## Introduction and Background

Scottsdale city is situated in the eastern Maricopa County, United States, state of Arizona and neighboring Phoenix. The city is situated next to a famous land mark of ‘Valley of the Sun’. The area taken by the city was historically occupied by prominent and ancient archiological societies called the Hohokam. Before 1400 AD, this prehistoric society occupied the region and created irrigation canals to carry out farming, which is still being practiced currently. The city’s population is approximated to be over 245,000, and it is predominantly concerned with Arizona wealth as it is a key tourism attraction. The forecasted employment for 2010 is 232,832, which accounts for 53% increase from 2000. The objective of this research is to find out the influence and relationship of factors such as pool/spa, bed space, gated community and fire space on the price of homes around the city. Regression model is used to help identify how the distinctive value of the dependent variable changes when any of the independent variables are manipulated.

## Model Specification and Data

### Specification of the Model

The independent variable, which accounts for the figures that are maneuvered or adjusted, includes bedrooms, fireplace, pool/spa, and community gate. Alternatively, the dependent variable, which is the observed consequence of the independent variable that is adjusted, is the price of buying home. This means that the four independent variables can cause manipulation on the dependent variable (price). I have chosen the four independent variables because their manipulation can be used to investigate whether there is any significant variation in the price. The aim is to determine whether the four independent variables have any correlation with the dependent variable, that is, the price of the homes sold around the city. Furthermore, the main objective is to find out the key factors that influence the prices of homes in the city, and the manner in which they influence these prices. I scaled down to these factors after establishing that they are the most influential among a set of multiple factors. The description dependent/independent is apparent in this context, because in case there exists a correlation, it cannot be that price influenced bedrooms, fireplace, pool/spa, or community gate, but an influence in the vice versa is feasible.

Regression analysis is also used in this model to help recognize how the distinctive value of the dependent variable gets adjusted when any of the independent variables are manipulated, while the other independent variables are kept constant. In particular, the regression analysis is used to determine the approximate values of the conditional expectation of the price variable, given the independent variables. In other words, this means the dependent variable’s average value given that the independent variables are held constant. This model can be used to speculate the prices of homes in Scottsdale city. In addition, the model is used to identify the independent variables that have some relationship with the dependent variable, as well as to find out the forms that these relationships take.

In this model, ANOVA has been used to present the observed variance in a certain variables, by dividing it into constituents that comes from varied sources of variation. At the same time, ANOVA is used to present statistical test of establishing whether different groups have similar means – if so, *t*-test is generalized to more than two groups. ANOVA is very useful in reducing type I error in this analysis because it involves multiple two-sample *t*-tests.

The analysis of variance conducted for the data, contains four, two-level factors: bedrooms, fireplace, pool/spa, and community gate. There are 30 observations in total. The ANOVA splits the variance into the following parts of sum of squares:

- The total sum of squares. This has DF as no. observations – 1
- Summation of factor squares. The DF = factor levels – 1.
- Residual squares summation. The degree of freedom is obtained by netting the total of the factor degrees of freedom from the total degrees of freedom.

The *F* test for the factor effects is provided by the ANOVA table shown below.

### Data Description

The data has been obtained from a survey carried out on 30 real estate professionals, who are involved in the business of selling homes in Scottsdale city. The degrees of freedoms are shown in the ‘DF’ line, and the mean square terms are shown in the “MS” line. The sum of squares terms is as shown in ‘SS’ line. As revealed from the table below, the model degree of freedom (MDF) can be assigned *p*, and the error degrees of freedom (DFE) assigned (*n-p-*1). The total degrees of freedom (DFE) are assigned (*n*-1), which is the total of DFE and DFM.

From the table, the F statistics is equal to 566324.2131/109582.4314 = 5.168020144. The likelihood of detecting a value ≥ 5.168020144 is 0.003564038. The test statistics is calculated by dividing the mean square model by the mean square error (MSM/MSE). When the mean square model is larger than the mean square error, then the ratio is huge and there is enough prove for the null hypothesis. In this regression, the test statistics MSM/MSE is represented by *F *(*p, n-p*-1) distribution; that is, *F* (4, 25).

The null hypothesis is stated as

β_{1} = β_{2} =… = β_{p} = 0, while the alternative hypothesis is at least one of the constraints β_{j}≠ 0, j = 1, 2, ,,, *p*. When the values of the test statistic are very large, then this is evidence against the null hypothesis. The square multiple correlation coefficient is represented by the ratio SSM/SST = R^{2}

## Presentation and Discussion of Results

### Results

Other than fireplace, all the other *t*-statistics are greater than 1.96, which means that they are significant at 5% level.

### Discussion

The *t*-statistics for all the variables, which is denoted by the coefficient divided by the standard error, is as shown in the table above. Bedrooms, fire place, pools, and gated community are significant, which means that they add statistically important descriptive power to SP. The coefficient of determination (R square) helps determine the correlation between the independent and the dependent variables. An R^{2 }value of 0.45 means that approximately 45% of the total price variation is explained by the four independent variables. This means that these variables are responsible for the value of comparison. In other words, this is academically known as the linear influence of the variables’ independent right-hand-side.

When finding the regression relationship of price of homes with each of the four independent variables individually, it is realized that each variable is strongly associated with the price of the homes. Both in the magnitude and direction, it means that the influence on the price of homes of each explanatory variable is not caused by human error, but has been approximated by the model. Every additional bedroom in the homes results in $ 100.11 increase in sales price. Just as we could expect, the bedroom coefficient is positive because it is natural that additional bedroom attracts additional cost. Surprisingly, the pool/spa coefficient is negative, denoting a negative relationship between pool/spa and price of the homes (Dubin, 1998).

This, rather unexpected relationship, attracted my attention and I researched around to find out why it was the case. Consequently, I found that the demand for houses with pools were losing demand, therefore, becoming less costly. Some of the factors that cause this reduction include the fact that most of the pools takes up more than 30% of the backyard, which could be used for other more economical activities; most of the pools are very old, hence requiring a lot of funds for repair; some pools do not have safety gate surrounding them; the swimming weather is very short; and because the area around the city is currently experiencing drought. All these factors have made the homes with pools very unattractive. Therefore, inclusion of a pool in a home leads to reduction of the price of the home by $ 209.2. The other two coefficients; that is, gated community and fireplace, are also directly related with the price because many home seekers are looking for security enhancement features and a fireplace that is separated from the rest of the home because of its convenience. Fireplace has the highest statistical significance, with inclusion of each fireplace leading to an increase of the price of home by a whopping $ 357.5. This is due to the fact that the area around the city is very cold, hence the strong need for a fireplace by the home buyers (Do & Gary, 1995).

The results of this analysis demonstrate the impact of homes’ characteristics on their selling prices. These characteristics have shown mixed results on the selling price. This analysis shows that some of those variations can be explained by variations in home prices. Specifically, the regression coefficients of some variables have varied behavior across varied home price levels. Those buyers who can afford to purchase very expensive homes have some home preferences, which are different from buyers who cannot afford expensive homes. This analysis has provided some interesting observations. For instance, gated community is frequently used to establish a home’s assessed value because it is probable that it will have a strong impact on the selling price. While this observation is borne in previous studies, it will be interesting to observe how home buyers coming from various price categories will attach relevance to this variable. This is demonstrated by the considerable variation between the coefficients at the highest and the lowest quartiles – this means that the extra price for the highest-priced homes is considerably more than the extra price for the lowest-priced homes. This analysis has presented an impression of the fundamentals of multiple regression and demonstrates its application in real estate appraisal for housing property. While the actual appraisers have applied this technique for a number of years, its application by assessors is somewhat new. The results of this analysis supplement the body of research demonstrating the prices of homes (Benson et al., 1998).

## Conclusion

The regression analysis has helped to confirm the expectations of the variations of prices of homes, which would have rather been considered spontaneous. Regression model is used to help identify how the distinctive value of the dependent variable changes when any of the independent variables are manipulated. In particular, the regression analysis is used to determine the approximate values of the conditional expectation of the price variable, given the independent variables. The analysis has demonstrated that there are some positive relationships between price as the dependent variable on one hand, and gated community, bedrooms, and fireplace on the other hand. This positive relationship shows that buyers are attracted by homes with fireplaces, and gated community because of different reasons. The most attractive variable, fireplaces, is preferred by many home buyers particularly because the city is characteristically cold and hence the high prices. However, the buyers appear to lack preference for homes with pools because of the multiple inconveniences associated with such luxurious facilities, including high cost of repair, draught season, and wasted space among others – this is why this variable has a strong negative regression coefficient. An R^{2 }value of 0.45 means that approximately 45% of the total price variation is explained by the four independent variables. This means that these variables are responsible for the value of comparison.

## References

Benson, E.D, et al. (1998). Pricing Residential Amenities: The Value of a View. *Journal of Real Estate Finance and Economics, *16(1), 55-73.

Do, A. Q., & Gary G. (1995). Golf Courses and Residential House Prices: An Empirical Examination. *Journal of Real Estate Finance and Economics, *10(3), 261-270.

Dubin, R. A. (1998). Predicting House Prices Using Multiple Listings Data. *Journal of Real Estate Finance and Economics *17(1), 35-59.