Answer :

Statistics Assignment

Report on Regression Analysis

1. Introduction

The following pages determine a regression model for the given data related to market prices for houses in Sydney. The variables included for analysis are: Market Price (in $000), Sydney price Index, Annual percentage change, Total number of square meters and Age of house (in years). Furthermore, the sample size is 18. 

As can be seen from above, there are multiple variables and hence, a multiple variable regression model will be appropriate. Further, the Ordinary Least Square technique, popularly known as OLS, will be used to arrive at an appropriate regression model for provided data.

This model will have one dependent variable the value of which will be predicted through regression model. This dependent variable is the Market Price (in $000). The remaining variables will be independent variables that impact the value of dependent variable, that is, Market Price (in $000). The independent variables include: Sydney price Index, Annual percentage change, Total number of square meters and Age of house (in years).

The Sydney price index is based on base value of 100 in 2011-12 and calculated accordingly. The Annual percentage change is in percentage form and reflects the increase or decrease in market prices in year-on-year basis. The Total number of square meters reflects the size of the land on which house is built and it is measured in square meters. The age of the house reflects the number of years since the house was built. 

The data for all the variables has been sourced from ABS6416.0, Residential Property Price Indexes for the Three Capital casitas.

To perform the regression analysis, Microsoft Excel will be utilised in order to generate the regression output.

2. Scatter Plots

The following graphs were generated using Microsoft Excel and represent relationship between the dependent variable and each of independent variables:

The above graph is a scatter plot for Sydney Price Index versus the Market Price. The independent variable (x) is the Sydney Price Index while the dependent variable (y) is the Market Price. It can be seen that there is a positive linear relationship between the two variables such that as the value of Sydney Price Index increases, the market price also increases.

The above graph is a scatter plot for Annual Percentage Change versus the Market Price. The independent variable (x) is the Annual Percentage Change while the dependent variable (y) is the Market Price. It can be seen that there is a positive linear relationship between the two variables such that as the value of Annual Percentage Change increases, the market price also increases. The trend line slope is not as steep as in case of Sydney Price Index, indicating that this variable has relatively lesser impact.

The above graph is a scatter plot for Total number of Square Meters versus the Market Price. The independent variable (x) is the Total number of Square Meters while the dependent variable (y) is the Market Price. It can be seen that there is a positive linear relationship between the two variables such that as the value of Total number of Square Meters increases, the market price also increases. The data is somewhat concentrated indicating close range values of the variables.

The above graph is a scatter plot for Age of House (years) versus the Market Price. The independent variable (x) is the Age of House (years) while the dependent variable (y) is the Market Price. It can be seen that there is a negative linear relationship between the two variables such that as the value of Age of House (years) increases, the market price decreases and vice versa. This can be related to real life where the older house will fetch a lower price and a newly constructed house will fetch a higher price.

3. Regression Model

As discussed earlier, Microsoft Excel was used to generate Regression output as follows:

4. Regression Equation

We can say that the Market Price (‘000) is regressed on Sydney price Index, Annual percentage change, Total number of square meters and Age of house (in years).

From the above output, we can interpret the least squares regression equation as follows:

y^{^} = 531.46 + 2.264 x₁ -6.310 x₂ + 0.507 x₃ – 2.637 x₄

where,

• Dependent variable, Market price ($000) is represented as: y^{^} 
• Independent variable, Sydney price index is represented as x₁ 
• Independent variable, Annual percentage change is represented as x₂ 
• Independent variable, Total number of Square Meters is represented as x₃ 
• Independent variable, Age of House (years) is represented as x₄ 

5. Coefficients & Sig Values

Coefficients

The above equation has a constant of 531.46 which is also known as Y-intercept coefficient. It is the minimum value for y even when all x values are zero. It is the point where regression line crosses the vertical axis and is also known as ‘β₀’ or ‘constant’. 

x₁ and x₃ are greater than zero indicating that the relationship is positive such that x and y increase or decrease together.

x₂ and x₄ are lesser than zero indicating that the relationship is negative such that as x increases, y decreases and vice versa.

Sig Value

The output above provides coefficients as well some other statistically significant information, such as p-values for each of the coefficients. These values help in determining whether the variable has statistically significant relationship with the dependent variable or not.

Each of the p-value entails a null hypothesis that the variable has no correlation with the dependent variable. The alternative hypothesis is that the variable has correlation with the dependent variable. This null hypothesis can be rejected if the p-value for a coefficient is less than the significance level in consideration. Hence, in such a case, we can conclude that the particular variable has a correlation with the dependent variable.

In our case, the significance level is 0.05. The p-values for various coefficients are:

• Intercept: 0.000, which is less than 0.05. Hence, the variable is statistically significant.
• x₁ : 0.001, which is less than 0.05 Hence, the variable is statistically significant.
• x₂ : 0.060, which is greater than 0.05. Hence, the variable is not statistically significant.
• x₃ : 0.132, which is greater than 0.05. Hence, the variable is not statistically significant.
• x₄ : 0.039, which is less than 0.05. Hence, the variable is statistically significant.

6. Coefficients of Determination

The values of R and R² help in determining the degree of relationship between independent and dependent variables. These are also called as Coefficients of Determination.

In the given case, R value is 0.900 which indicates that there is high degree of correlation between the variables in consideration. R² is at 0.810 with adjusted R² being 0.751. The value of R² is nothing but the proportion of change in dependent variable that can be explained through the independent variables. As can be seen, at 81.0%, a high proportion of change in value of ‘Market Price’ can be attributed to change in independent variables, namely, Sydney price Index, Annual percentage change, Total number of square meters and Age of house (in years).

7. Confidence Intervals

The confidence intervals for various parameters help in determining the change in values of dependent variable with a unit change in the parameter. In the given case, the confidence interval for various parameters can be stated as:

• x₁ : We can be 95% confident that with each increase in Sydney price index, the market price ($000) will increase between 1.054 and 3.473.
• x₂ : We can be 95% confident that with each decrease in Annual % change, the market price ($000) will increase between -12.919 and 0.300.
• x₃ : We can be 95% confident that with each increase in Total number of Square meters, the market price ($000) will increase between -0.174 and 1.187.
• x₄ : We can be 95% confident that with each increase in Age of House (years), the market price ($000) will decrease between -5.113 and -0.161.

Points that should be noted in above statements are:

• In each case, the first value is Lower 95% confidence level and the second number is Upper 95% confidence level.
• The change is in $000. Hence, for example, a change of 1.054 indicates an increase of $1,054.

8. Market Price versus Land Size

The following is the output when regression is done for single variable, that is, size in total square meters.

We can say that the Market Price (‘000) is regressed on Total number of square meters. From the above output, we can interpret the least squares regression equation as follows:

y^{^} = 659.26 + 0.64 x₁ 

where,

• Dependent variable, Market price ($000) is represented as: y^{^} 
• Independent variable, Total number of Square Meters is represented as x₁ 

It can be seen above that Total number of square meters explains around 9% of change in the dependent variable, that is, market price. Further, when we see the p-value for this variable, it is 0.23 which is greater than 0.05. Hence, we are unable to reject the null hypothesis that there is no correlation between the selected variables. In other words, this is not a statistically significant variable for our purpose at the significance level of 0.05.

9. Re-estimated Model

Basis above analysis, we can compare the two regression models:

In case of Model 1, R value is 0.900 which indicates that there is high degree of correlation between the variables in consideration. R² is at 0.810 with adjusted R² being 0.751. The value of R² is nothing but the proportion of change in dependent variable that can be explained through the independent variables. As can be seen, at 81.0%, a high proportion of change in value of ‘Market Price’ can be attributed to change in independent variables, namely, Sydney price Index, Annual percentage change, Total number of square meters and Age of house (in years).

In case of Model 2, R value is 0.300 which indicates that there is low degree of correlation between the variables in consideration. R² is at 0.09 with adjusted R² being 0.03. As can be seen, at 9.0%, a very low proportion of change in value of ‘Market Price’ can be attributed to change in independent variable, Total number of square meters.

Hence, it can be seen that Model 1 is a much better fit as compared to Model 2.

10. Prediction of Market Price

From above analysis, the regression equation in case of one variable that is, total square meters is:

y^{^} = 659.26 + 0.64 x₁ 

where,

• Dependent variable, Market price ($000) is represented as: y^{^} 
• Independent variable, Total number of Square Meters is represented as x₁ 

When value of independent variable is given at 400 square meters, we can plug the value of x and predict a value of y as follows:

 y^{^} = 659.26 + 0.64 x₁ 

y^{^} = 659.26 + 0.64 (400)

y^{^} = 659.26 + 256

y^{^} = 915.26

Hence, value of dependent variable is 915.26. In other words, when the total size is 400 square meters, the market price of the house in Sydney is expected to be $915,260.