Difference between Granger Causality and AR1

Question

Here are my questions:

1. is there a difference between "VAR(1)" and "AR(1)"?
2. Granger Causality inspects the direction of causality. In return, we receive a p-value on how much a time series is likely to contribute to a better prediction of the other.
3. Choi and Varian (2009) apply an "AR" model to look whether adding Google Search to the regression improves the prediction. Here, we can measure the magnitude of the effect.
4. Is it okay to apply Choi & Varian's AR method even though data is not mixed, but same frequency?
5. Why seems to be there no paper that applies both Granger and Choi/Varian's method? From my naive understanding, Granger can measure the direction of the effect while Choi & Varian's model can help to measure the magnitude of the effect. As such, both analysis might make sense in a paper?

Answer 1

1. VAR stands for Vectorautoregression - i.e., the lags of several variables depend on their own past as well as on the past of the others. In a univariate autoregression, a single variable only depends on its own past.
2. Granger causality indeed measures whether the past of one variable helps predicting another. But your interpretation of a p-value as p-value as measuring "how much a time series is likely to contribute to a better prediction of the other" is wrong. P-values cannot be interpreted that way.

As for 3.-5., I indeed agree the question is rather broad.

Answer 2

4.

Yes, why not? The concept is simple: you have a candidate variable that may be helpful in predicting some other variable (even after accounting for a bunch of other predictors such as own past values of the dependent variable). You test whether that is the case. If you reject the null, you conclude that the variable indeed helps.

5.

Because the two partly overlap. The difference is that the Granger test is used in a VAR model where there is more than one dependent variable and hence more room for applying the test (Does $y_1 \xrightarrow{Granger} y_2$?; Does $y_1 \xrightarrow{Granger} y_3$?; etc). Meanwhile, Choi and Varian has only one dependent variable.

Granger can measure the direction of the effect

You can test $H_0 \colon y_1 \xrightarrow{Granger} y_2$ and $H_0 \colon y_2 \xrightarrow{Granger} y_1$ and conclude which way (none, first, second or both) it goes. So in this sense you are right.

Choi & Varian's model can help to measure the magnitude of the effect

Yes. But also a VAR model allows measuring the magnitude of effects alongside Granger testing; hence, Choi and Varian are not superior in this sense.

Granger can measure the direction of the effect while Choi & Varian's model can help to measure the magnitude of the effect

As becomes clear from above, magnitude can be measured in both cases. Direction may or may not be of interest. If it is, do Granger tests. If it is not, a single Choi and Varian-style test is enough.