(1) Provide an introduction section on the rationale of your model , sample size, and the dependent and independent variables (including their unit of measurement) in this model.

(2) Plot the dependent variable against each independent variable using scatter plot/dot function in Excel. Describe the relationship from the plots.

(3) Present the full model in your assignment.

(4) Write down the least squares regression equation and correctly interpret the equation.

(5) Interpret the estimated coefficients of the regression model and discuss their sig values.

(6) What is the value of the coefficient of determination for the relationship between the dependent and independent variables. Interpret this value accurately and in a meaningful way.

(7) State the 95% confidence intervals for each parameters and interpret these intervals.

(8) Estimate the linear regression model to investigate the relationship between the market price and the land size in total number of square meters.

(9) Compare the original model (question 1) and re-estimated model (question 2) and evaluate the goodness of fit between them (Hint: Use R2and Coefficient of determination to evaluate the goodness of fit of the model).

(10) Predict the market price of a house (in $) with a building area of 400 square meters.

**Introduction and background:**

The undertaken business research report discusses about the linear significant associations and relationships between independent variables and dependent variable. The report is mainly based on statistical operation that is ‘Ordinary Least Square (OLS)’. The assignment questions based on regression model help to elaborate the strength of year-wise dependencies of economic factors and parameters on market values of housing and apartments in the cities of Australia (Waltl, 2016).

The business research report is the true reflection of economic factors on market price values of constructions in Sydney throughout 2002 to 2017. This explanatory research report helps to enhance the decision-making skills and abilities for the policy makers with the help of necessary calculations and visualizations. The trend and inherent aspects of market prices of a house or an apartment in this context of Sydney are discussed here.

**Rationale of the model:**

Rationale are very much required in a model to validate any operational statistic. The rationale of the ordinary least square model are followings:

- The observations of the parameters are collected from random sampling.
- The linear regression model should be linear in parameter (Sachindra et al., 2013).
- The conditional mean among the variables must be equal to 0.
- Independent variables of OLS are independent to each other; that is there is no multicollinearity among the variables (Searle & Gruber, 2016).
- There should be homoscedasticity and no autocorrelation between the variables.
- The error terms () of the OLS model must be normally distributed (~ N ()) (Myers et al., 2012).

**Sample size:**

The number of sample used for orthogonal least square models in this assignment is 16. The economic and other parameters of Sydney for 16 years are the experimental data for calculating the business research report. All the variables in this data set are numerical in nature. Therefore, the data analysis is absolutely quantitative.

**Dependent and Independent variables:**

The independent variables are those variables that act as ‘causes’ and dependent variable is the variable that is regarded as ‘effect’. In the first ordinary least square model, the dependent variable is ‘Sydney market price ($000)’ and the independent variables are ‘Sydney price index’, ‘Sydney annual % change’, ‘Total number of square meters’ and ‘Age of house (years)’. In the second ordinary least square model, the dependent variable is ‘Sydney market price ($000)’ and the independent variable is only ‘Total number of square meters’.

** **The dependent variable (Sydney market price ($000)) is plotted in the scatter chart along Y-axis assuming it to be the response factor. The independent variables (Sydney price index, Sydney annual % change, Total number of square meters and Age of house (years)) are plotted along X-axis assuming them as predictive factors.

The graph indicates that the dependent variable (Sydney market price ($000)) has negative association with ‘Age of house (years)’ and positive association with rest of the three independent variables. highest positive linear association with ‘Sydney price index’ and lowest positive linear association with ‘Total number of square meters’ that is land-size.

## Description of the Data Set

The least square regression equation between one dependent variable and more than one independent variables is given by

Here, y = the dependent variable,= the intercept of the least-square regression model, the coefficients of the independent variables/slope of the model, = the independent variables (Peng & Lai, 2012).

The least square regression model appropriately tests the statistical significant linear relationships between the dependent variable and the independent variables.

The full least square regression model is given by-

‘Sydney market price ($000)’ = 548.978 + 1.963 * ‘Sydney price Index’ – 5.622 * ‘Sydney annual % change’ + 0.519 * Total number of square meters’ – 2.488 * ‘Age of house (years)’.

The hypotheses of the ordinary least square (OLS) model:

Null hypothesis (H_{0}): There exists no statistical significant linear association among the dependent variable and independent variables.

Alternative hypothesis (H_{A}): There exists statistical significant linear association among the dependent variable and independent variables.

The estimated coefficients of the independent variables in the least-square regression equation signifies that-

- ‘Sydney Price Index’ has positive linear relationship with ‘Sydney market price ($000)’ and the association between these two variables is significant (p-value = 007160758 < 0.05). For 1 unit increase or decrease of ‘Sydney Price Index’, the ‘Sydney market price ($000)’ increases or decreases by 1.963 units (Kuznetsova, Brockhoff & Christensen, 2017).
- ‘Sydney annual % change’ has negative linear relationship with ‘Sydney market price ($000)’ and the association between these two variables is insignificant (p-value = 113361729>0.05). For 1 unit increase or decrease of ‘Sydney annual % change’, the ‘Sydney market price ($000)’ decreases or increases by 5.622 units.
- ‘Total number of square meters’ has positive linear relationship with ‘Sydney market price ($000)’ and the association between these two variables is insignificant (p-value = 140071458<0.05). For 1 unit increase or decrease of ‘Sydney annual % change’, the ‘Sydney market price ($000)’ increases or decreases by 0.519 units.
- ‘Age of house (years)’ has negative linear association with ‘Sydney market price ($000)’ and the association between these two variables is insignificant (p-value = 052251738<0.05). For 1 unit increase or decrease of ‘Sydney annual % change’, the ‘Sydney market price ($000)’ decreases or increases by 2.488 units.
- Overall, the p-value of F-statistic of the model is 0.002 that is less than 0.05. Therefore, the null hypothesis is rejected at 5% level of significance and alternative hypothesis is accepted. The OLS statistically signifies the association among dependent variable and independent variables.

Therefore, to predict the market price of any building in Sydney, the significant predictive factor is ‘Sydney Price Index’.

The value of co-efficient of determination (R^{2}) is 79.06% in this model. Hence, it could be inferred that the variation of dependent variable is 79.06% explained by the independent variables.

Also, the value of co-efficient of determination (R^{2}) is more justified than the adjusted R^{2} for finding the association in a multiple regression model as the statistic is free from intra-dependency among the independent variables. This intra-dependency among independent variables is known as ‘Multicollinearity’. The adjusted R^{2} is 70.69%. Correctly speaking, the independent variables can explain 70.69% variability of the dependent variable.

The 95% confidence intervals of each independent variables (Sydney price Index, Sydney annual % change, Total number of square meters and Age of house (years)) interprets that-

- For 1 unit change of ‘Sydney Price Index’, the change of ‘Sydney market price ($000)’ ranges in the interval of (664031125, 3.262956664) units with 95% probability.
- For 1 unit change of ‘Sydney annual % change’, the change of ‘Sydney market price ($000)’ ranges in the interval of (-12.84161778, 1.597209306) units with 95% probability.
- For 1 unit change of ‘Total number of square meters’, the change of ‘Sydney market price ($000)’ ranges in the interval of (-0.202568152, 1.240859409) units with 95% probability.
- For 1 unit change of ‘Age of house (years)’, the change of ‘Sydney market price ($000)’ ranges in the interval of (-5.005107781, 0.029375841) units with 95% probability.

The estimating equation of upper 95% confidence interval to predict ‘Sydney market price ($000)’ is given following-

‘Sydney market price ($000)’ = 729.750 + 3.263 * ‘Sydney price Index’ + 1.597 * ‘Sydney annual % change’ + 1.241 * Total number of square meters’ + 0.029 * ‘Age of house (years)’.

The estimating equation of lower 95% confidence interval to predict ‘Sydney market price ($000)’ is given following-

‘Sydney market price ($000)’ = 368.206 + 0.664 * ‘Sydney price Index’ - 12.842 * ‘Sydney annual % change’ - 0.203* Total number of square meters’ - 5.005* ‘Age of house (years)’.

## Dependent and Independent Variables

It could be noted that for the estimating equation of upper 95% confidence interval, no independent variable has negative association with the dependent variable. On the contrary, for the estimating equation of lower 95% confidence interval, all the independent variables except ‘Sydney price index’ have negative association with the dependent variable. For equal sets of the values of the independent variables, the value of dependent variable lies in the interval (‘Sydney market price ($000)’ (Upper limit), ‘Sydney market price ($000)’ (Lower limit)) with 95% confidence.

The re-estimated least square regression model is given by-

‘Sydney market price ($000)’ = 659.143 + 0.563 * ‘Total number of square meters’.

Note that, the p-value of F-statistic of the model is 0.256 that is greater than 5%. Hence, the null hypothesis of insignificant association of dependent and independent variables is established in this model.

The value of adjusted R^{2} is 70.686% in the first ordinary least square regression model. The statistic ‘adjusted R^{2}’ works as the ‘Co-efficient of determination’ (Lewbel, 2012). It is also a measure of ‘goodness of fit’. Therefore, the variation dependent variable ‘Sydney market price ($000)’ is explained by the independent variables ‘Sydney price Index’, ‘Sydney annual % change’, ‘Total number of square meters’ and ‘Age of house (years)’. The overall linear relationship of the dependent variable with respect to the independent variables is strong. It could be stated that the model is fitted good because of high value of adjusted R^{2} (Henseler & Sarstedt, 2013).

The value of R^{2} is 9.81% in the second ordinary least square regression model. The statistic (R^{2}) is known as the ‘Co-efficient of determination’ for the OLS model while the number of independent variable is only one. It finds the ‘goodness of fit’ of the model. Hence, the variation of the dependent variable ‘Sydney market Price ($000)’ is explained by the independent variable ‘Total number of square maters’. Hence, the statistical linear relationship is very week. The model is also not fitted good because of high value of R^{2}.

As per the least-square regression model, the predicted value of market price of a house (in $) of a building area of 400 square meters is calculated as-

‘Sydney market Price ($000)’ = 659.143 + 0.563 * 400 = 659.143 + 225.441 = 884.584.

Therefore, for the building area of 400 square meters, the market price of a house in Sydney would be 884.584 ($000).** **

**Conclusion:**

** **The analysis and discussion reveal that market price of Sydney throughout the years highly got influenced by price index. Neither annual percentage change of price nor land size are the significant estimators for deciding the market price of Sydney. However, age of the house is the insignificant but crucial predictor for estimating the market price of a house in Sydney.

It could be recommended that for keeping the market price of any building of Sydney in control, the economic and marketing policy-makers must concentrate on recent price index. Proper policy-making would be able to keep the price index of Sydney low that would keep the market price of any house or apartment of Sydney in control.

**References:**

Henseler, J., & Sarstedt, M. (2013). Goodness-of-fit indices for partial least squares path modeling. Computational Statistics, 28(2), 565-580.

Kuznetsova, A., Brockhoff, P. B., & Christensen, R. H. B. (2017). lmerTest package: tests in linear mixed effects models. Journal of Statistical Software, 82(13).

Lewbel, A. (2012). Using heteroscedasticity to identify and estimate mismeasured and endogenous regressor models. Journal of Business & Economic Statistics, 30(1), 67-80.

Myers, R. H., Montgomery, D. C., Vining, G. G., & Robinson, T. J. (2012). Generalized linear models: with applications in engineering and the sciences (Vol. 791). John Wiley & Sons.

Peng, D. X., & Lai, F. (2012). Using partial least squares in operations management research: A practical guideline and summary of past research. Journal of Operations Management, 30(6), 467-480.

Sachindra, D. A., Huang, F., Barton, A., & Perera, B. J. C. (2013). Least square support vector and multi?linear regression for statistically downscaling general circulation model outputs to catchment streamflows. International Journal of Climatology, 33(5), 1087-1106.

Searle, S. R., & Gruber, M. H. (2016). Linear models. John Wiley & Sons.

Waltl, S. R. (2016). Variation across price segments and locations: A comprehensive quantile regression analysis of the Sydney housing market. Real Estate Economics.

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