Econometrics

Econometrics:

1. Present the data in a table showing the region and crime alongside the values for each of the variables. Make sure the full definitions and sources of each variable are given

The data below is from the UK website www.crimestatistics.org.uk , we will choose robbery and theft and handling stolen goods, the data in for crime rates can be found in table 6.03 while the detection rates are in table 7.03.

crime rates

detection rates

Police force area and region

Burglary

Theft and handling stolen goods

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Burglary

Theft and handling stolen goods

Cleveland

17

46

13

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19

Durham

12

25

14

22

Northumbria

12

34

12

23

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Cheshire

11

31

16

18

Cumbria

9

27

13

24

Greater Manchester

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20

45

9

15

Lancashire

11

34

17

23

Merseyside

16

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41

16

19

Humberside

21

49

10

15

North Yorkshire

10

29

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13

21

South Yorkshire

17

40

16

20

West Yorkshire

19

44

15

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19

Derbyshire

12

30

12

18

Leicestershire

12

34

13

20

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Lincolnshire

11

30

9

18

Northamptonshire

17

42

12

21

Nottinghamshire

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24

55

8

13

Staffordshire

11

31

16

23

Warwickshire

12

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32

18

19

West Mercia

11

28

15

24

West Midlands

16

38

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10

15

Bedfordshire

10

32

15

19

Cambridgeshire

12

39

15

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16

Essex

9

31

12

17

Hertfordshire

11

34

21

19

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Norfolk

8

28

13

20

Suffolk

8

26

14

20

London, City of3

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+

+

25

17

Metropolitan Police

14

56

13

10

Hampshire

9

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32

17

19

Kent

10

31

10

17

Surrey

8

24

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11

14

Sussex

10

35

11

15

Thames Valley

12

40

14

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18

Avon and Somerset

14

40

10

12

Devon and Cornwall

9

29

13

19

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Dorset

9

32

18

19

Gloucestershire

13

35

12

20

Wiltshire

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8

25

15

20

Dyfed Powys

6

17

21

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2. Consider the following equation:

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CRi =  b0 + b1 DRi +  b2 TYPEDUMi + ui

Where i denotes region and the variables correspond to those listed above.

(i) What type of functional form is being used here?

The type function used in this case is a simple regression model known as a bivariate regression function, the reason why it is a bivariate regression function is because it contains an autonomous value b0 and one independent variables DRi although this will be the case where we will be considering burglary, when we consider theft and handling stolen goods the function will contain two independent variables because the dummy variable will be positive and therefore we will consider a multiple regression model.

(ii) What sign do you expect b0 to be?

b0 is likely to be positive because the crime rate will always be higher or equal; to the detection rate.

(iii) What sign do you expect b1 to be?

We expect b1 to be positive an this is because detection rate will be high and will be a positive value although lower than the crime rate.

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(iv) What sign do you expect b2 to be?

The sign for b2 will be neither negative nor positive because the dummy value when we consider one crime will be zero when considering burglary however when we will consider theft and

stolen goods we will consider the dummy variable as equal to 1, the value for b2 i n this case will be positive

(v) Why is the final term (u) included in the equation?

u is the disturbance term or the error term, the error is included in an equation because it represents variables that may be omitted in an equation which may influence the dependent variable.

3. Estimate the above equation by OLS and present the results in a suitable table (n.b. marks will be lost for simply copying and pasting the computer output)

CRi =  b0 + b1 DRi + ui

When we estimate the above function our function will be

CRi = 21.49288446 -0.682671503DRi

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(i) comment on the relationship between the coefficients and the answers you gave to question 2 parts (ii) to (iv)

the expected value of b1 did not match our expected values, however our expected autonomus value matched the expected value, therefore to this function the detection rates and the crime rate relationship is inverse, this is to say that as the value of detection rate increases the crime rate increases.

(ii) comment on the R2 statistic for this equation

the value of the R2 is

0.309958321, this correlation shows the strength of the relationship that exist between two variables, the value in this function is less than one and this means that the relationship that exist between our two variables is not strong.

(iii) calculate the elasticity of crime rates with respect to detection rates Elasticity Xi= bi . (Xi/Y)

Our elasticity is -0.791733188

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(iv) explain what assumptions need to be made about the u term in order to carry out ‘t’ tests on the parameters.

Assumption of the error term (u)

The disturbance term (u) is random variable

The disturbance term (u) has constant variance across all observations.

The disturbance term (u) and the parameters have zero covariance

The disturbance r term (u) has a mean value of zero [1]

4. You are now going to use the following equation

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CRi = b0DRi b1 expui+ b2 (TYPEDUM)

(i) use a suitable transformation and estimate this equation by OLS and present the results in a suitable table

We will use double log transformation and the result will be

our function will take the form

log CRi = log -5.08335 X 10-16-1.157779253log DRi

(ii) compare the detection rate elasticity from this equation with that from the previous equation

The elasticity in this function is -1.543705671, for our previous function the elasticity was

-0.791733188, therefore our current function is more elastic than our previous function.

(iii) work out the predicted crime rates when the detection rate for each crime is 25%

log CRi = log -5.08335 X 10-16-1.157779253log DRi

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If DR increases by 25 then or new crime rate will be log 1.49164 X 10-14

(iv) examine the residuals from this equation to see if you can identify any regions whose crime rates are being over or under-predicted for either crime

Residuals = total sum of squares – explained sum of squire

West Yorkshire has the highest value of residue while London city has the least residual value.

(v) Is the R2 from this equation higher than that from the previous equation? Is this information of any use? Explain.

Correlation of determination R2

Our current equation has an R2 value of 0.339161292, the previous equation had a value of

0.309958321, and therefore there is no great difference in the correlation of determination of the two equations.

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5. Carry out the following hypothesis tests for BOTH equations

(i) all coefficients are zero at the 10% level

Equation two

log CRi = log -5.08335 X 10-16-1.157779253log DRi

For b0

Null hypothesis B0 = 0

Alternative B0 ≠ 0

T critical at 10% level and at 2 degrees of freedom is 0.009852

T critical < t calculated

In this case we reject the null hypothesis and therefore our B0 is statistically significant.

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For b1

Null hypothesis B1 = 0

Alternative B1 ≠ 0

(i) b1=0 against the two sided alternative at the 5% level

The T value in the T table for 5% is 0.03775; the T calculated is greater than the T critical therefore we reject this null hypothesis.

(ii) b1 is less than zero against the alternative at the 20% level

the T value at 20% is 0.0025, for the hypothesis that

b1<0 where the T calculated is greater than the T critical therefore we reject this null hypothesis.

(iii) b1 is >-1 against the alternative at the 5% level

The T value in the T table for 5% is 0.03775, T calculated is greater than the T critical therefore we reject this null hypothesis.

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6. You should now write a short report of 450-600 words first explaining briefly what your results show and their limitations. Second, you should also include further exploration of the dataset and suggestions for improvement of the model you have estimated.

Our first estimated function is CRi = 21.49288446 -0.682671503DRi , this means that the autonomous crime level is 21.49 while the other parameter depicts that an increase in the level of detection by one unit will reduce the crime rate by 0.68, this means that crime rate will reduce if we increase the levl of detection rate. Our second equation is log CR

i

= log -5.08335 X 10

-16

-1.157779253log

DR

i

and this means that if we are to increase the detection rate by one unit than the crime rate will be reduced by log 1.1577, also in this function there exist an autonomous value of crime which is log 5.8083, this also shows that crime rate will reduce when we increase the detection rate.

From the above results we can conclude that the higher the level of detection rate the lower is the level of crime recorded, therefore an increased effort to increase the level of detection rate the more we can discourage crime and as a result we will get a reduction in crime occurrence. However our two equations show very little correlation between crime rate and detection which means that the relationship between crime rate and detection rate is not strong, however this does not mean that the estimated functions are not varied, the two equations are varied and they are in line with our assumptions first made that the level of crime rate depends on the detection rates.

There is a need to include other variables that influence crime detection, the failure to include this factors have resulted to a high level of the disturbance term, the equation therefore should be re specified with other factors that influence crime rate, one other factor that could be included is the occurrence of other crimes, other crimes will hinder the detection rates of a certain crime because may be the police will have a large work load to detect other crimes, therefore when we re specify the equation we must consider all the other crimes in our equation.

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However our estimated equations are varied but the second one is transformed meaning that it gives us a more clear understating on the relationship that exist between crime rate and detection rates, however there is need to re specify the equations, the hypothesis tests also proves to us that all the variables estimated are statistically significant when the null hypothesis are rejected meaning that our parameters are not equal to zero.

There is however a need for the police force to increase the level of detection rate because as our equations predict an increase in the level of detection rate reduces the occurrence of crime no matter which crime is taken into consideration, the increased detection rate will lead to a reduction in crime rate.

References:

England and Wales crimes (2004/2005) crime rate and detection rates, retrieved on 29th April, available at

www.crimestatistics.org.uk

P. Schmidt (1976) Econometrics, Marcel Decker publishers, USA

[1] Schmidt (1976)

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