐼0 (heteroskedasticity)

1.1.1. Exogenous Explanatory Variables Model Selection - Tests for Heteroskedascity

Stata regress Y 𝑌1 … 𝑌𝑙 estat hettest, rhs fstat

35 Advanced Econometrics I 2020/2021

Joaquim J.S. Ramalho

Chow Test for Structural Breaks:

Context:

▪ Two groups of individuals / firms / ...: 𝐻𝐵, 𝐻𝐶 ▪ It is suspected that the behaviour of the two groups in which regards the dependent variable may have different determinants

Implementation:

▪ Generate the dummy variable 𝐸 = ቊ1 if the individual belongs to 𝐻𝐵 0 if the individual belongs to 𝐻𝐶 ▪ Estimate the original model ‘duplicated’: 𝑍 = 𝜄0 + 𝜄1𝑌1 + ⋯ + 𝜄𝑙𝑌𝑙 + γ0𝐸 + γ1𝐸𝑌1 + ⋯ + γ𝑙𝐸𝑌𝑙 + 𝑤 ▪ Apply an F test for the significance of the terms where 𝐸 is present: 𝐼0: γ0 = ⋯ = γ𝑙 = 0 (no structural break) 𝐼1: Não 𝐼0 (with a structural break)

1.1.1. Exogenous Explanatory Variables Model Selection - Chow Test

Stata regress Y 𝑌1…𝑌𝑙 𝐸 𝐸𝑌1…𝐸𝑌𝑙 test 𝐸 𝐸𝑌1 … 𝐸𝑌𝑙

36 Advanced Econometrics I 2020/2021

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Endogeneity

Definition and Consequences Motivation: Omitted variables, Measurement Errors, Simultaneous Equations Solutions: Instrumental Variables, Panel Data

Methods based on Instrumental Variables

Two-Stage Least Squares Generalized Method of Moments (GMM)

Specification Tests

Test for the Exogeneity of an Explanatory Variable Test for the Exogeneity of the Instrumental Variables Tests for Correlation between Instrumental Variables and Explanatory Variables

1.1.2. Endogenous Explanatory Variables

37 Advanced Econometrics I 2020/2021

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Definitions:

Exogenous explanatory variables: 𝐹 𝑣|𝑌 = 0 → essential assumption in any regression model Endogenous explanatory variables: 𝐹 𝑣|𝑌 ≠ 0

Consequences:

OLS estimators become unbiased and inconsistent

Motivation:

Omitted variables Covariate measurement error Simultaneity 1.1.2. Endogenous Explanatory Variables Endogeneity: Definition and Consequences

38 Advanced Econometrics I 2020/2021

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Omitted variables - example:

True model: 𝑍 = 𝛾0 + 𝛾1𝑌1 + 𝛾2𝑌2 + 𝑤, 𝐹 𝑤|𝑌1, 𝑌2 = 0 Estimated model: 𝑍 = 𝛾0 + 𝛾1𝑌1 + 𝑣 As 𝑣 = 𝛾2𝑌2 + 𝑤:

▪ If 𝑑𝑝𝑤 𝑌1, 𝑌2 = 0, then 𝐹 𝑣 𝑌1 = 0 → 𝑌1 is exogenous ▪ If 𝑑𝑝𝑤 𝑌1, 𝑌2 ≠ 0, then 𝐹 𝑣 𝑌1 ≠ 0 → 𝑌1 is endogenous

1.1.2. Endogenous Explanatory Variables Endogeneity: Motivation

39 Advanced Econometrics I 2020/2021

Joaquim J.S. Ramalho

Covariate measurement error - example:

True model: 𝑍 = 𝛾0 + 𝛾1𝑌1

∗ + 𝑤,

𝐹 𝑤|𝑌1

∗ = 0

𝑓: measurement error Instead of 𝑌1

∗, it is observed 𝑌1= 𝑌1 ∗ + 𝑓

Estimated model: 𝑍 = 𝛾0 + 𝛾1𝑌1 + 𝑣 As 𝑣 = 𝑤 − 𝛾1𝑓:

▪ If 𝑑𝑝𝑤 𝑌1, 𝑓 = 0, then 𝐹 𝑣 𝑌1 = 0 → 𝑌1 is exogenous ▪ If 𝑑𝑝𝑤 𝑌1, 𝑓 ≠ 0, then 𝐹 𝑣 𝑌1 ≠ 0 → 𝑌1 is endogenous → most common case because the measurement error is on 𝑌1

1.1.2. Endogenous Explanatory Variables Endogeneity: Motivation

40 Advanced Econometrics I 2020/2021

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Measurement error on 𝑍 has less serious consequences:

True model: 𝑍∗ = 𝛾0 + 𝛾1𝑌1 + 𝑤 , 𝐹 𝑤|𝑌1 = 0 Instead of 𝑍∗, it is observed 𝑍 = 𝑍∗ + 𝑓 Estimated model: 𝑍 = 𝛾0 + 𝛾1𝑌1 + 𝑣 As 𝑣 = 𝑤 + 𝑓:

▪ In general, 𝑑𝑝𝑤 𝑌1, 𝑓 = 0 and 𝐹 𝑣 𝑌1 = 0, since the measurement error is on 𝑍 and not in 𝑌1 ▪ Hence, usually there are no endogeneity problems ▪ However, estimation is less precise, since the error term has now two components

1.1.2. Endogenous Explanatory Variables Endogeneity: Motivation

41 Advanced Econometrics I 2020/2021

Joaquim J.S. Ramalho

Simultaneity - example:

True model: ቊSupply: 𝑅 = 𝛾0 + 𝛾1𝑄 + 𝑣 Demand: 𝑅 = 𝛽0 + 𝛽1𝑄 + 𝑤 Estimated model: ቊSupply: 𝑅 = 𝛾0 + 𝛾1𝑄 + 𝑣 Demand: 𝑅 = 𝛽0 + 𝛽1𝑄 + 𝑤 As:

൞𝑄 = 𝛽0 − 𝛾0 𝛾1 − 𝛽1 + 𝑤 − 𝑣 𝛾1 − 𝛽1 𝑅 = ⋯

then 𝑄 is function of 𝑤 and 𝑣; hence:

▪ 𝐹 𝑣 𝑄 ≠ 0 in the supply equation → 𝑄 is endogenous ▪ 𝐹 𝑤 𝑄 ≠ 0 in the demand equation → 𝑄 is endogenous

1.1.2. Endogenous Explanatory Variables Endogeneity: Motivation

42 Advanced Econometrics I 2020/2021

Joaquim J.S. Ramalho

What to do in case of endogeneity:

Universal solution – methods based on ‘instrumental variables’:

▪ Two-Stage Least Squares ▪ Generalized Method of Moments (GMM)

When data is in panel form and the endogeneity problem is caused by omitted time-constant variables:

▪ Methods based on the removal of the ‘fixed effects’

1.1.2. Endogenous Explanatory Variables Endogeneity: Instrumental Variables

43 Advanced Econometrics I 2020/2021

Joaquim J.S. Ramalho

Instrumental variables:

Context:

▪ 𝑍 = 𝛾0 + 𝛾1𝑌1 + 𝛾2𝑌2 + ⋯ + 𝛾𝑙𝑌𝑙 + 𝑣 (structural model) ▪ 𝐹 𝑣 𝑌1 ≠ 0 → 𝑌1 is endogenous

Definitions of instrumental variable (𝐽𝑊

𝐵, … , 𝐽𝑊 𝑁):

▪ 𝐹 𝑣 𝐽𝑊

𝐵 = ⋯ = 𝐹 𝑣 𝐽𝑊 𝑁 = 0

▪ 𝑑𝑝𝑤 𝐽𝑊

𝐵, 𝑌1 ≠ 0, …,𝑑𝑝𝑤 𝐽𝑊 𝑁, 𝑌1 ≠ 0

Exogenous variables: 𝑎 = 1 𝑌2 ⋯ 𝑌𝑙 𝐽𝑊

𝐵⋯ 𝐽𝑊 𝑁

The number of instrumental variables must be equal or larger than the number of endogenous explanatory variables 1.1.2. Endogenous Explanatory Variables Endogeneity: Instrumental Variables

44 Advanced Econometrics I 2020/2021

Joaquim J.S. Ramalho

Implementation:

• 1. Estimate the reduced form of the model by OLS:

ด 𝑌1

• End. Expl. Var.

= 𝜌0 + 𝜌2𝑌2 + ⋯ + 𝜌𝑙𝑌𝑙

• Ex. Expl. Var.

+ 𝜌𝐵𝐽𝑊

𝐵 + ⋯ + 𝜌𝑁𝐽𝑊 𝑁

Instrumental Variables + 𝑥

and get ෠

𝑌1 = ො 𝜌0 + ො 𝜌2𝑌2 + ⋯ + ො 𝜌𝑙𝑌𝑙 + ො 𝜌𝐵𝐽𝑊

𝐵 + ⋯ + ො

𝜌𝑁𝐽𝑊

𝑁

2. Estimate the structural model, with 𝑌1 replaced by ෠ 𝑌1, by OLS:

𝑍 = 𝛾0 + 𝛾1 ෠ 𝑌1 + 𝛾2𝑌2 + ⋯ + 𝛾𝑙𝑌𝑙 + 𝑣

1.1.2. Endogenous Explanatory Variables

Methods based on Instrumental Variables: Two-Stage Least Squares (2SLS)

45 Advanced Econometrics I 2020/2021

Stata

(by default, variances are estimated in a standard way; to use another estimator, use the option vce(robust) or similar)

ivregress 2sls Y (𝑌1= 𝐽𝑊

𝐵… 𝐽𝑊 𝑁) 𝑌2 … 𝑌𝑙

Joaquim J.S. Ramalho

Very general estimation method:

▪ Applies to a large variety of cases ▪ Includes as particular cases OLS, maximum likelihood, etc.

Formulation:

▪ Moment conditions: 𝐹 𝑕 𝑍, 𝑌, 𝐽𝑊; 𝛾 = 0 ▪ 𝑛 moment conditions ▪ 𝑞 parameters: 𝛾0, 𝛾1, ⋯ , 𝛾𝑙 ▪ Optimization:

– 𝑛 = 𝑞 → just-identified model: 𝑕 𝑍, 𝑌, 𝐽𝑊; መ 𝛾 = 0 – 𝑛 > 𝑞 → overidentified model: min 𝐾 = 𝑕 𝑍, 𝑌, 𝐽𝑊; 𝛾 ′W𝑕 𝑍, 𝑌, 𝐽𝑊; 𝛾 where W is a weighting matrix; the first-order conditions are given by: 𝜖𝑕 𝑍, 𝑌, 𝐽𝑊; መ 𝛾

′

𝜖𝛾 W𝑕 𝑍, 𝑌, 𝐽𝑊; መ 𝛾 = 0

1.1.2. Endogenous Explanatory Variables Methods based on Instrumental Variables: GMM

46 Advanced Econometrics I 2020/2021

Joaquim J.S. Ramalho

Choosing 𝑋 in overidentified models:

To get efficient estimators, 𝑋 has to be defined as the inverse

• f the covariance matrix of the moment conditions:

𝑋 = Ω−1, where: Ω = 𝑊𝑏𝑠 𝑕 𝑍, 𝑌, 𝐽𝑊; 𝛾 = 𝐹 𝑕 𝑍, 𝑌, 𝐽𝑊; 𝛾 𝑕 𝑍, 𝑌, 𝐽𝑊; 𝛾 ′ 1.1.2. Endogenous Explanatory Variables Methods based on Instrumental Variables: GMM

Stata

(by default, variances are estimated in a standard way; to use another estimator, use the option vce(robust) or similar)

ivregress gmm Y (𝑌1= 𝐽𝑊

𝐵… 𝐽𝑊 𝑁) 𝑌2 … 𝑌𝑙 47 Advanced Econometrics I 2020/2021

Joaquim J.S. Ramalho

Particular case – OLS:

Assumption: 𝐹 𝑣|𝑌 = 0 ⟹ 𝐹 𝑌′𝑣 = 0 𝑕 𝑍, 𝑌, 𝐽𝑊; 𝛾 = 𝑌′𝑣 = 𝑌′ 𝑧 − 𝑌𝛾 𝑛 = 𝑞 ⟹ 𝑌′ 𝑧 − 𝑌 መ 𝛾 = 0

𝑌′ 𝑧 − 𝑌 መ 𝛾 = 0 𝑌′𝑧 − 𝑌′𝑌 መ 𝛾 = 0 𝑌′𝑌 መ 𝛾 = 𝑌′𝑧 መ 𝛾 = 𝑌′𝑌 −1𝑌′𝑧

1.1.2. Endogenous Explanatory Variables Methods based on Instrumental Variables: GMM

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Instrumental Variables (𝑛 = 𝑞):

Assumption: 𝐹 𝑣|𝑎 = 0 ⟹ 𝐹 𝑎′𝑣 = 0 Moment conditions: 𝑕 𝑍, 𝑌, 𝐽𝑊; 𝛾 = 𝑎′𝑣 = 𝑎′ 𝑧 − 𝑌𝛾 If 𝑛 = 𝑞 ⟹ 𝑎′ 𝑧 − 𝑌 መ 𝛾 = 0 መ 𝛾 = 𝑎′𝑌 −1𝑎′𝑧 In this case, GMM is identical to 2SLS estimation 1.1.2. Endogenous Explanatory Variables Methods based on Instrumental Variables: GMM

49 Advanced Econometrics I 2020/2021

Joaquim J.S. Ramalho

Instrumental Variables (𝑛 > 𝑞):

Optimization problem: min 𝐾 = 𝑧 − 𝑌𝛾 ′𝑎Ω−1𝑎′ 𝑧 − 𝑌𝛾 First-order conditions: −𝑌′𝑎෩ Ω−1𝑎′ 𝑧 − 𝑌 መ 𝛾 = 0 𝑌′𝑎෩ Ω−1𝑎′𝑧 = 𝑌′𝑎෡ Ω−1𝑎′𝑌 መ 𝛾 መ 𝛾 = 𝑌′𝑎෩ Ω−1𝑎′𝑌

−1𝑌′𝑎෡

Ω−1𝑎′𝑧 where ෩ Ω is a preliminar estimate of Ω = 𝐹 𝑎′𝑣𝑣′𝑎 , which requires a preliminar estimate for 𝛾; typically, ෨ 𝛾 is obtained from min 𝐾 = 𝑧 − 𝑌𝛾 ′𝑎𝑎′ 𝑧 − 𝑌𝛾 (assumes 𝑋 = 𝐽) 1.1.2. Endogenous Explanatory Variables Methods based on Instrumental Variables: GMM

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Tests relevant for 2SLS / GMM:

Tests for the exogeneity of an explanatory variable

▪ If the explanatory variable is exogenous, it is better to use OLS in order to get efficient estimators ▪ Methods based on IV’s should be used only if really necessary, since there may be a substantial loss in precision

Tests for the exogeneity of the instrumental variables

▪ To act as an IV, a variable has to be exogenous ⟶ when based on “IV’s” that actually are endogenous, 2SLS and GMM are inconsistent

Tests for correlation between instrumental variables and explanatory variables

▪ To act as an IV, a variable has to be correlated with the endogenous explanatory variable ⟶ when based on “IV’s” uncorrelated with the endogenous regressors, 2SLS and GMM are inconsistent ▪ If the IV’s are only weakly correlated with the endogenous regressors, then the model with be poorly identified ⟶ in such a case, 2SLS and GMM may display a huge variability

1.1.2. Endogenous Explanatory Variables Specification Tests

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2SLS - Wu-Hausman test:

• 1. Estimate the reduced model by OLS:

𝑌1 = 𝜌0 + 𝜌2𝑌2 + ⋯ + 𝜌𝑙𝑌𝑙 + 𝜌𝐵𝐽𝑊

𝐵 + ⋯ + 𝜌𝑁𝐽𝑊 𝑁 + 𝑥

• 2. Calculate the residuals ෝ

𝑥

• 3. Add ෝ

𝑥 to the strutural model and re-estimate it, by OLS: 𝑍 = 𝛾0 + 𝛾1𝑌1 + 𝛾2𝑌2 + ⋯ + 𝛾𝑙𝑌𝑙 + 𝜀ෝ 𝑥 + 𝑣

• 4. t test:

𝐼0: 𝜀 = 0 (𝑌1 is exogenous) 𝐼1: 𝜀 ≠ 0 (𝑌1 is endogenous)

GMM - Eichenbaum, Hansen e Singleton’s (1988) C test

Based on the difference of two J statistics (see the next page) 1.1.2. Endogenous Explanatory Variables Tests for the Exogeneity of an Explanatory Variable

Stata ivregress 2sls Y (𝑌1= 𝐽𝑊

𝐵… 𝐽𝑊 𝑁) 𝑌2 … 𝑌𝑙

estat endogenous Stata ivregress gmm Y (𝑌1= 𝐽𝑊

𝐵… 𝐽𝑊 𝑁) 𝑌2 … 𝑌𝑙

estat endogenous

52 Advanced Econometrics I 2020/2021

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These tests can be applied only when the model is

• veridentified

If the model is just-identified, then it is only possible to justify the exogeneity of the IV’s using theoretical arguments Hansen’s J test (also called test for overidentifying restrictions; this is an extension of the Sargan test, very common in the 2SLS framework under homoskedasticity):

• 1. Estimate the model by GMM
• 2. Test

𝐼0: 𝐹 𝑣|𝑎 = 0 (IV’s are exogenous) 𝐼1: 𝐹 𝑣|𝑎 ≠ 0 (IV’s are not exogenous) using the J statistic: መ 𝐾 = 𝑕 𝑍, 𝑌, 𝐽𝑊; መ 𝛾

′ ෡

W𝑕 𝑍, 𝑌, 𝐽𝑊; መ 𝛾 ~𝜓𝑟

2

1.1.2. Endogenous Explanatory Variables Tests for the Exogeneity of the Instrumental Variables

Stata

(applies also to the 2SLS estimator, with the obvious adaptation)

ivregress gmm Y (𝑌1= 𝐽𝑊

𝐵… 𝐽𝑊 𝑁) 𝑌2 … 𝑌𝑙

estat overid q: number of overidentifying restrictions

53 Advanced Econometrics I 2020/2021

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Alternatives:

F tests for the significance of the IV’s in the reduced form model Criteria / tests for ‘weak instruments’

F tests:

• 1. Estimate the reduced form model:

𝑌1 = 𝜌0 + 𝜌2𝑌2 + ⋯ + 𝜌𝑙𝑌𝑙 + 𝜌𝐵𝐽𝑊

𝐵 + ⋯ + 𝜌𝑁𝐽𝑊 𝑁 + 𝑥

• 2. Test the hypothesis:

𝐼0: 𝜌𝐵 = ⋯ = 𝜌𝑁 = 0 (𝐽𝑊’s and 𝑌1 are not correlated)

1.1.2. Endogenous Explanatory Variables

Tests for Correlation between Instrumental Variables and Explanatory Variables

54 Advanced Econometrics I 2020/2021

Stata

(applies also to the GMM estimator, with the obvious adaptation)

ivregress 2sls Y (𝑌1= 𝐽𝑊

𝐵… 𝐽𝑊 𝑁) 𝑌2 … 𝑌𝑙

estat firststage

Joaquim J.S. Ramalho

Tests for ‘weak instruments’:

Even when the previous tests reveal that 𝐽𝑊’s and 𝑌1 are correlated, that correlation may be so weak that 2SLS / GMM estimators are very little precise Criterium: 𝐺 < 10 → Suggest a high probability of having ‘weak instruments’ Tests: Cragg and Donald (2005), Kleibergen and Paap (2006) 1.1.2. Endogenous Explanatory Variables

Tests for Correlation between Instrumental Variables and Explanatory Variables

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1.2.1. Static Models 1.2.2. Dynamic Models

1.2. The Linear Regression Model with Panel Data

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Panel data:

𝑂 cross-sectional units: 𝑗 = 1, … , 𝑂 𝑈 time observations per unit: 𝑢 = 1, … , 𝑈

Econometric analysis more complex:

Cross-sectional data: different units ⇒ independent

• bservations

Panel data: same units ⇒ dependent observations over time 1.2.1. Static Models Definitions

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Advantages:

Make possible the analysis of the dynamics of individual behaviours Generate more efficient estimators, since samples are larger Allow for endogenous explanatory variables in some special cases It is simple to get instrumental variables

Limitations:

Prediction and calculation of partial effects not possible in some models 1.2.1. Static Models Definitions

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Temporal dimension:

Short panels:

▪ Sample comprises many individuals (𝑂 ⟶ ∞), but there is a reduced number of time observations (small 𝑈) ▪ The temporal correlation of the observations for each individual is an issue, but across individuals it is assumed independence

Long panels:

▪ Large number of time observations for each individual (𝑈 ⟶ ∞) ▪ Need to take into account time series issues (stationarity, cointegration, etc.)

1.2.1. Static Models Definitions

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Cross-sectional composition:

Balanced panels:

▪ Every individual is observed in all time periods (𝑈

𝑗 = 𝑈, ∀𝑗)

Unbalanced panels:

▪ Some individuals are not observed in some time periods (𝑈

𝑗 ≠ 𝑈)

▪ Motivation: some individuals refuse to continue providing information after some time periods ⟶ ‘attrition’ problem ▪ Most estimators work with unbalanced panels, provided that one may assume that there is no endogenous selection (the data are missing at random)

1.2.1. Static Models Definitions

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Variability over time and across individuals:

With panel data, the variance of 𝑍

𝑗𝑢 can be decomposed as

follows:

෍

𝑗=1 𝑂

෍

𝑢=1 𝑈

𝑍

𝑗𝑢 − ത

𝑍 2 = ෍

𝑗=1 𝑂

෍

𝑢=1 𝑈

𝑍

𝑗𝑢 − ത

𝑍

𝑗 + ത

𝑍

𝑗 − ത

𝑍 2 = ෍

𝑗=1 𝑂

෍

𝑢=1 𝑈

𝑍

𝑗𝑢 − ത

𝑍

𝑗 2 + ෍ 𝑗=1 𝑂

ത 𝑍

𝑗 − ത

𝑍 2

To obtain the the total, ‘within’ and ‘between’ variance, divide by, respectively, 𝑂𝑈 − 1, 𝑂 𝑈 − 1 and 𝑂 − 1 1.2.1. Static Models Definitions

Variability within the groups Variability between the groups

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Base Model – Model with individual effects:

𝑍

𝑗𝑢 = 𝛽𝑗 + 𝑦𝑗𝑢 ′ 𝛾 + 𝑣𝑗𝑢

𝑗 = 1, … , 𝑂; 𝑢 = 1, … , 𝑈

𝛽𝑗: individual effects, time-constant 𝑦𝑗𝑢 - explanatory variables, including:

▪ 𝑦𝑗𝑢: changes across individuals and over time ▪ 𝑦𝑗: time-constant ▪ 𝑒𝑢: temporal dummy variable ▪ 𝑢: trend (one may also have 𝑢2, 𝑢3, …) ▪ 𝑒𝑢.𝑦𝑗𝑢: interaction term

𝑣𝑗𝑢: idiosyncratic error term – changes randomly across individuals and

• ver time

1.2.1. Static Models Models

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Other models:

‘Pooled’ or ‘Population-averaged’ model: 𝑍

𝑗𝑢 = 𝛽 + 𝑦𝑗𝑢 ′ 𝛾 + 𝑣𝑗𝑢

▪ Direct extension of cross-sectional models ▪ Popular in the area of Statistics but not in Economics, where individual heterogeneity is always an issue

Random Coefficients Model: 𝑍

𝑗𝑢 = 𝛽𝑗 + 𝑦𝑗𝑢 ′ 𝛾𝑗 + 𝑣𝑗𝑢

▪ More complex than the base model, allowing for a different coefficient also for the explanatory variables ▪ Computationally-intensive model, hard to estimate

1.2.1. Static Models Models

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The individual effects model may be re-written as:

𝑍

𝑗𝑢 = 𝑦𝑗𝑢 ′ 𝛾 + 𝛽𝑗 + 𝑣𝑗𝑢

The error term has now two componentes, 𝛽𝑗 and 𝑣𝑗𝑢 The individual effects 𝛽𝑗 may be correlated, or not, with the explanatory variables

Fixed effects:

𝛽𝑗 and 𝑦𝑗𝑢 are correlated → 𝑦𝑗𝑢 is endogenous Direct estimation of the model is not possible

Random effects:

𝛽𝑗 and 𝑦𝑗𝑢 are not correlated → 𝑦𝑗𝑢 is exogenous Direct estimation of the model is possible 1.2.1. Static Models Fixed Effects versus Random Effects

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Most panel data estimators:

Are based on transformed versions of the base model Differ of the type of exogeneity required for the explanatory variables:

▪ Contemporaneous: 𝐹 𝑦𝑗𝑢𝑣𝑗𝑢 = 0 ▪ Weak (pre-determined variables): 𝐹 𝑦𝑗𝑢𝑣𝑗,𝑢+𝑘 = 0, 𝑘 ≥ 0 ▪ Strict: 𝐹 𝑦𝑗𝑢𝑣𝑗𝑡 = 0, ∀𝑡, 𝑢

1.2.1. Static Models Alternative Estimators

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List of estimators:

Estimators for the random effects model:

▪ Pooled OLS ▪ Between estimator ▪ Random effects estimator

Estimators for the fixed effects model:

▪ Fixed effects or Within estimator ▪ Least squares dummy variables (LSDV) estimator ▪ First-diferences estimator

Estimators based on instrumental variables:

▪ General estimators ▪ Hausman-Taylor estimator

1.2.1. Static Models Alternative Estimators

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Pooled OLS estimator:

Model: 𝑍

𝑗𝑢 = 𝛽 + 𝑦𝑗𝑢 ′ 𝛾 + 𝛽𝑗 − 𝛽 + 𝑣𝑗𝑢 𝑤𝑗𝑢

Assumption: 𝐹 𝑦𝑗𝑢 𝛽𝑗 + 𝑣𝑗𝑢 = 0

▪ Requires random effects: 𝛽𝑗 e 𝑦𝑗𝑢 must be uncorrelated ▪ Requires contemporaneous exogeneity between 𝑣𝑗𝑢 and 𝑦𝑗𝑢

Estimation:

▪ OLS with a cluster-type estimator for the variance

1.2.1. Static Models Estimators for the Random Effects Model

Stata regress Y 𝑌1 … 𝑌𝑙, vce(cluster clustvar)

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Between estimator:

Model: ത 𝑍

𝑗 = 𝛽 + ҧ

𝑦𝑗

′𝛾 + 𝛽𝑗 − 𝛽 + ത

𝑣𝑗

ത 𝑤𝑗

▪ ത 𝑍

𝑗 = 1 𝑈𝑗 σ𝑢=1 𝑈𝑗 𝑍 𝑗𝑢, etc.

Assumption: 𝐹 ҧ 𝑦𝑗 𝛽𝑗 + ത 𝑣𝑗 = 0

▪ Requires random effects: 𝛽𝑗 e 𝑦𝑗𝑢 must be uncorrelated ▪ Requires strict exogeneity between 𝑣𝑗𝑢 and 𝑦𝑗𝑢

Estimation:

▪ OLS

1.2.1. Static Models Estimators for the Random Effects Model

Stata xtreg Y 𝑌1 … 𝑌𝑙, be

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Random effects estimator:

Model: 𝑍

𝑗𝑢 = 𝛽 + 𝑦𝑗𝑢 ′ 𝛾 + 𝛽𝑗 − 𝛽 + 𝑣𝑗𝑢 𝑤𝑗𝑢

New assumptions:

▪ 𝑊𝑏𝑠 𝛽𝑗 = 𝜏𝛽

2

▪ 𝑊𝑏𝑠 𝑣𝑗𝑢 = 𝜏𝑣

2

Under the new assumptions:

𝑑𝑝𝑠 𝑤𝑗𝑢, 𝑤𝑗𝑡 = 𝜏𝛽

2/ 𝜏𝛽 2 + 𝜏𝑣 2

▪ Taking into account this result, more efficient estimators may be

• btained

▪ All the previous estimators did not fully exploit the panel nature of the data in the estimation process

1.2.1. Static Models Estimators for the Random Effects Model

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Estimation:

▪ Generalized least squares (GLS) ▪ It is equivalent to apply OLS to the following equation:

𝑍

𝑗𝑢 − ෠

𝜄𝑗 ത 𝑍

𝑗 = 1 − ෠

𝜄𝑗 𝛽 + 𝑦𝑗𝑢 − ෠ 𝜄𝑗 ҧ 𝑦𝑗

′𝛾 + 𝑤𝑗𝑢 – ෠ 𝜄𝑗 = 1 − ො 𝜏𝑣

2/ 𝑈𝑗 ො

𝜏𝛽

2 + ො

𝜏𝑣

2

– 𝑤𝑗𝑢 = 1 − ෠ 𝜄𝑗 𝛽𝑗 + 𝑣𝑗𝑢 − ෠ 𝜄𝑗 ത 𝑣𝑗

Assumption: 𝐹 ҧ 𝑦𝑗𝑤𝑗𝑢 = 0

▪ Requires random effects: 𝛽𝑗 e 𝑦𝑗𝑢 must be uncorrelated ▪ Requires strict exogeneity between 𝑣𝑗𝑢 and 𝑦𝑗𝑢

1.2.1. Static Models Estimators for the Random Effects Model

Stata xtreg Y 𝑌1 … 𝑌𝑙, re vce(cluster clustvar)

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Fixed effects / within estimator:

Model: 𝑍

𝑗𝑢−ത

𝑍

𝑗= 𝑦𝑗𝑢 − ҧ

𝑦𝑗 ′𝛾 + 𝑣𝑗𝑢 − ത 𝑣𝑗

▪ Corresponds to the subtraction of the model defining the between estimator from the base individual effects model

Assumption: 𝐹 𝑦𝑗𝑢 − ҧ 𝑦𝑗 𝑣𝑗𝑢 −ത 𝑣𝑗 = 0

▪ Requires strict exogeneity between 𝑣𝑗𝑢 and 𝑦𝑗𝑢

Estimation:

▪ OLS with a cluster-type estimator for the variance

1.2.1. Static Models Estimators for the Fixed Effects Model

Stata xtreg Y 𝑌1 … 𝑌𝑙, fe vce(cluster clustvar)

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Limitations:

▪ It is not possible to include in the model:

– Time-constant explanatory variables – (If the model includes time dummies) Explanatory variables with identical changes over time for all individuals (e.g. age)

▪ Prediction not possible ▪ Partial effects conditional not only on 𝑦𝑗𝑢 but also on 𝛽𝑗; how to calculate them?

Main advantage:

▪ Allow for (time-constant) unobserved individual heterogeneity that may be correlated with the explanatory variables, not requiring the use

• f instrumental variables

1.2.1. Static Models Estimators for the Fixed Effects Model

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LSDV estimator:

Model: 𝑍

𝑗𝑢 = ෍ 𝑘=1 𝑂

𝛽𝑘𝑒𝑗𝑘 + 𝑦𝑗𝑢

′ 𝛾 + 𝑣𝑗𝑢

where 𝑒𝑗𝑘 = 1 if 𝑗 = 𝑘 and 0 if 𝑗 ≠ 𝑘

Assumptions and estimates of 𝛾 identical to those of the fixed effects estimator Estimates of 𝛽𝑘:

▪ Given by ො 𝛽𝑗 = ത 𝑍

𝑗 − ҧ

𝑦𝑗′ መ 𝛾 ▪ Consistent only in case of a long panel, in which case it is also possible to make prediction and calculate partial effects conditional on both 𝑦𝑗𝑢 and 𝛽𝑗

1.2.1. Static Models Estimators for the Fixed Effects Model

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Stata areg Y 𝑌1 … 𝑌𝑙, absorb(clustvar) vce(cluster clustvar)

• r (time-constant variables need to be manually dropped in the second alternative)

regress Y 𝑌1 … 𝑌𝑙 i. 𝑑𝑚𝑣𝑡𝑢𝑤𝑏𝑠, vce(cluster clustvar)

Joaquim J.S. Ramalho

First-differences estimator:

Model: 𝑍

𝑗𝑢 − 𝑍 𝑗,𝑢−1 = 𝑦𝑗𝑢 − 𝑦𝑗,𝑢−1 ′𝛾 + 𝑣𝑗𝑢 − 𝑣𝑗,𝑢−1 ⟺

Δ𝑍

𝑗𝑢 = Δ𝑦𝑗𝑢 ′ 𝛾 + Δ𝑣𝑗𝑢

▪ Corresponds to the subtraction of the first-diferences equation from the base individual effects model

Assumption: 𝐹 Δ𝑦𝑗𝑢Δ𝑣𝑗𝑢 = 0

▪ Requires 𝐹 𝑦𝑗𝑢𝑣𝑗𝑢 = 𝐹 𝑦𝑗𝑢𝑣𝑗,𝑢−1 = 𝐹 𝑦𝑗𝑢𝑣𝑗,𝑢+1 = 0

Estimation:

▪ OLS

Comparison with the fixed-effects estimator:

▪ Identical when 𝑈 = 2 ▪ Does not require strict exogeneity

1.2.1. Static Models Estimators for the Fixed Effects Model

Stata regress D.Y D.𝑌1 … D.𝑌𝑙, vce(cluster clustvar) nocons

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Endogenous explanatory variables - 𝐹 𝑦𝑗𝑢𝑣𝑗𝑢 ≠ 0:

Possível IV’s for 𝑦𝑗𝑢:

▪ External instruments, as in the cross-sectional case ▪ Internal instruments (same explanatory variable but relative to other time periods)

Example of internal instruments:

▪ If 𝑦𝑗𝑢 is weakly exogenous (apart from the current period), then:

– All past values (lags) of 𝑦𝑗𝑢 may be used as IV’s – Possible IV’s: 𝑦𝑗,𝑢−1 or 𝑦𝑗,𝑢−1, 𝑦𝑗,𝑢−2 or 𝑦𝑗,𝑢−1, … , 𝑦𝑗,𝑢−5 , etc.

▪ If 𝑦𝑗𝑢 is strictly exogenous (apart from the current period), then:

– All past (lags) and future (leads) values of 𝑦𝑗𝑢 may be used as IV’s – Possible IV’s : 𝑦𝑗,𝑢−1 and/or 𝑦𝑗,𝑢+1, etc.

1.2.1. Static Models Instrumental Variables Estimators

Stata

(external instruments – 2SLS)

xtivreg Y (𝑌1= 𝐽𝑊

𝐵… 𝐽𝑊 𝑁) 𝑌2 … 𝑌𝑙, options (options: re, fe, be, fd)

Stata

(internal instruments – 2SLS)

xtivreg Y (𝑌1= 𝑀. 𝑌1 𝑀2. 𝑌1…) 𝑌2 … 𝑌𝑙, options

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Hausman-Taylor estimator:

Useful when interest lies on time-invariant explanatory variables which are correlated with 𝛽𝑗, since:

– Fixed effects estimators: drop those variables from the model – Random effects estimators: inconsistent

Intermediate case between the fixed and the random effects cases:

▪ Explanatory variables correlated with 𝛽𝑗 are dealt with under the assumption of fixed effects ▪ Explanatory variables uncorrelated with 𝛽𝑗 are dealt with under the assumption of random effects

1.2.1. Static Models Instrumental Variables Estimators

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Model: 𝑍

𝑗𝑢 = 𝑦1𝑗𝑢 ′ 𝛾1 + 𝑦2𝑗𝑢 ′ 𝛾2 + 𝑥1𝑗 ′ 𝛿1 + 𝑥2𝑗 ′ 𝛿2 + 𝛽𝑗 + 𝑣𝑗𝑢

▪ 𝑦1𝑗𝑢: time-varying explanatory variables, uncorrelated with 𝛽𝑗 ▪ 𝑦2𝑗𝑢: time-varying explanatory variables, correlated with 𝛽𝑗 ▪ 𝑥1𝑗: time-invariant explanatory variables, uncorrelated with 𝛽𝑗 ▪ 𝑥2𝑗: time-invariant explanatory variables, correlated with 𝛽𝑗

Assumptions:

▪ All explanatory variables are strictly exogenous relative to 𝑣𝑗𝑢 ▪ 𝑦1 has a larger dimension than 𝑥2 ▪ 𝑦1 and 𝑥2 are correlated

1.2.1. Static Models Instrumental Variables Estimators

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Estimation:

▪ 2SLS / GMM based on the following instruments for the explanatory variables correlated with 𝛽𝑗:

– 𝑦2𝑗𝑢: 𝑦2𝑗𝑢 − ҧ 𝑦2𝑗 – 𝑥2𝑗: ҧ 𝑦1𝑗

1.2.1. Static Models Instrumental Variables Estimators

Stata xthtaylorY 𝑌11 𝑌12 … 𝑌21 𝑌22… 𝑋

11 𝑋 12… 𝑋 21 𝑋 22 …, endog(𝑌21 𝑌22… 𝑋 21 𝑋 22 …) 78 Advanced Econometrics I 2020/2021

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Estimator Effects Efficiency Prediction Exogeneity Internal instruments for 𝒚𝒋𝒖 Random Fixed Pooled x x

Contemporaneous

Between x x

Strict Lags / Leads

Random Effects x x x

Strict Lags / Leads

Fixed Effects x

Only if long panel Strict Lags / Leads

First- Differences x

𝐹 𝑦𝑗𝑢𝑣𝑗𝑢 = 𝐹 𝑦𝑗𝑢𝑣𝑗,𝑢−1 = 𝐹 𝑦𝑗𝑢𝑣𝑗,𝑢+1 = 0

Lags / Leads (except 𝑢 − 1 and 𝑢 + 1)

1.2.1. Static Models Alternative Estimators

Static Panel Data Models - Summary

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Variance estimators – best alternatives:

Cluster-robust Bootstrap

Hausman test:

Random or fixed effects?

𝐼0: 𝐹 𝛽𝑗𝑦𝑗𝑢 = 0 (RE and FE consistent, RE also efficient) 𝐼1: 𝐹 𝛽𝑗𝑦𝑗𝑢 ≠ 0 (FE consistent, RE inconsistent) 𝐼 = መ 𝛾𝐺𝐹 − መ 𝛾𝑆𝐹 ′ 𝑊 መ 𝛾𝐺𝐹 − 𝑊 መ 𝛾𝑆𝐹

−1 መ

𝛾𝐺𝐹 − መ 𝛾𝑆𝐹 ∼ 𝜓𝑙

2

1.2.1. Static Models Inference and Model Evaluation

Stata

(models must be estimated using standard estimators of the variance)

xtreg Y 𝑌1 … 𝑌𝑙, fe estimates store ModelFE xtreg Y 𝑌1 … 𝑌𝑙 estimates store ModelRE hausman ModelFE ModelRE

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Models with lagged dependent variables:

Include 𝑍

𝑗,𝑢−1, 𝑍 𝑗,𝑢−2 , … as explanatory variables:

𝑍

𝑗𝑢 = 𝛿1𝑍 𝑗,𝑢−1 + ⋯ + 𝛿𝑞𝑍 𝑗,𝑢−𝑞 + 𝑦𝑗𝑢 ′ 𝛾 + 𝛽𝑗 + 𝑣𝑗𝑢, 𝑢 = 𝑞 + 1 , … 𝑈

All estimators for static panel data models are inconsistent

Example – 𝐵𝑆 1 model:

𝑍

𝑗𝑢 = 𝛿1𝑍 𝑗,𝑢−1 + 𝛽𝑗 + 𝑣𝑗𝑢

This equation holds for all time periods, so: 𝑍

𝑗,𝑢−1 = 𝛿1𝑍 𝑗,𝑢−2 + 𝛽𝑗 + 𝑣𝑗,𝑢−1

𝑍

𝑗,𝑢−2 = 𝛿1𝑍 𝑗,𝑢−3 + 𝛽𝑗 + 𝑣𝑗,𝑢−2

etc. 1.2.2. Dynamic Models Inconsistency of Estimators for Static Models

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Random effects estimator:

▪ Required assumption: 𝐹 𝑍

𝑗,𝑢−1𝛽𝑗 = 0

▪ However, 𝛽𝑗 is one of the components of 𝑍

𝑗,𝑢−1, so 𝐹 𝑍 𝑗,𝑢−1𝛽𝑗 ≠ 0

▪ All other lags of 𝑍

𝑗𝑢 are also correlated with 𝛽𝑗

Fixed effects model:

▪ Required assumption: 𝐹 𝑍

𝑗,𝑢−1 − ത

𝑍

𝑗

𝑣𝑗𝑢 − ത 𝑣𝑗 = 0 ▪ However, 𝑣𝑗,𝑢−1 (included in ത 𝑣𝑗) is one of the components of 𝑍

𝑗,𝑢−1, so

𝐹 𝑍

𝑗,𝑢−1 − ത

𝑍

𝑗

𝑣𝑗𝑢 − ത 𝑣𝑗 ≠ 0 ▪ All other lags of 𝑍

𝑗𝑢, which are included in ത

𝑍

𝑗, are also correlated with the

corresponding element of ത 𝑣𝑗

1.2.2. Dynamic Models Inconsistency of Estimators for Static Models

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First-diferences method:

▪ Required assumption: 𝐹 Δ𝑍

𝑗,𝑢−1Δ𝑣𝑗𝑢 = 0

▪ However, 𝑣𝑗,𝑢−1 (included in Δ𝑣𝑗𝑢) is one of the components of 𝑍

𝑗,𝑢−1

(included in Δ𝑍

𝑗,𝑢−1), so 𝐹 Δ𝑍 𝑗,𝑢−1Δ𝑣𝑗𝑢 ≠ 0

▪ Unlike the previous estimators, 𝑍

𝑗,𝑢−2, 𝑍 𝑗,𝑢−3, … are not correlated with

Δ𝑣𝑗𝑢, provided that there is no autocorrelation ▪ Solution: using 𝑍

𝑗,𝑢−2, 𝑍 𝑗,𝑢−3, … (ou functions of these variables) as

instruments for Δ𝑍

𝑗,𝑢−1

1.2.2. Dynamic Models Inconsistency of Estimators for Static Models

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Joaquim J.S. Ramalho

Base Dynamic Panel Data Model: ∆𝑍

𝑗𝑢= 𝛿1∆𝑍 𝑗,𝑢−1 + ∆𝑦𝑗𝑢 ′ 𝛾 + ∆𝑣𝑗𝑢, 𝑢 = 3, … , 𝑈

Assumption:

𝑣𝑗𝑢 has no autocorrelation ⟹ Δ𝑣𝑗𝑢 has first-order autocorrelation: 𝐷𝑝𝑤 ∆𝑣𝑗𝑢, ∆𝑣𝑗,𝑢−1 = 𝐷𝑝𝑤 𝑣𝑗𝑢 − 𝑣𝑗,𝑢−1, 𝑣𝑗,𝑢−1 − 𝑣𝑗,𝑢−2 = −𝐷𝑝𝑤 𝑣𝑗,𝑢−1, 𝑣𝑗,𝑢−1 ≠ 0

Main estimators:

Anderson-Hsiao (1981) Arellano-Bond (1991) – ‘Difference GMM’ Blundell-Bond (1998) – ‘System GMM’ 1.2.2. Dynamic Models Instrumental Variable Estimators

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Joaquim J.S. Ramalho

Anderson-Hsiao (1981):

Two alternative instruments:

▪ 𝑍

𝑗,𝑢−2

▪ ∆𝑍

𝑗,𝑢−2 (one observation is lost but in general seems to produce more

efficient estimators)

1.2.2. Dynamic Models Instrumental Variable Estimators

Stata ivregress gmm D.Y (𝐸𝑀. 𝑍 = 𝑀2. 𝑍) 𝐸. 𝑌1 … 𝐸. 𝑌𝑙

85 Advanced Econometrics I 2020/2021

Stata xtivreg D.Y (DL. 𝑍 = DL2. 𝑍) D. 𝑌1 … D.𝑌𝑙

• r

xtivreg Y (L. 𝑍 = L2. 𝑍) 𝑌1 … 𝑌𝑙, fd

Joaquim J.S. Ramalho

Arellano-Bond (1991):

Proposed using all available lags of 𝑍

𝑗,𝑢 as instruments:

▪ 𝑢 = 3: 𝑍

𝑗,1

▪ 𝑢 = 4: 𝑍

𝑗,2, 𝑍 𝑗,1

▪ … ▪ 𝑢 = 𝑈: 𝑍

𝑗,𝑈−2, … , 𝑍 𝑗,2, 𝑍 𝑗,1

Total number of instruments: 𝑈 − 1 𝑈 − 2 /2 It is possible to use only a subset of the available instruments More efficient than Anderson-Hsiao’s (1981) estimators 1.2.2. Dynamic Models Instrumental Variable Estimators

Stata xtabond Y 𝑌1 … 𝑌𝑙, maxldep(#) twostep vce(robust)

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Joaquim J.S. Ramalho

Blundell-Bond (1998):

Often, lags of 𝑍

𝑗,𝑢 are not good instruments for ∆𝑍 𝑗,𝑢

Proposed adding the following instruments: ∆𝑍

𝑗,2,…,∆𝑍 𝑗,𝑈−1

Total number of instruments:

𝑈−1 𝑈−2 2

+ 𝑈 − 2 It is possible to use only a subset of the available instruments More efficient than Arellano-Bond’s (1991) estimator Requires heavier assumptions than Arellano-Bond’s (1991) estimator 1.2.2. Dynamic Models Instrumental Variable Estimators

Stata xtdpdsys Y 𝑌1 … 𝑌𝑙, maxldep(#) twostep vce(robust)

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Joaquim J.S. Ramalho

Most common tests:

Hansen’s J test of overidentifying restrictions Test for autocorrelation

▪ All estimators for dynamic panel data models assume first-order autocorrelation:

– 𝐷𝑝𝑤 ∆𝑣𝑗𝑢, ∆𝑣𝑗,𝑢−𝑚 ≠ 0 – 𝐷𝑝𝑤 ∆𝑣𝑗𝑢, ∆𝑣𝑗,𝑢−𝑚 = 0, 𝑚>1

1.2.2. Dynamic Models Tests for Instruments Validity

Stata

(after xtabond or xtdpdsys, with variance estimated in a standard way)

estat sargan Stata

(after xtabond or xtdpdsys, with variance estimated in a standard way)

estat abond, artests(3)

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