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the goal of my research is to analyze if one variable X follows the movement of another variable Y over time. Meaning that Y is slightly ahead of X. The number of observations in each time series is the same and the variables would be price(logged). Can I use a regression model, in which the dependent variable is X and the independent variable is Y(lagged)? Thank you.

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I agree with the comment: Granger causality is the standard econometric way to test this kind of relationship.

In short, you have to start with understanding AutoRegressive (AR) models: explain $X_t$ thanks to its past a linear way:

$$X_t = c+ \sum_{\ell=1}^L a_\ell X_{t-\ell} +\epsilon_t.$$

Once it is done, you can add past information on $Y$, and thus test the model:

$$X_t = c+ \sum_{\ell=1}^L A_\ell X_{t-\ell}+ \sum_{n=1}^N B_\ell Y_{t-n} +\varepsilon_t.$$

The statistical significance of $B$ terms is what you look for, but first you need to have a good understanding of the model with $a$ only. It will be your benchmark model.

If you are more ''machine learning oriented'' and you do not like hypothesis testing:

  1. First: it is a bad idea; data scientists should be confrotable with hypothesis testing ;{)}
  2. Then, you can bootstrap or cross validate the regression with $A$ and $B$, using a LASSO, ridge of elastic net regression, and look if you have $Y$-variables in the result. But be careful: cross-validation of time series is subtle: you cannot break the time line, you should fold the dataset in consecutive blocks of observations.
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  • $\begingroup$ Should I use ln(price) or ln(return), does it make a difference regarding the outcome of my test? $\endgroup$ – user34031 Jun 26 '18 at 8:45
  • $\begingroup$ Ln(return) is a nonsense (returns can be negative!) $\endgroup$ – lehalle Jun 26 '18 at 21:23

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