I am trying to find the relationship of stock indices across the world. This has been done by the literature, however, I am wondering about the methods chosen.

I have decided to go with what I think is a general to specific approach.

I start with the most basic regression model,

$$Y_t = \alpha +\lambda X_{t-1}+\epsilon_t$$ (1)

I then move to another regression model, the pairwise granger causality test,

$$Y_t = \alpha +\lambda X_{t-1}+\theta Y_{t-1}+\epsilon_t$$ (2)

Finally, I include all of the potential $X's$ in the one regression to consider the links between the variables,

$$Y_t = \alpha +\lambda_1 X_{1,t-1}+...+\lambda_n X_{n,t-1}+\theta Y_{t-1}+\epsilon_t$$ (3)

I can justify why I use equations (2) and (3), but I am having a little bit more difficulty of why I would use equation 1.

My idea was that I should start with the most general framework and see if the relationships hold.

I am not sure if I should just exclude equation (1), is their any ever justification for starting at the most simple regression model as above?

$Y$ and $X$ are returns of different stock markets, i.e. the S&P 500 and FTSE 100.


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