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I'm pretty sure I'm misunderstanding something quite obvious here but I'm rather confused.

I have multiple time-series that have been smoothed with a gaussian kernel. My goal is to regress the time-series from a condition 3 ($y_t$) on the time-series from conditions 1 & 2 ($x^1_t$ and $x^2_t$).

More formally, this would be my regression setup:

$$ \tilde{y_t} = \beta_0 + \beta_1\tilde x^1_t + \beta_2\tilde x^2_t + \epsilon_t. $$

Both $\tilde x^1_t$ and $\tilde x^2_t$ are autocorrelated (because of the smoothing), but is it a problem when running the linear regression? To what extent will it influence the recovery of my $\beta$s knowing that my errors are therefore probably serially autocorrelated?

As both $\tilde x^1_t$ and $\tilde x^2_t$ are strongly correlated, I have been advised to remove the mean of my dependent and independent variables (i.e. $(\tilde x^1_t + \tilde x^2_t + \tilde y_t)/3$) from my independent and dependent variable before running the linear regression. For example, define $\tilde y^*$ as

$$ \tilde y^* = \tilde y_t - (\tilde x^1_t + \tilde x^2_t + \tilde{y_t})/3 $$

and run the regression on $\tilde y^*$. I don't see how this might help and find it strange to "inject" the dependent variable as one of my predictors.

This is my first question here and I hope it's not too confusing nor I missed anything. I have tried looking over other posts (such as this one) but couldn't really draw any link with my problem at hand.

Richard Hardy
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Thomas
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  • It could be helpful to distinguish between raw and smoothed values e.g. by writing $x$ and $\tilde x$ ($\tilde x$), respectively. Are you interested in the coefficients of $y_t=\beta_0+\beta_1 x_{1,t}+\beta_2 x_{2,t}+\epsilon_t$ but you are running $y_t=\tilde\beta_0+\tilde\beta_1\tilde x_{1,t}+\tilde\beta_2\tilde x_{2,t}+\tilde\epsilon_t$ instead? Also, I do not fully follow the part starting with I have been advised to..., especially to "inject" the dependent variable as one of my predictors. – Richard Hardy Mar 14 '24 at 17:30
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    There's some discussion and illustrative example of smoothing making spurious correlation (and hence, spurious regression) worse here: https://stats.stackexchange.com/questions/133155/how-to-use-pearson-correlation-correctly-with-time-series/133171#133171 – Glen_b Mar 15 '24 at 02:21
  • Thanks for this useful link. I also edited the question, it is hopefully clearer now. – Thomas Mar 15 '24 at 08:53
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    Your tildes were flying quite high, so I lowered them for you. I hope you do not mind. – Richard Hardy Mar 15 '24 at 09:03
  • I was thinking about looking at cointegration here, could this be one way to solve my issue ? – Thomas Mar 19 '24 at 09:02
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    Are your series integrated? If not, there is no place for cointegration. – Richard Hardy Mar 19 '24 at 09:10

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