Combining Disjoint OLS Regressions and Efficiently Updating the Model with New Variables

Question

# Combining Disjoint OLS Regressions and Efficiently Updating the Model with New Variables

1. Suppose you run two OLS regression models on disjoint datasets: one of **Y₁ (M×1)** onto **X₁ (M×1)** and the other of **Y₂ (N×1)** onto **X₂ (N×1)**.  
   Both models are **without intercept**.  
   Let the resulting coefficients be **β₁** and **β₂**, respectively.

Now, co-fit another OLS model on the **union** of the two datasets, also without intercept:  
   - You have a linear regression of **Y ((M+N)×1)** onto **X ((M+N)×1)**,  
     where **Y** is a concatenation of *Y₁* and *Y₂*, and **X** is a concatenation of *X₁* and *X₂*.

**Questions:**
   - (a) Given **β₁** and **β₂**, what is the possible range of the coefficient **β** if you run the OLS regression of *Y* onto *X*?
   - (b) What if all three regressions (of *Y₁* on *X₁*, *Y₂* on *X₂*, and *Y* on *X*) are run **with intercept**?  
     Let **β₁**, **β₂**, and **β** be the parameters associated with *X₁*, *X₂*, and *X*, respectively.  
     Given **β₁** and **β₂**, what is the possible range of **β**? (Assume the answer is an interval without gaps.)
   - (c) Again, assume **no intercepts** are fitted in any of the reg

Accepted Answer

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