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?
2 Answers
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).
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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5$\begingroup$ So what is your answer to the question? $\endgroup$ Commented Apr 10, 2019 at 19:14
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$\begingroup$ I answered the question in the title Richard $\endgroup$ Commented Apr 10, 2019 at 19:16
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$\begingroup$ Plug in 1 for $\rho$ if there's perfect correlation $\endgroup$ Commented Apr 10, 2019 at 19:17
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1$\begingroup$ 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 ♦Commented Apr 10, 2019 at 19:25