Discussion

The paper aims to provide a predictive model of the Market price of houses on the basis of the Annual % change, Price Index, Total number of square meters, and Age of house (years) using statistical analysis technique Regression. This multiple linear regression model is built using the Data Analysis Tool-Pak of Microsoft Excel Tool. The dependent variable (Y) is Market price and the rest of the four variables are predictor variables which are denoted as,

Price index -> X1, Annual % change -> X2, Total number of square meters -> X3, Age of house->X4. This analysis will also help to measure the strength of association of the variables and also estimates the Market price value from the values of the response variables for the three different cities in Australia- Sydney, Melbourne, and Brisbane. The responses are recorded for 15 financial years, starting from 2002-03 to 2016-17.

The multiple regression model is given by, Y = b₀ + b₁X₁ + b₂ X₂ + b₃X₃ + b₄X₄ + e where,

b₀ = the y-intercept, b₁ = partial regression coefficient of X₁, b₂ = partial regression coefficient of X₂, b₃ = partial regression coefficient of X₃, b₄ = partial regression coefficient of X₄, e = error of estimation.

A.2. Graphical representation

Firstly, the data is analysed with the help of scatter plot where the response variable Market price is plotted against the vertical axis and all the four explanatory variables are plotted along the horizontal axis. The scatter plot is shown below:

Figure 1

(Source: As created by the Author)

The data points of the explanatory variables are plotted along with trend-lines. The plot shows change of the response variable for change in the values of the explanatory variables graphically. The plot shows positive relationship for X₁ and X₃ variables and negative relationship for X₂ and X₄ variables.

The output of the regression analysis from Excel tool is shown below:

Regression Statistics
Multiple R	0.88916481
R Square	0.79061406
Adjusted R Square	0.70685968
Standard Error	43.8878261
Observations	15
ANOVA
	df	SS	MS	F	Significance F
Regression	4	72728.5872	18182.14679	9.43967451	0.001993481
Residual	10	19261.4128	1926.141284
Total	14	91990
	Coefficients	Standard Error	t Stat	P-value	Lower 95%
Intercept	548.978108	81.13153739	6.766519231	4.94032E-05	368.2057774
Sydney price Index	1.963493894	0.583205471	3.366727492	0.007160758	0.664031125
Annual % change	-5.622204236	3.240109357	-1.735189655	0.113361729	-12.84161778
Total number of square meters	0.519145629	0.3239088	1.60275247	0.140071458	-0.202568152
Age of house (years)	-2.48786597	1.129750872	-2.2021368	0.052251738	-5.005107781

Table 1

(Source: As created by the Author)

A.4. Regression equation

The estimated regression equation is written as,

There is negative linear association of Y with the Sydney price index and the Age of house. This association is positive for Total number of square meters and Annual % change.

 represents that estimated market price = 548.978 when values of all the explanatory variables are zero.  states that there will be 1.963 units of increment in the response variable for one unit increase in the X₁ variable. = -5.622 means that a one unit increase in the Annual % change will decrease the Market price by 5.622 units. The decrease of 2.488 units occurs in Market price for one unit increase in Age of house. There is an increase of 0.519 increase in the Y variable for single unit increase in the X₃ variable.

Regression analysis for Sydney

The coefficient of determination or R² is calculated using the formula below,

Here, R² = 0.7906. Thus it is a very good fit for the explained regression model. Approximately 79% of variation in response variable is explained by the explanatory variables.

The 95% CI of the regression coefficient ’s are calculated by the formula,

, i = 1 to 4.  standard error of estimate, and  = critical value of the t-statistic at 5% significance level (Zou, 2013). In the regression analysis output, these 95% CI’s are already calculated which are shown below.

CI for X₁ = (0.664031125, 3.262956664); it represents that the researcher is 95% confident that the Price index value of Sydney will lie between 0.664031125 and 3.262956664.

CI for X₂ = (-12.84161778, 1.597209306); here the value of Annual % change changes between the lower bound = 0.664031125 and upper bound = 3.262956664 with 95% confidence.

CI for X₃ = (-0.202568152, 1.240859409); the values of variable X₃ will vary within the interval values 1.24085 and -0.020257 with 95% confidence.

CI for X₄ = (-5.005107781, 0.029375841); the value of X₄ variable will lie within the calculated CI with 95% confidence level.

A.8. Regression model between Market price and Total number of square meters

The re-estimated linear regression model, where the single explanatory variable is Total number of square meters and the response variable is Market price, is given as,

Regression Statistics
Multiple R	0.313249771
R Square	0.098125419
Adjusted R Square	0.028750451
Standard Error	79.88618957
Observations	15
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	659.1430408	101.222089	6.511849797	1.9669E-05	440.466012	877.82007
Total number of square meters	0.563603274	0.4738972	1.189294376	0.2555933	-0.460189381	1.58739593

Table 2

(Source: As created by the Author)

The regression equation is

 = 659.143 + 0.5636

The original regression model has R² value = 0.790614058, which is greater than the R² value = 0.098125419 of the re-estimated model. Therefore, the original model or the multiple linear regression model is a better fit for the fitting the Market price value than the simple linear regression model that is the re-estimated model. Moreover, the original model explains 79% of the variation but the re-estimated explains only 9.81% of the variability of the Market price.

A.10. Predicted Market price value

Given that X₃ = 400. Thus, the predicted value of Market price = 659.143 + (0.5636  400) = 884.583 that is, $884583.

B.2. Scatter plot analysis

Before the regression analysis, the scatter plot is displayed to check the association among the variables.

Figure 2

(Source: As created by the Author)

From the scatter plot, it is seen that the Total number of square meters represent a positive linear relationship. All the other variables are not showing a good strength of association.

Graphical representation

Regression Statistics
Multiple R	0.801295
R Square	0.642074
Adjusted R Square	0.498904
Standard Error	19.79055
Observations	15
ANOVA
	df	SS	MS	F	Significance F
Regression	4	7025.999	1756.5	4.484691	0.024733446
Residual	10	3916.658	391.6658
Total	14	10942.66
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	82.70826	26.59056	3.110437	0.011052	23.46080361	141.955708
Melbourne Price Index	-0.67601	0.927016	-0.72923	0.482589	-2.74153307	1.38950858
Annual % change	1.851263	1.501679	1.232796	0.245847	-1.49468607	5.19721276
Total number of square meters	0.116315	0.145828	0.79762	0.443618	-0.20860965	0.44124034
Age of house (years)	-1.24947	0.451735	-2.76594	0.019926	-2.25599865	-0.2429419

Table 3

(Source: As created by the Author)

B.4. Regression equation

According to the above analysis, the regression equation is written as,

There is positive linear relationship with the Annual % change and Total number of sq-meters. However, negative association exits with the Melbourne price index and Age of house.

When all the explanatory variables are zero then the Market price = 82.708. If X1 increase one unit, then keeping other variables constant Y decreases by 0.676 units. When X2 increases one unit, then Y increase 1.851 units. There is decrease of 1.2497 units of Y for a single unit increase in X4 variable and increase of 0.116 units of Y for increase in X3 variable fixing the values of the other variables (Nimon & Oswald, 2013).

B.6. R²

Here R2 is 0.642074. Thus, the regression model gives a moderately good fitting. It states that 64.2% of the variation in the value of the Market price for Melbourne is explained by the explanatory variables.

B.7. Confidence Interval (CI)

CI for X1 = (-2.741533069, 1.389508577);at 95% confidence level, the Melbourne Price index value will lie between -2.741533069 and 1.389508577.

CI for X2 = (-1.494686069, 5.19721276); Here the value of Annual % change changes between the lower-bound = -1.494686069 and upper-bound = 5.19721276 with 95% confidence.

CI for X3 = (-0.208609648, 0.441240343); the values of variable X3 will lie within the given interval with 95% confidence.

CI for X4 = (-2.255998646, -0.242941926); it shows the upper-endpoint= -0.242941926 and lower-endpoint = -2.255998646 of the X4 variable within which the values will lie with 95% confidence.

B.8. Regression model between Market price and Total number of square meters

Regression Statistics
Multiple R	0.328763
R Square	0.108085
Adjusted R Square	0.039476
Standard Error	27.40006
Observations	15
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	50.09162	34.71803	1.442813	0.172733	-24.9121204	125.095356
Total number of square meters	0.204012	0.162541	1.255142	0.231522	-0.14713685	0.55516167

Table 4

(Source: As created by the Author)

The regression equation for the re-estimated model can be written as,

 = 50.092 + 0.204

B.9. Comparison of R²

From Table 3, the R² = 0.642074 and from Table 4, the value is 0.108085. Clearly, the multiple regression model is definitely a better fit for the response variable than the latter re-estimated regression model. In the re-estimated model, the X3 variable explains only 10.8% of variability. However, the original model explains 64.21% of variation.

B.10. Market price prediction

When X₃ = 400 then predicted value of the Market price = 50.09162 + (0.204012  400) = 131.69642 that is, $131696.

C.2. Graphical representation (Scatter plot)

Figure 3

(Source: As created by the Author)

The Excel output is shown in the below table

Regression Statistics
Multiple R	0.847362
R Square	0.7180224
Adjusted R Square	0.6052313
Standard Error	12.573407
Observations	15
ANOVA
	df	SS	MS	F	Significance F
Regression	4	4025.588	1006.397	6.365952	0.008182698
Residual	10	1580.906	158.0906
Total	14	5606.493
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	89.871245	16.91237	5.313936	0.000341	52.18814276	127.5543
Brisbane Price Index	-0.5082788	0.47905	-1.06101	0.313636	-1.575668349	0.559111
Annual % change	1.4620069	1.016153	1.438766	0.180771	-0.802123664	3.726137
Total number of square meters	0.056465	0.094729	0.59607	0.564376	-0.154603966	0.267534
Age of house (years)	-0.7797538	0.348462	-2.2377	0.049196	-1.556176484	-0.00333

Regression calculation

Table 5

(Source: As created by the Author)

Regression equation

The estimated regression equation can be written as,

The y-intercept suggests that the Market price will be 89.871 when all the explanatory variables are zero. The X₁ variable has negative coefficient value which suggests that one unit increase in X₁ variable will decrease Y value by 0.508 units. The value of Y variable increases 1.462 units for unit increase in the X₂ variable when rests of the variables are fixed. Similarly, there will be 0.056 units of increase and 0.780 units of decrease of the Market price value for a unit increase in the X₃ and X₄ variables respectively, keeping other variables constant.

C.6. R²

The above table shows that coefficient of determination is 0.718 and 71.8% variability of the response variable is explained by all the explanatory variables present in the regression model.

C.7. Confidence Interval

95% CI for X₁ = (-1.575668349, 0.559110736); at 95% confidence, the Brisbane Price index value will lie between -1.575668349 and 0.559110736.

95% CI for X₂ = (-0.802123664, 3.726137411); Here the value of Annual % change changes between the lower-bound = -0.802123664 and upper-bound = 3.726137411 with 95% confidence.

95% CI for X₃ = (-0.154603966, 0.267533947); the values of variable X₃ will lie within the given interval with 95% confidence.

95% CI for X₄ = (-1.556176484, -0.003331143) that shows the upper-bound and lower-bound of the X₄ variable within which the values will lie with 95% confidence.

C.8. Regression model between Market price and Total number of square meters

The regression model is shown below,

Regression Statistics
Multiple R	0.311985
R Square	0.097334
Adjusted R Square	0.027899
Standard Error	19.73047
Observations	15
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	65.98835	25.00006	2.639528	0.020414	11.979013	119.9977	11.97901	119.9977
Total number of square meters	0.138577	0.117044	1.183972	0.257621	-0.114282	0.391436	-0.11428	0.391436

Table 6

(Source: As created by the Author)

The fitted simple linear regression model is written as,

 = 65.988 + 0.139

C.9. Comparison of R²

Comparing Table 5 and Table 6, it is evident that the multiple linear regression model explains the variability more significantly as R-square of the original model = 0.718022376 and that of the re-estimated model is 0.097334455. Higher the value of R-square, better the fitting of the regression model.

C.10. Market price value (predicted)

 Putting the value of X₃ in the above regression equation, the predicted Y value is obtained as,

Market price = 65.988 + (0.139  400) = 121.588 that is, $121588.

Conclusion

The above linear regression analyses and interpretation of its components for the three cities of Australia clearly concludes that the multiple linear regression models are better fit of the response variable than the simple linear regression model for all the three cities. The above conclusion is drawn from the values of the coefficient of determination. In the course of this paper, strength of association is not only evaluated by the Excel tool, but also it is graphically represented with the help of the scatter plots. The paper has also discussed about the confidence intervals of all the explanatory variables at 95% level of significance. This study will help to understand how the Market price of house is dependent on the chosen four independent variables. Moreover, the evaluated multiple linear regression equation will help to predict the value of the Market price when the values of the other explanatory variables.

References

Nimon, K. F., & Oswald, F. L. (2013). Understanding the results of multiple linear regression: Beyond standardized regression coefficients. Organizational Research Methods, 16(4), 650-674.

Zou, G. Y. (2013). Confidence interval estimation for the Bland–Altman limits of agreement with multiple observations per individual. Statistical methods in medical research, 22(6), 630-642.