Granger causality interpretation using R

Question

I have three macro economic variables (ICS - consumer sentiment, ER - employment rate, DGO - durable goods order) and have run Granger causality tests in R on them. I don't really know how to interpret the results of a Granger test. Could anyone give me a hand with making some sense of the results?

I know that we are checking to see if one variable can be used to predict another and I understand that if that is true then there must be some lag in one of the variables and that the order of the Granger test has to do with the order. I don't know how to interpret the fact that 2 models are reported here. I can see that one model is with the regressor variable and the other model is without the regressor. I assume the Lags vector 1:3 means that we are testing 1, 2,and 3 month lags.

grangertest(ICS~ER, order = 3, data=modeling.mts)

Granger causality test

Model 1: ICS ~ Lags(ICS, 1:3) + Lags(ER, 1:3)
Model 2: ICS ~ Lags(ICS, 1:3)
  Res.Df Df      F Pr(>F)
1    258                 
2    261 -3 2.0352 0.1094

grangertest(ICS~DGO, order = 3, data=modeling.mts)

Granger causality test

Model 1: ICS ~ Lags(ICS, 1:3) + Lags(DGO, 1:3)
Model 2: ICS ~ Lags(ICS, 1:3)
   Res.Df Df     F   Pr(>F)   
1    258                      
2    261 -3 4.8621 0.002625 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

grangertest(DGO~ER, order = 3, data=modeling.mts)

Granger causality test

Model 1: DGO ~ Lags(DGO, 1:3) + Lags(ER, 1:3)
Model 2: DGO ~ Lags(DGO, 1:3)
  Res.Df Df      F  Pr(>F)  
1    258                    
2    261 -3 3.2704 0.02181 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Answer 1

The help page for the grangertest function is pretty clear, it should be of major help.

Model 1 is the unrestricted model that includes the Granger-causal terms.
Model 2 is the restricted model where the Granger-causal terms are omitted.
The test is a Wald test that assesses whether using the restricted Model 2 in place of Model 1 makes statistical sense (roughly speaking).

You interpret the results as follows:

• if Pr(>F)$ < \alpha$ (where $\alpha$ is your desired level of significance), you reject the null hypothesis of no Granger causality. This indicates that Model 2 is too restrictive as compared with Model 1.
• If the inequality is reversed, you do not reject the null hypothesis as the richer Model 1 is preferred to the restricted Model 2.

Note: you say we are checking to see if one variable can be used to predict another.
A more precise statement would be we are checking to see if including $x$ is useful for predicting $y$ when $y$'s own history is already being used for prediction. That is, do not miss the fact the $x$ has to be useful beyond (or extra to) the own history of $y$.