Question

This exercise considers monthly prices of shares in JC Penney from January 2000 through the end of 2011.9 The data table includes a column Month with values 1, 2 , ., 120 that can be used as the time index t.
(a) Fit a polynomial trend model to the time series of prices. Try polynomials of various degrees (i.e., try polynomials with t, t2, and other powers up to t6). Do any of these, with orders of 6 or less, capture the ups and downs of the prices?
(b) Compare the ability of a polynomial trend to follow the pattern in this time series to that of two exponentially weighted moving averages (with w = 0.5 and w = 0.9).
(c) Fit an auto regression to the time series of prices, using three lags of the price (yt -1, yt -2, and yt -3). Are all of these lagged predictors useful? Remove predictors that do not significantly improve the fit of the model and summarize your final model. (There’s a lot of collinearity among these predictors, so remove predictors one at a time.)
(d) Does the model you created in part (c) meet the conditions for the MRM?
(e) The data table includes the returns on Sears stock during this period. [Returns are the ratio of the change in price to the price in the earlier period, (price t – price t -1) / price t -1.] Does the sequence of returns appear simple, or do you find a pattern that can be used to forecast future returns?
(f) Compare using the model in part (c) for forecasting the price of this stock in the next month to a method that uses the simplicity of the returns. Which would you prefer to use (if either)? Explain your choice.

Answer 1

(a)

Sixth-degree polynomial claims statistically significant estimates, but model violates assumptions and does not track the peaks in prices. 

 

(b)

The EWMA with w = 0.5 is able to follow the gyrations better than any of the polynomials. It’s a bit rough, but that’s what you get in order to allow the EWMA to swing rapidly up and down later in the time series. With weight w = 0.9, the series lags behind changes in the prices and looks a bit like the 6th order polynomial. 

 

(c)

Only the first lag claims to be statistically significant (see d).

R2		0.958
se		4.417
n		143
Term	Estimate	Std Error	t Statistic	p-value
Intercept	1.1841	0.7947	1.49	0.1385
Lag 1 Price	0.9758	0.0171	57.00	<.0001*

 

(d)

Yes, although the data show some evidence of heteroscedasticity. No evident autocorrelation.

 

(e)

The returns vary less from the second half of 2003 through the first half of 2007, this pattern does not supply a prediction. Neither polynomial trends nor lags offer statistically significant predictors.

 

 

(f)

Returns offer a simple, easy-to-use approach. The forecast price next month is the current price.