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
9/62
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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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