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.

Automated Time-Series Forecasting at Google

Check out this piece on automated time-series forecasting at Google.  It's a fun and quick read. Several aspects are noteworthy.  

On the upside:

-- Forecast combination features prominently -- they combine forecasts from an ensemble of models.  

-- Uncertainty is acknowledged -- they produce interval forecasts, not just point forecasts.

On the downside:

-- There's little to their approach that wasn't well known and widely used in econometrics a quarter century ago (or more).  Might not something like Autobox, which has been around and evolving since the 1970's, do as well or better?

Varieties of RCT Extensibility

Even internally-valid RCT's have issues. They reveal the treatment effect only for the precise experiment performed and situation studied. Consider, for example, a study of the effects of fertilizer on crop yield, done for region X during a heat wave. Even if internally valid, the estimated treatment effect is that of fertilizer on crop yield in region X during a heat wave. The results do not necessarily generalize -- and in this example surely do not generalize -- to times of ``normal" weather, even in region X. And of course, for a variety of reasons, they may not generalize to regions other than X, even in heat waves.

Note the interesting time-series dimension to the failure of external validity (extensibility) in the example above. (The estimate is obtained during this year's heat wave, but next year may be "normal", or "cool". And this despite the lack of any true structural change. But of course there could be true structural change, which would only make matters worse.) This contrasts with the usual cross-sectional focus of extensibility discussions (e.g., we get effect e in region X, but what effect would we get in region Z?)

In essence, we'd like panel data, to account both for cross-section effects and time-series effects, but most RCT's unfortunately have only a single cross section.

Mark Rosenzweig and Chris Udry have a fascinating new paper, "Extenal Validity in a Stochastic World", that grapples with some of the time-series extensibility issues raised above.

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.)

Interesting New Work on Yield Curve Modeling

Loved last week's PIER lectures at Penn. Good people, good times, good spring weather.  (Please join us next year in May 2016! More information in due course.) On Thursday we did yield curves, which had me thinking about what's new that I like in that area. Not surprisingly, I'm a fan of dynamic Nelson-Siegel (DNS), arbitrage-Free Nelson-Siegel (AFNS), and the many variations.  (See the Diebold-Rudebusch 2013 book.) What's more surprising is that although Nelson-Siegel is almost thirty years old, and DNS/AFNS is almost a teenager, interesting and useful new variations keep coming along.

The most important new work concerns imposition of the zero lower bound (ZLB). Fischer Black's "shadow rate" approach has influenced me most. Recently it's been taken to new heights by Glenn Rudebusch and coauthors at the Federal Reserve Bank of San Francisco (e.g., Christensen and Rudebusch 2015 -- just published in Journal of Financial Econometrics), and Leo Krippner at the Reserve Bank of New Zealand (see his wonderful 2015 book). The amazing thing is that one can stay in the DNS/AFNS framework -- the key tractable subclass of Gaussian affine models -- and still respect the ZLB by appropriately truncating simple simulations. The figure below, assembled from some of Krippner's, says it all. Also see these slides.   

I'm also partial to shadow-rate ZLB work by Cynthia Wu and coauthors at Chicago and San Diego (e.g. Wu and Xia, 2014). (Thanks to Jim Hamilton, her Ph.D. advisor, for reminding me!) See the monthly Wu-Xia shadow short rate series, produced and published to the web by FRB Atlanta.

Last and not at all least is the recent "ARG0" work of Monfort et al., which imposes the ZLB in a very different and elegant way. Again see these slides.   

Another interesting strand of recent DNS/AFNS progress concerns modeling the interaction of bond yield factors, macro fundamentals, and central bank policy.  More on that sometime soon.

Vienna Workshop on High-Dimensional Time Series In Macroeconomics and Finance

Program looking good:  https://www.conftool.net/timeseries2015/sessions.php.  Presumably papers will be posted, or at least you can email the authors.

Measuring Predictability

A friend writes the following.  (I have edited very slightly for clarity.)

Based on forecasts you've seen, what would you say is a "reasonable" ratio of the standard deviation of the forecast error to the standard deviation of a covariance-stationary series being forecast? ... It would be great if you can tell me "I'd consider x reasonable and y too high." 

The problem is that the premise underlying the question (namely, that there is such a "reasonable" value of the ratio \(r\) of innovation variance to unconditional  variance) is false.  That is, there's no small value \(c\) of \(r\) such that \(r<c\) means that we've done a good forecasting job.  Equivalently, there's no large value \(c'\) of the predictive \( R^2~ (R^2 = 1 - r^2) \) such that \(R^2 > c'\) means that we've done a good forecasting job.  Instead, "good" \(c\) or \(c'\) values depend critically on the dynamic nature of the series being forecast.  Consider, for example, a covariance-stationary AR(1) process, \(y_t = \phi y_{t-1} + \varepsilon_t\), where \(\varepsilon_t \sim iid (0, \sigma^2)\). The innovation variance is \(\sigma^2\) and the unconditional variance is \(\sigma^2 / (1 - \phi^2)\), so the lower bound on  \(r\) (and hence the upper bound on  \(R^2\)) depends entirely on \(\phi\) and can be anywhere in the unit interval! This is an important lesson: "predictability" can (and does) differ greatly across economic series. For more than you ever wanted to know, see Diebold and Kilian (2001), "Measuring Predictability: Theory and Macroeconomic Applications".

Mostly Harmless Econometrics?

I've had Angrist-Pischke's Mostly Harmless Econometrics: An Empiricist's Companion (MHE) for a while, but I just got around to reading it. (By the way, a lower-level follow-up was just published.)

There's a lot to like about MHE. It's an insightful and fun treatment of micro-econometric regression-based causal effect estimation -- basically how to (try to) tease causal information from least-squares regressions fit to observational micro data. It's filled with wisdom, exploring many subtleties and nuances. In many ways it's written not for students at age 23, but rather for seasoned researchers at age 53. And it tells its story in a marvelously engaging conversational style.

But there's also a lot not to like about MHE. The problem isn't what it includes, but rather what it excludes. Starting with its title and continuing throughout, MHE promotes its corner of applied econometrics as all of applied econometrics, or at least all of the "mostly harmless" part (whatever that means). Hence it effectively condemns much of the rest as "harmful," and sentences it to death by neglect. It gives the silent treatment, for example, to anything structural -- whether micro-econometric or macro-econometric -- and anything involving time series. And in the rare instances when silence is briefly broken, we're treated to gems like "serial correlation [until recently was] Somebody Else's Problem, specifically the unfortunate souls who make their living out of time series data (macroeconomists, for example)" (pp. 315-316).

[Here's a rough parallel. Consider Hansen and Sargent's Recursive Models of Dynamic Linear Economies. It treats structural analysis and econometric estimation of dynamic macroeconomic models, and it naturally contains large doses of time series, state space, optimal filtering, etc. It's also appropriately titled and appropriately pitched. Now imagine that Hansen and Sargent had instead titled it Mostly Harmless Econometrics, declared its contents to be the central part of (the mostly harmless part of) applied econometrics, and pitched it as a general "empiricist's companion". Voilà!]

All told, Mostly Harmless Econometrics: An Empiricist's Companion is neither "mostly harmless" nor an "empiricist's companion." Rather, it's a companion for a highly-specialized group of applied non-structural micro-econometricians hoping to estimate causal effects using non-experimental data and largely-static, linear, regression-based methods. It's a novel treatment of that sub-sub-sub-area of applied econometrics, but pretending to be anything more is most definitely harmful, particularly to students, who have no way to recognize the charade as a charade.