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Possibly a basic question but am doubting myself.

I have iid samples $(Y_i,X_i,W_i,Z_i)$. I am interested in performing the two following regressions:

$Y = \alpha_0+\alpha_1 X+\alpha_2 Z$

$Z = \beta_0+\beta_1 X+\beta_2 W$

If I use sample splitting to calculate the estimates for the coefficients of each regression on non-overlappins datasets, will my determined coefficients $\hat{\alpha},\hat{\beta}$ be independent? I believe the answer is yes since they are determined using iid data. Will this also hold asymptotically? Am I allowed to use say that $cov(\hat{\alpha},\hat{\beta}) =0$?

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  • $\begingroup$ Hi: There may be other problems ( because of the random regressors ) but you've got Z as both a predictor and a response. In econometrics, you would need what is called a simultaneous equations model. I'm not sure what it's called in statistics but most likely it's also called that. Simultaneous equations is a huge topic so I won't try to explain it here. Any decent econometrics text will devote a chapter to it. This link gives a short introduction. www3.wabash.edu/econometrics/EconometricsBook/chap24.htm As I said, there may be other issues that others can see. $\endgroup$
    – mlofton
    Commented Aug 4 at 14:13
  • $\begingroup$ Hi, thank you for your answer. The problem I am working on is exactly that, but now I am wondering if the regression coefficients obtained using two sets of independent data yield independent coefficients themselves. I realize that if i were to use the same data I could have problems, but with non overlapping one? $\endgroup$
    – xcesc
    Commented Aug 4 at 14:41
  • $\begingroup$ I've never dealt with simultaneous equations but the lack of independence problem with Z being on both sides does not go away because of the non-overlapping data. I would first go through the theory of simultaneous equations ( it's non-trivial ) and get that straight. Note that the problem of endogeneity ( which is what simultaneous equations is really about ) is there with fixed regressors and you've got random regressors so there definitely can be other issues also. I'm no expert in this so hopefully someone else can say something more useful. $\endgroup$
    – mlofton
    Commented Aug 4 at 15:09

1 Answer 1

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Functions of independent variables are independent: $X\perp Y\implies f(X)\perp g(Y)$. If your samples are independent, estimates based on non-overlapping subsamples are thus independent.

This holds for any sample size, and thus holds asymptotically as well.

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