Negative fitted values in OLS regression I am running a regression where my dependent variable is a cross-section of variances. Therefore, I require my predicted values (fitted values) to be positive.
However, when running a simple OLS regression, a small percentage of my fitted values are negative, which is non-intuitive in this case (since variance cannot be negative).
Please note that approximately, my dependent variable is distributed according to a Chi-square distribution.
The output that I need from the regression are the fitted values in the original scale, as well a closed form expression of the MSE (Mean Square Error) of these fitted values.
Is there a way to impose a lower bound on the predicted values?
 A: 
I am running a regression where my dependent variable is a cross-section of variances. Therefore, I require my predicted values (fitted values) to be positive.

Then don't fit a model that doesn't obey such an obvious requirement...

However, when running a simple OLS regression,

... like, you know, OLS.

Please note that approximately, my dependent variable is distributed according to a Chi-square distribution.

Or rather, since population variances are usually not $1$, it should probably be approximately $\sigma^2$ times a chi-square -- so why not model it as, say a Gamma random variable (the distribution of a multiple of a chi-square)?
So why not use a GLM for this problem? All your fitted values are guaranteed to not go negative. See the example here (however, if you fit a straight line model, predicted values can - indeed, must - still go negative outside the data).

Is there a way to impose a lower bound on the predicted values?

If you fit a model for the mean such that the mean will remain positive (log-link, say, rather than identity-link) then out-of-sample predictions will obey the positivity restriction.
If you're modelling variances, the identity link usually won't make sense anyway. Choose one of the others, and the model - fitted and predicted - will stay positive.
A: The easiest way is to fit $z_t=\ln{y_t}$ instead of $y_t$. This way you get the fitted values as $\hat{y}_t=e^{\hat{z}_t}$, always positive.
UPDATE:
If you are assuming errors are normal $\zeta_t\sim\mathcal{N}(0,\sigma_z)$, then using lognormal properties, you can get $\sigma_{y,t}=(e^{\sigma^2}-1)e^{2\hat{z_t}+\sigma^2}$. Here, MSE will depend on the fitted value. You have to think carefully what is MSE in this case. This is asymmetric distribution, and also because of the log it compresses errors for higher values.
Without normal error assumption, you can estimate MSE by obtaining residual values $\hat r_t=y_t-e^{\hat z_t}$, then computing the MSE $\sigma^2_y=Var[\hat r_t]$. This is assuming that variance is constant.
Here's another interesting reference: Granger, Newbold, The Journal of the Royal Statistical Society B 38, 1976, 189–203. "Forecasting Transformed Series", it's Chapter 23 here. The method I described here is called "naive" in this paper.
Note, also, that this is all relevant to forecasting. If your goal is not forecasting but analysis then it's a bit different story.
