Notes on Structural Models (to Be Continued)
Empirical economics focus on data-driven problems
- Data contains endogenous Y, exogenous Z, unobserved exogenous U
Historical Background
- The words reduced-form and structural-form date back to the literature of estimation of simultaneous equations in macroeconomics (Hsiao, 1983)
- The equilibrium condition for Y on X and U is often written as: AY+BX=CU(1)
- Equation (1) implicitly determines Y
- Equation (1) is the structural-form
- If A is invertible, we can solve the equations for Y to obtain: Y=A−1CU−A−1BX(2)
- Equations (2) explicitly determine Y
- Equations (2) is the reduced-form
Structural Model
Structural models estimate features of a data generating process that are (assumed to be) invariant to the counterfactuals of interest, e.g., g(Y,X,Z,U;Θ)=0
- Models are derived from economic theories
- Possible to perform counterfactual simulations
A reduced form is a functional or stochastic mapping for which the inputs are (i) exogenous variables and (ii) unobservables (“structural errors”), and for which the outputs are endogenous variables, e.g., Y=f(X,Z,U)
- The main goal is to find a causal relationship between x and y
- Do not require knowledge of the underlying data generating process
Classic Models
1. Homogeneous Logit Model
Example:
- Logit models can be used to estimate market shares when we just have aggregate data.
- t=1,⋯,T markets, each with i=1,⋯,It customers; j=1,⋯,J products
- Utility: uitj=xtjβ−αptj+ξtj+ϵitj
- ϵitj follows Type I Extreme Value Distribution
- Independent of Irrelevant Alternatives (IIA) property
2. Random Coefficients Logit Model (BLP Model; Notes)
Example:
- Utility: uitj=xtjβi−αiptj+ξtj+ϵitj
- ϵitj follows Type I Extreme Value Distribution
- [βiαi]=[βˉαˉ]+ΠDi+Σvi
- [βˉ,αˉ]′: population mean
- Di: customer-level observables
- vi: customer-level unobservables
- Σ: variance-covariance scaling matrix
- Customers who like a certain product are more likely to like similar products
3. Dynamic Discrete Choice Model
- Agents are forward-looking and maximize their intertemporal payoff.
3.1. Single-Agent Dynamic Model
- Agents are independent of each other
Rust’s (1987) Bus Engine Replacement Problem
- N buses, i=1,2,⋯,N
- Decision dit∈{0,1}: whether to replace the engine of bus i in period t
- State variable xit: the cumulative mileage on bus i's engine at time t since the last replacement
- Single-period utility: u(dit,xit,ϵit;θ)={−cr−c(0,co)+ϵi1t−c(xit,co)+ϵi0tif dit=1if dit=0
- cr: replacement cost; c(xit,co): operating cost
3.2. Dynamic Game Model
- Agents interact with each other, leading to an equilibrium
- A player’s single-period utility: U(at,xt,ϵit)
- States at:={a1t,⋯,aant}; actions xt:={x1t,⋯,xnt}
4. Match Models
4. 1. One to One Matching
- Men M={m1,⋯,mk}; women: W={w1,⋯,wp}
- Preference: P(mi)=wk,wℓ,⋯,mi,wj,⋯
- A matching: μ=M∪W→M∪W
- w=μ(m) if and only if m=μ(w) (i.e., mutual)
- A matching is pairwise stable if there is no individual players or pairs of players who can profitably deviate from (block) it.
- Gale-Shapley Theorem: A stable matching exists for every marriage market (through the deferred acceptance algorithm).
4.2. Many to One Matching
- Colleges C={c1,⋯,cn} with number of positions, q1,⋯,qn and students S={s1,⋯,sp}
- Matching μ is a correspondence such that
- μ(s)∈C∪S (a student might be matched with no schools)
- μ(C)∈S (a school is matched to a group of student)
- μ(s)=C if and only if s∈μ(C) for every student s∈S and college c∈C
- There exists a stable matching in any many-to-one matching market.
Common Methods of Estimation
- Maximum likelihood estimation
- Generalized method of moments
- Simulation-based estimation
References
- Reiss, P. C., & Wolak, F. A. (2007). Structural econometric modeling: Rationales and examples from industrial organization. Handbook of econometrics, 6, 4277-4415.
- Hsiao, C. (1983). Chapter 4 Identification. Handbook of Econometrics, 1, 223–283.
- Huang, Y., Lee, S., & Tan, Y. (2019). Structural econometric models. Operations Research & Management Science in the Age of Analytics (pp. 17-43). INFORMS.
- http://www.econ.yale.edu/~pah29/intro.pdf
- https://kohei-kawaguchi.github.io/EmpiricalIO/assignment4.html
- https://github.com/woerman/ResEcon703