I entirely rewrote the question, since it was marked as unclear.
Suppose I have a linear regression model $Y=b_0+b_1 X_i+\epsilon$ for a time series dataset that consists of $Y$ and $X$.
I can now estimate $b_0$ and $b_1$ for different quantiles of $Y$: $bq_0$ and $bq_1$
Question 1: I have estimated $bq_0$ and $bq_1$ for the 0.1-quantile. Inserting new data for $X_i$ into the estimated regression model gives me with the 0.1 quantile of the new $Y$. Is this correct?
Question 2: To my understanding, quantile regression estimates the quantiles conditional on the specific quantile of $Y$. Is there a way to estimate effects conditional on the quantile of $X$? If so, one could construct a regression model that automatically changes it estimators, if $X$ reaches specific values. For example if $X$ is above the median, $bq_1$ is 1. If $X$ is below the median, $bq_1$ is 0.5.
Question 3: How do you cross validate the result from a quantile prediction for a time series? In the case of OLS for example, you could just compare the actual time series for year X, with the one you predicted for the year X. So if I had predicted one value for each month, I could compare this with the actual value for each month and see if it matches. How would you do this in case of quantile regression? If I predicted the 0.1 quantile for this case, I would have 12 data points that would represent the 0.1 quantile of the predicted distribution. Comparing this with the actual data does not make sense in my eyes.
I am sorry for being a bit confusing first. That is probably due to my lack of understanding of the topic. I hope it is clear what I am asking now.