Question 1.

1. (50 marks) Download the Excel file A2Q1.xlsx from Moodle. The spreadsheet contains observations for each of five variables for 90 American cities:

1. Use Excel to estimate the following relationship:
   \(Y_{i}=\beta_{0}+\beta_{1} X_{1 i}+\beta_{2} X_{2 i}+\beta_{3} X_{3 i}+\beta_{4} X_{4 i}+\varepsilon_{i}\)
   Attach the Excel output. (4 marks)
2. Use EViews to estimate the same relationship. Attach the EViews output. (4 marks)
3. The two outputs should be the same. Write out the estimated model, including the standard errors in parentheses under the estimated coefficients, \(n\), \(R^{2}\) and RSS (use Question 1 from Tutorial 3 as a template). ( 3 marks)
4. Do the coefficients have the signs that you expected? Make sure to explain if you felt any of the signs could have been ambiguous. (8 marks)
5. By comparing just the estimated coefficients, could you say that the mean number of rooms has a stronger influence on the mean market price for houses than the percentage of municipal tax revenue that comes from commercial property? (4 marks)
6. Predict the mean market price for houses for the following two scenarios and comment on the difference. (4 marks)
   Scenario 1: \(X_{1}=0.225, X_{2}=5.718, X_{3}=0.036\) and \(X_{4}=3678\)
   Scenario 2: \(X_{1}=0.099, X_{2}=5.237, X_{3}=0.018\) and \(X_{4}=3044\)
7. Develop and test appropriate hypotheses about the individual slope coefficients at the 10 percent significance level. (10 marks)
8. Develop and test an appropriate hypothesis about the overall fit of the model at the 5 percent significance level. (5 marks)
9. Suppose that all the "classical assumptions" hold. Does this mean that the true marginal effect of a one-unit increase in the mean number of rooms in houses is 50 (ie, \(\\) 50000)$ ? Explain. (4 marks)
10. The value of \(R^{2}\) is relatively low. Does this mean that the regression model cannot be used to predict mean house prices for a given city? Why or why not? (4 marks)

Question 2. An agent wants to find the relationship between the number of weeks houses are on the market prior to sale, $Y$, and the difference between the asking price and the municipal taxable value (in thousands of dollars), $X$. The following basic statistics based were calculated for 14 observations:

\(\begin{array}{lll}

\sum Y_{i}=168 & \sum X_{i}=1624 & \sum Y_{i}^{2}=2436.02 \\

\sum X_{i}^{2}=211856 & \sum X_{i} Y_{i}=22131.7 & \sum e_{i}^{2}=122.255

\end{array}\)

1. Calculate the estimated intercept coefficient, \(\hat{\beta}_{0}\), and slope coefficient, \(\hat{\beta}_{1}\). (6 marks)
2. Write down the estimated regression equation and interpret the value of the slope coefficient. (4 marks)
3. Calculate the estimated error variance, \(\hat{\sigma}^{2}\). ( 3 marks)
4. Calculate the coefficient of determination, \(R^{2}\). Briefly explain what it means. (5 marks)
5. Calculate the estimated standard errors for the intercept coefficient, \(s_{\hat{\beta}_{0}}\), and the slope coefficient, \(s_{\hat{\beta}_{1}}\). (Hint: \(s_{\hat{\beta}_{0}}=s_{\hat{\beta}_{1}} \sqrt{\frac{1}{n} \sum X_{i}^{2}}\).) (4 marks)
6. Develop and test an appropriate hypothesis about the slope coefficient at the 5 percent significance level. (4 marks)
7. Develop a 95-percent confidence interval for the true value of \(\beta_{1}\). (4 marks)

Question 3 (20 marks) The following table comes from a multiple regression model that was run on Excel. An avionics manufacturer wants to establish the relationship between the monthly sales of navigation units, \(Y\), and the number of ads placed in monthly magazines, \(X_{1}\), and the interest rate charged on credit customers, \(X_{2}\).

1. Write down the formulae for each of the highlighted cells. (8 marks)
2. Calculate the values for each of the highlighted cells. (8 marks)
3. Test for the significance of the individual coefficients. (2 marks)
4. Test for the significance of the overall model. (2 marks)

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Solution: The downloadable solution consists of 14 pages, 1287 words and 2 charts.
Deliverable: Word Document