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I'm looking for feedback for a design issue.

Say we have a simple OLS model like Y = X1 + X2. Then say we scale all vars by Z, such that the model is Y/Z = X1/Z + X2/Z. Now make a slight change to the model where we replace Z with Z', and the model becomes Y/Z' = X1/Z' + X2/Z'.

The goal is to compare the coefficients from the two models. The big picture is that I am considering creating two samples: one with the variables measured using Z and another using Z'. Then use a Chow test with a twist ... independence is an issue, so I would use an OLS procedure which considers the clusters of observations (in subject and in time, two-way cluster robust SEs). With adjustments for the clusters, I am thinking that the Chow test will accomplish my goal of comparing the coefficients.

I am grateful for any feedback.

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    $\begingroup$ How are $Z$ and $Z^\prime$ determined? In particular, are they estimates or somehow based on the data? Assuming not, the scaling won't affect estimation uncertainties and it won't change the coefficient estimates at all. $\endgroup$
    – whuber ♦
    Commented Dec 8, 2020 at 19:06
  • $\begingroup$ Assume the model is something like Interest Expense = Short term debt + long term debt. Then say Z is the market value of equity, and Z' is total assets. The coefficients are definitely different. That help at all? Thank you for your input! Rick $\endgroup$
    – Rick
    Commented Dec 8, 2020 at 20:00
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    $\begingroup$ That makes it more evident that $Z$ is another variable, not a constant. This raises significant questions about your models, because the implicit additive error terms are also scaled by $Z$ and $Z^\prime.$ Assuming you meant what you wrote about the "simple OLS model," this implies the scaled models are heteroscedastic. This unwelcome complication might be avoidable, depending on what $(X_1,X_2,Y,Z,Z^\prime)$ represent and what you're really trying to achieve. $\endgroup$
    – whuber ♦
    Commented Dec 8, 2020 at 20:25

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The Chow text compares linear regression coefficients for a cut in the data...points to the left...points to the right. This has a problem. We would first have to have a method of clustering the data to group it into left and right clusters. How, in that case, do we know that this clustering did not arise by chance alone? Well, we don't. And in particular there is no evidence or testing to insure that either side; the left cut, or the right cut data is linear.

So, should one be doing that? Consider Comparing methods of measurement: why plotting difference against standard method is misleading by Bland and Altman. The arguments in that paper suggest that local regression of clusters doesn't mean much.

I think that the first consideration is whether the data is plausibly linear or not. So tests of linearity, e.g. cusum seem to me to be a better starting point. Not that that is a necessary procedure. The first thing one should do with data is plot it. Subsequently, we try lots of different fit functions in an ANOVA partial probability sense for each regression coefficient parameter, or alternatively use AIC comparison of models (assuming the sample size is sufficiently large to even approach asymptotic convergence). Whatever best fits, we should investigate further to ascertain if our result is even plausible.

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