Suppose that $Y$ and $X$ have an unknown joint distribution with an arbitrarly complicated unknown C.E.F. $ \; E[Y|X]=\mu (X)$.

Suppose that we want to find the best (in MSE minimization terms) linear approximation of it and the we choose the well known linear projection model : $ \hat \mu (X)=\alpha + \beta X$ with $\alpha=E[Y]-\beta E[X] $ and $\beta = \frac{cov(X,Y)}{var(X)}$ .

Clearly, C.E.F. true error is : $$ e=Y-E[Y|X]$$ and we known that it satisfies the following condition: $$E[e|X]=0 $$ while the linear approximation error is : $$\epsilon=Y-\hat \mu(X) $$ and satisfies the less restrictive condition $ E[\epsilon X] =0$

$$ $$

Neither (1) nor (2) imply indipendence between $e,X$ and $\epsilon,X$ respectively. Also no homoschedastic assumption of errors is made or is necessary.

Now there's a clear difference between $\epsilon$ and $e$ that can be dramatic if C.E.F. is poorly aproximated by an affine function. Furthermore if we can demostrate in some ways that $E[\epsilon|X]=0 $ we meet the uniqueness condition of the $ Y = E[Y|X] + e $ decomposition ( further on this can be found in my previous question ) and then we can identify $\hat \mu (X) $ with C.E.F.

So clearly in the most general way we can write: $$1)\quad Y=\hat \mu(X) + \epsilon \qquad ; \qquad E[\epsilon X]=0$$

Then my question is:

  • Why is usually stated that 1) is an additive error model? We didn't any assumption on error in formulating 1) and it seems a natural way as it appears. Clearly $\epsilon $ and $X$ can be joined and $\epsilon = \epsilon (X)$ so where's the multiplicative error model comes from?

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