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• Rana Basheer

# Granger Causality

Updated: Apr 7

Granger causality is one of several tools that are extensively used to figure out cause and effect (causal) relationship from data. However there are some strong assumptions on data that limits the applicability of the Granger causality which will be listed later. Let us now formally introduce this Causality test for a simple linear model.

Let

$X$
$X$
$p$
$t$
$N$
$W$
$W_{p:t}$
$p$
$t$
$W_{p:t}$
$X_{p:t}$
$X$
$X_{t+1}$

$X_{t+1}=\sum_{i=p}^{t}{\left\{\sum_{j=1}^{N}{\left[\alpha_{ij}*W_{ij}\right]} + \beta_i*X_i\right\}}$

In the above equation,

$\alpha_{ij};i\in\left\{p,p+1,\cdots,t\right\}, j\in\left\{1,2,\cdots,N\right\}$
$\beta_i;i\in\left\{p,p+1,\cdots,t\right\}$
$Y$
$Y_{p:t}$
$X_{t+1}$
$\hat{X}_{t+1}=\sum_{i=p}^{t}{\left\{\sum_{j=1}^{N}{\left[\alpha_{ij}*W_{ij}\right]} + \beta_i*X_i + + \gamma_i*Y_i \right\}}$

where

$\gamma_i:i\in\left\{p,p+t,\cdots,t\right\}$
$\hat{X}_{t+1}$
$X_{t+1}$
$Y$
$X$
$\gamma_i=0;i\in\left\{p,p+1,\cdots,t\right\}$
$\gamma_i;\left\{p:t\right\}$

However, some of the caveats of the above method are:

1. The linear system model assumption for the output variable

$X$
1. The normality of the residuals from linear models. F-test is particularly sensitive to normality requirement. Non-normal residuals can skew the results from an F-test.

2. Finally, the minimum number of data samples that are required to verify this causality could be significantly large depending on the number of control variables

$N$
$\left(t-p\right)$

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