Long Memory / Scaling Laws in Return Volatility

The 25-year accumulation of evidence for long memory / fractional integration / self-similarity / scaling laws in financial asset return volatility continues unabated.  For the latest see this nice new paper from Bank of Portugal, in particular its key Table 6. Of course the interval estimates of the fractional integration parameter "d" are massively far from both 0 and 1 -- that's the well-known long memory. But what's new and interesting is the systematic difference in the intervals depending on whether one uses absolute or range-based volatility. The absolute d intervals tend to be completely below 1/2 (0<d<1/2 corresponds to covariance-stationary dynamics), whereas the range-based d intervals tend to include 1/2 (1/2<d<1 corresponds to mean-reverting but not covariance- stationary dynamics, due to infinite unconditional variance). 

Realized vol based on the range is less noisy than realized vol based on absolute returns. But least noisy of all, and not considered in the paper above, is realized vol calculated directly from high-frequency return data (HFD-vol), as done by numerous authors in recent decades. Interestingly, recent work for HFD-vol also reports d intervals that tend to poke above 1/2. See this earlier post.

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 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.

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.

Bayesian Random Projection (More on Terabytes of Economic Data)

Some additional thoughts related to Serena Ng's World Congress piece (earlier post here, with a link to her paper):

The key newish dimensionality-reduction strategies that Serena emphasizes are random projection and leverage score sampling.  In a regression context both are methods for optimally approximating an NxK "X matrix" with an Nxk X matrix, where k<<K. They are very different and there are many issues. Random projection delivers a smaller X matrix with columns that are linear combinations of those of the original X matrix, as for example with principal-component regression, which can sometimes make for difficult interpretation.  Leverage score sampling, in contrast, delivers a smaller X matrix with columns that are simply a subset of those of those of the original X matrix, which feels cleaner but has issues of its own.

Anyway, a crucial observation is that for successful predictive modeling we don't need deep interpretation, so random projection is potentially just fine -- if it works, it works, and that's an empirical matter.  Econometric extensions  (e.g., to VAR's) and evidence (e.g., to macro forecasting) are just now emerging, and the results appear encouraging.  An important recent contribution in that regard is Koop, Korobilis, and Pettenuzzo (in press), which significantly extends and applies earlier work of Guhaniyogi and Dunson (2015) on Bayesian random projection ("compression").  Bayesian compression fits beautifully in a MCMC framework (again see Koop et al.), including model averaging across multiple random projections, attaching greater weight to projections that forecast well.  Very exciting!

Analyzing Terabytes of Economic Data

Serena Ng's World Congress piece is out as an NBER w.p.  It's been floating around for a long time, but just in case you missed it, it's a fun and insightful read:

Opportunities and Challenges: Lessons from Analyzing Terabytes of Scanner Data
by Serena Ng  -  NBER Working Paper #23673.
http://papers.nber.org/papers/w23673

(Ungated copy at http://www.columbia.edu/~sn2294/papers/sng-worldcongress.pdf)

Abstract:

This paper seeks to better understand what makes big data analysis different, what we can and cannot do with existing econometric tools, and what issues need to be dealt with in order to work with the data efficiently.  As a case study, I set out to extract any business cycle information that might exist in four terabytes of weekly scanner data.  The main challenge is to handle the volume, variety, and characteristics of the data within the constraints of our computing environment. Scalable and efficient algorithms are available to ease the computation burden, but they often have unknown statistical properties and are not designed for the purpose of efficient estimation or optimal inference.  As well, economic data have unique characteristics that generic algorithms may not accommodate.  There is a need for computationally efficient econometric methods as big data is likely here to stay.

Regression Discontinuity and Event Studies in Time Series

Check out the new paper, "Regression Discontinuity in Time [RDiT]: Considerations for Empirical Applications", by Catherine Hausman and David S. Rapson.  (NBER Working Paper No. 23602, July 2017.  Ungated copy here.)

It's interesting in part because it documents and contributes to the largely cross-section regression discontinuity design literature's awakening to time series. But the elephant in the room is the large time-series "event study" (ES) literature, mentioned but not emphasized by Hausman and Rapson.  [In a one-sentence nutshell, here's how an ES works: model the pre-event period, use the fitted pre-event model to predict the post-event period, and ascribe any systematic forecast error to the causal impact of the event.]  ES's trace to the classic Fama et al. (1969).  Among many others, MacKinlay's 1997 overview is still fresh, and Gürkaynak and Wright (2013) provide additional perspective.

One question is what the RDiT approach adds to the ES approach, and related, what it adds to well-developed time-series toolkit of other methods for assessing structural change. At present, and notwithstanding the Hausman-Rapson paper, my view is "little or nothing".  Indeed in most respects it would seem that a RDiT study *is* an ES, and conversely.  So call it what you will, "ES" or "RDiT". 

But there are important open issues in ES / RDiT, and Hausman-Rapson correctly emphasize one of them, namely issues and difficulties associated with "wide" pre- and post-event windows, which is often the relevant case in time series.

Things are generally "easy" in cross sections, where we can usually take narrow windows (e.g., in the classic scholarship exam example, we use only test scores very close to the scholarship threshold).  Things are similarly "easy" in time series *IF* we can take similarly narrow windows (e.g., high-frequency asset return data facilitate taking narrow pre- and post-event windows in financial applications).  In such cases it's comparatively easy to credibly ascribe a post-event break to the causal impact of the event.

But in other time-series areas like macro and environmental, we might want (or need) to use wide pre- and post-event windows.  Then the trick becomes modeling the pre- and post-event periods successfully enough so that we can credibly assert that any structural change is due exclusively to the event -- very challenging, but not hopeless.

Hats off to Hausman and Rapson for beginning to bridge the ES and regression discontinuity literatures, and for implicitly helping to push the ES literature forward.

Time-Series Regression Discontinuity

I'll have something to say in next week's post.  Meanwhile check out the interesting new paper, "Regression Discontinuity in Time: Considerations for Empirical Applications", by Catherine Hausman and David S. Rapson, NBER Working Paper No. 23602, July 2017.  (Ungated version here.)

Big Data in Econometric Modeling

Here's a speakers' photo from last week's Penn conference, Big Data in Dynamic Predictive Econometric Modeling.  Click through to find the program, copies of papers and slides, a participant list, and a few more photos.  A good and productive time was had by all!

ML and Metrics VI: A Key Difference Between ML and TS Econometrics

[Click on "Machine Learning" at right for earlier "Machine Learning and Econometrics" posts.]

Continuing:

So then, statistical machine learning (ML) and time series econometrics (TS) have lots in common. But there's also an interesting difference: ML's emphasis on flexible nonparametric modeling of conditional-mean nonlinearity doesn't play a big role in TS. 

Of course there are the traditional TS conditional-mean nonlinearities: smooth non-linear trends, seasonal shifts, and so on. But there's very little evidence of important conditional-mean nonlinearity in the covariance-stationary (de-trended, de-seasonalized) dynamics of most economic time series. Not that people haven't tried hard -- really hard -- to find it, with nearest neighbors, neural nets, random forests, and lots more. 

So it's no accident that things like linear autoregressions remain overwhelmingly dominant in TS. Indeed I can think of only one type of conditional-mean nonlinearity that has emerged as repeatedly important for (at least some) economic time series: Hamilton-style Markov-switching dynamics.

[Of course there's a non-linear elephant in the room:  Engle-style GARCH-type dynamics. They're tremendously important in financial econometrics, and sometimes also in macro-econometrics, but they're about conditional variances, not conditional means.]

So there are basically only two important non-linear models in TS, and only one of them speaks to conditional-mean dynamics. And crucially, they're both very tightly parametric, closely tailored to specialized features of economic and financial data.

Now let's step back and assemble things:

ML emphasizes approximating non-linear conditional-mean functions in highly-flexible non-parametric fashion. That turns out to be doubly unnecessary in TS: There's just not much conditional-mean non-linearity to worry about, and when there occasionally is, it's typically of a highly-specialized nature best approximated in highly-specialized (tightly-parametric) fashion.

Machine Learning and Econometrics V: Similarities to Time Series

[Notice that I changed the title from "Machine Learning vs. Econometrics" to "Machine Learning and  Econometrics", as the two are complements, not competitors, as this post will begin to emphasize. But I've kept the numbering, so this is number five.  For others click on Machine Learning at right.]

Thanks for the overwhelming response to my last post, on Angrist-Pischke (AP).  I'll have more to say on AP a few posts from now, but first I need to set the stage.

A key observation is that statistical machine learning (ML) and time-series econometrics/statistics (TS) are largely about modeling, and they largely have the same foundational perspective. Some of the key ingredients are:

-- George Box got it right: "All models are false; some are useful", so search for good approximating models, not "truth".

-- Be explicit about the loss function, that is, about what defines a "good approximating model" (e.g., 1-step-ahead out-of-sample mean-squared forecast error)

-- Respect and optimize that loss function in model selection (e.g., BIC)

-- Respect and optimize that loss function in estimation (e.g., least squares)

-- Respect and optimize that loss function in forecast construction (e.g., Wiener-Kolmogorov-Kalman)

-- Respect and optimize that loss function in forecast evaluation, comparison, and combination (e.g., Mincer-Zarnowitz evaluations, Diebold-Mariano comparisons, Granger-Ramanathan combinations).

So time-series econometrics should embrace ML -- and it is.  Just look at recent work like this.

Predictive Loss vs. Predictive Regret

It's interesting  to contrast two prediction paradigms.

A.  The universal statistical/econometric approach to prediction:  
Take a stand on a loss function and find/use a predictor that minimizes conditionally expected loss.  Note that this is an absolute standard.  We minimize loss, not some sort of relative loss.

B.  An alternative approach to prediction, common in certain communities/literatures:
Take a stand on a loss function and find/use a predictor that minimizes regret.  Note that this is a relative standard.  Regret minimization is relative loss minimization, i.e., striving to do no worse than others.

Approach A strikes me as natural and appropriate, whereas B strikes me as as quirky and "behavioral".  That is, it seems to me that we generally want tools that perform well, not tools that merely perform no worse than others.

There's also another issue, the ex ante nature of A (standing in the present, conditioning on available information, looking forward) vs. the ex post nature of B (standing in the future, looking backward).  Approach A again seems more natural and appropriate.

Time-Series Econometrics and Climate Change

It's exciting to see time series econometrics contributing to the climate change discussion.  

Check out the upcoming CREATES conference, "Econometric Models of Climate Change", here.

Here are a few good examples of recent time-series climate research, in chronological order.  (There are many more.  Look through the reference lists, for example, in the 2016 and 2017 papers below.)

Jim Stock et al. (2009) in Climatic Change.

Pierre Perron et al. (2013) in Nature.

Peter Phillips et al. (2016) in Nature.

Proietti and Hillebrand (2017), forthcoming in Journal of the Royal Statistical Society.

Mixed-Frequency High-Dimensional Time Series

Notice that high dimensions and mixed frequencies go together in time series. (If you're looking at a huge number of series, it's highly unlikely that all will be measured at the same frequency, unless you arbitrarily exclude all frequencies but one.) So high-dim MIDAS vector autoregression (VAR) will play a big role moving forward. The MIDAS literature is starting to go multivariate, with MIDAS VAR's appearing; see Ghysels (2015, in press) and Mikosch and Neuwirth (2016 w.p.). 

But the multivariate MIDAS literature is still low-dim rather than high-dim. Next steps will be: 

(1) move to high-dim VAR estimation by using regularization methods (e.g. LASSO variants), 

(2) allow for many observational frequencies (five or six, say), 

(3) allow for the "rough edges" that will invariably arise at the beginning and end of the sample, and 

(4) visualize results using network graphics.

Indicator Saturation Estimation

In an earlier post, "Fixed Effects Without Panel Data",  I argued that you could allow for (and indeed estimate) fixed effects in pure cross sections (i.e., no need for panel data) by using regularization estimators like LASSO. The idea is to fit a profligately-parameterized model but then to recover d.f. by regularization.

Note that you can use the same idea in time-series contexts.  Even in a pure time series, you can allow for period-by-period time effects, broken polynomial trend with an arbitrary number of breakpoints, etc., via regularization.  

It turns out that a fascinating small literature on so-called "indicator saturation estimation" pursues this idea.  The "indicators" are things like period-by-period time dummies, break-date location dummies, etc., and "saturation" refers to the profligate parameterization.  Prominent contributors include David Hendry and Soren Johanssen; see this new paper and those that it cites.  (Very cool application, by the way, to detecting historical volcanic eruptions.)

The HAC Emperor has no Clothes

Well, at least in time-series settings. (I'll save cross sections for a later post.)

Consider a time-series regression with possibly heteroskedastic and/or autocorrelated disturbances, 

\( y_t = x_t' \beta + \varepsilon_t  \). 

A popular approach is to punt on the potentially non-iid disturbance, instead simply running OLS with kernel-based heteroskedasticity and autocorrelation consistent (HAC) standard errors.

Punting via kernel-HAC estimation is a bad idea in time series, for several reasons:

(1) [Kernel-HAC is not likely to produce good \(\beta\) estimates.] It stays with OLS and hence gives up on efficient estimation of \(\hat{\beta}\). In huge samples the efficiency loss from using OLS rather than GLS/ML is likely negligible, but time-series samples are often smallish. For example, samples like 1960Q1-2014Q4 are typical in macroeconomics -- just a couple hundred observations of highly-serially-correlated data.

(2) [Kernel-HAC is not likely to produce good \(\beta\) inference.] Its standard errors are not tailored to a specific parametric approximation to \(\varepsilon\) dynamics. Proponents will quickly counter that that's a benefit, not a cost, and in some settings the proponents may be correct. But not in time series settings. In time series, \(\varepsilon\) dynamics are almost always accurately and parsimoniously approximated parametrically (ARMA for conditional mean dynamics in \(\varepsilon\), and GARCH for conditional variance dynamics in \(\varepsilon\)). Hence kernel-HAC standard errors may be unnecessarily unreliable in small samples, even if they're accurate asymptotically. And again, time-series sample sizes are often smallish. 

(3) [Most crucially, kernel-HAC fails to capture invaluable predictive information.] Time series econometrics is intimately concerned with prediction, and explicit parametric modeling of dynamic heteroskedasticity and autocorrelation in \(\varepsilon\) can be used for improved prediction of \(y\). Autocorrelation can be exploited for improved point prediction, and dynamic conditional heteroskedasticity can be exploited for improved interval and density prediction. Punt on them and you're potentially leaving a huge amount of money on the table.

The clearly preferable approach is traditional parametric disturbance heteroskedasticty / autocorrelation modeling, with GLS/ML estimation. Simply allow for ARMA(p,q)-GARCH(P,Q) disturbances (say), with p,q, P and Q selected by AIC (say). (In many applications something like AR(3)-GARCH(1,1) or ARMA(1,1)-GARCH(1,1) would be more than adequate.) Note that the traditional approach is actually fully non-parametric when appropriately viewed as a sieve, and moreover it features automatic bandwidth selection.

Kernel-HAC people call the traditional strategy "pre-whitening," to be done prior to kernel-HAC estimation. But the real point is that it's all -- or at least mostly all -- in the pre-whitening.

In closing, I might add that the view expressed here is strongly supported by top-flight research. On my point (2) and my general recommendation, for example, see the insightful work of den Haan and Levin (2000). It fell on curiously deaf ears and remains unpublished many years later. (It's on Wouter den Haan's web site in a section called "Sleeping and Hard to Get"!) In the interim much of the world jumped on the kernel-HAC bandwagon. It's time to jump off.

Athey and Imbens on Machine Learning and Econometrics

Check out Susan Athey and Guido Imbens' NBER Summer Institute 2015 "Lectures on Machine Learning". (Be sure to scroll down, as there are four separate videos.) I missed the lectures this summer, and I just remembered that they're on video. Great stuff, reflecting parts of an emerging blend of machine learning (ML), time-series econometrics (TSE) and cross-section econometrics (CSE).

The characteristics of ML are basically (1) emphasis on overall modeling, for prediction (as opposed, for example, to emphasis on inference), (2) moreover, emphasis on non-causal modeling and prediction, (3) emphasis on computationally-intensive methods and algorithmic development, and (4) emphasis on large and often high-dimensional datasets.

Readers of this blog will recognize the ML characteristics as closely matching those of TSE! Rob Engle's V-Lab at NYU Stern's Volatility Institute, for example, embeds all of (1)-(4). So TSE and ML have a lot to learn from each other, but the required bridge is arguably quite short.

Interestingly, Athey and Imbens come not from the TSE tradition, but rather from the CSE tradition, which typically emphasizes causal estimation and inference. That makes for a longer required CSE-ML bridge, but it may also make for a larger payoff from building and crossing it (in both directions).

In any event I share Athey and Imbens' excitement, and I welcome any and all cross-fertilization of ML, TSE and CSE.