ADVANCED ECONOMETRICS I Theory (2/3) Instructor: Joaquim J. S. - PowerPoint PPT Presentation

ADVANCED ECONOMETRICS I Theory (2/3) 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

2. Nonlinear Regression Analysis 2.1. Model Estimation 2.1.1. Maximum Likelihood 2.1.2. Quasi-Maximum Likelihood Estimation 2.1.3. Generalized Method of Moments 2.2. Model Inference and Evaluation 2.3. Panel Data Models 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 2

2. Nonlinear Regression Analysis Motivation: Often, the dependent variable is discrete and/or bounded, in which case linear regression models cannot describe it appropriately Some continuous, bounded dependent variables may be transformed in such a way that linear regression models can still be used for their analysis; but in some cases such transformations are not available 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 3

2. Nonlinear Regression Analysis Quantities of interest: Linear models: ▪ 𝐹 𝑍 𝑌 Nonlinear models: ▪ 𝐹 𝑍 𝑌 ▪ If using a probabilistic model: 𝑄𝑠 𝑍 𝑌 ▪ In some models, there may be also interest on variants of the previous quantities: – Example: when modelling a nonnegative outcome, 𝑍 ≥ 0 , with lots of zeros, it may be interesting to estimate also: » 𝑄𝑠 𝑍 = 0 𝑌 » 𝐹 𝑍 𝑌, 𝑍 > 0 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 4

2. Nonlinear Regression Analysis Partial Effects: Linear models: ▪ Model: 𝐹 𝑍 𝑌 = 𝑌𝛾 ▪ Effects: ∆𝑌 𝑘 = 1 ⟹ ∆𝐹 𝑍 𝑌 = 𝛾 𝑘 Nonlinear models: ▪ Model: – 𝐹 𝑍 𝑌 = 𝐻 𝑌𝛾 – 𝑄𝑠 𝑍 𝑌 = 𝐺 𝑌𝛾 ▪ Effects: ∆𝑌 𝑘 = 1 ⟹ – ∆𝐹 𝑍 𝑌 = 𝜖𝐹 𝑍|𝑌 𝜖𝐻 𝑌𝛾 𝜖𝐻 𝑌𝛾 ′ 𝛾 = = 𝛾 𝑘 = 𝛾 𝑘 𝑕 𝑦 𝑗 𝜖𝑌 𝑘 𝜖𝑌 𝑘 𝜖𝑌𝛾 𝜖𝑄𝑠 𝑍|𝑌 𝜖𝐺 𝑌𝛾 𝜖𝐺 𝑌𝛾 ′ 𝛾 – ∆𝑄𝑠 𝑍 𝑌 = = = 𝛾 𝑘 = 𝛾 𝑘 𝑔 𝑦 𝑗 𝜖𝑌 𝑘 𝜖𝑌 𝑘 𝜖𝑌𝛾 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 5

2. Nonlinear Regression Analysis Partial effects may be compared across different models, but the values of 𝛾 cannot 𝜖𝐻 𝑌𝛾 𝜖𝐺 𝑌𝛾 > 0 and > 0 : However, because 𝜖𝑌𝛾 𝜖𝑌𝛾 ▪ The sign of the partial effect is given by the sign of 𝛾 𝑘 ▪ Testing the statistical significance of the partial effect is equivalent to test for 𝐼 0 : 𝛾 𝑘 = 0 To calculate the magnitude of the partial effects, there are three main alternatives: ▪ Calculate the partial effects for each individual in the sample and then obtain the mean of those effects Stata ▪ Replace x by its sample means (after estimating the model) margins , dydx(𝑤𝑏𝑠𝑚𝑗𝑡𝑢) at (…) ▪ Replace x by specific values margins , dydx(𝑤𝑏𝑠𝑚𝑗𝑡𝑢) atmeans margins , dydx(𝑤𝑏𝑠𝑚𝑗𝑡𝑢) 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 6

2. Nonlinear Regression Analysis 2.1. Model Estimation Estimation: Most common estimation methods: ▪ Maximum Likelihood (ML): more efficient ▪ Quasi-Maximum Likelihood (QML): more robust In both cases it is necessary to specify: ▪ The 𝐻 function in 𝐹 𝑍 𝑌 = 𝐻 𝑌𝛾 ▪ The 𝐺 function in 𝑄𝑠 𝑍 𝑌 = 𝐺 𝑌𝛾 Main assumptions: ▪ ML: – Correct specification of both 𝐻 and 𝐺 ▪ QML: – Correct specification of 𝐻 – 𝐺 does not need to be correctly specified but needs to belong to the linear exponential family (e.g. Normal, Bernoulli, Poisson, Exponencial, Gama, etc.) 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 7

2. Nonlinear Regression Analysis 2.1. Model Estimation ML / QML estimation - Statistics: Distribution function - 𝐺 𝑧 : gives the probability of the random variable 𝑍 taking a value less than or equal to 𝑧 : 𝐺 𝑧 = 𝑄𝑠 𝑍 ≤ 𝑧 Density function - 𝑔 𝑧 : ▪ Derivative of the distribution function: 𝑧 𝑔 𝑧 𝑒𝑍 𝜖𝐺 𝑧 𝑔 𝑧 = 𝐺 𝑧 = ׬ −∞ 𝜖𝑧 ▪ In the continuous case, describes the relative likelihood for the random variable 𝑍 being equal to 𝑧 (not the absolute likelihood) ▪ In the discrete case gives the probability of the random variable 𝑍 being equal to 𝑧 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 8

2. Nonlinear Regression Analysis 2.1. Model Estimation Likelihood function: ▪ In individual terms, it is the same as the density function ▪ Usually, it is calculated for the full sample, giving the likelihood of observing that sample under the assumption that the density function 𝑔 𝑧 describes appropriately the population behaviour ▪ Assuming independence across individuals and the same distribution for all of them, it is calculated as: 𝑂 𝑀 𝑧 = ς 𝑗=1 𝑔 𝑧 𝑗 , 0 ≤ 𝑀 𝑧 ≤ 1 Usually: ▪ 𝐺 𝑧 , 𝑔 𝑧 and 𝑀 𝑧 depend also on 1 or 2 parameters ▪ One of the parameters represents 𝐹 𝑧 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 9

2. Nonlinear Regression Analysis 2.1. Model Estimation Most popular density functions: 𝒁 𝒈 𝒛 Function 2𝜌𝜏 2 1/2 exp − 𝑧 − 𝜈 2 1 Normal 𝜈, 𝜏 2 ] − ∞, +∞[ 2𝜏 2 −𝑧 1 ]0, +∞[ Exponential 𝜈 𝜈 𝜈 𝑓 Γ 𝜀 Γ 𝜈𝜀 Γ 1 − 𝜈 𝜀 𝑧 𝜈𝜀−1 1 − 𝑧 1−𝜈 𝜀−1 ]0,1[ Beta 𝜈, 𝜀 𝜈 𝑧 1 − 𝜈 1−𝑧 {0,1} Bernoulli 𝜈 𝜈 𝑧 𝑓 −𝜈 {0,1,2, … } Poisson 𝜈 𝑧! In all cases: 𝐹 𝑧 = 𝜈 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 10

2. Nonlinear Regression Analysis 2.1. Model Estimation Econometrics: All the analysis is conditional on a set of explanatory variables The parameter 𝜈 ( = 𝐹 𝑧 in Statistics) is replaced by the function assumed for 𝐹 𝑧|𝑌 , for example 𝑌𝛾 (linear regression model) It is assumed that the likelihood function is known up to the set of parameters 𝛾 (and, in case the original function has 2 parameters, the other parameter) Density function to be considered: 𝑔 𝑧 𝑗 |𝑦 𝑗 ; 𝛾 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 11

2. Nonlinear Regression Analysis 2.1. Model Estimation Estimation: Given that: ▪ The density function 𝑔 ∙ is known, except for 𝛾 ▪ The probability that the sample values were in fact generated by the chosen density 𝑔 ∙ is measured by the likelihood function Then: ▪ We should choose for 𝛾 the value that maximizes 𝑀 𝑧 𝑗 |𝑦 𝑗 ; 𝛾 ▪ Optimization problem: 𝑂 max 𝛾 𝑀 𝑧|𝑌; 𝛾 = ෑ 𝑔 𝑧 𝑗 |𝑦 𝑗 ; 𝛾 𝑗=1 ▪ Actually, it is more common to maximize 𝑀𝑀 𝑌𝛾 = ln 𝑀 𝑧|𝑌; 𝛾 : 𝑂 max 𝛾 𝑀𝑀 𝑌𝛾 = ෍ ln 𝑔 𝑧 𝑗 |𝑦 𝑗 ; 𝛾 𝑗=1 – It is easier to maximize – It produces the same estimates for 𝛾 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 12

2. Nonlinear Regression Analysis 2.1. Model Estimation Properties of ML / QML Estimators: Asymptotic properties of ML estimators: ▪ Consistency ▪ Efficiency ▪ Normality Asymptotic properties of QML estimators: ▪ Consistency ▪ Normality ▪ Efficiency is lost; variance calculated in a robust way ▪ Not possible to predict 𝑄𝑠 𝑍 𝑌 and associated partial effects Finite sample properties for both estimators: ▪ Unknown 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 13

2. Nonlinear Regression Analysis 2.2. Model Inference and Evaluation Alternative forms for estimating parameter variances: Standard / Efficient ⟶ only available for ML Robust ⟶ only makes sense for QML Cluster-robust ⟶ panel data Bootstrap Classical tests: Likelihood Ratio (LR) ⟶ only available for ML Wald Score/LM 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 14

2. Nonlinear Regression Analysis 2.2. Model Inference and Evaluation Test for the joint significance of a set of parameters: Competing models: ▪ Restricted (smaller) model, based on 𝑀 𝑆 𝛾 0 + 𝛾 1 𝑦 1 + ⋯ + 𝛾 𝑕 𝑦 𝑕 ▪ Full (larger) model , based on 𝑀 𝐺 ൫ 𝛾 0 + 𝛾 1 𝑦 1 + ⋯ + 𝛾 𝑕 𝑦 𝑕 + 𝛾 𝑕+1 𝑦 𝑕+1 + ⋯ + 𝛾 𝑙 𝑦 𝑙 ൯ Hypotheses: 𝐼 0 : 𝛾 𝑕+1 = ⋯ = 𝛾 𝑙 = 0 (restricted model) 𝐼 1 : No 𝐼 0 (full model) 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 15

2. Nonlinear Regression Analysis 2.2. Model Inference and Evaluation LR test: Stata 2 𝑀𝑆 = 2 𝑀𝑀 𝐺 𝑌𝛾 𝐺 − 𝑀𝑀 𝑆 𝑌𝛾 𝑆 ~𝜓 𝑙−𝑕 (estimate one model) ▪ Available in most econometric packages estimates store Model1 (estimate the other model) ▪ Easy calculation estimates store Model2 lrtest 𝑁𝑝𝑒𝑓𝑚1 𝑁𝑝𝑒𝑓𝑚2 ▪ Both the competing models need to be estimated Wald test: −1 መ ′ Var መ 𝑋 = መ 2 𝛾 𝐸 𝛾 𝐸 𝛾 𝐸 ~𝜓 𝑙−𝑕 where መ 𝛾 𝑕+1 , … , መ መ 𝛾 𝐸 = 𝛾 𝑙 is estimated based on 𝑀𝑀 𝐺 𝑌𝛾 𝐺 ▪ When 𝐼 0 : 𝛾 𝑘 = 0 , 𝑋 simplifies to: መ 𝛾 𝑘 𝑢 = ~𝒪 0,1 𝜏 ෡ ො 𝛾 𝑘 Stata (after estimating the full model) ▪ Available in most econometric packages test 𝑌 𝑕+1 ⋯ 𝑌 𝑙 ▪ Only the full model needs to be estimated 2020/2021 Joaquim J.S. Ramalho Advanced Econometrics I 16