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Intuitive Question

Suppose I'm given a set of $k$ time-series $\{X_t^1,\dots, X_t^k\}$. Is there a way to determine how much of each series is dependent on the others.

Formal Question

More precisely if for each $j \in \{1,\ldots,k\}$, $X^j_t$ is a stochastic process with values in $\mathbb{R}$, $\varepsilon_t^j$ is a random variable in $\mathbb{R}$ and define the loss function: $$ \mathfrak{L}_j(a_1,\ldots,a_j):= \int \left( 1-\sum_{i\neq j} a_i \right) \sum_{i \neq j} (a_i X^i_t - X_t^j)^2 \, dt $$

Then how can we determine the best $a_i$ minimizing: $$ X_t^j = \sum_{i \neq j} a_i X_t^i + a_j\varepsilon_t^j, $$ where $\{a^1,\ldots,a^j\}$ are non-negative real numbers such that $$ \sum_i a_i =1 $$ is a convex combination; how can we find the best $a_i$; that is how can we determine the $a_i$ minimizing $\mathfrak{L}(a_1,\dots,a_k)^j$?.

If it makes things simpler we may assume that the processes are time-series...

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Just estimate regression equations, e.g. $$X_1=\theta_0+\theta_1X_2+...+\theta X_k+\epsilon.$$ Then change dependent variable and estimate $$X_2=\phi_0+\phi_1X_1+...+\phi X_k+\mu.$$ And so on.

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