In a multiple linear regression, why is it possible to have a highly significant F statistic (p<.001) but have very high p-values on all the regressor's t tests?

In my model, there are 10 regressors. One has a p-value of 0.1 and the rest are above 0.9



For dealing with this problem see the follow-up question.

As Rob mentions, this occurs when you have highly correlated variables. The standard example I use is predicting weight from shoe size. You can predict weight equally well with the right or left shoe size. But together it doesn't work out.

Brief simulation example

RSS = 3:10 #Right shoe size
LSS = rnorm(RSS, RSS, 0.1) #Left shoe size - similar to RSS
cor(LSS, RSS) #correlation ~ 0.99

weights = 120 + rnorm(RSS, 10*RSS, 10)

##Fit a joint model
m = lm(weights ~ LSS + RSS)

##F-value is very small, but neither LSS or RSS are significant
summary(m)

##Fitting RSS or LSS separately gives a significant result. 
summary(lm(weights ~ LSS))

