Presented 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
Burglary
Theft and handling stolen goods
Cleveland
17
46
13
19
Durham
12
25
14
22
Northumbria
12
34
12
23
Cheshire
11
31
16
18
Cumbria
9
27
13
24
Greater Manchester
20
45
9
15
Lancashire
11
34
17
23
Merseyside
16
41
16
19
Humberside
21
49
10
15
North Yorkshire
10
29
13
21
South Yorkshire
17
40
16
20
West Yorkshire
19
44
15
19
Derbyshire
12
30
12
18
Leicestershire
12
34
13
20
Lincolnshire
11
30
9
18
Northamptonshire
17
42
12
21
Nottinghamshire
24
55
8
13
Staffordshire
11
31
16
23
Warwickshire
12
32
18
19
West Mercia
11
28
15
24
West Midlands
16
38
10
15
Bedfordshire
10
32
15
19
Cambridgeshire
12
39
15
16
Essex
9
31
12
17
Hertfordshire
11
34
21
19
Norfolk
8
28
13
20
Suffolk
8
26
14
20
London, City of 3
+
+
25
17
Metropolitan Police
14
56
13
10
Hampshire
9
32
17
19
Kent
10
31
10
17
Surrey
8
24
11
14
Sussex
10
35
11
15
Thames Valley
12
40
14
18
Avon and Somerset
14
40
10
12
Devon and Cornwall
9
29
13
19
Dorset
9
32
18
19
Gloucestershire
13
35
12
20
Wiltshire
8
25
15
20
Dyfed Powys
6
17
21
27
2. Consider the following equation:
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.
(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 in 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
(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 ourexpected 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
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 zer
4. You are now going to use the following equation
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 10-16-1.157779253logDRi
(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 10-16-1.157779253logDRi
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.
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 10-16-1.157779253logDRi
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.
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.
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 CRi = log -5.08335 X 10-16-1.157779253logDRi 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.
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
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