# Do residuals depend on the equation used for estimation?

I am considering a simple decomposition of share prices, in an attempt to calculate idiosyncratic price movements.

Suppose we have a stock with price $$S$$. Then, one can think about the price as follows:

$$\Delta S = \beta_1 * \Delta M + \varepsilon_1$$

Where $$M$$ represents "market-wide factors" (say the change in the index).

One can also decompose the price as follows:

$$\Delta S = \beta_2 * \Delta M' + \beta_3 * \Delta Sec + \varepsilon_2$$

Where $$\Delta Sec$$ is the change in the sectoral index, and $$\Delta M'$$ is the residual change in the market index, accounting for changes in the sector (say $$\Delta M = \beta_4 * \Delta Sec + \Delta M'$$)

My question is: Do the residuals only equate under certain conditions and certain estimation methods?

• You can answer this yourself simply by comparing the two expressions for $\Delta S$ under the assumption $\varepsilon_1=\varepsilon_2.$ Think about what that implies for the coefficients and variables in the model.
– whuber
Dec 19, 2022 at 14:58
• @whuber after a bit of thought, I got what you have written. But I'm now curious about the state of the residuals if we were to estimate the two aforementioned equations using OLS. Dec 19, 2022 at 15:00
• Little can be said about that due to the absence of any real information about the data and the generality of the models. If we take your "say..." assumption to heart, plug that into the first model and equate the two.
– whuber
Dec 19, 2022 at 15:07