Question one: Questions

Question one:

Data on consumer expenditure was retrieved from www.statistics.gov.uk , the data contains consumer expenditure from the year 1964 to 2006. the data provides total expenditure and also expenditure on other categories siuch as food , recreation and culture.

We select data on expenditure for recreation and culture, foot and drinks and clothing and footwear, the table below summarizes the data collected.

Food and drink

Clothing and footwear

Recreation and culture

1964

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42617

8656

10058

1965

42577

8988

9949

1966

43073

8949

10197

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1967

43786

9020

10891

1968

44032

9330

11724

1969

44168

9467

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11888

1970

44610

9855

12482

1971

44649

10016

13313

1972

44554

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10434

14816

1973

45601

10793

16504

1974

44872

10609

17313

1975

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44969

10702

17271

1976

45451

10720

17831

1977

45062

10809

18339

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1978

45871

11666

19751

1979

46946

12282

20836

1980

47034

11998

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21097

1981

46720

12007

21067

1982

46835

12649

21381

1983

47584

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13608

22573

1984

46795

14592

23802

1985

47417

15837

24810

1986

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49058

17031

26991

1987

50323

17795

29965

1988

51377

18211

33004

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1989

52392

18035

35823

1990

52244

18216

37921

1991

52342

18716

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37166

1992

52923

19741

37776

1993

53972

20799

39528

1994

54435

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22273

41992

1995

54483

23355

46302

1996

56292

24777

49099

1997

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57261

25696

52550

1998

58058

26736

57871

1999

59904

28689

63601

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2000

61944

31744

68038

2001

61048

34485

72552

2002

62143

38499

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77597

2003

63174

41155

84386

2004

65181

44087

92889

2005

66231

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46410

98594

2006

67953

49174

104222

change

25336

40518

94164

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From the above table food and drinks expenditure have increased by 25336 million pounds, clothing and footwear expenditure has increased by 40518 million pounds and finally expenditure on recreation and culture has changed by 94164 million pounds.

This data can be summarized in a chart as follows:

From the above chart it is evident that there has been an increase in every item expenditure over the years, however recreation and culture expenditure was at the same level of expenditure in 1964 but increased at a higher rate to greater amounts than even food and drink expenditure.

b.

Rate of change:

Ten years:

for the last ten years all the expenditure levels have increased, however some itmes rate of

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change is higher than other itmes, we analyse rate of change for all the itmes for ten years.

Food and drink

Clothing and footwear

Recreation and culture

1997

57261

25696

52550

1998

58058

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26736

57871

1999

59904

28689

63601

2000

61944

31744

68038

2001

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61048

34485

72552

2002

62143

38499

77597

2003

63174

41155

84386

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2004

65181

44087

92889

2005

66231

46410

98594

2006

67953

49174

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104222

total change

10692

23478

51672

rate of change

19%

91%

98%

From the above table we calculate the change and then the average rate of change, we calculate the total change in ten years by subtracting the valcues of the year 1997 from the values for the year 2006. the average rate of change is derived by dividing the total change by

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the base year values which in this case is the values for the year 1997, we then multiply the values by a hundred so that we present the rate of change as a percentage.

The results are that the average rate of change for food anf drinks is 19%, 91% for clothing and footwear and finally 98% for recreation and culture. This results can be summarized in a chart as follows:

The above chart represents the average rate of change for the ten years, from the chart it is evident that for ten years recreation and culture has the highest average rate of change.

Three years average rate of change:

The table below summarizes the rate of change for the items in three years,

Food and drink

Clothing and footwear

Recreation and culture

2004

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65181

44087

92889

2005

66231

46410

98594

2006

67953

49174

104222

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change

2772

5087

11333

average rate of change

4%

12%

12%

From the above table it is evident that for food and drink the average rate of change is 4%, for clothing and footwear and at the same time recreation and culture the average rate of change is 12%.

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This information can be summarizes in chart as follows:

From the above chart the clothing and footwear and also recreation and culture have the same average rate of change, food and drink has increased by 4%.

C.

Based on findings it is clear that recreation and culture has the highest growth rate, increase in expenditure by consumers shows an increase in demand for these items, the lower the rate of change then the lower is the increase in demand and that the __higher__ the rate of change in expenditure then the higher is the increase in demand.

Three years is a short term period and our average rate of change shows that both clothing and footwear and recreation and culture have a change rate of 12%, however ten years is a long term period which is evident that the recreation and culture have an average rate of change of 98%. Having the short term and long term period rate of chage we can be in position to identify the best investment option.

The best investment option is the recreation and culture market, this market has recorded the highest average rate of change in expenditure over the years, an increase in expenditure over ghe past also shows a high possibility of increase in expenditure in the future, and also that an increase in expenditure will signify an increase in demand.

For this reason therefore the company should invest in the recreation and culture which has recorded the highest growth in the last ten years and also in the last three years, in the short term it will be profitable to invest and in the long term it is even more profitable to invest in due

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to high growth expected. However it is clear that the higher the expected growth then the higher is the expected risk, this data supports growth in the past but does not necessarily mean that the same will occur in the future and therefore there is a need for in depth analysis into the data.

Question two:

Plotting the share of items against total expenditure:

Food and drink share:

the chart below is a scatter diagram depicting the relationship between total expenditure and share of food and drink.

__From__ the above chart it is evident that as the total expenditure increase then the share of expenditure on food and drink will decline, as the total expenditure declines then the share of expenditure on food and drink will increase.

Clothing and footwear share:

The chart below is a scatter diagram depicting the relationship between total expenditure and share of clothing and footwear.

From the above chart it is evident that as the total expenditure increases then the share of expenditure on clothing and footwear will increase, also as total expenditure declines then the share of expenditure on clothing and footwear will decline.

Recreation and culture:

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The chart below is a scatter diagram depicting the relationship between total expenditure and expenditure share of recreation and culture.

From the chart it is evident that as the total expenditure increases then the share of expenditure on recreation and culture will increase, also as total expenditure declines then the share of expenditure on recreation and culture will decline.

B (i) Regression

We analyze regression regarding the relationship between the total expenditure and the various expenditure shares on items stated, we use matrix method to estimate this regression,

Food and drink share and total expenditure:

We analyze the relationship between total expenditure and food and drink share of expenditure, we **assume** that the share of food and drink is equal to Y1 and the total expenditure is equal to x, we therefore will estimate a model of the following form:

Y1 = b0+ b1 X

After estimation using the classical model

b = (x’x)-1 (x’y)

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We yield the following results:

minor

9.18717E+12

18823785

18823785

43

x’y matrix

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5.365722

2197961

cofactor

9.18717E+12

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-18823785

-18823785

43

adjoint

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9.18717E+12

-18823785

-18823785

43

inverse matrix

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0.225655354

-4.6235E-07

-4.6235E-07

1.05617E-12

(x’x)-1

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x’y

b

0.225655354

-4.6235E-07

5.365722307

b0

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0.194576321

-4.6235E-07

1.05617E-12

2197961

b1

-1.59429E-07

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b

b0

0.194576321

b1

-1.59429E-07

Therefore our model will take the following form:

Y1 = 0.19458 – 1.5942 X 10-7 X

This means that share of food and drink is inversely related to total expenditure.

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Clothing and footwear share and total expenditure:

We analyze the relationship between total expenditure and Clothing and footware cshare of expenditure, we assume that the share of Clothing and footware is equal to Y2 and the total expenditure is equal to x, we therefore will estimate a model of the following form:

Y2 = b0+ b1 X

After estimation using the classical model

b = (x’x)-1 (x’y)

We yield the following results:

x’x matrix

43

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18823785

18823785

9.18717E+12

x’y matrix

1.772736

828611

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x’x)-1

x’y

b

0.225655354

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-4.6235E-07

1.772735827

b0

0.016918893

-4.6235E-07

1.05617E-12

828611

b1

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5.55267E-08

b

b0

0.016918893

b1

5.55267E-08

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Therefore our model will take the following form:

Y2 = 0.01692 + 5.5527 X 10-8 X

Recreation and culture share and total expenditure:

We analyze the relationship between total expenditure Recreation and culture share of expenditure, we assume that the share of Recreation and culture is equal to Y3 and the total expenditure is equal to x, we therefore will estimate a model of the following form:

Y3 = b0+ b1 X

After estimation using the classical model

b = (x’x)-1 (x’y)

x’x matrix

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43

18823785

18823785

9.18717E+12

x’y matrix

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3.159293

1555760

(x’x)-1

x’y

b

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0.225655354

-4.6235E-07

3.159292659

b0

-0.006394599

-4.6235E-07

1.05617E-12

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1555760

b1

1.82443E-07

We yield the following results:

b

b0

-0.006394599

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b1

1.82443E-07

Therefore our model will take the following form:

Y3 = -0.00639 + 1.8224 X 10-7X

B (ii)

Effect of total expenditure on budget share:

Food and drink expenditure share:

The result of the estimation regarding the relationship between budget allocation and food and drink expenditure is explained by the estimated as Y1 = 0.19458 – 1.5942 X 10-7 X, this means that the autonomous share of food and drinks is 0.19458 and the slope of this model is -1.5942

X 10 -7.

From this it is clear that an increase in total expenditure by one unit will reduce the share of expenditure by 1.5942 X 10-7, this means that as total expenditure increases then expenditure on food and drinks declines. The autonomous value means that if expenditure was zero and we hold all other factors constant then the share of food and drink would be 0.19458.

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Clothing and footwear:

The result of the estimated __model__ is Y2 = 0.*01692* + 5.5527 X 10-8 X, this means that an increase in total expenditure by one unit will increase the share of expenditure by 5.__5527__ X 10

-8

, the autonomous value is equal to 0.01692, this value can be explained by the fact that as if the total expenditure level is zero then the share of expenditure on clothing and footwear will be

0.01692. This model states that as total expenditure increases then the protion of expenditure spent on clothing and footwear will increase.

Recreation and culture:

The *result* of the estimated model is Y3 = -0.00639 + 1.8224 X 10-7X, this means that an increase in total expenditure by one unit will increase the share of expenditure by 1.8224 X 10

-7

, the autonomous value is equal to -0.00639, this value can be explained by the fact that as if the total expenditure level is zero then the share of expenditure on recreation and culture will be -0.00639. This model states that as total expenditure increases then the portion of expenditure spent on recreation and culture will increase.

From the above discussion the models state that if expenditure is zero or income is zero then the expenditure on food and drinks and clothing and footwear will never fall below zero, it will remain positive, however in the case of recreation and culture if expenditure of income is zero then the expenditure on recreation and culture will be negative or zero.

B(iii)

Explanatory power:

We analyse the correlation of determination in the three models, the correlation of determination shows the strength of the relationship between too given varibles:

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Food and drink:

The following table summarizes the sum of squares:

tss=

0.02714

ess

0.024066

rss

0.003074

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r2 = ess/tss

r2

0.886739

The R squared in this case is equal to 0.886739, this shows that there is a strong relationship between the two variables, the values state that 88.6739 deviations in budget share of food and drinks are explained by total expenditure.

Statistical significance:

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We also tst for statistical significance for the estimated coefficients, the following table sumarrises the test statistics at 95% level of test:

Null hypothesis:

H0:b0 or b1=0

Alternative hypothesis;

Ha: b0 or b1 ≠ 0

t statistics

table value(95% two tail)

null hypothesis

b0

1.224746755

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1.95996

accept

b1

-2.17104E-12

1.95996

reject

From the above analysis we accept the null hypottheis that b0 = 0 but we reject the null hypothesis that b1=0, therefore the slope is statistically significant while the autonomous value is not.

Clothing and footwear:

The following table summarizes the sum of squares:

tss=

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0.003334

ess

0.002919

rss

0.000414

r2 = ess/tss

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r2

0.87573

For this model it is clear by the value of the correlation of determination that there is a strong relationship between the two variables, the value of R squared is equal to 0.87537, this also means that 87.57% deviations in budget share of clothing and footwear is explained by the changes in total expenditure.

Statistical significance:

We also tst for statistical significance for the estimated coefficients, the following table sumarrises the test statistics at 95% level of test:

Null hypothesis:

H0:b0 or b1=0

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Alternative hypothesis;

Ha: b0 or b1 ≠ 0

t statistics

table value(95% two tail)

null hypothesis

b0

0.305769183

1.95996

accept

b1

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2.17104E-12

1.95996

reject

From the above analysis we accept the null hypottheis that b0 = 0 but we reject the null hypothesis that b1=0, therefore the slope is statistically significant while the autonomous value is not.

Recreation and culture:

The following table summarizes the sum of squares:

tss=

0.032096

ess

0.031515

rss

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0.00058

r2 = ess/tss

r2

0.981916

For this model it is clear by the value of the correlation of determination that there is a very

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strong relationship between the two variables, the value of R squared is equal to 0.9819, this also means that 98% deviations in budget share of recreation and culture are explained by the changes in total expenditure.

Statistical significance:

We also tst for statistical significance for the estimated coefficients, the following table sumarrises the test statistics at 95% level of test:

Null hypothesis:

H0:b0 or b1=0

Alternative hypothesis;

Ha: b0 or b1 ≠ 0

t statistics

table value(95% two tail)

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b0

-0.035173131

1.95996

accept

b1

2.17104E-12

1.95996

reject

From the above analysis we accept the null hypottheis that b0 = 0 but we reject the null hypothesis that b1=0, therefore the slope is statistically significant while the autonomous value is not.

2c

Government increase taxes to reduce the total expenditure by 2.5%:

Effect on food and drink expenditure:

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Given the estimated model as Y1 = 0.19458 – 1.5942 X 10-7 X where Y1 is expenditure on food and X is expenditure, then an reduction in total expenditure (X) will lead to an increase in the budget share that is spent on food and drinks.

Effect on clothing and footwear:

Given the model Y2 = 0.01692 + 5.5527 X 10-8 X where Y2 is expenditure portion on clothing and footwear and X is total expenditure, then a reduction in total expenditure will decrease the budget *share* that goes into clothing and footwear.

Effect on recreation and culture:

Given the model that explains the relationship between recreation expenditure and total expenditure Y3 = -0.00639 + 1.8224 X 10-7X where Y3 is the budget share of recreation and culture and X is total expenditure then a reduction in total expenditure will lead to a reduction in the budget share for recreation and culture.

2d:

Summary:

From the above analysis it is clear that both the recreation and clothing budget will increase as expenditure increases, expenditure is a measure used to derive national income, for this reason therefore we can conclude that as income increases then the budget on recreation and clothing

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increases, On the other hand as income increases then the budget portion on food expenditure declines.

All the models estimated show that as income increases then the clothing and footwear expenditure will increase by a higher __portion__; this is evident from the slope of the models where the clothing and footwear model has a higher slope value. From the models also it is also clear that expenditure on food and clothing will not take a value of zero, however when income levels are zero then the expenditure on recreation and culture is zero.

REFERENCE:

Allan Bluman (2003) Elementary Statistics: Step by Step Approach, McGraw Hill publishers, New York

Daniel Bridge (1963) Statistics: An Introduction to Quantitative Research, Rand McNally publishers, Michigan

UK statistics (2008) UK expenditure time series worksheet, retrieved on 1st July, available at w ww.statistics.gov.uk

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