Question

Q1. The dataset jj.txt contains quarterly earnings per share for Johnson & Johnson.
There are 84 quarters (21 years) measured from the first quarter of 1960 to the last quarter of 1980.
1. Make a time plot of the Johnson & Johnson data by using R. Describe the data based on the
graph.

2. Take a logarithm of the Johnson & Johnson data. Make a time plot of the log-
transformed data and describe it.

3. Denote the Johnson & Johnson data at time t as vt. Let xt = log(vt). Fit the regression
model xt = βt + α1Q1t + α2Q2t + α3Q3t + α4Q4t + µt;
where Qit = 1 if time t corresponds to quarter i = 1; 2; 3; 4, and zero otherwise. The Qit's are
called indicator variables. We will assume for now that µt is a Gaussian white noise sequence.
(Hint: Below is an example of R codes for this regression model t. You are welcome to use these
to make your own codes. You are strongly recommended to run each line one at a time and verify
the result.)

4. What is the interpretation of the parameters β, α1, α2, α3, and α4 in the following
regression model?
xt = βt + α1Q1t + α2Q2t + α3Q3t + α4Q4t + µt;

5. What are the least-squares estimates of the parameters β, α1, α2, α3, and α4? Interpret
the values of these estimates in the context of the study.

6. What happens if you include an intercept term in the model in part 3? Write an identifiable regression model in this case. Also, what is the interpretation of the regression parameters in this model?

7. Graph the log-transformed data xt, and superimpose the fitted values, say ^xt, on the
graph.

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

Answer 1

We plot the data in Johnson & Johnson using the following code :

Then we get the graph as following:

From the graph, we can conclude that the time series shows an increasing trend with fluctuations increasing with time thus indicating cyclical fluctuations.

Now by taking logarithm of the data we plot the new series and obtain the following graph:

From the above time series plot it is observed that there is only linear trend, seasonal variation and irregular fluctuation present in the series.

The regression is then obtained as the following code:

The regression equation then obtained as :

Xt=1.3819+0.2285*Q1t+0.2463*Q2t+0.3058*Q3t+0.2201*Q4t

Where, βt= intercept of the regression line

α1= amount of change in the data xt due to an unit change in value of quarter 1

α2= amount of change in the data xt due to an unit change in value of quarter 2

α3= amount of change in the data xt due to an unit change in value of quarter 3

α4= amount of change in the data xt due to an unit change in value of quarter 4