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Suppose that I have data $\left\{ (x_i, y_i, z_i ) : i=1, 2, \dots, N\right\} $. I have fitted two linear models: $$ \left[\begin{matrix} z_1\\ z_2\\ \vdots\\ z_N \end{matrix}\right]=\left[\begin{matrix} 1&x_1\\ 1&x_2\\ \vdots&\vdots\\ 1&x_N \end{matrix}\right]\left[\begin{matrix} a_1\\ b_1 \end{matrix}\right] $$ and $$ \left[\begin{matrix} z_1\\ z_2\\ \vdots\\ z_N \end{matrix}\right]=\left[\begin{matrix} 1&x_1&y_1\\ 1&x_2&y_2\\ \vdots&\vdots&\vdots\\ 1&x_N&y_N \end{matrix}\right]\left[\begin{matrix} a_2\\ b_2\\ c_2 \end{matrix}\right].$$ That is, the first model uses $x_i$ values and the second one uses both $x_i$ and $y_i$ values to explain $z_i$.

Now I am considering whether the estimated coefficients $b_1$ and $b_2$ ('slopes') are statistically significant (that is, are the $x_i$:s significant to the model). First of all, I am not sure how to get started with that that problem. Secondly, is it possible that $b_1$ would be significant and $b_2$ would not, or vice versa?

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    $\begingroup$ Sure they can, if you search our site, you'll find many examples of people asking about cases when exactly this happened. $\endgroup$
    – Tim
    Nov 2, 2022 at 19:00
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    $\begingroup$ It seems like this is a really trivial question. Testing the significance of a regression coefficient is covered in most elementary stats books - i.e. Kutner, Nachtsheim, Wasserman, Neter "Applied Linear Regression Models". $\endgroup$
    – AdamO
    Nov 2, 2022 at 20:15

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If the columns of the design matrix $X_{N\times(p+1)}$ are linearly independent, $b_{1}$ and $b_{2}$ coefficients indicate the amount of change in the responses $z_{i}$ while other predictors are held fixed. In the case where there is only a single predictor, this amount of change can be interpreted as the slope of the 2D regression line. In the case where there are 2 predictors, $a_{2}$ and $b_{2}$ in your coefficient vector together represent the 3D regression plane. Thus, $b_{1}$ and $b_{2}$ are surely significant and both help to explain the response vector $z$.

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  • $\begingroup$ what is $p$ ? Here there are only 2 or three covariates... Besides, your conclusion "b1 and b2 are surely significant" seems to disagree with @Tim's comment under the post. Can you elaborate on this? $\endgroup$
    – utobi
    Nov 2, 2022 at 20:25
  • $\begingroup$ $p$ is the number of columns aside from the 1 vector. I am not sure what he means by statistically "significant" but changing the values of $b_{1}$ and $b_{2}$ will change the response, indeed. $\endgroup$ Nov 2, 2022 at 20:28

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