Multicollinearity in simple linear regression (not multiple)?

Question

I am doing a simple linear regression analysis with 1 independent variable. I am checking data against assumptions. As I am checking against Tolerance and VIF level, I get the their values equal to 1 (both case). Therefore, I guess I shouldn't check against multicollinearity, right?

Answer 1

There is not much reason to expect multicollinearity in simple regression indeed. Multicollinearity arises when some regressor can be written as a linear combination of the other regressors. If the only other regressor is the constant term, the only way this can be the case is if $x_i$ has no variation, i.e. $\sum_i(x_i-\bar{x})^2=0$. In that case, $x_i=c$ for all $i$, so that the regressor is just $c$ times the constant, so you get multicollinearity.

Then, the OLSE of the slope parameter is not well-defined: $$ \hat\beta_1=\frac{\sum_i(x_i-\bar{x})(y_i-\bar{y})}{\sum_i(x_i-\bar{x})^2} $$ Such a sample might look as follows: Heuristically, it is quite intuitive that OLS breaks down here - $\hat\beta_1$ is supposed to tell you what happens to $y$ if $x$ changes. If your sample does not contain any variation in $x$, it is just not informative about that question.

To give an applied example, suppose you want to investigate the effect of additional labor market experience on earnings but all you observe is employees with exactly 15 years of experience - that sample is not going to tell you anything about the relationship of interest.