# Variance of OLS Coefficients with Omitted Variables

I've read in several sources, for example here page 51, that if you omit a relevant variable from an OLS regression the resulting standard errors are smaller. Their arguments make sense but I'm trying to reconcile them with the following, which to me should make the standard errors higher.

Suppose the real model is

$$y = X\beta + Z\delta + \epsilon$$

$$\tilde y = X\beta + \mu$$

where $$\epsilon, \mu$$ are IID white noise.

My argument is that by not including the relevant variable $$Z$$ your mean squared error will be greater. The MSE is also your estimator of the variance of the noise term. The variance of the noise term is part of the estimate of the standard error of the coefficients, therefore the estimated standard error should be higher (and variance estimates will also be biased high).

The other thing is suppose I actually calculate the standard error, which is

$$Var\big((X'X)^{-1}X'\tilde y\big) = (X'X)^{-1}X'Var(\tilde y)X(X'X)^{-1}$$

$$Var(\tilde y) = Var(X\beta + \mu) = Var(\mu)$$

Since $$\mu = Z \delta + \epsilon$$ we have that

$$Var(\tilde y) = Var(Z\delta + \epsilon) = \delta^2Var(Z) + Var(\epsilon)$$

I used the fact that $$Z$$ and $$\epsilon$$ are independent and that since I don't know $$Z$$, it's a random variable.

Either way both of these things point me to the fact that if you omit a variable the standard error of the coefficients should be greater, not lesser. How do I make sense of this?