Is multicolinearity testing altogether useless if the p-value on each regressor is less than 0.01?

Isn't the effect of multicolinearity to artificially increase the standard errors and thus artificially decrease the t-statistics, and thus artificially increase the p-values? If so, if all coefficients of the model have p-values less than 0.01, can we conclude that the model does not suffer from multicolinearity, at least not so severely that we'd care about it? Does the direction of correlation between the multicolinear variables matter in the above conclusion?

• Short answer is you still have to worry about multicollinearity. For one thing, remember that with a large enough sample size almost anything will have p<.01. – zbicyclist Nov 9 '18 at 21:32
• Isn't the reason we test for multicolinearity that we are concerned that we may erroneously (due to multicolinearity) conclude that a variable is not statistically significant when it really is and would be in the absence of multicolinearity? If so, why would we still worry about multicolinearity if all our p-values are less than 0.01 with a very large sample? Which part am I am getting wrong? – ColorStatistics Nov 9 '18 at 21:39
• The primary concern of an analysis is to validly estimate the contributions of various variables. Statistical significance is one of the tools along the way. As you note, multicollinearity increases the error around the beta coefficients (and can even lead to a sign reversal). – zbicyclist Nov 10 '18 at 3:01
• @zbicyclist: I believe you are assuming that the context is causal analysis. It is my fault for not stating explicitly that I was asking the question in a forecasting context. Thus, as I see it the primary concern is that the model provide as accurate a forecast as possible. Whether the coefficients have causal interpretations is not important to me. – ColorStatistics Nov 10 '18 at 3:12
• In a forecasting context, multicollinearity arguably matters less -- but statistical significance of individual variables doesn't matter much at all. The primary criterion is whether the complete forecast model "works", which is given by the performance on a holdout (e.g. one period ahead forecast). – zbicyclist Nov 11 '18 at 18:06