Question

Assume the following regression model with two regressors (this model is called the \"true model\"): $Y_i = \beta_1 X_{i1} + \beta_2 X_{i2} + \epsilon_i$, $i = 1, ..., n$. Consider an alternative model, where the variable $X_2$ is omitted: $Y_i = \beta_1 X_{i1} + u_i$, $i = 1, ..., n$. Let $\hat{\beta}_j$ denote the OLS-estimator of $\beta_j$ in model (2) for $j = 1, 2$ and let $\tilde{\beta}_1$ denote the OLS-estimator of $\beta_1$ in model (3). a) In model (3), express $u_i$ as a function of $X_{i2}$ and $\epsilon_i$. Derive $E[u_i|X]$ given that model (2) fulfills the Full Ideal Conditions (FIC). b) Which FIC is violated in model (3)? c) Show that the conditional bias of $\tilde{\beta}_1$ is given by $Bias(\tilde{\beta}_1|X) = \beta_2 \hat{\gamma}$, where $\hat{\gamma}$ is the slope estimate in the regression of $X_{i2}$ on $X_{i1}$ without a constant.

          Assume the following regression model with two regressors (this model is called the \"true model\"):
$Y_i = \beta_1 X_{i1} + \beta_2 X_{i2} + \epsilon_i$,  $i = 1, ..., n$.
Consider an alternative model, where the variable $X_2$ is omitted:
$Y_i = \beta_1 X_{i1} + u_i$,  $i = 1, ..., n$.
Let $\hat{\beta}_j$ denote the OLS-estimator of $\beta_j$ in model (2) for $j = 1, 2$ and let $\tilde{\beta}_1$ denote the OLS-estimator of $\beta_1$ in model (3).
a) In model (3), express $u_i$ as a function of $X_{i2}$ and $\epsilon_i$. Derive $E[u_i|X]$ given that model (2) fulfills the Full Ideal Conditions (FIC).
b) Which FIC is violated in model (3)?
c) Show that the conditional bias of $\tilde{\beta}_1$ is given by
$Bias(\tilde{\beta}_1|X) = \beta_2 \hat{\gamma}$,
where $\hat{\gamma}$ is the slope estimate in the regression of $X_{i2}$ on $X_{i1}$ without a constant.

Show more…

Assume the following regression model with two regressors (this model is called the ẗrue model)̈:
Yi = β1 Xi1 + β2 Xi2 +, i = 1, ..., n.
Consider an alternative model, where the variable X2 is omitted:
Yi = β1 Xi1 + ui, i = 1, ..., n.
Let β̂j denote the OLS-estimator of in model (2) for j = 1, 2 and let β̃1 denote the OLS-estimator of β1 in model (3).
a) In model (3), express ui as a function of Xi2 and . Derive E[ui|X] given that model (2) fulfills the Full Ideal Conditions (FIC).
b) Which FIC is violated in model (3)?
c) Show that the conditional bias of β̃1 is given by
Bias(β̃1|X) = β2 γ̂,
where γ̂ is the slope estimate in the regression of Xi2 on Xi1 without a constant.

Added by Stacey W.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

08/24/2023

Step 1

Rearranging the equation, we get ui = Yi - beta1Xi1. Therefore, u_i = beta2Xi2 + epsilon_i - beta1Xi1. Show more…

Show all steps

lock

Thanks for your feedback!

please show full working with reasoningAssume the following regression model with two regressors (this model is called the "true model"): Y_(i)=eta _(1)x_(i1)+eta _(2)x_(i2)+epsi lon_(i),i=1,dots,n. Consider an alternative model, where the variable x_(2) is omitted: Y_(i)=eta _(1)x_(i1)+u_(i),i=1,dots,n. Let hat(eta )_(j) denote the OLS-estimator of eta _(j) in model (2) for j=1,2 and let tilde(eta )_(1) denote the OLS-estimator of eta _(1) in model (3). a) In model (3), express u_(i) as a function of x_(i2) and epsi lon_(i). Derive E[u_(i)|x] given that model (2) fulfills the Full Ideal Conditions (FIC). b) Which FIC is violated in model (3)? c) Show that the conditional bias of tilde(eta )_(1) is given by Bias(tilde(eta )_(1)|x)=eta _(2)hat(gamma ), where hat(gamma ) is the slope estimate in the regression of x_(i2) on x_(i1) without a constant. Assume the following regression model with two regressors (this model is called the "true model"): Y=BXi1+32Xi2+ei i=1,...,n. (2) Consider an alternative model, where the variable X2 is omitted: Yi=3Xi1+ui i=1,...,n. (3) Let 3; denote the OLS-estimator of ; in model (2) for j = 1,2 and let 3i denote the OLS-estimator of 1 in model (3). a) In model (3), express u; as a function of X;2 and e;. Derive E[u;|X] given that model (2) fulfills the Full Ideal Conditions (FIC). 2 P b) Which FIC is violated in model (3)? 2 P c) Show that the conditional bias of 31 is given by Bias(31|X)=32Y (4) where is the slope estimate in the regression of Xi2 on Xi1 without a constant. 6 P

Play audio

Feedback

Powered by NumerAI

Sri K and 63 other subject Intro Stats / AP Statistics educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Recommended Videos

Recommended Textbooks

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later