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Let's address the two questions in the images you uploaded:

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### Problem 2: Showing Serial Uncorrelation of \( e_{it} \)

In the random effects model, the composite error is given by:

\[
u_{it} = \mu_i + v_{it}
\]
where:
- \( \mu_i \) is uncorrelated with \( v_{it} \),
- \( v_{it} \) has constant variance \( \sigma^2_v \) and is serially uncorrelated.

You are asked to show that the error term \( e_{it} \), defined as:

\[
e_{it} = u_{it} - \lambda \bar{u}_i
\]
is serially uncorrelated, where:

\[
\lambda = 1 - \left[\frac{\sigma^2_v}{\sigma^2_v + T \sigma^2_\mu}\right]^{1/2}
\]

**Solution steps**:
1. **Definition of \( e_{it} \)**: Notice that \( e_{it} \) removes some of the average \( \bar{u}_i \), where:
   \[
   \bar{u}_i = \frac{1}{T} \sum_{t=1}^{T} u_{it}
   \]
2. **Serial correlation structure**: You need to compute the covariance between \( e_{it} \) and \( e_{is} \) (for \( t \neq s \)) and show that it is zero.

To do this:
- Use the fact that \( \mu_i \) is constant over time and that \( v_{it} \) is serially uncorrelated. This simplifies your task because you only need to check whether the inclusion of the term \( \lambda \bar{u}_i \) introduces any correlation between \( e_{it} \) and \( e_{is} \).
- Verify that all cross-products of \( v_{it} \) and \( v_{is} \) result in 0 due to serial uncorrelation, and the terms involving \( \mu_i \) cancel each other out.

Thus, you will show that the error terms \( e_{it} \) and \( e_{is} \) are indeed uncorrelated.

---

### Problem 3: Estimating the Airfare Model

We are asked to estimate the following model:

\[
\log(fare_{it}) = \eta_t + \beta_1 \text{concen}_{it} + \beta_2 \log(\text{dist}_{i}) + \beta_3 [\log(\text{dist}_{i})]^2 + a_i + v_{it}
\]
where \( \eta_t \) allows for different year intercepts. The steps are:

1. **(i) Pooled OLS Estimate**:
   - Estimate the model by Pooled OLS including year dummies to account for \( \eta_t \).
   - If \( \Delta \text{concen} = 0.10 \), the estimated percentage increase in \( \log(fare) \) is approximately \( \beta_1 \times 0.10 \), as the coefficient \( \beta_1 \) gives the semi-elasticity of \( \log(fare) \) with respect to \( \text{concen} \).

2. **(ii) Random Effects (RE) Estimation**:
   - Estimate the equation using Random Effects. The RE estimator allows for variation between groups (airlines, cities, etc.) and within groups over time.
   - Compare the estimate of \( \beta_1 \) from RE to the OLS estimate. Typically, RE accounts for both time-invariant and time-varying effects, so the estimate may be different.

3. **(iii) Fixed Effects (FE) Estimation**:
   - Now, estimate the equation using Fixed Effects (FE), which controls for all time-invariant heterogeneity.
   - The FE estimator of \( \beta_1 \) should focus on within-unit (within-airline, within-city) variations.
   - Typically, FE and RE estimates can be similar if the between-group variation is not significant.

4. **(iv) Hausman Test**:
   - Use the Hausman test to compare the RE and FE models. The Hausman test checks whether the RE estimates are consistent, or if FE should be preferred. If the p-value is small, it suggests FE is better suited as RE violates the assumption that the individual effects are uncorrelated with the explanatory variables.

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### Next Steps:

1. Would you like more detailed steps for calculating the variance and correlation for \( e_{it} \)?
2. Do you need help with the regression in part 3?
3. Are you interested in more details on how to conduct a Hausman test?
4. Should we explore specific interpretations of \( \beta_1, \beta_2 \), and \( \beta_3 \) in this context?
5. Would you like assistance with coding the regression in a software package like Stata or R?

**Tip**: In econometric models, always check assumptions about error terms (like serial correlation or homoskedasticity) to ensure unbiased, efficient estimates.

2. In a random effects model, define the composite error uit = μi + vit, where μi is uncorrelated with vit and vit have constant variance σv^2 and are serially uncorrelated. Define eit = uit − λūi, where λ is λ = 1 − [σv^2 / (σv^2 + Tσμ^2)]^(1/2). Show that the eit are serially uncorrelated.

3. Use the data in airfare.dta for this exercise. We are interested in estimating the model log(fareit) = ηt + β1 concenit + β2 log(disti) + β3 [log(disti)]^2 + ai + vit, where ηt means that we allow for different year intercepts.

Learn how to prove the serial uncorrelation of errors in a random effects model with a detailed explanation. Explore the estimation of airfare models using Pooled OLS, Random Effects, Fixed Effects, and the Hausman test.

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