# Determining the shape of the relationship between $y$ and $x$ after passing the Granger causality test

Let's say you have two time series, and you have already established a Granger-causal relationship between them. E.g. when testing if $X$ Granger-causes $Y$ (with a lag of 1), we calculated a $p$-value of 0.01. The results of our test mean that the $\beta X_{t-1}$ term in the equation

$$Y_t = \alpha Y_{t-1} + \beta X_{t-1} + \varepsilon$$

provides statistically significant information about $Y_t$, compared to the original equation

$$Y_t = \alpha Y_{t-1} + \varepsilon.$$

Here's my question. The equation $Y_t = \alpha Y_{t-1} + \beta X_{t-1} + \varepsilon$ has an $R^2$ value of 0.2. This means that the data does not fit well to this regression line. Is there a better way to determine a function to calculate $Y_t$?

• I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? – Richard Hardy Feb 24 '17 at 14:24

Also, an $R^2=0.20$ may be considered either low or high depending on the context. For example, if the noise is much stronger than the signal in the true data generating process (as if often the cases with financial returns, for example), you would not expect a high $R^2$ from a good model. (Actually, a high $R^2$ would then be a sign of overfitting.)
Given this, the remaining question is how to determine a good model for $y_t$ based on $x_t$. This is a very general question, probably too general to have a concise yet useful answer.