FIN728 Financial Ecn Modelling Assessment Answer

Solution 1

The data table is given as follows:

Table 1

Table 1 presents a fictitious example comprising 8 regions within the U.K. for the year 2022. The underlying hypothesis suggests that regions with a higher proportion of unemployed individuals would likely exhibit an elevated crime rate.

Solution a

We are given the data set of Unemployment rate and crime rate for the different regions in the United Kingdom (U.K.)

The crime rate per thousand population for Yorkshire and the Humber region is 171. It means that per thousand people there are 171 crimes occurred. This rate is higher as compare to the other regions crime rate.

The data type in the crime rate per thousand population variable is the quantitative discrete. As the crime rate values are clearly countable values.

Solution b

The sample means and sample standard deviation for both variables is calculated as follows:

Unemployment rate:

Total values = 8

Sum = 27

Sample mean = Sum/total observations

= 27/8

= 3.3750

Standard Deviation, s: 1.0607

Steps

For Crime rate per thousand population:

Total values = 8

Sum = 693

Sample mean = Sum/total observations

= 693/8

= 86.6250

Standard Deviation, s: 38.1386

Steps

Solution c

Imagine that the crime rate per one thousand individuals follows a pattern akin to being drawn from a normal distribution, characterized by a mean and a standard deviation.

The probability that the crime rate per thousand population is greater than 100 is calculated as follows:

P(X > 100) = P(X – mu/sigma > 100 – 86.6250/38.1386)

= P(z > 0.3507)

= 0.3517

The probability is calculated using the normal table.

Hence, the probability that the crime rate per thousand population is 0.3517.

Solution 2

The regression equation is calculated using the above data table 1.

The summary statistic values are calculated as follows:

Let unemployment rate is x variable and crime rate is y variable. Here Unemployment rate is independent variable and crime rate per thousand population is the dependent variable.

The slope estimate is calculated as follows:

The intercept term is calculated as follows:

Thus, the regression line is given as follows:

Crime rate (y) = 81.8571 + 1.4127 * Unemployment rate

Interpretation of intercept term:

The intercept estimate is 81.8571 indicate that the expected crime rate per thousand population is 81.8571 for a region where the unemployment rate is 0. That is for a region where the unemployment rate is 0, the predicted value of crime rate per thousand population is 81.8571.

Interpretation of slope:

The slope estimate is 1.4127. It indicates that with the 1 unit increase in the unemployment rate, the crime rate per thousand population increases by 1.4127.

Solution 3

a)

In this part, we need to calculate the residual sum of square (RSS) using the regression equation.

First, we need to find the predicted values for each value of unemployment rate and then subtract the predicted value from the original value and take the square of the error value.

Now add all the values, the final value is the RSS value.

The calculations are shown as follows:

For Unemployment 4, predicted crime rate = 81.8571 + 1.4127 * 4 = 87.5079

For Unemployment 3, predicted crime rate = 81.8571 + 1.4127 * 3 = 86.0952

For Unemployment 4, predicted crime rate = 81.8571 + 1.4127 * 4 = 87.5079

For Unemployment 5, predicted crime rate = 81.8571 + 1.4127 * 5 = 88.9206

For Unemployment 3, predicted crime rate = 81.8571 + 1.4127 * 3 = 86.0952

For Unemployment 2, predicted crime rate = 81.8571 + 1.4127 * 2 = 84.6825

For Unemployment 4, predicted crime rate = 81.8571 + 1.4127 * 4 = 87.5079

For Unemployment 2, predicted crime rate = 81.8571 + 1.4127 * 2 = 84.6825

Now the residual values are calculated as follows:

Residual value = Original y – predicted crime rate

For Unemployment 4, Residual = 96 - 87.5079 = 8.4921

For Unemployment 3, Residual = 171 – 86.0952 = 84.9048

For Unemployment 4, Residual = 78 - 87.5079 = -9.5079

For Unemployment 5, Residual = 95 – 88.9206 = 6.0794

For Unemployment 3, Residual = 70 – 86.0952 = -16.0952

For Unemployment 2, Residual = 79 – 84.6824 = -5.6824

For Unemployment 4, Residual = 45 - 87.5079 = -42.5079

For Unemployment 2, Residual = 59 – 84.6825 = -25.6825

Now the take square of the residuals. RSS = 10166.1576

The TSS is calculated by subtracting the mean of y variable from the original value and take the square and then sum.

For crime rate the mean value is 86.6250.

b) The r-square is calculated as follows:

The total variation explained by the Unemployment rate in crime rate per thousand population is 0.15%. The value is close to 0, hence the regression model created is not useful.

c) The hypothesis test is conducted as follows:

The significance level is 0.05.

The test-statistic is calculated as follows:

The p-value for the test-statistic value 0.0066 with n- 2 = 8 -2 = 6 degrees of freedom is calculated using t-distribution table as follows:

P-value = 2* P(t>0.0066)

= 0.9949

The p-value is greater than the level of significance (0.9949 > 0.05). Hence, the null hypothesis is not rejected.

There is significant evidence that the coefficient associated with the “Unemployment Rate” variable is zero.

d) The regression equation is given as follows:

????rimeRate???? = ????0 + ????1 Unemployment rate + ????i

Since, there are total 8 observations. The matrix are given as follows:

Substitute the values in the above matrix form,

The slope is calculated using the following formula,

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