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In a typical regression scenario, when two independent variables are collinear, this will cause standard errors of the coefficients to be inflated. What happens if there is a perfect correlation between one of your independent variables and response variable?

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Both regression and correlation implicitly assume the following linear model: $$ y = X\beta + \varepsilon $$

Where y is your dependent variable, the columns of X are the independent variables, $\beta$ is the vector of coefficients and $\varepsilon$ is noise.

The only way that you can have a perfect correlation is if there is no noise, i.e. $\varepsilon=0$. And since you have no noise, that also means that you can have no uncertainty about your coefficient estimates in this case, so your standard errors are all 0 (assuming X is full-rank).

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If you're talking about regression with one predictor:

$$ E(Y_i | X = x_i) = \alpha + \beta x_i$$

Then the least squares regression coefficient estimate is

$$ \hat {\beta} = \frac{ \rho \times \sigma_y }{ \sigma_x } $$

$\rho$ is the correlation between x and y, and the sigmas are the two standard deviations.

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    $\begingroup$ So what is your answer to the question? $\endgroup$ – Richard Hardy Apr 10 '19 at 19:14
  • $\begingroup$ I answered the question in the title Richard $\endgroup$ – beta1_equals_beta2 Apr 10 '19 at 19:16
  • $\begingroup$ Plug in 1 for $\rho$ if there's perfect correlation $\endgroup$ – beta1_equals_beta2 Apr 10 '19 at 19:17
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    $\begingroup$ The question appears to be about the standard errors of the parameter estimates. (If it were only about the relationship between correlation and regression it would be closed as a duplicate: that discussion is well covered on this site.) $\endgroup$ – whuber Apr 10 '19 at 19:25
  • $\begingroup$ Thanks for the info whuber $\endgroup$ – beta1_equals_beta2 Apr 10 '19 at 19:32

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