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I am trying to reconcile an apparent contradiction in the results of my regression against three independent variables $X_1,X_2, X_3 $:

1) For each of the $X_i's$, I regressed $Y|X_i $ and in every case,the slope coefficient $b_i$ in $Y= b_0+b_iX_i$ was negative.

2) When I regressed $Y|X_1, X_2, X_3 $, only two of the (non-constant, i.e., not including $b_0$ ) $b_i$ coefficients came out negative.

But don't we interpret these $b_i's$ to mean that a change in one unit in in $X_i$ in $Y=b_0+b_1X_1+b_2X_2+b_3X_3$ , while holding other $X_i$ to be zero, leads to a change in $b_i$ units of output in $Y$, apparently contradicting 1) ? Also, shouldn't the coefficients in $Y= b_0+ b_1X_1+ b_2X_2+ b_3X_3$ equal to Cov(Y,x_i) , or does this last require that the $X_i's$ are standardized ( so that we would have $b_0=0$, so that the standardized coefficients should all be negative in order to agree with the condition in 1)? Thanks.

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  • $\begingroup$ The issues in your question are essentially those of the marked duplicate. If you have additional questions after that you may want to ask a different question. (Note that the coefficients in the multiple regression will not be the covariance between Y and the corresponding predictor even when standardized) $\endgroup$ – Glen_b -Reinstate Monica Aug 20 '16 at 7:20
  • $\begingroup$ Ok, sorry, I will delete but could you please tell me if the statements in my last paragraph re standardization are correct $\endgroup$ – MSIS Aug 20 '16 at 14:05
  • $\begingroup$ Please don't delete! Duplicates are important to keep, as you asked the question in a different way (i.e. using some different keywords) using a different title. So some people will find your question rather than the linked duplicate -- and then this being here helps to direct them to a place with some good answers. If you delete, those people may not find out there's a good answer. As I read it, the first half of your last paragraph is addressed by the link, and the rest of it is directly addressed in the last sentence of my previous comment. $\endgroup$ – Glen_b -Reinstate Monica Aug 20 '16 at 17:29