Estimating Regime-Dependent Regression Coefficients with Separate Fits

Summary

The document explains how to estimate coefficients in a regression whose intercept and slope depend on a binary state indicator. The proposed method splits observations according to the indicator’s lagged value and fits the same regression separately to each subset. The fit for the zero state estimates the baseline intercept and slope; the fit for the one state estimates the sums of the baseline and state-specific coefficients. Subtracting the baseline estimates from the latter gives the state-specific adjustments.

This is a practical way to recover the parameters of a two-regime linear model, such as one used in momentum research. The answer gives no sample data, statistical results, or comparison with a single regression containing interaction terms. Separate fits may also be less precise when one state has few observations, and the method assumes the model is otherwise appropriately specified.

Key ideas

• Split observations by the lagged value of the binary state indicator.
• Fit the same regression separately within each state.
• The zero-state regression estimates the baseline intercept and slope.
• Subtract baseline estimates from the one-state estimates to recover the state-specific adjustments.

Tags

# Don't know if it is obvious, but how do I fit the following model in R?


# Don't know if it is obvious, but how do I fit the following model in R?












From the Paper "momentum crashes", Daniel and Moskowitz

$I_B$ is a dummy Variable which could be either one or zero

Is it possible to regress on two intercepts? or do i get something wrong ? Are there options to create my own linear regression model ?

## Answer by Alex C (score 0, accepted)

https://quant.stackexchange.com/a/46244

What I would do: run the regression twice.

The first time use only the time periods with $I_{t-1} = 0$

The resulting Alpha and Beta will be estimates of $\alpha_0,\beta_0$

Now run the regression a second time, using the other data points, those with $I_{t-1} = 1$. The resulting Alpha and Beta are estimates of $\alpha_0+\alpha_B,\beta_0+\beta_B$. By subtracting the already known $\alpha_0,\beta_0$, you can find $\alpha_B,\beta_B$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.