Without a doubt, impact analysis is an essential field of statistics. Many people associate the effect with a lot of work - designing an experiment, collecting the data, etc. However, in a world full of data, we can understand the impact with clever methods. Meet Granger causality.
Granger Causality
Clive Granger, the author of the idea, writes:
“The definition [of causality] is based entirely on the predictability of some series, say Xt. If some other series Yt contains information in past terms that helps in the prediction of Xt and if this information is contained in no other series used in the predictor, then Yt is said to cause Xt. The flow of time clearly plays a central role in these definitions. In the author’s opinion, there is little use in the practice of attempting to discuss causality without introducing time, although philosophers have tried to do so.”
This translates to “cause precedes effect”.
I like the snarky comment at the end. For one, isn’t it interesting how scientists used to write papers? The main reason is how it points to the fact that there are other opinions. If you ask me, I would say that this is just common sense. Of course, cause precedes effect. I can’t first break a window and then throw a ball that breaks it, right?
There are many ways to think about causality. For example, Bertrand Russell, a well-known philosopher, was sceptical about the importance of causality and compared it to “a relic of a bygone age”. The idea suggests that since fundamental theories of how the world works (e.g. physics), the concept of causation is not very useful. Physical theories use equations that are symmetrical - both sides will change depending on what variable you are solving for. But causal relationships are asymmetrical.
I took this example from the Internet Encyclopedia of Philosophy. If you haven’t scratched your head in a while, you might enjoy it.
The fundamental truth about causality is less relevant when it comes to practice. However, we still call it Granger causality because it is one of many interpretations of causality.
Granger Causality vs Natural Experiments
Both Granger causality and natural experiments are approaches to causal analysis without performing standard experiments. Traditional experiments first require design, and the second step is data collection. Only then do scientists proceed to analyse the data. I wrote a story about natural experiments and how they relate to traditional experiments; it’s a nuanced topic. Long story short, the main difference is the level of certainty.
Performing an experiment in a controlled environment gives us more certainty that the observed effect is really something. When we design an experiment, we think about controlling as many factors (i.e. other possible causes) as possible. We only change one factor (i.e. the cause we wish to study) at the time. At least that’s the theory; it’s not easy to do this in social sciences.
Natural experiments and granger causality are alternatives and could be classified as quasi-experimental approaches. The distinction is that natural experiments require a bit of luck. At the same time, Granger causality can be performed on any stationary time series (I will explain what that is in the next section).
Granger Causality in Practice
The primary application of Granger’s methods was econometrics, but it goes beyond that. All the areas capturing information in time can make use of Granger causality.
Your data needs to be collected in regular intervals, e.g. daily, weekly, monthly, etc. The method assumes stationarity of the data. You can think of stationarity as data free from time-related biases. These are often present in practice. However, there are tools to handle them, and the simplest one is differencing.
Differenced time-series are differences between consequent time points. In other words, instead of using the actual values such as [5, 9, 6, 4], you would use their differences [ 4, -3, -2]. Sometimes, you need to difference twice to achieve stationarity.
Let’s say you want to know whether people become more interested in climate change after a heat wave. We can download data from Google trends.
Ok, so how do you go about this?
By looking at the graph, we could think that the increase in the relative search volumes about heat waves comes before the rise in interest in climate change. But to do this properly, we need to first make sure that the data is stationary. For instance, we can see that interest in heat waves increases in summer - this is seasonality which could bias our analysis, so that’s one of the corrections we need to make. We test for stationarity with a statistical test, e.g. Augmented Dickey-Fuller.
Next, feed this data into a multivariate regression model called Vector Autoregressive (VAR) Model. Autoregression means that past values of one variable, e.g. interest in climate change, predict the current value of the same variable, i.e. interest in climate change. In causal analysis, we include past values of another variable together with the autoregressive predictors, e.g. past interest in climate change + past interest in heat waves.
Finally, we compare whether the simple autoregressive model (without past interest in heat waves) performs “significantly” worse. As you can never be sure which variable is granger-causing which, it is common to complete the procedure both ways: heat waves (H) → climate change (C), and climate change (C) → heat waves (H).
Here is a visual representation of the process for one direction:
Based on the results, the answer for UK data is no. Working with time series is tricky, and you can easily fool yourself by looking at it. Therefore, something like Granger causality is handy.
If you want to try out the analysis for yourself, here is the R code I used to get the answer.