# Testing for a causal effect (with 2 time series)

A few days ago, I came back on a sentence I found (in a French newspaper), where someone was claiming that

“… an old variable explains 85% of the change in a new variable. So we can talk about causality”

and I tried to explain that it was just stupid : if we consider the regression of the temperature on day $t+1$ against the number of cyclist on day $t$, the $R^2$ exceeds 80%… but it is hard to claim that the number of cyclists on specific day will actually cause the temperature on the next day…

Nevertheless, that was frustrating, and I was wondering if there was a clever way to test for causality in that case. A popular one is Granger causality (I can mention a paper we published a few years ago where we use such a test, Tents, Tweets, and Events: The Interplay Between Ongoing Protests and Social Media). To explain that test, consider a bivariate time series (just like the one we have here), $\boldsymbol{z}_t=(x_t,y_t)$, and consider some bivariate autoregressive model
$${\displaystyle {\begin{bmatrix}x_{t}\\y_{t}\end{bmatrix}}={\begin{bmatrix}c_{1}\\c_{2}\end{bmatrix}}+{\begin{bmatrix}a_{1,1}&\textcolor{red}{a_{1,2}}\\\textcolor{blue}{a_{2,1}}&a_{2,2}\end{bmatrix}}{\begin{bmatrix}x_{t-1}\\y_{t-1}\end{bmatrix}}+{\begin{bmatrix}u_{t}\\v_{t}\end{bmatrix}}}$$where $\boldsymbol{\varepsilon}_t=(u_t,v_t)$ is some bivariate white noise, in the sense that (i) ${\displaystyle \mathbb{E} (\boldsymbol{\varepsilon}_{t})=\boldsymbol{0}}$ (the noise is centered) (ii) ${\displaystyle \mathbb{E} (\boldsymbol{\varepsilon}_{t}\boldsymbol{\varepsilon}_{t}^\top)=\Omega }$, so the variance matrix is constant, but possibly non-diagonal (iii) ${\displaystyle \mathbb{E} (\boldsymbol{\varepsilon}_{t}\boldsymbol{\varepsilon}_{t-h}^\top)=\boldsymbol{0} }$ for all $h\neq 0$. Note that we can use the simplified expression$${\displaystyle {\boldsymbol{z}_t=\boldsymbol{c}+\boldsymbol{A}\boldsymbol{z}_{t-1}+\boldsymbol{\varepsilon}_t}}$$Now, Granger test is based on several quantities. With off-diagonal terms of matrix $\Omega$, we have a so-called instantaneous causality, and since $\Omega$ is symmetry, we will write $x\leftrightarrow y$. With off-diagonal terms of matrix $\boldsymbol{A}$, we have a so-called lagged causality, with either $\textcolor{blue}{x\rightarrow y}$ or $\textcolor{red}{x\leftarrow y}$ (and possibly both, if both terms are significant).

So I wanted to try on my two-variable problem.

df = read.csv("cyclistsTempHKI.csv") dfts = cbind(C=ts(df$cyclists,start = c(2014, 1,2), frequency = 365), T=ts(df$meanTemp,start = c(2014, 1,2), frequency = 365)) library(vars)

I now have “time series” objects, and we can fit a VAR model,

var2 = VAR(dfts, p = 1, type = "const") coefficients(var2) $C Estimate Std. Error t value Pr(&gt;|t|) C.l1 0.8684009 0.02889424 30.054460 8.080226e-107 T.l1 70.3042012 20.07247411 3.502518 5.102094e-04 const 807.6394001 187.75472482 4.301566 2.110412e-05$T Estimate Std. Error t value Pr(&gt;|t|) C.l1 0.0003865391 6.257596e-05 6.177118 1.540467e-09 T.l1 0.6611135594 4.347074e-02 15.208241 6.086394e-42 const -1.6413074565 4.066184e-01 -4.036481 6.446018e-05

For instant, we can run a causality, to test if the number of cyclists can cause the temperature (on the next day)

causality(var2, cause = "C") $Granger Granger causality H0: C do not Granger-cause T data: VAR object var2 F-Test = 38.157, df1 = 1, df2 = 842, p-value = 1.015e-09 Here, we should clearly reject $H_0$, which is that there is no causal effect. Which is the way statisticians say that there should be some causal effect between the number of cyclist and the temperature… So clearly, something is wrong here. Either it is some sort of superpower that cyclists are not aware of. Or this test that was used for forty years (Clive Granger even got a Nobel price for it) is not working. Or we missed something. Actually… I think we missed something here. Possibly because the series are not stationary. We can almost see it with Phi = matrix(c(coefficients(var2)$C[1:2,1],coefficients(var2)$T[1:2,1]),2,2) eigen(Phi) eigen() decomposition$values [1] 0.9594810 0.5700335

where the highest eigenvalue is very close to one. But actually, we look here at the temperature…

plot(dfts)

so, at least, we should expect some seasonal unit root here. So let us use two techniques. The first one is a classical one-year difference, $\Delta_{365}\boldsymbol{z}_t=\boldsymbol{z}_t-\boldsymbol{z}_{t-365}$

var2 = VAR(diff(dfts,365), p = 1, type = "const") coefficients(var2) $C Estimate Std. Error t value Pr(&gt;|t|) C.l1 0.8376424 0.07259969 11.537823 1.993355e-16 T.l1 42.2638410 28.58783276 1.478386 1.449076e-01 const -507.5514795 219.40240747 -2.313336 2.440042e-02$T Estimate Std. Error t value Pr(&gt;|t|) C.l1 0.000518209 0.0003277295 1.5812096 1.194623e-01 T.l1 0.598425288 0.1290511945 4.6371154 2.162476e-05 const 0.547828079 0.9904263469 0.5531235 5.823804e-01

The test on the fited VAR model yields

causality(var2, cause = "C") $Granger Granger causality H0: C do not Granger-cause T data: VAR object var2 F-Test = 2.5002, df1 = 1, df2 = 112, p-value = 0.1167 i.e., with a 11% $p$-value, we should reject the assumption that the number of cyclists cause the temperature (on the next day), and actually, we should also reject the other way causality(var2, cause = "T")$Granger   Granger causality H0: T do not Granger-cause C   data: VAR object var2 F-Test = 2.1856, df1 = 1, df2 = 112, p-value = 0.1421

Nevertheless, if we look at the instantaneous causality, this one makes more sense

$Instant H0: No instantaneous causality between: T and C data: VAR object var2 Chi-squared = 13.081, df = 1, p-value = 0.0002982 The second idea would be to use a one day difference, $\Delta_{1}\boldsymbol{z}_t=\boldsymbol{z}_t-\boldsymbol{z}_{t-1}$ and to fit a VAR model on that one VARselect(diff(dfts,1), lag.max = 4, type="const")$selection AIC(n) HQ(n) SC(n) FPE(n) 3 3 2 3

but on that one, a VAR(1) model – with only one lag – might not be sufficient. It might be better to consider a VAR(3)

var2 = VAR(diff(dfts,1), p = 3, type = "const")

and on that one, one more time, we should reject the causal effect of the number of cyclists on the temperature (on the next day)

causality(var2, cause = "C") $Granger Granger causality H0: C do not Granger-cause T data: VAR object var2 F-Test = 0.67644, df1 = 3, df2 = 828, p-value = 0.5666 and this time, there could be a (lagged) causal effect of the temperature on the number of cyclists causality(var2, cause = "T")$Granger   Granger causality H0: T do not Granger-cause C   data: VAR object var2 F-Test = 7.7981, df1 = 3, df2 = 828, p-value = 3.879e-05   \$Instant   H0: No instantaneous causality between: T and C   data: VAR object var2 Chi-squared = 55.83, df = 1, p-value = 7.905e-14

but nothing instantaneously… So it looks like Granger causality performs well on that one !

# Generalization, stereotypes and clichés

In Thinking, Fast and Slow, Daniel Kahneman discusses at length the importance of stereotypes in understanding many decision-making processes. A so-called System 1 is used for quick decision-making: it allows us to recognize people and objects, helps us focus our attention, and encourages us to fear spiders. It is based on knowledge stored in memory and accessible without intention, and without effort. It can be contrasted with System 2, which allows for more complex decision-making, requiring discipline and sequential reflection. In the first case, our brain uses the stereotypes that govern judgments of representativeness, and uses this heuristic to make decisions. If I cook a fish for friends who have come to eat, I open a bottle of white wine. The cliché “fish goes well with white wine” allows me to make a decision quickly, without having to think about it. Stereotypes are statements about a group that are accepted (at least provisionally) as facts about each member. Whether correct or not, stereotypes are the basic tools for thinking about categories in System 1. But in many cases, a more in-depth, more sophisticated reflection – corresponding to System 2 – will make it possible to make a more judicious, even optimal decision. Without choosing any red wine, a pinot noir could perhaps also be perfectly suitable for roasted red mullets.

To generalize is to be an idiot, to particularize is the alone distinction of merit” wrote William Blake around 1800, annotating speeches by the painter Joshua Reynolds. Stigmatizing an entire population because of a minority in a decision-making process is a misleading generalization, often punished by society. Moral punishment, but sometimes also legal (when hiring for example) in a society that tends to be civilized, asking not to draw erroneous conclusions about an individual from the statistics of a group to which he would be attached. But isn’t that what the actuary does every day?

# The usual suspects

For Schauer (2009), this “generalization“, condemned by William Blake, is probably the actuary’s raison d’être: “to be an actuary is to be a specialist in generalization, and actuaries engage in a form of decision-making that is sometimes called actuarial“. If I decide to insure a sports car, I have I am given risky driving characteristics that probably belong to the majority of sports car owners, attributes that I may not share. And as we noted in the introduction, insurance companies, of course, are not the only ones that operate actuarially, according to Schauer’s definition. We all do it, much more often than most of us would probably recognize. We do this when we choose airlines based on their safety record, punctuality or lost luggage. We do this when we associate personal characteristics (a visible tattoo, black or brightly coloured clothing) with behavioural characteristics (such as a propensity for violence) that these personal characteristics would seem to indicate. And we operate in this way when we engage in stereotypes that may be harmless on the basis of nationality, for example by calling French people are rude, or Scots all wear kilts, while at the same time acknowledging that more pernicious stereotypes on the basis of ethnic origin, gender, sexual orientation are too widespread today! As the misconception of the word “prejudice” indicates, many people believe that it is unfair to make individual decisions based on non-universal group characteristics. Even if group allocations have a solid statistical basis. Because the big difference between actuarial science and everyday life is that actuaries have to use a large number of observations. On a personal level, I can thus decide not to travel with such an airline anymore because on three trips, I have experienced two bad experiences. Before deciding that travel insurance deserves a higher premium when flying with this company, it takes more than three observations!

In fact, the question is often whether an insurance company’s refusal to provide coverage, or to increase the premiums it charges for the same coverage, is an injustice when it is based on an actuarially justified (but perhaps not universal) generalization. As Leemens (2000) noted, the question was asked of the legislator when insurers observed that Jewish women from Eastern Europe were particularly vulnerable to breast and ovarian cancer. At the end of 2012, the European Court of Justice put an end to all discrimination based on the gender of policyholders: insurers were no longer able to differentiate between insurance product prices according to whether the member was male or female. But the use of age is still allowed. Indeed, age is often an indicator of a possible decrease in vision or hearing, slower reaction time (and increased risk of sudden disability), etc. And although there are many individual variations, the available data provide important empirical justification.

# Machines, causality, and stereotypes

A major criticism of machine learning models is the lack of interpretation. But very often, the validation of econometric models requires a narrative built around stereotypes. And this narrative is essential, as Pearl & Mackenzie (2018) reminds us. Indeed, in the “The Ladder of Causation“, there are three levels. At the first level, we find the notion of association (or correlation), or even conditional probability, which serve as a basis for the constitution of stereotypes: if we observe

P[carries | brushing your teeth] < P[carries | don’t brush your teeth]

brushing teeth will be associated with a decrease in the probability of having carries. It is also the basis for regression methods, which are based on correlations between the variable of interest and others, wrongly called explanatory. In Figure 1, we can see the daily cycling traffic in Helsinki, and the average temperature. We will tend to prefer the one on the left, showing the evolution of the number of cyclists as a function of temperature, suggesting that temperature could explain the number of cyclists, and not the other way around. But the stereotype doesn’t necessarily focus on the causal link: if I see a lot of cyclists passing through the window, I’ll tell myself it must be hot, or at least warm.

Figure 1: Näytä Data – Author’s visualization

The first level answers the question “what if I see…?“(e.g. “what cycling traffic to expect if the temperature reaches 20°C? “) and this task can be perfectly accomplished by a machine. The second level is the one that makes it possible to understand an effect, an intervention. The question is then “what if I do…? “. To use our example, we are trying to understand the importance of brushing our teeth on the appearance of cavities. What if brushing your teeth is more natural for children with good teeth? We see the third level of the scale coming up, asking the question “what if I had done…?“and based on the idea of a counterfactual model. We are no longer content to measure correlations, we will build a model explaining what would happen by making a change in the causal variables: what would really happen if the child who did not brush his teeth began to do so? For Pearl & Mackenzie (2018) a human being (maybe even an actuary) can make these more advanced arguments than a machine can (yet) do. And very often, these causal patterns are stereotyped. As Charpentier & Diago Barry (2015) points out, in epidemiology, researchers have long questioned the explanation to be given to the fact that small babies of smokers have a higher probability of survival than babies of non-smoking mothers. The intuition that something is wrong comes from prejudices, stereotypes that we have, and that a machine cannot have.

# When actuaries tell each other stories

As Antonio & Charpentier (2017) noted, the European “gender directive” has confused many insurers who used gender to construct their rates, as the latter was highly correlated with the frequency of claims. But by introducing telematic data, gender was no longer significant in the regression. Gender has long been used as a proxy to capture an effect that can be observed using telematic data, giving rise to many sexist stereotypes and other stereotypes.

But the stories also make it possible to decide between a false correlation (“spurious correlation“) and a correlation that could be interpreted. In Figure 2, we have life expectancy at birth, a variable that we could try to explain in a pension study context, for example, by French department. On the right, two variables taken at random: the number of licenses of a tennis club, and the number of advertising agencies. Stereotypes are what will allow us to construct a causal graph, allowing us to understand why there is such a strong correlation between these variables and life expectancy.

Figure 2: Life expectancy at birth for men, left. At the centre, number of tennis licenses per 100,000 inhabitants (source FFT). On the right, number of advertising agencies per 100,000 inhabitants (source INSEE, code NAF 7311Z). Visualization of the author.

While William Blake condemned stereotypes by saying “to generalize is to be an idiot“, he also clearly went further, continuing with “to particularize is the alone distinction of merit“. This individualisation is also advocated by more and more insurers, and even desired by many insureds. But as Grace & Terry (2002) pointed out, many policyholders suffer from a significant optimism bias – “if I have an accident, it will not be my fault” – leading them to doubt the insurer’s classification – “I’m less risky than the others“. And morality seems to prove them right, against actuaries. Yet, not only is generality not, in general, unjust, but justice itself can have considerable elements of generality. To the extent that justice is centred on equity and to the extent that equity itself is closely linked to equality, then equity, and therefore justice, can now be seen as itself based on the idea of generality. The just society is not necessarily a society in which each individual is treated as an isolated set of unique attributes, requiring individualized attention. On the contrary, in some cases, the just society is a society in which generality is not only unavoidable, but also necessary for justice itself. And pooling risks together is the natural response in an insurance context. And it might not be such a big deal if that class is not as homogenous at it could be, or as we would have expected it to be…

Antonio, K. & Charpentier, A. (2017).  La tarification par genre en assurance, corrélation ou causalité ?. Risques. 110 : 107-110.

Charpentier, A. & Diago Barry, A. (2015). Big data : passer d’une analyse de corrélation à une interprétation causale. Risques, 101: 107-111.

Grace, J. & Terry, M. (2002). Exploring the Causes of Comparative Optimism. Psychologica Belgica. 42: 65–98

Kahneman, D. (2011).Thinking, Fast and Slow. FSG Eds.

Leemens, T. (2000). Selective Justice, Genetic Discrimination, and Insurance: Should We Single Out Genes in Our Laws? McGill law journal. Revue de droit de McGill 45(2):347-412.

Pearl, J. & Mackenzie, D. (2018). The Book of Why: The New Science of Cause and Effect. Basic Books.

Schauer, F.F. (2009). Profiles, Probabilities, and Stereotypes. Harvard University Press.

# On the interpretation of a regression model

Yesterday, (aka @NaytaData ) posted a nice graph on reddit, with bicycle traffic and mean air temperature, in Helsinki, Finland, per day,

I found that graph interesting, so I did ask for the data ( kindly sent them to me tonight).

df=read.csv("cyclistsTempHKI.csv") library(ggplot2) ggplot(df, aes(meanTemp, cyclists)) + geom_point() + geom_smooth(span = 0.3)

But as mentioned by someone on twitter, the interpretation is somehow trivial : people get out on their bike when the weather is nice. The hotter, the more cyclists on the road. Which is interpreted here in a causal way…

But actually, we can also visualize the data as follows, as suggested by Antoine Chambert-Loir

 ggplot(df, aes(cyclists, meanTemp)) + geom_point() + geom_smooth(span = 0.3)

The interpretation would be, somehow, that the more cyclists on the road, the hotter it is. Why not consider this causal interpretation here ? Like cyclists go so fast, or sweat so much, that they increase temperature…

Of course, it is the standard (recurrent) discussion “correlation is not causality”, but in regression models, we like to tell a story, to pretend that we have some sort of a causal story. But we do not prove it. Here, we know that the first one is more credible than the second one, but how do we know that ? To go further, how can we use machine learning techniques to prove causal relationships ? How could a machine choose between the first and the second story ?

# Talk on Causality with Non-Gaussian Time Series, at Paris 7 Diderot

This Monday, I will be giving a talk at Paris 7, room 1016 of the Sophie Germain building, on causality with non-Gaussian time series. Slides are now online.

# Tents, Tweets, and Events: Ongoing Protests and Social Media

Our paper, entitled Tents, Tweets, and Events: The Interplay Between Ongoing Protests and Social Media, written with Marco Toledo Bastos (aka ) and Dan Mercea (aka) just appeared in the Journal of Communication

Recent protest movements have fuelled deliberations about the extent to which social media ignite protests. In this paper we compare time-series data of Twitter, Facebook, and onsite protest activity to test the hypothesis of Granger-causality between social media streams and protestors attending demonstrations during the Indignados in Spain, the Occupy movement in the U.S., and the Vinegar protests in Brazil. After applying a Gaussianization procedure to the time series, we confirmed the hypothesis that contentious communication on Twitter and Facebook was Granger-causal of onsite protest activity during the Indignados and the Occupy protests, with bidirectional causality between online and onsite protest activity in the Occupy series. The Vinegar protests in Brazil presented Granger-causality only between Facebook and Twitter and between protestors and injured or arrested protestors. The results indicate that the causal relationship between online and onsite political varies considerably across different socioeconomic contexts with different levels of Internet penetration.