# Interpreting coefficients in Linear Regression Y|X1, X2, X3 [duplicate]

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.

• 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) – Glen_b -Reinstate Monica Aug 20 '16 at 7:20
• Ok, sorry, I will delete but could you please tell me if the statements in my last paragraph re standardization are correct – MSIS Aug 20 '16 at 14:05