Problem 1:

1. Let's revisit a research on the US Phillips curve (the relationship between the unemployment rates and inflation rates). We estimate the following regression model from 1948 to 1970 . Using the regression output, answer all questions (show your work for full credit).

1. Write the regression equation. What are the economic meanings of the estimated coefficients? Does it make sense.
2. Explain how to calculate the TSS (Total Sum of Square), ESS (Explained Sum of Square), RSS (Residual Sum o Square) and their degrees of freedom, and find them from the output.
3. Calculate the R square and adjusted R square, and explain them. What can you say about the goodness of fit?
4. Calculate the sample variance of the estimated residuals \(\left(\mathrm{e}_{\mathrm{i}}\right)\) and the sample variance of the unemployment rate (UNEMP ).
5. Calculate the sample variances of estimated coefficients and explain the sample distribution of the estimated coefficients.
6. According to the model, what is the predicted unemployment rate if the inflation rates are \(10 \%, 5 \%\), and \(1 \%\) ? Show your work how to calculate them?
7. Perform the $t$ tests of the coefficients based on the null hypothesis of zero and alternative hypothesis of nonzero. Are they significant?
8. According to the Phillips curve, the relationship between unemployment and inflation rates should be negative. Prove the theory using a t test.

Problem 2:

2. We obtain the yearly gasoline consumption data from 1975 to 1999 . We have following variables :

REALGAS = Annual real gasoline demand

REALINC $=$ Annual real income

GASPRICE = Annual gasoline price index in major cities

PTRANS = Annual price index of public transportation

POP = Population

Use the data from. Estimate the following regression models using Excel (Excel output must be included with your answer).

Regression model 1 : REALGAS \(=\hat{\beta}_{0}+\hat{\beta}_{1} G A S P R I C E_{i}+e_{i}\)

Regression model 2 : REAIGAS \(S_{i}=\hat{\beta}_{0}+\hat{\beta}_{1} R E A L I N C_{j}+\hat{\beta}_{2} G A S P R I C E_{j}+e_{i}\)

Regression model $3:$ REALGAS \(S_{i}=\hat{\beta}_{0}+\hat{\beta}_{1}\) REALINC \(+\hat{\beta}_{2}\) \[GASPRIC{{E}_{j}}\] \(+{{\hat{\beta }}_{3}}PTRAN{{S}_{j}}+{{\hat{\beta }}_{4}}PO{{P}_{i}}+{{e}_{i}}\)

1. Plot your data (REALGAS vs. REALINC, GASPRICE, PTRANS, POP) and explain them. Calculate the correlation coefficients between REALGAS vs. all independent variables, explain their meanings, and predict the best variable to explain the annual gasoline demands.
2. Estimate above three models using Excel, write the regression equations, and explain the meaning of estimated coefficients for each model.
3. Write the TSS (Total Sum of Square), ESS (Explained Sum of Square), RSS (Residual Sum of Square), R squares and adjusted \(\mathrm{R}\) squares for each model, and compare them.
4. Perform the \(t\) tests based on the null hypothesis of zero and alternative hypothesis of nonzero for each coefficient in model 3 at the significant level of \(90 \%\) and \(99 \%\). Explain what you find from the tests.
5. Explain the relationship between R square and the correlation coefficient between REALGAS and GASPRICE from the Regression Model 1.
6. Does the second model improve the overall fit from the first one? Does the third model improve the overall fit from the second one? Explain which model is the best one.
7. Perform the \(\mathrm{F}\) tests for each model. You need to setup the hypotheses, and describe your verdict.
8. We estimate the regression models using OLS under the 7 classical assumptions. List the assumptions with brief descriptions.

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