# Econometrics of Big Data

**Course Description**

As in many other fields, economists are increasingly making use of high-dimensional models – models with many unknown parameters that need to be inferred from the data. Such models arise naturally in modern data sets that include rich information for each unit of observation (a type of “big data”) and in nonparametric applications where researchers wish to learn, rather than impose, functional forms. High-dimensional models provide a vehicle for modeling and analyzing complex phenomena and for incorporating rich sources of confounding information into economic models.

Our goal in this course is two-fold. First, we wish to provide an overview and introduction to several modern methods, largely coming from statistics and machine learning, which are useful for exploring high-dimensional data and for building prediction models in high-dimensional settings. Second, we will present recent proposals that adapt high-dimensional methods to the problem of doing valid inference about model parameters and illustrate applications of these proposals for doing inference about economically interesting parameters.

**Course prerequisites**

The course is a PhD level course. Basic knowledge of parametric statistical models and associated asymptotic theory is expected.

Preliminary Outline

Lecture 1: Introduction to High-Dimensional Modeling

- Breiman, L. (1996), “Bagging Predictors,” Machine Learning 26: 123-140
- Friedman, J., T. Hastie, and R. Tibshirani (2000), “Additive logistic regression: A statistical view of boosting (with discussion),” Annals of Statistics, 28, 337-407
- Hastie, T., R. Tibshirani, and J. Friedman (2009), The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer. [Elements from Chapters 2, 5, 7, 8.7, 10]
- James, G., D. Witten, T. Hastie, and R. Tibshirani (2014), An Introduction to Statistical Learning with Applications in R, Springer. [Elements from Chapters 2, 3, 5, 7, 8.2]
- Li, Q. and J. S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press. [Elements from Chapters 2, 14]
- Schapire, R. (1990), “The strength of weak learnability,” Machine Learning, 5, 197-227

Lecture 2: Introduction to Distributed Computing for Very Large Data Sets

Lecture 3: Tree-based Methods

- Athey, S. and G. Imbens (2015), “Machine Learning Methods for Estimating Heterogeneous Causal Effects,” working paper, http://arxiv.org/abs/1504.01132
- Hastie, T., R. Tibshirani, and J. Friedman (2009), The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer. [Chapters 9, 10, 15, 16]
- James, G., D. Witten, T. Hastie, and R. Tibshirani (2014), An Introduction to Statistical Learning with Applications in R, Springer. [Chapter 8]
- Wager, S. and S. Athey (2015), “Estimation and Inference of Heterogeneous Treatment Effects using Random Forests,” working paper, http://arxiv.org/abs/1510.04342
- Wager, S. and G. Walther (2015), “Uniform Convergence of Random Forests via Adaptive Concentration,” working paper, http://arxiv.org/abs/1503.06388
- Wager, S., T. Hastie, and B. Efron (2014), “Confidence Intervals for Random Forests: The Jackknife and the Infinitesimal Jackknife,” Journal of Machine Learning Research, 15, 1625−1651

Lecture 4: An Overview of High-Dimensional Inference

- Belloni, A. and V. Chernozhukov (2013), “Least Squares After Model Selection in High-dimensional Sparse Models,” Bernoulli, 19(2), 521-547.
- Belloni, A., D. Chen, V. Chernohukov, and C. Hansen (2012), “Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain,” Econometrica, 80(6), 2369-2430
- Belloni, A., V. Chernozhukov, and C. Hansen (2014), “High-Dimensional Methods and Inference on Structural and Treatment Effects,” Journal of Economic Perspectives, 28(2), 29-50
- Belloni, A., V. Chernozhukov, and C. Hansen (2014), “Inference on Treatment Effects after Selection amongst High-Dimensional Controls,” Review of Economic Studies, 81(2), 608-650
- Belloni, A., V. Chernozhukov, and C. Hansen (2015), “Inference in High Dimensional Panel Models with an Application to Gun Control,” forthcoming Journal of Business and Economic Statistics
- Belloni, A., V. Chernozhukov, I. Fernández-Val, and C. Hansen (2013), “Program Evaluation with High-Dimensional Data,” working paper, http://arxiv.org/abs/1311.2645
- Chernozhukov, V., C. Hansen, and M. Spindler (2015), “Post-Selection and Post-Regularization Inference in Linear Models with Many Controls and Instruments,” American Economic Review, 105(5), 486-490
- Chernozhukov, V., C. Hansen, and M. Spindler (2015), “Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach,” Annual Review of Economics, 7, 649-688

Lecture 5: Penalized Estimation Methods

- Belloni, A. and V. Chernozhukov (2013), “Least Squares After Model Selection in High-dimensional Sparse Models,” Bernoulli, 19(2), 521-547.
- Fan, J. and J. Lv (2008), “Sure independence screening for ultrahigh dimensional feature space,” Journal of the Royal Statistical Society, Series B, 70(5), 849-911
- Hastie, T., R. Tibshirani, and J. Friedman (2009), The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer. [Chapters 3, 4, 5, 18]
- James, G., D. Witten, T. Hastie, and R. Tibshirani (2014), An Introduction to Statistical Learning with Applications in R, Springer. [Chapter 6]

Lecture 6: Moderate p Asymptotics

Lecture 7: Examples

- Belloni, A., D. Chen, V. Chernohukov, and C. Hansen (2012), “Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain,” Econometrica, 80(6), 2369-2430
- Belloni, A., V. Chernozhukov, and C. Hansen (2014), “High-Dimensional Methods and Inference on Structural and Treatment Effects,” Journal of Economic Perspectives, 28(2), 29-50
- Belloni, A., V. Chernozhukov, and C. Hansen (2014), “Inference on Treatment Effects after Selection amongst High-Dimensional Controls,” Review of Economic Studies, 81(2), 608-650
- Belloni, A., V. Chernozhukov, and C. Hansen (2015), “Inference in High Dimensional Panel Models with an Application to Gun Control,” forthcoming Journal of Business and Economic Statistics
- Belloni, A., V. Chernozhukov, I. Fernández-Val, and C. Hansen (2013), “Program Evaluation with High-Dimensional Data,” working paper, http://arxiv.org/abs/1311.2645
- Chernozhukov, V., C. Hansen, and M. Spindler (2015), “Post-Selection and Post-Regularization Inference in Linear Models with Many Controls and Instruments,” American Economic Review, 105(5), 486-490
- Chernozhukov, V., C. Hansen, and M. Spindler (2015), “Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach,” Annual Review of Economics, 7, 649-688
- Gentzkow, M., J. Shapiro, and M. Taddy (2015), “Measuring Polarization in High-Dimensional Data: Method and Application to Congressional Speech,” working paper, http://www.brown.edu/Research/Shapiro/
- Hansen, C. and D. Kozbur (2014), “Instrumental Variables Estimation with Many Weak Instruments Using Regularized JIVE,” Journal of Econometrics, 182(2), 290-308
- Kleinberg, J., J. Ludwig, S. Mullainathan, and Z. Obermeyer (2015), “Prediction Policy Problems,” American Economic Review: Papers and Proceedings, 105(5), 491-495

Lecture 8: Inference: Computation

Lecture 9: Introduction to Unsupervised Learning

- Blei, D., A. Ng, and M. Jordan (2003), Lafferty, J., ed. “Latent Dirichlet allocation,” Journal
- of Machine Learning Research, 3 (4-5), 993-1022
- Hastie, T., R. Tibshirani, and J. Friedman (2009), The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer. [Chapter 14]
- James, G., D. Witten, T. Hastie, and R. Tibshirani (2014), An Introduction to Statistical Learning with Applications in R, Springer. [Chapter 10]
- Li, Q. and J. S. Racine (2007), Nonparametric Econometrics: Theory and Practice, Princeton University Press. [Chapter 1]
- Stock J. H and Watson M. W (2002), “Forecasting using principal components from a large number of predictors,” Journal of the American Statistical Association, 97, 1167-1179

Lecture 10: Very Large p Asymptotics

- Belloni, A., D. Chen, V. Chernozhukov, and C. Hansen (2012): “Sparse Models and Methods for Optimal Instruments with an Application to Eminent Domain,” Econometrica, 80, 2369–2429. (ArXiv, 2010)
- Belloni, A., and V. Chernozhukov (2011): “`1-penalized quantile regression in high-dimensional sparse models,” Annals of Statistics, 39(1), 82–130. (ArXiv, 2009)
- Belloni, A., and V. Chernozhukov (2013): “Least Squares After Model Selection in High-dimensional Sparse Models,” Bernoulli, 19(2), 521–547. (ArXiv, 2009)
- Belloni, A., V. Chernozhukov, and C. Hansen (2010) “Inference for High-Dimensional Sparse Econometric Models,” Advances in Economics and Econometrics. 10th World Congress of Econometric Society, Shanghai, 2010. (ArXiv, 2011).
- Belloni, A., V. Chernozhukov, and C. Hansen (2014), “Inference on Treatment Effects after Selection amongst High-Dimensional Controls,” Review of Economic Studies, 81(2), 608-650
- Belloni, A., V. Chernozhukov, K. Kato (2013): “Uniform Post Selection Inference for LAD Regression Models,” arXiv:1304.0282. (ArXiv, 2013)
- Belloni, A., V. Chernozhukov, L. Wang (2011a): “Square-Root-LASSO: Pivotal Recovery of Sparse Signals via Conic Programming,” Biometrika, 98(4), 791–806. (ArXiv, 2010).
- Belloni, A., V. Chernozhukov, L. Wang (2011b): “Square-Root-LASSO: Pivotal Recovery of Nonparametric Regression Functions via Conic Programming,” (ArXiv, 2011)
- Belloni, A., V. Chernozhukov, Y. Wei (2013): “Honest Confidence Regions for Logistic Regression with a Large Number of Controls,” arXiv preprint arXiv:1304.3969 (ArXiv, 2013)
- Bickel, P., Y. Ritov and A. Tsybakov, “Simultaneous analysis of Lasso and Dantzig selector”, Annals of Statistics, 2009.
- Candes E. and T. Tao, “The Dantzig selector: statistical estimation when p is much larger than n,” Annals of Statistics, 2007.
- Donald S. and W. Newey, “Series estimation of semilinear models,” Journal of Multivariate Analysis, 1994.
- Tibshirani, R, “Regression shrinkage and selection via the Lasso,” J. Roy. Statist. Soc. Ser. B, 1996.
- Frank, I. E., J. H. Friedman (1993): “A Statistical View of Some Chemometrics Regression Tools,” Technometrics, 35(2), 109–135.
- Gautier, E., A. Tsybakov (2011): “High-dimensional Instrumental Variables Rergession and Confidence Sets,” arXiv:1105.2454v2
- Hahn, J. (1998): “On the role of the propensity score in efficient semiparametric estimation of average treatment effects,” Econometrica, pp. 315–331.
- Heckman, J., R. LaLonde, J. Smith (1999): “The economics and econometrics of active labor market programs,” Handbook of labor economics, 3, 1865–2097.
- Imbens, G. W. (2004): “Nonparametric Estimation of Average Treatment Effects Under Exogeneity: A Review,” The Review of Economics and Statistics, 86(1), 4–29.
- Leeb, H., and B. M. Potscher (2008): “Can one estimate the unconditional distribution of post-model-selection estimators?,” Econometric Theory, 24(2), 338–376.
- Robinson, P. M. (1988): “Root-N-consistent semiparametric regression,” Econometrica, 56(4), 931–954.
- Rudelson, M., R. Vershynin (2008): “On sparse reconstruction from Foruier and Gaussian Measurements”, Comm Pure Appl Math, 61, 1024-1045.
- Jing, B.-Y., Q.-M. Shao, Q. Wang (2003): “Self-normalized Cramer-type large deviations for independent random variables,” Ann. Probab., 31(4), 2167–2215.

**Course literature**

Course notes and a list of readings provided at the beginning of the course.

**Examination **

Written examination (100%). Participants get a take-home final exam on the last day to be completed over the next couple of weeks.

**Examination content**

Content of the lectures

**Examination relevant literature**

To be discussed in class