Simultaneous Equations Models

Nội dung

Section 1: What are Simultaneous Equations Models (SEM)?

• Definition of Simultaneous Equations Models

  • Overview of SEM: Systems of equations where endogenous variables are determined simultaneously.

  • Key distinction: SEM vs. single-equation models.

  • Examples of SEM in practice: Supply and demand models, macroeconomic policy models.

• Simultaneity and Endogeneity

  • Definition of simultaneity: Situations where one variable affects another, and vice versa.

  • Endogeneity in SEM: How endogenous variables are correlated with the error term.

  • Examples of simultaneity in economics: Price and quantity determination in the market.

• The Importance of SEM in Econometrics

  • Why SEM is used: Addressing simultaneity bias and recursive relationships.

  • Applications of SEM in real-world settings: Labor supply and demand, consumption and investment models, financial markets.

Section 2: Structural Form and Reduced Form in SEM

• Structural Form of SEM

  • What is the structural form? Representation of relationships between endogenous variables and exogenous variables.

  • Example: Structural equations in a supply-demand system.

• Reduced Form of SEM

  • Definition of reduced form: Expressing endogenous variables as a function of only exogenous variables.

  • Converting structural equations into reduced form equations.

  • Example: Reduced form of a simultaneous demand and supply system.

• Comparison Between Structural Form and Reduced Form

  • Role of structural form in representing economic theory.

  • Role of reduced form in prediction and policy analysis.

  • How to interpret reduced form coefficients.

Section 3: Simultaneity Bias and Endogeneity

• Understanding Simultaneity Bias

  • Definition of simultaneity bias: How the correlation between endogenous variables and error terms leads to bias in Ordinary Least Squares (OLS) estimation.

  • Illustration of bias using a simple two-equation model (e.g., supply and demand).

• Endogeneity in SEM

  • Sources of endogeneity: Omitted variables, measurement error, simultaneity.

  • How endogeneity impacts the consistency of OLS estimates in SEM.

• The Need for Special Estimation Methods

  • Why OLS is inappropriate for SEM.

  • Motivation for using alternative estimation methods: Instrumental variables (IV), Two-Stage Least Squares (2SLS), and other methods.

Nội dung

Section 1: Introduction to Identification in Simultaneous Equations Models

• What is Identification?

  • Definition of identification: The ability to uniquely estimate the parameters of the structural model.

  • Importance of identification in SEM: Ensuring the model is well-defined and estimable.

  • Difference between identified, under-identified, and over-identified models.

  • Consequences of identification issues: Bias, inconsistency, and inability to make valid inferences.

  • How identification differs between single-equation models and simultaneous equations models.

• Reduced Form and Its Implications for Identification

• Definition and derivation of the reduced form from the structural form.

• How the reduced form helps express endogenous variables as functions of only exogenous variables.

• Example: Reduced form equations for a basic SEM.

Section 2: The Order Condition for Identification

• Order Condition: A Necessary but Not Sufficient Condition

  • Explanation of the order condition for identification.

  • Formula: The number of excluded exogenous variables must be at least equal to the number of endogenous variables minus one.

  • Example: Applying the order condition to a system of two equations (e.g., supply and demand).

• Examples of Order Condition in Simple SEM

  • Case of just-identified, over-identified, and under-identified models.

  • Practical illustration using a two-equation model (e.g., investment and GDP).

Section 3: The Rank Condition for Identification

• Rank Condition: A Necessary and Sufficient Condition

  • Definition and importance of the rank condition.

  • Rank condition in matrix terms: The matrix of excluded instruments must have full rank.

• Application of the Rank Condition

  • Step-by-step process to check the rank condition.

  • Examples: Testing the rank condition for both just-identified and over-identified models.

• Practical Implications of the Rank Condition

  • The role of instrumental variables in satisfying the rank condition.

  • How failure to meet the rank condition leads to identification problems.

Section 4: Identified, Under-Identified, and Over-Identified Models

• Identified Models

  • Definition of just-identified models: When the number of excluded exogenous variables satisfies both the order and rank conditions.

  • Example: Identified model in a simple macroeconomic system (e.g., income determination).

• Under-Identified Models

  • Definition and consequences of under-identification: Not enough information to uniquely estimate structural parameters.

  • Examples of under-identified models: Failure to exclude enough exogenous variables.

• Over-Identified Models

  • Definition of over-identification: More excluded exogenous variables than required for identification.

  • Implications for estimation: Testing over-identifying restrictions.

  • Example: Over-identified model in a monetary policy system (e.g., interest rate targeting and inflation control).

Section 5: Testing for Identification

• Tests for Over-Identification

  • Sargan-Hansen test for over-identifying restrictions.

  • Practical steps for applying the test in SEM.

• Testing for Rank Condition

  • Methods for testing the rank condition in practice.

  • How to interpret test results for identifying whether the system is under-identified, just-identified, or over-identified.

Section 6: The Role of Instruments in Identification

• Definition and Purpose of Instrumental Variables

  • What makes a valid instrument: Exogeneity and relevance.

  • Role of instruments in satisfying identification conditions.

• Choosing and Testing Instruments for Identification

  • Guidelines for selecting appropriate instruments.

  • Tests for instrument validity: Relevance (F-statistics) and exogeneity (Sargan test).

Sách

• Wooldridge, J. M. (2010). Econometric analysis of cross section and panel data. MIT press. 

• Hsiao, C. (2022). Analysis of panel data (No. 64). Cambridge university press. 

• Baltagi, B. H., & Baltagi, B. H. (2008). Econometric analysis of panel data (Vol. 4, pp. 135-145). Chichester: Wiley. 

• Cameron, A. C., & Trivedi, P. K. (2010). Microeconometrics using stata (Vol. 2). College Station, TX: Stata press. 

• Greene, W. (2012) Econometric Analysis. 7th Edition 

• Schmidt, P. (1976). Econometrics Marcel Dekker. New York. 

• Davidson, R., & MacKinnon, J. G. (1993). Estimation and inference in econometrics (Vol. 63). New York: Oxford. 

Bài báo

• Klein, L. R. (1950). Economic fluctuations in the United States, 1921-1941.

Nội dung

Section 1: Introduction to Estimation in Simultaneous Equations Models

• • Why standard Ordinary Least Squares (OLS) is inappropriate in SEM due to endogeneity and simultaneity bias.

  • The need for specialized estimation techniques to account for simultaneity.

• Role of Identification in Estimation

  • Importance of identification in SEM estimation.

  • Brief review of order and rank conditions as prerequisites for estimation.

• Estimation Methods Overview

  • Summary of the key estimation techniques for SEM: Instrumental Variables (IV), Two-Stage Least Squares (2SLS), Three-Stage Least Squares (3SLS), Generalized Method of Moments (GMM), and Maximum Likelihood (ML) methods.

Section 2: Ordinary Least Squares (OLS) and Its Limitations in SEM

• OLS in the Context of SEM

  • Why OLS leads to biased and inconsistent estimates in simultaneous equations models.

• Simultaneity Bias

  • Illustration of simultaneity bias using an example of supply and demand estimation.

  • Mathematical demonstration of why OLS fails when endogenous variables are correlated with error terms.

• Practical Example of OLS Failures in SEM

  • Case study: Estimation of supply and demand using OLS and the resulting bias.

Section 3: Instrumental Variables (IV) Estimation

• The Role of Instrumental Variables in SEM

  • Definition of instrumental variables (IV): How they solve the endogeneity problem in SEM.

  • Conditions for valid instruments: Relevance and exogeneity.

• Estimating SEM with IV

  • Steps for implementing IV estimation in SEM.

  • Example: Using IV to estimate a supply-demand system with price as an endogenous variable.

• Testing Instrument Validity

  • Methods for testing the relevance (F-statistics) and exogeneity (Sargan test) of instruments.

  • Practical example: Applying validity tests to a real-world SEM model.

Section 4: Two-Stage Least Squares (2SLS)

• What is Two-Stage Least Squares (2SLS)?

  • Overview of the 2SLS method: A solution to endogeneity in SEM.

  • How 2SLS works: First stage (predicting endogenous variables using instruments) and second stage (estimating the structural equation).

• Step-by-Step Procedure for 2SLS

  • First stage: Regressing endogenous variables on exogenous variables and instruments.

  • Second stage: Estimating the structural equation using predicted values from the first stage.

• Examples of 2SLS Estimation

  • Case study: Applying 2SLS to estimate the effect of education on wages (labor supply and demand model).

• Properties of the 2SLS Estimator

  • Consistency, asymptotic normality, and efficiency of the 2SLS estimator.

• Limitations of 2SLS

  • Issues such as weak instruments, small sample bias, and inefficient estimates.

Section 5: Three-Stage Least Squares (3SLS)

• Introduction to Three-Stage Least Squares (3SLS)

  • Definition of 3SLS: Simultaneous estimation of all equations in a system, accounting for cross-equation correlations.

  • The advantage of 3SLS over 2SLS: Efficiency gains from joint estimation and accounting for error term correlations.

• Steps in 3SLS Estimation

  • First stage: Estimating reduced form equations using instrumental variables.

  • Second stage: Computing the covariance matrix of residuals.

  • Third stage: Generalized least squares (GLS) estimation using the covariance matrix.

• Example of 3SLS Estimation

  • Case study: Applying 3SLS to estimate a system of equations for monetary policy (interest rate, inflation, and output).

• Properties of the 3SLS Estimator

  • Efficiency of 3SLS compared to 2SLS.

  • When to use 3SLS: Advantages and potential limitations.

Section 6: Limited Information Maximum Likelihood (LIML)

• Overview of LIML Estimation

  • What is LIML? Estimation method for single equation systems within SEM using a likelihood approach.

  • How LIML differs from 2SLS: Its ability to handle small sample bias better than 2SLS.

• Theoretical Framework of LIML

  • How to derive the LIML estimator.

• Example of LIML Estimation

  • Applying LIML to a supply and demand system: Comparison with 2SLS results.

• Advantages and Limitations of LIML

  • When to prefer LIML over 2SLS: Small sample efficiency.

  • Limitations: Computational complexity and availability of alternative estimators.

Section 7: Full Information Maximum Likelihood (FIML)

• Introduction to Full Information Maximum Likelihood (FIML)

  • What is FIML? Estimation method that uses the entire system of equations in SEM simultaneously.

  • Difference between FIML and LIML: FIML uses information from all equations in the system.

• FIML Estimation Process

  • Estimating structural equations by maximizing the likelihood function of the entire system.

• Example of FIML Estimation

  • Case study: Applying FIML to a multi-equation system for economic growth and inflation.

• Properties of the FIML Estimator

  • Efficiency of FIML compared to 2SLS and 3SLS.

  • FIML as a small-sample efficient estimator.

• Limitations of FIML

  • Computational challenges: FIML can be complex and may not converge in certain cases.

Section 8: Generalized Method of Moments (GMM) in SEM

• What is Generalized Method of Moments (GMM)?

  • Introduction to GMM: A flexible estimation technique that generalizes the method of moments for SEM.

  • Why use GMM in SEM? Advantages in handling heteroskedasticity and instrument proliferation.

• GMM Estimation in SEM

  • How GMM works: Using moment conditions to estimate parameters.

• GMM Example in SEM

  • Case study: Estimating a system of equations for firm investment and output using GMM.

• Properties of GMM Estimators

  • Consistency and efficiency of GMM estimators under different assumptions.

• Practical Challenges in GMM

  • Weak instruments, overfitting with too many moment conditions, and the curse of dimensionality.

Section 9: Choosing the Right Estimation Technique for SEM

• Criteria for Choosing Estimation Methods

  • When to use IV, 2SLS, 3SLS, LIML, FIML, or GMM.

  • Practical considerations: Sample size, instrument quality, error correlations, and computational complexity.

• Trade-offs Between Efficiency and Complexity

  • Balancing efficiency (FIML, 3SLS) with simplicity and computational feasibility (2SLS, LIML).

Section 10: Practical Applications of SEM Estimation Techniques

• Estimating Demand and Supply Systems

  • Example: Using 2SLS and 3SLS to estimate a simultaneous supply-demand system.

• Policy Evaluation Using SEM

  • Example: Estimating the effects of fiscal and monetary policies using FIML and GMM.

• Estimating Macroeconomic Models

  • Example: Joint estimation of inflation, output, and unemployment using 3SLS or FIML.

Section 11: Software Implementation of SEM Estimation Techniques

• Implementing 2SLS and 3SLS in Stata

  • Stata commands: ivregress, reg3, and sem for 2SLS and 3SLS estimation.

• LIML and FIML in R

  • Using systemfit package in R for LIML and FIML estimation.

• GMM Implementation in Stata and MATLAB

  • GMM estimation using Stata (gmm) and MATLAB (gmmest).

Section 12: Conclusion and Key Takeaways

• Summary of Estimation Techniques

  • Recap of key estimation methods: IV, 2SLS, 3SLS, LIML, FIML, and GMM.

• Best Practices for Estimation in SEM

  • Guidelines for choosing the appropriate estimation technique based on the problem at hand.

• Future Directions in SEM Estimation

  • Emerging techniques and computational advancements in the field of SEM estimation.

Sách

• Greene, W. (2012) Econometric Analysis. 7th Edition 

• Davidson, R., & MacKinnon, J. G. (1993). Estimation and inference in econometrics (Vol. 63). New York: Oxford. 

• Klein, L. R. (1950). Economic fluctuations in the United States, 1921-1941.

• Baldwin, S. A. (2019). Psychological statistics and psychometrics using Stata (p. 454). College Station, TX: Stata Press.

Bài báo

• Zellner, A. (1962). Three stage least squares: simultaneous estimation of simultaneous equtions. Econometrica, 30, 63-68.

• Iwasaki, I., Ma, X., & Mizobata, S. (2024). Board structure in emerging markets: A simultaneous equation modeling. Journal of Economics and Business, 128, 106160.

• Salvatore, D. (2023). A simultaneous equations model of the relationship between international trade, and economic growth and development with dynamic policy simulations. Journal of Policy Modeling, 45(4), 789-805.

Nội dung

Section 1: Introduction to Hypothesis Testing in SEM

• • Why hypothesis testing is crucial in SEM: Validating model assumptions and testing relationships between variables.

  • Types of hypotheses in SEM: Testing for structural relationships, over-identification, endogeneity, and more.

    1. 1. 1. 1. • Testing the exclusion of variables (zero restrictions).

                • Testing for structural parameter constraints across equations.

                • Tests related to endogeneity, over-identification, and validity of instruments.

Section 2: Testing for Endogeneity in SEM

• 2.2 Hausman Test for Endogeneity

  • Overview of the Hausman test: Comparing OLS and IV/2SLS estimators to detect endogeneity.

  • Steps to perform the Hausman test in SEM.

  • Interpretation of results: Accepting or rejecting the null hypothesis of exogeneity.

• 2.3 Practical Example of the Hausman Test

  • Case study: Testing for endogeneity in a simultaneous system of demand and supply.

Section 3: Over-Identification and Testing the Validity of Instruments

• Over-Identification in SEM

  • Definition of over-identification: More instruments than necessary for identification.

  • The importance of testing over-identifying restrictions to ensure valid estimation.

• Sargan Test for Over-Identifying Restrictions

  • Overview of the Sargan test: A test of the validity of the over-identifying restrictions.

  • Steps to implement the Sargan test after IV/2SLS estimation.

  • Interpretation of results: Whether the instruments are valid or invalid.

• Hansen’s J-Test for Over-Identification

  • Introduction to Hansen's J-test: A robust version of the Sargan test for over-identifying restrictions.

  • Practical implementation of Hansen’s J-test in the presence of heteroskedasticity.

• Practical Example of Sargan and Hansen Tests

  • Case study: Testing for the validity of instruments in an investment and output model.

Section 4: Hypothesis Testing for Cross-Equation Restrictions

• Testing Cross-Equation Parameter Constraints

  • Definition of cross-equation restrictions: When parameters are expected to be the same across different equations.

  • Examples of cross-equation constraints in SEM: Testing if coefficients are equal across equations (e.g., income elasticity across demand and supply equations).

• Wald Test for Cross-Equation Restrictions

  • Overview of the Wald test: Testing joint hypotheses about parameter restrictions across equations.

  • Steps to implement the Wald test in SEM: Formulating the null and alternative hypotheses.

  • Interpreting Wald test results: Whether to reject or fail to reject cross-equation restrictions.

• Likelihood Ratio Test for Cross-Equation Constraints

  • Using the Likelihood Ratio (LR) test to evaluate restrictions across equations.

  • Steps for implementing the LR test: Estimating the unrestricted and restricted models.

• Lagrange Multiplier (LM) Test

  • Application of the Lagrange Multiplier (LM) test to test for cross-equation restrictions in SEM.

  • Practical advantages and limitations of the LM test in SEM.

• Example of Cross-Equation Restriction Testing

  • Case study: Testing for parameter equality in a labor supply and demand system.

Section 5: Testing for Exclusion Restrictions

• Exclusion Restrictions in SEM

  • What are exclusion restrictions? Hypotheses about whether certain variables should be excluded from one or more equations.

  • The importance of exclusion restrictions for identification and model specification.

• Wald Test for Exclusion Restrictions

  • Implementing the Wald test to check if certain variables should be excluded from the SEM.

  • Example: Testing whether interest rates should be excluded from a consumption equation.

• Likelihood Ratio Test for Exclusion Restrictions

  • Likelihood Ratio test as a means to test exclusion restrictions: Comparing restricted and unrestricted models.

• Example of Exclusion Restriction Testing

  • Case study: Testing exclusion restrictions in a model of inflation and unemployment.

Section 6: Testing for Structural Breaks in SEM

• Structural Breaks in SEM

  • Definition and examples of structural breaks: Policy changes, economic shocks, or regime changes that may alter the relationships between variables.

• Chow Test for Structural Breaks

  • Overview of the Chow test: A test to detect structural breaks at known breakpoints.

  • Steps to implement the Chow test: Splitting the data and testing for parameter stability across sub-samples.

• CUSUM and CUSUMSQ Tests for Parameter Stability

  • Application of CUSUM (Cumulative Sum) and CUSUMSQ (Cumulative Sum of Squares) tests to detect parameter instability over time.

• Example of Structural Break Testing

  • Case study: Testing for structural breaks in a model of monetary policy before and after a financial crisis.

Section 7: Testing for Heteroskedasticity and Serial Correlation in SEM

• Heteroskedasticity in SEM

  • The importance of detecting heteroskedasticity in SEM: Potential bias in standard errors and hypothesis tests.

• Breusch-Pagan Test for Heteroskedasticity

  • Implementing the Breusch-Pagan test to check for heteroskedasticity in the SEM.

• Testing for Serial Correlation

  • Durbin-Watson and Breusch-Godfrey tests for serial correlation in SEM.

  • Importance of detecting and addressing serial correlation in time series SEM.

• Example of Testing for Heteroskedasticity and Serial Correlation

  • Case study: Testing for heteroskedasticity and serial correlation in a system of macroeconomic equations.

Section 8: Goodness-of-Fit Tests in SEM

• Measuring the Overall Fit of SEM

  • Assessing the goodness-of-fit in SEM models: R-squared, adjusted R-squared, and root mean squared error (RMSE).

• Chi-Square Test for Model Fit

  • Application of the chi-square test for evaluating the overall fit of SEM models.

• Example of Goodness-of-Fit Testing

  • Case study: Evaluating the fit of a SEM model of economic growth and trade.

Section 9: Software Implementation of Hypothesis Testing in SEM

• Hypothesis Testing in Stata

• Hypothesis Testing in R

• Hypothesis Testing in EViews

Section 10: Conclusion and Key Takeaways

• Recap of Hypothesis Testing Methods in SEM

  • Summary of key hypothesis testing methods: Endogeneity, over-identification, exclusion restrictions, cross-equation restrictions, and structural breaks.

• Importance of Rigorous Hypothesis Testing in SEM

  • Why hypothesis testing is essential for validating SEM models and ensuring robust estimation.

• Looking Ahead to Diagnostic Testing

  • Transition to the next chapter on diagnostic testing and model validation in SEM.

Sách

• Greene, W. (2012) Econometric Analysis. 7th Edition 

• Schmidt, P. (1976). Econometrics Marcel Dekker. New York. 

• Davidson, R., & MacKinnon, J. G. (1993). Estimation and inference in econometrics (Vol. 63). New York: Oxford. 

• Acock, A. C. (2013). Discovering structural equation modeling using Stata. Stata Press Books.

Bài báo

• Thomson, E., & Williams, R. (1982). Beyond wives' family sociology: A method for analyzing couple data. Journal of Marriage and the Family, 999-1008.

• Iwasaki, I., Ma, X., & Mizobata, S. (2024). Board structure in emerging markets: A simultaneous equation modeling. Journal of Economics and Business, 128, 106160.

• Salvatore, D. (2023). A simultaneous equations model of the relationship between international trade, and economic growth and development with dynamic policy simulations. Journal of Policy Modeling, 45(4), 789-805.

Nội dung

Section 1: Introduction to Recursive and Block-Recursive Systems

• What Are Recursive Systems?

  • Definition of recursive systems in SEM: Systems where each equation can be solved sequentially without feedback between equations.

  • Key characteristics: No simultaneity; one-way causality between variables.

  • Examples: Income determination models, production function models.

• What Are Block-Recursive Systems?

  • Definition of block-recursive systems: Systems of equations where blocks of equations can be estimated separately, with interactions only within blocks.

  • Example: A multi-sector economy with independent sectoral relationships within each block.

• Comparison Between General SEM, Recursive Systems, and Block-Recursive Systems

  • Differences in complexity, estimation techniques, and feedback structures.

  • When to use recursive and block-recursive systems over fully simultaneous models.

Section 2: Theoretical Foundations of Recursive Systems

• 2.1 Structure of Recursive Systems

  • General form of recursive systems.

  • Sequential relationship between endogenous variables: One-way causality from higher-order to lower-order equations.

• Identification in Recursive Systems

  • Why identification is easier in recursive systems: No simultaneity means that standard OLS can be applied.

  • Order of identification: Estimation proceeds step by step, solving each equation sequentially.

• Economic Applications of Recursive Systems

  • Examples of recursive models:

    • Keynesian income determination model.

    • Recursive models of consumption and savings behavior.

Section 3: Estimation Techniques for Recursive Systems

• Ordinary Least Squares (OLS) for Recursive Systems

  • Why OLS is consistent and efficient in recursive systems.

  • Steps for estimating recursive systems using OLS.

• Instrumental Variables (IV) Estimation in Recursive Systems

  • When IV is needed in recursive systems: Dealing with potential endogeneity of exogenous variables.

  • Example: Estimating a recursive consumption model with endogenous income.

• Maximum Likelihood Estimation (MLE) in Recursive Systems

  • Overview of MLE as an estimation technique for recursive systems.

  • When MLE offers advantages over OLS.

Section 4: Practical Examples of Recursive Systems

• Recursive Macroeconomic Models

  • Example: A recursive model of consumption, investment, and output determination.

• Recursive Labor Market Models

  • Example: A recursive labor supply and demand model where wage determination influences employment.

• Recursive Financial Models

  • Example: Recursive models of asset pricing and investment, where past returns influence current investment decisions.

Section 5: Theoretical Foundations of Block-Recursive Systems

• Structure of Block-Recursive Systems

  • General form of block-recursive systems: Systems of equations grouped into blocks, with equations within each block being interdependent, but no feedback between blocks.

  • Definition of blocks: A set of equations where endogenous variables interact within the block but are not influenced by other blocks.

• Identification in Block-Recursive Systems

  • Identification within blocks: How identification works within a block of equations.

  • Estimation within blocks vs. across blocks: Simpler estimation due to no feedback across blocks.

• Economic Applications of Block-Recursive Systems

  • Examples of block-recursive models:

    • Multi-sector models of the economy where different sectors (e.g., agriculture, industry) are modeled as separate blocks.

    • Block-recursive models of education and labor markets.

Section 6: Estimation Techniques for Block-Recursive Systems

• Estimating Each Block Independently

  • Step-by-step procedure for estimating each block of equations separately using OLS, IV, or 2SLS.

  • Efficiency of estimation within blocks compared to fully simultaneous systems.

• Generalized Least Squares (GLS) Estimation in Block-Recursive Systems

  • Using GLS to estimate block-recursive systems when error terms are correlated within blocks.

  • Example: Applying GLS to estimate the relationship between different sectors of the economy in a block-recursive system.

• Maximum Likelihood Estimation (MLE) in Block-Recursive Systems

  • MLE techniques for estimating block-recursive systems.

  • Advantages of MLE over OLS when error terms are correlated across blocks or when there are complex interactions within blocks.

Section 7: Practical Examples of Block-Recursive Systems

• Block-Recursive Macroeconomic Models

  • Example: Estimating a block-recursive model of the economy where production and consumption are modeled as separate blocks.

• Block-Recursive Sectoral Models

  • Example: A multi-sector model where agriculture, manufacturing, and services are modeled as separate blocks.

• Block-Recursive Financial Models

  • Example: Estimating a block-recursive system for different financial markets (e.g., equity market, bond market, and currency market).

Section 8: Hypothesis Testing in Recursive and Block-Recursive Systems

• Testing for Exogeneity in Recursive Systems

  • Using Hausman tests and other techniques to check for endogeneity in recursive systems.

• Testing Cross-Equation Restrictions in Block-Recursive Systems

  • Wald tests, Likelihood Ratio tests, and Lagrange Multiplier tests for testing cross-equation restrictions within blocks.

• Example of Hypothesis Testing in Recursive and Block-Recursive Models

  • Practical example: Testing parameter restrictions in a recursive model of consumption and savings.

Section 9: Recursive and Block-Recursive Systems with Time Series Data

• Dynamic Recursive Systems

  • Incorporating lagged variables into recursive systems to capture dynamic relationships.

  • Example: A dynamic recursive model of consumption and investment over time.

• Block-Recursive Time Series Models

  • Applying block-recursive systems to time series data: Estimating different blocks over time.

  • Example: A block-recursive system of economic sectors with time-varying effects.

• Testing for Stability and Structural Breaks in Recursive Systems

  • Using Chow tests, CUSUM tests, and CUSUMSQ tests to check for parameter stability and structural breaks in recursive and block-recursive systems.

Section 10: Software Implementation for Recursive and Block-Recursive Systems

• Estimating Recursive Systems in Stata

• Estimating Block-Recursive Systems in Stata

• Estimating Recursive and Block-Recursive Systems in R

Section 11: Applications and Extensions of Recursive and Block-Recursive Systems

• Recursive Systems in Policy Evaluation

  • Example: Using recursive systems to evaluate the impact of fiscal and monetary policy.

• Block-Recursive Systems in Development Economics

  • Example: Estimating a block-recursive system to study interactions between education, health, and labor markets.

• Extensions of Recursive and Block-Recursive Systems

  • Extensions to nonlinear models: Recursive systems with nonlinear relationships.

  • Example: Estimating a nonlinear recursive model in environmental economics.

Section 12: Conclusion and Key Takeaways

• Summary of Key Concepts in Recursive and Block-Recursive Systems

  • Recap of the benefits and limitations of recursive and block-recursive systems compared to general SEM.

• Best Practices for Estimation and Model Specification

  • Guidelines for specifying and estimating recursive and block-recursive systems in practice.

• Future Directions in Recursive and Block-Recursive Systems

  • Discussion of emerging trends and advanced applications in recursive and block-recursive SEM.

Sách

• Wooldridge, J. M. (2010). Econometric analysis of cross section and panel data. MIT press. 

• Hsiao, C. (2022). Analysis of panel data (No. 64). Cambridge university press. 

• Baltagi, B. H., & Baltagi, B. H. (2008). Econometric analysis of panel data (Vol. 4, pp. 135-145). Chichester: Wiley. 

• Cameron, A. C., & Trivedi, P. K. (2010). Microeconometrics using stata (Vol. 2). College Station, TX: Stata press. 

• Greene, W. (2012) Econometric Analysis. 7th Edition 

• Schmidt, P. (1976). Econometrics Marcel Dekker. New York. 

• Davidson, R., & MacKinnon, J. G. (1993). Estimation and inference in econometrics (Vol. 63). New York: Oxford. 

Bài báo

Nội dung

Section 1: Introduction to Generalized Structural Equation Models (GSEM)

• • Definition and overview of GSEM: A flexible extension of structural equation models (SEM) that accommodates non-normal distributions, categorical variables, and more complex relationships.

  • Key differences between SEM and GSEM: GSEM as a more general framework that allows for non-Gaussian distributions and mixed models.

  • Motivation for GSEM: When traditional SEM assumptions of normality and linearity are violated.

  • Practical applications of GSEM: Educational testing, healthcare, latent class analysis, and finance.

  • Examples of GSEM in economics, social sciences, and other disciplines: Modeling categorical and count data, and handling latent variables.

Section 2: Theoretical Foundations of GSEM

• Generalized Linear Models (GLM) and GSEM

  • Review of GLM: Framework for modeling non-normal distributions (e.g., logistic regression, Poisson regression).

  • Extension of GLM to GSEM: Allowing for latent variables and simultaneous equations in models with non-normal data.

• Mixed Effects Models and GSEM

  • Incorporating mixed models into GSEM: Fixed and random effects in a structural equation framework.

  • Examples: Random intercepts and slopes in hierarchical data (e.g., panel data or longitudinal studies).

• Latent Variables and GSEM

  • Incorporating latent variables in GSEM: Modeling unobserved constructs using indicators.

  • Applications: Latent traits in psychology, risk preferences in finance.

Section 3: Model Specification in GSEM

• Structural and Measurement Equations in GSEM

  • Defining structural and measurement equations in GSEM.

  • Specifying relationships between endogenous and exogenous variables in generalized models.

• Nonlinear and Non-Normal Models

  • Handling nonlinear relationships in GSEM: Modeling interactions and quadratic terms.

  • Example: Nonlinear effects of income on health outcomes using GSEM.

• Model Setup for Categorical and Count Data

  • How to specify models with categorical, binary, and count data in GSEM.

  • Example: A GSEM model with logistic regression for binary outcomes and Poisson regression for count data.

Section 4: Estimation Techniques for GSEM

• Maximum Likelihood Estimation (MLE) in GSEM

  • Overview of MLE for GSEM: Estimating parameters in models with non-normal data.

  • Challenges and advantages of MLE in GSEM: Dealing with categorical variables, non-normality, and mixed models.

• Quasi-Maximum Likelihood Estimation (QMLE)

  • Introduction to QMLE: A flexible estimation technique when traditional likelihood assumptions do not hold.

  • Example: Applying QMLE to a GSEM with count and categorical variables.

• Bayesian Estimation in GSEM

  • Overview of Bayesian estimation for GSEM: Incorporating prior distributions and obtaining posterior estimates.

  • Applications of Bayesian methods: Handling complex models, missing data, and latent variables in GSEM.

Section 5: Goodness-of-Fit and Model Diagnostics in GSEM

• Assessing Goodness-of-Fit in GSEM

  • Criteria for model fit: Likelihood ratio tests, Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC).

  • Assessing fit for models with non-normal data: Adjusted chi-square tests, pseudo R-squared, and deviance statistics.

• Residual Diagnostics in GSEM

  • Analyzing residuals for model adequacy: Checking for overdispersion, misspecification, and non-linearity.

  • Example: Residual analysis in a Poisson-based GSEM for count data.

• Testing for Model Misspecification

  • Testing hypotheses in GSEM: Wald tests, Likelihood Ratio tests, and Lagrange Multiplier tests.

  • How to handle potential model misspecifications in GSEM: Detecting omitted variables and incorrect functional forms.

Section 6: Practical Examples of GSEM

• GSEM for Binary Data

  • Example: Logistic regression in a GSEM framework for modeling binary outcomes, such as the probability of a household defaulting on a loan.

• GSEM for Count Data

  • Example: Using Poisson regression within a GSEM to model the number of doctor visits as a function of socioeconomic factors.

• GSEM with Latent Variables

  • Example: Estimating a GSEM with latent variables for consumer satisfaction using multiple indicators and categorical outcomes.

Section 7: Advanced Topics in GSEM

• Multilevel and Hierarchical GSEM

  • Incorporating multilevel data into GSEM: Handling clustered data with random effects.

  • Example: A hierarchical GSEM for modeling school performance across different regions.

• Handling Missing Data in GSEM

  • Methods for handling missing data in GSEM: Full Information Maximum Likelihood (FIML), multiple imputation, and Bayesian methods.

  • Example: A GSEM model for health outcomes with missing covariate data.

• Longitudinal Data in GSEM

  • Extending GSEM to model longitudinal data: Accounting for time-varying covariates and outcomes.

  • Example: GSEM applied to panel data of labor market outcomes over time.

• GSEM for Complex Survey Data

  • Accounting for survey weights and design effects in GSEM.

  • Example: GSEM with stratified survey data on household income and spending behavior.

Section 8: Hypothesis Testing in GSEM

• Testing Parameter Constraints

  • Wald tests and Likelihood Ratio tests for testing constraints on parameters in GSEM.

  • Example: Testing for the equality of effects across groups in a GSEM.

• Cross-Equation Hypothesis Testing

  • Testing cross-equation constraints in GSEM: Applying joint hypothesis tests to multiple equations.

  • Example: Testing whether income affects health outcomes and education simultaneously in a GSEM.

• Testing for Interaction Effects

  • Specifying and testing for interaction terms in GSEM: Understanding nonlinear relationships between variables.

  • Example: Interaction between education and income in predicting health status.

Section 9: Software Implementation of GSEM

• Implementing GSEM in Stata

  • Stata commands for estimating GSEM models: Using the gsem command.

  • Example: Step-by-step guide to estimating a GSEM with logistic and Poisson models in Stata.

• GSEM in R

  • Using R packages (lavaan, MplusAutomation, brms) to estimate GSEM.

  • Example: Implementing a GSEM for longitudinal data using lavaan in R.

Section 10: Applications of GSEM in Different Fields

• GSEM in Economics

  • Example: Estimating a GSEM for household consumption decisions with categorical outcomes and latent variables.

• GSEM in Education

  • Example: Modeling student performance with multilevel data and latent traits using GSEM.

• GSEM in Health and Social Sciences

  • Example: Using GSEM to model health outcomes with latent health status indicators and count data on medical visits.

Section 11: Conclusion and Key Takeaways

• Recap of GSEM Key Concepts

  • Summary of GSEM advantages: Flexibility in handling categorical data, mixed models, and latent variables.

• Best Practices for GSEM Estimation and Interpretation

  • Practical guidelines for specifying, estimating, and interpreting GSEM models.

• Future Directions in GSEM Research

  • Emerging trends: GSEM with machine learning, deep learning extensions, and applications in big data contexts.

Sách

• Rabe-Hesketh, S., & Skrondal, A. (2008). Multilevel and longitudinal modeling using Stata. STATA press.

• Acock, A. C. (2013). Discovering structural equation modeling using Stata. Stata Press Books.

• Corrado, L. (2005). Generalized Latent Variable Modeling: Multilevel, Longitudinal, and Structural Equation Models.

• Bollen, K. A. (2014). Structural equations with latent variables. John Wiley & Sons.

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• Skrondal, A., & Rabe-Hesketh, S. (2009). Prediction in multilevel generalized linear models. Journal of the Royal Statistical Society Series A: Statistics in Society, 172(3), 659-687.

• Goodman, L. A. (2002). Latent class analysis: The empirical study of latent types, latent variables, and latent structures. Applied latent class analysis, 3-55.

• Masyn, K. E. (2013). Latent class analysis and finite mixture modeling. In The Oxford Handbook of Quantitative Methods, ed. T. D. Little, vol. 2, 551–610. New York: Oxford University Press.

• Bentler, P. M., & Raykov, T. (2000). On measures of explained variance in nonrecursive structural equation models. Journal of Applied Psychology, 85(1), 125.

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Section 1: Introduction to Nonlinear Simultaneous Equations Models

• • Definition of NSEM: Simultaneous equations where at least one equation is nonlinear in parameters or variables.

  • Importance of NSEM: Addressing real-world phenomena that involve nonlinearity in economic relationships.

  • Comparison of linear SEM and NSEM: Nonlinearity in functional form, estimation complexity, and model behavior.

  • Practical examples of nonlinear relationships: Production functions, demand curves, utility maximization.

• Applications of NSEM

  • Common applications in economics: Demand and supply systems, production functions, dynamic optimization models.

  • Case study: Nonlinear production and cost functions in firm behavior analysis.

Section 2: Theoretical Foundations of NSEM

• Structure of Nonlinear Simultaneous Equations

  • General form of NSEM

  • Types of nonlinearity: Nonlinearity in variables, nonlinearity in parameters, or both.

• Nonlinearities in Economic Theory

  • Overview of common sources of nonlinearity in economics:

    • Production functions: Cobb-Douglas, CES functions.

    • Consumption and utility functions: Logarithmic utility, CRRA (Constant Relative Risk Aversion) utility.

    • Investment models: Nonlinear adjustment costs.

• Identification in NSEM

  • Identification issues in nonlinear models: Order and rank conditions for NSEM.

  • Importance of instrument choice: Dealing with the nonlinear structure for proper identification.

Section 3: Estimation Techniques for NSEM

• Nonlinear Two-Stage Least Squares (NL2SLS)

  • Introduction to NL2SLS: Extending the two-stage least squares (2SLS) method to nonlinear equations.

  • Estimation procedure for NL2SLS: First stage for instrumenting endogenous variables, second stage for nonlinear estimation.

• Nonlinear Three-Stage Least Squares (NL3SLS)

  • Extension of 3SLS to nonlinear models: Simultaneous estimation accounting for nonlinearity and cross-equation error correlations.

  • Steps for NL3SLS: Simultaneous estimation using generalized least squares (GLS) in the context of nonlinear systems.

• Maximum Likelihood Estimation (MLE) for NSEM

  • Using MLE for NSEM: Theoretical framework for estimating nonlinear models.

  • Advantages of MLE in NSEM: Efficient estimation under correct specification.

  • Example: Estimating a nonlinear demand-supply model using MLE.

• eneralized Method of Moments (GMM) for NSEM

  • GMM in the context of nonlinear models: Estimating parameters using moment conditions derived from the nonlinear structure.

  • Two-step GMM and efficient GMM for NSEM: Step-by-step application of GMM to nonlinear systems.

• Bayesian Estimation for NSEM

  • Bayesian techniques for estimating NSEM: Incorporating prior information and computing posterior distributions.

  • Applications of Bayesian methods to complex, nonlinear economic models.

Section 4: Hypothesis Testing and Diagnostics in NSEM

• Testing for Nonlinearity

  • Wald, Likelihood Ratio, and Lagrange Multiplier tests for testing nonlinearity in the system.

  • How to determine if a model is better represented as nonlinear rather than linear.

• Nonlinear Two-Stage Least Squares 

  • Sargan-Hansen test for over-identifying restrictions in the context of nonlinear models.

  • Practical examples: Testing instrument validity in nonlinear demand and supply systems.

• Residual Diagnostics in NSEM

  • Checking for model adequacy: Residual analysis, heteroskedasticity tests, and serial correlation tests.

  • Example: Testing for autocorrelation and heteroskedasticity in a nonlinear system of equations.

Section 5: Practical Examples of Nonlinear Simultaneous Equations Models

• Nonlinear Demand and Supply Models

  • Case study: Estimating a nonlinear demand-supply system for agricultural commodities using NL2SLS.

  • Interpretation of results: Nonlinear price effects and elasticity estimation.

• Nonlinear Production Functions

  • Estimating nonlinear production functions: CES (Constant Elasticity of Substitution) and Cobb-Douglas production models.

  • Example: Applying NSEM to estimate the effect of labor and capital on output in different sectors.

• Nonlinear Dynamic Investment Models

  • Example: Estimating nonlinear models of firm investment behavior with adjustment costs using GMM.

  • Analysis of investment decisions and policy implications.

Section 6: Dynamic Nonlinear Simultaneous Equations Models

• Dynamic Extensions of NSEM

  • Incorporating time dynamics into nonlinear models: Lagged variables, autoregressive terms, and time-varying coefficients.

  • Example: Nonlinear dynamic models of consumption and investment.

• Nonlinear Error Correction Models (ECM)

  • Introduction to ECM in the context of nonlinearity: Modeling long-run relationships with nonlinear short-run adjustments.

  • Example: Nonlinear ECM for modeling trade balances and currency exchange rates.

Section 7: Nonlinear Systems with Latent Variables

• Latent Variable Models in NSEM

  • Incorporating unobserved (latent) variables into nonlinear simultaneous equations.

  • Example: Modeling consumer preferences with latent factors and nonlinear demand equations.

• Structural Equation Models with Nonlinearities

  • Extension of structural equation models (SEM) to incorporate nonlinear relationships between variables.

  • Applications in psychology, health economics, and education.

Section 8: Software Implementation for NSEM

• 8.1 Implementing NSEM in Stata

  • Commands for estimating NSEM in Stata

  • Example: Step-by-step guide for estimating a nonlinear demand system in Stata.

• NSEM Estimation in R

  • Using the nlsystemfit package for nonlinear simultaneous equation estimation in R.

  • Example: Estimating a nonlinear production function with R.

• NSEM Estimation in MATLAB

  • MATLAB’s fminunc and fmincon for estimating nonlinear systems with maximum likelihood and GMM.

  • Example: Dynamic nonlinear investment models using MATLAB.

Section 9: Applications of Nonlinear SEM in Different Fields

• NSEM in Finance

  • Example: Estimating nonlinear risk-return relationships in financial markets using NSEM.

• NSEM in Environmental Economics

  • Example: Nonlinear modeling of pollution and output with nonlinear production functions and environmental regulations.

• NSEM in Health Economics

  • Example: Estimating nonlinear health production functions with latent health status and treatment effects.

Section 10: Conclusion and Key Takeaways

• Summary of Key Concepts in NSEM

  • Recap of nonlinearities in simultaneous equations, estimation techniques, and practical applications.

• Best Practices for Estimation and Interpretation

  • Guidelines for choosing the appropriate estimation technique for NSEM and interpreting nonlinear relationships.

• Future Directions in NSEM Research

  • Emerging trends: NSEM in machine learning, deep learning frameworks, and large-scale economic models.

Sách

• Dhrymes, P. J. (2012). Topics in advanced econometrics: Volume II linear and nonlinear simultaneous equations. Springer Science & Business Media.

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• Wall, M. M., & Amemiya, Y. (2007). Nonlinear structural equation modeling as a statistical method. In Handbook of latent variable and related models (pp. 321-343). North-Holland.

• Adeline, A., & Moussa, R. K. (2020). Simultaneous Equations Model with Non-Linear and Linear Dependent Variables on Panel Data. Theoretical Economics Letters, 10(1), 69-89.

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• Wooldridge, J. M. (2010). Econometric analysis of cross section and panel data. MIT press. 

• Hsiao, C. (2022). Analysis of panel data (No. 64). Cambridge university press. 

• Baltagi, B. H., & Baltagi, B. H. (2008). Econometric analysis of panel data (Vol. 4, pp. 135-145). Chichester: Wiley. 

• Cameron, A. C., & Trivedi, P. K. (2010). Microeconometrics using stata (Vol. 2). College Station, TX: Stata press. 

• Greene, W. (2012) Econometric Analysis. 7th Edition 

• Schmidt, P. (1976). Econometrics Marcel Dekker. New York. 

• Davidson, R., & MacKinnon, J. G. (1993). Estimation and inference in econometrics (Vol. 63). New York: Oxford. 

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Nội dung

Section 1: Introduction to Structural Equation Models (SEM)

• • Definition and overview of SEM: Combining factor analysis and multiple regression to model complex relationships between observed and unobserved (latent) variables.

  • The role of SEM in econometrics: Simultaneous estimation of multiple equations.

  • Use cases in various fields: Economics, psychology, sociology, marketing, and education.

  • Examples: Modeling consumer behavior, economic growth, psychological traits, and social relationships.

• Advantages of SEM Over Traditional Models

  • Ability to model latent variables and measurement error.

  • Simultaneous estimation of interdependent relationships between multiple variables.

  • Greater flexibility and richer data analysis.

Section 2: Components of Structural Equation Models

• The Measurement Model

  • Definition: Describing the relationships between latent variables and their observed indicators.

  • Reflective and formative indicators: Differences in how observed variables relate to latent constructs.

• The Structural Model

  • Definition: Modeling relationships between endogenous and exogenous variables, whether observed or latent.

  • Path analysis: Direct and indirect effects between variables.

• Exogenous vs. Endogenous Variables

  • Differentiating between exogenous (independent) variables and endogenous (dependent) variables in SEM.

• Combining the Measurement and Structural Models

  • Integration of the measurement model and the structural model into a single system.

  • Example: A complete SEM for consumer satisfaction, including both measurement of satisfaction (latent variable) and its effect on purchase behavior.

Section 3: Model Specification in SEM

• Specifying the Measurement Model

  • Defining relationships between observed indicators and latent variables.

  • Example: Measurement model for latent health status based on observed health indicators.

• pecifying the Structural Model

  • Identifying relationships between variables in the system.

  • Structural paths: Direct and indirect effects in SEM.

  • Example: Specifying a structural model for firm performance based on latent organizational factors and external economic factors.

• 3.3 Model Identification

  • Order and rank conditions for identification in SEM.

  • Rules for ensuring that the model can be uniquely estimated.

  • Example: Identifying an over-identified vs. under-identified SEM.

Section 4: Estimation Techniques in SEM

• • 1. • Maximum Likelihood Estimation (MLE)

• Description: The most commonly used estimation technique in SEM. MLE finds parameter estimates that maximize the likelihood of the observed data given the model.

• Assumptions: Data is normally distributed; Large sample size is required for consistency.

• Advantages: Produces efficient and asymptotically unbiased estimates under correct model assumptions.

• Applications: Widely used in most SEM software like Stata, R (lavaan), and Mplus.

  • 1. • • Generalized Least Squares (GLS)

• Description: Minimizes a weighted sum of squared differences between the observed and predicted covariance matrices, accounting for heteroscedasticity.

• Assumptions: Normally distributed data; Homoscedasticity (constant variance across observations).

• Advantages: Can be used to address heteroskedasticity in SEM models ; Applications: SEM with large sample sizes, especially in models with measurement error.

  • 1. 1. • Weighted Least Squares (WLS)

• Description: A method that uses weights to account for differences in the precision of different estimates, suitable for non-normal data.

• Assumptions: Often used when dealing with ordinal or non-normally distributed data.

• Advantages: More robust to violations of normality.

• Applications: SEM with non-normally distributed data, particularly in models with categorical or ordinal variables.

  • 1. 1. • Two-Stage Least Squares (2SLS)

• Description: A method used for SEM with endogenous variables. It first estimates predicted values for the endogenous variables and then uses these predicted values to estimate the structural equation.

• Assumptions: Requires the identification of valid instrumental variables.

• Advantages: Corrects for endogeneity in SEM models.

• Applications: SEM models with simultaneous equations where endogeneity is present.

  • 1. • Three-Stage Least Squares (3SLS)

• Description: An extension of 2SLS that accounts for cross-equation error correlations. It estimates all the equations in the system simultaneously.

• Assumptions: Assumes joint normality of errors across equations.

• Advantages: Increases efficiency by considering correlations in the residuals between different equations.

• Applications: Used in models with multiple equations where errors are correlated across equations.

  • 1. • Limited Information Maximum Likelihood (LIML)

• Description: A variation of MLE that deals with systems of simultaneous equations but focuses on estimating single structural equations.

• Assumptions: Data is normally distributed.

• Advantages: LIML is less biased than 2SLS when there are weak instruments.

• Applications: Models with simultaneous equations and weak instruments.

  • 1. • Full Information Maximum Likelihood (FIML)

• Description: A method that estimates all model parameters simultaneously using all available data, even with missing values.

• Assumptions: Requires the data to follow a multivariate normal distribution.

• Advantages: Handles missing data more efficiently compared to traditional methods like listwise deletion.

• Applications: SEM with incomplete datasets or missing data.

  • 1. • Bayesian Estimation

• Description: A method that combines prior information with the observed data to estimate model parameters using Bayesian principles.

• Assumptions: Requires specification of prior distributions for the parameters.

• Advantages: Can handle small sample sizes, complex models, and missing data; Allows for greater flexibility in model specification and parameter estimation.

• Applications: SEM with small samples, non-normal data, or models that involve latent variables and hierarchical structures.

  • 1. • Generalized Method of Moments (GMM)

• Description: A flexible estimation technique that uses moment conditions derived from the model’s equations to estimate parameters.

• Assumptions: Does not require normality of errors.

• Advantages: Suitable for SEM with heteroskedasticity and other non-standard data distributions.

• Applications: Used in SEM models dealing with panel data, heteroskedasticity, and endogeneity.

  • 1. • Asymptotically Distribution-Free (ADF) Estimation (also known as Weighted Least Squares)

• Description: A robust technique used when data does not follow a normal distribution. It uses the asymptotic covariance matrix of the sample moments to perform estimation.

• Assumptions: Assumes large sample sizes.

• Advantages: Robust to violations of normality in the data.

• Applications: SEM for ordinal or non-normal data, particularly when dealing with small samples.

  • 1. • Partial Least Squares (PLS)

• Description: An alternative estimation technique to MLE, used primarily when the sample size is small, or the data does not meet distributional assumptions.

• Assumptions: Does not assume normality or large sample sizes.

• Advantages: Can handle complex models with many latent variables and small sample sizes.

• Applications: Used in exploratory research or when theory development is still in its early stages. Widely used in marketing, management, and social sciences.

  • 1. • Robust Maximum Likelihood (MLR)

• Description: A variation of MLE that provides robust standard errors and fit indices in the presence of non-normality or missing data.

• Assumptions: Relaxes the assumption of multivariate normality.

• Advantages: Accounts for non-normality and missing data without requiring the complete data matrix.

• Applications: Suitable for SEM with non-normal data or missing values.

• Summary of Estimation Techniques for SEM:

  • MLE, GLS, WLS: Suitable for large sample sizes, normally distributed data, or heteroscedasticity issues.

  • 2SLS, 3SLS, LIML, FIML: Used for simultaneous equations, dealing with endogeneity, or missing data.

  • Bayesian Estimation, GMM: Useful for handling small samples, complex models, and non-normal data.

  • PLS, ADF, MLR: Alternatives to MLE for small samples, non-normal data, and robust estimation.

Section 5: Goodness-of-Fit and Model Diagnostics in SEM

• Assessing Overall Model Fit

  • Goodness-of-fit measures: Chi-square, RMSEA (Root Mean Square Error of Approximation), CFI (Comparative Fit Index), TLI (Tucker-Lewis Index).

  • Interpretation of goodness-of-fit indices and their thresholds.

• Residual Diagnostics in SEM

  • Analyzing residuals to check for model adequacy and misspecification.

  • Checking for outliers, omitted variables, and incorrect functional forms.

• Testing Hypotheses in SEM

  • Wald test, Likelihood Ratio test, and Lagrange Multiplier test for hypothesis testing in SEM.

  • Example: Testing the impact of a latent variable on an observed outcome.

Section 6: Practical Examples of SEM

• Example 1: SEM for Consumer Behavior

  • Modeling consumer satisfaction and its effect on loyalty and repeat purchasing behavior.

• Example 2: SEM for Economic Growth

  • Modeling economic growth using latent factors like innovation, education, and institutional quality, and their effects on GDP.

• Example 3: SEM for Health Outcomes

  • Estimating the effects of latent health status on healthcare utilization and spending using multiple observed health indicators.

Section 7: Handling Latent Variables in SEM

• Measurement of Latent Variables

  • Defining how latent variables are measured using observed indicators.

  • Example: Measuring latent risk preference using survey data.

• Testing for Construct Validity

  • Convergent and discriminant validity: Ensuring that the model accurately reflects the latent variables being measured.

• Example: Latent Variables in Educational Research

  • Using latent variables to model student engagement and academic performance based on survey data.

• SEM in Economics

  • Example: Using SEM to model latent innovation and its impact on firm productivity.

• SEM in Marketing

  • Example: Estimating brand loyalty as a latent construct and linking it to consumer behavior.

• SEM in Psychology and Social Sciences

  • Example: Measuring latent psychological traits like depression and anxiety and their impact on life satisfaction.

Section 8: SEM for Panel Data

• Extending SEM to Panel Data

  • Incorporating time-varying effects and individual-specific latent variables in SEM.

  • Example: Using SEM to analyze firm-level productivity over time.

• Dynamic SEM

  • Introduction to dynamic SEM: Modeling temporal relationships between latent and observed variables.

  • Example: Estimating the dynamic relationship between financial risk and investment decisions using panel data.

Section 9: Multilevel SEM

• Multilevel SEM Models

  • Modeling data at different levels (e.g., individuals within firms, students within schools).

  • Applications of multilevel SEM in educational and organizational research.

• Estimation Techniques for Multilevel SEM

  • Maximum likelihood estimation for multilevel SEM.

  • Example: Multilevel SEM to analyze the effects of school-level factors on student academic outcomes.

Section 10: SEM with Missing Data

• Handling Missing Data in SEM

  • Full Information Maximum Likelihood (FIML) and Multiple Imputation techniques to deal with missing data.

  • Example: Estimating SEM with incomplete survey data.

• Example: Dealing with Missing Health Indicators in SEM

  • Using FIML to estimate a health model where some indicators are missing for certain respondents.

Section 11: Extensions and Advanced Topics in SEM

• Nonlinear SEM

  • Introduction to nonlinear SEM: Modeling non-linear relationships between latent and observed variables.

  • Example: Nonlinear SEM for modeling diminishing returns to education on income.

• SEM with Interaction Effects

  • Modeling interaction terms between latent variables and observed variables in SEM.

  • Example: Estimating the interaction between job satisfaction (latent) and work hours on productivity.

Section 12: Software Implementation for SEM

• SEM in Stata

  • Using Stata's sem and gsem commands to estimate SEM models.

  • Step-by-step guide to estimating a structural equation model in Stata.

• SEM in R

  • Using the lavaan package in R to estimate SEM models.

  • Example: Estimating a SEM for customer satisfaction using R.

• SEM in Mplus

  • Advanced modeling of SEM with latent variables and complex data structures in Mplus.

  • Example: Estimating a multilevel SEM in Mplus for educational research.

Section 13: Conclusion and Key Takeaways

• Summary of Key Concepts in SEM

  • Recap of measurement and structural models, estimation techniques, and hypothesis testing in SEM.

• Best Practices for SEM Estimation and Interpretation

  • Guidelines for specifying and interpreting SEM models.

• Future Directions in SEM Research

  • Discussion of emerging trends: SEM with machine learning, big data, and new statistical advancements.

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Nội dung

Section 1: Introduction to Simultaneous Equations Models (SEM) in Panel Data

• Why Use SEM with Panel Data?

  • Benefits of using SEM with panel data: Accounting for dynamic relationships, individual heterogeneity, and complex feedback loops.

  • Example: Simultaneous modeling of investment and output across firms over time.

Section 2: Model Specification for Simultaneous Equations in Panel Data

• Structural Equations in Panel Data

  • Defining structural equations for panel data: Simultaneous determination of multiple endogenous variables.

  • Example: A model of demand and supply for labor across multiple regions over several years.

• Fixed Effects vs. Random Effects in SEM

  • Explanation of fixed effects (FE) and random effects (RE) models in panel SEM.

  • Decision-making criteria: When to use FE or RE models, using the Hausman test to decide.

• Individual and Time Effects in Panel SEM

  • Understanding how individual (cross-sectional) and time-specific effects are treated in SEM.

  • Example: Incorporating both firm-specific and time-specific effects in an investment model.

• Error Components in Panel SEM

  • Structure of error terms: Cross-sectional and time-series error components.

  • Example: Modeling errors with firm-level shocks and time-varying economic conditions.

Section 3: Identification of Simultaneous Equations in Panel Data

• Identification in Panel SEM

  • Conditions for identification: Rank and order conditions.

  • Why identification is crucial in SEM: Ensuring unique parameter estimates.

• Over-Identification in Panel SEM

  • Dealing with over-identification: The need for valid instrumental variables.

  • Example: Identifying simultaneous relationships between wages and employment in panel data.

Section 4: Estimation Techniques for Simultaneous Equations in Panel Data

• Two-Stage Least Squares (2SLS) in Panel Data

  • Introduction to 2SLS for addressing endogeneity: Using instrumental variables.

  • Application to panel data: Estimating relationships when some variables are endogenous.

  • Example: Estimating labor supply and demand equations using 2SLS in a panel setting.

• Three-Stage Least Squares (3SLS) in Panel Data

  • Explanation of 3SLS: Simultaneous estimation of equations while accounting for cross-equation correlations.

  • Application to panel data: Incorporating both fixed effects and simultaneous feedback loops.

  • Example: Estimating the simultaneous determination of wages and productivity across firms using 3SLS.

• Generalized Method of Moments (GMM) for Panel Data

  • Overview of GMM for dynamic SEM in panel data: Moment conditions and instrumenting endogenous variables.

  • Difference GMM and system GMM approaches in dynamic panel SEM.

  • Example: Estimating investment and output using system GMM with panel data from multiple firms.

• Maximum Likelihood Estimation (MLE) for Panel SEM

  • Using MLE for estimating simultaneous equations in panel data: Accounting for heteroskedasticity and cross-equation correlations.

  • Example: Maximum likelihood estimation of a model of firm-level output and labor costs across multiple years.

Section 5: Dynamic Simultaneous Equations in Panel Data

• Introduction to Dynamic Panel SEM

  • Incorporating time dynamics: Lagged endogenous variables and feedback effects over time.

  • Why dynamic SEM is useful in panel data: Capturing long-term and short-term effects in the system.

  • Example: Dynamic SEM of output, investment, and productivity over time.

• Arellano-Bond Estimation for Dynamic Panel Data SEM

  • Using Arellano-Bond for dynamic SEM in panel data: Addressing endogeneity and unobserved heterogeneity.

  • Example: Dynamic modeling of GDP growth and investment using lagged variables.

• System GMM for Dynamic SEM

  • Explanation of system GMM: Estimating dynamic relationships with better handling of endogeneity.

  • Example: Dynamic SEM for export and production decisions across countries using system GMM.

Section 6: Dealing with Endogeneity in Panel SEM

• Endogeneity in Panel Data SEM

  • Sources of endogeneity in panel data: Simultaneity, omitted variables, and measurement error.

  • Example: Endogeneity between labor demand and wages in a multi-year panel of firms.

• Instrumental Variables in Panel SEM

  • Choosing valid instruments: Ensuring exogeneity of instruments for panel SEM.

  • Example: Instrumenting wages with industry-level shocks in a wage determination system.

• Testing for Endogeneity in Panel SEM

  • Using the Hausman test and Sargan-Hansen test to detect endogeneity in panel SEM models.

  • Example: Testing endogeneity in a simultaneous model of inflation and unemployment with panel data.

Section 7: Hypothesis Testing and Diagnostics in Panel SEM

• Hypothesis Testing in Simultaneous Equations Models for Panel Data

  • Wald test, Likelihood Ratio test, and Lagrange Multiplier test for model diagnostics.

  • Example: Testing cross-equation restrictions in a simultaneous model of firm output and labor input.

• Testing Over-Identification in Panel SEM

  • Using Sargan and Hansen tests to assess the validity of instruments in panel SEM.

  • Example: Over-identification tests in a panel model of consumption and income across countries.

• Testing for Serial Correlation and Heteroskedasticity

  • Breusch-Godfrey test for serial correlation and Breusch-Pagan test for heteroskedasticity.

  • Example: Checking for serial correlation in a panel SEM of healthcare expenditure and economic growth.

Section 8: Practical Applications of Simultaneous Equations Models in Panel Data

• Application 1: Investment and Production in Firms

  • Example: Estimating the simultaneous relationship between investment and production decisions across firms using panel data.

• Application 2: Supply and Demand in Agriculture

  • Example: Modeling supply and demand in the agricultural sector using panel data from multiple regions over time.

• Application 3: Wage Determination and Employment

  • Example: Estimating a simultaneous system of wage determination and employment levels across industries using panel data.

Section 9: Software Implementation for Panel SEM

• Implementing Panel SEM in Stata

  • Using Stata’s xtivreg, xtabond, and reg3 commands for estimating simultaneous equations in panel data.

  • Step-by-step guide for estimating a dynamic SEM in Stata using 2SLS and GMM.

• Implementing Panel SEM in R

  • Using the plm, systemfit, and gmm packages in R for panel SEM estimation.

  • Example: Estimating a dynamic simultaneous equation system for firm growth and investment in R.

• Implementing Panel SEM in EViews

  • Tools in EViews for estimating panel SEM: Dynamic SEM and simultaneous equations modeling.

  • Example: Estimating a simultaneous system of output and investment using EViews.

Section 10: Conclusion and Key Takeaways

• Summary of Key Concepts in Simultaneous Equations Models in Panel Data

  • Recap of identification, estimation methods, and dealing with endogeneity in panel SEM.

• Best Practices for Panel SEM Estimation and Model Validation

  • Guidelines for choosing estimation techniques, validating models, and addressing model diagnostics in panel SEM.

• Future Trends in Panel SEM Research

  • Discussion of advanced topics: Non-linear panel SEM, Bayesian methods in panel SEM, and applications in machine learning.

Sách

Bài báo

• Hsiao, C., & Zhou, Q. (2015). Statistical inference for panel dynamic simultaneous equations models. Journal of Econometrics, 189(2), 383-396.

• Alsaleh, M., Abdul-Rahim, A. S., & Mohd-Shahwahid, H. O. (2017). An empirical and forecasting analysis of the bioenergy market in the EU28 region: Evidence from a panel data simultaneous equation model. Renewable and Sustainable Energy Reviews, 80, 1123-1137.

• Cai, L. (2010). The relationship between health and labour force participation: Evidence from a panel data simultaneous equation model. Labour Economics, 17(1), 77-90.

• Tripathi, J. S. (2023). Trade-growth nexus: A study of G20 countries using simultaneous equations model with dynamic policy simulations. Journal of Policy Modeling, 45(4), 806-816.

 Nội dung

Section 1: Introduction to Simultaneous Equations Models (SEM) with Spatial Data

• • Definition of SEM: A system of equations where multiple endogenous variables are simultaneously determined; Importance of SEM in econometrics: Dealing with endogeneity and capturing feedback loops.

  • Definition of spatial data: Data that includes a geographical or spatial dimension (e.g., location-specific variables)

  • Examples: Regional economic data, neighborhood data, geographical data, and environmental variables.

• Why Use SEM with Spatial Data?

  • Benefits of combining SEM and spatial data: Capturing spatial dependencies, spillover effects, and local interactions.

  • Example: Simultaneously modeling regional unemployment and wage determination while accounting for spatial dependencies.

Section 2: Spatial Econometrics Basics

• Overview of Spatial Dependence and Spatial Heterogeneity

  • Definition of spatial dependence: When the value of a variable in one location depends on values in neighboring locations.

  • Spatial heterogeneity: Variation in relationships between variables across different regions.

• Spatial Weights Matrix

  • Definition of the spatial weights matrix: A matrix representing the spatial relationship between different units (e.g., regions, countries).

  • Types of spatial weights matrices: Contiguity matrix, distance matrix, inverse distance matrix.

  • Example: Constructing a spatial weights matrix for counties or provinces.

• Key Spatial Models

  • Spatial autoregressive model (SAR), Spatial error model (SEM), and Spatial Durbin model (SDM).

  • Example: Estimating a SAR model for regional housing prices.

Section 3: Simultaneous Equations Models (SEM) with Spatial Data

• Spatial Simultaneous Equations Models

  • Extending SEM to include spatial dependencies: Simultaneously modeling spatial and endogenous relationships.

  • Example: Modeling spatial spillovers in the housing market using SEM with spatial data.

• Endogeneity in Spatial Models

  • The issue of endogeneity in spatial models: Simultaneity in spatial interactions.

  • Example: Endogeneity between regional GDP and population growth with spatial spillovers.

• Model Specification for Spatial SEM

  • Structural form and reduced form for spatial SEM: Incorporating spatial lags and spatial error terms.

  • Example: Simultaneously modeling urban land prices and crime rates with spatial dependencies.

Section 4: Identification in Spatial SEM

• Identification of Spatial SEM

  • Order and rank conditions for identification in spatial simultaneous equations models.

  • Importance of correct identification in spatial SEM to avoid biased parameter estimates.

  • Example: Identifying a simultaneous system of spatial housing demand and supply equations.

• Over-Identification in Spatial SEM

  • Handling over-identification: The need for valid instruments in spatial SEM.

  • Example: Identifying and testing for over-identification in a spatial model of regional economic growth.

Section 5: Estimation Techniques for Spatial SEM

• Two-Stage Least Squares (2SLS) for Spatial SEM

  • Using 2SLS to estimate spatial SEM: Instrumenting endogenous spatial variables.

  • Example: Estimating a spatial SEM for pollution levels and economic activity using 2SLS.

• Three-Stage Least Squares (3SLS) for Spatial SEM

  • Simultaneous estimation of multiple equations and spatial dependencies using 3SLS.

  • Example: Simultaneous estimation of spatial demand and supply systems in the housing market using 3SLS.

• Generalized Method of Moments (GMM) for Spatial SEM

  • Introduction to GMM in spatial SEM: Estimating models with spatial lag and spatial error terms.

  • Example: Using GMM to estimate spatial spillover effects in a system of regional economic development and infrastructure investment.

• Maximum Likelihood Estimation (MLE) for Spatial SEM

  • Using MLE for spatial SEM: Estimating parameters in systems with spatial autocorrelation and cross-equation dependence.

  • Example: MLE estimation of a spatial SEM for neighborhood crime rates and housing values.

Section 6: Dynamic Spatial Simultaneous Equations Models

• Dynamic Spatial SEM

  • Incorporating temporal dynamics into spatial SEM: Using lagged dependent variables and spatial lags over time.

  • Example: Dynamic spatial SEM for regional GDP and investment growth.

• Estimating Dynamic Spatial SEM with GMM

  • Using GMM to estimate dynamic spatial models: Addressing both time dynamics and spatial spillovers.

  • Example: Dynamic modeling of spatial wage disparities and employment growth across regions using GMM.

Section 7: Testing and Diagnostics in Spatial SEM

• Testing for Spatial Dependence

  • Moran’s I test, Lagrange Multiplier tests, and robust LM tests for detecting spatial dependence.

  • Example: Testing for spatial dependence in a simultaneous system of crime rates and property prices.

• Testing for Endogeneity in Spatial SEM

  • Hausman test for endogeneity in spatial SEM.

  • Example: Testing for endogeneity in a spatial model of pollution and economic output.

• Goodness-of-Fit and Model Diagnostics for Spatial SEM

  • R-squared, log-likelihood, AIC/BIC, and spatial diagnostics for model evaluation.

  • Example: Evaluating model fit in a spatial SEM for education outcomes and labor market performance.

Section 8: Practical Applications of Spatial SEM

• Application 1: Urban Housing Markets

  • Example: Estimating the simultaneous relationship between housing demand and supply across cities with spatial dependencies.

• Application 2: Regional Economic Growth

  • Example: Simultaneous modeling of regional economic growth and infrastructure investment with spatial spillover effects.

• Application 3: Environmental Pollution and Economic Activity

  • Example: Estimating a spatial SEM for the relationship between pollution and economic activity across neighboring regions.

• Application 4: Public Policy and Crime Rates

  • Example: Modeling the simultaneous relationship between public policy interventions and crime rates with spatial interactions.

Section 9: Software Implementation for Spatial SEM

• Implementing Spatial SEM in Stata

  • Using Stata’s spreg and ivreg2 commands to estimate spatial SEM.

  • Step-by-step guide for estimating a spatial SEM for regional GDP and trade.

• Implementing Spatial SEM in R

  • Using the spdep and systemfit packages in R for spatial SEM estimation.

  • Example: Estimating a spatial SEM for regional housing prices and income levels in R.

• Implementing Spatial SEM in MATLAB

  • MATLAB functions for estimating spatial SEM: Using spatialEconometrics and GMM packages.

  • Example: Dynamic spatial SEM for crime and economic disparities using MATLAB.

Section 10: Advanced Topics in Spatial SEM

• Nonlinear Spatial SEM

  • Extending SEM to nonlinear models with spatial interactions.

  • Example: Nonlinear spatial SEM for modeling spatial effects in pollution control policies.

• Spatial SEM with Panel Data

  • Combining spatial SEM with panel data techniques: Accounting for time dynamics and spatial heterogeneity.

  • Example: Panel data spatial SEM for regional unemployment and wage determination across multiple time periods.

• Bayesian Spatial SEM

  • Bayesian estimation methods for spatial SEM: Incorporating prior information in the estimation process.

  • Example: Bayesian spatial SEM for environmental quality and regional economic growth.

Section 11: Conclusion and Key Takeaways

• Summary of Key Concepts in Spatial SEM

  • Recap of spatial dependence, estimation methods, and identification in spatial simultaneous equations models.

• Best Practices for Spatial SEM Estimation and Interpretation

  • Guidelines for model specification, spatial weight matrix selection, and interpretation of spatial spillover effects.

• Future Directions in Spatial SEM Research

  • Discussion on emerging topics: Machine learning with spatial SEM, nonparametric methods, and applications in big data.

Sách

Bài báo

• Cui, J., Zhang, Q., & Wang, Q. (2024). Investigating the interactive effects between venture capital and urban innovation capabilities: New evidence from a spatial simultaneous equations model. Finance Research Letters, 67, 105957.

• Yang, K., & Lee, L. F. (2019). Identification and estimation of spatial dynamic panel simultaneous equations models. Regional Science and Urban Economics, 76, 32-46.

• Jeanty, P. W., Partridge, M., & Irwin, E. (2010). Estimation of a spatial simultaneous equation model of population migration and housing price dynamics. Regional Science and Urban Economics, 40(5), 343-352.

• Liu, Y., & Hao, Y. (2023). How does coordinated regional digital economy development improve air quality? New evidence from the spatial simultaneous equation analysis. Journal of Environmental Management, 342, 118235.

• Rosburg, A., Isakson, H., Ecker, M., & Strauss, T. (2017). Beyond standardized test scores: The impact of a public school closure on house prices. Journal of Housing Research, 26(2), 119-135.