*Applied Economic Forecasting Using Time Series Methods*, by Eric Ghysels and Massimilliano Marcellino. It will be published this week by Oxford University Press. It has a very nice modern awareness of Big Data with emphasis on reduced-rank structure, regularization methods -- LASSO appears as early as p. 23! -- , structural change, mixed-frequencies, etc. It's also very tastefully done in terms of what's included and what's excluded, emphasizing what's most important and de-emphasizing the rest. As regards non-linearity, for example, volatility dynamics and regime-switching are in, and most of the rest is out.

## Monday, April 23, 2018

### Ghysels and Marcellino on Time-Series Forecasting

If you're teaching a forecasting course and want a good text, or if you're just looking for an informative and modern treatment, see

## Monday, April 16, 2018

### The History of Forecasting Competitions

Check out Rob Hyndman's "Brief History of Time Series Forecasting Competitions". I'm not certain whether the title's parallel to Hawking's Brief History of Time is intentional. At any rate, even if Hyndman's focus is rather more narrow than the origin and fate of the universe, his post is still fascinating and informative. Thanks to Ross Askanasi for bring it to my attention.

## Monday, April 9, 2018

### An Art Market Return Index

Rare and collectible goods, from fine art to fine wine, have many interesting and special aspects. Some are shared and some are idiosyncratic.

From the vantage point of alternative investments (among other things), it would be useful to have high-frequency indices for those asset markets, just as we do for traditional "financial" asset markets like equities.

Along those lines, in "Monthly Art Market Returns" Bocart, Ghysels, and Hafner develop a high-frequency measurement approach, despite the fact that art sales generally occur very infrequently. Effectively they develop a mixed-frequency repeat-sales model, which captures the correlation between art prices and other liquid asset prices that are observed much more frequently. They use the model to extract a monthly art market return index, as well as sub-indices for contemporary art, impressionist art, etc.

Quite fascinating and refreshingly novel.

From the vantage point of alternative investments (among other things), it would be useful to have high-frequency indices for those asset markets, just as we do for traditional "financial" asset markets like equities.

Along those lines, in "Monthly Art Market Returns" Bocart, Ghysels, and Hafner develop a high-frequency measurement approach, despite the fact that art sales generally occur very infrequently. Effectively they develop a mixed-frequency repeat-sales model, which captures the correlation between art prices and other liquid asset prices that are observed much more frequently. They use the model to extract a monthly art market return index, as well as sub-indices for contemporary art, impressionist art, etc.

Quite fascinating and refreshingly novel.

## Monday, April 2, 2018

### Econometrics, Machine Learning, and Big Data

Here's a useful slide deck by Greg Duncan at Amazon, from a recent seminar at FRB San Francisco (powerpoint, ughhh, sorry...). It's basically a superset of the keynote talk he gave at Penn's summer 2017 conference, Big Data in Predictive Dynamic Econometric Modeling. Greg understands better than most the close connection between "machine learning" and econometrics / statistics, especially between machine learning and the predictive perspective emphasized in time series for a century or so.

## Monday, March 26, 2018

### Classic Jacod (1994) Paper

*J. Financial Econometrics*will soon publish Jean Jacod's brilliant and beautiful 1994 paper, "Limit of Random Measures Associated with the Increments of a Brownian Semimartingale", which I just had the pleasure of reading for the first time. (Ungated version here.) Along with several others, I was asked to supply some comments for the issue's introduction. What follows is adapted from those comments, providing some historical background. (Except that it's not really historical background -- keep reading...)

Jacod's paper effectively lays the foundation for the vast subsequent econometric "realized volatility" (empirical quadratic variation) literature of the past twenty years. Reading it leads me to recall my early realized volatility work with Torben Andersen and Tim Bollerslev in the late 1990's and early 2000's. It started in the mid-1990's at a meeting of the NBER Asset Pricing Program, where I was the discussant for a paper of theirs, eventually published as Andersen and Bollerslev (1998). They were using realized volatility as the "realization" in a study of GARCH volatility forecast accuracy, and my discussion was along the lines of, "That's interesting, but I think you've struck gold without realizing it -- why not skip the GARCH and instead simply characterize, model, and forecast realized volatility directly?".

So we decided to explore realized volatility directly. Things really took off with Andersen et al. (2001) and Andersen et al. (2003). The research program was primarily empirical, but of course we also wanted to advance the theoretical foundations. We knew some relevant stochastic integration theory, and we made progress culminating in Theorem 2 of Andersen et al. (2003). Around the same time, Ole Bardorff-Nielsen and Neil Shephard were also producing penetrating and closely-related results (most notably Barndorff-Nielsen and Shephard, 2002). Very exciting early times.

Now let's return to Jacod's 1994 paper, and consider it against the above historical background of early econometric realized volatility papers. Doing so reveals not only its elegance and generality, but also its prescience: It was written well

*before*the "historical background"!! One wonders how it went unknown and unpublished for so long.

References

Andersen, T. G. and T. Bollerslev (1998), "Answering the Skeptics: Yes, Standard Volatility Models do Provide Accurate Forecasts,"

*International Economic Review*, 39, 885-905.

Andersen, T.G., T. Bollerslev, F.X. Diebold, and P. Labys (2001), "The Distribution of Realized Exchange Rate Volatility,"

*Journal of the American Statistical Association*, 96, 42-55.

Andersen, T.G., T. Bollerslev, F.X. Diebold, and P. Labys (2003), "Modeling and Forecasting Realized Volatility,"

*Econometrica*, 71, 579-625.

Barndorff-Nielsen, O. and N. Shephard (2002), "Econometric Analysis of Realized Volatility and its Use in Estimating Stochastic Volatility Models,"

*Journal of the Royal Statistical Society*, 64,

253-280.

Jacod, J. (1994), "Limit of Random Measures Associated with the Increments of a Brownian Semimartingale," Manuscript, Institute de Mathematiques de Jussieu, Universite Pierre et Marie Curie, Paris.

## Monday, March 19, 2018

### Big Data and Economic Nowcasting

Check out this informative paper from the Federal Reserve Bank of New York: "Macroeconomic Nowcasting and Forecasting with Big Data", by Brandyn Bok, Daniele Caratelli, Domenico Giannone, Argia Sbordone, and Andrea Tambalotti.

Key methods for confronting big data include (1) imposition of restrictions (for example, (a) zero restrictions correspond to "sparsity", (b) reduced-rank restrictions correspond to factor structure, etc.), and (2) shrinkage (whether by formal Bayesian approaches or otherwise).

Bok et al. provide historical perspective on use of (1)(b) for macroeconomic nowcasting; that is, for real-time analysis and interpretation of hundreds of business-cycle indicators using dynamic factor models. They also provide a useful description of FRBNY's implementation and use of such models in policy deliberations.

It is important to note that the Bok et al. approach nowcasts current-quarter GDP, which is different from nowcasting "the business cycle" (as done using dynamic factor models at FRB Philadelphia, for example), because GDP alone is not the business cycle. Hence the two approaches are complements, not substitutes, and both are useful.

Key methods for confronting big data include (1) imposition of restrictions (for example, (a) zero restrictions correspond to "sparsity", (b) reduced-rank restrictions correspond to factor structure, etc.), and (2) shrinkage (whether by formal Bayesian approaches or otherwise).

Bok et al. provide historical perspective on use of (1)(b) for macroeconomic nowcasting; that is, for real-time analysis and interpretation of hundreds of business-cycle indicators using dynamic factor models. They also provide a useful description of FRBNY's implementation and use of such models in policy deliberations.

It is important to note that the Bok et al. approach nowcasts current-quarter GDP, which is different from nowcasting "the business cycle" (as done using dynamic factor models at FRB Philadelphia, for example), because GDP alone is not the business cycle. Hence the two approaches are complements, not substitutes, and both are useful.

## Monday, March 12, 2018

### Sims on Bayes

Here's a complementary and little-known set of slide decks from Chris Sims, deeply insightful as always. Together they address some tensions associated with Bayesian analysis and sketch some resolutions. The titles are nice, and revealing. The first is "Why Econometrics Should Always and Everywhere Be Bayesian". The second is "Limits to Probability Modeling" (with Chris' suggested possible sub-title: "Why are There no Real Bayesians?").

## Thursday, March 8, 2018

### H-Index for Journals

In an earlier rant, I suggested that journals move from tracking inane citation "impact factors" to citation "H indexes" or similar, just as routinely done when evaluating individual authors. It turns out that RePEc already does it, here. There are literally many thousands of journals ranked. I show the top 25 below. Interestingly, four "field" journals actually make the top 10, effectively making them "super (uber?) field journals" (J. Finance, J. Financial Economics, J. Monetary Economics, and J. Econometrics). For example, J. Econometrics is basically indistinguishable from Review of Economic Studies.

## The rankings

## Wednesday, February 28, 2018

### The Rate of Return on Everything

Jorda, Knoll, Kuvshinov, Schularick and Taylor deliver more than just a memorable title, "The Rate of Return on Everything, 1870-2015". (Dec 2017 NBER version here; earlier ungated June 2017 version here.) Their paper is a fascinating exercise in data construction and analysis. It goes well beyond, say, the earlier and also-fascinating Dimson et al. (2002) book, by including housing, among other things.

*Caveat emptor*: In this case two words suffice -- survivorship bias. Jorda et al. are well aware of it, and they work hard to assess and address it. But still.## Monday, February 26, 2018

### STILL MORE on NN's and ML

I recently discussed how the nonparametric consistency of wide NN's proved underwhelming, which is partly why econometricians lost interest in NN's in the 1990s.

The other thing was the realization that NN objective surfaces are notoriously bumpy, so that arrival at a local optimum (e.g., by the stochastic gradient descent popular in NN circles) offered little comfort.

So econometricians' interest declined on both counts. But now both issues are being addressed. The new focus on NN depth as opposed to width is bearing much fruit. And recent advances in "reinforcement learning" methods effectively promote global as opposed to just local optimization, by experimenting (injecting randomness) in clever ways. (See, e.g., Taddy section 6, here.)

All told, it seems like quite an exciting new time for NN's. I've been away for 15 years. Time to start following again...

The other thing was the realization that NN objective surfaces are notoriously bumpy, so that arrival at a local optimum (e.g., by the stochastic gradient descent popular in NN circles) offered little comfort.

So econometricians' interest declined on both counts. But now both issues are being addressed. The new focus on NN depth as opposed to width is bearing much fruit. And recent advances in "reinforcement learning" methods effectively promote global as opposed to just local optimization, by experimenting (injecting randomness) in clever ways. (See, e.g., Taddy section 6, here.)

All told, it seems like quite an exciting new time for NN's. I've been away for 15 years. Time to start following again...

## Wednesday, February 21, 2018

### Larry Brown

Larry Brown has passed away. Larry was a giant of modern statistics and a towering presence at Penn. Simultaneously, everyone who knew him liked him, immensely. He will be missed dearly, both professionally and personally.

I received the obituary below from Penn's Statistics Department.

I received the obituary below from Penn's Statistics Department.

**Lawrence David Brown**Lawrence D. Brown died peacefully at 6:30 a.m. on Feb. 21, 2018, at the age of 77. Larry preserved his unfailing fortitude and good humor to his last day. Larry was born on Dec. 16, 1940, in Los Angeles, California. His parents moved to Alexandria, VA, during World War II, then returned to California. His father, Louis Brown, was a successful tax lawyer and later a professor of law at the University of Southern California, where he worked tirelessly on behalf of client services and conflict prevention, for which he coined the phrase preventive law. His mother, Hermione Kopp Brown, studied law in Virginia and then in Los Angeles and became one of the leading women lawyers in Los Angeles in the field of entertainment law, with emphasis on estate planning. Larry inherited their dedication for service, their mental acuity and resourcefulness, and their selfless good spirits. Larry graduated from Beverly Hills High School in 1957 and from the California Institute of Technology in 1961 and earned his Ph.D. in mathematics from Cornell University three years later. Initially hired at the University of California, Berkeley, he then taught in the mathematics department at Cornell University from 1966-72 and 1978-94 and in the statistics department at Rutgers University from 1972-78; he moved to the Wharton School at the University of Pennsylvania in 1994 and taught his last course there as the Miers Busch Professor of Statistics in the fall of 2017. One of the leading statisticians of his generation, he was the recipient of many honors, including devoted service as a member of the National Academy of Sciences, election to the American Academy of Arts and Sciences, the presidency of the Institute of Mathematical Statistics, and an honorary doctorate from Purdue University. He was much loved by his colleagues and his students, many of whom hold leading positions in the United States and abroad. His passion for his work was matched by his devotion to his family. His wife Linda Zhao survives him, as do their sons Frank and Louie, their daughter Yiwen Zhao, his daughters from his first marriage, Yona Alpers and Sarah Ackman, his brothers Marshall and Harold and their wives Jane and Eileen, and 19 grandchildren.## Monday, February 19, 2018

### More on Neural Nets and ML

I earlier mentioned Matt Taddy's "The Technological Elements of Artificial Intelligence" (ungated version here).

Among other things the paper has good perspective on the past and present of neural nets. (Read: his views mostly, if not exactly, match mine...)

Here's my personal take on some of the history vis a vis econometrics:

Econometricians lost interest in NN's in the 1990's. The celebrated Hal White et al. proof of NN non-parametric consistency as NN width (number of neurons) gets large at an appropriate rate was ultimately underwhelming, insofar as it merely established for NN's what had been known for decades for various other non-parametric estimators (kernel, series, nearest-neighbor, trees, spline, etc.). That is, it seemed that there was nothing

But the non-parametric consistency focus was all on NN

Here are some questions/observations on the new "deep learning":

1. Adding NN depth often seems helpful, insofar as deep learning often seems to "work" in various engineering applications, but where/what are the

2. Taddy emphasizes what might be called two-step deep learning. In the first step, "pre-trained" hidden layer nodes are obtained based on unsupervised learning (e.g., principle components (PC)) from various sets of variables. And then the second step proceeds as usual. That's very similar to the age-old idea of PC regression. Or, in multivariate dynamic environments and econometrics language, "factor-augmented vector autoregression" (FAVAR), as in Bernanke et al. (2005). So, are modern implementations of deep NN's effectively just nonlinear FAVAR's? If so, doesn't that also seem underwhelming, in the sense of -- dare I say it -- there being nothing really new about deep NN's?

3. Moreover, PC regressions and FAVAR's have issues of their own relative to one-step procedures like ridge or LASSO. See this and this.

Among other things the paper has good perspective on the past and present of neural nets. (Read: his views mostly, if not exactly, match mine...)

Here's my personal take on some of the history vis a vis econometrics:

Econometricians lost interest in NN's in the 1990's. The celebrated Hal White et al. proof of NN non-parametric consistency as NN width (number of neurons) gets large at an appropriate rate was ultimately underwhelming, insofar as it merely established for NN's what had been known for decades for various other non-parametric estimators (kernel, series, nearest-neighbor, trees, spline, etc.). That is, it seemed that there was nothing

*special*about NN's, so why bother?But the non-parametric consistency focus was all on NN

*width*; no one thought or cared much about NN*depth*. Then, more recently, people noticed that adding NN depth (more hidden layers) could be seriously helpful, and the "deep learning" boom took off.Here are some questions/observations on the new "deep learning":

1. Adding NN depth often seems helpful, insofar as deep learning often seems to "work" in various engineering applications, but where/what are the

*theorems*? What can be said rigorously about depth?2. Taddy emphasizes what might be called two-step deep learning. In the first step, "pre-trained" hidden layer nodes are obtained based on unsupervised learning (e.g., principle components (PC)) from various sets of variables. And then the second step proceeds as usual. That's very similar to the age-old idea of PC regression. Or, in multivariate dynamic environments and econometrics language, "factor-augmented vector autoregression" (FAVAR), as in Bernanke et al. (2005). So, are modern implementations of deep NN's effectively just nonlinear FAVAR's? If so, doesn't that also seem underwhelming, in the sense of -- dare I say it -- there being nothing really new about deep NN's?

3. Moreover, PC regressions and FAVAR's have issues of their own relative to one-step procedures like ridge or LASSO. See this and this.

## Tuesday, February 13, 2018

### Neural Nets, ML and AI

"The Technological Elements of Artificial Intelligence", by Matt Taddy, is packed with insight on the development of neural nets and ML as related to the broader development of AI. I have lots to say, but it will have to wait until next week. For now I just want you to have the paper. Ungated version at http://www.nber.org/chapters/c14021.pdf.

Abstract:

We have seen in the past decade a sharp increase in the extent that companies use data to optimize their businesses. Variously called the `Big Data' or `Data Science' revolution, this has been characterized by massive amounts of data, including unstructured and nontraditional data like text and images, and the use of fast and flexible Machine Learning (ML) algorithms in analysis. With recent improvements in Deep Neural Networks (DNNs) and related methods, application of high-performance ML algorithms has become more automatic and robust to different data scenarios. That has led to the rapid rise of an Artificial Intelligence (AI) that works by combining many ML algorithms together - each targeting a straightforward prediction task - to solve complex problems.

We will define a framework for thinking about the ingredients of this new ML-driven AI. Having an understanding of the pieces that make up these systems and how they fit together is important for those who will be building businesses around this technology. Those studying the economics of AI can use these definitions to remove ambiguity from the conversation on AI's projected productivity impacts and data requirements. Finally, this framework should help clarify the role for AI in the practice of modern business analytics and economic measurement.

We have seen in the past decade a sharp increase in the extent that companies use data to optimize their businesses. Variously called the `Big Data' or `Data Science' revolution, this has been characterized by massive amounts of data, including unstructured and nontraditional data like text and images, and the use of fast and flexible Machine Learning (ML) algorithms in analysis. With recent improvements in Deep Neural Networks (DNNs) and related methods, application of high-performance ML algorithms has become more automatic and robust to different data scenarios. That has led to the rapid rise of an Artificial Intelligence (AI) that works by combining many ML algorithms together - each targeting a straightforward prediction task - to solve complex problems.

We will define a framework for thinking about the ingredients of this new ML-driven AI. Having an understanding of the pieces that make up these systems and how they fit together is important for those who will be building businesses around this technology. Those studying the economics of AI can use these definitions to remove ambiguity from the conversation on AI's projected productivity impacts and data requirements. Finally, this framework should help clarify the role for AI in the practice of modern business analytics and economic measurement.

## Monday, February 12, 2018

### ML, Forecasting, and Market Design

Nice stuff from Milgrom and Tadelis. Improved forecasting via improved machine learning in turn helps improve our ability to design effective markets -- better anticipating consumer/producer demand/supply movements, more finely segmenting and targeting consumers/producers, more accurately setting auction reserve prices, etc. Presumably full density forecasts, not just the point forecasts on which ML tends to focus, should soon move to center stage.

http://www.nber.org/chapters/c14008.pdf

http://www.nber.org/chapters/c14008.pdf

## Monday, February 5, 2018

### Big Data, Machine Learning, and Economic Statistics

Greetings from a very happy Philadelphia celebrating the Eagles' victory!

The following is adapted from the "background" and "purpose" statements for a planned 2019 NBER/CRIW conference, "Big Data for 21st Century Economic Statistics". Prescient and fascinating reading. (The full call for papers is here.)

These naturally-occurring data include not only administrative data maintained by government agencies but also scanner data, data scraped from the Web, credit card company records, data maintained by payroll providers, medical records, insurance company records, sensor data, and the Internet of Things. If the challenges associated with their use can be satisfactorily resolved, these emerging sorts of data could allow the statistical agencies not only to supplement or replace the survey data on which they currently depend, but also to introduce new statistics that are more granular, more up-to-date, and of higher quality than those currently being produced.

The following is adapted from the "background" and "purpose" statements for a planned 2019 NBER/CRIW conference, "Big Data for 21st Century Economic Statistics". Prescient and fascinating reading. (The full call for papers is here.)

__Background__: The coming decades will witness significant changes in the production of the social and economic statistics on which government officials, business decision makers, and private citizens rely. The statistical information currently produced by the federal statistical agencies rests primarily on “designed data” -- that is, data collected through household and business surveys. The increasing cost of fielding these surveys, the difficulty of obtaining survey responses, and questions about the reliability of some of the information collected, have raised questions about the sustainability of that model. At the same time, the potential for using “big data” -- very large data sets built to meet governments’ and businesses’ administrative and operational needs rather than for statistical purposes -- in the production of official statistics has grown.These naturally-occurring data include not only administrative data maintained by government agencies but also scanner data, data scraped from the Web, credit card company records, data maintained by payroll providers, medical records, insurance company records, sensor data, and the Internet of Things. If the challenges associated with their use can be satisfactorily resolved, these emerging sorts of data could allow the statistical agencies not only to supplement or replace the survey data on which they currently depend, but also to introduce new statistics that are more granular, more up-to-date, and of higher quality than those currently being produced.

__Purpose__: The purpose of this conference is to provide a forum where economists, data providers, and data analysts can meet to present research on the use of big data in the production of federal social and economic statistics. Among other things, this involves discussing (1) Methods for combining multiple data sources, whether they be carefully designed surveys or experiments, large government administrative datasets, or private sector big data, to produce economic and social statistics; (2) Case studies illustrating how big data can be used to improve or replace existing statistical data series or create new statistical data series; (3) Best practices for characterizing the quality of big data sources and blended estimates constructed using data from multiple sources.## Monday, January 29, 2018

### Structural VAR Analysis

Kilian and Lutkepohl's

*Structural Vector Autoregressive Analysis*is now out. The back-cover blurbs below are not hyperbole. Indeed Harald Uhlig's is an understatement in certain respects -- to his list of important modern topics covered I would certainly add the "external instrument" approach. For more on that, beyond K&L, which went to press some time ago, see Stock and Watson's masterful 2018 external-instrument survey and extension, just now released as an NBER working paper. (Ungated K&L draft here; ungated S&W draft here.)## Sunday, January 21, 2018

### Averaging for Prediction in Econometrics and ML

Random thought. At the risk of belaboring the obvious, it's interesting to heighten collective awareness by thinking about the many appearances of averaging in forecasting, particularly in forecast combination. Some averages are weighted, and some are not. Most are linear, some are not.

- The "equal weights puzzle" in forecast combination
- Random forests, and ensemble averaging algorithms more generally
- Bootstrap aggregation ("bagging")
- Boosting
- Best subset averaging
- Survey averages
- k-nearest-neighbor forecasts
- Amisano-Geweke equally-weighted prediction pools
- "1/N" portfolios
- Bayesian model averaging
- Bates-Granger-Ramanathan frequentist model averaging
- Any forecasts extracted from markets (the ultimate information aggregator), ranging from "standard" markets (e.g., volatility forecasts extracted from options prices, interest rate forecasts extracted from the current yield curve, etc.), to explicit so-called "prediction markets" (e.g., sports betting markets).

## Sunday, January 14, 2018

### Comparing Interval Forecasts

Here's a new one, "On the Comparison of Interval Forecasts".

You'd think that interval forecast evaluation would be easy. After all, point forecast evaluation is (more or less) well understood and easy, and density forecast evaluation is (more or less) well understood and easy, and interval forecasts seem somewhere in between, so by some sort of continuity argument you'd think that their evaluation would also be well understood and easy. But no. In fact it's quite difficult, maybe impossible...

You'd think that interval forecast evaluation would be easy. After all, point forecast evaluation is (more or less) well understood and easy, and density forecast evaluation is (more or less) well understood and easy, and interval forecasts seem somewhere in between, so by some sort of continuity argument you'd think that their evaluation would also be well understood and easy. But no. In fact it's quite difficult, maybe impossible...

## Monday, January 8, 2018

### 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

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