The Cochrane-Orcutt method is a great preliminary methodthat estimates the parameters for the simple linear regression model with first-order autocorrelated errors given in Eq. (14.2). The attached data set gives a hypothetical time series of market share percentage and price. We expect that as price increases, fewer people buy, making the market share decrease, and and vice versa. What would be great to know is how to quantify this change and correlation. This is where you come in.

Problem 1: Fit a simple linear regression model to these data. Plot the residuals versus time. Is there any indication of autocorrelation?

Problem 2: Use the Durbin-Watson test to determine if there is positive autocorrelation in the errors. What are your conclusions?

Problem 3: Use one iteration of the Cochrane-Orcutt procedure to estimate the regression coefficients.

Problem 4: Find the standard errors of these regression coefficients. Is there positive autocorrelation remaining after the first iteration? Would you conclude that the iterative parameter estimation technique has been successful?

Problem 5 [Theoretical]: Given the process x(t) = a + bt + e(t), where e(t) is a white noise process with known variance, show that y(t) = x(t) - x(t-1) is a stationary time series.