* Laptop required In order to participate in practical sessions, you must bring your own portable computer.

Overview

The objective of the course is to teach student how to use state of the art Bayesian methods to estimate and analyze modern macroeconomic models. The course will cover the most popular methods to construct posterior estimates of structural model parameters and probability intervals for arbitrary model outputs (such as impulse response functions and variance decompositions). Special attention will be given to recent advances in empirically analyzing the role of news, noise and imperfect information in business cycle models. The course aim to give students a good understanding of both advantages and limitations of the current generation of DSGE models.

Matlab programs to implement the theoretical methods and replicate the applications studied in class will be made available to students.

Course Outline

1. Macro models as data generating processes
   1. Models for forecasting and for policy making
   2. The building blocks of modern macro models
   3. Linearized macro models as state space systems
   4. Limitations of empirical models
2. State Space Models and Likelihood based estimation
   1. The Kalman filter
   2. The likelihood function for linear Gaussian models
   3. Numerical maximization of the likelihood
3. Bayesian Estimation of DSGE models
   1. Bayesian and frequentist methods
   2. Priors, data, posteriors
   3. Choosing priors for macro models
   4. Estimating DSGE models using the Metropolis-Hastings algorithm
4. Bayesian analysis of DSGE models
   1. Constructing probability intervals
   2. Prior predictive analysis
   3. Bayesian model comparison
5. Structural empirical models of news, noise and imperfect information
   1. News and noise: Identifying the effect of sentiment shocks
   2. Using survey data in likelihood based estimation
   3. The Kalman simulation smoother

References

• Canova, F. 2007, Methods for Applied Macroeconomic Research, Princeton University Press.
• Geweke, J. 2005, Contemporary Bayesian Econometrics and Statistics, Wiley-Interscience.
• Hamilton J. D. 1994, Time Series Analysis, Princeton University Press.
• Koop, G. 2003, Bayesian Econometrics, Wiley.

In addition to the text book references above, a list of the relevant research articles will be provided.

About the Instructor

Kristoffer Nimark was Researcher at the Center for Research on International Economics (CREI), Adjunct Professor at Universitat Pompeu Fabra, and Affiliated Professor of the Barcelona GSE until 2014. Previously he was a Visiting Assistant Professor at New York University and Senior Research Manager at the Reserve Bank of Australia.

* Laptop required In order to participate in practical sessions, you must bring your own portable computer.

Overview

Most time series in economics and finance display a large degree of persistence. Also, since economic and financial systems are known to go through both structural and behavioral changes, many of them exhibit characteristics that are not compatible with linearity.

The goal of this course is to introduce some popular non-stationary and non-linear time series models that have been found to be effective at modeling macroeconomic and financial time series data.

The course is mainly designed for practitioners and students who will use time series data in empirical analysis.

Topics Covered

1. Linear models for nonstationary data. Integration and fractional integration. Univariate and panel unit root tests, non-stationary dynamic factor models.
2. Cointegration and Fractional cointegration.
3. Measuring persistence. Impulse response functions and related methods. Aggregation and persistence.
4. Non-linear methods. Markov Switching and Threshold models.
5. Univariate and multivariate analysis
6. Dynamic non linear factor models
7. Real time assessment of recession probabilities

Prerequisites

Statistics and econometrics at an advanced undergraduate level are recommended. Computer applications will be based on MATLAB so some knowledge of basic programming will be useful although not necessary.

About the instructors

Laura Mayoral is a Barcelona GSE affiliated professor and works at the Institute for Economic Analysis (IAE) since 2006. Before joining the institute she was Assistant Professor and Ramón y Cajal Felow at Universitat Pompeu Fabra and Universitat Autonoma de Barcelona. She has also been visiting professor at New York University (Abu Dhabi), the Paris School of Economics and the Department of Economics of the University of Gothenburg.

Gabriel Pérez-Quirós has a B.A. in Economics from Universidad de Murcia (1989), Master in Economics and Finance from CEMFI (1991), and PhD in Economics from the University of California San Diego (1996). He is currently the Unit Head of Macroeconomic Research at the Research Department of the Bank of Spain. He previously worked on business cycle research at the Federal Reserve Bank of New York and the European Central Bank. He also worked as an advisor in the Economic Bureau of the Spanish Prime Minister and has been consultant for the European Commission, the European Central Bank, United Nations and the World Bank. He was a member of the Scientific Committee of the Euro Area Business Cycle Network. He is a Research Affiliate of the Centre for Economic Policy Research (CEPR) and was co-editor of SERIES, Journal of the Spanish Economic Association. He has published extensively on applications of non-linear models to the analysis of economic and financial variables over the b usiness cycle. He teaches PhD courses at the Universidad de Alicante where he has supervised several dissertations on these topics.

Overview

The objective of this first course is twofold. First, to present some of the most popular time series models designed to analyze the propagation mechanisms and measure the effects of economic shocks. In particular we will cover in details Structural Vector Autoregressive models with a special focus on several identification schemes used in the literature. We also present several extension like Smooth transition VARs, Threshold VAR, Time-Varying Coefficients VAR and FAVARs. The second objective is to discuss some recent applications of these models in economics. The focus will be on monetary and fiscal policy shocks, news shocks, technology shocks and stock market bubbles. Matlab programs to implement the theoretical methods and replicate the applications studied in class will be made available to students.

Requirement: basic knowledge of univariate time series models (ARMA models).

Contents

I. Structural VAR (SVAR)

a) The model: Representation, estimation.

b) Tools: Impulse response functions, variance decomposition, historical decomposition.

c) Identification: short and long-run zero, sign restrictions, penalty function approach, mixed restrictions, external instruments, narrative approach.

d) Applications: Monetary and fiscal policy shocks, technology shocks, news shocks, uncertainty shocks.

e) Local Projections

II. Nonlinear SVAR

a) Threshold VAR, Smooth Transition VAR.

b) Applications: Fiscal policy shocks in booms and recessions.

c) Nonlinear MA and local projections.

III. Time-Varying Coefficients VAR

a) The model: representation and estimation.

b) Applications: the Great Moderation and monetary policy.

IV. Factor Augmented VAR (FAVAR) Representation, estimation, identification of structural shocks. Application: Monetary policy.

About the Instructor

Luca Gambetti is Associate Professor of Economics at UAB and Barcelona GSE Associate Research Professor. He is a research fellow of MOVE (Markets, Organizations and Votes in Economics) and an external member of RECent. He obtained his PhD in Economics from Universitat Pompeu Fabra in 2006. Luca's research focuses on quantitative macroeconomics and applied time series analysis. His research has been published among others in the Journal of Monetary Economics, the Economic Journal, the Journal of Applied Econometrics and the American Economic Journal: Macro.

* Laptop required In order to participate in practical sessions, you must bring your own portable computer.

Overview

This is a course in introductory Bayesian econometrics with a focus on models used in empirical macroeconomics. It begins with a brief introduction to Bayesian econometrics, describing the main concepts underlying Bayesian theory and seeing how Bayesian methods work in the familiar context of the re- gression model. Computational methods are of great importance in modern Bayesian econometrics and these are discussed in detail.

In macroeconomics, we often have Big Data and work with models where the number of parameters to be estimated is large relative to the number of observations in the data set. A range of Bayesian methods have been derived for dealing with Big Data including Bayesian model averaging (BMA), stochastic search variable selection (SSVS) and the least absolute shrinkage and selection operator (LASSO). The second part of the course covers these methods and shows how they are applied in the context of the regression model. Subsequently, the course turns to state space models and discusses estimation of several state space models popularly used in macroeconomics. These include time series models where parameters change over time, models with regime change and stochastic volatility models.

The models and methods covered in this course are of direct use in many macroeconomic applications. But they also represent the groundwork that underlies popular multivariate macroeconomic models such as Vector Autoregres- sions (VARs), time-varying parameter VARs (TVP-VARs), factor and Dynamic Stochastic General Equilibirum (DSGE) models. 

Background

The course assumes that participants have some background knowledge of econometrics (e.g. obtained from a graduate level or senior undergraduate level econometrics course), but will assume no prior knowledge of Bayesian methods. I will assume that participants have a basic knowledge of probability (i.e. definitions and rules relating to conditional, marginal and joint probabilities and definitions and properties of common distributions such as the Normal and t- distributions). In addition, the participant should have a knowledge of basic matrix algebra. The Appendices to Koop (2003) provide a summary of the probability theory and matrix algebra used in this course.

Course Content

Code for references: K = Koop (2003), KPT = Koop, Poirier and Tobias (2007), Ko = Korobilis (2013)

Topic 1: Bayesian Basics


i) An Overview of Bayesian Econometrics (Reading: K, Chapter 1).


ii) Bayesian inference in the Normal linear regression model. Computational topic: Monte Carlo integration
Reading: K, Chapters 2 and 3.


iii) Bayesian treatment of the regression model with general error covariance matrix (Reading: K, pages 117-121 and 130-137)
 Computational topic: Gibbs sampling (Reading: K pages 62-68) and the Metropolis-Hasting algorithm (Reading, K pages 92-99).

Topic 2: Bayesian Model Averaging and Model Selection

i) Bayesian Model Averaging (BMA) (K, Chapter 11 and KPT, pages 281- 287) Computational topic: Markov Chain Monte Carlo Model Composition (MC3)

ii) Prior shrinkage: empirical Bayes and training sample priors (Ko)


iii) Stochastic Search Variable Selection (KPT, pages 287-289)


iv) The Bayesian Lasso (Ko)

Topic 3: Bayesian State Space Modelling

Reading: KK, section 3 and K, chapter 8.


i) The Normal linear state space model


ii) Linearized DSGE models as state space models


iii) Dynamic mixture models for modelling breaks and regime change


iv) Stochastic volatility


v) Forecasting with TVP regression models (including dynamic model averaging)

Reading 

Most of the material is taken from the textbook: Koop, G. (2003). Bayesian Econometrics. I also have a book of solved exercises that I will draw on occasionally: Koop, G., Poirier, D. and Tobias, J. (2007). Bayesian Econometric Methods, Cambridge University Press, (Volume 7 in the Econometrics Exercises Series edited by Karim Abadir, Jan Magnus and P.C.B Phillips). A good reference for the Big Data material covered in the course is: Korobilis, D. (2013). "Hierarchical shrinkage priors for dynamic regressions with many predictors," International Journal of Forecasting, 29, 43-59.

I end the course with an application to inflation forecasting based on:

Koop, G. and Korobilis, D. (2012). Forecasting inflation using dynamic model averaging, International Economic Review

About the Instructor

Gary Koop is a Professor in the Department of Economics at the University of Strathclyde. He received his PhD from the University of Toronto in 1989. He has held university posts in the UK, the US and Canada. His research interests lie in the field of Bayesian econometrics with a particular focus on macroeconometrics. He has a wide range of publications of theoretical and empirical work within this field. He has written several textbooks including Bayesian Econometrics and Bayesian Econometric Methods (co-authored with Dale Poirier and Justin Tobias). He is co-editor (with John Geweke and Herman van Dijk) of the Oxford Handbook of Bayesian Econometrics.

Overview

This course deals with factor models for large cross-sections of time-series (large N environment). We build the argument in steps, starting from the simplest multivariate technique, principal components.

We then discuss “small N” factor models for cross-sectional data and study how to estimate factor models with the EM algorithm.

We then review dynamic “small N” models for time-series, the associated state-space form and the Kalman filter and smoother, which are typically used to estimate those models.

Moving to the “large N” environment, first with cross-sectional data and then with time-series data, we discuss the link between factors and principal components. We clarify the distinction between static and dynamic factors and highlight how “large N” dynamic factor models can be used to perform structural analysis by using techniques similar to those used in Structural VAR models.

In this context, we review several applications, among others, factor augmented VAR models (FAVAR), the construction of business cycle indicators, how to handle the jagged nature of macroeconomic data releases in nowcasting and forecasting exercises, the analysis of monetary policy in real time, the identification of the monetary transmission mechanism, the identification of news shocks to technology.

If time permits, we discuss non-invertibilities and the relation to factor models.

Matlab programs to implement the theoretical methods and replicate the applications studied in class will be made available to students.

Requirements: good knowledge of time series econometrics, in particular VAR analysis.

Contents

Factor Models

1. Principal components estimator.
2. Small N, i.i.d. and dynamic, the EM algorithm, Kalman filter/smoother.
3. Large N, i.i.d. and dynamic. Consistency at large: a law of large numbers in the cross-section.
4. Applications: Commonality in European regions, new Eurocoin, monetary policy in real time, nowcasting, measuring macroeconomic uncertainty.

Structural Factor model (SFM)

1. Specification and estimation.
2. Tools: Impulse response functions, variance decomposition, historical decomposition.
3. Identification: Short and long-run zero, sign restrictions, penalty function approach.
4. DSGE and Factor models.
5. Applications: Monetary policy shocks, house prices, disaggregated prices.

Factor augmented VAR (FAVAR)

1. Applications: Monetary Policy, news shocks.
2. Testing non-invertibility.

About the instructor

Luca Sala is Associate Professor at the Ettore Bocconi Department of Economics and Research Fellow of IGIER (Innocenzo Gasparini Institute for Economic Research). He took part in the Graduate Research Program of the European Central Bank and was Visiting Student at Tel Aviv University. He has been a visiting scholar at the Department of Economics, New York University. He taught at the Università Nova de Lisboa and University of Oslo. He did research at the European Central Bank and Norges Bank. He has a PhD from the European Center for Advanced Research in Economics and Statistics (ECARES), at the Université Libre de Bruxelles (ULB).

* Laptop required In order to participate in practical sessions, you must bring your own portable computer.

Course overview

Introduced to econometrics by Nobel laureate Chris Sims and his students, Bayesian VAR methods have recently became the workhorse models for forecasting macroeconomic variables and are routinely used by central banks to inform policy decisions.

The two key characteristics of these methods is the possibility of handling very large cross-section of data – thereby including a large information set to base forecasts on- and the possibility of specifying a-priori beliefs on the behaviour of macroeconomic time series. More recent developments extended these models to account for time variation in the coefficients and volatilities, which dramatically improve the accuracy of density forecasts and now-casts.

This course aims at introducing state of the art methods for structural analysis and forecasting with Bayesian Vector Autoregressions. The course has a hands-on philosophy and Matlab code will be provided for each of the topics covered.

At the end of the course students will be able to specify and estimate a variety of multivariate linear models featuring drifting coefficients and volatilities, to produce real-time forecasts and now-casts, and to assess forecast uncertainty via fan charts.

Who should attend this course

Students should be familiar with the concept of linear regression models, the least squares estimator, and the definition of the likelihood function.

Deep understanding of asymptotic theory, test statistics, GMM, EM algorithms or other classical concepts is NOT needed for this course. However, good knowledge of basics of Bayesian computation and linear regression using conjugate priors would be beneficial. 

We will need to rely heavily on distributions such as the Normal, Gamma, and Wishart so students should be familiar with the concept of a p.d.f., a c.d.f., and their basic functional forms. Computations are in MATLAB. I will provide all the code in a very accessible form, so that even students with no knowledge of programming can attend this class. Nevertheless, students who are serious about using Bayesian macroeconometrics are expected to have some basic MATLAB skills (e.g. know how to estimate a VAR with OLS using basic commands).

Readings and resources

1. Banbura, M., Giannone, D., and Reichlin, L., 2010. Large Bayesian Vector Autoregressions, Journal of Applied Econometrics 25, 71-92

2. Carriero A., Clark, T. and Marcellino, M., 2019. Large Bayesian VARs with time varying volatility and non-conjugate priors. Journal Econometrics, forthcoming.

3. Cogley, T., and Sargent, T., 2005. Drifts and Volatilities: Monetary Policies and Outcomes in the post-WWII US, Review of Economic Dynamics 8, 262-302.

4. Del Negro, Marco, and Schorfheide Frank, Priors from General Equilibrium Models for VARS, International Economic Review, Volume 45, Number 2, p.643–673, (2004) 

5. Jacquier, E., Polson, N.G., Rossi, P. E., 1994, Bayesian Analysis of Stochastic Volatility Models. Journal of Business & Economic Statistics 20(1), 69-87.

6. Kadiyala, K., and Karlsson, S., 1997. Numerical Methods for Estimation and Inference in Bayesian VAR-Models, Journal of Applied Econometrics 12, 99-132.

7. Kim, S., Shephard, N. and S. Chib, 1998. Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models. Review of Economic Studies 65, 361-393.

8. Litterman, R., 1986. Forecasting with Bayesian Vector Autoregressions-Five Years of Experience, Journal of Business and Economic Statistics 4, 25-38.

9. Primiceri, G., 2005. Time Varying Structural Vector Autoregressions and Monetary Policy, Review of Economic Studies 72, 821-852.

Lecture Outline

1. Introduction to Bayesian methods and VARs. VAR models. Forecasting formulae. Bayes formula. The likelihood principle. Common misconceptions about unit roots and cointegration. The Minnesota prior.
2. Bayesian Vector Autoregressions. The independent Normal-Inverse Wishart prior. Gibbs sampling. Curse of dimensionality. The conjugate Normal-Inverse Wishart prior. MC sampling. Marginal likelihood. Hierarchical priors. Priors from DSGE models. Density forecasting and fan charts. 
3. State space models. The autoregressive model with time varying coefficients and volatilities. Carter-Kohn algorithm. Kim, Shepard, and Chib algorithm.
4. The Jacquier-Polson-Rossi approach. Metropolis Hastings algorithm. Exact simulation of an AR model with time varying volatilities. Volatility in mean model. Leverage model.
5. VARs with drifting coefficients and volatilities. Triangularization. Large Bayesian VARs with drifting volatilities.

About the Instructor

Andrea Carriero is Professor of Economics at Queen Mary University of London. He has been a consultant for the U.K. Treasury the Debt Management Office, and has previously worked in the Monetary Policy Strategy division of the European Central Bank. He has been a research visitor at the Federal Reserve Banks of New York and a visiting scholar at the University of Pennsylvania. He has an extensive experience in teaching a variety of hand-on courses on applied and financial econometrics in universities and central banks. Andrea’s research focuses on empirical macroeconomics and forecasting, with a particular emphasis in Bayesian methods and large datasets. He has published in several peer-reviewed international journals including the Journal of Econometrics, the Review of Economics and Statistics, The Journal of Business and Economics Statistics, the International Economic Review, the Journal of Applied Econometrics, and the Journal of the Royal Statistical Society.

The course provides an introduction to the state-of-the-art econometric and statistical techniques used for the analysis of large panels of economic and financial time series.

The course begins by reviewing the properties of the classic linear regression model in a large dimensional environment. It then introduces some of the most popular methodologies used to carry out estimation in such a setting, namely regularized estimation techniques such as Ridge, LASSO and Elastic-net. The course then focuses on showing how this methodology can be used for forecasting economic and financial time series using large panels. These techniques are applied to carry out forecasting using the FRED-MD dataset.

The second topic of the course is covariance matrix estimation in large dimensions. It is shown that the performance of the classic sample covariance estimator is poor when the dimensionality of the covariance is large. This motivates a large literature that proposes to regularize the sample covariance using a number of different strategies. In particular, the course focuses on the class of shrinkage estimators proposed by Ledoit and Wolf. These methods are illustrated with an application to asset allocation.

The third and final topic of the course is the estimation of large dimesional network models. It is shown how the estimation of these models can be cast as either a large covariance or a large Vector Autoregression estimation problem subject to appropriate sparsity constrainst. These network techiniques are then applied to estimate the CDS credit risk network of the European financial system as well to estimate the Granger volatility risk network of the US financial system.

Requirement: basic knowledge of matrix algebra, econometrics and time series econometrics.

Course Outline

1. Estimation and Regularization of Large Dimensional Regression Models

Linear regression model estimation and regularization in large dimensions, Ridge regression, LASSO regression, Elastic-net.

2. Forecasting Using Large Dimensional Panels of Time Series

Forecasting using factor, shrinkage and hybrid methods.

Application: Forecasting using the FRED-MD dataset

3. Large Dimensional Covariance Estimation

Covariance matrix estimation in large dimensions, the Marchenko–Pastur law, Ledoit & Wolf shrinkage estimators.

Application: Asset allocation for large dimensional panels of assets

4. Large Dimensional Network Estimation

Network models, contemporaneous Network models and covariance estimation, Granger Network models and VAR estimation.

Application: Estimation of the CDS credit risk network of the European financial system, Estimation the Granger volatility risk network of the US financial system.

About the instructor

Christian Brownlees is Assistant Professor in the Department of Economics and Business at Universitat Pompeu Fabra and Barcelona GSE Associate Research Professor. He obtained his PhD in Statistics in 2007 from the University of Florence and was a Post-Doc Research Fellow at NYU Stern until 2011. Christian’s research focuses on time-series analysis for financial and macro applications. His research has been published among others in the Journal of Econometrics, Annals of Statistics and the Review of Financial Studies.

References

• Bai, J. and Ng, S. (2002). Determining the number of factors in approximate factor models. Econometrica, 70, 191-221.
• Bai, J. and Ng, S. (2008). Forecasting economic time series using targeted predictors. Journal of Econometrics, 146, 304-317.
• Bai, J. and Ng, S. (2009). Boosting di¤usion indices. Journal of Applied Econometrics, 24, 607-629.
• Barigozzi, M. and Brownlees, C. NETS: Network Estimation for Time Series. Journal of Applied Econometrics, 2019, 34, 347-364
• Brownlees, C. Nualart, E. and Yucheng, S. Realized Networks. Journal of Applied Econometrics 2018, 33, 986-1006
• Bühlmann, P. and S. van de Geer (2011). Statistics for High–Dimensional Data: Methods, Theory and Applications. New York: Springer.
• Dahlhaus, R. (2000). Graphical Interaction Models for Multivariate Time Series. Metrika 51, 157–172.
• De Mol, C., D. Giannone, and L. Reichlin (2008). Forecasting Using a Large Number of Predictors: Is Bayesian Shrinkage a Valid Alternative to Principal Components? Journal of Econometrics 146, 318–328.
• Diebold, F. and K. Yilmaz (2009). Measuring financial asset return and volatility spillovers, with application to global equity markets. Economic Journal, 158–171.
• Diebold, F. and K. Yilmaz (2015). Financial and Macroeconomic Connectedness: A Network Approach to Measurement and Monitoring. Oxford University Press.
• Fan, J., Fan, Y., and Lv, J. (2008). High Dimensional Covariance Matrix Estimation using a Factor Model. Journal of Econometrics, 147, 186-197
• Friedman, J., T. Hastie, and R. Tibshirani (2008). Sparse Inverse Covariance Estimation with the Graphical Lasso. Biostatistics 9, 432–441.
• Hautsch, N., J. Schaumburg, and M. Schienle (2012). Financial Network Systemic Risk Contributions. Technical report, Discussion Paper 2012-053, CRC 649, Humboldt-Universität zu Berlin.
• Kock, A. B. (2012). On the Oracle Property of the Adaptive Lasso in Stationary and Nonstationary Autoregressions. CREATES Research Papers 2012-05, School of Economics and Management, University of Aarhus.
• Ledoit, O. and Wolf, M. (2004). A Well-Conditioned Estimator for Large-Dimensional Covariance Matrices. Journal of Multivariate Analysis, 88, 365-411
• Ledoit, O. and Wolf, M. (2012). Nonlinear Shrinkage Estimation of Large-Dimensional Covariance Matrices. Annals of Statistics, 40, 1024-1060
• 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.
• Stock, J. H. and Watson, M. W. (2004). Combination forecasts of output growth in a sevencountry data set. Journal of Forecasting, 23, 405–430.