9. The vice president of purchasing for a large national retailer has asked you to prepare
an analysis of retail sales by state. Data are available for the following variables:
Y (retsal) = Per capita retail sales in $
X1 (perinc) = Per capita personal income in $
X2 (unempl) = Unemployment rate in %
X3 (totpop) = State population in 000s
Excel regression output of a potential model is:
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.673063624
R Square 0.453014642
Adjusted R Square 0.430223585
Standard Error 612.871189
Observations 50
ANOVA
df SS MS F Significance F
Regression 2 14931938.3 7465969.149 19.87686003 5.14537E-07
Residual 47 18029332.53 375611.0943
Total 49 32961270.82
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 3054.280348 724.3827234 4.216390382 0.000109151 1597.811292 4510.749404
unempl -86.25168104 40.20459701 -2.14531888 0.037015057 -167.0884398 -5.414922307
perinc 0.253683705 0.048149492 5.268668342 3.2101E-06 0.156872664 0.350494746
(a) Comment on the effects of unemployment and per capita personal income.
(b) You think the prediction equation can be improved by adding state population
as an additional explanatory variable. You obtained the following output:
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.687861707
R Square 0.473153727
Adjusted R Square 0.439525242
Standard Error 607.8480121
Observations 50
ANOVA
df SS MS F Significance F
Regression 3 15595748.15 5198582.716 14.07002785 1.12578E-06
Residual 46 17365522.67 369479.2058
Total 49 32961270.82
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 2828.429295 737.9401735 3.832870736 0.000375307 1343.885177 4312.973414
unempl -71.32832666 41.40025242 -1.722895936 0.091481858 -154.6148904 11.95823702
perinc 0.272491364 0.049773611 5.474615138 1.66335E-06 0.172359776 0.372622951
totpop -0.024730373 0.018450316 -1.340376626 0.186566538 -0.061847621 0.012386875
4
i. Is this model better? Why/why not?
ii. For this model, write out an expression for sales.
iii. For this model, calculate a 95% confidence interval for predicted sales,
if unemployment is 8.1%, per capita income is $15,000 and the state’s
population is 6 million. Use a z-value of 1.96.
(c) Write down two additional explanatory variables which you think could help to
explain sales. Give a brief justification for each.
(25 marks)
10. (a) Time series are usually considered to have a combination of four components.
What are these components? For each of them, give one example of data for
which you would expect that component to be present.
(b) The following table gives average UK household electricity demand in kilowatt
hours (kWh) over the last five years. Quarter 1 represents Spring.
Year Q1 Q2 Q3 Q4
2005 4.5 4.1 4.4 5.1
2006 4.9 4.6 4.6 5.3
2007 5.0 4.7 4.8 5.5
2008 5.2 5.0 5.1 5.6
2009 5.3 5.1 5.2 5.7
i. State two features about household electricity demand that are apparent
from these data.
ii. Show that the 4-point centred moving average for Quarter 3 in 2007 is
5.025.
iii. Calculate the ratio-to-moving-average (R2MA) for Quarter 3 in 2007.
iv. Compute the four seasonal indices using the following table of R2MA
v. The estimated trend line is found to be:
^y = 4:461 + 0:050x;
where x is the Quarter number (Q1 of 2005 corresponds to x = 1). Provide a forecast, to three decimal places, for average UK household electricity demand for the summer of 2015. Do you have any comment to make about this forecast?