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Dynamic Panel Models and System GMM Estimation

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Hausman Test: Fixed Effects Versus Random Effects Instrumental Variables +1 more→→Vector Autoregression (VAR) Models and Impulse Responses

Core Idea

Dynamic panels include lagged dependent variables, which correlate with fixed effects, violating strict exogeneity. Arellano-Bond and Blundell-Bond GMM estimators use internal instruments (lags of dependent variable and regressors) for consistent estimation. Arellano-Bond (difference GMM) assumes mean stationarity; Blundell-Bond (system GMM) relaxes this.

Explainer

Start from what you know about panel fixed effects. The within estimator demeans the data to eliminate unobserved entity-specific effects (αᵢ), giving consistent estimates when strict exogeneity holds — meaning regressors are uncorrelated with the idiosyncratic error in all time periods. The problem with a dynamic panel is the inclusion of the lagged dependent variable yᵢ,ₜ₋₁ on the right-hand side. This lag contains information about all past values of y, which in turn are functions of αᵢ. So the regressor is mechanically correlated with the fixed effect. The within estimator is inconsistent, and the bias is large in short panels (small T) even as N grows.

The Arellano-Bond solution (difference GMM) first-differences the model to remove αᵢ, just like the within transformation. But now the equation is Δyᵢₜ = ρΔyᵢ,ₜ₋₁ + ΔXᵢₜβ + Δεᵢₜ. The problem is that Δyᵢ,ₜ₋₁ = yᵢ,ₜ₋₁ − yᵢ,ₜ₋₂ is still correlated with Δεᵢₜ = εᵢₜ − εᵢ,ₜ₋₁ (because Δyᵢ,ₜ₋₁ depends on εᵢ,ₜ₋₁). This is precisely the IV problem you studied: endogenous regressor in the differenced equation. The key insight is that levels of y dated t−2 and earlier are valid instruments: they are correlated with Δyᵢ,ₜ₋₁ (relevance) but uncorrelated with Δεᵢₜ provided errors are not serially correlated (exclusion). The estimator stacks these moment conditions and uses GMM to exploit all of them efficiently.

Blundell-Bond (system GMM) addresses a weakness of difference GMM: when y is highly persistent (ρ close to 1), lagged levels are weak instruments for the differenced equation — the correlation between yᵢ,ₜ₋₂ and Δyᵢ,ₜ₋₁ is near zero. Blundell-Bond adds the original levels equations back to the system, using lagged differences as instruments for the levels (Δyᵢ,ₜ₋₁ is a valid instrument for yᵢ,ₜ₋₁ in the levels equation if the initial conditions satisfy a stationarity restriction). The combined system gains precision, especially for persistent variables like firm size or GDP.

Two specification tests are essential for credibility. The Sargan/Hansen test checks whether the instruments are jointly valid (overidentification test); rejection suggests instrument proliferation or model misspecification. The Arellano-Bond AR(2) test checks for second-order serial correlation in the differenced residuals — if AR(2) is present, the t−2 lags are no longer valid instruments. A common failure mode is using too many instruments ("instrument proliferation"), which weakens the Hansen test and can bias coefficients. The rule of thumb is to keep the instrument count below the number of entities. Dynamic panel GMM is powerful for studying firm investment, growth regressions, and any setting where past outcomes causally determine current outcomes, but the instrument construction requires careful thought about the underlying error structure.

Practice Questions 5 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → Counting to 20 → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Number Bonds to 10 → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Dividing Integers → Unit Rates → Proportions → Percent Concept → Converting Between Fractions, Decimals, and Percents → Operations with Rational Numbers → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Sine, Cosine, and Tangent Ratios → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Indefinite Integrals → Basic Integration Rules → Riemann Sums → Definite Integral Definition → Probability Density Functions and Continuous Distributions → Cumulative Distribution Functions → Continuous Random Variables → Probability Density Functions → Expected Value → Weak Law of Large Numbers → Probability Axioms and Rules → Conditional Probability → Independence of Events → Sampling Distributions → Standard Error of Estimators → Hypothesis Testing: Framework and Logic → P-values and Statistical Significance → Effect Size and Practical Significance → Hypothesis Testing: Framework and Logic → Z-Tests and T-Tests for Means → One-Sample Z-Test for Means → One-Sample and Two-Sample T-Tests → Inference in Linear Regression → Prediction Intervals in Regression → Linear Regression Basics → Residuals and Goodness of Fit (R²) → Simple (Bivariate) OLS Regression → Classical OLS Assumptions (Gauss-Markov) → Multiple Regression → Interpreting Regression Coefficients → Hypothesis Testing in Regression → F-Test and Joint Significance → R-Squared and Model Fit → Omitted Variable Bias → Causal Inference and the Identification Problem → Potential Outcomes and the Rubin Causal Model → Selection Bias → Instrumental Variables → Dynamic Panel Models and Arellano-Bond/Blundell-Bond Estimation → Dynamic Panel Models: Arellano-Bond Estimator → First-Difference Estimator for Panel Data → Within Estimator (Fixed Effects) for Panel Data → Between and Random Effects Estimators for Panel Data → Hausman Test: Fixed Effects Versus Random Effects → Dynamic Panel Models and System GMM Estimation

Longest path: 126 steps · 903 total prerequisite topics

Prerequisites (3)

Hausman Test: Fixed Effects Versus Random Effectshard Instrumental Variableshard Dynamic Panel Models: Arellano-Bond Estimatorsoft

Leads To (1)

Vector Autoregression (VAR) Models and Impulse Responsessoft