In time series, we typically use univariate data. That is, we use a single series to predict its future values. Let's say that we are studying Google's stock price data, and we are asked to forecast the future values of stock prices. In this case, we will need historic data of Google's stock prices. Based on that, we will make predictions.

However, at times, we need multiple time series to make a forecast. But why is it that we need multiple time series? Any guesses?

The following graph shows Google's stock price data:

The answer is that we need to understand and explore the relationship between multiple time series as this can improve our forecast. For example, we have got correlated time series of GDP Deflator: Services and WPI: All Commodities, as follows:

It is quite evident that these two seem to carry a relationship. When we have to forecast GDP Deflator: Services, we can use WPI: All Commodities time series data as input. This is called Granger causality.

To put it more aptly, Granger causality is one of the ways to investigate causality between two variables in a time series. This method is a probabilistic account of causality.

Even though we are talking about causality here, it isn't exactly the same. Typically, causality is associated with a situation where variable 1 is the cause of variable 2 or vice versa. However, with Granger causality, we do not test a true cause and effect relationship. The fundamental reason to know is whether a particular variable comes before another in the time series. 

You aren't testing a true cause and effect relationship. What you want to know is whether a particular variable comes before another in the time series. That is to say, if we find Granger causality in the data, then there isn't any causal link in the true sense of the word. 

Let's look at one more example. Here, we have GDP per capita for OECD nations. We can see that the number of OECD countries have a similar growth and pattern in GDP. We can assume that these countries are responsible for each other's GDP growth due to dependencies.

The following is a graph that shows GDP per capita for OECD countries:

We can utilize the series of one country to forecast for another one. More often than not, this kind of relationship is prevalent more in financial time series. The stock market of India, NSE/BSE, and so on may have an impact on NYSE. Therefore, NYSE indices can be used to make a forecast for NSE indices.

Let's infuse a bit of mathematics into this. Let's say there are two time series, X(t) and Y(t). If the past values of X(t) are helping to predict the future values of Y(t), it is said that X(t) Granger causes Y(t).

So, Y(t) is a function of the lag of Y(t) and also of the lag of X(t). It can be expressed as as follows:

Y(t) = f(Yt-p, Xt-p)

However, this only holds true in the following situations:

• Cause takes place prior to effect. What this means is that Y(t) = f(Xt-1),
• Cause has got significant information about the future values of its effect, for example:

Y(t) = a1 Yt-1 + b1 Xt-1 + error

Xt-1 is adding an extra effect on Y(t). The magnitude of effect is decided by b1.

Let's say we have two equations:

Yt= a0+ a1* Yt-1

Yt= a0 + a1*Yt-1 + a2*Xt-1

The null hypothesis here is as follows:

• H0: a2=0: What this means is that there is no effect of other series, where Xt-1 on Yt

The alternate hypothesis here is as follows::

• H1: a2≠ 0: What this means is that there is a significant effect of other series, where Xt-1 on Yt

We run a t-test to determine whether there is a significant effect of other series of Xt-1 on Yt.

If the null hypothesis is rejected, we can say that it is a case of Granger causality.