0

I have two observed time series $x_i$ and $y_i$ and I want to test if $x_i$ is a good predictor of of $y_i$. So I run a simple linear regression Y ~ X and use $R^2$ as a measure of goodness of fit.

However $x_i$ and $y_i$ are both noisy observations and don't have constant variance: for each entry i, the uncertainty in $x_i$ is $w_i = var(x_i)$ and uncertainty in $y_i$ is $v_i = var(y_i)$.

So when I look at $R^2$, I want to make sure that observations with the smallest variance $var(x_i)$ and $var(y_i)$ have a higher weight. Intuitively, I want the samples where there is less variance in the observation to have a bigger weight in the regression. How can I achieve this? What should be the right metric / $R^2$ for this problem?

I tried to look into Weighted Least Squares but it seems this is more about interpreting the error residuals and make the regressor BLUE. I also tried dividing both sides by the variance Y/var(Y) ~ X/var(X) but I am not sure of the statistical soundness of this approach.

Ray
  • 1
  • How do you know how large the uncertainty associated with each observation is? – Ryan Volpi Apr 21 '22 at 18:41
  • For example each observation is observed with a different tool so different precision. I want the observations made with the tool with highest precision to have more weight, without discarding all other observations – Ray Apr 21 '22 at 20:11

0 Answers0