Is a huge F-statistic OK? I ran a multivariate regression model in R and got an overall F-statistic of 3,525.690 and three stars attached to it. This seems to be quite good from a statistical point of view.
However, should I be concerned about something or is this really that good?

Let $n$ be the sample size.
Define $RSS$ to be the residual sum of squares for the model, and define $TSS$ to be the total sum of squares. That is:
$$
RSS = \sum_{i=1}^n (y_i - \hat y_i)^2\\
TSS = \sum_{i=1}^n (y_i - \bar y)^2
$$
Now define $p$ to be the number of parameters in the model (including the intercept).
Then: $$F = \dfrac{
\dfrac{
TSS-RSS
}{
p-1
}
}{
\dfrac{
RSS
}{
n-p
}
} = \dfrac{(TSS-RSS)(n-p)}{RSS(p-1)}
$$
If $n$ is large and $p$ is small, then even a small difference between $TSS$ and $RSS$ (so low $R^2$) would correspond to a large $F$-statistic.
