**3. Problem Statement:** Suppose that a population equation is given by: \[ \mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\boldsymbol{\gamma} + \mathbf{u} \] Instead, you estimated the following linear regression model: \[ \mathbf{Y} = \mathbf{Xb} + \boldsymbol{e} \] Assume: \[ \text{E}[\mathbf{u} \mid \mathbf{X}, \mathbf{Z}] = 0 \] It is well known that the estimator of \(\boldsymbol{\beta}\) typically will suffer from omitted variables bias. Derive the bias and state the condition(s) under which the estimator will be still unbiased.

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**3. Problem Statement:**

Suppose that a population equation is given by:

\[ \mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\boldsymbol{\gamma} + \mathbf{u} \]

Instead, you estimated the following linear regression model:

\[ \mathbf{Y} = \mathbf{Xb} + \boldsymbol{e} \]

Assume: 

\[ \text{E}[\mathbf{u} \mid \mathbf{X}, \mathbf{Z}] = 0 \]

It is well known that the estimator of \(\boldsymbol{\beta}\) typically will suffer from omitted variables bias. Derive the bias and state the condition(s) under which the estimator will be still unbiased.
Transcribed Image Text:**3. Problem Statement:** Suppose that a population equation is given by: \[ \mathbf{Y} = \mathbf{X}\boldsymbol{\beta} + \mathbf{Z}\boldsymbol{\gamma} + \mathbf{u} \] Instead, you estimated the following linear regression model: \[ \mathbf{Y} = \mathbf{Xb} + \boldsymbol{e} \] Assume: \[ \text{E}[\mathbf{u} \mid \mathbf{X}, \mathbf{Z}] = 0 \] It is well known that the estimator of \(\boldsymbol{\beta}\) typically will suffer from omitted variables bias. Derive the bias and state the condition(s) under which the estimator will be still unbiased.
Expert Solution
Step 1

Given that,

The population equation is given by,

Y=Xβ+Zγ+u

Y=Xb+e

Eu|X,Z=0 is assumed.

 

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