﻿ Structural_Econometric_Model

# Notes on Structural Models (to Be Continued)

## Structural VS. Reduced Form in Empirical Economics

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: $\color{blue} AY+BX=CU \hspace{4em} (1)$
• Equation (1) implicitly determines $Y$
• Equation (1) is the $\textbf{\color{blue}structural-form}$
• If $A$ is invertible, we can solve the equations for $Y$ to obtain: $\color{purple}Y=A^{-1}CU-A^{-1}BX \hspace{4em} (2)$
• Equations (2) explicitly determine $Y$
• Equations (2) is the $\textbf{\color{purple}reduced-form}$

### Structural Model

Structural models estimate features of a data generating process that are (assumed to be) invariant to the counterfactuals of interest, e.g., $g(Y, X, Z, U; \Theta) = 0$

• Models are derived from economic theories
• Possible to perform counterfactual simulations

### Reduced-Form

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.
• $t=1, \cdots, T$ markets, each with $i = 1, \cdots, I_t$ customers; $j=1, \cdots, J$ products
• Utility: $u_{itj}=x_{tj}\beta-\alpha p_{tj}+\xi_{tj}+\epsilon_{itj}$
• $\epsilon_{itj}$ follows Type I Extreme Value Distribution
• Independent of Irrelevant Alternatives (IIA) property

### 2. Random Coefficients Logit Model (BLP Model; Notes)

Example:

• Utility: $u_{itj}=x_{tj}\beta_i-\alpha_i p_{tj}+\xi_{tj}+\epsilon_{itj}$
• $\epsilon_{itj}$ follows Type I Extreme Value Distribution
• $\begin{bmatrix} \beta_i \\ \alpha_i\end{bmatrix}=\begin{bmatrix} \bar{\beta} \\ \bar{\alpha}\end{bmatrix}+\Pi D_i+\Sigma v_i$
• $[\bar{\beta}, \bar{\alpha}]'$: population mean
• $D_i$: customer-level observables
• $v_i$: customer-level unobservables
• $\Sigma$: 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, $i=1,2,\cdots,N$
• Decision $d_{it}\in \{0,1\}$: whether to replace the engine of bus $i$ in period $t$
• State variable $x_{it}$: the cumulative mileage on bus $i$'s engine at time $t$ since the last replacement
• Single-period utility: $u(d_{it}, x_{it}, \epsilon_{it}; \theta)=\begin{cases}-c_r - c(0, c_o) + \epsilon_{i1t} & \textit{if } d_{it}=1 \\ - c(x_{it}, c_o) + \epsilon_{i0t} & \textit{if } d_{it}=0 \end{cases}$
• $c_r$: replacement cost; $c(x_{it}, c_o)$: operating cost

#### 3.2. Dynamic Game Model

• Agents interact with each other, leading to an equilibrium
• A player’s single-period utility: $U(a_t, x_t, \epsilon_{it})$
• States $a_t:=\{a_{1t},\cdots, a_{a_{nt}}\}$; actions $x_t:=\{x_{1t},\cdots, x_{nt}\}$

### 4. Match Models

#### 4. 1. One to One Matching

• Men $M=\{m_1, \cdots, m_k\}$; women: $W=\{w_1,\cdots,w_p\}$
• Preference: $P(m_i)=w_k, w_{\ell}, \cdots, m_i, w_j, \cdots$
• A matching: $\mu=M\cup W \to M\cup W$
• $w=\mu(m)$ if and only if $m=\mu(w)$ (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 $C=\{c_1,\cdots,c_n\}$ with number of positions, $q_1,\cdots,q_n$ and students $S=\{s_1,\cdots,s_p\}$
• Matching $\mu$ is a correspondence such that
• $\mu(s)\in C\cup S$ (a student might be matched with no schools)
• $\mu(C)\in S$ (a school is matched to a group of student)
• $\mu(s)=C$ if and only if $s\in \mu(C)$ for every student $s\in S$ and college $c\in C$
• 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