Introduction to Structural Econometrics

... background material available at https://github.com/softecon/talks

Scientifically well-posed models make explicit the assumptions used by analysts regarding preferences, technology, the information available to agents, the constraints under which they operate, and the rules of interaction among agents in market and social settings and the sources of variability among agents. (Heckman, 2008)

Why?

• identify deep parameters and mechanisms governing economic behavior

• provide counterfactual policy predictions

• determine optimal policies

• integrate separate results in unified framework

Areas of Application

• Labor Economics

• Industrial Organization

• Health Economics

• Political Economy

• Marketing

• ...

Why not?

• requires skills in economics, numerical methods, and software engineering

• scepticism within (parts) of the profession

• long project durations

• need for large computational resources

• ... pretence of knowledge?

Usefulness of Structural Econometric Models

• Verification How accurately does the computational model solve the underlying equations of the model for the quantities of interest?

• Validation How accurately does the model represent the reality for the quantities of interest?

• Uncertainty Quantification How do the various sources of error and uncertainty feed into uncertainty in the model-based prediction of the quantities of interest?

Developing, Estimating, and Validating Dynamic Microeconometric Models

Research Example

Robust Human Capital Investment under Risk and Ambiguity

• Plausible

  • better description of agent decision problem

• Meaningful

  • reinterpretation of economic phenomenon
  • reevaluation of policy interventions

• Tractable

Starting point ...

Keane, M. and Wolpin, K. (1994). The Solution and Estimation of Discrete Choice Dynamic Programming Models by Simulation and Interpolation: Monte Carlo Evidence. The Review of Economics and Statistics, 76(4):648–672.

Basic Model with Risk

Notation

$$\begin{align*}\begin{array}{ll} &\\ k = 1, ..., K & \text{Alternative}\\ t = 1, ..., T & \text{Time}\\ S(t) & \text{State Space at Time $t$}\\ R_k(S(t),t) & \text{Rewards for Alternative $k$ at Time $t$}\\ d_k(t) & \text{Indicator for Alternative $k$ at Time $t$}\\ \delta & \text{Discount Factor} \end{array}\end{align*}$$

Timing of Information

Decision Tree

Individual's Objective under Risk

$$\begin{align*} \\ V(S(t),t) = \max_{\{d_k(t)\}_{k \in K}} E\left[ \sum_{\tau = t}^T \delta^{\tau - t} \sum_{k\in K}R_k(\tau)d_k(\tau)\Bigg| S(t)\right]\\ \\ \end{align*}$$

• What economic concepts are incorporated, which are missing?

Dynamic Programming

$$\begin{align*} \\ V(S(t),t) = \max_{k \in K}\{V_k(S(t),t)\}, \end{align*}$$

where for all but the final period:

$$\begin{align*} \\ V_k(S(t),t) = R_k(S(t),t) + \delta E\left[V(S(t + 1), t + 1)\mid S(t), d_k(t) = 1\right] \end{align*}$$

• What is the link beteen this solution method and dynamic decision making under uncertainty?

Individual Characteristics

$$ \begin{align*}\begin{array}{ll} &\\ x_{1,t} & \text{Experience in Occupation A at Time $t$}\\ x_{2,t} & \text{Experience in Occupation B at Time $t$}\\ s_{t} & \text{Years of Schooling at Time $t$} \end{array}\end{align*} $$

Reward Functions

$$ \begin{align*}\begin{array}{ll} &\\ R_1(t) & = \exp\{\alpha_{10} + \alpha_{11}s_t + \alpha_{12}x_{1,t} - \alpha_{13}x_{1,t}^2 + \alpha_{14}x_{2,t} - \alpha_{15}x_{2,t}^2 + \epsilon_{1,t} \}\\ R_2(t) & = \exp\{\alpha_{20} + \alpha_{21}s_t + \alpha_{22}x_{1,t} - \alpha_{23}x_{1,t}^2 + \alpha_{24}x_{2,t} - \alpha_{25}x_{2,t}^2 + \epsilon_{2,t} \}\\ R_3(t) & = \left(\beta_0 - \beta_1(1 - d_3(t - 1))\right) - \beta_2 I \left[s_t \geq 12 \right] + \epsilon_{3,t}\\ R_4(t) & = \gamma_0 + \epsilon_{4,t}\\ & \\ \epsilon &\sim N(0, \Sigma_\epsilon)\\ \\ \end{array}\end{align*} $$

• What economic concepts are incorporated, which are missing?

Underlying Model for Reward Function

$$ \begin{align*} \\ \\ R_j(t) = w_j(t) = r_j e_j(t) \qquad j = 1, 2\\ \\ \\ \end{align*} $$

Technology of Skill Production

$$ \begin{align*}\begin{array}{ll} &\\ e_j(t) = \exp\{ \alpha_{j1}s_t + \alpha_{j2}x_{1,t} - \alpha_{j3}x_{1,t}^2 + \alpha_{j4}x_{2,t} - \alpha_{j5}x_{2,t}^2 + \epsilon_{j,t} \}\\ \\ \end{array}\end{align*} $$

Wage Equation $$ \begin{align*}\begin{array}{ll} &\\ \ln w_j(t) = \underbrace{\ln r_j }_{\alpha_{j0}} + \alpha_{j1}s_t + \alpha_{j2}x_{1,t} - \alpha_{j3}x_{1,t}^2 + \alpha_{j4}x_{2,t} - \alpha_{j5}x_{2,t}^2 + \epsilon_{j,t} \\ \end{array}\end{align*} $$

Wages and Schooling

Wages and Experience

State Space

• at time $t$

$$\begin{align*} \\ S(t) & = \{s_t, x_{1,t}, x_{2,t}, d_3(t-1), \epsilon_{1,t}, \epsilon_{2,t}, \epsilon_{3,t}, \epsilon_{4,t}\}\\ \end{align*}$$

• laws of motion

$$\begin{align*} &\\ x_{j, t + 1} & = x_{j, t} + d_j(t) & \forall\quad j \in \{1, 2\} \\[1em] s_{t + 1} & = s_t + d_3(t) & \\[1em] f(\epsilon_{t + 1}\mid S(t), d_k(t)) & = f(\epsilon_{t + 1}\mid \bar{S}(t), d_k(t)) \end{align*} $$

Choices over Time

Basic Model with Risk and Ambiguity

Robust Decision Making

Set of Admissible Beliefs

$$\begin{align*} \\ \\ N = \{\mathcal{N} \in \mathrm{Q}: D_{KL}(\mathcal{N}_0\mid \mathcal{N}) \leq \theta\}\\ \end{align*} $$

Distribution of Labor Market Shocks

$$ \begin{eqnarray*} \\ \begin{pmatrix} \epsilon_{1,t}\\ \epsilon_{2,t} \end{pmatrix} & \sim & \mathcal{N}\left[\left(\begin{array}{c} \mu_{\epsilon_{1, t}}\\ \mu_{\epsilon_{2, t}} \end{array}\right),\left(\begin{array}{cccc} 16\times10^{-4} & 0.00 \\ 0.00 & 25\times10^{-2} \end{array}\right)\right] \end{eqnarray*}$$

Exploring Set of Admissible Beliefs

Admissible Values

Individual's Objective under Ambiguity

$$\begin{align*} &\\ V^*(S(t),t) = \max_{\{d_k(t)\}_{k \in K}} \left\{ \min_{{\mathcal{N}} \in {\mathbb{N}}}E_{{\mathcal{N}}}\left[ \sum_{\tau = t}^T \delta^{\tau - t} \sum_{k\in K}R_k(\tau)d_k(\tau)\Bigg| S(t)\right]\right\}\\ \\ \end{align*}$$

• What economic concepts are incorporated, which are missing?

Robust Dynamic Programming

$$\begin{align*} \\ V^*(S(t),t) = \max_{k \in K}\{V^*_k(S(t),t)\}, \\ \end{align*}$$

where for all but the final period:

$$\begin{align*} \\ V^*_k(S(t),t) = R_k(S(t),t) + \delta \min_{\mathcal{N} \in \mathbb{N}}E_\mathcal{N}\left[V^*(S(t + 1), t + 1)\mid\cdot\right] \end{align*} $$

Quantifying Level of Ambiguity

Changing Schooling Investment

Changing Occupational Sorting

Assessing Model Misspecification

Psychic Costs

The estimated costs of post-secondary education in dynamic life-cycle models of educational choice are usually way too large to be due to tuition costs alone. The importance of psychic costs as a determinant of educational choices is a common finding in the literature.

Modeling Trade-off

Model Misspecification and Psychic Costs Estimates

Ex-Ante Evaluation of Tuition Policy

Contact

Philipp Eisenhauer

• Mail eisenhauer@policy-lab.org


• Web http://eisenhauer.io


• Repository https://github.com/peisenha

Software Engineering for Economists Initiative

• Overview http://softecon.github.io


• Repository https://github.com/softEcon