A to B or B to A? That is the Question: Be Careful about Reverse Causality

Douglas Adams, the author of The Hitchhiker’s Guide to the Galaxy, had said, “The complexities of cause and effect defy analysis.”  Human beings have, throughout history, often been stumped by these complexities.

In the middle ages, many Europeans believed that lice and good health were correlated. The reasoning was that sick people rarely had any lice on them.  Today we know, of course, that lice are extremely sensitive to body temperature and would leave the body of sick individuals. Thus, it was not that fewer lice caused sickness, but that sickness caused fewer lice.

Even Economics has suffered from these issues. Alfred Marshall, in the third edition of his Principles of Microeconomics, had stated the Giffen Paradox – the curious exception to the law of demand,
“There are however some exceptions. For instance, Mr Giffen has pointed out, a rise in the price of bread makes so large a drain on the resources of the poorer labouring families and raises so much the marginal utility of money to them, that they are forced to curtail their consumption of meat and the more expensive farinaceous foods: and, bread being still the cheapest food which they can get and will take, they consume more, and not less of it.”

However, for over a century, economists have struggled to get comprehensive empirical evidence in support of this hypothesis. The key issue, as several studies have pointed out, is that it could be higher consumption leading to higher prices – which may give the illusion of an upward sloping demand curve. Indeed, in a market economy, the price is an equilibrium of two equations – the demand equation and the supply equation. We must not forget that demand and price are positively correlated on the supply side of the equation.

Closer to the present, the issue continues to haunt us. In 2010, Harvard University economists Carmen Reinhart and Kenneth Rogoff published a paper titled ‘Growth in a Time of Debt’. One of their key findings was

..median growth rates for countries with public debt over 90 percent of GDP are roughly one percent lower than otherwise;….. Countries with debt-to-GDP ratios above 90 percent have a slightly negative average growth rate, in fact.”

Their research was extremely influential – it was used by many fiscal hawks in the west to make arguments about reigning in debt levels. George Osborne, former Chancellor of the Exchequer in the UK, had famously said, “As Rogoff and Reinhart demonstrate convincingly, all financial crises ultimately have their origins in one thing.”

Alas, research by other economists into their works showed glaring oversights. Technical errors (yes, there was one) aside, economists pointed out to how there could simply be a case of reverse causality – i.e., it was low growth that caused high debt levels instead (this was acknowledged by Reinhart and Rogoff).

So why is reverse causality an issue in econometrics? In econometrics, we try to avoid endogeneity as that gives us biased or inconsistent estimates. What endogeneity means is that the explanatory variable and the error terms are correlated.

So how does reverse causality lead to this situation? Picture this:

I am using Telecom Regulatory Authority of India quarterly data which has price (Average Revenue per User) and quantity (Minutes of Usage) data.

Suppose I have the following equations:

MoU = a + B*ARPU + error_1  ….. (i)

I would expect a negative correlation between MoU and ARPU based on the law of demand (ignoring omitted variables for the moment). However, I forget that higher consumption of minutes that leads to higher prices as well, i.e.:

ARPU = b + C*MoU + error_2 ……(ii)

So why is ARPU endogenous in the first equation? Simple, imagine a shock to, error_1, which leads to higher MoU. Since MoU would rise, by equation (ii), this would also lead to higher ARPU. As a result, a change in error_1 leads to a change in ARPU in equation (i). Thus, there is endogeneity.

Visually –
error_1 goes up -> MoU goes up -> ARPU goes up
Thus, error_1 and MoU are correlated!

Here, I try to run a simple OLS regression of ARPU on MoU and get the following results:-

Balanced Panel: n=22, T=28, N=616 
Residuals :    Min.  1st Qu.   Median  3rd Qu.     Max. 
              -102.730  -33.519  -12.179   26.099  250.750  
Coefficients :   Estimate Std. Error t-value  Pr(>|t|)   
 df_subset2$ARPU 1.371101   0.076506  17.921 < 2.2e-16 ***

So a simple OLS regression reveals that an increase in price has led to an increase in minutes of usage per user. But we now know the reason behind this counter-intuitive result.

Similar to my results,  Steven D. Levitt from the University of Chicago found that studies that look at the impact of greater police deployment at crime rates may wrongly get a positive correlation between the two. This is because it may be higher amounts of crime that lead to greater police deployment in an area, and thus, the results would be flawed.

In conclusion, it is very important to know that we’ve got the direction of our causation right in our model. Otherwise, it is very likely that are making incorrect inferences. However, in some cases, it may be unavoidable to get data with reverse causality or a simultaneity. What do we do then?  There is a technique known as Instrumental Variables estimation which is ideal for these kinds of situations. Steven D. Levitt used this technique to accurately study the effect of greater police deployment on crime rates.


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    Available at: https://www.newyorker.com/news/john-cassidy/the-reinhart-and-rogoff-controversy-a-summing-up
    (Accessed 20 October 2017)
  2. Europeans in the Middle Ages Believed Lice were a Sign of Good Health (2017), The Vintage News.
    Available at: https://www.thevintagenews.com/2017/03/23/europeans-in-the-middleages-believed-that-lice-were-a-sign-of-good-health/
    (Accessed 20 October 2017)
  3. Jensen R and Miller N, Giffen Behavior and Subsistence Consumption, The American Economic Review,  Vol. 98, No. 4 (Sep., 2008), pp. 1553-1577
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    Available at: https://krugman.blogs.nytimes.com/2010/08/11/reinhart-and-rogoff-are-confusing-me/
    (Accessed 20 October 2017)
  5. Krugman P (2013), Reinhart-Rogoff, Continued, The New York Times.
    Available at: https://krugman.blogs.nytimes.com/2013/04/16/reinhart-rogoff-continued/?_r=0
    (Accessed 20 October 2017)
  6. Performance Indicator Reports, Telecom Regulatory Authority of India.
    Available at: http://www.trai.gov.in/release-publication/reports/performance-indicators-reports
    (Accessed 19 October 2017)
  7. Levitt S (1997), Using Electoral Cycles in Police Hiring to Estimate the Effect of Police on Crime, The American Economic Review, Volume 87, Issue 3 (Jun., 1997), 270-290.
  8. Lyons J (2013), George Osborne favourite “godfathers of austerity” economists agree to making error in research, Mirror
    Available at: http://www.mirror.co.uk/news/uk-news/george-osbornes-favourite-economists-reinhart-1838219
    (Accessed 20 October 2017)
  9. Reinhart C and Rogoff K (2010), Growth in a Time of Debt, working paper at National Bureau of Economic Research
    Available at: http://www.nber.org/papers/w15639
    (Accessed 20 October 2017)
  10. Reinhart C and Rogoff K (2013), Reinhart-Rogoff Response to Critique, The Wall Street Journal.
    Available at: https://blogs.wsj.com/economics/2013/04/16/reinhart-rogoff-response-to-critique/
    (Accessed 20 October 2017)
  11. Stigler G (1947), Notes on the History of the Giffen Paradox, The University of Chicago Press Journals, Vol. 55, No. 2 (Apr., 1947), pp. 152-156


  1. Very good insights! Looking forward in reading some more!

  2. This was a good read. Very informative!

  3. Very good content. You have a very good take on the subject!

  4. The other reason why the estimated coefficient is positive is maybe because you are dealing with average data. In this article (https://www.cornerstone.com/Publications/Articles/Guidelines-Quantitative-Techniques-for-Competition), Card et al. give an example: suppose in a given month the price of cheaper cars falls AND people buy more expensive cars. Analyzing the relationship between these two variables could lead to the conclusion that higher prices caused higher sales. According to the authors, this is a measurement error problem as you are not dealing a fixed weight price index.

    • Sujan

      October 26, 2017 at 1:34 am

      Hey! This is a very interesting point that you raise, and it is definitely pertinent. With the lack of granular data, unfortunately, I would be unable to check if there is such a phenomenon occurring in this data set. However, my intuition is that this kind of phenomenon may be more likely for goods that have a high degree of differentiation between them. Mobile network services are a highly standardized good and it seems to be unlikely that price movements for plans with more talk time don’t move in tandem with plans that offer less talk time.

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