Identifying a $\min\{x_1,x_2\}$ in a regression

Question

How does one go about identifying if a complementary relationship (in the economic sense) between $x_1 $and $x_2$ exists in the determination of $y$.

i.e. $$y=\beta_o+\beta_1\min\{x_1,x_2\}+\mu$$

what tests have to be done here and can this be estimated using OLS?

_{*R code would be appreciated.}

Answer 1

This answer is going to build on the suggestions made by @jbowman and @AlexR in the comments on the question.

1. As @jbowman pointed out, you can simply define $x_3=\min\{x_1,x_2\}$ and estimate the model $y=\beta_0+\beta_1x_3+\mu$.
   • OLS is a linear estimation technique and cannot directly incorporate non-linear relationships. E.g. if your regression is $\ln{y}=\beta_0+\beta_1x^2+\mu$, then the way to actually estimate the model is by regressing $\ln{y}$ on $x^2$, where the logarithm of $y$ and the square of $x$ are computed before estimating the model.
   • You might want to think about omitted-variable bias. If you define $x_3=\min\{x_1,x_2\}$, then $x_3$ will likely be correlated to some other functions of $x_1$ and/or $x_2$. If you don't account for these other functions in your model, they will be "left" in the error term and can cause your estimates to be biased. I would try to regress $y$ on $x_3$, $x_1$, $x_2$ and $x_1x_2$. Then you can test (i) whether the coefficients for $x_1$, $x_2$ and $x_1x_2$ are individually or jointly significant and (ii) whether the coefficient for $x_3$ is still significant after including these additional variables.
2. @AlexR. then suggested to break down the minimum function into its components, $\min\{x_1,x_2\}=\frac{1}{2}(x_1+x_2-\vert{x_1-x_2}\vert)$, and to hence regress $y$ on $x_1$, $x_2$ and $\vert{x_1-x_2}\vert$.
   • Note that this regression will not estimate the $\beta_1$ coefficient in your original model directly. Since $$\begin{align} y&=\beta_0+\beta_1\min\{x_1,x_2\}+\mu \\ & =\beta_0+\beta_1\frac{1}{2}x_1+\beta_1\frac{1}{2}x_2+\beta_1(-\frac{1}{2})\vert{x_1-x_2}\vert+\mu \\ & = \gamma_0+\gamma_1x_1+\gamma_2x_2+\gamma_3\vert{x_1-x_2}\vert+\mu \end{align}$$ the estimated coefficients for $x_1$ and $x_2$ ($\hat\gamma_1$ and $\hat\gamma_2$) need to be multiplied by $2$ and the estimated coefficient for $\vert{x_1-x_2}\vert$ (that is, $\hat\gamma_3$) needs to be multiplied by $-2$ in order to obtain the estimate for $\beta_1$.
   • I can't formally prove this right now, but this specification is less likely to suffer from omitted-variable bias than the one suggested by @jbowman since it already contains the components that would be correlated with the output of the minimum function.
   • You can then test the following hypotheses either jointly or individually $$\begin{align} H_0:\;&\gamma_1=\gamma_2 \\ & \gamma_1=-\gamma_3 \end{align}$$ If your model is specified correctly*, then you would expect the tests to have a hard time rejecting the null.
3. I wrote above "if your model is specified correctly". Yes, that is one aspect that is necessary for the hypothesis test to avoid rejecting the null. However, the estimation on this particular model can also be very sensitive to the data in your model.
   • E.g. if you have $x_2>>x_1$ in 99% of your sample, then $\min\{x_1,x_2\}=x_1$ in 99% of your sample. Thus, OLS will likely "figure out" that the way to minimize the sum of squared residuals is to simply set $\hat\gamma_2\approx0,\hat\gamma_3\approx0$ and to essentially treat the model as if it was simply linear in $x_1$ and "live with the consequences" in the 1% of the sample where this is not the case. In order for your estimation to work as expected, you will need a healthy share of both cases.
   • Another example: If you have $x_1 \approx x_2$ in a large portion of your sample, then the results of your estimation might be numerically unstable and have very high standard errors due to multicollinearity (since $x_1 \approx x_2$ will lead not only to $corr(x_1,x_2) \approx 1$, but also to $\vert{x_1-x_2}\vert \approx 0 = const.$ and thus correlated with the intercept). In order for your estimation to work as expected, you will thus also need a healthy number of cases in your sample where $x_1$ and $x_2$ significantly differ from each other (in both directions, as explained in the previous bullet).