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Suppose I am running a regression

$$x_t = \alpha + b_1y_{1t} + \dots + b_m y_{mt} + \varepsilon_t$$

where the $y_{i}$ are potentially linearly correlated (Some have an IVF bigger than 4; generally lower than 5 though)

Is the estimation and standard error of $\alpha$ affected by the multicollinearity problem? Clearly the estimates for $b_i$ will be, but I don't know about $\alpha$.

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    $\begingroup$ What happens when, say, $y_1$ is a constant? $\endgroup$
    – whuber ♦
    Commented Jan 16, 2016 at 14:56
  • $\begingroup$ @whuber, doesn't that illustrate a different point? Multicollinearity between $y_i$s themselves and multicollinearity between $\alpha$ and $y_1$ is not quite the same, is it? $\endgroup$
    – Richard Hardy
    Commented Jan 16, 2016 at 15:28
  • $\begingroup$ @whuber Uhm I don't know.. maybe $b_1 = 0$ and it works out? I am not an expert on this but if you regress on a constant you should not find any relationship right? $\endgroup$
    – Ant
    Commented Jan 16, 2016 at 15:28
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    $\begingroup$ @Richard On the contrary, it is exactly the same thing. $\alpha$ merely is the coefficient of a constant regressor (equal to $1$). If there is a linear combination of the other regressors that is close to being a constant, then there will be collinearity problems. The most blatant form of this phenomenon is perfect collinearity, which will be exhibited when one of the regressors itself is a constant, which is why I suggested contemplating that situation. $\endgroup$
    – whuber ♦
    Commented Jan 16, 2016 at 17:24
  • $\begingroup$ @whuber, oh, you are right and spot on, as usual! $\endgroup$
    – Richard Hardy
    Commented Jan 16, 2016 at 17:33

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