APPLIED ECONOMIC MODELLING Theory (Chapter 1) Instructor: Joaquim - - PowerPoint PPT Presentation

Theory (Chapter 1)

APPLIED ECONOMIC MODELLING

Instructor: Joaquim J. S. Ramalho E-mail: jjsro@iscte-iul.pt Personal Website: http://home.iscte-iul.pt/~jjsro Office: D5.10 Course Website: http://home.iscte-iul.pt/~jjsro/appliedeconomicmodelling.htm Fénix: https://fenix.iscte-iul.pt/disciplinas/03178-3/2017-2018/2- semestre

Joaquim J.S. Ramalho

This course provides an introduction to the modern econometric techniques used in applied economic studies:

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

Pre-requisites (recommended):

▪ Microeconomics, Macroeconomics, Econometrics

Course Description

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Continuous assessment:

▪ Test (50% of the grade) ▪ Problem Set (50% of the grade) ▪ Approval requires:

– Weighted mean of at least 10 – Minimum grade at the test of 7 – Minimum attendance: 80% of classes

• r

Final examination (100% of the grade):

Grading

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• 1. Microeconomic Models

1.1. Linear Regression Models for Cross-Sectional and Panel Data 1.2. Demand Analysis 1.3. Production and Cost Functions 1.4. Efficiency and Stochastic Frontier Models

• 2. Macroeconomic Models

2.1. Linear Regression Models for Time Series Data 2.2. The Demand for Money 2.3. The Philips Curve

• 3. Econometric Models for Policy Analysis

3.1. Hedonic Regression Models 3.2. Differences-in-Differences Estimator

Software: Stata

Contents

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Recommended:

▪ Intriligator, M., Bodkin, R. and Hsiao, C. (1995), Econometric Models, Techniques and Applications, 2nd Ed., Prentice Hall. ▪ Patterson, K. (2000), An Introduction to Applied Econometrics: A Time Series Approach, Palgrave. ▪ Verbeek, M. (2017), A Guide to Modern Econometrics, 5th Ed., Wiley.

Others:

▪ Berndt, E.R. (1991), The Practice of Econometrics – Classic and Comtemporary, Addison-Wesley. ▪ Coelli, T.J., D.S.P. Rao, C.J. O’Donnell, G.E. Battese (2005), An Introduction to Efficiency and Productivity Analysis, 2nd Ed., Springer. ▪ Deaton, A., J. Muellbauer (1980), Economics and Consumer Behavior, Cambridge University Press. ▪ Wooldridge, J.M. (2015), Introductory Econometrics: a Modern Approach, 6ª Ed., South-Western Publishers.

Textbooks

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Model Specification:

𝑍

𝑗 = 𝛾0 + 𝛾1𝑌𝑗1 + ⋯ + 𝛾𝑙𝑌𝑗𝑙 + 𝑣𝑗

𝑗 = 1, ⋯ , 𝑂 𝑍 = 𝑌𝛾 + 𝑣 𝐹 𝑍|𝑌 = 𝑌𝛾

Assumptions:

Random 1. sampling

• 2. 𝐹 𝑣|𝑌

= 0 No 3. perfect colinearity Homoskedasticity 4. : 𝑊𝑏𝑠 𝑣|𝑌 = 𝜏2 Normality 5.

: 𝑣~𝑂𝑝𝑠𝑛𝑏𝑚 0, 𝜏2

1.1. Linear Regression Models for Cross-Sectional and Panel Data Cross-Sectional Data

𝑣: error term 𝛾: parameters 𝑙: n. explanatory variables 𝑞: n. parameters 𝑂: n. observations 𝜏2: error variance

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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 መ 𝛾 = 𝑌′𝑌 −1𝑌′𝑧

Estimator properties:

Finite samples:

▪ Assumptions 1-3: Unbiasedness ▪ Assumptions 1-4: Unbiasedness and efficiency ▪ Assumptions 1-5: Unbiasedness, efficiency and normality

Asymptotically:

▪ Assumptions 1-3: Consistency ▪ Assumptions 1-4: Consistency, efficiency and normality

Cross-Sectional Data Estimation

ො 𝑣: residual ෠ 𝑍: fitted value of 𝑍 መ 𝛾: estimate for 𝛾 Stata regress Y 𝑌1 ⋯ 𝑌𝑙

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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:

▪ 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 ▪ If significant, analyzing the sign of the effect (positive, negative) ▪ If significant, calculating and analyzing the magnitude of the effect

Cross-Sectional Data Partial / Marginal Effects

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Variance of the parameter estimators:

Needed for performing hypothesis tests Alternative estimators:

▪ Standard – assumes homoskedasticity 𝑊𝑏𝑠 መ 𝛾 = ො 𝜏2 𝑌′𝑌 −1 ▪ Robust – valid under both homoskedasticity and heteroskedasticity 𝑊𝑏𝑠 መ 𝛾 = 𝑌′𝑌 −1𝑌′෡ Ω𝑌 𝑌′𝑌 −1, where ෡ Ω = 𝑒𝑗𝑏𝑕 ො 𝑣𝑗

2

▪ Cluster-robust – specific for panel data 𝑊𝑏𝑠 መ 𝛾 = σ𝑗=1

𝑂

𝐘𝑗

′𝐘𝑗 −1 σ𝑗=1 𝑂

𝐘𝑗

′ෝ

𝐯𝑗ෝ 𝐯𝑗

′𝐘𝑗 σ𝑗=1 𝑂

𝐘𝑗

′𝐘𝑗 −1, where 𝐘𝑗 and ෝ

𝐯𝑗 are based on the 𝑈 observations available for individual 𝑗

Cross-Sectional Data Inference

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

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Test for the individual significance of a parameter – t test:

𝐼0: 𝛾𝑘 = 0 𝐼1: 𝛾𝑘 ≠ 0

• r 𝐼1: 𝛾𝑘 > 0
• r 𝐼1: 𝛾𝑘 < 0

𝑢 = መ 𝛾𝑘 ො 𝜏෡

𝛾𝑘

~𝑢𝑂−𝑞

Cross-Sectional Data Inference

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

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Test for the joint significance of a set of parameters – F / Wald tests:

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 𝑋 = 𝑂 መ 𝛾∗

′ Var መ

𝛾∗

−1 መ

𝛾∗~𝜓𝑟

2 → general formula

Cross-Sectional Data Inference

𝑆2: coefficient of determination of the model under H1 𝑆∗

2: coefficient of determination of the model

under H1 𝑟: n. of parameters being tested መ 𝛾∗: vector of parameters to be tested

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

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RESET test:

Intuition:

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

∞

γj 𝑌 መ 𝛾

𝑘+1

Implementation:

▪ Estimate the original model ▪ 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 / Wald test for the significance of the added variables: 𝐼0: γ1 = γ2 = γ3 = ⋯ = 0 (suitable model functional form) 𝐼1: No 𝐼0 (unsuitable model functional form)

Cross-Sectional Data RESET Test

Stata (only test version based on three fitted powers) regress Y 𝑌1 … 𝑌𝑙 estat ovtest

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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 inconsistent

Motivation:

Omitted variables Simultaneity

Endogeneity Definition and Consequences

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Example:

True model: 𝑍 = 𝛾0 + 𝛾1𝑌1 + 𝛾2𝑌2 + 𝑤 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

Endogeneity Omitted Variables

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Example:

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

Endogeneity Simultaneity

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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’

Endogeneity Solutions

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Instrumental variables:

Context:

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

Definition 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

The number of instrumental variables must be equal or larger than the number of endogenous explanatory variables

Endogeneity Instrumental Variables

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Implementation:

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

ด 𝑌1

• End. Expl. Var.

= 𝜌0 + 𝜌2𝑌2 + ⋯ + 𝜌𝑙𝑌𝑙

• Exog. Expl. Var.

+ 𝜌𝐵𝐽𝑊

𝐵 + ⋯ + 𝜌𝑁𝐽𝑊 𝑁

Instrumental Variables + 𝑥

and get ෠

𝑌1 = ො 𝜌0 + ො 𝜌2𝑌2 + ⋯ + ො 𝜌𝑙𝑌𝑙 + ො 𝜌𝐵𝐽𝑊

𝐵 + ⋯ + ො

𝜌𝑁𝐽𝑊

𝑁

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

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

Endogeneity Two-Stage Least Squares

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 18 Applied Economic Modelling 2017/2018

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Tests relevant for 2SLS:

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 IVs 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 ‘IVs’ that actually are endogenous, 2SLS is 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 ‘IVs’ uncorrelated with the endogenous regressors, 2SLS is inconsistent

Endogeneity Specification Tests

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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 + 𝜀 ෝ 𝑥 + 𝜁

• 4. t test:

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

Endogeneity Test for the Exogeneity of an Explanatory Variable

Stata ivregress 2sls Y (𝑌1= 𝐽𝑊

𝐵… 𝐽𝑊 𝑁) 𝑌2 … 𝑌𝑙

estat endogenous

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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 Test for overidentifying restrictions:

• 1. Estimate the original model by 2SLS:

𝑍 = 𝛾0 + 𝛾1𝑌1 + 𝛾2𝑌2 + ⋯ + 𝛾𝑙𝑌𝑙 + 𝑣

• 2. Calculate the residuals ො

𝑣

• 3. Estimate, by OLS, the auxiliary model:

ො 𝑣 = 𝛿0 + 𝛿2𝑌2 + ⋯ + 𝛿𝑙𝑌𝑙 + 𝛿𝐵𝐽𝑊

𝐵 + ⋯ + 𝛿𝑁𝐽𝑊 𝑁 + 𝜁

• 4. Test:

𝐼0: 𝐹 𝑣|𝑎 = 0 (IV’s are exogenous) 𝐼1: 𝐹 𝑣|𝑎 ≠ 0 (IV’s are not exogenous) using an 𝐺 / 𝑋𝑏𝑚𝑒 statistic (degrees of freedom: 𝜓𝑟

2)

Endogeneity Test for the Exogeneity of the Instrumental Variables

Stata ivregress 2sls Y (𝑌1= 𝐽𝑊

𝐵… 𝐽𝑊 𝑁) 𝑌2 … 𝑌𝑙

estat overid q: number of overidentifying restrictions

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F / Wald tests for the significance of the IV’s in the reduced form model:

• 1. Estimate the reduced form model:

𝑌1 = 𝜌0 + 𝜌2𝑌2 + ⋯ + 𝜌𝑙𝑌𝑙 + 𝜌𝐵𝑊𝐽𝐵 + ⋯ + 𝜌𝑁𝑊𝐽𝑁 + 𝑥

• 2. Test the hypothesis:

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

Endogeneity

Tests for the Correlation between Instrumental Variables and Explanatory Variables

Stata ivregress 2sls Y (𝑌1= 𝐽𝑊

𝐵… 𝐽𝑊 𝑁) 𝑌2 … 𝑌𝑙

estat firststage

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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

Some advantages:

Allow for endogenous explanatory variables in some special cases It is simpler to define instrumental variables

Panel Data Definitions

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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 ▪ 𝑦𝑢: individual-constant

𝑣𝑗𝑢: idiosyncratic error term – changes randomly across individuals and

• ver time

Panel Data Base Model

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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

Panel Data Fixed Effects versus Random Effects

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Model: 𝑍

𝑗𝑢 = 𝛽 + 𝑦𝑗𝑢 ′ 𝛾 + 𝛽𝑗 − 𝛽 + 𝑣𝑗𝑢 𝑤𝑗𝑢

Main assumption: 𝛽𝑗 e 𝑦𝑗𝑢 must be uncorrelated (random effects) Estimation:

▪ Pooled OLS estimator: OLS with a cluster-type estimator for the variance ▪ Random effects estimator: generalized least squares (GLS), possibly with a cluster-type estimator for the variance

Panel Data Estimators for Random Effects Models

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

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

Joaquim J.S. Ramalho

Alternative OLS-based estimators:

▪ Fixed effects / within estimator (𝛽𝑗’s are eliminated): 𝑍

𝑗𝑢−ത

𝑍

𝑗= 𝑦𝑗𝑢 − ҧ

𝑦𝑗 ′𝛾 + 𝑣𝑗𝑢 − ത 𝑣𝑗 ▪ LSDV (𝛽𝑗’s are estimated)

𝑍

𝑗𝑢 = ෍ 𝑘=1 𝑂

𝛽𝑘𝑒𝑗𝑘 + 𝑦𝑗𝑢

′ 𝛾 + 𝑣𝑗𝑢

where 𝑒𝑗𝑘 = 1 if 𝑗 = 𝑘 and 0 if 𝑗 ≠ 𝑘

Estimates for 𝛾 are identical

Panel Data Estimators for Fixed Effects Models

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

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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)

ത 𝑍

𝑗 = σ𝑢=1 𝑈

𝑍

𝑗𝑢, etc.

Joaquim J.S. Ramalho

Main limitation - it is not possible to include in fixed effects models:

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

Main advantage - allow for (time-constant) unobserved individual heterogeneity that may be correlated with the explanatory variables, not requiring the use of instrumental variables

Panel Data Estimators for Fixed Effects Models

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Endogenous explanatory variables - 𝐹 𝑦𝑗𝑢𝑣𝑗𝑢 ≠ 0:

Possible IV’s for 𝑦𝑗𝑢:

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

Examples of internal instruments:

▪ If 𝑦𝑗𝑢 is weakly exogenous, 𝐹 𝑦𝑗𝑢𝑣𝑗,𝑢+𝑘 = 0, 𝑘 > 0, 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, 𝐹 𝑦𝑗𝑢𝑣𝑗𝑡 = 0, ∀𝑡, 𝑢, 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.

Panel Data Instrumental Variables Estimators

Stata (external instruments – 2SLS) xtivreg Y (𝑌1= 𝐽𝑊

𝐵… 𝐽𝑊 𝑁) 𝑌2 … 𝑌𝑙, options

(options: re, fe) Stata (internal instruments – 2SLS) xtivreg Y (𝑌1= 𝑀. 𝑌1 𝑀2. 𝑌1…) 𝑌2 … 𝑌𝑙, options (options: re, fe)

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Hausman test:

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

Panel Data Random or Fixed Effects?

Stata (models must be estimated using standard estimators of the variance) xtreg Y 𝑌1 … 𝑌𝑙, fe estimates store ModelFE xtreg Y 𝑌1 … 𝑌𝑙, re estimates store ModelRE hausman ModelFE ModelRE

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𝑟𝑗 level of consumption of good i 𝑟′ = 𝑟1 … 𝑟𝑜 n goods 𝑞𝑗 price of good i 𝑞′ = 𝑞1 … 𝑞𝑜 𝑣 𝑟 utility function 𝐹 = 𝑞′𝑟 household’s budget / total expenditure 𝑥𝑗 =

𝑞𝑗𝑟𝑗 𝐹

expenditure share of good i

1.2. Demand Analysis Notation

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Original problem Dual problem (utility maximization) (cost minimization)

Duality Solve Solve first-order first-order conditions conditions Substitute Substitute 𝑔 𝑞, 𝐹 𝑔∗ 𝑞, 𝑣 into 𝑣 𝑟 into 𝑞′𝑟 Solve 𝑣 = 𝑕 𝑞, 𝐹 for 𝐹 Inversion Solve 𝑕∗ 𝑞, 𝑣 for 𝑣

1.2. Demand Analysis Consumer Theory

32 Applied Economic Modelling 2017/2018

max𝑟 𝑣 𝑟 subject to 𝑞′𝑟 = 𝐹 Marshalian demands 𝑟 = 𝑔 𝑞, 𝐹 Indirect utility function 𝑣 = 𝑕 𝑞, 𝐹 m𝑗𝑜𝑟 𝑞′𝑟 subject to 𝑣 𝑟 = 𝑣 Hicksian demands 𝑟 = 𝑔∗ 𝑞, 𝑣 Cost function 𝐹 = 𝑕∗ 𝑞, 𝑣

Choose consumption levels to minimize the cost

• f obtaining a fixed level of utility, given prices

Choose consumption levels to maximize utility given a budget constraint and prices

Joaquim J.S. Ramalho

Shephard’s Lemma: Roy’s identity: 𝑔

𝑗 ∗ 𝑞, 𝑣 = 𝜖𝑕∗ 𝑞,𝑣 𝜖𝑞𝑗

𝑔

𝑗 𝑞, 𝐹 = − ൗ

𝜖𝑣 𝑞,𝐹 𝜖𝑞𝑗

ൗ

𝜖𝑣 𝑞,𝐹 𝜖𝐹

Substitute 𝑣 = 𝑕 𝑞, 𝐹 into 𝑔∗ 𝑞, 𝑣 Substitute 𝐹 = 𝑕∗ 𝑞, 𝑣 into 𝑔 𝑞, 𝐹

Theory:

▪ The analysis may start with the original or dual problem or with the specification of the cost or indirect utility functions

Empirics:

▪ Total focus on Marshalian demands ▪ The econometric specification for the Marshalian demand should respect its theoretical properties

1.2. Demand Analysis Reverse Process

33 Applied Economic Modelling 2017/2018

Marshalian demands 𝑟 = 𝑔 𝑞, 𝐹 Indirect utility function 𝑣 = 𝑕 𝑞, 𝐹 Hicksian demands 𝑟 = 𝑔∗ 𝑞, 𝑣 Cost function 𝐹 = 𝑕∗ 𝑞, 𝑣

Joaquim J.S. Ramalho

• 1. Adding up:

𝑞′𝑔 𝑞, 𝐹 = 𝑞′𝑔∗ 𝑞, 𝑣 = 𝐹 ⇔ ෍

𝑗=1 𝑜

𝑥𝑗 = 1

• 2. Homogeneity of degree 0 in 𝑞 and 𝐹 (Marshall) or in 𝑞 (Hicks):

𝑔 𝑞, 𝐹 = 𝑔 𝜇𝑞, 𝜇𝐹 = 𝑔∗ 𝜇𝑞, 𝑣 = 𝑔∗ 𝑞, 𝑣 , 𝜇 > 0

• 3. Symmetry and Negativity – the substitution or Slustky matrix of

compensated price responses, which is formed by the elements 𝑡𝑗𝑘 =

𝜖𝑔

𝑗 ∗ 𝑞,𝑣

𝜖𝑞𝑘

(Hicksian demands) or 𝑡𝑗𝑘 =

𝜖𝑔𝑗 𝑞,𝐹 𝜖𝐹

𝑟𝑘 +

𝜖𝑔𝑗 𝑞,𝐹 𝜖𝑞𝑘

(Marshalian demands), is negative semidefinite and symmetric:

• 𝑡𝑗𝑗 ≤ 0
• 𝑡𝑗𝑘 = 𝑡

𝑘𝑗,

𝑗 ≠ 𝑘

1.2. Demand Analysis Properties of Demands (integrability conditions)

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Main aims of empirical studies - estimating:

▪ Expenditure elasticities:

 𝜁𝑗 > 0 ⇒ Normal goods  𝜁𝑗 < 0 ⇒ Inferior goods

▪ (Own) price elasticities:

 𝜁𝑗𝑗 < −1 ⇒ Price elastic  −1 < 𝜁𝑗𝑗 < 0 ⇒ Price inelastic  𝜁𝑗𝑗 > 0 ⇒ Giffen good

▪ Cross-price elasticities:

 𝜁𝑗𝑘 > 0 ⇒ Substitutes  𝜁𝑗𝑘 = 0 ⇒ Independents  𝜁𝑗𝑘 < 0 ⇒ Complements

1.2. Demand Analysis Empirical Studies

35 Applied Economic Modelling 2017/2018

𝜁𝑗 = %∆𝑟𝑗 %∆𝐹 ≅ 𝜖𝑔

𝑗 𝑞, 𝐹

𝜖𝐹 𝐹 𝑟𝑗 𝜁𝑗𝑗 = %∆𝑟𝑗 %∆𝑞𝑗 ≅ 𝜖𝑔

𝑗 𝑞, 𝐹

𝜖𝑞𝑗 𝑞𝑗 𝑟𝑗 𝜁𝑗𝑘 = %∆𝑟𝑗 %∆𝑞𝑘 ≅ 𝜖𝑔

𝑗 𝑞, 𝐹

𝜖𝑞𝑘 𝑞𝑘 𝑟𝑗

Joaquim J.S. Ramalho

Level of aggregation:

▪ Market data ▪ Household data

Data structure:

▪ Cross-sectional data ▪ Panel data ▪ Time-series data

1.2. Demand Analysis Data

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Dependent variable:

▪ 𝑟𝑗 or log 𝑟𝑗 , sometimes per capita ▪ 𝐹𝑗 = 𝑞𝑗𝑟𝑗 or log 𝐹𝑗 , sometimes per capita ▪ 𝑥𝑗 ▪ Market shares

Explanatory variables:

▪ Price (except when all households face the same price) ▪ Total expenditure / Income ▪ Household characteristics (micro data) ▪ Product characteristics (differentiated goods) ▪ Firm characteristics (differentiated goods)

1.2. Demand Analysis Variables

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Single equation models:

▪ Goods are modelled individually ▪ Theory plays a minor role: adding up restriction not relevant, only homogeneity and

𝜖𝑔𝑗 𝑞,𝐹 𝜖𝑞𝑗

< 0 are checked ▪ Even the homogeneity restriction is often ignored, with models chosen based on goodness of fit

Systems of equations:

▪ Goods are modelled simultaneously ▪ Based entirely on theory – two approaches:

– Restrictions are imposed algebraically before model estimation, often starting with indirect utility or cost functions – Restrictions are tested statistically after model estimation

▪ Hard to estimate, with nonlinearity in the parameters and too many parameters to be estimated

1.2. Demand Analysis Econometric Models

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Discrete choice models for differentiated goods:

Also based entirely on theory, with restrictions imposed algebraically ▪ before model estimation Market demand is derived from discrete choice models of consumer ▪ behavior: product market shares are the aggregate outcome of consumer decisions The number of model parameters is substantially reduced by placing ▪ a priori restrictions on the pattern of cross-price elasticities (some are zero, some are equal to each other, etc.) Main disadvantage: the purchase of multiple goods is ruled out ▪

1.2. Demand Analysis Econometric Models

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Linear: 𝑟𝑗 = 𝛽𝑗 + 𝛾𝑗𝐹 + ෍

𝑘=1 𝑜

𝛾𝑗𝑘 𝑞𝑘 + ⋯ + 𝑣𝑗 Double logarithmic / Log-linear / Cobb-Douglas: 𝑟𝑗 = 𝑓𝛽𝑗𝐹𝜁𝑗𝑞1

𝜁𝑗1 … 𝑞𝑜 𝜁𝑗𝑜 … 𝑓𝑣𝑗

log 𝑟𝑗 = 𝛽𝑗 + 𝜁𝑗 log 𝐹 + ෍

𝑘=1 𝑜

𝜁𝑗𝑘log 𝑞𝑘 + ⋯ + 𝑣𝑗 Semi-logarithmic: 𝑟𝑗 = 𝛽𝑗 + 𝛾𝑗 log 𝐹 + ෍

𝑘=1 𝑜

𝛾𝑗𝑘log 𝑞𝑘 + ⋯ + 𝑣𝑗

1.2. Demand Analysis Functional Forms for Single Equation Models

40 Applied Economic Modelling 2017/2018

Homogeneity restriction not satisfied Homogeneity restriction requires 𝜁𝑗 + σ𝑘=1

𝑜

𝜁𝑗𝑘 = 0 Homogeneity restriction requires 𝛾𝑗 + σ𝑘=1

𝑜

𝛾𝑗𝑘 = 0

Joaquim J.S. Ramalho

Quadratic: 𝑟𝑗 = 𝛽𝑗 + 𝛾𝑗𝐹 + ෍

𝑘=1 𝑜

𝛾𝑗𝑘𝑞𝑘 + ෍

𝑘=1 𝑜

෍

𝑛≥1 𝑜

𝛾𝑗𝑘𝑛𝑞𝑘𝑞𝑛 + ⋯ + 𝑣𝑗 Translog: log 𝑟𝑗 = 𝛽𝑗 + 𝛾𝑗 log 𝐹 + ෍

𝑘=1 𝑜

𝛾𝑗𝑘log 𝑞𝑘 + ෍

𝑘=1 𝑜

෍

𝑛≥𝑘 𝑜

𝛾𝑗𝑘𝑛log 𝑞𝑘 log 𝑞𝑛 + ⋯ + 𝑣𝑗

(may also include quadratic and interaction terms with logged prices for log 𝐹)

Generalized Leontief Constant elasticity of substitution (CES) ...

1.2. Demand Analysis Functional Forms for Single Equation Models

41 Applied Economic Modelling 2017/2018

Homogeneity restriction not satisfied Homogeneity restriction requires 𝛾𝑗 + σ𝑘=1

𝑜

𝛾𝑗𝑘 = 0 and other conditions

Joaquim J.S. Ramalho

Seminal paper:

▪ Berry, S. (1994), Estimating discrete-choice models of product differentiation, RAND Journal of Economics, 25(2), 242–262

Main assumptions:

▪ Each firm produces a single good (𝑔 = 0, … , 𝐺) ▪ Consumers (𝑘 = 1, … , 𝐾) can only buy one product, including the outside good (𝑔 = 0) ▪ Product / firm characteristics:

– 𝑞𝑔: observed price of product f (treated as endogenous due to correlation with 𝜚𝑔) – 𝑌𝑔: other observed characteristics of product f (treated as exogenous) – 𝜚𝑔: unobserved characteristics of product f (included in the model error term) – 𝑡𝑔: market share of firm f

▪ Consumer characteristics:

– 𝑌

𝑘: unobserved characteristics of consumer j

– 𝑧𝑘: unobserved choice of consumer j

1.2. Demand Analysis Discrete Choice Models for Differentiated Goods

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If 𝑌

𝑘 and 𝑧𝑘 were observed, and price was exogenous (no 𝜚𝑔),

a standard discrete choice probability model could be applied:

▪ Main model for discrete choices – multinomial logit model ▪ 𝑉

𝑘𝑔 - utility that consumer 𝑘 derives from the product 𝑔:

𝑉

𝑘𝑔 = 𝛽 + 𝛾𝑞𝑔 + 𝜄′𝑌𝑔 + 𝜇′𝑌 𝑘

▪ Normalization: 𝑉

𝑘0 = 0

▪ Probability of consumer 𝑘 choosing product 𝑔: 𝑄𝑠 𝑧𝑘 = 𝑔|𝑞𝑔, 𝑌𝑔, 𝑌

𝑘 =

exp 𝑉

𝑘𝑔

σ𝑛=0

𝐺

exp 𝑉

𝑘𝑛

▪ For each consumer, the sum of the probabilities of choosing the available products is equal to 1: ෍

𝑔=0 𝐺

𝑄

𝑘𝑔 = 1

1.2. Demand Analysis Multinomial Logit Model

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Under our assumptions:

▪ Individual consumer decisions are not observed, so we have to work with an estimate of the aggregate choice probabilities, the market shares 𝑡𝑔 ▪ The mean utility function, which omits consumer characteristics (assumed to be a zero mean random disturbance), are given by: ഥ 𝑉

𝑔 = 𝛽 + 𝛾𝑞𝑔 + 𝜄′𝑌𝑔 + 𝜚𝑔

▪ The aggregate model is therefore given by: 𝑡𝑔 = exp ഥ 𝑉

𝑔

σ𝑛=0

𝐺

exp ഥ 𝑉𝑛 which cannot be directly estimated due to endogeneity of 𝑞𝑔

1.2. Demand Analysis Discrete Choice Models for Differentiated Goods

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Estimation:

Using ▪ the outside good as reference, the model may be written in relative terms: 𝑡𝑔 𝑡0 = exp ഥ 𝑉

𝑔

𝑡𝑔 𝑡0 = exp 𝛽 + 𝛾𝑞𝑔 + 𝜄′𝑌𝑔 + 𝜚𝑔 Then ▪ , it can be linearized: ln 𝑡𝑔 − ln 𝑡0 = 𝛽 + 𝛾𝑞𝑔 + 𝜄′𝑌𝑔 + 𝜚𝑔 Finally ▪ , it may be estimated by 2SLS

1.2. Demand Analysis Discrete Choice Models for Differentiated Goods

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Price elasticities:

▪ Own-price elasticities: Ƹ 𝜁𝑔𝑔 = 𝜖 Ƹ 𝑡𝑔 𝜖𝑞𝑔 𝑞𝑔 Ƹ 𝑡𝑔 = 𝛾𝑞𝑔 1 − Ƹ 𝑡𝑔 ▪ Cross-price elasticities: Ƹ 𝜁𝑔𝑕 = 𝜖 Ƹ 𝑡𝑔 𝜖𝑞𝑕 𝑞𝑕 Ƹ 𝑡𝑔 = −𝛾𝑞𝑔 Ƹ 𝑡𝑔

1.2. Demand Analysis Discrete Choice Models for Differentiated Goods

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Endogeneity Zeros Nonlinear budgets:

▪ Multi-part tariffs ▪ Informational problems

1.2. Demand Analysis Other econometric Issues

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𝑟

• utput

𝑞

• utput price

𝑦𝑗 input i 𝑦′ = 𝑦1 … 𝑦𝑜 n inputs 𝑥𝑗 price of input i 𝑥′ = 𝑥1 … 𝑥𝑜 𝑑 = 𝑥′𝑦 production cost 𝑠 = 𝑞𝑟 revenue π = 𝑠 − 𝑑 profits

1.3. Production and Cost Functions Notation and Identities

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Production function or production frontier:

▪ 𝑟 = 𝑔 𝑦 ▪ Maximum attainable output for alternative combinations of inputs

General optimization problem:

▪ Maximizing profits subject to a given technology: max𝑟,𝑦 π ∙ = 𝑞𝑟 − 𝑥′𝑦 subject to 𝑟 = 𝑔 𝑦

– Output and input prices are given (perfect competition) – Solution gives optimal levels of inputs and output:

» Unconditional input demand functions: 𝑦 𝑞, 𝑥 » Unconditional output supply function: 𝑟 𝑞, 𝑥

– Profit function: π 𝑞, 𝑥 – Hotteling’s Lemma:

» 𝑦𝑗 𝑞, 𝑥 = −

𝜖π 𝑞,𝑥 𝜖𝑥𝑗

» 𝑟 𝑞, 𝑥 =

𝜖π 𝑞,𝑥 𝜖𝑞

– Profit maximization implies both cost minimization and revenue maximization

1.3. Production and Cost Functions The Theory of the Firm

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Specific optimization problems:

▪ Minimizing cost: min𝑦 𝑑 ∙ = 𝑥′𝑦 subject to 𝑟 = 𝑔 𝑦

– Solution gives the optimal level of inputs - conditional input demand functions: 𝑦 𝑟, 𝑥 – Cost function: 𝑑 𝑟, 𝑥 – Shephard’s Lemma: 𝑦𝑗 𝑟, 𝑥 = 𝜖𝑑 𝑟, 𝑥 𝜖𝑥𝑗

▪ Maximizing revenue: max𝑟 𝑠 ∙ = 𝑞𝑟 subject to 𝑟 = 𝑔 𝑦

– Solution gives the optimal level of output - conditional output supply function: 𝑟 𝑞, 𝑦 – Revenue function: 𝑠 𝑞, 𝑦 – Also possible to work backwards from the revenue function to the conditional output supply function: 𝑟 𝑞, 𝑦 = 𝜖𝑠 𝑞, 𝑦 𝜖𝑞

1.3. Production and Cost Functions The Theory of the Firm

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Production function:

▪ Marginal product: 𝑁𝑄𝑗 =

𝜖𝑔 𝑦 𝜖𝑦𝑗

▪ Marginal rate of technical substitution: 𝑁𝑆𝑈𝑇𝑗𝑘 =

𝑁𝑄𝑗 𝑁𝑄𝑘

(rate at which 𝑦𝑘 must be reduced when one extra unit of 𝑦𝑗 is used so that output remains constant) ▪ Output elasticity: 𝜁𝑗 =

%∆𝑟 %∆𝑦𝑗 ≅ 𝜖𝑔 𝑦 𝜖𝑦𝑗 𝑦𝑗 𝑟

▪ Returns to scale (𝜚 > 1):

– Decreasing: 𝑔 𝜚𝑦 < 𝜚𝑔 𝑦 – Constant: 𝑔 𝜚𝑦 = 𝜚𝑔 𝑦 – Increasing: 𝑔 𝜚𝑦 > 𝜚𝑔 𝑦

1.3. Production and Cost Functions Quantities of Interest

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Cost function:

Average ▪ costs: 𝐵𝐷 =

𝑑 𝑟,𝑥 𝑟

Marginal ▪ costs: 𝑁𝐷 =

𝜖𝑑 𝑟,𝑥 𝜖𝑟

Cost ▪ elasticity / Economies of scale: 𝜁𝑑 = %∆𝐷 %∆𝑟 ≅ 𝜖𝑑 𝑟, 𝑥 𝜖𝑟 𝑟 𝑑 𝑟, 𝑥 = 𝑁𝐷 𝐵𝐷 = 1 𝐹𝑇

– 𝜁𝑑 > 1 / 𝐹𝑇 < 1 ⇒ Diseconomies of scale – 𝜁𝑑 = 1 / 𝐹𝑇 = 1 ⇒ Constant returns to scale – 𝜁𝑑 < 1 / 𝐹𝑇 > 1 ⇒ Economies of scale

Revenue and profit functions:

Less ▪ used in applied economics Maximizing ▪ revenue mirrors the problem of minimizing cost

1.3. Production and Cost Functions Quantities of Interest

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Functional forms:

▪ Similar to those used with demand functions, with the obvious adaptations in terms of variables ▪ To measure technological changes, include a time trend in the model ▪ Main restrictions:

– Production, cost, revenue and profit are nonnegative functions – Production (in 𝑦), cost (in 𝑥 and 𝑟) and revenue (in 𝑞 and 𝑦) are nondecreasing functions – Cost (in 𝑥), revenue (in 𝑞) and profit (in 𝑥 and 𝑞) are homogeneous functions of degree 1

Typically, due to endogeneity of the inputs, the production function is not estimated directly; instead, a cost function is estimated and then the parameters of the production function are recovered

1.3. Production and Cost Functions Econometrics

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Measurement of variables:

▪ Output:

– Physical units – (Added / Gross) monetary values

▪ Inputs:

– Physical units (e.g. number of employees, labour hours employed per year, number of tractors) – Monetary values (e.g. capital stock)

▪ In time series, monetary values are deflated for price changes

1.3. Production and Cost Functions Econometrics

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Joaquim J.S. Ramalho

The theory of the firm assumes that all firms are efficient:

The production function is based on ▪ engineering (not economic) considerations, giving the maximum attainable output for alternative combinations of inputs – actually, this is a production frontier and all firms are assumed to produce on that frontier The ▪ cost function gives the minimum cost of producing a given level of

• utput for a given set of input prices – actually, this is a cost frontier

and all firms are assumed to be on that frontier Similarly ▪ , one may have profit and revenue frontiers

1.4. Efficiency and Stochastic Frontier Models

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In practice, focussing on costs, observed cost may differ from minimum cost due to:

▪ Random shocks (𝑤), which are not under the control of firms ▪ Inefficiency (𝑣), which is related to the quality of the firm’s production factors and their utilization: two firms using the same levels of inputs may produce different levels of outputs

Cost frontier model:

▪ Deterministic cost frontier: 𝐸𝐺 = 𝑑 𝑟, 𝑥 ▪ Stochastic cost frontier (incorporates random shocks): 𝑇𝐷𝐺 = 𝑑 𝑟, 𝑥, 𝑤 ▪ Observed cost (incorporates also inefficiency): 𝑑 = 𝑑 𝑟, 𝑥, 𝑤, 𝑣

𝑑 𝑟, 𝑥, 𝑤 may be above or below 𝑑 𝑟, 𝑥 but in all cases 𝑑 𝑟, 𝑥, 𝑤, 𝑣 ≥ 𝑑 𝑟, 𝑥, 𝑤 :

▪ 𝑑 𝑟, 𝑥, 𝑤, 𝑣 = 𝑑 𝑟, 𝑥, 𝑤 : efficient firm ▪ 𝑑 𝑟, 𝑥, 𝑤, 𝑣 > 𝑑 𝑟, 𝑥, 𝑤 : inefficient firm

1.4. Efficiency and Stochastic Frontier Models Cost Frontier

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Joaquim J.S. Ramalho

1.4. Efficiency and Stochastic Frontier Models Cost Frontier

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y c Deterministic frontier

• ▪

▪

Random shock effect Random shock effect Inefficiency effect Inefficiency effect

The random shock effect may be positive or negative The inefficiency effect is always nonnegative

𝑧𝐵 𝑧𝐶 𝑑𝐶 𝑑𝐵

Joaquim J.S. Ramalho

Typical formulation for the cost function:

▪ Stochastic cost funtion: 𝑇𝐷𝐺 = 𝑑 𝑟, 𝑥 ∗ exp 𝑤 ▪ Observed cost function: 𝑑 = 𝑇𝐷𝐺 ∗ exp 𝑣 where:

– 𝑤 is a symmetrical error term – 𝑣 ≥ 0 is an asymmetric, positively skewed error term that represents firm- specific cost inefficiency

Cost efficiency: 𝐷𝐹 =

𝑇𝐷𝐺 𝑑 = exp −𝑣 , 0 ≤ 𝐷𝐹 ≤ 1

• r 𝐷𝐹∗ =

𝑑 𝑇𝐷𝐺 = exp 𝑣 , 1 ≤ 𝐷𝐹∗ ≤ +∞ (Stata)

▪ 𝐷𝐹 or 𝐷𝐹∗ = 1 → the firm is cost efficient, operating on the 𝑇𝐷𝐺 and producing at the minimum cost ▪ 𝐷𝐹 or 𝐷𝐹∗ ≠ 1 → the firm is cost inefficient, operating above the 𝑇𝐷𝐺 and not producing as efficiently as it might

1.4. Efficiency and Stochastic Frontier Models Cost Frontier

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Joaquim J.S. Ramalho

Linearized model: ln 𝑑 = ln 𝑑 𝑟, 𝑥 + 𝑤 + 𝑣 Main assumptions:

▪ 𝑤, 𝑣 uncorrelated ▪ 𝐹 𝑤 = 0 ▪ 𝐹 𝑣 ≠ 0 ▪ Homoskedasticity: 𝑊𝑏𝑠 𝑤 = 𝜏𝑤

2, 𝑊𝑏𝑠 𝑣 = 𝜏𝑣 2

Properties:

▪ Estimators for the 𝛾’s associated to the explanatory variables consistent ▪ Estimator for 𝛾0 inconsistent (it is consistent for 𝛾0 + 𝐹 𝑣 ) ▪ Impossible to estimate 𝐷𝐹 or 𝐷𝐹∗, since the error terms cannot be separated

1.4. Efficiency and Stochastic Frontier Models OLS Estimation

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Joaquim J.S. Ramalho

Further (distributional) assumptions:

▪ 𝑤~𝒪 0, 𝜏𝑤

2

▪ Regarding 𝑣:

– 𝑣 = 𝑉 and 𝑉~𝒪 0, 𝜏𝑣

2 ⟶ Half-normal distribution

– 𝑣~𝐹𝑦𝑞𝑝𝑜𝑓𝑜𝑢𝑗𝑏𝑚 𝜏𝑣 ⟶ Exponential distribution – 𝑣 = 𝑉 and 𝑉~𝒪 𝜈, 𝜏𝑣

2 ⟶ Truncated normal distribution (𝜈 may be

explained by a set of variables) – Others

According to the distribution assumed for 𝑣, different the resultant expressions for 𝐹 exp −𝑣 |𝑤, 𝑣

• r 𝐹 exp 𝑣 |𝑤, 𝑣 ,

which are then used for obtaining estimates for 𝐷𝐹 or 𝐷𝐹∗

1.4. Efficiency and Stochastic Frontier Models Maximum Likelihood Estimation

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Stata frontier Y 𝑌1 … 𝑌𝑙, cost distribution(hnormal) frontier Y 𝑌1 … 𝑌𝑙, cost distribution(exponential) frontier Y 𝑌1 … 𝑌𝑙, cost distribution(tnormal) frontier Y 𝑌1 … 𝑌𝑙, cost distribution(tnormal) cm(𝑋

1 … 𝑋 𝑘)

Joaquim J.S. Ramalho

Parameters to be estimated:

▪ 𝛾, 𝜏𝑤

2, 𝜏𝑣 2

Other quantities:

▪ 𝜏2 = 𝜏𝑤

2 + 𝜏𝑣 2

▪ 𝜇 =

𝜏𝑣

2

𝜏𝑤

2

▪ 𝛿 =

𝜏𝑣

2

𝜏2

Testing for inefficient effects:

▪ 𝐼0: 𝜇 = 0 versus 𝐼0: 𝜇 > 0 or ▪ 𝐼0: 𝛿 = 0 versus 𝐼0: 𝛿 > 0

1.4. Efficiency and Stochastic Frontier Models Maximum Likelihood Estimation

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Joaquim J.S. Ramalho

Estimates for costs (needed for computing marginal costs, average costs, economies of scale, etc.):

𝑑 = 𝑑 𝑟, 𝑥 ∗ exp 𝑤 ∗ exp 𝑣 Ƹ 𝑑 = Ƹ 𝑑 𝑟, 𝑥 ∗ ෣ exp 𝑣

• r

Ƹ 𝑑 = Ƹ 𝑑 𝑟, 𝑥 ∗ ෢ 𝐷𝐹∗

• r

Ƹ 𝑑 = Ƹ 𝑑 𝑟, 𝑥 ෢ 𝐷𝐹

1.4. Efficiency and Stochastic Frontier Models Maximum Likelihood Estimation

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