Interpretation of regression coefficient if independent variable is substracted from dependent variable

Question

What is the correct interpretation for an OLS regression coefficient $b$ if I substract the independent variable $X$ from $Y$ as in:

$$(Y-X)= a + bX + e$$

($Y$ and $X$ follow a normal distribution)?

The full story is the following:

$X$ and $Y$ are two performance measures (e.g., the performance of a firm). The relation between the two measures is $Y$ = $X$ + $DIFF$. All three values are observable.

$X$ contains an unobservable bias $B$ (e.g., the firm wants to show a good performance and exaggerates $X$ using a positive $B$). $Y$ does not contain a bias (however, there still are some unobservable "natural" differences between $Y$ and unbiased $X$, so $DIFF$ is not $B$).

I want to use the observable variables $Y$, $X$, and $DIFF$ to figure out how much of the bias $B$ in $X$ is undone in $Y$ (and, thus, ends up in $DIFF$). In other words, $Y$ = ($X0$ + $B$) + ($DIFF0$ - $B$), where $X0$ and $DIFF0$ are the unobservable “true” values of $X$ and $DIFF$.

In the following OLS regression $$ DIFF = a + bX + e, $$ if the coefficient $b$ moves toward 0, I would interpret that as less bias being excluded from $Y$ (and $b$ should be -1 if the whole bias is excluded from $Y$)

However, I am not sure if my regression approach/interpretation is right here. Therefore, any help is very much appreciated (I am also happy about literature suggestions).

Answer 1

Your proposed model is one which includes an offset: a term in a regression model which has a fixed coefficient. The negative X term on the LHS is equivalent to a positive X term (coefficient +1) on the RHS. But having already adjusted for X, this effectively only subtracts 1 from the b of your regression model, making it moot.

If you wish to show, using a single regression slope coefficient, how one rating "scales up or down" to match a gold standard, the approach you are after is the Concordance Correlation Coefficient from Lin et al 1989. Basically a calibration plot is produced with X and Y scaled to 0,1 intervals (if they have disparate ranges), and a calibration line (regression through the origin) is produced. This is a graphical tool as well. One can often see non-linear effects as well, and describe and report them: consider e.g. that a 5 star expert rating will translate to a 5 star user rating, but a 3 star expert rating may still translate to a 5 star user rating for the favorable reviewers were inclined to share their opinion whereas lukewarm reviewers didn't bother.

Alternately: your formulation of DIFF suggests a measurement error model is needed. You do not describe how DIFF may be a scalar difference or a function of X. Measurement error models have been described in great detail both on this site and online or in other resources, especially as they relate to ratings. I would start with the CCC as an intuitive and useful tool.