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.