The data gives the monthly sales for a cosmetic manufacturer (yt) and the corresponding monthly sales for the entire industry (xt).

Cosmetic Sales Data

Month    x_i        y_i
1          5      0.318
2        5.06    0.33
3        5.12   0.356
4        5.1     0.334
5       5.35    0.386
6       5.57    0.455
7       5.61     0.46
8        5.8     0.527
9       6.04    0.598
10     6.16     0.65
11     6.22    0.655
12     6.31    0.713
13     6.38    0.724
14     6.54    0.775
15     6.68     0.78
16     6.73    0.796
17     6.89    0.859
18     6.97     0.88

The units of both variables are millions of dollars.

(a) Build a simple linear regression model relating company sales to industry sales.

Apply OLS method to the model and obtain the residuals. Plot the residuals against time. Is there any indication of autocorrelation?

(b) Apply the Durbin-Watson test to detect whether there is positive autocorrelation in the errors. What is your conclusion?

(c) Apply a suitable transformation on the data to rectify the problem of the autocorrelation. Examine the procedure and verify that the autocorrelation problem has been rectified after the transformation.