# Modeling a time series quantity by modeling its constituent time series

I have a time series target, let's say $$Y_1$$. This quantity depends on two other time-series quantities deterministically, $$Y_2 \text{ and } Y_3$$. That is, we have some function which takes $$Y_2$$ and $$Y_3$$ and computes $$Y_1$$. This function was verified by carrying out simultaneous lab sampling for $$Y_1, Y_2,$$ and $$Y_3$$.

Now, I have created a Dynamic Partial Least Squares Regression model for determining the values of $$Y_2$$ and $$Y_3$$ in between lab samples, using lab sample data for training the model. The R-squared of fit for $$Y_1$$ was 0.78 and for $$Y_2$$ was 0.59.

Next, I try to calculate $$Y_1$$ using the predicted values of $$Y_2$$ and $$Y_3$$ by the function mentioned above (Modelling $$Y_1$$ directly gives bad results). However, the R-squared for $$Y_1$$ for even this type of calculated model was only 0.05, even though the calculation is deterministic and accurate. I am failing to understand why this is happening. Is this simply a matter of propagation of errors for the two models for $$Y_2$$ and $$Y_3$$? Based on the advice here, I am looking to either completely abandon this approach of modeling $$Y_1$$ indirectly, or look into it further.