Question 18
The Board of Directors investigates whether synergy gains from corporate acquisitions (dependent variable measured in
millions of ) depend on the acquisition premium paid ( ) and the listing status of the target firm (a dummy variable Dp
that takes the value of 1 if the target is a privately held firm and zero if the target is a publicly listed firm). Which of the
following models gives the mean synergy gain if a privately held firm is acquired?
A.
B.
C.
D.

Ask by Gross Davies. in the United Kingdom

Jan 10,2025

Question

Question 18
The Board of Directors investigates whether synergy gains from corporate acquisitions (dependent variable measured in
millions of ) depend on the acquisition premium paid ( ) and the listing status of the target firm (a dummy variable Dp
that takes the value of 1 if the target is a privately held firm and zero if the target is a publicly listed firm). Which of the
following models gives the mean synergy gain if a privately held firm is acquired?
A.
B.
C.
D.

Ask by Gross Davies. in the United Kingdom

Jan 10,2025

UpStudy AI · Accepted Answer

To determine the appropriate model for the mean synergy gain when a privately held firm is acquired, let's analyze the given information and the provided options.

### Given:
- **Dependent Variable**: Synergy gains (measured in millions of $$ £ $$).
- **Independent Variables**:
  - $$ x_1 $$: Acquisition premium paid.
  - $$ D_p $$: Dummy variable where $$ D_p = 1 $$ if the target is a privately held firm and $$ D_p = 0 $$ if the target is a publicly listed firm.

### Objective:
Identify the regression model that accurately represents the mean synergy gain when $$ D_p = 1 $$ (i.e., when a privately held firm is acquired).

### Analysis of Model Options:

1. **Model A: $$ \hat{y} = \beta_0 + \beta_1 x_1 $$**
   - **Interpretation**: This model includes only the acquisition premium ($$ x_1 $$) as a predictor and omits the dummy variable $$ D_p $$.
   - **Implication**: It does not account for whether the firm is privately held or publicly listed.
   - **Conclusion**: **Incorrect**.

2. **Model B: $$ \hat{y} = \beta_0 + \beta_1 x_1 + \beta_2 D_p $$**
   - **Interpretation**: This model includes both the acquisition premium ($$ x_1 $$) and the dummy variable ($$ D_p $$) as predictors.
   - **Implication**: 
     - When $$ D_p = 1 $$ (privately held), the mean synergy gain becomes $$ \beta_0 + \beta_1 x_1 + \beta_2 $$.
     - This correctly captures the additional effect ($$ \beta_2 $$) of acquiring a privately held firm on the synergy gains.
   - **Conclusion**: **Correct**.

3. **Model C: $$ \hat{y} = \beta_0 + \beta_1 D_p $$**
   - **Interpretation**: This model includes only the dummy variable ($$ D_p $$) and omits the acquisition premium ($$ x_1 $$).
   - **Implication**: It does not account for the variability in synergy gains due to the acquisition premium.
   - **Conclusion**: **Incorrect**.

4. **Model D: $$ \hat{y} = \beta_0 + \beta_2 D_p - \beta_1 x_1 $$**
   - **Interpretation**: This model includes both the dummy variable ($$ D_p $$) and the acquisition premium ($$ x_1 $$), but applies a negative coefficient to $$ x_1 $$.
   - **Implication**: The negative relationship with $$ x_1 $$ may not align with theoretical expectations unless specifically justified.
   - **Conclusion**: **Incorrect** unless there’s a strong theoretical rationale for the negative coefficient, which isn’t provided.

### Conclusion:

**Model B** appropriately includes both the acquisition premium ($$ x_1 $$) and the dummy variable ($$ D_p $$), allowing for the calculation of the mean synergy gain when a privately held firm is acquired.

**Answer:**  
**B.** $$ \hat{y}=\beta_{0}+\beta_{1} x_{1}+\beta_{2} D_{P} $$
The correct model is B: $$ \hat{y} = \beta_{0} + \beta_{1} x_{1} + \beta_{2} D_{P} $$.

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