1. Study on Energy Demand:

Let's consider following energy demand equation:

Model 1 \(: Q_{i}=\hat{\beta}_{0}+\hat{\beta}_{1} I_{i}+\hat{\beta}_{2} P_{i}+e_{i}\)

where \(\mathrm{Q}=\) Demand for Energy (Real Energy Demand, 1973=100)

\(\begin{aligned}

&\mathrm{I}=\text { Income }(\text { Real GDP }, 1973=100) \\

&\mathrm{P}=\text { Energy Price }(\text { Real Price, } 1973=100)

\end{aligned}\)

1. Estimate the above model and explain the economic meanings of estimated coefficients. Which variables are significant?
   Some researches show after oil shock in 1974 , the demand for energy has been changed. To make the intercept dummy variable, you put 0 from 1965 to 1973 and 1 from 1974 to 1982 . To make the slope dummy, multiply the intercept dummy by the oil price. Let's include a slope dummy and an intercept dummy to Model 1. Therefore the model with dummy variables is:
   Model 1a: \(Q_{i}=\hat{\beta}_{0}+\hat{\beta}_{0 d} D_{i}+\hat{\beta}_{1} I_{i}+\hat{\beta}_{2} P_{i}+\hat{\beta}_{3}\left(P_{i}^{*} D_{i}\right)+e_{i}\)
2. Estimate the above model, write the estimated regression equation, and explain the coefficients before and after 1974 . Carefully explain the price and income effects on demand for energy before and after 1974.
3. Perform the t test on \(\hat{\beta}_{0 d}\) and \(\hat{\beta}_{3}\) at \(5 \%\) significant level (include your hypothesis and critical values). Are they significant? What are that meanings?
4. Perform the F test based on null hypothesis of zero for the coefficients of dummy variables, which is \(H o: \beta_{o d}=\beta_{3}=0 .\) State your finding and its meaning.
5. Perform the Chow test for structural changes before and after 1974. Does it consistent with what you find from question 4)?
   Let's consider following different functional forms:
   Model 2: Double log: \(\quad \ln \left(Q_{i}\right)=\hat{\beta}_{0}+\hat{\beta}_{1} \ln \left(I_{i}\right)+\hat{\beta}_{2} \ln \left(P_{i}\right)+e_{i}\)
   Model 3: Semilog: \(\quad Q_{i}=\hat{\beta}_{0}+\hat{\beta}_{1} \ln \left(I_{i}\right)+\hat{\beta}_{2} \ln \left(P_{i}\right)+e_{i}\)
   Model 4: Semilog: \(\quad \ln \left(Q_{i}\right)=\hat{\beta}_{0}+\hat{\beta}_{1} I_{i}+\hat{\beta}_{2} P_{i}+e_{i}\)
6. Estimate above three models using Excel, and write the estimated regression equations and carefully explain the meanings of the estimated coefficients \(\beta_{1}\) and \(\beta_{2}\) for each model.
7. Calculate the price elasticity of the demand for energy for each model including Model 1 and COMPARE the elasticity. If it is necessary, use the mean PRICE \((\mathrm{P}=\bar{P})\) and the mean DEMAND \((\mathrm{Q}=\bar{Q})\).
8. Calculate the income elasticity of the demand for energy for each model including Model 1 , and compare the results. If it is necessary, use the mean INCOME \((\mathrm{I}=\bar{I})\) and the mean \(\operatorname{DEMAND}(\mathrm{Q}=\bar{Q})\).
9. Compare the \(R^{2}\) and adjusted \(R^{2}\) for all models. Which is better by \(R^{2}\) and adjusted \(R^{2}\) ? Any problems to compare the models using \(R^{2}\) or adjusted \(R^{2}\) ? If so, explain how to compare the models.

2. Compare the consequences when we have multicollinearity among independent variables, omitted, and irrelevant variables in OLS.

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