In the Wiley review book under Multicollinearity, it states that “Multicollinearity does not affect the consistency of OLS estimates and regression coefficients, but makes them inaccurate and unreliable.”
Question: what is definition of ‘consistency’ of estimates vs. accuracy and reliability?
Thanks!
Multicollinearity will be consistently innacurate, that’s how I’d treat it
Sorry, that doesn’t help, but thanks for trying. To ask another way, what makes two regression coefficients consistent but not accurate? Example?
Multicollinearity –> [Higher] Standard errors –> low t- stats – > A variable that is really significant appears Insignificant, therefore making it not accurate.
However, the the defining characteristic of a consistent parameter is the fact that larger sample sizes tend to produce more accurtate estimates (the sample parameter converges to the population parameter), which is still exhibited by the regression coefficients w/ multicollinearty.
That is how I would treat it, but I’d also be interested to perhaps see an example.
The idea of the T-tests is to see if the coefficients are statistically significant explaining the dependent variable. So, to test this we need 2 values, the coefficient per se and the standard error associated to that coefficient.
A regression with multicollinearity problems has big standard errors, so the T-Tests reveal that those coefficients are not statistically significant (some or all of them perhaps) while they are, in fact, significant (error type II).
Why big standard errors? Because, since the independent variables are correlated toguether, the OLS estimate takes into account for each variable more standard error that it deserves (like a contamination from the neighbors) giving an inaccurate and an unreliable calculations. This is like creating an excesive standard error from the nothing.
Thats why we must not violate the OLS assumptions.
Consistency is when the probability distribution of a parameter’s estimator collapses to a single point as the sample size increases. In other words, the standard error for the sampling distribution (of the given estimator) shrinks towards zero as the sample size grows. Therefore, it’s less likely that the value of our estimator is far away from the true parameter value when we increase the sample size (if the estimator is consistent).
Reliability refers to the magnitude and sign of the estimate being correct, and accuracy refers to the standard error being small (more or less, precision of our estimates).
When multicollinearity is a problem, the consistency property of OLS still holds (there’s some math that can be used to show why, but you won’t need it for the exam).
However, the correlation between the (group) of independent variables makes it hard to determine the partial effect of any one variable on the dependent variable. The coefficient estimates are likely to be incorrect in terms of magnitude and possibly in direction.
For example, in a regression of salary (y) on years of education (x1) and age (x2), it is likely that x1 and x2 are correlated. For our purposes, let’s assume the correlation is high enough to pose the issue of multicollinearity (there’s no clear cutoff for “too high” of a correlation, by the way). We would guess that x1 and x2 each have a positive effect on y and both should be significant in a regression (and research tends to show this in some capacity). However, our (hypothetical) estimated regression shows that x1 is actually negative, small, and not significant. This would tip you off that something isn’t right (since the estimate is different than the common logic and other research). This is an example of how multicollinearity could affect the reliability of estimated coefficients (small effect of education and it’s estimated as negative). This also shows how the coefficients are not “accurate”-- the standard error was inflated, meaning we had less precision in our estimates (insignificant t-test).
Hope this helps!