### Course Description

The use of quantitative methods in financial markets has experienced an extraordinary growth over the past three decades. Nowadays finance professionals routinely use sophisticated statistical techniques, many of which are at the frontier of academic research. The purpose of this course is to present some of the most important econometric methods usually employed in financial markets. In particular, it contains a thorough analysis of some of the statistical techniques applied to portfolio management, financial consulting, and risk control. In addition to the morning lectures, students will conduct empirical and Monte Carlo simulation exercises in the computer room in the afternoons. Although the lectures will be to a large extent self-contained, some background in both Econometrics and Finance Theory at the graduate level would be convenient.

**Day 1:** We will begin with an introduction to the elements of financial decision theory under uncertainty, the portfolio section problem and portfolio and stochastic discount factor mean variance frontiers. This will be followed by an introduction to the Generalized Method of Moments approach to estimation and inference, which will be central to the first part of the week’s material. We will discuss identification, asymptotic properties, classical hypothesis tests and underidentification tests, as well as extensions to a countable or uncountable infinite number of moments.

**Day 2:** On the second day we will consider inference about portfolio and stochastic discount factor mean frontiers for both arbitrage portfolios and gross returns. After a quick revision of the material covered the first day, we will consider unrestricted estimation of mean-variance frontiers for both types of financial assets, followed by restricted estimation with tangency restrictions for gross returns, and spanning restrictions for both types of returns. We will also consider estimation of mean-variance frontiers imposing asset pricing restrictions. Importantly, in all cases we jointly discuss restricted estimation and hypothesis testing using a unifying GMM approach.

**Day 3:** On the third day we introduce some distributions that are useful in financial modelling. They cover discrete and positive random variables, distributions defined over the entire real line and multivariate distributions. This allows us to formally develop likelihood methods in the classical case, together with pseudo maximum likelihood estimation and sequential methods, and study their asymptotic properties. In addition, we will introduce semiparametric likelihood methods, and study the resulting efficiency/consistency trade-offs in the context of mean-variance efficiency tests. We conclude with some distributional tests based either on orthogonal polynomials or the Lagrange multiplier principle more generally.

**Day 4**: On the fourth day, we address covariance structures typically used for financial returns, which include index models and single and multiple factor models. We will also discuss factor mimicking portfolios and the estimation and inference issues associated with those models. We then go beyond mean and variance and address financial risk management, systemic risk measures and mean – variance – skewness analysis.

**Day 5**: The last day will cover time series models, including univariate models (ARMA, GARCH and stochastic volatility) and multivariate ones (VARMA, dynamic factor models, MGARCH. and conditionally heteroskedastic factor models). We will consider both likelihood-based estimation and indirect inference approaches, along with serial dependence tests for both univariate and factor models. Other specification tests will also be discussed. We will then use the dynamic framework to address the role of conditioning information in financial decisions. Specifically, we will combine least squares predictions and mean-variance analysis, and discuss in detail how conditioning information affects portfolio choice by looking at conditional, unconditional and extended mean-variance frontiers for asset returns and stochastic discount factors, as well as passive frontiers with managed portfolios.

**Lab Work: **In the computer lab in the afternoon sessions we will implement most of the methods discussed in lecture, using data sets covering a variety of areas in economics and finance. We will use R as the statistical software package.

### Self references

Amengual, D. and Sentana, E. (2010): “A comparison of mean-variance efficiency tests”, *Journal of Econometrics* 154 (1), pp. 16-34.

http://dx.doi.org/10.1016/j.jeconom.2009.06.006

Amengual, D. and Sentana, E. (2011): “Inference in multivariate dynamic models with elliptical innovations, mimeo, CEMFI.

Amengual, D., Fiorentini, G. and Sentana, E. (2012): “Sequential estimators of shape parameters in multivariate dynamic models”, CEMFI Working Paper 1201.

http://www.cemfi.es/ftp/wp/1201.pdf

Arellano, M., Hansen, L.P. and Sentana, E. (2012): “Underidentification?”, *Journal of Econometrics* 170 (2), pp. 256-280.

http://dx.doi.org/10.1016/j.jeconom.2012.05.005

Calzolari, G., Fiorentini, G. and Sentana, E. (2004): “Constrained indirect estimation”, *Review of Economic Studies* 71 (4), pp. 945-973.

http://dx.doi.org/10.1111/0034-6527.00310

Diez de los Rios, A. and Sentana, E. (2011): “Testing uncovered interest parity: a continuous time approach”, *International Economic Review* 52 (4), pp. 1215-1251.

http://dx.doi.org/10.1111/j.1468-2354.2011.00665.x

Fiorentini, G. and Sentana, E. (2009): “Dynamic specification tests for static factor models”, CEMFI Working Paper 0912.

ftp://ftp.cemfi.es/wp/09/0912.pdf

Fiorentini, G. and Sentana, E. (2010a): “New testing approaches for mean-variance predictability, mimeo, CEMFI.

ftp://ftp.cemfi.es/pdf/papers/es/mvpredictability.pdf

Fiorentini, G. and Sentana, E. (2010a): “On the efficiency and consistency of likelihood estimation in multivariate conditionally heteroskedastic dynamic regression models”, CEMFI Working Paper 0713, revised.

ftp://ftp.cemfi.es/pdf/papers/es/tmlepmle.pdf

Fiorentini, G. and Sentana, E. (2012): “Tests for serial dependence in static, non-Gaussian factor models”, CEMFI Working Paper 1211.

http://www.cemfi.es/ftp/wp/1211.pdf

Fiorentini, G., Sentana, E. and Calzolari, G. (2003): “Maximum likelihood estimation and inference in multivariate conditionally heteroskedastic dynamic regression models with student t innovations”, *Journal of Business and Economic Statistics* 21 (4), pp. 532-546.

http://dx.doi.org/10.1198/073500103288619232

León , A., Mencía, J. and Sentana, E. (2009): “Parametric properties of seminonparametric distributions, with applications to option valuation”, *Journal of Business and Economic Statistics* 27 (2), pp. 176-192.

http://dx.doi.org/10.1198/jbes.2009.0013

Mencía, J. and Sentana, E. (2009): “Multivariate location-scale mixtures of normals and mean-variance-skewness portfolio allocation, *Journal of Econometrics* 153 (2), pp. 105-121.

http://dx.doi.org/10.1016/j.jeconom.2009.05.001

Mencía, J. and Sentana, E. (2012): “Distributional tests in multivariate dynamic models with Normal and Student t innovations”, *Review of Economics and Statistics* 94 (1), pp. 133-152.

http://dx.doi.org/10.1162/REST_a_00141

Peñaranda, F. and Sentana, E. (2010): “A unifying approach to the empirical evaluation of asset pricing models”, CEMFI Working Paper 1004, revised.

ftp://ftp.cemfi.es/pdf/papers/es/apevaluation1011.pdf

Peñaranda, F. and Sentana, E. (2011a): “Duality in mean-variance frontiers with conditioning information”, CEMFI Working Paper 0715, revised.

ftp://ftp.cemfi.es/pdf/papers/es/duality1105.pdf

Peñaranda, F. and Sentana, E. (2011b): “Inference about portfolio and stochastic discount factor mean variance frontiers”, mimeo, CEMFI.

ftp://ftp.cemfi.es/pdf/papers/es/tangency1011.pdf

Peñaranda, F. and Sentana, E. (2012): “Spanning tests in return and stochastic discount factor mean-variance frontiers: a unifying approach”, *Journal of Econometrics* 170 (2), pp. 303-324.

http://dx.doi.org/10.1016/j.jeconom.2012.05.007

Sentana, E. (2003): “Mean-variance portfolio allocation with a value at risk constraint, *Revista de Economía Financiera* 1, pp. 4-14.

http://www.aefin.es/AEFIN_data/muestra.asp?art=A1-1.pdf

Sentana, E. (2004): “Factor representing portfolios in large asset markets, *Journal of Econometrics* 119 (2), pp. 257-289.

http://dx.doi.org/10.1016/S0304-4076(03)00197-0

Sentana, E. (2005): “Least squares predictions and mean-variance analysis, *Journal of Financial Econometrics* 3 (1), pp. 56-78.

http://dx.doi.org/10.1093/jjfinec/nbi002

Sentana, E. (2009): “The econometrics of mean-variance efficiency tests: a survey”, *Econometrics Journal* 12 (3), pp. C65-C101.

http://dx.doi.org/10.1111/j.1368-423X.2009.00295.x

Sentana, E., Calzolari, E. and G. Fiorentini (2008): “Indirect estimation of large conditionally heteroskedastic factor models, with an application to the Dow 30 stocks”, *Journal of Econometrics* 146 (1), pp. 10-25.

http://dx.doi.org/10.1016/j.jeconom.2008.06.001

### Other references

Bontemps, C. and Meddahi, N. (2005): “Testing normality: A GMM approach”, *Journal of Econometrics*, 124 (1), 149-186.

Bontemps, C. and Meddahi, N. (2011): “Testing distributional assumptions: A GMM approach”, *Journal of Applied Econometrics* 27 (6) pp. 1099-1255.

http://dx.doi.org/10.1002/jae.1250

Carrasco, M. and Florens, J.P. (2000): “Generalization of GMM to a continuum of moment conditions”, *Econometric Theory* 16 (06), pp 797-834, December 2000

Lo, A.W. (2002): “The Statistics of Sharpe Ratios”, *Financial Analyst Journal* July/August, 36-52.

López, J.A. (1999): “Methods for Evaluating Value-at-Risk Estimates”, *Federal Reserve Bank of San Francisco Economic Review* 2, 3-17.

Monfardini, C. (1998): “Estimating Stochastic Volatility Models through Indirect Inference”, *Econometrics Journal* 1, C113-C128.

### Books

Alexander, C. (2001): *Market Models. A Guide to Financial Data Analysis*, Wiley.

Hansen, L.P. and Sargent, T.J. (1991): “Lecture notes on least squares prediction theory”, in L.P. Hansen and T.J. Sargent, eds. *Rational Expectations Econometrics*, Westview Press.

Ingersoll, J.E. (1987): *Theory of financial decision making*, Rowan and Littlefield.

Campbell, J.Y., Lo, A.W. and MacKinlay, A.C. (1997):* The Econometrics of Financial Markets*, Princeton University Press (sections 5.3-5.6, 6.1-6.2 and 8.2).

Cochrane, J. (2005), *Asset Pricing*, Princeton University Press.

Hamilton, J.D. (1994): *Time Series Analysis*, Princeton University Press.

Johnson, R.A. and Wichern, D.W. (1992): *Applied Multivariate Statistical Analysis*, 3rd edition, Prentice-Hall.

Ogaki, M. (1993): “Generalized method of moments: econometric applications”, in G.S. Maddala, C.R. Rao and H.D. Vinod, *Handbook of Statistics *vol. 11, Elsevier Science Publishers.

Poirier, D.J. (1995): *Intermediate Statistics and Econometrics. A Comparative Approach*, MIT Press.