*19*

The **Granger Causality test** is used to determine whether or not one time series is useful for forecasting another.

This test uses the following null and alternative hypotheses:

**Null Hypothesis (H _{0}):** Time series

*x*does not Granger-cause time series

*y*

**Alternative Hypothesis (H _{A}):** Time series

*x*Granger-causes time series

*y*

The term “Granger-causes” means that knowing the value of time series *x* at a certain lag is useful for predicting the value of time series *y* at a later time period.

This test produces an F test statistic with a corresponding p-value. If the p-value is less than a certain significance level (i.e. α = .05), then we can reject the null hypothesis and conclude that we have sufficient evidence to say that time series *x* Granger-causes time series y.

To perform a Granger-Causality test in R, we can use the **grangertest()** function from the lmtest package, which uses the following syntax:

**grangertest(x, y, order = 1)**

where:

**x:**The first time series**y:**The second time series**order:**The number of lags to use in the first time series. Default is 1.

The following step-by-step example shows how to use this function in practice.

**Step 1: Define the Two Time Series**

For this example, we’ll use the **ChickEgg** dataset that comes pre-loaded in the lmtest package. This dataset contains values for the number of eggs manufactured along with the number of chickens in the U.S. from 1930 to 1983:

#load lmtest package library(lmtest) #load ChickEgg dataset data(ChickEgg) #view first six rows of dataset head(ChickEgg) chicken egg [1,] 468491 3581 [2,] 449743 3532 [3,] 436815 3327 [4,] 444523 3255 [5,] 433937 3156 [6,] 389958 3081

**Step 2: Perform the Granger-Causality Test**

Next, we’ll use the **grangertest()** function to perform a Granger-Causality test to see if the number of eggs manufactured is predictive of the future number of chickens. We’ll run the test using three lags:

#perform Granger-Causality test grangertest(chicken ~ egg, order = 3, data = ChickEgg) Granger causality test Model 1: chicken ~ Lags(chicken, 1:3) + Lags(egg, 1:3) Model 2: chicken ~ Lags(chicken, 1:3) Res.Df Df F Pr(>F) 1 44 2 47 -3 5.405 0.002966 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Here is how to intrepret the output:

**Model 1:**This model attempts to predict the number of chickens using the number of chickens in the previous three years and the number of eggs in the previous three years as predictor variables.**Model 2:**This model attempts to predict the number of chickens using only the number of chickens in the previous three years as predictor variables.**F:**This is the F test statistic. It turns out to be 5.405.**Pr(>F):**This is the p-value that corresponds to the F test statistic. It turns out to be .002966.

Since the p-value is less than .05, we can reject the null hypothesis of the test and conclude that knowing the number of eggs is useful for predicting the future number of chickens.

**Step 3: Perform the Granger-Causality Test in Reverse**

Although we rejected the null hypothesis of the test, it’s actually possible that there is a case of reverse causation happening. That is, it’s possible that the number of chickens is causing the number of eggs to change.

To rule out this possibility, we need to perform the Granger-Causality test in reverse, using chickens as the predictor variable and eggs as the response variable:

#perform Granger-Causality test in reverse grangertest(egg ~ chicken, order = 3, data = ChickEgg) Granger causality test Model 1: egg ~ Lags(egg, 1:3) + Lags(chicken, 1:3) Model 2: egg ~ Lags(egg, 1:3) Res.Df Df F Pr(>F) 1 44 2 47 -3 0.5916 0.6238

The p-value of the test is 0.6238. Since this isn’t less than .05, we can’t reject the null hypothesis. That is, the number of chickens isn’t predictive of the future number of eggs.

Thus, we can conclude that knowing the number of eggs is useful for predicting the future number of chickens.