Tag Archives: Time series analysis

Oil and Gas Prices II

October 26, 2014 Clive Jones

One of the more interesting questions in applied forecasting is the relationship between oil and natural gas prices in the US market, shown below.

Up to the early 1990’s, the interplay between oil and gas prices followed “rules of thumb” – for example, gas prices per million Btu were approximately one tenth oil prices.

There is still some suggestion of this – for example, peak oil prices recently hit nearly $140 a barrel, at the same time gas prices were nearly $14 per million Btu’s.

However, generally, ratio relationships appear to break down around 2009, if not earlier, during the first decade of the century.

A Longer Term Relationship?

Perhaps oil and gas prices are in a longer term relationship, but one disturbed in many cases in short run time periods.

One way economists and ecommetricians think of this is in terms of “co-integrating relationships.” That’s a fancy way of saying that regressions of the form,

Gas price in time t = constant + α(oil price in time t) + (residual in time t)

are predictive. Here, α is a coefficient to be estimated.

Now this looks like a straight-forward regression, so you might say – “what’s the problem?”

Well, the catch is that gas prices and oil prices might be nonstationary – that is, one or another form of a random walk.

If this is so – and positive results on standard tests such as the augmented Dickey Fuller (ADR) and Phillips-Peron are widely reported – there is a big potential problem. It’s easy to regress one completely unrelated nonstationary time series onto another, getting an apparently significant result, only to find this relationship disappears in the forecast. In other words two random series can, by chance, match up to each other over closely, but that’s no guarantee they will continue to do so.

Here’s where the concept of a co-integrating relationship comes into play.

If you can show, by various statistical tests, that variables are cointegrated, regressions such as the one above are more likely to be predictive.

Well, several econometric studies show gas and oil prices are in a cointegrated relationship, using data from the 1990’s through sometime in the first decade of the 2000’s. The more sophisticated specify auxiliary variables to account for weather or changes in gas storage. You might download and read, for example, a study published in 2007 under the auspices of the Dallas Federal Reserve Bank – What Drives Natural Gas Prices?

But it does not appear that this cointegrated relationship is fixed. Instead, it changes over time, perhaps exemplifying various regimes, i.e. periods of time in which the underlying parameters switch to new values, even though a determinate relationship can still be demonstrated.

Changing parameters are shown in the excellent 2012 study by Ramberg and Parsons in the Energy Journal – The Weak Tie Between Natural Gas and Oil Prices.

The Underlying Basis

Anyway, there are facts relating to production and use of oil and natural gas which encourage us to postulate a relationship in their prices, although the relationship may shift over time.

This makes sense since oil and gas are limited or completely substitutes in various industrial processes. This used to be more compelling in electric power generation, than it is today. According to the US Department of Energy, there are only limited amounts of electric power still produced by generators running on oil, although natural gas turbines have grown in importance.

Still, natural gas is often produced alongside of and is usually dissolved in oil, so oil and natural gas are usually joint products.

Recently, technology has changed the picture with respect to gas and oil.

On the demand side, the introduction of the combined-cycle combustion turbine made natural gas electricity generation more cost effective, thereby making natural gas in electric power generation even more dominant.

On the demand side, the new technologies of extracting shale oil and natural gas – often summarized under the rubric of “fracking” or hydraulic fracturing – have totally changed the equation, resulting in dramatic increases in natural gas supplies in the US.

This leaves the interesting question of what sort of forecasting model for natural gas might be appropriate.

business cycle, cycles, time series forecasting

Random Cycles

July 30, 2014 Clive Jones

In 1927, the Russian statistician Eugen Slutsky wrote a classic article called ‘The summation of random causes as the source of cyclic processes,’ a short summary of which is provided by Barnett –

If the variables that were taken to represent business cycles were moving averages of past determining quantities that were not serially correlated – either real-world moving averages or artificially generated moving averages – then the variables of interest would become serially correlated, and this process would produce a periodicity approaching that of sine waves

It’s possible to illustrate this phenomena with rolling sums of the digits of pi (π). The following chart illustrates the wave-like result of charting rolling sums of ten consecutive digits of pi.

So to be explicit, I downloaded the first 450 digits of pi, took them apart, and then graphed the first 440 rolling sums.

The wave-like pattern Illustrates a random cycle.

Forecasting Random Cycles

If we consider this as a time series, each element x_k is the following sum,

x_k = d_k+d_k-1+..+d_k-10

where dj is the jth digit in the decimal expansion of pi to the right of the initial value of 3.

Now, apparently, it is not proven that the digits of pi are truly random, although one can show that, so far as we can compute, these digits are described by a uniform distribution.

As far as we know, the probability that the next digit will be any digit from 0 to 9 is 1/10=0.1

So as one moves through the digits of pi, generating rolling sums, each new sum means the addition of a new digit, which is unknown and can only be predicted up to its probability. And, at the same time, a digit at the beginning of the preceding sum drops away in the new sum.

Note also that we can always deduce what the series of original digits is, given a series of these rolling sums up to some point.

So the issue is whether the new digit added to the next sum is greater than, equal to, or less than the leading digit of the current sum – which is where we now stand in this sort of analysis. This determines whether the next rolling sum will be greater than, equal to, or less than the current sum.

Here’s where the forecasts can be produced. If the rolling sum is large enough, approaching or equal to 90, there is a high probability that the next rolling sum will be lower, leading to this wave-like pattern. Conversely, if the rolling sum is near zero, the chances are the subsequent sum will be larger. And all this arm-waving can be complemented by exact probabilistic calculations.

Some Ultimate Thoughts

It’s interesting we are really dealing here with a random cycle. That’s proven by the fact that, at any time, the series could go flat-line or trace out some other kind of weird movement.

Thus, the quasi-periodic aspect can be violated for as many periods as you might choose, if one arrives at a run of the same digit in the expansion of pi.

This reminds me of something George Gamow wrote in one of his popular books, where he discusses thermodynamics and the random movement of atoms and molecules in the air of a room. Gamow observes it is entirely possible all the air by chance will congregate in one corner, leaving a vacuum elsewhere. Of course, this is highly improbable.

The only difference would be that there are a finite number of atoms and molecules in the air of any room, but, presumably, an infinite number of digits in the expansion of pi.

The morale of the story is, in any case, to be cautious in imposing a fixed cycle on this type of series.

autoregressive model, business cycle, cointegration in time series, cycles, Granger causality, technology forecasting

Semiconductor Cycles

July 27, 2014 Clive Jones

I’ve been exploring cycles in the semiconductor, computer and IT industries generally for quite some time.

Here is an exhibit I prepared in 2000 for a magazine serving the printed circuit board industry.

The data come from two sources – the Semiconductor Industry Association (SIA) World Semiconductor Trade Statistics database and the Census Bureau manufacturing series for computer equipment.

This sort of analytics spawned a spate of academic research, beginning more or less with the work of Tan and Mathews in Australia.

One of my favorites is a working paper released by DRUID – the Danish Research Unit for Industrial Dynamics called Cyclical Dynamics in Three Industries. Tan and Mathews consider cycles in semiconductors, computers, and what they call the flat panel display industry. They start with quoting “industry experts” and, specifically, some of my work with Economic Data Resources on the computer (PC) cycle. These researchers went on to publish in the Journal of Business Research and Technological Forecasting and Social Change in 2010. A year later in 2011, Tan published an interesting article on the sequencing of cyclical dynamics in semiconductors.

Essentially, the appearance of cycles and what I have called quasi-cycles or pseudo-cycles in the semiconductor industry and other IT categories, like computers, result from the interplay of innovation, investment, and pricing. In semiconductors, for example, Moore’s law – which everyone always predicts will fail at some imminent future point – indicates that continuing miniaturization will lead to periodic reductions in the cost of information processing. At some point in the 1980’s, this cadence was firmly established by introductions of new microprocessors by Intel roughly every 18 months. The enhanced speed and capacity of these microprocessors – the “central nervous system” of the computer – was complemented by continuing software upgrades, and, of course, by the movement to graphical interfaces with Windows and the succession of Windows releases.

Back along the supply chain, semiconductor fabs were retooling periodically to produce chips with more and more transitors per volume of silicon. These fabs were, simply put, fabulously expensive and the investment dynamics factors into pricing in semiconductors. There were famous gluts, for example, of memory chips in 1996, and overall the whole IT industry led the recession of 2001 with massive inventory overhang, resulting from double booking and the infamous Y2K scare.

Statistical Modeling of IT Cycles

A number of papers, summarized in Aubrey deploy VAR (vector autoregression) models to capture leading indicators of global semiconductor sales. A variant of these is the Bayesian VAR or BVAR model. Basically, VAR models sort of blindly specify all possible lags for all possible variables in a system of autoregressive models. Of course, some cutoff point has to be established, and the variables to be included in the VAR system have to be selected by one means or another. A BVAR simply reduces the number of possibilities by imposing, for example, sign constraints on the resulting coefficients, or, more ambitiously, employs some type of prior distribution for key variables.

Typical variables included in these models include:

WSTS monthly semiconductor shipments (now by subscription only from SIA)
Philadelphia semiconductor index (SOX) data
US data on various IT shipments, orders, inventories from M3
data from SEMI, the association of semiconductor equipment manufacturers

Another tactic is to filter out low and high frequency variability in a semiconductor sales series with something like the Hodrick-Prescott (HP) filter, and then conduct a spectral analysis.

Does the Semiconductor/Computer/IT Cycle Still Exist?

I wonder whether academic research into IT cycles is a case of “redoubling one’s efforts when you lose sight of the goal,” or more specifically, whether new configurations of forces are blurring the formerly fairly cleanly delineated pulses in sales growth for semiconductors, computers, and other IT hardware.

“Hardware” is probably a key here, since there have been big changes since the 1990’s and early years of this brave new century.

For one thing, complementarities between software and hardware upgrades seem to be breaking down. This began in earnest with the development of virtual servers – software which enabled many virtual machines on the same hardware frame, in part because the underlying circuitry was so massively powerful and high capacity now. Significant declines in the growth of sales of these machines followed on wide deployment of this software designed to achieve higher efficiencies of utilization of individual machines.

Another development is cloud computing. Running the data side of things is gradually being taken away from in-house IT departments in companies and moved over to cloud computing services. Of course, critical data for a company is always likely to be maintained in-house, but the need for expanding the number of big desktops with the number of employees is going away – or has indeed gone away.

At the same time, tablets, Apple products and Android machines, created a wave of destructive creation in people’s access to the Internet, and, more and more, for everyday functions like keeping calendars, taking notes, even writing and processing photos.

But note – I am not studding this discussion with numbers as of yet.

I suspect that underneath all this change it should be possible to identify some IT invariants, perhaps in usage categories, which continue to reflect a kind of pulse and cycle of activity.

autoregressive model, cycles, data science

Some Cycle Basics

July 25, 2014 Clive Jones

A Fourier analysis is one of the first steps in analyzing cycles.

Take sunspots, for example,

There are extensive historic records on the annual number of sunspots, dating back to 1700. The annual data shown in the following graph dates back to 1700, and is currently maintained by the Royal Belgium Observatory.

This series is relatively stationary, although there may be a slight trend if you cut this span of data off a few years before the present.

In any case, the kind of thing you get with a Fourier analysis looks like this.

This shows the power or importance of the cycles/year numbers, and maxes out at around 0.09.

These data can be recalibrated into the following chart, which highlights the approximately 11 year major cycle in the sunspot numbers.

Now it’s possible to build a simple regression model with a lagged explanatory variable to make credible predictions. A lag of eleven years produces the following in-sample and out-of-sample fits. The regression is estimated over data to 1990, and, thus, the years 1991 through 2013 are out-of-sample.

It’s obvious this sort of forecasting approach is not quite ready for prime-time television, even though it performs OK on several of the out-of-sample years after 1990.

But this exercise does highlight a couple of things.

First, the annual number of sunspots is broadly cyclical in this sense. If you try the same trick with lagged values for the US “business cycle” the results will be radically worse. At least with the sunspot data, most of the fluctuations have timing that is correctly predicted, both in-sample (1990 and before) and out-of-sample (1991-2013).

Secondly, there are stochastic elements to this solar activity cycle. The variation in amplitude is dramatic, and, indeed, the latest numbers coming in on sunspot activity are moving to much lower levels, even though the cycle is supposedly at its peak.

I’ve reviewed several papers on predicting the sunspot cycle. There are models which are more profoundly inspired by the possible physics involved – dynamo dynamics for example. But for my money there are basic models which, on a one-year-ahead basis, do a credible job. More on this forthcoming.

accuracy of forecasts, identifying seasonal effects, macroeconomic forecasting, sales forecasting, time series forecasting

Seasonal Variation

July 13, 2014 Clive Jones

Evaluating and predicting seasonal variation is a core competence of forecasting, dating back to the 1920’s or earlier. It’s essential to effective business decisions. For example, as the fiscal year unfolds, the question is “how are we doing?” Will budget forecasts come in on target, or will more (or fewer) resources be required? Should added resources be allocated to Division X and taken away from Division Y? To answer such questions, you need a within-year forecast model, which in most organizations involves quarterly or monthly seasonal components or factors.

Seasonal adjustment, on the other hand, is more mysterious. The purpose is more interpretive. Thus, when the Bureau of Labor Statistics (BLS) or Bureau of Economic Analysis (BEA) announce employment or other macroeconomic numbers, they usually try to take out special effects (the “Christmas effect”) that purportedly might mislead readers of the Press Release. Thus, the series we hear about typically are “seasonally adjusted.”

You can probably sense my bias. I almost always prefer data that is not seasonally adjusted in developing forecasting models. I just don’t know what magic some agency statistician has performed on a series – whether artifacts have been introduced, and so forth.

On the other hand, I take the methods of identifying seasonal variation quite seriously. These range from Buys-Ballot tables and seasonal dummy variables to methods based on moving averages, trigonometric series (Fourier analysis), and maximum likelihood estimation.

Identifying seasonal variation can be fairly involved mathematically.

But there are some simple reality tests.

Take this US retail and food service sales series, for example.

Here you see the highly regular seasonal movement around a trend which, at times, is almost straight-line.

Are these additive or multiplicative seasonal effects? If we separate out the trend and the seasonal effects, do we add them or are the seasonal effects “factors” which multiply into the level for a month?

Well, for starters, we can re-arrange this time series into a kind of Buys-Ballot table. Here I only show the last two years.

The point is that we look at the differences between the monthly values in a year and the average for that year. Also, we calculate the ratios of each month to the annual total.

The issue is which of these numbers is most stable over the data period, which extends back to 1992 (click to enlarge).

Now here Series N relates to the Nth month, e.g. Series 12 = December.

It seems pretty clear that the multiplicative factors are more stable than the additive components in two senses. First, some additive components have a more pronounced trend; secondly, the variability of the additive components around this trend is greater.

This gives you a taste of some quick methods to evaluate aspects of seasonality.

Of course, there can be added complexities. What if you have daily data, or suppose there are other recurrent relationships. Then, trig series may be your best bet.

What if you only have two, three, or four years of data? Well, this interesting problem is frequently encountered in practical applications.

I’m trying to sort this material into posts for this coming week, along with stuff on controversies that swirl around the seasonal adjustment of macro time series, such as employment and real GDP.

Stay tuned.

Top image from http://www.livescience.com/25202-seasons.html

ARIMA models, artificial intelligence (AI), gold price forecasting, predictive analytics

More Blackbox Analysis – ARIMA Modeling in R

July 8, 2014 Clive Jones

Automatic forecasting programs are seductive. They streamline analysis, especially with ARIMA (autoregressive integrated moving average) models. You have to know some basics – such as what the notation ARIMA(2,1,1) or ARIMA(p,d,q) means. But you can more or less sidestep the elaborate algebra – the higher reaches of equations written in backward shift operators – in favor of looking at results. Does the automatic ARIMA model selection predict out-of-sample, for example?

I have been exploring the Hyndman R Forecast package – and other contributors, such as George Athanasopoulos, Slava Razbash, Drew Schmidt, Zhenyu Zhou, Yousaf Khan, Christoph Bergmeir, and Earo Wang, should be mentioned.

A 76 page document lists the routines in Forecast, which you can download as a PDF file.

This post is about the routine auto.arima(.) in the Forecast package. This makes volatility modeling – a place where Box Jenkins or ARIMA modeling is relatively unchallenged – easier. The auto.arima(.) routine also encourages experimentation, and highlights the sharp limitations of volatility modeling in a way that, to my way of thinking, is not at all apparent from the extensive and highly mathematical literature on this topic.

Daily Gold Prices

I grabbed some data from FRED – the Gold Fixing Price set at 10:30 A.M (London time) in London Bullion Market, based in U.S. Dollars.

Now the price series shown in the graph above is a random walk, according to auto.arima(.).

In other words, the routine indicates that the optimal model is ARIMA(0,1,0), which is to say that after differencing the price series once, the program suggests the series reduces to a series of independent random values. The automatic exponential smoothing routine in Forecast is ets(.). Running this confirms that simple exponential smoothing, with a smoothing parameter close to 1, is the optimal model – again, consistent with a random walk.

Here’s a graph of these first differences.

But wait, there is a clustering of volatility of these first differences, which can be accentuated if we square these values, producing the following graph.

Now in a more or less textbook example, auto.arima(.) develops the following ARIMA model for this series

Thus, this estimate of the volatility of the first differences of gold price is modeled as a first order autoregressive process with two moving average terms.

Here is the plot of the fitted values.

Nice.

But of course, we are interested in forecasting, and the results here are somewhat more disappointing.

Basically, this type of model makes a horizontal line prediction at a certain level, which is higher when the past values have been higher.

This is what people in quantitative finance call “persistence” but of course sometimes new things happen, and then these types of models do not do well.

From my research on the volatility literature, it seems that short period forecasts are better than longer period forecasts. Ideally, you update your volatility model daily or at even higher frequencies, and it’s likely your one or two period ahead (minutes, hours, a day) will be more accurate.

Incidentally, exponential smoothing in this context appears to be a total fail, again suggesting this series is a simple random walk.

Recapitulation

There is more here than meets the eye.

First, the auto.arima(.) routines in the Hyndman R Forecast package do a competent job of modeling the clustering of higher first differences of the gold price series here. But, at the same time, they highlight a methodological point. The gold price series really has nonlinear aspects that are not adequately commanded by a purely linear model. So, as in many approximations, the assumption of linearity gets us some part of the way, but deeper analysis indicates the existence of nonlinearities. Kind of interesting.

Of course, I have not told you about the notation ARIMA(p,d,q). Well, p stands for the order of the autoregressive terms in the equation, q stands for the moving average terms, and d indicates the times the series is differenced to reduce it to a stationary time series. Take a look at Forecasting: principles and practice – the free forecasting text of Hyndman and Athanasopoulos – in the chapter on ARIMA modeling for more details.

Incidentally, I think it is great that Hyndman and some of his collaborators are providing an open source, indeed free, forecasting package with automatic forecasting capabilities, along with a high quality and, again, free textbook on forecasting to back it up. Eventually, some of these techniques might get dispersed into the general social environment, potentially raising the level of some discussions and thinking about our common future.

And I guess also I have to say that, ultimately, you need to learn the underlying theory and struggle with the algebra some. It can improve one’s ability to model these series.

accuracy of forecasts, analytical software, ARIMA models, data science, gold price forecasting, time series forecasting

More on Automatic Forecasting Packages – Autobox Gold Price Forecasts

April 29, 2014 Clive Jones

Yesterday, my post discussed the statistical programming language R and Rob Hyndman’s automatic forecasting package, written in R – facts about this program, how to download it, and an application to gold prices.

In passing, I said I liked Hyndman’s disclosure of his methods in his R package and “contrasted” that with leading competitors in the automatic forecasting market space –notably Forecast Pro and Autobox.

This roused Tom Reilly, currently Senior Vice-President and CEO of Automatic Forecast Systems – the company behind Autobox.

Reilly, shown above, wrote –

You say that Autobox doesn’t disclose its methods. I think that this statement is unfair to Autobox. SAS tried this (Mike Gilliland) on the cover of his book showing something purporting to a black box. We are a white box. I just downloaded the GOLD prices and recreated the problem and ran it. If you open details.htm it walks you through all the steps of the modeling process. Take a look and let me know your thoughts. Much appreciated!

AutoBox Gold Price Forecast

First, disregarding the issue of transparency for a moment, let’s look at a comparison of forecasts for this monthly gold price series (London PM fix).

A picture tells the story (click to enlarge).

So, for this data, 2007 to early 2011, Autobox dominates. That is, all forecasts are less than the respective actual monthly average gold prices. Thus, being linear, if one forecast method is more inaccurate than another for one month, that method is less accurate than the forecasts generated by this other approach for the entire forecast horizon.

I guess this does not surprise me. Autobox has been a serious contender in the M-competitions, for example, usually running just behind or perhaps just ahead of Forecast Pro, depending on the accuracy metric and forecast horizon. (For a history of these “accuracy contests” see Markridakis and Hibon’s article on M3).

And, of course, this is just one of many possible forecasts that can be developed with this time series, taking off from various ending points in the historic record.

The Issue of Transparency

In connection with all this, I also talked with Dave Reilly, a founding principal of Autobox, shown below.

Among other things, we went over the “printout” Tom Reilly sent, which details the steps in the estimation of a final time series model to predict these gold prices.

A blog post on the Autobox site is especially pertinent, called Build or Make your own ARIMA forecasting model? This discussion contains two flow charts which describe the process of building a time series model, I reproduce here, by kind permission.

The first provides a plain vanilla description of Box-Jenkins modeling.

The second flowchart adds steps revised for additions by Tsay, Tiao, Bell, Reilly & Gregory Chow (ie chow test).

Both start with plotting the time series to be analyzed and calculating the autocorrelation and partial autocorrelation functions.

But then additional boxes are added for accounting for and removing “deterministic” elements in the time series and checking for the constancy of parameters over the sample.

The analysis run Tom Reilly sent suggests to me that “deterministic” elements can mean outliers.

Dave Reilly made an interesting point about outliers. He suggested that the true autocorrelation structure can be masked or dampened in the presence of outliers. So the tactic of specifying an intervention variable in the various trial models can facilitate identification of autoregressive lags which otherwise might appear to be statistically not significant.

Really, the point of Autobox model development is to “create an error process free of structure.” That a Dave Reilly quote.

So, bottom line, Autobox’s general methods are well-documented. There is no problem of transparency with respect to the steps in the recommended analysis in the program. True, behind the scenes, comparisons are being made and alternatives are being rejected which do not make it to the printout of results. But you can argue that any commercial software has to keep some kernel of its processes proprietary.

I expect to be writing more about Autobox. It has a good track record in various forecasting competitions and currently has a management team that actively solicits forecasting challenges.

ARIMA models, data science, Granger causality, predictive analytics, time series forecasting

Granger Causality

February 11, 2014 Clive Jones

After review, I have come to the conclusion that from a predictive and operational standpoint, causal explanations translate to directed graphs, such as the following:

And I think it is interesting the machine learning community focuses on causal explanations for “manipulation” to guide reactive and interactive machines, and that directed graphs (or perhaps a Bayesian networks) are a paramount concept.

Keep that thought, and consider “Granger causality.”

This time series concept is well explicated in C.W.J. Grangers’ 2003 Nobel Prize lecture – which motivates its discovery and links with cointegration.

An earlier concept that I was concerned with was that of causality. As a postdoctoral student in Princeton in 1959–1960, working with Professors John Tukey and Oskar Morgenstern, I was involved with studying something called the “cross-spectrum,” which I will not attempt to explain. Essentially one has a pair of inter-related time series and one would like to know if there are a pair of simple relations, first from the variable X explaining Y and then from the variable Y explaining X. I was having difficulty seeing how to approach this question when I met Dennis Gabor who later won the Nobel Prize in Physics in 1971. He told me to read a paper by the eminent mathematician Norbert Wiener which contained a definition that I might want to consider. It was essentially this definition, somewhat refined and rounded out, that I discussed, together with proposed tests in the mid 1960’s.

The statement about causality has just two components: 1. The cause occurs before the effect; and 2. The cause contains information about the effect that that is unique, and is in no other variable.

A consequence of these statements is that the causal variable can help forecast the effect variable after other data has first been used. Unfortunately, many users concentrated on this forecasting implication rather than on the original definition. At that time, I had little idea that so many people had very fixed ideas about causation, but they did agree that my definition was not “true causation” in their eyes, it was only “Granger causation.” I would ask for a definition of true causation, but no one would reply. However, my definition was pragmatic and any applied researcher with two or more time series could apply it, so I got plenty of citations. Of course, many ridiculous papers appeared.

When the idea of cointegration was developed, over a decade later, it became clear immediately that if a pair of series was cointegrated then at least one of them must cause the other. There seems to be no special reason why there two quite different concepts should be related; it is just the way that the mathematics turned out

In the two-variable case, suppose we have time series Y={y1,y2,…,yt} and X = {x1,..,xt}. Then, there are, at the outset, two cases, depending on whether Y and X are stationary or nonstationary. The classic case is where we have an autoregressive relationship for yt,

y_t = a₀+a₁y_t-1+..+a_ky_t-k

and this relationship can be shown to be a weaker predictor than

yt = a₀+a₁y_t-1+..+a_ky_t-k + b₀+b₁x_t-1+..+b_mx_t-m

In this case, we say that X exhibits Granger causality with respect to Y.

Of course, if Y and X are nonstationary time series, autoregressive predictive equations make no sense, and instead we have the case of cointegration of time series, where in the two-variable case,

y_t=φx_t-1+u_t

and the series of residuals u_tare reduced to a white noise process.

So these cases follow what good old Wikipedia says,

A time series X is said to Granger-cause Y if it can be shown, usually through a series of t-tests and F-tests on lagged values of X (and with lagged values of Y also included), that those X values provide statistically significant information about future values of Y.

There are a number of really interesting extensions of this linear case, discussed in a recent survey paper.

Stern points out that the main enemies or barriers to establishing causal relations are endogeneity and omitted variables.

So I find that margin loans and the level of the S&P 500 appear to be mutually interrelated. Thus, it is forecasts of the S&P 500 can be improved with lagged values of margin loans, and you can improve forecasts of the monthly total of margin loans with lagged values of the S&P 500 – at least over broad ranges of time and in the period since 2008. The predictions of the S&P 500 with lagged values of margin loans, however, are marginally more powerful or accurate predictions.

Stern gives a colorful example where an explanatory variable is clearly exogenous and appears to have a significant effect on the dependent variable and yet theory suggests that the relationship is spurious and due to omitted variables that happen to be correlated with the explanatory variable in question.

Westling (2011) regresses national economic growth rates on average reported penis lengths and other variables and finds that there is an inverted U shape relationship between economic growth and penis length from 1960 to 1985. The growth maximizing length was 13.5cm, whereas the global average was 14.5cm. Penis length would seem to be exogenous but the nature of this relationship would have changed over time as the fastest growing region has changed from Europe and its Western Offshoots to Asia. So, it seems that the result is likely due to omitted variables bias.

Here Stern notes that Westling’s data indicates penis length is lowest in Asia and greatest in Africa with Europe and its Western Offshoots having intermediate lengths.

There’s a paper which shows stock prices exhibit Granger causality with respect to economic growth in the US, but vice versa does not obtain. This is a good illustration of the careful ste-by-step in conducting this type of analysis, and how it is in fact fraught with issues of getting the number of lags exactly right and avoiding big specification problems.

Just at the moment when it looks as if the applications of Granger causality are petering out in economics, neuroscience rides to the rescue. I offer you a recent article from a journal in computation biology in this regard – Measuring Granger Causality between Cortical Regions from Voxelwise fMRI BOLD Signals with LASSO.

Here’s the Abstract:

Functional brain network studies using the Blood Oxygen-Level Dependent (BOLD) signal from functional Magnetic Resonance Imaging (fMRI) are becoming increasingly prevalent in research on the neural basis of human cognition. An important problem in functional brain network analysis is to understand directed functional interactions between brain regions during cognitive performance. This problem has important implications for understanding top-down influences from frontal and parietal control regions to visual occipital cortex in visuospatial attention, the goal motivating the present study. A common approach to measuring directed functional interactions between two brain regions is to first create nodal signals by averaging the BOLD signals of all the voxels in each region, and to then measure directed functional interactions between the nodal signals. Another approach, that avoids averaging, is to measure directed functional interactions between all pairwise combinations of voxels in the two regions. Here we employ an alternative approach that avoids the drawbacks of both averaging and pairwise voxel measures. In this approach, we first use the Least Absolute Shrinkage Selection Operator (LASSO) to pre-select voxels for analysis, then compute a Multivariate Vector AutoRegressive (MVAR) model from the time series of the selected voxels, and finally compute summary Granger Causality (GC) statistics from the model to represent directed interregional interactions. We demonstrate the effectiveness of this approach on both simulated and empirical fMRI data. We also show that averaging regional BOLD activity to create a nodal signal may lead to biased GC estimation of directed interregional interactions. The approach presented here makes it feasible to compute GC between brain regions without the need for averaging. Our results suggest that in the analysis of functional brain networks, careful consideration must be given to the way that network nodes and edges are defined because those definitions may have important implications for the validity of the analysis.

So Granger causality is still a vital concept, despite its probably diminishing use in econometrics per se.

Let me close with this thought and promise a future post on the Kaggle and machine learning competitions on identifying the direction of causality in pairs of variables without context.

Correlation does not imply causality—you’ve heard it a thousand times. But causality does imply correlation.

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