# Resources for learning about spurious time series regression

"Spurious regression" (in the context of time series) and associated terms like unit root tests are something I've heard a lot about, but never understood.

Why/when, intuitively, does it occur? (I believe it's when your two time series are cointegrated, i.e., some linear combination of the two is stationary, but I don't see why cointegration should lead to spuriousness.) What do you do to avoid it?

I'm looking for a high-level understanding of what cointegration/unit root tests/Granger causality have to do with Spurious regression (those three are terms I remember being associated with spurious regression somehow, but I don't remember what exactly), so either a custom response or a link to references where I can learn more would be great.

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These concepts have been created to deal with regressions (for instance correlation) between non stationary series.

Clive Granger is the key author you should read.

Cointegration has been introduced in 2 steps:

1/ Granger, C., and P. Newbold (1974): "Spurious Regression in Econometrics,"

In this article, the authors point out that regression among non stationary variables should be conducted as regressions among changes (or log changes) of the variables. Otherwise you might find high correlation without any real significance. (= spurious regression)

2/ Engle, Robert F., Granger, Clive W. J. (1987) "Co-integration and error correction: Representation, estimation and testing", Econometrica, 55(2), 251-276.

In this article (for which Granger has been rewarded by the Nobel jury in 2003), the authors go further, and introduce cointegration as a way to study the error correction model that can exist between two non stationary variables.
Basically the 1974 advice to regress the change in the time series may lead to unspecified regression models. You can indeed have variables whose changes are uncorrelated, but which are connected through an "error correction model".

Hence, you can have correlation without cointegration, and cointegration without correlation. The two are complementary.

If there was only one paper to read, I suggest you start with this one, which is a very good and nice introduction:

(Murray 1993) Drunk and her dog

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Let's start with the spurious regression. Take or imagine two series which are both driven by a dominant time trend: for example US population and US consumption of whatever (it doesn't matter what item you think about, be it soda or licorice or gas). Both series will be growing because of the common time trend. Now regress aggregate consumption on aggregate population size and presto, you have a great fit. (We could simulate that quickly in R too.)

But it means nothing. There is no relationship (as we as the modelers know) -- yet the linear model sees a fit (in the minimizing sum of squares sense) as both series happen to both be uptrending without a causal link. We fell victim to a spurious regression.

What could or should be modeled is change in one series on change in the other, or maybe per capita consumption, or ... All those changes make the variables stationary which helps to alleviate the issue.

Now, from 30,000 feet, unit roots and cointegration help you with formal inference in these case by providing rigorous statistical underpinning (Econometrica publications and a Nobel don't come easily) where none was available.

As for question in good resources: it's tricky. I have read dozens of time series books, and most excel at the math and leave the intuition behind. There is nothing like Kennedy's Econometrics text for time series. Maybe Walter Enders text comes closest. I will try to think of some more and update here.

Other than books, software for actually doing this is important and R has what you need. The price is right too.

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