Predicting the Market Over Short Time Horizons

Google “average time a stock is held.” You will come up with figures that typically run around 20 seconds. High frequency trades (HFT) dominate trading volume on the US exchanges.

All of which suggests the focus on the predictability of stock returns needs to position more on intervals lasting seconds or minutes, rather than daily, monthly, or longer trading periods.

So, it’s logical that Michael Rechenthin, a newly minted Iowa Ph.D., and Nick Street, a Professor of Management, are getting media face time from research which purportedly demonstrates the existence of predictable short-term trends in the market (see Using conditional probability to identify trends in intra-day high-frequency equity pricing).

Here’s the abstract –

By examining the conditional probabilities of price movements in a popular US stock over different high-frequency intra-day timespans, varying levels of trend predictability are identified. This study demonstrates the existence of predictable short-term trends in the market; understanding the probability of price movement can be useful to high-frequency traders. Price movement was examined in trade-by-trade (tick) data along with temporal timespans between 1 s to 30 min for 52 one-week periods for one highly-traded stock. We hypothesize that much of the initial predictability of trade-by-trade (tick) data is due to traditional market dynamics, or the bouncing of the price between the stock’s bid and ask. Only after timespans of between 5 to 10 s does this cease to explain the predictability; after this timespan, two consecutive movements in the same direction occur with higher probability than that of movements in the opposite direction. This pattern holds up to a one-minute interval, after which the strength of the pattern weakens.

The study examined price movements of the exchange traded fund SPY, during 2005, finding that

.. price movements can be predicted with a better than 50-50 accuracy for anywhere up to one minute after the stock leaves the confines of its bid-ask spread. Probabilities continue to be significant until about five minutes after it leaves the spread. By 30 minutes, the predictability window has closed.

Of course, the challenges of generalization in this world of seconds and minutes is tremendous. Perhaps, for example, the patterns the authors identify are confined to the year of the study. Without any theoretical basis, brute force generalization means riffling through additional years of 31.5 million seconds each.

Then, there are the milliseconds, and the recent blockbuster written by Michael Lewis – Flash Boys: A Wall Street Revolt.

I’m on track for reading this book for a bookclub to which I belong.

As I understand it, Lewis, who is one of my favorite financial writers, has uncovered a story whereby high frequency traders, operating with optical fiber connections to the New York Stock Exchange, sometimes being geographically as proximate as possible, can exploit more conventional trading – basically buying a stock after you have put in a buy order, but before your transaction closes, thus raising your price if you made a market order.


The LA Times  has a nice review of the book and ran the above photo of Lewis.

More on the Predictability of Stock and Bond Markets

Research by Lin, Wu, and Zhou in Predictability of Corporate Bond Returns: A Comprehensive Study suggests a radical change in perspective, based on new forecasting methods. The research seems to me to of a piece with a lot of developments in Big Data and the data mining movement generally. Gains in predictability are associated with more extensive databases and new techniques.

The abstract to their white paper, presented at various conferences and colloquia, is straight-forward –

Using a comprehensive data set, we find that corporate bond returns not only remain predictable by traditional predictors (dividend yields, default, term spreads and issuer quality) but also strongly predictable by a new predictor formed by an array of 26 macroeconomic, stock and bond predictors. Results strongly suggest that macroeconomic and stock market variables contain important information for expected corporate bond returns. The predictability of returns is of both statistical and economic significance, and is robust to different ratings and maturities.

Now, in a way, the basic message of the predictability of corporate bond returns is not news, since Fama and French made this claim back in 1989 – namely that default and term spreads can predict corporate bond returns both in and out of sample.

What is new is the data employed in the Lin, Wu, and Zhou (LWZ) research. According to the authors, it involves 780,985 monthly observations spanning from January 1973 to June 2012 from combined data sources, including Lehman Brothers Fixed Income (LBFI), Datastream, National Association of Insurance Commissioners (NAIC), Trade Reporting and Compliance Engine (TRACE) and Mergents Fixed Investment Securities Database (FISD).

There also is a new predictor which LWZ characterize as a type of partial least squares (PLS) formulation, but which is none other than the three pass regression filter discussed in a post here in March.

The power of this PLS formulation is evident in a table showing out-of-sample R2 of the various modeling setups. As in the research discussed in a recent post, out-of-sample (OS) R2 is a ratio which measures the improvement in mean square prediction errors (MSPE) for the predictive regression model over the historical average forecast. A negative OS R2 thus means that the MSPE of the benchmark forecast is less than the MSPE of the forecast by the designated predictor formulation.


Again, this research finds predictability varies with economic conditions – and is higher during economic downturns.

There are cross-cutting and linked studies here, often with Goyal’s data and fourteen financial/macroeconomic variables figuring within the estimations. There also is significant linkage with researchers at regional Federal Reserve Banks.

My purpose in this and probably the next one or two posts is to just get this information out, so we can see the larger outlines of what is being done and suggested.

My guess is that the sum total of this research is going to essentially re-write financial economics and has huge implications for forecasting operations within large companies and especially financial institutions.

Stock Market Predictability – Controversy

In the previous post, I drew from papers by Neeley, who is Vice President of the Federal Reserve Bank of St. Louis, David Rapach at St. Louis University and Goufu Zhou at Washington University in St. Louis.

These authors contribute two papers on the predictability of equity returns.

The earlier one – Forecasting the Equity Risk Premium: The Role of Technical Indicators – is coming out in Management Science. Of course, the survey article – Forecasting the Equity Risk Premium: The Role of Technical Indicators – is a chapter in the recent volume 2 of the Handbook of Forecasting.

I go through this rather laborious set of citations because it turns out that there is an underlying paper which provides the data for the research of these authors, but which comes to precisely the opposite conclusion –

The goal of our own article is to comprehensively re-examine the empirical evidence as of early 2006, evaluating each variable using the same methods (mostly, but not only, in linear models), time-periods, and estimation frequencies. The evidence suggests that most models are unstable or even spurious. Most models are no longer significant even insample (IS), and the few models that still are usually fail simple regression diagnostics.Most models have performed poorly for over 30 years IS. For many models, any earlier apparent statistical significance was often based exclusively on years up to and especially on the years of the Oil Shock of 1973–1975. Most models have poor out-of-sample (OOS) performance, but not in a way that merely suggests lower power than IS tests. They predict poorly late in the sample, not early in the sample. (For many variables, we have difficulty finding robust statistical significance even when they are examined only during their most favorable contiguous OOS sub-period.) Finally, the OOS performance is not only a useful model diagnostic for the IS regressions but also interesting in itself for an investor who had sought to use these models for market-timing. Our evidence suggests that the models would not have helped such an investor. Therefore, although it is possible to search for, to occasionally stumble upon, and then to defend some seemingly statistically significant models, we interpret our results to suggest that a healthy skepticism is appropriate when it comes to predicting the equity premium, at least as of early 2006. The models do not seem robust.

This is from Ivo Welch and Amit Goyal’s 2008 article A Comprehensive Look at The Empirical Performance of Equity Premium Prediction in the Review of Financial Studies which apparently won an award from that journal as the best paper for the year.

And, very importantly, the data for this whole discussion is available, with updates, from Amit Goyal’s site now at the University of Lausanne.


Where This Is Going

Currently, for me, this seems like a genuine controversy in the forecasting literature. And, as an aside, in writing this blog I’ve entertained the notion that maybe I am on the edge of a new form of or focus in journalism – namely stories about forecasting controversies. It’s kind of wonkish, but the issues can be really, really important.

I also have a “hands-on” philosophy, when it comes to this sort of information. I much rather explore actual data and run my own estimates, than pick through theoretical arguments.

So anyway, given that Goyal generously provides updated versions of the data series he and Welch originally used in their Review of Financial Studies article, there should be some opportunity to check this whole matter. After all, the estimation issues are not very difficult, insofar as the first level of argument relates primarily to the efficacy of simple bivariate regressions.

By the way, it’s really cool data.

Here is the book-to-market ratio, dating back to 1926.


But beyond these simple regressions that form a large part of the argument, there is another claim made by Neeley, Rapach, and Zhou which I take very seriously. And this is that – while a “kitchen sink” model with all, say, fourteen so-called macroeconomic variables does not outperform the benchmark, a principal components regression does.

This sounds really plausible.

Anyway, if readers have flagged updates to this controversy about the predictability of stock market returns, let me know. In addition to grubbing around with the data, I am searching for additional analysis of this point.

Evidence of Stock Market Predictability

In business forecast applications, I often have been asked, “why don’t you forecast the stock market?” It’s almost a variant of “if you’re so smart, why aren’t you rich?” I usually respond something about stock prices being largely random walks.

But, stock market predictability is really the nut kernel of forecasting, isn’t it?

Earlier this year, I looked at the S&P 500 index and the SPY ETF numbers, and found I could beat a buy and hold strategy with a regression forecasting model. This was an autoregressive model with lots of lagged values of daily S&P returns. In some variants, it included lagged values of the Chicago Board of Trade VIX volatility index returns. My portfolio gains were compiled over an out-of-sample (OS) period. This means, of course, that I estimated the predictive regression on historical data that preceded and did not include the OS or test data.

Well, today I’m here to report to you that it looks like it is officially possible to achieve some predictability of stock market returns in out-of-sample data.

One authoritative source is Forecasting Stock Returns, an outstanding review by Rapach and Zhou  in the recent, second volume of the Handbook of Economic Forecasting.

The story is fascinating.

For one thing, most of the successful models achieve their best performance – in terms of beating market averages or other common benchmarks – during recessions.

And it appears that technical market indicators, such as the oscillators, momentum, and volume metrics so common in stock trading sites, have predictive value. So do a range of macroeconomic indicators.

But these two classes of predictors – technical market and macroeconomic indicators – are roughly complementary in their performance through the business cycle. As Christopher Neeley et al detail in Forecasting the Equity Risk Premium: The Role of Technical Indicators,

Macroeconomic variables typically fail to detect the decline in the actual equity risk premium early in recessions, but generally do detect the increase in the actual equity risk premium late in recessions. Technical indicators exhibit the opposite pattern: they pick up the decline in the actual premium early in recessions, but fail to match the unusually high premium late in recessions.

Stock Market Predictors – Macroeconomic and Technical Indicators

Rapach and Zhou highlight fourteen macroeconomic predictors popular in the finance literature.

1. Log dividend-price ratio (DP): log of a 12-month moving sum of dividends paid on the S&P 500 index minus the log of stock prices (S&P 500 index).

2. Log dividend yield (DY): log of a 12-month moving sum of dividends minus the log of lagged stock prices.

3. Log earnings-price ratio (EP): log of a 12-month moving sum of earnings on the S&P 500 index minus the log of stock prices.

4. Log dividend-payout ratio (DE): log of a 12-month moving sum of dividends minus the log of a 12-month moving sum of earnings.

5. Stock variance (SVAR): monthly sum of squared daily returns on the S&P 500 index.

6. Book-to-market ratio (BM): book-to-market value ratio for the DJIA.

7. Net equity expansion (NTIS): ratio of a 12-month moving sum of net equity issues by NYSE-listed stocks to the total end-of-year market capitalization of NYSE stocks.

8. Treasury bill rate (TBL): interest rate on a three-month Treasury bill (secondary market).

9. Long-term yield (LTY): long-term government bond yield.

10. Long-term return (LTR): return on long-term government bonds.

11. Term spread (TMS): long-term yield minus the Treasury bill rate.

12. Default yield spread (DFY): difference between BAA- and AAA-rated corporate bond yields.

13. Default return spread (DFR): long-term corporate bond return minus the long-term government bond return.

14. Inflation (INFL): calculated from the CPI (all urban consumers

In addition, there are technical indicators, which are generally moving average, momentum, or volume-based.

The moving average indicators typically provide a buy or sell signal based on a comparing two moving averages – a short and a long period MA.

Momentum based rules are based on the time trajectory of prices. A current stock price higher than its level some number of periods ago indicates “positive” momentum and expected excess returns, and generates a buy signal.

Momentum rules can be combined with information about the volume of stock purchases, such as Granville’s on-balance volume.

Each of these predictors can be mapped onto equity premium excess returns – measured by the rate of return on the S&P 500 index net of return on a risk-free asset. This mapping is a simple bi-variate regression with equity returns from time t on the left side of the equation and the economic predictor lagged by one time period on the right side of the equation. Monthly data are used from 1927 to 2008. The out-of-sample (OS) period is extensive, dating from the 1950’s, and includes most of the post-war recessions.

The following table shows what the authors call out-of-sample (OS) R2 for the 14 so-called macroeconomic variables, based on a table in the Handbook of Forecasting chapter. The OS R2 is equal to 1 minus a ratio. This ratio has the mean square forecast error (MSFE) of the predictor forecast in the numerator and the MSFE of the forecast based on historic average equity returns in the denominator. So if the economic indicator functions to improve the OS forecast of equity returns, the OS R2 is positive. If, on the other hand, the historic average trumps the economic indicator forecast, the OS R2 is negative.


(click to enlarge).

Overall, most of the macro predictors in this list don’t make it.  Thus, 12 of the 14 OS R2 statistics are negative in the second column of the Table, indicating that the predictive regression forecast has a higher MSFE than the historical average.

For two of the predictors with a positive out-of-sample R2, the p-values reported in the brackets are greater than 0.10, so that these predictors do not display statistically significant out-of-sample performance at conventional levels.

Thus, the first two columns in this table, under “Overall”, support a skeptical view of the predictability of equity returns.

However, during recessions, the situation is different.

For several the predictors, the R2 OS statistics move from being negative (and typically below -1%) during expansions to 1% or above during recessions. Furthermore, some of these R2 OS statistics are significant at conventional levels during recessions according to the  p-values, despite the decreased number of available observations.

Now imposing restrictions on the regression coefficients substantially improves this forecast performance, as the lower panel (not shown) in this table shows.

Rapach and Zhou were coauthors of the study with Neeley, published earlier as a working paper with the St. Louis Federal Reserve.

This working paper is where we get the interesting report about how technical factors add to the predictability of equity returns (again, click to enlarge).


This table has the same headings for the columns as Table 3 above.

It shows out-of-sample forecasting results for several technical indicators, using basically the same dataset, for the overall OS period, for expansions, and recessions in this period dating from the 1950’s to 2008.

In fact, these technical indicators generally seem to do better than the 14 macroeconomic indicators.

Low OS R2

Even when these models perform their best, their increase in mean square forecast error (MSFE) is only slightly more than the MSFE of the benchmark historic average return estimate.

This improved performance, however, can still achieve portfolio gains for investors, based on various trading rules, and, as both papers point out, investors can use the information in these forecasts to balance their portfolios, even when the underlying forecast equations are not statistically significant by conventional standards. Interesting argument, and I need to review it further to fully understand it.

In any case, my experience with an autoregressive model for the S&P 500 is that trading rules can be devised which produce portfolio gains over a buy and hold strategy, even when the Ris on the order of 1 or a few percent. All you have to do is correctly predict the sign of the return on the following trading day, for instance, and doing this a little more than 50 percent of the time produces profits.

Rapach and Zhou, in fact, develop insights into how predictability of stock returns can be consistent with rational expectations – providing the relevant improvements in predictability are bounded to be low enough.

Some Thoughts

There is lots more to say about this, naturally. And I hope to have further comments here soon.

But, for the time being, I have one question.

The is why econometricians of the caliber of Rapach, Zhou, and Neeley persist in relying on tests of statistical significance which are predicated, in a strict sense, on the normality of the residuals of these financial return regressions.

I’ve looked at this some, and it seems the t-statistic is somewhat robust to violations of normality of the underlying error distribution of the regression. However, residuals of a regression on equity rates of return can be very non-normal with fat tails and generally some skewness. I keep wondering whether anyone has really looked at how this translates into tests of statistical significance, or whether what we see on this topic is mostly arm-waving.

For my money, OS predictive performance is the key criterion.


I’ve been reading about the bootstrap. I’m interested in bagging or bootstrap aggregation.

The primary task of a statistician is to summarize a sample based study and generalize the finding to the parent population in a scientific manner..

The purpose of a sample study is to gather information cheaply in a timely fashion. The idea behind bootstrap is to use the data of a sample study at hand as a “surrogate population”, for the purpose of approximating the sampling distribution of a statistic; i.e. to resample (with replacement) from the sample data at hand and create a large number of “phantom samples” known as bootstrap samples. The sample summary is then computed on each of the bootstrap samples (usually a few thousand). A histogram of the set of these computed values is referred to as the bootstrap distribution of the statistic.

These well-phrased quotes come from Bootstrap: A Statistical Method by Singh and Xie.

OK, so let’s do a simple example.

Suppose we generate ten random numbers, drawn independently from a Gaussian or normal distribution with a mean of 10 and standard deviation of 1.


This sample has an average of 9.7684. We would like to somehow project a 95 percent confidence interval around this sample mean, to understand how close it is to the population average.

So we bootstrap this sample, drawing 10,000 samples of ten numbers with replacement.

Here is the distribution of bootstrapped means of these samples.


The mean is 9.7713.

Based on the method of percentiles, the 95 percent confidence interval for the sample mean is between 9.32 and 10.23, which, as you note, correctly includes the true mean for the population of 10.

Bias-correction is another primary use of the bootstrap. For techies, there is a great paper from the old Bell Labs called A Real Example That Illustrates Properties of Bootstrap Bias Correction. Unfortunately, you have to pay a fee to the American Statistical Association to read it – I have not found a free copy on the Web.

In any case, all this is interesting and a little amazing, but what we really want to do is look at the bootstrap in developing forecasting models.

Bootstrapping Regressions

There are several methods for using bootstrapping in connection with regressions.

One is illustrated in a blog post from earlier this year. I treated the explanatory variables as variables which have a degree of randomness in them, and resampled the values of the dependent variable and explanatory variables 200 times, finding that doing so “brought up” the coefficient estimates, moving them closer to the underlying actuals used in constructing or simulating them.

This method works nicely with hetereoskedastic errors, as long as there is no autocorrelation.

Another method takes the explanatory variables as fixed, and resamples only the residuals of the regression.

Bootstrapping Time Series Models

The underlying assumptions for the standard bootstrap include independent and random draws.

This can be violated in time series when there are time dependencies.

Of course, it is necessary to transform a nonstationary time series to a stationary series to even consider bootstrapping.

But even with a time series that fluctuates around a constant mean, there can be autocorrelation.

So here is where the block bootstrap can come into play. Let me cite this study – conducted under the auspices of the Cowles Foundation (click on link) – which discusses the asymptotic properties of the block bootstrap and provides key references.

There are many variants, but the basic idea is to sample blocks of a time series, probably overlapping blocks. So if a time series yt  has n elements, y1,..,yn and the block length is m, there are n-m blocks, and it is necessary to use n/m of these blocks to construct another time series of length n. Issues arise when m is not a perfect divisor of n, and it is necessary to develop special rules for handling the final values of the simulated series in that case.

Block bootstrapping is used by Bergmeir, Hyndman, and Benıtez in bagging exponential smoothing forecasts.

How Good Are Bootstrapped Estimates?

Consistency in statistics or econometrics involves whether or not an estimate or measure converges to an unbiased value as sample size increases – or basically goes to infinity.

This is a huge question with bootstrapped statistics, and there are new findings all the time.

Interestingly, sometimes bootstrapped estimates can actually converge faster to the appropriate unbiased values than can be achieved simply by increasing sample size.

And some metrics really do not lend themselves to bootstrapping.

Also some samples are inappropriate for bootstrapping.  Gelman, for example, writes about the problem of “separation” in a sample

[In} example of a poll from the 1964 U.S. presidential election campaign, … none of the black respondents in the sample supported the Republican candidate, Barry Goldwater… If zero black respondents in the sample supported Barry Goldwater, then zero black respondents in any bootstrap sample will support Goldwater as well. Indeed, bootstrapping can exacerbate separation by turning near-separation into complete separation for some samples. For example, consider a survey in which only one or two of the black respondents support the Republican candidate. The resulting logistic regression estimate will be noisy but it will be finite.

Here is a video doing a good job of covering the bases on boostrapping. I suggest sampling portions of it first. It’s quite good, but it may seem too much going into it.