class: title-slide <br><br><br> # Lecture 7 ## Dynamic Discrete Choice Models ### Tyler Ransom ### ECON 6343, University of Oklahoma --- # Attribution Many of these slides are based on slides written by Peter Arcidiacono. I use them with his permission. --- # Plan for the day 1. Optimal Stopping Problems 2. Finite Horizon Problems - backwards recursion - expectations over how the observed states transition - calculation of expected future utility 3. Infinite Horizon Problems - Solving a fixed-point problem - Rust (1987) bus engine problem --- # Optimal stopping - Today we'll get back to the dynamic models we discussed a few weeks ago - To start with, let's discuss the theory of .hi[optimal stopping] - Gives mathematical context for maximizing rewards or minimizing costs - Optimal stopping problems are by definition dynamic --- # Examples of optimal stopping problems - Many economic problems involve some sort of optimal stopping: - The Secretary Problem (when to hire from a sequence of job candidates) - Seach theory more generally (job search, spousal search, house search, ...) - "Buy/sell/hold" problems (e.g. stock/options trading) - Replacement problems (e.g. infrastructure; see Gillingham, Iskhakov, Munk-Nielsen et al. (2022)) - Optimal stopping problems inherently have a tension between costs and benefits: - It is costly to interview job candidates - But it is also costly to miss out on the best candidate --- # Mathematics of optimal stopping - In a discrete choice setting, .hi[dynamic programming] is the best solution method - Within a discrete choice setting, time can be either continuous or discrete - If continuous time, use Hamiltonians and Differential Equations - If discrete time, use recursive methods - Solution method also depends on the time horizon - If the time horizon is finite, then we can use dynamic programming - If the time horizon is infinite, then need to (also) solve for a fixed point - We will discuss these details today --- # Finite horizon problems - Individual `\(i\)`'s .hi[flow utility] for option `\(j\)` at time `\(t\)` is: `\begin{align*} U_{ijt}&=u_{ijt}+\epsilon_{ijt}\\ &=X_{it}\alpha_j+\epsilon_{ijt} \end{align*}` - `\(i\)` chooses `\(d_{it}\)` to maximize her .hi[expected lifetime utility] `\begin{align*} \max \mathbb{E}\sum_{\tau=t}^T\sum_{j}\beta^{\tau-t}1\left[d_{it}=j\right]U_{ijt}\left(X_{it},\epsilon_{ijt}\right) \end{align*}` --- # Expectations and assumptions - `\(i\)` takes expectations over `\(X\)`'s (future states) and `\(\epsilon\)`'s (future errors) - `\(\epsilon\)`'s are assumed to be iid over time - Future states are not affected by `\(\epsilon\)`'s except through choices: `\begin{align*} \mathbb{E}(X_{t+1}|d_t,...,d_1,\epsilon_t,...,\epsilon_{1})&=\mathbb{E}(X_{t+1}|d_t,...,d_1) \end{align*}` --- # Two-period example - Consider the utility of choice `\(j\)` in the last period: `\begin{align*} U_{ijT}&=u_{ijT}+\epsilon_{ijT}\\ &=X_{iT}\alpha_j+\epsilon_{ijT} \end{align*}` - Define the .hi[conditional valuation function] for choice `\(j\)` as the flow utility of `\(j\)` minus the associated `\(\epsilon\)` plus the expected value of future utility conditional on `\(j\)`: `\begin{align*} v_{ijT-1}&=u_{ijT-1}+\beta \mathbb{E}\max_{k\in J}\left\{u_{ikT}+\epsilon_{ikT}|d_{iT-1}=j\right\} \end{align*}` where `\(\beta\)` is the discount factor - Suppose `\(X_{iT}\)` was deterministic given `\(X_{iT-1}\)` and `\(d_{iT-1}\)` and the `\(\epsilon\)`'s are T1EV - What would the `\(\mathbb{E}\max\)` expression be? --- # When Dynamics Don't Matter - As in static models, we need to normalize with respect to some alternative - Suppose we normalize with respect to `\(j'\)`: `\begin{align*} v_{ijT-1}-v_{ij'T-1}&=u_{ijT-1}+\beta \mathbb{E}\max_{k\in J}\left\{u_{ikT}+\epsilon_{ikT}|d_{iT-1}=j\right\}-\\ &\phantom{\text{-}-}u_{ij'T-1}-\beta \mathbb{E}\max_{k\in J}\left\{u_{ikT}+\epsilon_{ikT}|d_{iT-1}=j'\right\} \end{align*}` - If the two expected future value terms are equal, we get a cancellation `\(\implies\)` choices have to (at least probabilistically) affect the future states - The simplest way to satisfy this condition is to have switching costs in the model - Intuition: switching costs make one think carefully about changing course --- # Finite Horizon Dynamics In period `\(T-1\)` we have: `\begin{align*} v_{ijT-1}=u_{ijT-1}+\beta \mathbb{E}\max_{k\in J}\left\{u_{ikT}+\epsilon_{ikT}|d_{iT-1}=j\right\} \end{align*}` Rolling back one more period, `\begin{align*} v_{ijT-2}=u_{ijT-2}+\beta \mathbb{E}\max_{k\in J}\left\{v_{ikT-1}+\epsilon_{ikT-1}|d_{iT-2}=j\right\} \end{align*}` Keep going back and the `\(\mathbb{E}\max\)` operator can always be expressed as functions of the next period conditional value functions: `\begin{align*} v_{ijt}=u_{ijt}+\beta \mathbb{E}\max_{k\in J}\left\{v_{ikt+1}+\epsilon_{ikt+1}|d_{it}=j\right\} \end{align*}` Another name for `\(\mathbb{E}\max_{k\in J}\left\{v_{ikt+1}+\epsilon_{ikt+1}\right\}\)` is.... --- # Stochastic `\(X\)`'s - Let `\(f_{jt}(X_{it+1}|X_{it})\)` be the pdf associated with moving from `\(X_{it}\)` to `\(X_{it+1}\)` given choice `\(j\)` at time `\(t\)` .hi[Example:] suppose we were interested in Covid on OU's campus. The choice set is {close campus, open campus but online classes, in-person classes}. The transitions on the `\(X\)`'s would be the Covid case (or fatality) counts associated with each of the choices. Since these Covid case probabilities do not depend on the `\(\epsilon\)`'s, it is convenient to integrate them out of the future utility term The conditional value function is then: `\begin{align*} v_{jt}(X_{it})&=u_{jt}(X_{it})+\beta \int_{X_{it+1}}\mathbb{E}_{\epsilon}\left\{\max_{k\in J} v_{kt+1}(X_{it+1})+\epsilon_{ikt+1}\right\}dF_{jt}(X_{it+1}|X_{it}) \end{align*}` --- # Stochastic `\(X\)`'s 2 If the `\(\epsilon\)`'s are distributed Type 1 extreme value, what is the expression for the conditional value function? What about the general GEV case? We can then start at the last period and work our way backwards to obtain all of the relevant conditional value functions --- # Choice Probabilities - The choice probabilities are then calculated in the same way as in the static case - The only difference is now we use `\(v\)`'s instead of `\(u\)`'s - In the multinomial logit case we have: `\begin{align*} p_{jt}(X_{it})&=\frac{\exp(v_{jt}(X_{it}))}{\sum_{k\in J}\exp(v_{kt}(X_{it}))} \end{align*}` --- # Estimation - The likelihood of the data is: `\begin{align*} \mathcal{L}(\alpha,\beta,\gamma)&=\prod_i\prod_t\prod_j\left[p_{jt}(X_{it},\alpha,\beta,\gamma)f_{jt}(X_{it+1}|X_{it},\gamma)\right]^{d_{it}=j} \end{align*}` where `\(\gamma\)` governs the transitions of the `\(X\)`'s - The log likelihood is then given by: `\begin{align*} \ell(\alpha,\beta,\gamma)&=\sum_i\sum_t\sum_j (d_{it}=j)\left\{\ln[p_{jt}(X_{it},\alpha,\beta,\gamma)]+\ln[f_{jt}(X_{it+1}|X_{it},\gamma)]\right\} \end{align*}` Since the log likelihood function is additively separable, we can estimate `\(\gamma\)` in a first stage --- # Infinite Horizon `\begin{align*} v_{j}(X_{i})&=u_{j}(X_{i})+\beta \int_{X'}V(X')dF_{j}(X'|X_{i})\\ &=u_{j}(X_{i})+\beta \int_{X'}E_{\epsilon'}\left(\max_{k\in J} v_{k}(X')+\epsilon'_{ik}\right)dF_{j}(X'|X_{i})\\ \end{align*}` which in the Type 1 extreme value case for the `\(\epsilon\)`'s yields: `\begin{align*} v_j(X_i)=u_j(X_i)+\beta\int_{X'}\ln\left(\sum_{k\in J}\exp[v_{k}(X')]\right)dF_j(X'|X_i)+\beta c \end{align*}` Now, stack the conditional value functions for each possible state and choice. Because the `\(v\)`'s are on both sides of the stacked equations, we need to solve for a fixed point (This works because it is a contraction mapping) --- # Infinite Horizon 2 Let `\(\mathcal{X}\)` denote the number of states `\(X\)` can take on The stacked equations are then: .smallest[ `\begin{align*} \left[\begin{array}{c}v_1(X_1)\\ v_1(X_2)\\ \vdots \\ v_1(X_{\mathcal{X}})\\ \vdots\\ v_{J}(X_{\mathcal{X}})\end{array}\right]= \left[\begin{array}{c}u_1(X_1)+\beta\int_{X'}\ln\left(\sum_{k\in J}\exp[v_{k}(X')]\right)dF_1(X'|X_1)+\beta c\\ u_1(X_2)+\beta\int_{X'}\ln\left(\sum_{k\in J}\exp[v_{k}(X')]\right)dF_1(X'|X_2)+\beta c\\ \vdots\\ u_1(X_{\mathcal{X}})+\beta\int_{X'}\ln\left(\sum_{k\in J}\exp[v_{k}(X')]\right)dF_1(X'|X_{\mathcal{X}})+\beta c\\ \vdots\\ u_J(X_{\mathcal{X}})+\beta\int_{X'}\ln\left(\sum_{k\in J}\exp[v_{k}(X')]\right)dF_J(X'|X_{\mathcal{X}})+\beta c\\ \end{array}\right] \end{align*}` ] - Plug in values for the parameters and take a guess at the `\(v\)`'s - Substitute in for the `\(v\)`'s on the right hand side which gives us a new set of `\(v\)`'s - Repeat until convergence --- # Optimal stopping in Rust (1987) - Rust analyzes the decision to replace a bus engine `\((d=1)\)` or not `\((d=0)\)` - How is this an optimal stopping problem? - The maintenance superintendent Harold Zurcher wants to minimize costs - But he also doesn't want buses to break down while in service - Premature replacement can be very costly, but so is in-service breakdown - The goal is then to figure out when to optimally replace engines - Especially when some buses might happen to get driven more than others --- # Rust (1987) - Replacement decision depends upon the mileage on the engine, `\(x\)`, the cost of replacing the engine `\(\overline{P}\)` and the scrap value of the current engine `\(\underline{P}\)` - The payoffs net of the error term are given by: `\begin{align*} u_0(x_i,\theta)&=-c(x_i,\theta)\\ u_1(x_i,\theta)&=-[\overline{P}-\underline{P}+c(0,\theta)] \end{align*}` - Mileage is discrete and transitions according to some process `\(f(x_{t+1}|x_t)\)` - Example: some probability of staying at the current mileage, some probability of moving up one mileage state and some probability of moving up two mileage states --- # Estimation 1. Calculate the mileage transitions, i.e. get `\(f(x_{t+1}|x_t)\)` 2. Maximize the log likelihood of the choices: `\begin{align*} \ell(\theta)=\sum_i\sum_t \sum_j (d_{it}=j)\ln(p_{jt}(x_{it},\theta)) \end{align*}` 3. Within the maximization routine, solve a fixed point problem in the `\(v\)`'s each time the log likelihood function is evaluated - For your problem set, I'll walk you through how to estimate a model similar to Rust's --- # Nested Fixed Point (NFXP) Algorithm .center[ <img src="https://editorialexpress.com/jrust/nfxp.gif" width="50%" /> Source: https://editorialexpress.com/jrust/nfxp.html ] --- # References .smaller[ Ackerberg, D. A. (2003). "Advertising, Learning, and Consumer Choice in Experience Good Markets: An Empirical Examination". In: _International Economic Review_ 44.3, pp. 1007-1040. DOI: [10.1111/1468-2354.t01-2-00098](https://doi.org/10.1111%2F1468-2354.t01-2-00098). Adams, R. P. (2018). _Model Selection and Cross Validation_. Lecture Notes. Princeton University. URL: [https://www.cs.princeton.edu/courses/archive/fall18/cos324/files/model-selection.pdf](https://www.cs.princeton.edu/courses/archive/fall18/cos324/files/model-selection.pdf). Ahlfeldt, G. M., S. J. Redding, D. M. Sturm, et al. (2015). "The Economics of Density: Evidence From the Berlin Wall". In: _Econometrica_ 83.6, pp. 2127-2189. DOI: [10.3982/ECTA10876](https://doi.org/10.3982%2FECTA10876). Altonji, J. G., T. E. Elder, and C. R. Taber (2005). "Selection on Observed and Unobserved Variables: Assessing the Effectiveness of Catholic Schools". In: _Journal of Political Economy_ 113.1, pp. 151-184. DOI: [10.1086/426036](https://doi.org/10.1086%2F426036). Altonji, J. G. and C. R. Pierret (2001). "Employer Learning and Statistical Discrimination". In: _Quarterly Journal of Economics_ 116.1, pp. 313-350. DOI: [10.1162/003355301556329](https://doi.org/10.1162%2F003355301556329). Angrist, J. D. and A. B. Krueger (1991). "Does Compulsory School Attendance Affect Schooling and Earnings?" In: _Quarterly Journal of Economics_ 106.4, pp. 979-1014. DOI: [10.2307/2937954](https://doi.org/10.2307%2F2937954). Angrist, J. D. and J. Pischke (2009). _Mostly Harmless Econometrics: An Empiricist's Companion_. Princeton University Press. ISBN: 0691120358. Arcidiacono, P. (2004). "Ability Sorting and the Returns to College Major". In: _Journal of Econometrics_ 121, pp. 343-375. DOI: [10.1016/j.jeconom.2003.10.010](https://doi.org/10.1016%2Fj.jeconom.2003.10.010). Arcidiacono, P., E. Aucejo, A. Maurel, et al. (2016). _College Attrition and the Dynamics of Information Revelation_. Working Paper. Duke University. URL: [https://tyleransom.github.io/research/CollegeDropout2016May31.pdf](https://tyleransom.github.io/research/CollegeDropout2016May31.pdf). Arcidiacono, P., E. Aucejo, A. Maurel, et al. (2025). "College Attrition and the Dynamics of Information Revelation". In: _Journal of Political Economy_ 133.1. DOI: [10.1086/732526](https://doi.org/10.1086%2F732526). Arcidiacono, P. and J. B. Jones (2003). "Finite Mixture Distributions, Sequential Likelihood and the EM Algorithm". In: _Econometrica_ 71.3, pp. 933-946. DOI: [10.1111/1468-0262.00431](https://doi.org/10.1111%2F1468-0262.00431). Arcidiacono, P., J. Kinsler, and T. Ransom (2022b). "Asian American Discrimination in Harvard Admissions". In: _European Economic Review_ 144, p. 104079. DOI: [10.1016/j.euroecorev.2022.104079](https://doi.org/10.1016%2Fj.euroecorev.2022.104079). Arcidiacono, P., J. Kinsler, and T. Ransom (2022a). "Legacy and Athlete Preferences at Harvard". In: _Journal of Labor Economics_ 40.1, pp. 133-156. DOI: [10.1086/713744](https://doi.org/10.1086%2F713744). Arcidiacono, P. and R. A. Miller (2011). "Conditional Choice Probability Estimation of Dynamic Discrete Choice Models With Unobserved Heterogeneity". In: _Econometrica_ 79.6, pp. 1823-1867. DOI: [10.3982/ECTA7743](https://doi.org/10.3982%2FECTA7743). Arroyo Marioli, F., F. Bullano, S. Kucinskas, et al. (2020). _Tracking R of COVID-19: A New Real-Time Estimation Using the Kalman Filter_. Working Paper. medRxiv. DOI: [10.1101/2020.04.19.20071886](https://doi.org/10.1101%2F2020.04.19.20071886). Ashworth, J., V. J. Hotz, A. Maurel, et al. (2021). "Changes across Cohorts in Wage Returns to Schooling and Early Work Experiences". In: _Journal of Labor Economics_ 39.4, pp. 931-964. DOI: [10.1086/711851](https://doi.org/10.1086%2F711851). Attanasio, O. P., C. Meghir, and A. Santiago (2011). "Education Choices in Mexico: Using a Structural Model and a Randomized Experiment to Evaluate PROGRESA". In: _Review of Economic Studies_ 79.1, pp. 37-66. DOI: [10.1093/restud/rdr015](https://doi.org/10.1093%2Frestud%2Frdr015). Aucejo, E. M. and J. James (2019). "Catching Up to Girls: Understanding the Gender Imbalance in Educational Attainment Within Race". In: _Journal of Applied Econometrics_ 34.4, pp. 502-525. DOI: [10.1002/jae.2699](https://doi.org/10.1002%2Fjae.2699). Baragatti, M., A. Grimaud, and D. Pommeret (2013). "Likelihood-free Parallel Tempering". In: _Statistics and Computing_ 23.4, pp. 535-549. DOI: [ 10.1007/s11222-012-9328-6](https://doi.org/%2010.1007%2Fs11222-012-9328-6). Bayer, P., R. McMillan, A. Murphy, et al. (2016). "A Dynamic Model of Demand for Houses and Neighborhoods". In: _Econometrica_ 84.3, pp. 893-942. DOI: [10.3982/ECTA10170](https://doi.org/10.3982%2FECTA10170). Begg, C. B. and R. Gray (1984). "Calculation of Polychotomous Logistic Regression Parameters Using Individualized Regressions". In: _Biometrika_ 71.1, pp. 11-18. DOI: [10.1093/biomet/71.1.11](https://doi.org/10.1093%2Fbiomet%2F71.1.11). Beggs, S. D., N. S. Cardell, and J. Hausman (1981). "Assessing the Potential Demand for Electric Cars". In: _Journal of Econometrics_ 17.1, pp. 1-19. DOI: [10.1016/0304-4076(81)90056-7](https://doi.org/10.1016%2F0304-4076%2881%2990056-7). Berry, S., J. Levinsohn, and A. Pakes (1995). "Automobile Prices in Market Equilibrium". In: _Econometrica_ 63.4, pp. 841-890. URL: [http://www.jstor.org/stable/2171802](http://www.jstor.org/stable/2171802). Blass, A. A., S. Lach, and C. F. Manski (2010). "Using Elicited Choice Probabilities to Estimate Random Utility Models: Preferences for Electricity Reliability". In: _International Economic Review_ 51.2, pp. 421-440. DOI: [10.1111/j.1468-2354.2010.00586.x](https://doi.org/10.1111%2Fj.1468-2354.2010.00586.x). Blundell, R. (2010). "Comments on: ``Structural vs. Atheoretic Approaches to Econometrics'' by Michael Keane". In: _Journal of Econometrics_ 156.1, pp. 25-26. DOI: [10.1016/j.jeconom.2009.09.005](https://doi.org/10.1016%2Fj.jeconom.2009.09.005). Bresnahan, T. F., S. Stern, and M. Trajtenberg (1997). "Market Segmentation and the Sources of Rents from Innovation: Personal Computers in the Late 1980s". In: _The RAND Journal of Economics_ 28.0, pp. S17-S44. DOI: [10.2307/3087454](https://doi.org/10.2307%2F3087454). Brien, M. J., L. A. Lillard, and S. Stern (2006). "Cohabitation, Marriage, and Divorce in a Model of Match Quality". In: _International Economic Review_ 47.2, pp. 451-494. DOI: [10.1111/j.1468-2354.2006.00385.x](https://doi.org/10.1111%2Fj.1468-2354.2006.00385.x). Card, D. (1995). "Using Geographic Variation in College Proximity to Estimate the Return to Schooling". In: _Aspects of Labor Market Behaviour: Essays in Honour of John Vanderkamp_. Ed. by L. N. Christofides, E. K. Grant and R. Swidinsky. Toronto: University of Toronto Press. Cardell, N. S. (1997). "Variance Components Structures for the Extreme-Value and Logistic Distributions with Application to Models of Heterogeneity". In: _Econometric Theory_ 13.2, pp. 185-213. URL: [https://www.jstor.org/stable/3532724](https://www.jstor.org/stable/3532724). Caucutt, E. M., L. Lochner, J. Mullins, et al. (2020). _Child Skill Production: Accounting for Parental and Market-Based Time and Goods Investments_. Working Paper 27838. National Bureau of Economic Research. DOI: [10.3386/w27838](https://doi.org/10.3386%2Fw27838). Chen, X., H. Hong, and D. Nekipelov (2011). "Nonlinear Models of Measurement Errors". In: _Journal of Economic Literature_ 49.4, pp. 901-937. DOI: [10.1257/jel.49.4.901](https://doi.org/10.1257%2Fjel.49.4.901). Chintagunta, P. K. (1992). "Estimating a Multinomial Probit Model of Brand Choice Using the Method of Simulated Moments". In: _Marketing Science_ 11.4, pp. 386-407. DOI: [10.1287/mksc.11.4.386](https://doi.org/10.1287%2Fmksc.11.4.386). Cinelli, C. and C. Hazlett (2020). "Making Sense of Sensitivity: Extending Omitted Variable Bias". In: _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ 82.1, pp. 39-67. DOI: [10.1111/rssb.12348](https://doi.org/10.1111%2Frssb.12348). Coate, P. and K. Mangum (2019). _Fast Locations and Slowing Labor Mobility_. Working Paper 19-49. Federal Reserve Bank of Philadelphia. Cunha, F., J. J. Heckman, and S. M. Schennach (2010). "Estimating the Technology of Cognitive and Noncognitive Skill Formation". In: _Econometrica_ 78.3, pp. 883-931. DOI: [10.3982/ECTA6551](https://doi.org/10.3982%2FECTA6551). Cunningham, S. (2021). _Causal Inference: The Mixtape_. Yale University Press. URL: [https://www.scunning.com/causalinference_norap.pdf](https://www.scunning.com/causalinference_norap.pdf). Delavande, A. and C. F. Manski (2015). "Using Elicited Choice Probabilities in Hypothetical Elections to Study Decisions to Vote". In: _Electoral Studies_ 38, pp. 28-37. DOI: [10.1016/j.electstud.2015.01.006](https://doi.org/10.1016%2Fj.electstud.2015.01.006). Delavande, A. and B. Zafar (2019). "University Choice: The Role of Expected Earnings, Nonpecuniary Outcomes, and Financial Constraints". In: _Journal of Political Economy_ 127.5, pp. 2343-2393. DOI: [10.1086/701808](https://doi.org/10.1086%2F701808). Diegert, P., M. A. Masten, and A. Poirier (2025). _Assessing Omitted Variable Bias when the Controls are Endogenous_. arXiv. DOI: [10.48550/ARXIV.2206.02303](https://doi.org/10.48550%2FARXIV.2206.02303). Erdem, T. and M. P. Keane (1996). "Decision-Making under Uncertainty: Capturing Dynamic Brand Choice Processes in Turbulent Consumer Goods Markets". In: _Marketing Science_ 15.1, pp. 1-20. DOI: [10.1287/mksc.15.1.1](https://doi.org/10.1287%2Fmksc.15.1.1). Evans, R. W. (2018). _Simulated Method of Moments (SMM) Estimation_. QuantEcon Note. University of Chicago. URL: [https://notes.quantecon.org/submission/5b3db2ceb9eab00015b89f93](https://notes.quantecon.org/submission/5b3db2ceb9eab00015b89f93). Farber, H. S. and R. Gibbons (1996). "Learning and Wage Dynamics". In: _Quarterly Journal of Economics_ 111.4, pp. 1007-1047. DOI: [10.2307/2946706](https://doi.org/10.2307%2F2946706). Fu, C., N. Grau, and J. Rivera (2020). _Wandering Astray: Teenagers' Choices of Schooling and Crime_. Working Paper. University of Wisconsin-Madison. URL: [https://www.ssc.wisc.edu/~cfu/wander.pdf](https://www.ssc.wisc.edu/~cfu/wander.pdf). Gillingham, K., F. Iskhakov, A. Munk-Nielsen, et al. (2022). "Equilibrium Trade in Automobiles". In: _Journal of Political Economy_. DOI: [10.1086/720463](https://doi.org/10.1086%2F720463). Haile, P. (2019). _``Structural vs. Reduced Form'' Language and Models in Empirical Economics_. Lecture Slides. Yale University. URL: [http://www.econ.yale.edu/~pah29/intro.pdf](http://www.econ.yale.edu/~pah29/intro.pdf). Haile, P. (2024). _Models, Measurement, and the Language of Empirical Economics_. Lecture Slides. Yale University. URL: [https://www.dropbox.com/s/8kwtwn30dyac18s/intro.pdf](https://www.dropbox.com/s/8kwtwn30dyac18s/intro.pdf). Heckman, J. J., J. Stixrud, and S. Urzua (2006). "The Effects of Cognitive and Noncognitive Abilities on Labor Market Outcomes and Social Behavior". In: _Journal of Labor Economics_ 24.3, pp. 411-482. DOI: [10.1086/504455](https://doi.org/10.1086%2F504455). Hotz, V. J. and R. A. Miller (1993). "Conditional Choice Probabilities and the Estimation of Dynamic Models". In: _The Review of Economic Studies_ 60.3, pp. 497-529. DOI: [10.2307/2298122](https://doi.org/10.2307%2F2298122). Hurwicz, L. (1950). "Generalization of the Concept of Identification". In: _Statistical Inference in Dynamic Economic Models_. Hoboken, NJ: John Wiley and Sons, pp. 245-257. Ishimaru, S. (2022). _Geographic Mobility of Youth and Spatial Gaps in Local College and Labor Market Opportunities_. Working Paper. Hitotsubashi University. James, J. (2011). _Ability Matching and Occupational Choice_. Working Paper 11-25. Federal Reserve Bank of Cleveland. James, J. (2017). "MM Algorithm for General Mixed Multinomial Logit Models". In: _Journal of Applied Econometrics_ 32.4, pp. 841-857. DOI: [10.1002/jae.2532](https://doi.org/10.1002%2Fjae.2532). Jin, H. and H. Shen (2020). "Foreign Asset Accumulation Among Emerging Market Economies: A Case for Coordination". In: _Review of Economic Dynamics_ 35.1, pp. 54-73. DOI: [10.1016/j.red.2019.04.006](https://doi.org/10.1016%2Fj.red.2019.04.006). Keane, M. P. (2010). "Structural vs. Atheoretic Approaches to Econometrics". In: _Journal of Econometrics_ 156.1, pp. 3-20. DOI: [10.1016/j.jeconom.2009.09.003](https://doi.org/10.1016%2Fj.jeconom.2009.09.003). Keane, M. P. and K. I. Wolpin (1997). "The Career Decisions of Young Men". In: _Journal of Political Economy_ 105.3, pp. 473-522. DOI: [10.1086/262080](https://doi.org/10.1086%2F262080). Koopmans, T. C. and O. Reiersol (1950). "The Identification of Structural Characteristics". In: _The Annals of Mathematical Statistics_ 21.2, pp. 165-181. URL: [http://www.jstor.org/stable/2236899](http://www.jstor.org/stable/2236899). Kosar, G., T. Ransom, and W. van der Klaauw (2022). "Understanding Migration Aversion Using Elicited Counterfactual Choice Probabilities". In: _Journal of Econometrics_ 231.1, pp. 123-147. DOI: [10.1016/j.jeconom.2020.07.056](https://doi.org/10.1016%2Fj.jeconom.2020.07.056). Krauth, B. (2016). "Bounding a Linear Causal Effect Using Relative Correlation Restrictions". In: _Journal of Econometric Methods_ 5.1, pp. 117-141. DOI: [10.1515/jem-2013-0013](https://doi.org/10.1515%2Fjem-2013-0013). Lang, K. and M. D. Palacios (2018). _The Determinants of Teachers' Occupational Choice_. Working Paper 24883. National Bureau of Economic Research. DOI: [10.3386/w24883](https://doi.org/10.3386%2Fw24883). Lee, D. S., J. McCrary, M. J. Moreira, et al. (2020). _Valid t-ratio Inference for IV_. Working Paper. arXiv. URL: [https://arxiv.org/abs/2010.05058](https://arxiv.org/abs/2010.05058). Lewbel, A. (2019). "The Identification Zoo: Meanings of Identification in Econometrics". In: _Journal of Economic Literature_ 57.4, pp. 835-903. DOI: [10.1257/jel.20181361](https://doi.org/10.1257%2Fjel.20181361). Mahoney, N. (2022). "Principles for Combining Descriptive and Model-Based Analysis in Applied Microeconomics Research". In: _Journal of Economic Perspectives_ 36.3, pp. 211-22. DOI: [10.1257/jep.36.3.211](https://doi.org/10.1257%2Fjep.36.3.211). McFadden, D. (1978). "Modelling the Choice of Residential Location". In: _Spatial Interaction Theory and Planning Models_. Ed. by A. Karlqvist, L. Lundqvist, F. Snickers and J. W. Weibull. Amsterdam: North Holland, pp. 75-96. McFadden, D. (1989). "A Method of Simulated Moments for Estimation of Discrete Response Models Without Numerical Integration". In: _Econometrica_ 57.5, pp. 995-1026. DOI: [10.2307/1913621](https://doi.org/10.2307%2F1913621). URL: [http://www.jstor.org/stable/1913621](http://www.jstor.org/stable/1913621). Mellon, J. (2020). _Rain, Rain, Go Away: 137 Potential Exclusion-Restriction Violations for Studies Using Weather as an Instrumental Variable_. Working Paper. University of Manchester. URL: [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3715610](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3715610). Miller, R. A. (1984). "Job Matching and Occupational Choice". In: _Journal of Political Economy_ 92.6, pp. 1086-1120. DOI: [10.1086/261276](https://doi.org/10.1086%2F261276). Mincer, J. (1974). _Schooling, Experience and Earnings_. New York: Columbia University Press for National Bureau of Economic Research. Ost, B., W. Pan, and D. Webber (2018). "The Returns to College Persistence for Marginal Students: Regression Discontinuity Evidence from University Dismissal Policies". In: _Journal of Labor Economics_ 36.3, pp. 779-805. DOI: [10.1086/696204](https://doi.org/10.1086%2F696204). Oster, E. (2019). "Unobservable Selection and Coefficient Stability: Theory and Evidence". In: _Journal of Business & Economic Statistics_ 37.2, pp. 187-204. DOI: [10.1080/07350015.2016.1227711](https://doi.org/10.1080%2F07350015.2016.1227711). Pischke, S. (2007). _Lecture Notes on Measurement Error_. Lecture Notes. London School of Economics. URL: [http://econ.lse.ac.uk/staff/spischke/ec524/Merr_new.pdf](http://econ.lse.ac.uk/staff/spischke/ec524/Merr_new.pdf). Ransom, M. R. and T. Ransom (2018). "Do High School Sports Build or Reveal Character? Bounding Causal Estimates of Sports Participation". In: _Economics of Education Review_ 64, pp. 75-89. DOI: [10.1016/j.econedurev.2018.04.002](https://doi.org/10.1016%2Fj.econedurev.2018.04.002). Ransom, T. (2022). "Labor Market Frictions and Moving Costs of the Employed and Unemployed". In: _Journal of Human Resources_ 57.S, pp. S137-S166. DOI: [10.3368/jhr.monopsony.0219-10013R2](https://doi.org/10.3368%2Fjhr.monopsony.0219-10013R2). Rudik, I. (2020). "Optimal Climate Policy When Damages Are Unknown". In: _American Economic Journal: Economic Policy_ 12.2, pp. 340-373. DOI: [10.1257/pol.20160541](https://doi.org/10.1257%2Fpol.20160541). Rust, J. (1987). "Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher". In: _Econometrica_ 55.5, pp. 999-1033. URL: [http://www.jstor.org/stable/1911259](http://www.jstor.org/stable/1911259). Shalizi, C. R. (2019). _Advanced Data Analysis from an Elementary Point of View_. Cambridge University Press. URL: [http://www.stat.cmu.edu/~cshalizi/ADAfaEPoV/ADAfaEPoV.pdf](http://www.stat.cmu.edu/~cshalizi/ADAfaEPoV/ADAfaEPoV.pdf). Smith Jr., A. A. (2008). "Indirect Inference". In: _The New Palgrave Dictionary of Economics_. Ed. by S. N. Durlauf and L. E. Blume. Vol. 1-8. London: Palgrave Macmillan. DOI: [10.1007/978-1-349-58802-2](https://doi.org/10.1007%2F978-1-349-58802-2). URL: [http://www.econ.yale.edu/smith/palgrave7.pdf](http://www.econ.yale.edu/smith/palgrave7.pdf). Stinebrickner, R. and T. Stinebrickner (2014a). "Academic Performance and College Dropout: Using Longitudinal Expectations Data to Estimate a Learning Model". In: _Journal of Labor Economics_ 32.3, pp. 601-644. DOI: [10.1086/675308](https://doi.org/10.1086%2F675308). Stinebrickner, R. and T. R. Stinebrickner (2014b). "A Major in Science? Initial Beliefs and Final Outcomes for College Major and Dropout". In: _Review of Economic Studies_ 81.1, pp. 426-472. DOI: [10.1093/restud/rdt025](https://doi.org/10.1093%2Frestud%2Frdt025). Su, C. and K. L. Judd (2012). "Constrained Optimization Approaches to Estimation of Structural Models". In: _Econometrica_ 80.5, pp. 2213-2230. DOI: [10.3982/ECTA7925](https://doi.org/10.3982%2FECTA7925). Train, K. (2009). _Discrete Choice Methods with Simulation_. 2nd ed. Cambridge; New York: Cambridge University Press. ISBN: 9780521766555. Wiswall, M. and B. Zafar (2018). "Preference for the Workplace, Investment in Human Capital, and Gender". In: _Quarterly Journal of Economics_ 133.1, pp. 457-507. DOI: [10.1093/qje/qjx035](https://doi.org/10.1093%2Fqje%2Fqjx035). Young, A. (2020). _Consistency without Inference: Instrumental Variables in Practical Application_. Working Paper. London School of Economics. ]