Monthly rental price of a comparable home and the pricing of a two-bedroom home

© 2017 The Authors. This is an open access article under the terms of the Creative Commons Attribution License 4.0, which allows use, distribution and reproduction in any medium, provided the original work is properly cited. DOI: http://dx.doi.org/10.18533/jefs.v5i04.257 ISSN 2379-9463(Print), ISSN 2379-9471(Online)


Introduction
The purpose of this study is to predict the price of a two bedroom home located in one of the United States' most populated cities1 .Specifically, we wish to see how the listing price of a two bedroom home is affected by the monthly rental price of a comparable home2 in the same city.
Our specific choice of real-estate (two bedroom home) and location (major United States city) fits the needs of our target purchaser: a young family3 looking to plant roots in a metropolis where job opportunities are abundant 4 .This family can either be moving from a non-major city within the United States, or coming from a different country.We have decided to focus on renting versus purchasing because a large down payment, or insufficient credit history, may prevent a young family from purchasing a home, and therefore renting versus purchasing becomes a legitimate decision.

Data selection
To predict how the listing price of a two bedroom home is affected by the monthly rental price of a comparable home in the same city, we began by extracting data from Zillow's online database (Zillow, 2016).Zillow's database included real estate data for the majority of cities throughout the United States, measuring multiple spectrums such as listing price, rental price, and changes in home value.The variables we selected for our analysis are shown in Table 1.The median listing price of a two bedroom home.

Explanatory variables (per city of interest) Med_RnP_1B
The median rental price per month of a one bedroom home.

Med_RnP_2B
The median rental price per month of a two bedroom home.

Med_RnP_3B
The median rental price per month of a three bedroom home.

Med_RnP_Condo
The median rental price per month of a condo.

PR_Ratio
The price-to rent-ratio for all homes.The price-to rent-ratio is defined as the median listing price for all homes, divided by twelve times the median monthly rental price for all similar homes.The lower the ratio, the smaller the gap between rental costs and listing prices, and therefore the decision to buy a home versus renting a similar one becomes more attractive (Humphries,  2010).

Inv_Measure
The inventory measure.The inventory measure is defined as the number of single-family, condominium and cooperative housing units for sale.This data has been smoothed and seasonally adjusted.Incr_Val050; Incr_Val5070; Incr_Val7090; Incr_Val90100 The percentage of homes increasing in value (expressed as four dummy variables). 5ere are three important details worth noting about our data.First, for all variables, we used the average of the twelve month period spanning April 2015 to March 2016.Second, as previously noted, our data focused on the one most populated city of each state.Third, due to data availability, we have omitted four states: Maine, North Dakota, Vermont, and Wyoming.Therefore, our sample size is forty-six rather than fifty.

Descriptive statistics
Table 2 provides the mean, standard deviation, minimum and maximum for both our dependent and explanatory variables.Interestingly, the maximum listing price of a two bedroom home is roughly ten times the minimum listing price; however, the maximum rental price for all units is roughly five times the minimum rental price.Therefore, one could argue that the housing price disparity across states is greater for purchasing a home than it is for renting one.
To capture the pricing differences across cities, figure 1 graphs each city's listing price for a two bedroom home.The cost of housing in New York, New York is substantially greater than that of the United States' other major cities (roughly 30% more expensive than its predecessor, Boston, Massachusetts); therefore, it is on the verge of becoming an outlier6 .

Journal of Economic and Financial Studies (JEFS)
Page 58  As seen in table 1, to account for housing that is comparable to a two bedroom home, we included the monthly rental price of a one bedroom home, two bedroom home, three bedroom home and condominium, under the assumption that these are realistic alternatives to purchasing a two bedroom home.To further capture the difference between purchasing and renting, we included the price-to-rent ratio for all homes.Additionally, we added two explanatory variables to sketch the health of each city's real estate market.These included each city's inventory measure, and percentage of homes increasing in value.
Our decision to include these variables were not only to account for rental costs and to measure the health of each city's real estate market, but also to alleviate omitted variable bias.Our regressors help us to alleviate omitted variable bias for two reasons.Each of them, intuitively, are "a determinant of the dependent variable", and all of them are correlated with at least one other regressor, as shown in table 3 (Stock & Watson, 2015).
Moving on, as shown earlier in table 1, our explanatory variable explaining "the percentage of homes increasing in value" was transformed into four dummy variables, now explained in table 4. Indicator = 1 if the percentage of homes increasing in value is greater than or equal to 90%, but less than or equal to 100%.
These four dummy variable "buckets" indicate the strength of the housing market.The higher the percentage of homes increasing in value, the stronger the housing market per city.For our analysis, in order to avoid perfect multicollinearity, we will drop the fourth bucket representing 90 ≥ [% of homes increasing in value] ≤ 100%.We chose this bucket because it represents the least amount of cities7 .

OLS assumptions and limitations
Our project attempts to follow the four ordinary least squares (OLS) assumptions for multivariate regression (Stock & Watson, 2015).The first OLS assumption states that the conditional distribution of the error term is expected to equal zero.Naturally, this is a fairly unrealistic assumption, and therefore our analysis is slightly limited in this sense; however, we have done our best to include a number of variables8 that realistically capture the complete error term.Second, we attempted to keep our dependent and explanatory variables independently and identically disrupted.Nonetheless, the variables of our analysis were not particularly selected independently (i.e.randomly).In order to pinpoint specific cities (those most populated per state), we "cherrypicked" specific cities, and therefore our analysis was not random.However, our variables were identically distributed (e.g.all coming from a single large population, the United States).
The third OLS assumption claims that "large outliers are unlikely" (Stock & Watson, 2015).The graphs in figure 2 plot the median listing price of a two-bedroom home on the y-axis, and the median monthly rental price of a one-bedroom home, two-bedroom home, three bedroom home, and condo on the x-axis (starting from the top left, going clockwise).
Based on figure 2, we argue that our data maintains the third OLS assumption9 .As seen in figure 2, there are a few data points that are a bit "out there" (for example, New York, New York being at the far upper right of each graph); however, we have decided to include these points because they are relevant to the discussion at hand -we want to predict how the listing price of a two-bedroom home is affected by the monthly rental price of a comparable home in the same city across the United States, not within an isolated geography or city economy.
The fourth and final OLS assumption says that there can be no perfect multicollinearity.Our regression holds this assumption -no regressor is a linear function of another regressor (Wei, 2016).Regarding our complete set of dummy variables, as mentioned earlier, in order to avoid perfect multicollinearity we chose to drop the dummy variable representing 90 ≥ [% of homes increasing in value] ≤ 100%.
Figure 2: Median listing price of a two bedroom home against the median rental price of a comparable home Aside from our OLS assumptions, there are two important limitations worth noting.First, our sample size of forty-six is rather small; therefore, we do not gain the statistical benefits of a large sample size such as data variability.Second, even though each city is the most populated of its' state, they all have meaningfully different economies and living standards.In light of this, we are attempting to capture a significantly broad range of data, which is damaging because it becomes difficult to pinpoint a meaningful finding.However, that is the nature of our study; as mentioned earlier, we want to predict how the listing price of a two bedroom home is affected by the monthly rental price of a comparable home in the same city across the United States, not within a geographically isolated economy.

Model specification
We use OLS single and multivariate regression analysis to predict how the listing price of a two bedroom home is affected by the monthly rental price of a comparable home in the same city, paired with the health of the real estate market for that city.
Our dependent variable is always the median listing price of a two bedroom home, and the explanatory variables that we use vary with our analysis. 10Robust standard errors are used in each equation in order to control for heteroscedasticity.With all variables included, our regression builds as follows: +  5 (_) +  6 (_) +  7 (_050) +  8 (_5070) +  9 (_7090)

Data analysis and results
Table 5 presents the results of our regression.According to regressions (1) through (4), the monthly rental price of a residence comparable to a two-bedroom home is a reasonable measure of a two-bedroom home's listing price -at least when used as a single variate regression.For each of our first four regressions, as the monthly rental price of a comparable home increases by one dollar, the listing price of a two-bedroom home increases by an amount in the $200 to $300 dollar range.However, as shown in regressions (5) through (8), when the prices of our four comparable living spaces are simultaneously used to determine our listing price, the majority (aside from condos) become insignificant.However, even though regressions (5) through (8) are fairly insignificant, both the rental price of a condo and the price-to-rent ratio stay consistently significant at the 99% level.Moving on, we found it rather odd that our binary buckets expressing the percentage of homes increasing in value were proven insignificant in regression (8).Intuitively, the health of the housing market should be a strong determinant of housing prices.As shown in table 7, they proved insignificant as a stand-alone multivariate regression explaining the price of a two bedroom home.In fact, even as tested single variate regressions, all but one 12 of our four binary buckets were insignificant.
Our argument for this phenomena was that this statistic captured all homes 13 (not only two bedroom homes), which may or may not be increasing in value in that particular city.

Conclusions
According to our results, the purveyor who wishes to predict the price of a two bedroom home, based on the monthly rental payments of a comparable home in the same city, should either use one of the single variate regressions with monthly rental payments as the explanatory variable (expressed in regressions (1) through (4) in table 5); or multivariate regression (8), bearing only the rental price of a condo and the price-to-rent ratio into consideration as their significant explanatory variables.Generally speaking, multivariate regression (8) is the best estimator for our study primarily to alleviate omitted variable bias.
Conferring with our multivariate regression reveals that as the monthly rental price of a condo increases by $1.00 (holding all other variables constant), the listing price of a two bedroom home will increase by $160.65.Likewise, holding all other variables constant, as the price-to-rent ratio increases by one, the listing price of a two bedroom home will increase by $16,479.19. 14Two additional, broad conclusions can be drawn from our analysis.First, if the rental price of a condo is going up, so is the listing price of a two bedroom home.Therefore, a frugal home buyer should keep an eye on the rental price of condos; when rental prices decrease, that is when they should consider purchasing a home.Second, as the price-to-rent ratio decreases 15 , so will the price of a two bedroom home.

Figure 1 :
Figure 1: Listing price of a two bedroom home

Table 1 :
Selected variables Dependent variable (per city of interest) Med_LstP_2B

Table 3 :
Explanatory variable correlation

Table 4 :
Percentage of homes increasing in value (per city of interest)

Table 5 :
Regression results

Table 7 :
Regression results **The individual coefficient is statistically significant at the 1% significance level using a two-sided test.