class: title-slide <br><br><br> # Lecture 12 ## Factor Models ### Tyler Ransom ### ECON 6343, University of Oklahoma --- # Plan for the Day 1. Discuss proxy variables & measurement error 2. Methods for dimensionality reduction 3. Economic content of factor models 4. Examples of factor models --- # Attribution I gratefully acknowledge Esteban Aucejo for sharing his slides on factor models, some of which I incorporated in what follows. I also based some content on Shalizi (2019) which is an excellent textbook on data analysis. --- # What are proxy variables? .center[  ] [Source](https://xkcd.com/2652/) --- # Proxy variables - Suppose we have a simple linear regression model `\begin{align*} y &= X\beta + \varepsilon \end{align*}` - If `\((y,X)\)` come from observational data, the model is likely confounded - That is, our OLS estimate of `\(\hat{\beta}\)` would be biased (because `\(X\)` is correlated with `\(\varepsilon\)`) - One potential way to remove the bias is to include a .hi[proxy variable] - This is a variable that we can observe and that is related to something in `\(\varepsilon\)` - Example: unobserved ability biases returns to schooling. IQ might be a viable proxy --- # A brief review of measurement error - What if our proxy is measured with error? This can cause econometric problems - In linear regression, under "classical measurement error" (CEV) assumption: - OLS estimates are attenuated to zero (i.e. "attenuation bias") - OLS t-stats are biased downwards - There are other "non-classical" forms of measurement error as well - See Pischke (2007) for a good treatment of ME - Naturally, measurement error is a beast in non-linear models - See Chen, Hong, and Nekipelov (2011) for a complete treatment --- # Proxy variables & measurement error - Unless they happen to resolve the endogeneity problem, proxy variables won't work - (Need `\(\mathbb{E}\left(\varepsilon \vert X, proxy\right) = \mathbb{E}\left(\varepsilon \vert proxy\right)\)`) - And usually proxies don't satisfy this requirement - You can use instrumental variables to solve the ME problem - But only in linear models - And, of course, instrument validity is almost always in question - So it seems you have to choose between omitted variable bias and attenuation bias --- # What if we have many correlated proxies? - For the unobserved ability question, we might have many different proxies - e.g. individuals might take multiple standardized tests - How do we know which test scores to attempt to use as proxies? - What if each test itself suffers from measurement error? - What if the test scores are highly correlated with each other? - Today we'll talk about how to handle this situation - The application focuses on measuring ability - But this approach is generally applicable when we have many noisy measurements --- # Dimensionality reduction - .hi[Dimensionality reduction] is a common task in data analysis - If 3 variables all give the same information, why not just have 1? - There are two related methods for reducing dimensionality 1. .hi[Principle Components Analysis (PCA)] 2. .hi[Factor Analysis] --- # PCA - PCA is one way to reduce dimensionality. Let `\(M\)` be an `\(N\times J\)` matrix of data - Decompose `\(M\)` as follows: `\begin{align*} M &= \boldsymbol{\theta}\Lambda\\ [N\times J] &= [N\times J] [J\times J] \end{align*}` - `\(\Lambda\)` are stacked eigenvectors - `\(\boldsymbol{\theta}\)` is an .hi[orthogonalized transformation] of `\(M\)` (columns of `\(\boldsymbol{\theta}\)` are uncorrelated) - `\(\Lambda\)` indicates the rotation angle to get from `\(\boldsymbol{\theta}\)` back to `\(M\)` - If `\(M\)` were orthogonal to begin with, `\(\Lambda = I\)` and `\(M=\boldsymbol{\theta}\)` --- # PCA 2 - Nothing on the previous slide helps us with dimensionality reduction per se - We reduce dimensionality by choosing the eigenvectors with the largest magnitudes - These represent the dimensions `\(\boldsymbol{\theta}\)` with the greatest variance - We say that we "select the first `\(K\)` .hi[principal components] of `\(M\)`" - Mathematically, we "reduce" (i.e. "approximate") `\(M\)` by choosing a subset of `\(\boldsymbol{\theta}\)` and `\(\Lambda\)` `\begin{align*} \widetilde{M} &= \boldsymbol{\theta}_k\Lambda_k\\ [N\times J] &= [N\times K] [K\times J]\\ &\\ M &= \boldsymbol{\theta}_k\Lambda_k + \boldsymbol{\varepsilon} \end{align*}` where `\(\boldsymbol{\varepsilon}\equiv M - \widetilde{M}\)` is a `\(N\times J\)` matrix --- # Visual depiction of PCA .center[] - The arrows are the eigenvectors; longer arrows correspond to more variance (image source: [Wikipedia](https://upload.wikimedia.org/wikipedia/commons/f/f5/GaussianScatterPCA.svg)) --- # Factor Analysis - Factor Analysis comes in two forms: Exploratory (EFA) and Confirmatory (CFA) - EFA: see what factors might be in the data - CFA: write down a model and use the data to test it - In economics, we pretty much only do CFA - FA is used extensively in psychometrics - It is a natural tool for analyzing cognitive or behavioral tests - Each test measures some set of skills, but does so noisily - Tests tend to measure the same set of skills, so they are correlated --- # How factor analysis works - Suppose our `\(J\)` columns of `\(M\)` correspond to measurements (e.g. test scores) - FA tries to find some underlying unobservables that commonly affect `\(M\)` - We assume that we cannot observe `\(\boldsymbol{\theta}\)` - If we assume that `\(M\)` is standardized (mean-zero, unit-variance), then `\begin{align*} M &= \underbrace{\boldsymbol{\theta}_k\Lambda_k + \boldsymbol{\varepsilon}}_{\boldsymbol{u}} \end{align*}` - `\(u\)` is a composite error term (since both `\(\boldsymbol{\theta}\)` and `\(\varepsilon\)` are unobservable) - In FA, we call `\(\boldsymbol{\theta}\)` .hi[factors], and we call `\(\Lambda\)` .hi[factor loadings] and `\(\boldsymbol{\varepsilon}\)` .hi[uniquenesses] --- # PCA vs. FA - Clearly, PCA and FA are related, but there are important differences - The `\(\boldsymbol{\theta}_k\)` we get from PCA and FA are going to be different - But the `\(\Lambda_k\)` are identical (and hence the `\(\boldsymbol{\varepsilon}\)` are different) - `\(\boldsymbol{\theta}_k^{PCA}\)` has larger variance than `\(\boldsymbol{\theta}_k^{FA}\)` - This is because PCA treats `\(M\)` as not measured with error, but FA does the opposite - For many more excellent details, see Shalizi (2019) [here](http://www.stat.cmu.edu/~cshalizi/uADA/12/lectures/ch19.pdf) --- # Extensions of FA - We can extend FA to allow for `\(X\)`'s that affect our measurements `\begin{align*} M &= X\boldsymbol{\beta} + \underbrace{\boldsymbol{\theta}_k\Lambda_k + \boldsymbol{\varepsilon}}_{\boldsymbol{u}} \\ [N\times J] &= [N\times L] [L\times J] + [N\times K] [K\times J] + [N\times J] \end{align*}` - If `\(X\)` is `\(N\times L\)` then `\(\boldsymbol{\beta}\)` is a `\(L\times J\)` matrix - However, we do need to make more assumptions for econometric identification! --- # Identifying assumptions for a 2-factor model - We need to make the following assumptions: `\begin{align*} \mathbb{E}\left(\boldsymbol{\varepsilon}\right) &= \mathbf{0}_{J\times 1}\\ \mathbb{V}\left(\boldsymbol{\varepsilon}\right) &\equiv \mathbb{E}\left(\boldsymbol{\varepsilon}'\boldsymbol{\varepsilon}\right)=\Omega_{J\times J}\\ \Omega_{[j,j]} &= \sigma^2_j, \Omega_{[j,k]} = 0\\ \mathbb{E}\left(\boldsymbol{\theta}\right) &= \mathbf{0}_{2\times 1}\\ \mathbb{V}\left(\boldsymbol{\theta}\right) &= \Sigma_\boldsymbol{\theta} \end{align*}` - Let `\(u \equiv M - X\boldsymbol{\beta}\)`, then `\begin{align*} \mathbb{E}\left(u\right) &= \mathbf{0}_{J\times 1}\\ \mathbb{V}\left(u\right) &= \Lambda\Sigma_\boldsymbol{\theta}\Lambda' + \Omega\\ \Sigma_\boldsymbol{\theta} &= \left[\begin{array}{cc} \sigma^2_{\theta_1} & \sigma_{\theta_1 \theta_2}\\ \sigma_{\theta_1 \theta_2} & \sigma^2_{\theta_2}\\ \end{array}\right] \end{align*}` --- # Identification of a 2-factor model - Our only data source to estimate `\(\Lambda\)` and `\(\Sigma_\boldsymbol{\theta}\)` is `\(\mathbb{V}\left(M-X\boldsymbol{\beta}\right)\equiv \mathbb{V}\left(u\right)\)` - Let's look at the variance-covariance matrix of `\(u\)`: - This has `\(J\)` diagonal elements and `\(\frac{J(J-1)}{2}\)` unique off-diagonal elements - With these `\(J+\frac{J(J-1)}{2}\)` moments in the data, we want to estimate: - The `\(J\)` diagonal elements of `\(\Omega\)` (i.e. the `\(\sigma^2_j\)`'s) - `\(2J\)` elements of `\(\Lambda\)` - four elements of `\(\Sigma_\boldsymbol{\theta}\)` - We have `\(3J+4\)` parameters, but only `\(J+\frac{J(J-1)}{2}\)` data moments - In general, the model is not identified. We will need to impose further assumptions. --- # Additional identifying assumptions - The following are common assumptions, but you could impose others 1. `\(\theta_1 \perp \theta_2\)` (so `\(\Sigma_\boldsymbol{\theta}\)` is diagonal) 2. The scale of each factor is arbitrary. 2 ways to normalize the scale: - `\(\Sigma_\boldsymbol{\theta} = I_{2\times 2}\)` - or - Set one element of each row of `\(\Lambda=1\)` - With these two assumptions, we achieve identification if `\begin{align*} 2J + J \leq J+\frac{J(J-1)}{2} \end{align*}` - So `\(J\geq 5\)` is necessary (but not sufficient) for identification --- # Other identification considerations - For .hi[model interpretability], also need to put more structure on `\(\Lambda\)` - For example, suppose we have 6 measurements: - 3 from a cognitive test and 3 from a personality test - In this case, the first row of `\(\Lambda\)` should be 0 for the personality measures - Likewise, the second row of `\(\Lambda\)` should be 0 for the cognitive measures - If all 6 measurements come from a cog. test, can't identify a non-cog. factor - Could possibly identify `\(\sigma_{\theta_1\theta_2}\)` if one measurement measures both factors --- # Estimation of factor models - Typically, we impose distributional assumptions on `\(\boldsymbol{\theta}\)` and `\(\boldsymbol{\varepsilon}\)` - e.g. assume `\(\boldsymbol{\theta}\)` and `\(\boldsymbol{\varepsilon}\)` are each MVN with 0 covariance and `\(\boldsymbol{\theta} \perp \boldsymbol{\varepsilon}\)` - Then we estimate `\((\Lambda, \Sigma_\boldsymbol{\theta}, \Omega)\)` by maximum likelihood - The likelihood function will need to be integrated, since `\(\boldsymbol{\theta}\)` is unobserved - Can use quadrature, simulated method of moments, MCMC, or the EM algorithm - As you know, these vary in their ease of use --- # Using factor models to `\(\downarrow\)` bias of regression estimates - The whole reason we use a factor model is to reduce bias - Let's go back to the log wage example from the start of today - `\(\beta\)`'s are biased if we omit cognitive ability (omitted variable bias) - `\(\beta\)`'s are also biased if we include IQ score (attenuation bias from meas. err.) - We know cognitive ability affects wages, and we have (noisy) measurements of it - We can estimate the log wage parameters by maximum likelihood - We combine together the log wage and factor model likelihoods - I'll walk you through how to do this in the Problem Set (due next time) --- # Factor models and dynamic selection - Factor models can also be used to account for dynamic selection - Intuition: `\(\uparrow\)` cog. abil `\(\Rightarrow \uparrow\)` schooling `\(\Rightarrow \uparrow\)` wages - Schooling is endogenous, so we can add a schooling choice model to our likelihood - When the ability factor enters choice of schooling, this induces a correlation between schooling choices and wages - But conditional on the factor, we have separability of the likelihood components - `\(\mathcal{L} = \int_A \underbrace{\mathcal{L}_1(A)}_{\text{measurements}}\underbrace{\mathcal{L}_2(A)}_{\text{choices}}\underbrace{\mathcal{L}_3(A)}_{\text{wages}} dF(A)\)` - We covered a variant of this case back when we discussed Mixed Logit --- # Seminal papers applying factor analysis - Heckman, Stixrud, and Urzua (2006) - first paper to apply this method to an econometric model - show that this method works - 2 latent factors impact a variety of outcomes - Cunha, Heckman, and Schennach (2010) - develop a dynamic factor model of early childhood skill production - latent ability in one period affects investment in subsequent periods --- # Recent papers - Aucejo and James (2019) - Why do women attain more education than men, especially among Blacks? - Use 59 measures of early student information - 3 factors: family background, math/verbal skills, externalizing behavior - Family background differences drive most of the observed gaps - Ashworth, Hotz, Maurel et al. (2021) - Estimate wage returns to schooling and different types of work experience - 2 factors: cognitive and "not" cognitive - Accounting for selection matters a lot for calculation of returns to schooling --- # References .smallest[ 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. ]