How to test if $\hat\beta_1$ changes significantly when $X_2$ is added to the model? I set up a (multinominal) logit model (model 1) and then, holding all else constant, set up a second model (model 2) in which I add a control variable ($X_2$) that did not feature in model 1. The rationale is to test whether the strength of an independent variable of interest, $\hat\beta_1$, changes significantly between model 1 and model 2. I find that adding ($X_2$) in model 2 does decrease the strength of $\hat\beta_1$ compared with model 1, but I am unsure of how to test whether this difference is significant. Until now I am using a crude method: examining whether the confidence intervals for $\hat\beta_1$ overlap between model 1 and model 2. If not, I am interpreting the difference as significant. Yet this test seems harsh: if the confidence intervals marginally overlap, we can surely be more than 95% confident that the true value of $\hat\beta_1$ does not lie at the extremes of both estimates. My question: is there a better way of testing for a significant between $\hat\beta_1$ in model 1 an model 2?
(PS. I estimated the models using the mlogit function in R.)
 A: So there is a way (a couple ways actually) to do this, but it is kind of a pain, so ask yourself how important this really is and how much sense it makes in the context of your research before proceeding.  
Let $\hat \beta^{(1)}_1$ and $\hat \beta^{(2)}_1$ be the coefficients from the the model without and with $X_2$ respectively.  Your hypothesis test is the following:
$$
H_0:\beta^{(1)}_1 - \beta^{(2)}_1 = 0 $$ $$H_1:\beta^{(1)}_1 - \beta^{(2)}_1 \neq 0
$$
Less degrees of freedom adjustments, the z-score test statistic would be of the form
$$
z = \frac{\hat \beta^{(1)}_1 - \hat \beta^{(2)}_1}{\sqrt{\mathrm{var}(\hat \beta^{(1)}_1 - \hat \beta^{(2)}_1)/n}}
$$
To run the test, it is necessary to estimate the variance of $(\hat \beta^{(1)}_1 - \hat \beta^{(2)}_1)$, which can be broken down below as
$$
\mathrm{var}(\hat \beta^{(1)}_1 - \hat \beta^{(2)}_1) = \mathrm{var}(\hat \beta^{(1)}_1) + \mathrm{var}(\hat \beta^{(2)}_1) -2\mathrm{cov}(\hat \beta^{(1)}_1,\hat \beta^{(2)}_1)
$$
$\mathrm{var}(\hat \beta^{(1)}_1)$ and $\mathrm{var}(\hat \beta^{(2)}_1)$ are easily estimated using the standard error's of the coefficients, the hard part is estimating the covariance term, $\mathrm{cov}(\hat \beta^{(1)}_1,\hat \beta^{(2)}_1)$ .  There are two ways I know of:


*

*Estimate both mlogit models simultaneously and use the resulting hessian matrix to estimate covariance:  This would most likely require you to program your own bivariate multinominal likelihood function and optimize over it (not as hard as it sounds). I do not have time to write a likelihood for you, explain how to use the optim function in r, get standard errors from the hessian matrix, and so forth, but this is what you will have to do in this approach.  Perhaps look here for the multinominal log likelihood and here for an example of estimating a logit model using a negative log-likelihood function and optim in R.

*Use a bootstrap to estimate $\mathrm{var}(\hat \beta^{(1)}_1 - \hat \beta^{(2)}_1)$ directly: This will allow you to use the same mlogit you were using before.  However, you will need to employ a bootstrap in which you estimate both models and collect $\hat \beta^{(1)}_{1,g} - \hat \beta^{(2)}_{1,g}$ for $g=1,2,...,G$ iterations .  This will result in a vector of bootstrap estimates from which you can estimate $\mathrm{var}(\hat \beta^{(1)}_1 - \hat \beta^{(2)}_1)$.


There are caveats to both techniques which I refrain from diving into depth here.  The bootstrap is generally considered more robust since it does not rely on asymptotic normally quiet so much, but you need to make sure you are employing the correct bootstrap (i.e. is your data cross-sectional?, auto-correlated?, etc.... these will imply different bootstrap techniques), it is also computationally expensive since it requires you to re-estimate the model thousands of times.  In the end, both techniques are asymptotically equivalent, so given enough data the difference would be inconsequential.
