Since the introduction of Generalized Method of Moments (GMM) in 1982, there has been considerable interest in the estimation of parameters of econometric models based on the information in population moment conditions. GMM itself has been widely applied in diverse areas spanning all areas of economics. These applications have both stimulated and been facilitated by the development of numerous statistical inference techniques based on GMM estimators. With this interest, there has also come close scrutiny of the method that has revealed circumstances in which standard GMM inferences can be unreliable. The recognition of these limitations has stimulated the development of alternative moment-based estimation methods, leading examples of which are Continuous Updating GMM (CUGMM), Empirical Likelihood (EL), and Exponential Tilting (ET). CUGMM, EL and ET are all members of the class of Generalized Empirical Likelihood (GEL) estimators.

In summary the course has covered the following topics:

• Sources of endogeneity and examples of moment conditions in economic models
• Frameworks for moment based estimation: Generalized Method of Moments, Minimum Discrepancy and Generalized Empirical Likelihood.
• Large Sample properties of moment-based estimators.
• Model misspecification; weak identification; many moment conditions.
• Microeconometric models: panel and pseudo-panel data. Time series; testing for structural stability.

The course provided an introduction to estimation and inference within both the GMM and GEL frameworks. Computer labs provide practical experience of GMM and GEL estimation of economic models using toolboxes developed in MATLAB. Prior knowledge of MATLAB is desirable but not necessary; a MATLAB tutorial was provided.

Course outline

Day 1: The first lecture covers endogeneity in econometric models and various principles for moment-based estimation. Using specific examples from the empirical literature, we discuss sources of endogeneity and also the formulation of moment conditions that can be used as a basis for estimation of the parameters of interest. We introduce the principles behind GMM and EL/ET/CUE estimation, and show that EL/ET/CUE can be viewed as special cases of both minimum discrepancy and GEL estimation principles.

Day 2: The second lecture focuses on GMM estimation. We present the large sample properties of the estimator, leading to discussion of the so-called two-step and iterated GMM estimators. We also describe how GMM affects a decomposition of the original moment condition into identifying and overidentifying restrictions, and show how the latter can form a basis for a test of the model specification.

Day 3: The third lecture will focus on GEL estimation. We present the large sample properties of the estimator and a number of model specification tests. GMM and GEL are compared and contrasted, both in terms of their (first order and second order asymptotic) statistical properties and also computational requirements.

Day 4:  Lecture 4 discusses applications of GMM/GEL methodologies to estimation of econometric models based on panel, pseudo-panel and time series data. For panel data, the discussion focuses on linear dynamic models and the use of moments based on the equation of interest in both levels and differences. For time series data, the discussion also includes tests for structural stability.

 Day 5: Lecture 5 provides an overview of recent developments in the GMM/GEL literature, in particular on methods for moment selection, (weak) identification robust inference and estimation based on “many moments”. As time permits, we also briefly discuss some other moment related methods such as (the simulation-based) Indirect Inference and estimation based on moment inequalities.

Preliminary reading list

Endogeneity and moment conditions:

Acemoglue, D., Johnson, S. and Robinson, J. A. (2001). ‘The Colonial Origins of Comparative Development: An Empirical Investigation’. The American Economic Review, Vol. 91, No. 5.

Angrist, J. D., and Krueger, A. B. (1991). ‘Does compulsory school attendance affect schooling and earnings?’, The Quarterly Journal of Economics, 87: 979–1014.

Clarida, R., J. Gali, and M. Gertler. (2000). ‘Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory’, The Quarterly Journal of Economics, 115, 147-180.

Hansen, L. P., and Singleton, K. S. (1982). ‘Generalized instrumental variables estimation of nonlinear rational expectations models’, Econometrica, 50: 1269–1286.

GMM:

Hall, A. R., 2005, Generalized Method of Moments, Oxford University Press.

Hansen, L. P. (1982). ‘Large sample properties of Generalized Method of Moments estimators’, Econometrica, 50: 1029–1054.

GEL and Minimum Discrepancy:

Corcoran, S. (1998). ‘Bartlett adjustment of empirical discrepancy statistics’, Biometrika, 85: 965–972.

Kitamura, Y. (2007). ‘Empirical likelihood methods in econometrics: theory and practice’, in R. Blundell, W. K. Newey, and T. Personn (eds.), Advances in Economics and Econometrics: Ninth World Congress of the Econometric Society. Cambridge University Press, Cambridge, UK.

Kitamura, Y., and Stutzer, M. (1997). ‘An information-theoretic alternative to generalized method of moments estimation’, Econometrica, 65: 861–874.

Parente, P. M. D. C., and Smith, R. J. (2014). ‘Recent developments in empirical likelihood and related methods’, Annual Review of Economics, 6: forthcoming.

Smith, R. J. (1997). ‘Alternative semi-parametric likelihood approaches to generalized method of moments estimation’, Economics Journal, 107: 503–519.

Newey, W. K., and Smith, R. J. (2004). ‘Higher order properties of GMM and generalized empirical likelihood estimators’, Econometrica, 72: 219–256.

Andrews, M., O. Elamin, A. R. Hall, K. Kyriakoulis, and M. Sutton (2014), ‘Inference in the Presence of Redundant Moment Conditions and the Impact of Government Health Expenditure on Health Outcomes in England’, Discussion paper, University of Manchester Economics Discussion Paper Series: EDP-1401.

Panel data:

Arellano, M. and Bond, S. (1991), ‘Some tests of specification for panel data: Monte carlo evidence and an application to employment equations.’ Review of Economic Studies 58(2), 277.

Arellano, M. and Bover, O. (1995), ‘Another look at the instrumental variable estimation of error-components models’, Journal of Econometrics, 68, 29–51.

Pseudo- panel data:

Angrist, J. (1991). ‘Grouped-data estimation and testing in simple labor-supply models’, Journal of Econometrics, 47, 243–266.

Andrews, M., A. R. Hall and R. Khatoon, (2013), ‘Inference based on repeated cross-section data: a Generalized Empirical Likelihood approach with application to the returns to education for minorities’, unpublished mimeo, University of Manchester.

Time series and structural stability testing:

Andrews, D. W. K. (1991). ‘Heteroscedasticity and autocorrelation consistent covariance matrix estimation’, Econometrica, 59: 817–858.

Kitamura, Y. (1997). ‘Empirical likelihood methods with weakly dependent processes’, Annals of Statistics, 25: 2084–2102.

Smith, R. J. (2011). ‘GEL criteria for moment condition models’, Econometric Theory, 27: 1192–1235.

Andrews, D. W. K. (1993). ‘Tests for parameter instability and structural change with unknown change point’, Econometrica, 61: 821–856.

Sowell, F. (1996). ‘Optimal tests of parameter variation in the Generalized Method of Moments framework’, Econometrica, 64: 1085–1108.

Hall, A. R., and Sen, A. (1999). ‘Structural stability testing in models estimated by Generalized Method of Moments’, Journal of Business and Economic Statistics, 17: 335–348.

Hall, A. R., Li, Y., Orme, C. D., and Sinko, A. (2013). ‘Testing for Structural Instability in Moment Restriction Models: an Info-metric Approach’, Discussion paper, University of Manchester Economics Discussion Paper Series: EDP-1326.

Moment selection:

Andrews, D. W. K. (1999). ‘Consistent moment selection procedures for Generalized Method of Moments estimation’, Econometrica, 67: 543–564.

Belloni, D., Chen, D., Chernozhukov, V., and Hansen, C. (2012). ‘Sparse models and methods for optimal instruments with an application to eminent domain’, Econometrica, 80: 2369–2430.

Carrasco, M. (2012). ‘A regularization approach to the many instruments problem’, Journal of Econometrics, 170: 383–398.

Donald, S. G., and Newey, W. K. (2001). ‘Choosing the number of instruments’, Econometrica, 69: 1161–1192.

Hall, A. R., Inoue, A., Jana, K., and Shin, C. (2007). ‘Information in Generalized Method of Moments Estimation and Entropy Based Moment Selection’, Journal of Econometrics, 138: 488–512.

Liao, Z. (2013). ‘Adaptive GMM shrinkage estimation with consistent moments election’, Econometric Theory, 29: 1–48.

Weak identification robust inference:

Andrews, D. W. K., and Stock, J. H. (2006). ‘Testing with many weak instruments’, in R. Blundell, W. Newey, and T. Persson (eds.), Advances in Economics and Econometrics, Theory and Applications: Ninth World Congress of the Econometric Society, pp. 122–173. Cambridge University Press, Cambridge, UK.

Hall, A. R., Rudebusch, G., and Wilcox, D. (1996). ‘Judging instrument relevance in instrumental variables estimation’, International Economic Review, 37: 283–298.

Kleibergen, F. (2002). ‘Pivotal statistics for testing structural parameters in instrumental variables regression’, Econometrica, 70: 1781–1803.

Kleibergen, F. (2005). ‘Testing parameters in GMM without assuming that they are identified’, Econometrica, 73: 1103–1124.

Staiger, D., and Stock, J. (1997). ‘Instrumental variables regression with weak instruments’, Econometrica, 65: 557–586.

Stock, J., and Wright, J. (2000). ‘GMM with weak identification’, Econometrica, 68: 1055–1096.

Stock, J. H., Wright, J. H., and Yogo, M. (2002). ‘A survey of weak instruments and weak identification in generalized method of moments’, Journal of Business and Economic Statistics, 20: 518–529.

Many moments:

Chao, J., and Swanson, N. (2005). ‘Consistent estimation with a large number of weak instruments’, Econometrica, 73: 1673–1692.

Han, C., and Phillips, P. C. B. (2006). ‘GMM with many moment conditions’, Econometrica, 74: 147–192.

Newey, W. K., and Windmeijer, F. (2009). ‘Generalized Method of Moments with many weak moment conditions’, Econometrica, 77: 687–719.