# Regression in different regimes

I am interested in studying the intercept of a multivariate linear regression model such as:

$$y_t = a + b_1x_{1,t} + b_2x_{2,t} + ... + b_mx_{m,t} + u_t$$ with $$t = 1, 2, ..., n$$

Under two possible scenarios that depend on the value of a variable $$z$$, which is not included in the above model:

• Scenario "High": includes all periods $$t$$ such that $$z_t > \bar{z}$$
• Scenario "Low": includes all periods $$t$$ such that $$z_t \leqq \bar{z}$$

I have thought of simply performing two regressions, one for each scenario, on two subsets of the full sample selected based on $$t$$.

I would like to know whether the above methodology would be acceptable or there are some better alternatives.

## 1 Answer

I assume you are interested in the change of the intercept only the two scenarios. If the variable z is observed with certainty (I.e. you know what variable is and is given within the dataset, as opposed to the second case mentioned below), then use a single regression with intercept (let’s say a) where you have a Dummy that takes value 1 for scenario 1 and 0 for scenario 2. The resulting estimated intercept a will be the one for scenario 2. Then the coefficient of the dummy will denote the additive constant that you add to a to get to the value of the intercept a when scenario 1 occurs. The difference between this approach and yours is that this approach assumes that all the other coefficients are not sensitive to the change of scenario (instead of fitting two different models with two different datasets as you did). So it is fine if, as I anticipated, you just want to study the differential intercept across the two scenario. For more than 2 scenarios, just add more dummies than a single one.

If instead you assume that the regimes are not given and the variable/state driving the changes in scenarios is not observable, then you should look at Dynamic MS regressions like this but, reading your question, I do not think it is your case. So I just attach it in the unlikely case you were referring to this.

• Just in case I need to research this further, is the methodology you described equivalent to a threshold regression? – Gianluca Aug 31 '19 at 13:45
• None of them coincide with a threshold regression, much less the first. The second neither, however, just to give you a sense of what the second is (a general intuitive rough sense), we could compre it to a sort of threshold regression where the threshold depends on a variable that you do not know or observe. In the common threshold regression the variable setting the threshold is a given exogenous variable (not a random state with a certain prob described endogenously by the model). – Fr1 Aug 31 '19 at 13:58