time series data : Predict $Y$ with $X$, where $X$ is dependent on $Y$
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Let $Y$ be a target variable which you want to predict on using $X$ (e.g with a statistical model), where $X,Yin mathbb{R}$ . You are given data which looks like this : $$ data_t = (X_t, Y_t), tin {0,1,2,...,n} $$ You want to predict $Y$ using $X$ , but you know that $X$ is not independent, in fact you know that a change in $Y_{t}$ has a causal effect on $X_{t+j}$ , where $j$ is not known but $jgeq 1$ $(1)$ $$ X_t = hat{f}(Y_{t-j}, args) $$ You also know that a change in $X_{t}$ can have a non-causal effect on $Y_{t+j}$ which is not random and not necessary due to a former change in $Y_{t-j}$ $(2)$ $$ Y_t = hat{f}(X_{t-j}, args) $$ It is very likely that patterns in $X_{ito j}$ can determine $Y_{j}$ or $Y_{j+x}$ $x geq 1$ $(3)$ My Questions : $(a)$ Given $(1)...
