# Determinants of House Prices: A Quantile Regression Approach

## Authors

- First Online:

DOI: 10.1007/s11146-007-9053-7

- Cite this article as:
- Zietz, J., Zietz, E.N. & Sirmans, G.S. J Real Estate Finance Econ (2008) 37: 317. doi:10.1007/s11146-007-9053-7

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## Abstract

OLS regression has typically been used in housing research to determine the relationship of a particular housing characteristic with selling price. Results differ across studies, not only in terms of size of OLS coefficients and statistical significance, but sometimes in direction of effect. This study suggests that some of the observed variation in the estimated prices of housing characteristics may reflect the fact that characteristics are not priced the same across a given distribution of house prices. To examine this issue, this study uses quantile regression, with and without accounting for spatial autocorrecation, to identify the coefficients of a large set of diverse variables across different quantiles. The results show that purchasers of higher-priced homes value certain housing characteristics such as square footage and the number of bathrooms differently from buyers of lower-priced homes. Other variables such as age are also shown to vary across the distribution of house prices.

### Keywords

Hedonic price functionQuantile regressionSpatial lag### JEL Classification

R31C21C29## Introduction

The published real estate literature has put forth a number of housing characteristics to explain house prices. Hedonic regression analysis is typically used to identify the marginal effect on house price of each of these housing characteristics. Sirmans et al. (2005) examine hedonic pricing models for more than 125 empirical studies and find that studies often disagree on both the magnitude and direction of the effect of certain characteristics. For example, their analysis shows that, of 40 empirical studies examining the number of bedrooms, 21 studies find that bedrooms have a positive impact on house price, nine studies identify a negative relationship, and ten studies report no significant relationship between house price and the number of bedrooms.

Different estimation results for a given variable, in particular disagreement on the direction of the effect, can be confusing to market participants. In addition, there may be reason to believe that housing characteristics are not valued the same across a given distribution of house prices. Malpezzi et al. (1980) acknowledge the problem in valuing individual house features and note that the impact on price of individual features cannot be easily quantified. Malpezzi (2003) also notes that different consumers may value housing characteristics differently. To alleviate some of the confusion, this study examines the extent to which conflicting results may be attributed to differences in the effect of housing characteristics across the distribution of house prices. For example, if a particular housing characteristic is priced differently for houses in the upper-price range as compared to houses in the lower-price range, the typical OLS regression may not provide useful information for either price range since it is based on the mean of the entire price distribution.

As an alternative to OLS regression, this study uses quantile regression to identify the implicit prices of housing characteristics for different points in the distribution of house prices. This explicitly allows higher-priced houses to have a different implicit price for a housing characteristic than lower-priced houses. Since quantile regression uses the entire sample, the problem of truncation is avoided (Heckman 1979). This will eliminate the problem of biased estimates that is created when OLS is applied to house price sub-samples (e.g., Newsome and Zietz 1992).

## The Implicit Pricing of Housing Characteristics

Variables with predominantly consistent results across studies

Variable | Appearances | No. times positive | No. times negative | No. times non-significant |
---|---|---|---|---|

Lot size | 52 | 45 | 0 | 7 |

Square feet | 69 | 62 | 4 | 3 |

Brick | 13 | 9 | 0 | 4 |

No. bathrooms | 40 | 34 | 1 | 5 |

No. rooms | 14 | 10 | 1 | 3 |

Full baths | 37 | 31 | 1 | 5 |

Fireplace | 57 | 43 | 3 | 11 |

Air-conditioning | 37 | 34 | 1 | 2 |

Basement | 21 | 15 | 1 | 5 |

Garage spaces | 61 | 48 | 0 | 13 |

Pool | 31 | 27 | 0 | 4 |

Variables with predominantly inconsistent results across studies

Variable | Appearances | No. times positive | No. times negative | No. times not significant |
---|---|---|---|---|

Age | 78 | 7 | 63 | 8 |

Bedrooms | 40 | 21 | 9 | 10 |

Distance | 15 | 5 | 5 | 5 |

Time on market | 18 | 1 | 8 | 9 |

A key question is the cause of this parameter uncertainty. Based on the findings of Sirmans et al. (2005), it seems unlikely that parameter variation for housing characteristics can be fully explained by regional differences, different specifications, or alternative data sets. In addition, as suggested by Newsome and Zietz (1992), housing characteristics may not be valued the same across a given distribution of housing prices. Specifically, the marginal value, percentage contribution, or elasticity value of a certain housing characteristic may be different across the range of house prices. In fact, would one expect to find that owners of high-end houses and low-end houses attach the same value to every housing characteristic? This would require that the preference structure of all homeowners be identical and that the owners of low-end and high-end homes differ only in the income constraint they face.

As discussed by Rosen (1974), Epple (1987), and Bartik (1987), the demand and supply functions that underlie hedonic price equations can be very difficult to identify empirically. The general acceptance of hedonic pricing models in real estate application rests on the assumption that the underlying supply function of housing characteristics is vertical in price/quantity space. The supply of housing characteristics is fixed at any given point in time and is independent of the implicit price of a characteristic. The intersection of the downward sloping demand curve for a housing characteristic with the given vertical supply curve of that characteristic identifies the implicit price of the housing characteristic. This implicit price is identical to the one generated by the hedonic pricing model. Assuming that all consumers are equal, then the implicit price of a characteristic is the implied valuation of that characteristic by the representative consumer. OLS estimation fits nicely into this representative agent framework since it identifies those implicit prices that optimally predict the mean house price for a given sample.

A problem arises when the relevance of the representative agent paradigm is questioned.^{1} For the sake of argument, assume that there are two consumers: a “poor” one who is income and credit constrained and a “rich” one who is not. The poor consumer is not in the market for an expensive house because no bank will underwrite the needed loan and the rich household would not think of buying a poor man’s house because it does not provide the desired amenities and may negatively affect his/her desire for social status. Thus, in essence, there are two segmented markets. Segmentation may not only imply that the rich and the poor occupy houses of different values but they may also develop group-specific likes and dislikes of certain housing characteristics.^{2} Builders, aware of this situation, would build houses to fit the perceived needs of the groups. What results is not one set of supply curves of housing characteristics but two, one for the “rich” household and one for the “poor” household. Similarly, there are two sets of demand curves for each housing characteristic resulting in two sets of implicit prices for housing characteristics.

The above argument suggests that there may be marked differences in the elasticity of house price with respect to housing characteristics across the distribution of housing prices. A seemingly logical approach would be to tie the different segments to the house price. A high house price rations “poor” households out of the market intended for “rich” households and a low housing price is a sufficient deterrent for entry by a “rich” household. The major task is to identify the different market segments and their implicit prices. In this regard, the usefulness of OLS regression may be questioned and a more appropriate approach may be quantile regression.

## Quantile Regression Methodology

Quantile regression is based on the minimization of weighted absolute deviations (also known as L_1 method) to estimate conditional quantile (percentile) functions (Koenker and Bassett 1978; Koenker and Hallock 2001). For the median (quantile = 0.5), symmetric weights are used, and for all other quantiles (e.g., 0.1, 0.2 ....., 0.9) asymmetric weights are employed. In contrast, classical OLS regression (also known as L_2 method) estimates conditional mean functions. Unlike OLS, quantile regression is not limited to explaining the mean of the dependent variable. It can be employed to explain the determinants of the dependent variable at any point of the distribution of the dependent variable. For hedonic price functions, quantile regression makes it possible to statistically examine the extent to which housing characteristics are valued differently across the distribution of housing prices.

One may argue that the same goal may be accomplished by segmenting the dependent variable, such as house price, into subsets according to its unconditional distribution and then applying OLS on the subsets, as done, for example, in Newsome and Zietz (1992). However, as clearly argued by Heckman (1979), this “truncation of the dependent variable” may create biased parameter estimates and should be avoided. Since quantile regression employs the full data set, a sample selection problem does not arise.

*y*

_{i}is the dependent variable at observation

*i*,

*x*

_{j,i}the

*j*th regressor variable at observation

*i*, and

*b*

_{j}an estimate of the model’s

*j*th regression coefficient. By contrast, quantile regression minimizes a weighted sum of the absolute deviations,

*h*

_{i}is defined as

*i*th observation is strictly positive or as

*i*th observation is negative or zero. The variable

*q*(0 <

*q*< 1) is the quantile to be estimated or predicted.

The standard errors of the coefficient estimates are estimated using bootstrapping as suggested by Gould (1992, 1997). They are significantly less sensitive to heteroskedasticity than the standard error estimates based on the method suggested by Rogers (1993).^{3}

Quantile regression analyzes the similarity or dissimilarity of regression coefficients at different points of the distribution of the dependent variable, which is sales price in our case. It does not consider spatial autocorrelation that may be present in the data. Because similarly priced houses are unlikely to be all clustered geographically, one cannot expect that quantile regression will remove the need to account for spatial autocorrelation.

In this paper, spatial autocorrelation is incorporated into the quantile regression framework through the addition of a spatial lag variable. The spatial lag variable is defined as **W***y*, where **W** is a spatial weight matrix of size *T* × *T*, where *T* is the number of observations, and where *y* is the dependent variable vector, which is of size *T* × 1. Any spatial weight matrix can be employed, for example, one based on the *i*th nearest neighbor method, contiguity, or some other scheme. In the present application, a contiguity matrix is used.^{4}

Adding a spatial lag to an OLS regression is well known to cause inference problems owing to the endogeneity of the spatial lag (Anselin 2001). This is not any different for quantile regression than for OLS. We follow the approach suggested by Kim and Muller (2004) to deal with this endogeneity problem in quantile regression. As instruments we employ the regressors and their spatial lags.^{5} However, instead of using a density function estimator for the derivation of the standard errors, we follow the well established route of bootstrapping the standard errors (Greene 2000, pp. 400–401).^{6}

## Data and Estimation Results

This study uses multiple listing service (MLS) data from the Orem/Provo, Utah area.^{7} The data consist of 1,366 home sales from mid-1999 to mid-2000. Table 3 provides a description of the variables. Most are standard housing characteristics while some are specific to the region. The data also include a number of geographic and neighborhood variables, which are derived by geo-coding all observations. An objective is to measure the effect of quantile regression on a large number of diverse variables. Table 4 gives summary statistics for the explanatory variables and the dependent variable, sale price. The quantile values reported in Table 4 for the independent variables are averages of the values that are associated with the sale prices found in a 5% confidence interval around a given quantile point of the dependent variable (*sp*). For example, the sale price associated with quantile point 0.2 is $123,000. A 5% confidence interval of this quantile point covers the price range from $121,902 to $124,526 and the houses with sale price in this range have on average square footage of 1,760.6.

Variable definitions

Variable | Definition |
---|---|

| Sale price in 1,000 dollars; ln( |

| Spatial lag variable, based on normalized contiguity weight matrix |

| Size of house in square feet, divided by 1,000 |

| Lot size in acres |

| Year in which the property was built |

| Number of bedrooms |

| Number of full bathrooms |

| Number of 3/4 bathrooms (shower, no tub) |

| Number of half baths |

| Number of decks |

| Number of patios |

| Number of garage places |

| Percentage of house covered by finished basement |

| 1 if pool is present, 0 otherwise |

| 1 if air conditioning is evaporator, roof type, 0 otherwise |

| 1 if air conditioning is evaporator, window type, 0 otherwise |

| 1 if air conditioning is electric, 0 otherwise |

| 1 if air conditioning is gas, 0 otherwise |

| 1 if hardwood flooring is present in house, 0 otherwise |

| 1 if tile flooring is present in house, 0 otherwise |

| 1 if exterior is made of stucco |

| 1 if exterior is made of brick |

| 1 if exterior is made of aluminum |

| 1 if exterior is of type frame |

| 1 if full landscaping |

| 1 if partial landscaping |

| 1 if lot contains a sprinkler system |

| 1 if lot has mountain view |

| Distance to interstate Highway 15, in miles (US Topographical map) |

| Distance to city center of Orem, in miles (US Topographical map) |

| Magnitude of largest earthquake, on Richter Scale (EPA data) |

| Percentage of population classified as non-white, by census tract |

| Percentage of all vacant housing units for rent, by census tract |

Basic statistics and quantiles of individual variables, 1,366 observations

| Mean | Min. | Max. | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

| 146.649 | 90.000 | 247.000 | 115.000 | 123.000 | 129.900 | 135.980 | 141.350 | 148.000 | 158.990 | 169.900 | 188.590 |

5% conf. | 113.574 | 121.902 | 128.049 | 134.000 | 139.900 | 146.500 | 156.000 | 166.805 | 185.000 | |||

Interval | 116.961 | 124.526 | 131.500 | 137.675 | 142.500 | 150.000 | 160.000 | 173.000 | 191.000 | |||

| 2.2020 | 0.792 | 4.800 | 1.4709 | 1.7606 | 1.9829 | 2.0385 | 2.1117 | 2.1824 | 2.3931 | 2.5893 | 3.0604 |

| 0.2478 | 0.010 | 2.100 | 0.2598 | 0.2228 | 0.2392 | 0.2193 | 0.2669 | 0.2388 | 0.2664 | 0.3096 | 0.2794 |

| 1975 | 1877 | 2000 | 1957 | 1960 | 1973 | 1976 | 1984 | 1986 | 1984 | 1985 | 1985 |

| 3.76 | 1 | 7 | 3.0455 | 3.5614 | 3.5077 | 3.8714 | 3.7385 | 3.7500 | 3.7705 | 3.9455 | 4.4000 |

| 1.63 | 0 | 5 | 1.1364 | 1.4561 | 1.4615 | 1.5000 | 1.6154 | 1.7143 | 1.8689 | 1.9818 | 2.1714 |

| 0.37 | 0 | 3 | 0.2500 | 0.2281 | 0.3385 | 0.5000 | 0.3692 | 0.3393 | 0.4426 | 0.4364 | 0.4286 |

| 0.21 | 0 | 3 | 0.2500 | 0.1754 | 0.2615 | 0.0714 | 0.2462 | 0.1429 | 0.1639 | 0.2182 | 0.4857 |

| 0.27 | 0 | 3 | 0.1818 | 0.1404 | 0.2154 | 0.2571 | 0.3385 | 0.2500 | 0.4098 | 0.2545 | 0.4000 |

| 0.47 | 0 | 2 | 0.3636 | 0.4737 | 0.5077 | 0.4429 | 0.4615 | 0.5179 | 0.5902 | 0.4727 | 0.6000 |

| 1.39 | 0 | 5 | 0.7500 | 0.7719 | 1.0615 | 1.3143 | 1.6923 | 1.7679 | 1.9016 | 1.7818 | 1.8286 |

| 0.44 | 0 | 1 | 0.2045 | 0.3660 | 0.3708 | 0.5067 | 0.4886 | 0.4609 | 0.4161 | 0.4695 | 0.4957 |

| 0.01 | 0 | 1 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0364 | 0.0000 |

| 0.39 | 0 | 1 | 0.4091 | 0.4035 | 0.3846 | 0.5714 | 0.3846 | 0.4107 | 0.3770 | 0.4000 | 0.2286 |

| 0.11 | 0 | 1 | 0.1136 | 0.2105 | 0.1538 | 0.1000 | 0.1077 | 0.0179 | 0.0492 | 0.0182 | 0.0286 |

| 0.26 | 0 | 1 | 0.1591 | 0.1404 | 0.2000 | 0.1429 | 0.2615 | 0.3571 | 0.2787 | 0.2727 | 0.4571 |

| 0.07 | 0 | 1 | 0.1136 | 0.0351 | 0.0769 | 0.0571 | 0.0615 | 0.0536 | 0.0984 | 0.1636 | 0.1143 |

| 0.28 | 0 | 1 | 0.2500 | 0.3158 | 0.2154 | 0.2429 | 0.1231 | 0.2321 | 0.1967 | 0.2364 | 0.5714 |

| 0.25 | 0 | 1 | 0.1364 | 0.2982 | 0.2154 | 0.2429 | 0.1385 | 0.2679 | 0.2787 | 0.2545 | 0.4571 |

| 0.15 | 0 | 1 | 0.0000 | 0.0702 | 0.1538 | 0.0429 | 0.0462 | 0.1429 | 0.2295 | 0.2909 | 0.3429 |

| 0.69 | 0 | 1 | 0.4773 | 0.5965 | 0.5692 | 0.7286 | 0.8462 | 0.8214 | 0.7377 | 0.7455 | 0.7714 |

| 0.54 | 0 | 1 | 0.5227 | 0.4386 | 0.4000 | 0.5429 | 0.5692 | 0.7143 | 0.6557 | 0.5273 | 0.4857 |

| 0.06 | 0 | 1 | 0.1364 | 0.1053 | 0.1077 | 0.1143 | 0.0615 | 0.0179 | 0.0328 | 0.0727 | 0.0571 |

| 0.71 | 0 | 1 | 0.7045 | 0.6842 | 0.6615 | 0.7571 | 0.7231 | 0.6607 | 0.6721 | 0.6000 | 0.7143 |

| 0.10 | 0 | 1 | 0.1591 | 0.1053 | 0.1077 | 0.1429 | 0.0769 | 0.1250 | 0.0656 | 0.1091 | 0.1143 |

| 0.54 | 0 | 1 | 0.2955 | 0.3860 | 0.4000 | 0.5143 | 0.5231 | 0.6964 | 0.6393 | 0.5273 | 0.7429 |

| 0.64 | 0 | 1 | 0.4318 | 0.5614 | 0.6000 | 0.7143 | 0.7385 | 0.6964 | 0.7705 | 0.6727 | 0.6286 |

| 1.56 | 0.01 | 11.13 | 1.1520 | 1.4553 | 1.8746 | 1.2814 | 1.4517 | 1.8157 | 1.6734 | 1.9460 | 1.7200 |

| 6.53 | 0.26 | 22.30 | 7.0700 | 6.5818 | 7.7612 | 4.6543 | 5.5723 | 6.5064 | 6.7964 | 6.5222 | 6.4486 |

| 1.55 | 0.12 | 4.08 | 1.8268 | 1.5537 | 1.5674 | 1.6931 | 1.6769 | 1.4861 | 1.4915 | 1.3985 | 1.5229 |

| 0.07 | 0.02 | 0.23 | 0.0966 | 0.0792 | 0.0721 | 0.0812 | 0.0806 | 0.0638 | 0.0633 | 0.0646 | 0.0646 |

| 0.23 | 0.00 | 0.76 | 0.3238 | 0.2768 | 0.2105 | 0.2420 | 0.2509 | 0.1995 | 0.1984 | 0.1953 | 0.1939 |

*sp*) is expressed in logged form,

*α*is a constant term,

*β*

_{i}is the regression coefficient for the

*i*th housing characteristic,

*X*

_{i}, and

*ɛ*is the residual error term.

*p*values).

*P*values of less than 0.05 indicate statistical significance of a coefficient estimate at the 5% level or better.

^{8}Both Tables 5 and 6 present the results of the standard OLS regression in the leftmost column and the estimates of the quantile regressions in the remainder of the tables.

^{9}The points on which the quantile regressions are centered are provided in the first row of Table 4. Tables 7 and 8 present the quantile regression results when spatial autocorrelation is taken into account.

Coefficient estimates, OLS and by quantile

| OLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|---|

| 1.9733 | 0.5323 | 1.1311 | 1.3725 | 1.6984 | 1.8964 | 1.9134 | 2.2572 | 2.3973 | 2.7520 |

| 0.1179 | 0.0896 | 0.1025 | 0.1073 | 0.1083 | 0.1228 | 0.1254 | 0.1324 | 0.1346 | 0.1376 |

| 0.1549 | 0.1362 | 0.1432 | 0.1400 | 0.1408 | 0.1858 | 0.1824 | 0.1917 | 0.2292 | 0.3223 |

| 0.0013 | 0.0020 | 0.0017 | 0.0015 | 0.0014 | 0.0013 | 0.0013 | 0.0011 | 0.0011 | 0.0009 |

| 0.0052 | 0.0090 | 0.0071 | 0.0137 | 0.0115 | 0.0088 | 0.0068 | 0.0043 | 0.0063 | 0.0020 |

| 0.0510 | 0.0372 | 0.0473 | 0.0457 | 0.0511 | 0.0479 | 0.0462 | 0.0468 | 0.0506 | 0.0694 |

| 0.0221 | 0.0161 | 0.0169 | 0.0145 | 0.0210 | 0.0166 | 0.0202 | 0.0203 | 0.0261 | 0.0477 |

| 0.0268 | −0.0041 | 0.0220 | 0.0207 | 0.0192 | 0.0215 | 0.0300 | 0.0390 | 0.0469 | 0.0441 |

| 0.0051 | 0.0036 | 0.0076 | 0.0060 | 0.0045 | 0.0099 | 0.0111 | 0.0031 | −0.0011 | 0.0031 |

| 0.0050 | 0.0073 | 0.0041 | 0.0062 | 0.0042 | 0.0029 | 0.0035 | 0.0077 | 0.0015 | 0.0039 |

| 0.0268 | 0.0284 | 0.0263 | 0.0268 | 0.0272 | 0.0265 | 0.0275 | 0.0254 | 0.0235 | 0.0246 |

| 0.0002 | 0.0015 | 0.0020 | −0.0054 | −0.0062 | −0.0067 | 0.0023 | 0.0015 | −0.0037 | −0.0081 |

| 0.0106 | 0.0542 | 0.0560 | 0.0428 | 0.0451 | 0.0090 | 0.0068 | −0.0036 | −0.0035 | 0.0086 |

| −0.0045 | 0.0016 | −0.0065 | −0.0079 | −0.0092 | −0.0103 | −0.0066 | −0.0081 | −0.0149 | 0.0006 |

| −0.0060 | 0.0191 | 0.0073 | 0.0085 | −0.0006 | −0.0139 | −0.0125 | −0.0199 | −0.0192 | −0.0030 |

| 0.0283 | 0.0400 | 0.0205 | 0.0175 | 0.0199 | 0.0189 | 0.0237 | 0.0247 | 0.0200 | 0.0359 |

| −0.0023 | −0.0247 | −0.0117 | −0.0013 | −0.0025 | −0.0046 | 0.0041 | 0.0103 | 0.0073 | 0.0015 |

| 0.0290 | 0.0261 | 0.0249 | 0.0255 | 0.0292 | 0.0296 | 0.0338 | 0.0351 | 0.0341 | 0.0319 |

| 0.0174 | 0.0062 | 0.0111 | 0.0158 | 0.0197 | 0.0191 | 0.0217 | 0.0258 | 0.0200 | 0.0069 |

| 0.0724 | 0.0691 | 0.0640 | 0.0736 | 0.0744 | 0.0745 | 0.0722 | 0.0711 | 0.0652 | 0.0546 |

| 0.0119 | 0.0209 | 0.0150 | 0.0172 | 0.0157 | 0.0123 | 0.0112 | 0.0127 | 0.0081 | −0.0128 |

| 0.0207 | 0.0303 | 0.0295 | 0.0254 | 0.0273 | 0.0264 | 0.0227 | 0.0227 | 0.0156 | 0.0092 |

| 0.0158 | 0.0304 | 0.0144 | 0.0097 | 0.0060 | 0.0048 | 0.0082 | 0.0178 | 0.0025 | 0.0154 |

| 0.0023 | 0.0286 | 0.0152 | 0.0116 | 0.0108 | −0.0016 | −0.0071 | −0.0064 | −0.0090 | −0.0250 |

| −0.0114 | 0.0066 | 0.0096 | −0.0064 | −0.0148 | −0.0398 | −0.0422 | −0.0236 | −0.0142 | −0.0109 |

| 0.0238 | 0.0335 | 0.0236 | 0.0230 | 0.0231 | 0.0203 | 0.0175 | 0.0228 | 0.0237 | 0.0243 |

| 0.0136 | 0.0266 | 0.0214 | 0.0191 | 0.0135 | 0.0123 | 0.0090 | 0.0030 | −0.0023 | −0.0109 |

| 0.0046 | 0.0062 | 0.0020 | 0.0030 | 0.0019 | 0.0033 | 0.0074 | 0.0100 | 0.0102 | 0.0086 |

| −0.0018 | −0.0027 | −0.0028 | −0.0021 | −0.0018 | −0.0025 | −0.0019 | −0.0016 | −0.0018 | −0.0017 |

| 0.0025 | −0.0011 | −0.0017 | −0.0030 | −0.0041 | −0.0041 | −0.0001 | 0.0043 | 0.0072 | 0.0182 |

| −0.2315 | −0.2552 | −0.1894 | −0.1462 | −0.1695 | −0.1832 | −0.2398 | −0.2114 | −0.2206 | −0.2992 |

| −0.0199 | −0.0189 | −0.0179 | −0.0237 | −0.0135 | −0.0125 | −0.0131 | −0.0084 | −0.0091 | −0.0316 |

| 0.7648 | 0.5225 | 0.5307 | 0.5428 | 0.5476 | 0.5539 | 0.5648 | 0.5684 | 0.5688 | 0.5485 |

*P* values of coefficient estimates, OLS and by quantile

| OLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|---|

| 0.000 | 0.352 | 0.009 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.000 | 0.010 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.007 |

| 0.171 | 0.040 | 0.080 | 0.001 | 0.013 | 0.043 | 0.020 | 0.312 | 0.193 | 0.812 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.000 | 0.051 | 0.021 | 0.040 | 0.000 | 0.028 | 0.022 | 0.047 | 0.008 | 0.000 |

| 0.000 | 0.717 | 0.041 | 0.003 | 0.006 | 0.006 | 0.002 | 0.001 | 0.000 | 0.000 |

| 0.362 | 0.669 | 0.242 | 0.257 | 0.547 | 0.150 | 0.035 | 0.592 | 0.868 | 0.704 |

| 0.310 | 0.270 | 0.536 | 0.371 | 0.533 | 0.633 | 0.486 | 0.160 | 0.823 | 0.560 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 |

| 0.982 | 0.817 | 0.750 | 0.447 | 0.441 | 0.289 | 0.765 | 0.876 | 0.740 | 0.493 |

| 0.581 | 0.068 | 0.112 | 0.187 | 0.088 | 0.742 | 0.815 | 0.860 | 0.906 | 0.812 |

| 0.561 | 0.909 | 0.555 | 0.418 | 0.244 | 0.227 | 0.538 | 0.484 | 0.138 | 0.957 |

| 0.555 | 0.261 | 0.590 | 0.412 | 0.948 | 0.076 | 0.274 | 0.138 | 0.110 | 0.817 |

| 0.000 | 0.003 | 0.082 | 0.109 | 0.026 | 0.004 | 0.016 | 0.028 | 0.059 | 0.004 |

| 0.841 | 0.324 | 0.412 | 0.916 | 0.835 | 0.731 | 0.830 | 0.617 | 0.618 | 0.911 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.002 | 0.406 | 0.143 | 0.035 | 0.001 | 0.007 | 0.006 | 0.002 | 0.054 | 0.547 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 |

| 0.052 | 0.043 | 0.016 | 0.001 | 0.000 | 0.002 | 0.139 | 0.175 | 0.416 | 0.326 |

| 0.001 | 0.001 | 0.001 | 0.000 | 0.000 | 0.000 | 0.001 | 0.010 | 0.070 | 0.466 |

| 0.105 | 0.016 | 0.333 | 0.494 | 0.506 | 0.569 | 0.487 | 0.188 | 0.854 | 0.486 |

| 0.768 | 0.061 | 0.139 | 0.143 | 0.303 | 0.864 | 0.521 | 0.470 | 0.415 | 0.107 |

| 0.323 | 0.771 | 0.509 | 0.472 | 0.197 | 0.001 | 0.002 | 0.206 | 0.459 | 0.589 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.002 | 0.002 | 0.003 | 0.035 |

| 0.013 | 0.009 | 0.002 | 0.000 | 0.066 | 0.047 | 0.103 | 0.666 | 0.800 | 0.253 |

| 0.075 | 0.215 | 0.540 | 0.473 | 0.662 | 0.411 | 0.042 | 0.001 | 0.001 | 0.028 |

| 0.021 | 0.018 | 0.002 | 0.003 | 0.036 | 0.002 | 0.030 | 0.059 | 0.099 | 0.294 |

| 0.535 | 0.860 | 0.690 | 0.510 | 0.397 | 0.411 | 0.986 | 0.326 | 0.267 | 0.017 |

| 0.010 | 0.007 | 0.037 | 0.022 | 0.013 | 0.004 | 0.048 | 0.150 | 0.138 | 0.091 |

| 0.177 | 0.253 | 0.302 | 0.181 | 0.491 | 0.545 | 0.313 | 0.596 | 0.575 | 0.078 |

Coefficient estimates of spatial lag model, 2SLS and by quantile

| 2SLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|---|

| 1.9014 | 0.4768 | 1.1333 | 1.3286 | 1.6242 | 1.8207 | 1.8231 | 2.2428 | 2.4257 | 2.7359 |

| 0.0137 | 0.0077 | 0.0158 | 0.0159 | 0.0127 | 0.0107 | 0.0096 | 0.0062 | 0.0097 | 0.0148 |

| 0.1173 | 0.0905 | 0.1025 | 0.1052 | 0.1078 | 0.1217 | 0.1256 | 0.1318 | 0.1328 | 0.1370 |

| 0.1533 | 0.1363 | 0.1420 | 0.1340 | 0.1420 | 0.1854 | 0.1830 | 0.1968 | 0.2301 | 0.3190 |

| 0.0013 | 0.0020 | 0.0016 | 0.0015 | 0.0014 | 0.0013 | 0.0013 | 0.0011 | 0.0010 | 0.0009 |

| 0.0051 | 0.0087 | 0.0075 | 0.0132 | 0.0109 | 0.0086 | 0.0065 | 0.0046 | 0.0058 | 0.0017 |

| 0.0513 | 0.0382 | 0.0464 | 0.0486 | 0.0512 | 0.0488 | 0.0471 | 0.0476 | 0.0534 | 0.0696 |

| 0.0230 | 0.0177 | 0.0181 | 0.0171 | 0.0221 | 0.0177 | 0.0203 | 0.0215 | 0.0281 | 0.0482 |

| 0.0275 | −0.0031 | 0.0215 | 0.0232 | 0.0198 | 0.0204 | 0.0304 | 0.0401 | 0.0488 | 0.0432 |

| 0.0053 | 0.0049 | 0.0073 | 0.0063 | 0.0042 | 0.0107 | 0.0109 | 0.0031 | −0.0015 | 0.0048 |

| 0.0051 | 0.0047 | 0.0047 | 0.0061 | 0.0050 | 0.0037 | 0.0028 | 0.0069 | −0.0008 | 0.0041 |

| 0.0265 | 0.0279 | 0.0261 | 0.0263 | 0.0264 | 0.0261 | 0.0266 | 0.0259 | 0.0248 | 0.0253 |

| −0.0001 | 0.0007 | 0.0044 | −0.0057 | −0.0056 | −0.0059 | 0.0011 | −0.0001 | −0.0017 | −0.0082 |

| 0.0109 | 0.0567 | 0.0583 | 0.0391 | 0.0482 | 0.0115 | 0.0065 | −0.0037 | −0.0041 | 0.0040 |

| −0.0047 | −0.0004 | −0.0088 | −0.0107 | −0.0097 | −0.0090 | −0.0062 | −0.0100 | −0.0157 | 0.0032 |

| −0.0065 | 0.0185 | 0.0041 | 0.0043 | −0.0015 | −0.0142 | −0.0125 | −0.0221 | −0.0204 | −0.0032 |

| 0.0274 | 0.0369 | 0.0185 | 0.0141 | 0.0182 | 0.0192 | 0.0235 | 0.0229 | 0.0183 | 0.0359 |

| −0.0034 | −0.0274 | −0.0130 | −0.0058 | −0.0033 | −0.0032 | 0.0020 | 0.0081 | 0.0072 | 0.0004 |

| 0.0287 | 0.0284 | 0.0241 | 0.0282 | 0.0287 | 0.0293 | 0.0347 | 0.0342 | 0.0338 | 0.0300 |

| 0.0174 | 0.0049 | 0.0112 | 0.0145 | 0.0195 | 0.0174 | 0.0205 | 0.0263 | 0.0205 | 0.0085 |

| 0.0722 | 0.0667 | 0.0707 | 0.0766 | 0.0744 | 0.0717 | 0.0725 | 0.0704 | 0.0634 | 0.0534 |

| 0.0121 | 0.0195 | 0.0152 | 0.0159 | 0.0159 | 0.0139 | 0.0132 | 0.0121 | 0.0084 | −0.0110 |

| 0.0206 | 0.0268 | 0.0310 | 0.0268 | 0.0272 | 0.0259 | 0.0234 | 0.0233 | 0.0156 | 0.0085 |

| 0.0152 | 0.0279 | 0.0145 | 0.0108 | 0.0042 | 0.0060 | 0.0091 | 0.0184 | 0.0050 | 0.0132 |

| 0.0027 | 0.0262 | 0.0160 | 0.0162 | 0.0086 | −0.0011 | −0.0054 | −0.0051 | −0.0109 | −0.0260 |

| −0.0110 | 0.0046 | 0.0036 | −0.0037 | −0.0156 | −0.0395 | −0.0382 | −0.0195 | −0.0150 | −0.0082 |

| 0.0235 | 0.0344 | 0.0253 | 0.0229 | 0.0234 | 0.0186 | 0.0174 | 0.0227 | 0.0249 | 0.0245 |

| 0.0140 | 0.0267 | 0.0219 | 0.0191 | 0.0145 | 0.0120 | 0.0103 | 0.0042 | −0.0035 | −0.0127 |

| 0.0043 | 0.0057 | 0.0016 | 0.0026 | 0.0008 | 0.0039 | 0.0061 | 0.0102 | 0.0100 | 0.0071 |

| −0.0017 | −0.0026 | −0.0023 | −0.0018 | −0.0019 | −0.0023 | −0.0020 | −0.0016 | −0.0020 | −0.0017 |

| 0.0027 | −0.0022 | −0.0002 | −0.0030 | −0.0038 | −0.0038 | −0.0018 | 0.0047 | 0.0073 | 0.0191 |

| −0.2198 | −0.2269 | −0.1529 | −0.1385 | −0.1637 | −0.1627 | −0.2331 | −0.1800 | −0.2180 | −0.3411 |

| −0.0188 | −0.0195 | −0.0202 | −0.0245 | −0.0100 | −0.0131 | −0.0118 | −0.0090 | −0.0085 | −0.0294 |

*P* Values of coefficients of spatial lag model, 2SLS and by quantile

| 2SLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|---|

| 0.000 | 0.177 | 0.006 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.010 | 0.008 | 0.003 | 0.005 | 0.064 | 0.021 | 0.037 | 0.306 | 0.006 | 0.000 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.000 | 0.007 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.142 | 0.018 | 0.051 | 0.000 | 0.000 | 0.003 | 0.016 | 0.197 | 0.070 | 0.669 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.000 | 0.001 | 0.004 | 0.000 | 0.000 | 0.001 | 0.000 | 0.001 | 0.000 | 0.000 |

| 0.000 | 0.671 | 0.002 | 0.000 | 0.000 | 0.004 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.334 | 0.395 | 0.142 | 0.134 | 0.391 | 0.035 | 0.021 | 0.534 | 0.753 | 0.440 |

| 0.301 | 0.354 | 0.286 | 0.125 | 0.255 | 0.349 | 0.431 | 0.158 | 0.884 | 0.481 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.983 | 0.922 | 0.520 | 0.376 | 0.299 | 0.203 | 0.825 | 0.993 | 0.816 | 0.330 |

| 0.735 | 0.007 | 0.017 | 0.073 | 0.083 | 0.574 | 0.695 | 0.831 | 0.863 | 0.892 |

| 0.512 | 0.954 | 0.139 | 0.044 | 0.110 | 0.178 | 0.351 | 0.178 | 0.027 | 0.796 |

| 0.490 | 0.145 | 0.665 | 0.555 | 0.814 | 0.070 | 0.111 | 0.034 | 0.026 | 0.813 |

| 0.000 | 0.000 | 0.002 | 0.020 | 0.008 | 0.005 | 0.000 | 0.001 | 0.008 | 0.002 |

| 0.756 | 0.221 | 0.332 | 0.555 | 0.722 | 0.741 | 0.834 | 0.467 | 0.427 | 0.973 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.002 | 0.423 | 0.038 | 0.001 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.313 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.035 | 0.010 | 0.001 | 0.000 | 0.000 | 0.001 | 0.005 | 0.043 | 0.087 | 0.140 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.019 | 0.235 |

| 0.137 | 0.004 | 0.087 | 0.116 | 0.604 | 0.439 | 0.201 | 0.092 | 0.564 | 0.467 |

| 0.723 | 0.014 | 0.034 | 0.022 | 0.227 | 0.890 | 0.432 | 0.476 | 0.148 | 0.011 |

| 0.266 | 0.740 | 0.667 | 0.608 | 0.065 | 0.000 | 0.000 | 0.106 | 0.162 | 0.585 |

| 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

| 0.008 | 0.000 | 0.000 | 0.000 | 0.003 | 0.007 | 0.012 | 0.396 | 0.518 | 0.050 |

| 0.032 | 0.020 | 0.333 | 0.185 | 0.759 | 0.127 | 0.027 | 0.002 | 0.000 | 0.074 |

| 0.013 | 0.000 | 0.001 | 0.002 | 0.001 | 0.001 | 0.002 | 0.032 | 0.003 | 0.072 |

| 0.506 | 0.628 | 0.948 | 0.361 | 0.329 | 0.244 | 0.620 | 0.182 | 0.065 | 0.000 |

| 0.013 | 0.036 | 0.080 | 0.027 | 0.024 | 0.011 | 0.002 | 0.088 | 0.005 | 0.011 |

| 0.242 | 0.112 | 0.149 | 0.043 | 0.507 | 0.371 | 0.362 | 0.521 | 0.526 | 0.025 |

*sqft*and

*acres*from Table 4.

Price effect of unit increase in characteristic, 2SLS and by quantile

| 2SLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|---|

| 17,206 | 10,410 | 12,608 | 13,672 | 14,661 | 17,205 | 18,594 | 20,953 | 22,558 | 25,836 |

| 22,478 | 15,672 | 17,471 | 17,407 | 19,306 | 26,208 | 27,081 | 31,290 | 39,090 | 60,163 |

| 185 | 227 | 200 | 199 | 190 | 185 | 195 | 177 | 174 | 162 |

| 747 | 996 | 917 | 1,715 | 1,483 | 1,220 | 967 | 724 | 985 | 322 |

| 7,526 | 4,396 | 5,703 | 6,311 | 6,966 | 6,892 | 6,967 | 7,568 | 9,071 | 13,133 |

| 3,370 | 2,030 | 2,226 | 2,218 | 3,012 | 2,508 | 3,010 | 3,411 | 4,781 | 9,093 |

| 4,030 | −355 | 2,650 | 3,010 | 2,694 | 2,880 | 4,500 | 6,368 | 8,283 | 8,147 |

| 775 | 558 | 901 | 819 | 569 | 1,508 | 1,615 | 498 | −247 | 904 |

| 743 | 546 | 583 | 790 | 682 | 522 | 416 | 1,097 | −143 | 775 |

| 3,883 | 3,209 | 3,211 | 3,420 | 3,589 | 3,692 | 3,936 | 4,122 | 4,207 | 4,771 |

| −22 | 79 | 546 | −737 | −759 | −840 | 158 | −9 | −294 | −1,541 |

| 630 | 657 | 202 | 342 | 110 | 550 | 908 | 1,626 | 1,703 | 1,344 |

| −243 | −300 | −279 | −230 | −255 | −332 | −295 | −262 | −334 | −327 |

| 390 | −247 | −28 | −389 | −520 | −544 | −261 | 740 | 1,247 | 3,603 |

| −32,232 | −26,089 | −18,809 | −17,993 | −22,254 | −23,004 | −34,494 | −28,615 | −37,040 | −64,320 |

| −2,756 | −2,247 | −2,490 | −3,184 | −1,359 | −1,845 | −1,747 | −1,429 | −1,440 | −5,549 |

Price effect of characteristic, 2SLS and by quantile

| 2SLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|---|

| 1,599 | 6,520 | 7,172 | 5,079 | 6,550 | 1,628 | 968 | −592 | −695 | 761 |

| −688 | −50 | −1,080 | −1,391 | −1,317 | −1,269 | −919 | −1,594 | −2,667 | 597 |

| −957 | 2,125 | 506 | 555 | −197 | −2,009 | −1,846 | −3,512 | −3,458 | −609 |

| 4,015 | 4,239 | 2,276 | 1,830 | 2,474 | 2,712 | 3,477 | 3,633 | 3,110 | 6,779 |

| −496 | −3,155 | −1,595 | −757 | −444 | −454 | 298 | 1,291 | 1,219 | 71 |

| 4,208 | 3,265 | 2,962 | 3,668 | 3,899 | 4,144 | 5,129 | 5,436 | 5,738 | 5,654 |

| 2,556 | 566 | 1,372 | 1,889 | 2,653 | 2,458 | 3,029 | 4,187 | 3,482 | 1,596 |

| 10,584 | 7,665 | 8,692 | 9,944 | 10,122 | 10,137 | 10,736 | 11,189 | 10,769 | 10,064 |

| 1,780 | 2,247 | 1,875 | 2,064 | 2,166 | 1,960 | 1,951 | 1,920 | 1,421 | −2,077 |

| 3,025 | 3,086 | 3,811 | 3,484 | 3,705 | 3,665 | 3,469 | 3,711 | 2,655 | 1,610 |

| 2,230 | 3,203 | 1,784 | 1,405 | 565 | 854 | 1,342 | 2,920 | 854 | 2,487 |

| 397 | 3,013 | 1,964 | 2,100 | 1,173 | −151 | −792 | −805 | −1,853 | −4,897 |

| −1,607 | 527 | 439 | −486 | −2,116 | −5,589 | −5,655 | −3,099 | −2,541 | −1,547 |

| 3,451 | 3,954 | 3,115 | 2,974 | 3,182 | 2,632 | 2,579 | 3,609 | 4,228 | 4,615 |

| 2,047 | 3,067 | 2,688 | 2,481 | 1,970 | 1,691 | 1,519 | 662 | −592 | −2,397 |

Price elasticities of square footage and acres, 2SLS and by quantile

| 2SLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|---|

| 0.258 | 0.133 | 0.180 | 0.209 | 0.220 | 0.257 | 0.274 | 0.315 | 0.344 | 0.419 |

| 0.038 | 0.035 | 0.032 | 0.032 | 0.031 | 0.049 | 0.044 | 0.052 | 0.071 | 0.089 |

There is very little difference in the results of Tables 5 and 7, although the spatial lag variable of Table 7 is statistically significant for most but not for all quantiles. In comparing the *p* values of Tables 6 and 8, it appears that those of Table 8 are on average slightly lower, especially for some variables, such as *airel*, *exbri*, *laful*, or *dorem*. The similarity in results between Tables 5 and 7 for the regression coefficients and between Tables 6 and 8 for the *p* values of the coefficient estimates suggests that the quantile effects dominate the spatial autocorrelation effects. Put differently, for the given model and data set, it is more important for the results to account for quantile effects than for spatial autocorrelation effects. Whether this result holds in general awaits further research on other models and data sets..

Tables 5 and 7 both show that the coefficients of a number of variables vary considerably across quantiles. For example, there is more than a 50% difference between the square footage coefficient for the 0.1 quantile and the 0.9 quantile. This is economically significant. The dollar price effects reported for variable *sqft* in Table 9 attest to that: the marginal price of a square foot for quantile point 0.9 is close to 150% above that of quantile point 0.1; yet, the sale price for quantile point 0.9 (Table 4) is only 64% above that of quantile point 0.1. Table 11 shows a similar effect for the price elasticity of square footage: the price elasticity for the 0.9 quantile of housing prices is three times as high as that for the 0.1 quantile. The 2SLS estimate of variable *sqft* clearly overstates the contribution of a square foot to the sale price of lower-price houses but understates the contribution for higher-priced houses. The results are very similar, although more dramatic, for the variable *acres*.

The variable *year* is a proxy for age.^{10} A 1-year increase reduces the age of the house by 1 year. The positive sign reported in Tables 5 and 7 suggests that newer houses sell for relatively more. This is a standard result. However, the coefficients of Table 7 and the corresponding marginal effects of Table 9 reveal that there is a lower premium for newness for higher-priced homes. Lower-priced homes have the highest premium for newness (or discount for age).

The 2SLS coefficient for the number of bedrooms, *bedr*, is not significant in Table 7, which is not surprising given what is reported in Table 2. However, the quantile regressions provide a somewhat different picture. The regression coefficients for *bedr* are statistically significant primarily in the lower and middle price ranges and are not significant in the upper price range. The underlying economic reason for this result may be tied to the fact that lower- and medium-priced houses tend to have fewer bedrooms than expensive houses, yet will often contain as many or more occupants. As a result, an additional bedroom will have a higher marginal value in the lower-priced ranges.

The bathroom variables show a similar result: additional bathrooms have a much higher value-added impact in higher-priced homes than in lower-priced ones.

The relationship between explanatory variables and selling price as shown by the quantile regressions

Regression coefficient increases as selling price increases | Regression coefficient decreases as selling price increases | Regression coefficient remains relatively constant as selling price increases | Regression coefficient shows no definite pattern as selling price increases | Regression |
---|---|---|---|---|

Square feet | Year built | Garage | Bedrooms | Deck |

Acres | Mountain view lot | Electric AC | Percent of population nonwhite | Patio |

Full baths | Stucco exterior | Basement | ||

Three-quarter baths | Brick exterior | Pool | ||

Half baths | Aluminum exterior | Evaporator AC | ||

Hardwood floors | Sprinkler system | Window evaporator AC | ||

Tile floors | Distance to interstate | Gas AC | ||

Distance to city center | Frame exterior | |||

Full landscaping | ||||

Partial landscaping | ||||

Earthquake magnitude | ||||

Percent rental houses |

## Conclusions

One of the most popular areas of research in real estate economics and finance has been the pricing of residential real estate. Empirical research has primarily focused on identifying house characteristics that most influence selling price. The results from this body of literature have often been in conflict regarding the impact of a variable on selling price. This study seeks to clarify some of the confusion by using quantile regression to measure the effect of various housing characteristics on house prices.

Results of this study show that the effect of housing characteristics on selling price can be better explained by estimating quantile regressions across price categories. For example, previous studies that have examined the effect of characteristics such as square footage or age on selling price have found mixed results in terms of both the level and the direction of change. This study shows that some of those differences may be explained by differences in house prices. In particular, the regression coefficients of some variables behave differently across different house price levels, or quantiles. Buyers of higher-priced homes appear to price certain housing characteristics differently from buyers of lower-priced homes.

For the given data set, it is shown that the quantile effects dominate any effects on coefficient size and statistical significance that arise from spatial autocorrelation. In fact, taking explicit account of spatial autocorrelation in the quantile regressions, adds very little information. Whether this is a general result or particular to the data set that is being used in this study is an open question that awaits further research.

This study produces some interesting results. For example, square footage is often used to determine the appraised value of a home since it is expected to have a significant effect on the selling price. While previous studies bear this out, it is interesting to see how buyers in different price ranges value this variable. This is shown by the significant difference between the coefficients at the lowest and the highest quantiles where the additional price of a square foot for the highest priced homes is two and a half times the additional price per square foot for the lowest-priced homes. Clearly, traditional methodologies such as OLS or models that take into account spatial autocorrelation can overstate the value of a marginal square foot for lower-priced homes but understate the effect on higher-priced homes.

The quantile results provide some valuable insights to the different relationships that the explanatory variables have with selling price. For example, some variables such as square footage, lot size, bathrooms, and floor type have a greater impact as selling price increases. Other variables have a relatively constant effect on selling price across different price ranges. These include garage, exterior siding, sprinkler system, and distance to city center. Some other variables such as bedrooms and percentage of nonwhite population have a significant effect on selling price but there is no clear pattern of the effect across different price ranges. Lastly, the quantile regressions confirm that most variables showing no statistical significance under OLS or 2SLS remain not significant across the different price ranges.

These results add to the body of research explaining house prices. Even though variations in the value of housing characteristics across different price ranges may have been considered intuitive beforehand, quantile regression provides a way to confirm these expectations.

The articles in Durlauf and Young (2001) provide a good idea of the social dynamics that may evolve and why they may evolve.

The Matlab program xy2cont.m of J.LeSage’s Econometrics Toolbox is employed, which is an adaptation of the Matlab program fdelw2.m of Kelley Pace’s Spatial Statistics Toolbox 2.0.

If **X** identifies the data matrix, then the spatial lags of the regressors are computed as **WX**, where **W** is the spatial weight matrix used for the construction of the spatial lag of the dependent variable.

Variance inflation factors (VIF) are calculated for all variables. The maximum VIF is 2.51, the mean VIF is 1.54. This does not suggest that the regressions suffer from multicollinearity.

The *p* values of the OLS estimates are based on an estimate of the variance–covariance matrix that is robust to heteroskedasticity.

The variable *year* can be converted to measure the age of a house by simply subtracting the value of *year* from 2000 for a given observation. This linear transformation does not affect the coefficients of any variable other than *year* or *age* and the constant.