Coefficient Decreases but Standard Errors stay the Same with Inclusion of Control Variables

Question

I estimate 2 models in OLS.

$Y=\hat{\beta} X+e$ and $Y=\tilde{\beta} X+\gamma W +u$

The inclusion of the $W$ variable decreases the size of $\beta$ but does not change the $Var(\beta)$. $X$ and $W$ are not very collinear, which may explain why the standard errors did not change, but why then does the coefficient on $X$ decrease?

I read this post and was a bit confused on whether it explained my question or not: How are standard errors affected in a multivariate regression?

Edit: To be more specific: The sample size is 86,000. The variance decreases by less than 1% while the coefficient decreases by more than 50%.

Answer 1

The most plausible candidate seems omitted variable bias in that case. If both $X$ and $W$ have a positive effect on $Y$ and are positively correlated, then regressing $Y$ on $X$ only will force $X$ to also "do the work" for $W$, i.e. also reflect its positive impact on $Y$. Once you also include $W$, $X$ can concentrate on its own effect, which brings down its coefficient.

That the standard errors hardly chance in such a large sample reflects the fact that coefficients are estimated very precisely (although very precisely wrong in the first case).

Consider this little example in which the regressors in x are generated as mean zero multivariate normals with positive covariance of 0.4. $X$ has a positive coefficient of 0.3, while that of $W$ is 0.6. The results roughly replicate your findings.

library(mvtnorm)
n <- 86000
sigma <- matrix(c(1,.4,.4,1), ncol=2)
x <- rmvnorm(n=n, mean=c(1,2), sigma=sigma)
y <- .3*x[,1]+.6*x[,2]+rnorm(n,1)
summary(lm(y~x[,1]))
summary(lm(y~x))