Introduction

Paul Schrimpf

2026-02-23

Some Motivating Examples

To give some motivation and a common thread for the material that we will cover, we will focus on learning computational techniques useful for structural econometrics. Most of these techniques will be applicable to computational economics and scientific computing more broadly.

\[ \def\Er{{\mathrm{E}}} \def\En{{\mathbb{E}_n}} \def\cov{{\mathrm{Cov}}} \def\var{{\mathrm{Var}}} \def\R{{\mathbb{R}}} \def\arg{{\mathrm{arg}}} \newcommand\norm[1]{\left\lVert#1\right\rVert} \def\rank{{\mathrm{rank}}} \newcommand{\inpr}{ \overset{p^*_{\scriptscriptstyle n}}{\longrightarrow}} \def\inprob{{\,{\buildrel p \over \rightarrow}\,}} \def\indist{\,{\buildrel d \over \rightarrow}\,} \DeclareMathOperator*{\plim}{plim} \DeclareMathOperator*{\argmax}{argmax} \DeclareMathOperator*{\argmin}{argmin} \newcommand{\etae}{{\boldsymbol{\eta}}} \def\ke{{\mathbf{k}}} \def\xe{{\mathbf{x}}} \]

Extremum Estimators

\[ \begin{align*} \hat{\theta} \in \argmin_{\theta \in \Theta} Q_n(\theta) \end{align*} \]

• Parameter \(\theta \in \Theta\)
• Data-dependent objective function \(Q_n: \Theta \to \R\)

Examples

Least-Squares

• Model: \[ y_i = x_i'\beta + \epsilon_i \]
• \(Q_n(\beta) = \frac{1}{n} \sum_{i=1} (y_i - x_i'\beta)^2\)

Maximum Likelihood

• Model: \(y_i\) has conditional pdf \(f(y|x_i;\theta)\), independent across \(i\)
• \(Q_n(\theta) = -\frac{1}{n} \sum_{i=1}^n \log\left(f(y_i | x_i; \theta)\right)\)

Generalized Method of Moments

• Model: \[ \Er\left[g(y_i,x_i,\theta_0) \right] = 0 \] for moment function \(g(y_i,x_i,\cdot): \Theta \to \R^k\)
• \(Q_n(\theta) = \left(\frac{1}{n}\sum_{i=1}^n g(y_i,x_i,\theta)\right)' W \left(\frac{1}{n}\sum_{i=1}^n g(y_i,x_i,\theta)\right)\)

Moment function or conditional density come from some economic model. We will briefly look at some more detailed examples to illustrate.

Example: Consumption and Assets

Consumption and Assets

• Hansen and Singleton (1982)
• Model \[ \begin{align*} \max_{c_t, q_t} & \Er\left[ \sum_{t=0}^\infty \beta^t u(c_t) \Biggm| \mathcal{I}_0 \right] \\ \text{s.t. } & \;\; p_t q_t + c_t \leq (p_t + d_t)q_{t-1} + y_t \end{align*} \]

Consumption and Assets

• Cleverly rearrange first order conditions: \[ \Er\left[\beta \frac{u'(c_{t+1})}{u'(c_t)} \underbrace{\frac{p_{t+1} + d_{t+1}}{p_t}}_{R_t} \Biggm| \mathcal{I}_s \right] = 1 \text{ for } s \leq t \]

Consumption and Assets

• Assume \(u(c) = \frac{c^{1-\gamma}}{1-\gamma}\) \[ \Er\left[\beta \frac{c_{t+1}^{-\gamma}}{c_t^{-\gamma}} R_t \Biggm| \mathcal{I}_s \right] = 1 \text{ for } s \leq t \]
• Model implies \[ \Er\left[\left(\beta \frac{c_{t+1}^{-\gamma}}{c_t^{-\gamma}} R_t -1 \right)Z_t \right] = 0 \] for any \(Z_t \in \mathcal{I}_t\)
• I.e. \[ g(\overbrace{X_t}^{(c_t,c_{t+1},R_t,Z_t)}, \underbrace{\theta}_{(\beta,\gamma)}) = \left(\beta \frac{c_{t+1}^{-\gamma}}{c_t^{-\gamma}} R_t -1 \right)Z_t \]

Example: Random Coefficients Demand

Random Coefficients Demand

• Berry, Levinsohn, and Pakes (1995)
• Consumers choose product: \[ j = \argmax_{j \in \{0, ..., J\}} x_{jt}' (\bar{\beta} + \Sigma \nu_i) + \xi_{jt} + \epsilon_{ijt} \]
  • \(\nu_i \sim N(0,I_k)\), \(\epsilon_{ijt} \sim\) Type I Extreme Value
  • Unobserved demand shock \(\xi_{jt}\)

Random Coefficients Demand

• Aggregate demand: \[ s_{jt} = \int \frac{e^{x_{jt}'(\bar{\beta} + \Sigma \nu) + \xi_{jt}}} {\sum_{k = 0}^J e^{x_{kt}'(\bar{\beta} + \Sigma \nu) + \xi_{kt}} } dF\nu \]

Random Coefficients Demand

• Instruments \(Z_{jt}\) with \(E[\xi_{jt} Z_{jt}] = 0\)
• \(g(s_{jt},x_{jt}, Z_{jt}, \bar{\beta},\Sigma) = \left(\delta_{jt}(s_{\cdot t}, x_{\cdot t},\beta,\Sigma) - x_{jt}'\bar{\beta}\right) Z_{jt}\)
• where \(\delta_{jt}\) solves \[ s_{jt} = \int \frac{e^{\delta_{jt} + x_{jt}'\Sigma \nu}} {\sum_{k = 0}^J e^{\delta_{kt} + x_{kt}'\Sigma \nu}} dF\nu \]

Example: Insurance and Drug Demand

Insurance and Drug Demand

• Einav, Finkelstein, and Schrimpf (2015)
• Risk-neutral forward-looking individual faces uncertain health shocks, choose whether or not to fill prescriptions
• Prescriptions are defined by \((\theta ,\omega )\)

  • \(\theta >0\) is the prescription’s (total) cost

  • \(\omega >0\) is the monetized cost of not taking the drug

  • Arrive at weekly rate \(\lambda\), drawn from \(G(\theta ,\omega)=G_{2}(\omega |\theta )G_{1}(\theta )\)

  • \(\lambda\) follows a Markov process \(H(\lambda |\lambda ^{\prime })\)

• Insurance defines \(c(\theta ,x)\) – the out-of-pocket cost associated with a prescription that costs \(\theta\) when total spending so far is \(x\)

Insurance and Drug Demand

• Flow utility \[ u(\theta ,\omega ;x)=\left \{ \begin{array}{ll} -c(\theta ,x) & if\text{ }filled \\ -\omega & if\text{ }not\text{ }filled% \end{array}% \right. \]

• Bellman equation: \[ \begin{eqnarray*} v(x,t,\lambda _{t+1}) &=&E_{\lambda |\lambda _{t+1}} \\ &&\hspace{-1.35in}\left[ \begin{array}{c} (1-\lambda )\delta v(x,t-1,\lambda )+ \\ \lambda \int \max\left \{ \begin{array}{l} -c(\theta ,x)+\delta v(x+\theta ,t-1,\lambda ), \\ -\omega +\delta v(x,t-1,\lambda )% \end{array}% \right \} dG(\theta ,\omega )% \end{array}% \right] \end{eqnarray*} \] with terminal condition \(v(x,0)=0\) for all \(x\)

Insurance and Drug Demand

• Estimate by simulated method of moments
  • Simulate model
  • Minimize difference between observed summary statistics and summary statistics in simulated data

Example: Pipeline Investment Incentives

Pipeline Investment

• Solimine and Schrimpf (2025)
• Pipeline \(i\), year \(t\), chooses investment \(i_{it}\)
• Capital evolves as \(k_{it+1} = k_{it} + i_{it}\)
• Common exogenous state variable \(s_t\)
• Private investment cost shock \(\eta_{it}\)
• Profits \(\pi(k_{it},\ke_{-it}, s_t)\)
• Investment cost \(\tilde{c}(i_{it},k_{it},\ke_{-it},\eta_{it})\)
• Others strategies \(\sigma_{-i}(\ke_t,s_t,\etae_{-it})\)

Pipeline Investment: Bellman Equation

• Bellman equation \[ \begin{align} V(k_{it},\ke_{-it},s_{t},\eta_{it}) = \max_{i_{it}} & \pi(k_{it},\ke_{-it}, s_{t}) - \tilde{c}(i_{it}, k_{it}, s_t, \eta_{it}) + \\ & + \beta \Er\left[ V(k_{it} + i_{it},\ke_{-it} + \sigma_{-i}(\ke_t,s_t,\etae_{-it}),s_{t+1}) \bigm| s_{t}, \ke_{-it}, k_{it} + i_{it}, \eta_{it} \right] \end{align} \]

Pipeline Investment: First Order Condition and Envelope Theorem

• First order condition \[ \frac{\partial \tilde{c}}{\partial i}(i_{it},\xe_{t}, \eta_{it}) = \beta \Er\left[ \frac{\partial V}{\partial k_i}(\xe_{t+1}, \eta_{it+1}) \biggm| i_{it}, \xe_{t}, \eta_{it} \right] \]

• Envelope theorem

\[ \begin{align} \frac{\partial V}{\partial k_i}(\xe_{t}, \eta_{it}) = & \frac{\partial \pi}{\partial k_i}(\xe_{t},\eta_{it}) - \frac{\partial \tilde{c}}{\partial k_i}(i_{it}, \xe_{t}, \eta_{it}) + \notag \\ & + \beta \Er\left[ \frac{\partial V}{\partial k_i}(\xe_{t+1}, \eta_{it+1}) + \frac{\partial V}{\partial \ke_{-i}}(\xe_{t+1}, \eta_{it+1}) \frac{\partial \sigma_{-i}}{\partial k_i}(\xe_{t}, \etae_{-it}) \biggm | i_{it}, \xe_{t}, \eta_{it} \right] \notag \\ = & \frac{\partial \pi}{\partial k_i}(\xe_{t},\eta_{it}) - \frac{\partial \tilde{c}}{\partial k_i}(i_{it},\xe_{t}, \eta_{it}) + \frac{\partial \tilde{c}}{\partial i}(i_{it}, \xe_{it}, \eta_{it}) + \beta \Er\left[ \frac{\partial V}{\partial \ke_{-i}}(\xe_{t+1},\eta_{it+1}) \frac{\partial \sigma_{-i}}{\partial k_i}(\xe_{t}, \etae_{-it}) \biggm| i_{it}, \xe_t, \eta_{it} \right] \end{align} \]

Pipeline Investment: Euler-like equation

\[ \begin{aligned} \frac{\partial \tilde{c}}{\partial i}(i_{it}, \xe_{t}, \eta_{it}) = & \beta \Er\left[ \frac{\partial \pi}{\partial k}(\xe_{t+1}, \eta_{it+1}) + \frac{\partial \tilde{c}}{\partial i}(i_{it+1}, \xe_{t+1}, \eta_{it+1}) - \frac{\partial \tilde{c}}{\partial k}(i_{it+1}, \xe_{t+1}, \eta_{it+1}) \biggm| i_{it}, \xe_{t}, \eta_{it} \right] + \\ & + \beta^2 \Er\left[\frac{\partial V}{\partial \ke_{-i}}(\xe_{t+2}, \eta_{it+2}) \frac{\partial \sigma_{-i}}{\partial k_i}(\xe_{t+1}, \etae_{-it+1}) \biggm| i_{it}, \xe_{t}, \eta_{it} \right] \end{aligned} \]

• Estimation: \(\pi\) and \(\Er[\cdot|\cdot]\) can be estimated from observed data, use those to recover \(\tilde{c}\)

Pipeline Investment: Identifying Investment Cost

• Conditional expectation operators: \[ \mathcal{E}_{\sigma}(f)(\xe_t,\eta_{it}) = \Er\left[f(\xe_{t+1},\eta_{it+1}) \left( I + \frac{\partial \sigma_{-i}}{\partial \ke_{-i}}(\xe_t, \etae_{-it}) \right)\bigm|i^*(\xe_t,\eta_t),\xe_t,\eta_{it}\right]. \] and \[ \mathcal{E}(f)(\xe_t, \eta_{it}) = \Er[f(\xe_{t+1},\eta_{it+1})| i^*(\xe_t,\eta_{it}),\xe_t, \eta_{it}] \]
• Investment cost solves \[ \frac{\partial \tilde{c}}{\partial i} = \beta \mathcal{E} \left( \frac{\partial \pi}{\partial k} + \frac{\partial \tilde{c}}{\partial i} - \frac{\partial \tilde{c}}{\partial k} + \beta\mathcal{E}\left( \left(\left(I - \beta \mathcal{E}_\sigma \right)^{-1} \left(\frac{\partial \pi}{\partial \ke_{-i}} - \frac{\partial \tilde{c}}{\partial \ke_{-i}} \right)\right) \times \frac{\partial \sigma_{-i}}{\partial k} \right) \right) \]

Implementing

Implementing

• Need to go from mathematical description of model to code
• All examples need to minimize an objective function
  • Helpful to compute both objective function and its derivatives
• Random coefficients demand model also needs to solve nonlinear equations and compute integrals
• Drug example needs numeric dynamic programming - function approximation and numeric integration
• Pipeline example involves differential equation, conditional expectation operators

References

Berry, Steven, James Levinsohn, and Ariel Pakes. 1995. “Automobile Prices in Market Equilibrium.” Econometrica 63 (4): pp. 841–890. http://www.jstor.org/stable/2171802.

Einav, Liran, Amy Finkelstein, and Paul Schrimpf. 2015. “The Response of Drug Expenditure to Nonlinear Contract Design: Evidence from Medicare Part d.” The Quarterly Journal of Economics 130 (2): 841–99. https://doi.org/10.1093/qje/qjv005.

Hansen, Lars Peter, and Kenneth J. Singleton. 1982. “Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models.” Econometrica 50 (5): 1269–86. http://www.jstor.org/stable/1911873.

Solimine, Philip, and Paul Schrimpf. 2025. “Investment and Misallocation in Infrastructure Networks: The Case of u.s. Natural Gas Pipelines.” In Proceedings of the 26th ACM Conference on Economics and Computation, 2. New York, NY, USA: Association for Computing Machinery. https://doi.org/10.1145/3736252.3742482.