# Relationship between Regression and Correlation

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?

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.

• So what is your answer to the question? – Richard Hardy Apr 10 '19 at 19:14
• I answered the question in the title Richard – beta1_equals_beta2 Apr 10 '19 at 19:16
• Plug in 1 for $\rho$ if there's perfect correlation – beta1_equals_beta2 Apr 10 '19 at 19:17
• 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.) – whuber Apr 10 '19 at 19:25
• Thanks for the info whuber – beta1_equals_beta2 Apr 10 '19 at 19:32