Including a variable in a regression but NOT estimating its coefficient

Question

I came across a curious problem trying to replicate a paper. The results in the paper were estimated using Eviews, which I am not familiar with. I noticed the author specified (by formula) and estimated an equation (using 2SLS) as follows:

Y = C(1) + X*Z + C(2)*H + ...

I have never encountered such a thing before. No coefficient is estimated for the interaction X*Z, but it is included in the equation. I do not fully understand how this pans out econometrically. I have succesfully replicated the above estimation using Eviews, but I am trying to do the same in Stata. But Stata automatically calculates coefficients for included variables in the regression. I think I am missing something crucial here. What am I looking at here and how could I do this equivalently in Stata?

Here's the paper I am trying to replicated: https://mitcre.mit.edu/wp-content/uploads/2014/03/The-Quarterly-Journal-of-Economics-2010-Saiz-1253-96.pdf

Specifically, we're estimating Equation 3. As is seen, the interaction term includes no coefficient. This is because, according to the author, the interaction term is "known and calibrated in the model".

Answer 1

Let's consider a linear model under the usual assumptions (Gauss-Markov, normal error term, etc):

$$y_i = \beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + \epsilon_i$$

The way we solve for the OLS estimate of $\beta = (\beta_0,\beta_1,\beta_2)^T$ is by solving a multivariate minimization problem where we minimize square loss over all $n$ observations.

$$L(y,\hat{y}) = \sum_{i=1}^n(y_i - \hat{y}_i)^2 = \sum_{i=1}^n(y_i - (\beta_0 + \beta_1x_{i1} + \beta_2x_{i2}))^2$$

We do the usual calculus to find $\beta$ that minimizes $L$.

But maybe we want to set $\beta_1=2$. No problem! We just modify the loss function.

$$L(y,\hat{y}) = \sum_{i=1}^n(y_i - \hat{y}_i)^2 = \sum_{i=1}^n(y_i - (\beta_0 + 2x_{i1} + \beta_2x_{i2}))^2$$

The $\beta = (\beta_0,\beta_2)^T$ that minimizes this loss function is whatever the calculus says minimizes the loss.

Therefore, there is no inherent problem with specifying a coefficient.

However, we want to do more than just say that the minimum is achieved at $\underset{\beta}{\text{argmin}} \sum_{i=1}^n(y_i - (\beta_0 + 2x_{i1} + \beta_2x_{i2}))^2$. We want to calculate the coefficients.

Yes, I would say to follow Nick Cox's advice and minimize $L(z,\hat{z}) = \sum_{i=1}^n(z_i - (\beta_0 + \beta_2x_{i2}))^2$ for $z_i = y_i - 2x_{i1}$. (In other words, make a new response variable.) I am not sure, however, how parameter inference would go in this situation.