ECOM90009 Quantitative Methods For Business

ECOM90009 Quantitative Methods For Business

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ECOM90009 Quantitative Methods For Business

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ECOM90009 Quantitative Methods For Business

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Course Code: ECOM90009
University: The University Of Melbourne

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Country: Australia

Questions:
1.  Obtain the descriptive statistics for the Margin (Y) variable and for the 7 independent variables Number (X1), Nearest (X2), OfficeSpace (X3), Enrolment (X4), Income (X5), Distance (X6) and Quality (X7). If we only know the values of the descriptive statistics what will our estimate of the typical value of the Margin (Y) variable be equal to. Please note whether or not the value that you obtain has any limitations or problems.
Comment on which if any of the 8 variables in your model do not have a symmetrical distribution.What does the Sum of the Quality (X7) dummy variable tell us?
2.  Find the set of correlation coefficients for all possible combinations of the 8 variables i.e. the Margin or Y variable and the 7 independent variables Number (X1), Nearest (X2), OfficeSpace (X3), Enrolment (X4), Income (X5), Distance (X6) and Quality (X7). Use the hypothesis testing procedure where the null hypothesis is H0: ? = 0 and the alternative hypothesis is HA: ? ≠ 0 and the testing statistic is z = r. to identify which pairs of the 8 variables have a significant linear relationship. 
Answer:
Preliminary analysis of the regression model

Descriptive statistics

From the descriptive statistics the appropriate estimate of the Margin will be obtained from the mean, this gives a value of 46.4175. This value can be affected with presence of outliers hence leading to inaccurate estimation of the margin value.
The skewness of the Income data is 0.0084, this indicates that the data is symmetrically distributed.
The sum of the quality variable indicates the number of motels that have been built or extensively renovated in the past 6 years.

Set of correlation coefficient

Testing the hypothesis
Vs
At 5% significance level
We reject the null hypothesis if the value of the z statistics is less than 0.05 and fail to reject otherwise.  
Margin and (Nearest, Office space, Enrolment, Income, Quality)
Number and (Nearest, Income, Distance)
Nearest and (Office space, Enrolment, distance, Quality)
Office space and (Enrolment, Income, Distance, Quality)
Enrolment and (Distance, Quality)
Income and quality 

Scatter diagrams

Margin vs Number 
The regression coefficient of margin and number of given by -0.4876, when this is compared against the scatter diagram it can be visualised that the two variables have a weak negative correlation. 
Margin vs Nearest
The regression coefficient of margin and Nearest is obtained as 0.1885.
From the scatter plot below, it can be deduced that the variables have a weak positive association a factor which is supported by the regression coefficient.  
Margin vs Office space  
The regression coefficient of margin and office space is obtained as 0.5884, this supports the scatter plot display below to indicate that the variables have a strong positive correlation
Margin vs Enrolment  
Margin and enrolment have a correlation coefficient of 0.1284, this is reflected in the scatter plot as a form of weak positive association.
Margin vs Income
Margin and income have a coefficient of 0.2081. from the scatter plot it can be visualised that the variables have weak positive association. 
Margin vs Distance
Margin and distance have a correlation coefficient of -0.1305, from the scatter plot we can conclude that this is a weak negative association 
Margin vs Quality
The correlation coefficient of margin and quality is 0.3997, The scatter plot displayed does not give a visual display that can enable support of the nature of association.  
Except for the plot of margin and quality, the other scatter plots show information which is expected based on the earlier calculated correlation coefficients. 
Question 1

Regression model  

From the regression output the Margin can be predicted by the linear model 
The coefficient of  -0.00754, this shows that the margin reduces by -0.00754 whenever the value of Number increases by a single unit. On the other hand, the coefficient of  is 0.2386 indicating that whenever Office space increases by a single variable margin increases by 0.2386.

The error terms are assumed to be normally distributed with a mean of zero and a standard deviation of . By looking at the outputs of the standard residual, it is observed that the values are clustered around the line indicating that relevant assumptions are met.

The assumptions are relevant as they assist determine the general impact of the error term in the developed model.

The values;  

The listed values above prove that the model is generally significance and can be applied efficiently to predict the values of margin. 
From the excel output, the p-value obtained indicates that the variable Number, Income and Office space significantly affects the values of the margin.
The model is generally significant hence indicating no evidence of multicollinearity. 

The parsimonious regression model  

Using this model, the estimated model can be derived as; 
By checking the residual graphs its visible that some of the graphs are not equally clustered around the line
The variable  is removed, the resulting model is given by   
Repeating the process, we eliminate  the remaining variables have p values less than 0.05 hence the final model is indicated as 
Parsimonious model, when compared to the earlier model the Parsimonious model gives more accurate prediction of the margin as it considers all the significant variables. 

Estimating the margin values  

The original model makes abetter forecast when compared to the parsimonious model.  
The values of the Victorian housing charts is symmetrically distributed. 

Average forecasts 

As the value of the smoothing parameter goes up the value of the forecast increases.
Out of sample period forecasts  
Combined line chart for the out of sample period 

Multiplicative time series model

12 months moving average 
The values give 12 months forecast of the housing starts for the Victoria region 

The linear forecast is given by the model 

The quadratic forecast is given by  
From the two trendlines the most appropriate one to represent the housing starts values is the quadratic model.
Combined line graph to represent the forecast and the actual values   

Drawing a chart to represent the values 

The most appropriate forecast to apply in the future predictions is the seasonal indices model as it breaks down the data to monthly variables which can easily be predicted with much accuracy. 

References
Good, P. I., & Hardin, J. W. (2009). Common Errors in Statistics (3rd ed.). Hoboken, New Jersey: Wiley.
Malakooti, B. (2013). Operations and Production Systems with Multiple Objectives. John Wiley & Sons.
Willem, W., De Baets, B., & Boullart, L. (2008). ROC analysis in ordinal regression learning. Pattern Recognition Letters, 29, 1-9.

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