Factor Model w Time-Varying Loadings

Markus Pelger has a nice paper on factor modeling with time-varying loadings in high dimensions. There are many possible applications. He applies it to level-slope-curvature yield-curve models. 

For me another really interesting application would be measuring connectedness in financial markets, as a way of tracking systemic risk. The Diebold-Yilmaz (DY) connectedness framework is based on a high-dimensional VAR with time-varying coefficients, but not factor structure. An obvious alternative in financial markets, which we used to discuss a lot but never pursued, is factor structure with time-varying loadings, exactly in Pelger! 

It would seem, however, that any reasonable connectedness measure in a factor environment would need to be based not only time-varying loadings but also time-varying idiosynchratic shock variances, or more precisely a time-varying noise/signal ratio (e.g., in a 1-factor model, the ratio of the idiosyncratic shock variance to the factor innovation variance). That is, connectedness in factor environments is driven by BOTH the size of the loadings on the factor(s) AND the amount of variation in the data explained by the factor(s). Time-varying loadings don't really change anything if the factors are swamped by massive noise. 

Typically one might fix the factor innovation variance for identification, but allow for time-varying idiosyncratic shock variance in addition to time-varying factor loadings. It seems that Pelger's framework does allow for that. Crudely, and continuing the 1-factor example, consider y_t  =  lambda_t  f_t  +  e_t. His methods deliver estimates of the time series of loadings lambda_t and factor f_t, robust to heteroskedasticity in the idiosyncratic shock e_t. Then in a second step one could back out an estimate of the time series of e_t and fit a volatility model to it. 
Then the entire system would be estimated and one could calculate connectedness measures based, for example, on variance decompositions as in the DY framework. 

Yield-Curve Modeling

Happy New Year to all!

Riccardo Rebonato's Bond Pricing and Yield-Curve Modeling: A Structural Approach will soon appear from Cambridge University Press. It's very well done -- a fine blend of  theory, empirics, market sense, and good prose.  And not least, endearing humility, well-captured by a memorable sentence from the acknowledgements: "My eight-year-old son has forgiven me, I hope, for not playing with him as much as I would have otherwise; perhaps he has been so understanding because he has had a chance to build a few thousand paper planes with the earlier drafts of this book."

TOC below.  Pre-order here. 

Contents

Acknowledgements page ix
Symbols and Abbreviations xi

Part I The Foundations
1 What This Book Is About 3
2 Definitions, Notation and a Few Mathematical Results 24
3 Links among Models, Monetary Policy and the Macroeconomy 49
4 Bonds: Their Risks and Their Compensations 63
5 The Risk Factors in Action 81
6 Principal Components: Theory 98
7 Principal Components: Empirical Results 108

Part II The Building Blocks: A First Look
8 Expectations 137
9 Convexity: A First Look 147
10 A Preview: A First Look at the Vasicek Model 160

Part III The Conditions of No-Arbitrage
11 No-Arbitrage in Discrete Time 185
12 No-Arbitrage in Continuous Time 196
13 No-Arbitrage with State Price Deflators 206
14 No-Arbitrage Conditions for Real Bonds 224
15 The Links with an Economics-Based Description of Rates 241

Part IV Solving the Models
16 Solving Affine Models: The Vasicek Case 263
17 First Extensions 285
18 A General Pricing Framework 299
19 The Shadow Rate: Dealing with a Near-Zero Lower Bound 329

Part V The Value of Convexity
20 The Value of Convexity 351
21 A Model-Independent Approach to Valuing Convexity 371
22 Convexity: Empirical Results 391

Part VI Excess Returns
23 Excess Returns: Setting the Scene 415
24 Risk Premia, the Market Price of Risk and Expected Excess Returns 431
25 Excess Returns: Empirical Results 449
26 Excess Returns: The Recent Literature – I 473
27 Excess Returns: The Recent Literature – II 497
28 Why Is the Slope a Good Predictor? 527
29 The Spanning Problem Revisited 547

Part VII What the Models Tell Us
30 The Doubly Mean-Reverting Vasicek Model 559
31 Real Yields, Nominal Yields and Inflation: The D’Amico–Kim–Wei Model 575
32 From Snapshots to Structural Models: The Diebold–Rudebusch Approach 602
33 Principal Components as State Variables of Affine Models: The PCA Affine Approach 618
34 Generalizations: The Adrian–Crump–Moench Model 663
35 An Affine, Stochastic-Market-Price-of-Risk Model 688

36 Conclusions 714

Bibliography 725

index 000

Regression on Term Structures

An important insight regarding use of dynamic Nelson Siegel (DNS) and related term-structure modeling strategies (see here and here) is that they facilitate regression on an entire term structure.  Regressing something on a curve might initially sound strange, or ill-posed.  The insight, of course, is that DNS distills curves into level, slope, and curvature factors; hence if you know the factors, you know the whole curve.  And those factors can be estimated and included in regressions, effectively enabling regression on a curve.

In a stimulating new paper, “The Time-Varying Effects of Conventional and Unconventional Monetary Policy: Results from a New Identification Procedure”, Atsushi Inoue and Barbara Rossi put that insight to very good use. They use DNS yield curve factors to explore the effects of monetary policy during the Great Recession.  That monetary policy is often dubbed "unconventional" insofar as it involved the entire yield curve, not just a very short "policy rate".

I recently saw Atsushi present it at NBER-NSF and Barbara present it at Penn's econometrics seminar.  It was posted today, here.

Conference on Bond Markets and Yield Curve Modeling

Fantastic job by Bank of Canada and FRBSF. Kudos to both for successfully assembling such talent.    It's just ending as I write.  It was all good, but the papers/discussants that resonated most with me were:

Session 4: Predicting Interest Rates

Robust Bond Risk Premia
Michael Bauer, Federal Reserve Bank of San Francisco
James Hamilton, University of California at San Diego
Discussant: John Cochrane, Hoover Institute at Stanford University

Loss Functions for Forecasting Treasury Yields
Hitesh Doshi, University of Houston
Kris Jacobs, University of Houston
Rui Liu, University of Houston
Discussant: Frank Diebold, University of Pennsylvania

Session 5: Term Structure Modeling and the Zero Lower Bound

Session Chair: Antonio Diez de los Rios, Bank of Canada
Tractable Term Structure Models: A New Approach
Bruno Feunou, Bank of Canada
Jean-Sebastien Fontaine, Bank of Canada
Anh Le, Kenan-Flagler Business School, University of North Carolina at Chapel Hill
Discussant: Greg Duffee, Johns Hopkins University

Staying at Zero with Affine Processes: An Application to Term Structure Modelling
Alain Monfort, Banque de France
Fulvio Pegoraro, Banque de France
Jean-Paul Renne, Banque de France
Guillaume Roussellet, Banque de France
Discussant: Marcel Priebsch, Board of Governors of the Federal Reserve System

Here's the whole thing:

5th Conference on Fixed Income Markets
Recent Advances in Fixed Income Research and Implications for Monetary Policy
Bank of Canada and Federal Reserve Bank of San Francisco
Yellen Conference Center
November 5-6, 2015
Thursday, November 5
8:00 – 8:45 a.m. Breakfast
8:45 – 9:00 a.m. Welcoming Remarks
Timothy Lane, Deputy Governor, Bank of Canada
9:00 – 10:30 a.m. Session 1: The Effects of Quantitative Easing
Session Chair: Michael Bauer, Federal Reserve Bank of San Francisco
A Lesson from the Great Depression that the Fed Might have Learned: A
Comparison of the 1932 Open Market Purchases with Quantitative Easing
Michael Bordo, Rutgers University, Hoover Institute at Stanford University, NBER
Arunima Sinha, Fordham University
Discussant: Annette Vissing-Jorgensen, Berkeley Haas
Transmission of Quantitative Easing: The Role of Central Bank Reserves
Jens Christensen, Federal Reserve Bank of San Francisco
Signe Krogstrup, Swiss National Bank
Discussant: Arvind Krishnamurthy, Stanford Graduate School of Business
10:30 – 11:00 a.m. Break
11:00 a.m. – 12:30 p.m. Session 2: Macroeconomic Risks and the Yield Curve
Economic Policy Uncertainty and the Yield Curve
Markus Leippold, Swiss Financial Institute and University of Zurich
Felix Matthys, Princeton University
Discussant: Anna Cieslak, Duke University
Macro Risks and the Term Structure
Geert Bekaert, Columbia University and NBER
Eric Engstrom, Board of Governors of the Federal Reserve System
Andrey Ermolov, Columbia University
Discussant: Mikhail Chernov, University of California at Los Angeles
12:30 p.m. Lunch, Market Street Dining Room, Fourth Floor
1:45 – 3:15 p.m. Session 3: Bond Prices in Equilibrium
Session Chair: Michael Ehrmann, Bank of Canada
A Macroeconomic Model of Equities and Real, Nominal, and Defaultable Debt
Eric Swanson, University of California at Irvine
Discussant: Hanno Lustig, Stanford Graduate School of Business
Bond Risk Premia in Consumption-based Models
Drew Creal, University of Chicago Booth School of Business
Jing Cynthia Wu, University of Chicago Booth School of Business and NBER
Discussant: Ivan Shaliastovich, Wharton School of the University of Pennsylvania
3:15 – 3:45 p.m. Break
3:45 – 5:15 p.m. Session 4: Predicting Interest Rates
Robust Bond Risk Premia
Michael Bauer, Federal Reserve Bank of San Francisco
James Hamilton, University of California at San Diego
Discussant: John Cochrane, Hoover Institute at Stanford University
Loss Functions for Forecasting Treasury Yields
Hitesh Doshi, University of Houston
Kris Jacobs, University of Houston
Rui Liu, University of Houston
Discussant: Frank Diebold, University of Pennsylvania
5:15 – 6:00 p.m. Reception, Salons A&B, Fourth Floor
6:00 – 8:00 p.m. Dinner, Market Street Dining Room, Fourth Floor
Introduction: John C. Williams, President, Federal Reserve Bank of San Francisco
Keynote Speaker: Athanasios Orphanides, Massachusetts Institute of Technology
Friday, November 6
8:00 – 8:45 a.m. Breakfast
8:45 – 10:15 a.m. Session 5: Term Structure Modeling and the Zero Lower Bound
Session Chair: Antonio Diez de los Rios, Bank of Canada
Tractable Term Structure Models: A New Approach
Bruno Feunou, Bank of Canada
Jean-Sebastien Fontaine, Bank of Canada
Anh Le, Kenan-Flagler Business School, University of North Carolina at Chapel Hill
Discussant: Greg Duffee, Johns Hopkins University
Staying at Zero with Affine Processes: An Application to Term Structure
Modelling
Alain Monfort, Banque de France
Fulvio Pegoraro, Banque de France
Jean-Paul Renne, Banque de France
Guillaume Roussellet, Banque de France
Discussant: Marcel Priebsch, Board of Governors of the Federal Reserve System
10:15 – 10:45 a.m. Break
10:45 – 12:15 p.m. Session 6: Financial Stability in Bond Markets
Reaching for Yield by Corporate Bond Mutual Funds
Jaewon Choi, University of Illinois at Urbana-Champaign
Matias Kronlund, University of Illinois at Urbana-Champaign
Discussant: Francis Longstaff, University of California at Los Angeles
Collateral, Central Bank Repos, and Systemic Arbitrage
Falko Fecht, Frankfurt School of Finance & Management
Kjell Nyborg, University of Zurich, Swiss Finance Institute, and CEPR
Jorg Rocholl, ESMT European School of Management and Technology
Jiri Woschitz, University of Zurich
Discussant: Stefania D’Amico, Federal Reserve Bank of Chicago
12:15 – 1:30 p.m. Lunch
1:30 p.m. Adjourn
Program Committee:
Antonio Diez de los Rios, Bank of Canada
Jean-Sebastien Fontaine, Bank of Canada
Michael Bauer, Federal Reserve Bank of San Francisco
Jens Christensen, Federal Reserve Bank of San Francisco

Christensen on Term Structure Modeling at EUI

Jens Christensen will give a Euro-Area Business Cycle Network three-day course at EUI, September 7-9. Jens is fantastic. Plus it's Florence in early September. It's not too late to register!

Course Syllabus

Recommended Reading List

Course Schedule

List of Participants (as of 05.08.2015)

Logistical Information