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 ErrorNNN-NN-NNNN

Observations 50

ANOVA

df SS MS F Significance F

Regression 2 14931938.3(NNN) NNN-NNNN149 19.87686003 5.14537E-07

Residual 47 18029332.53(NNN) NNN-NNNN

Total 49 32961270.82

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 3054.280348(NNN) NNN-NNNN4.216390382 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 e ects 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(NNN) NNN-NNNN

Observations 50

ANOVA

df SS MS F Significance F

Regression 3 15595748.15(NNN) NNN-NNNN716 14.07002785 1.12578E-06

Residual 46 17365522.67(NNN) NNN-NNNN

Total 49 32961270.82

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 2828.429295(NNN) NNN-NNNN3.832870736 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

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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% con dence 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 justi cation 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 ve 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

values. Replace `?’ with your answer to part `iii.’.

Season Autumn Winter Spring Summer

0.961749 1.088000 1.026178 0.953368

0.946015 1.084399 1.015228 0.944724

? 1.081081 1.007264 0.959233

0.973747 1.064133 1.002364 0.960000

Seasonal averages * 1.079403 1.012759 0.954331

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

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demand for the summer of 2015. Do you have any comment to make about

this forecast?