A Structural Model For the Johnson & Johnson data, say yt

, shown in Figure 1.1, let xt = log(yt). In this problem, we are going to fit a special type of structural model, xt = Tt + St + Nt where Tt is a trend component, St is a seasonal component, and Nt is noise. In our case, time t is in quarters (1960.00, 1960.25, . . . ) so one unit of time is a year.

(a) Fit the regression model xt = βt

|{z}

trend

+ α1Q1(t) + α2Q2(t) + α3Q3(t) + α4Q4(t)

| {z }

seasonal

+ wt

|{z}

noise where Qi(t) = 1 if time t corresponds to quarter i = 1, 2, 3, 4, and zero otherwise.

The Qi(t)’s are called indicator variables. We will assume for now that wt is a Gaussian white noise sequence. Hint: Detailed code is given in Code R.4, the last example of Section R.4.

(b) If the model is correct, what is the estimated average annual increase in the logged earnings per share?

(c) If the model is correct, does the average logged earnings rate increase or decrease from the third quarter to the fourth quarter? And, by what percentage does it increase or decrease?

(d) What happens if you include an intercept term in the model in (a)? Explain why there was a problem.

(e) Graph the data, xt

, and superimpose the fitted values, say xˆt

, on the graph.

Examine the residuals, xt − xˆt

, and state your conclusions. Does it appear that the model fits the data well (do the residuals look white)?