Introduction

Paul Schrimpf

2024-09-23

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

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.

Extremum Estimators

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

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

Example: Least-Squares

• Model: \[ y_i = x_i'\beta + \epsilon_i \]
• $Q_n() = _{i=1} (y_i - x_i’)^2

Example: Maximum Likelihood

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

Example: 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() = ({i=1}^n g(y_i,x_i,))’ W ({i=1}^n g(y_i,x_i,))

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

Example: Consumption and Assets

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

Example: Consumption and Assets

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

Example: Consumption and Assets

• Assume \(u(c) = \frac{c^{1-\gamma}}{1-\gamma}\) \[ \Er\left[\beta \frac{c_{t+1}^{-\gamma}}{c_t^{-\gamma}} R_t | \mathcal{I}_s \right] = 1 \text{ for } s \leq t \]
• 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

Example: 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 \(\xi_{jt}\)

Example: Random Coefficients Demand

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

Example: 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

Example: 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 $$ when total spending so far is \(x\)

Example: 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 \(x\)

Example: Insurance and Drug Demand

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

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

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.