Same SE for all regression coefficients I am using R to run a linear regression with interaction terms on a small data set(~40 observations, 5 explanatory variables).
My explanatory variables are coded in a way that they can take values -1,0 or 1.
There are different conditions(combinations of input variables) that are more common than others. (f.e. the case [v1=0,v2=0,v3=0,v4=0,v5=0] has 7, the case [-1,-1,1,1,-1] has only one occurrence in the data)
After inspecting the standard errors of the regression summary, I noticed that all the SE's are roughly the same. 
For example:
Variable 1           0.13716
Variable 2           0.13943
Variable 3           0.14018
...
This is the first time that I encountered such similar SE's. I assume it may be related to the low numbers of observations and possible values for each variable. So i might not actually be that unexpected.
But are there any general interpretations for such cases? I was especially asked whether or not it isn't strange for the SE's to be similar if the number of occurrences of conditions differ.
 A: With linear regression, the covariance matrix for the estimated effects is used to extract the standard errors. This has a very simple formula:
$ \text{Cov}(\hat \beta) = \hat \sigma^2 (X^TX)^{-1} $
We have $\hat \sigma^2$ as the scalar estimated residual variance and $X^TX$ is covariance of the predictors multiplied by $n$, the number of samples. We compute the standard errors for each coefficient by simply taking the square root of the diagonal of $\text{Cov}(\hat \beta)$.
How does this relate to your question? Well, if the standard errors of all the regression parameters is very similar, this implies that the entries of the diagonal of $\text{Cov}(\hat \beta)$ are all very close, which is the same as the diagonals of $(X^T X)^{-1}$ being similar values. Why might this happen? One scenario in which this might happen (but not the only one!) is if all the columns of $X$ are very similarly distributed, both in terms of variance of each column and the covariance with other columns. 
At this point, I'm merely speculating about what your data might be, but given that all the variables take on the same values (-1, 0, 1), it seems quite likely that the "similarly distributed predictors" hypothesis is true. 
