Solved – Granger causality in a trivariate system

confoundinggranger-causalityvector-autoregression

I am working on testing Granger causality, in particular with the Toda & Yamamoto approach as in Dave Giles blog post "VAR or VECM When Testing for Granger Causality?".

Can I consider more than two variables in the VAR model?

For now, I have $y$ and $x$, with causality running from $y$ to $x$. I want to test if a third variable (confounding variable) might influence this causality. I'm unsure whether to test another model with a second predictor and simply compare the results or if there's any other way to measure the influence of a third variable.

For example, I find that sentiments Granger cause the stock price but I'd like to check whether interest rates have an influence on this VAR.

Best Answer

  1. Can I consider more than two variables in the VAR model?

Yes, of course. You would not find a restriction like "no more than two variables" in a textbook, and you would find applied work with more than two series. Also, here is an explicit answer by Dave Giles from his blog post "Questions About Granger Causality Testing - The Fine Print":

Although the illustrative examples that I used in my previous posts involved just two variables, the methodology I outlined applies equally if we a VAR model for three or more variables. Indeed, the discussion in Toda and Yamamoto (1995) is for the case of an arbitrary number of variables, $n$.

  1. I find that sentiments Granger cause the stock price but I'd like to check whether interest rates have an influence on this VAR.

Call stock price $p$, sentiments $s$, and interest rate $i$. Take a VAR(1) model

$$ \begin{aligned} p_t &= \beta_{10} + \beta_{11} p_{t-1} + \beta_{12} s_{t-1} + \beta_{13} i_{t-1} + \varepsilon_{1,t} \\ s_t &= \beta_{20} + \beta_{21} p_{t-1} + \beta_{22} s_{t-1} + \beta_{23} i_{t-1} + \varepsilon_{2,t} \\ i_t &= \beta_{30} + \beta_{31} p_{t-1} + \beta_{32} s_{t-1} + \beta_{33} i_{t-1} + \varepsilon_{3,t} \end{aligned} $$

Below are some examples of how to test different hypotheses.

  • Does $s \not\xrightarrow{Granger} p$? Test $H_0\colon \ \beta_{12}=0$.
  • Does $i \not\xrightarrow{Granger} p$? Test $H_0\colon \ \beta_{13}=0$.
  • Does $\{s,i\} \not\xrightarrow{Granger} p$? Test $H_0\colon \ \beta_{12}=\beta_{13}=0$.
  • Does $i \not\xrightarrow{Granger} \{p,s\}$? Test $H_0\colon \ \beta_{13}=\beta_{23}=0$.

The last one tells you whether interest rates play a role in determining stock prices and sentiment (at least one of them), which seems to be what you are interested in.

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