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If $y_1$, $y_2$, and $y_3$ are time series such that:

$$y_1=y_2+y_3$$

Suppose all those variables were regressed against index $x$, so we get coefficients $r_{y_1}$, $r_{y_2}$, and $r_{y_3}$. In general, it should then hold that:

$$r_{y_1}=r_{y_2}+r_{y_2}$$

I expect they should not equate since variance itself is not additive, as discussed in this post here. If so, does that mean that regression is not a linear operator?

If $y_1$, $y_2$, and $y_3$ are time series such that:

$$y_1=y_2+y_3$$

Suppose all those variables were regressed against index $x$, so we get coefficients $r_{y_1}$, $r_{y_2}$, and $r_{y_3}$. In general, it should then hold that:

$$r_{y_1}=r_{y_2}+r_{y_3}$$

I expect they should not equate since variance itself is not additive, as discussed in this post here. If so, does that mean that regression is not a linear operator?

If $y_1$, $y_2$, and $y_3$ are time series such that:

$$y_1=y_2+y_3$$

Suppose all those variables were regressed against index $x$, so we get coefficients $ry_1$, $ry_2$, and $ry_3$. In general, it should then hold that:

$$ry_1=ry_2+ry_2$$

I expect they should not equate since variance itself is not additive, as discussed in this post here. If so, does that mean that regression is not a linear operator?

If $y_1$, $y_2$, and $y_3$ are time series such that:

$$y_1=y_2+y_3$$

Suppose all those variables were regressed against index $x$, so we get coefficients $r_{y_1}$, $r_{y_2}$, and $r_{y_3}$. In general, it should then hold that:

$$r_{y_1}=r_{y_2}+r_{y_2}$$

I expect they should not equate since variance itself is not additive, as discussed in this post here. If so, does that mean that regression is not a linear operator?