On a study of consumption, we have GDP and the prime rate as independent variables from the first quarter of 1978 to fourth quarter' of 2001 . The regression model is:

Consumption \(_{t}=\beta_{0}+\beta_{1} G D P_{t}+\beta_{2}\) \[Primerat{{e}_{t}}\]

Data from:/prob4data,

1. Estimate the model using \(\mathrm{OLS}\), and explain the regression output including estimated coefficients, t test and \(F\) test. Find out which variables are significant.
2. Plot the residual by time, and explain the residual plot whether you find any problem. Do we violate any 7 assumptions of OLS? If so, what are the consequences?
3. Let's assume that we have first order serial correlation. Express the equation for the error terms for both population and sample.
4. Carry out the DW test and state whether you find serial correlation. For the Durbin-Watson test, setup the hypothesis, write the ranges of critical values at 5 percent significant level (explain the rejection, inconclusive, and accept area), and explain the result.
5. Based on your conclusion, is the OLS estimator biased or unbiased, consistent or inconsistent, efficient or inefficient?
6. Give three cases in which DW test is not valid.
7. What are the possible remedies if you find the serial correlation? Explain how to estimate the GLS using the above example.
8. Estimate the model using GLs and compare the results with the first regression model. Plot the residual by time, and check the serial correlation.
9. Let's consider a first differenced model. Estimate the following regression model using first differenced variables;

\(Consumption_{t}-\) \(Consumption_{t-1}\) \(=\beta_{0}+\beta_{1}\left(GDP_{t}-GDP_{\mathrm{t}-1}\right)+\beta_{2}(Primerate_{t}-Primerate_{t-1})+\varepsilon_{t}\)

Estimate the above model, calculate the Durbin-Watson statistic, perform the test, and explain your finding for this model. Has the first differencing dealt with the autocorrelation problem in OLS?

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