Repeat Sales Methods for Growing Cities and Short Horizons

Transcription

1 Repeat Sales Methods for Growing Cities and Short Horizons Karl L. Guntermann, Crocker Liu and Adam D. Nowak * July 9, 2014 Abstract The accurate estimation of real estate indices is important for many purposes. A common method to estimate these indices is to use a repeat sales procedure. Although this does not require property attributes, this method discards a large amount of sales. This paper proposes a method that can be used to incorporate a significant percentage of these discarded observations without requiring additional data collection by the researcher. We apply our method to the metropolitan statistical areas of Phoenix and Seattle and find that standard errors are on average two thirds the value of standard errors from a repeat sales procedure in Seattle and three eights the value in Phoenix. *Please do not quote without permission from the author. JEL classification: G11, G14, R20, R21, R22 and R32 Keywords: Repeat sales index, spatial errors, hybrid methods, nuisance parameters * W.P. Carey School of Business, Arizona State (karl.guntermann@asu.edu), (480) , School of Hotel Administration, Cornell University (chl62@cornell.edu), (607) , and West Virginia Business School, West Virginia University (adam.d.nowak@gmail.com), (219) , respectively. Address all correspondence to Adam Nowak, West Virginia Business School, West Virginia University. Morgantown, WV Phone: (219) , Fax: (304) adam.d.nowak@gmail.com. We wish to thank the Center for Real Estate Theory and Practice at ASU for providing us with the necessary transaction data.. All errors are our own.

2 INTRODUCTION Accurate estimation of real estate indices is important for many purposes. First, accurate real estate indices are important to lenders in the real estate market. Accurate measurement of house prices is important to determining the value of assets on the lender s balance sheet. Second, policy makers require accurate measurement of market-wide prices in order to comment on local price trends and provide insight to housing policy. Unlike stock indexes, infrequent trading of heterogeneous assets in the real estate market requires using special econometric techniques in order to estimate latent market prices. In one setting, house prices are modeled using hedonic methods that incorporate property attribetus and time. An alternative method is the repeat-sales regression estimator (hereafter RSR) developed by Bailey, Muth and Nourse (1963) and further advanced by Case and Shiller (1987). These methods estimate house price indices using repeat sales by regressing changes in prices on dummy variables corresponding to time periods. Sales without a previous same-house sale, unfortunately, are discarded. Discarded sales provide no information regarding the price index; in some data sets, the loss in information can be significant. Because of this, researchers have proposed hybrid methods that utilize repeat-sales in conjunction with sales without a previous same-house sale. The proposed method is one such hybrid method. We augment the RSR method while not requiring any significant data collection. In doing so, we propose a method that utilizes nearly 100% of the observations. Meese and Wallace (1997) compare hedonic and RSR and indicate drawbacks when observations are discarded. Case and Quigley (1995), Quigley (1995), Hill, Knight & Sirmans (1997), Englund, Quigley & Redfearn (1998) and Hwang & Quigley (2002) find using both repeat-sales and non repeat-sales decreases standard errors. Despite the advantages of the hybrid techniques, Bourassa, Cantoni & Hoesli (2013) point out that the requisite hedonic variables are not always available to the researcher. As a remedy, Bourassa, Hoesli & Sun (2006) and Clapp & Giacotto (1998) provide methods that incorporate appraisals as an alternative to collecting hedonic attributes. This analysis emphasizes the correlation in real estate prices for properties located within a particular location. Location can be defined as a zip code, school district, census block, subdivision, etc. Often, the choice of location is determined by the researcher s question on hand. When studying schools, it is prudent to use school districts. Rockoff (2004), Clapp,

3 Nanda & Ross (2007) and Gibbons, mach & Silva (2013) all find effects that school districts are capitalized into house prices. However, absent this precise notion of location, it is not always apparent what category should be used. This paper proposes using a nearest neighbor approach. We assume any two properties located in close proximity to each other have positively correlated values do to a common flow of services associated with location. Examples of these McMillen & McDonald (1998) and McMillen (2001) that find evidence of employment subcenters in Chicago metro. The nearest neighbors approach is used in spatial correlation models. Spatial error models are designed to capture correlation due to geographic proximity and have been used in real estate research 1. Most relevant to this study, Gelfand, Ecker, Knight & Sirmans (2004) model microlevel spati0-temporal effects alongside broader market wide price effects. Conway, Li, Wolch & Jerrett (2008) examine the effects of greenspace on property values where there are spillover effects due to location. Osland (2010) applies spatial errors to a hedonic model and notes that spatial effects are almost always present in data. McMillen (2010) finds that mis-specification produces spatial errors and suggests non-parametric methods as a curative measure. Pace, Barry, Willey & Sirmans (2010) use spatial effects to forecast real estate prices and Pavlov (2000) uses spatially-varying coefficients in a hedonic regression. Our measure of spatial weighting is based on the geographic distance between two points. In our data, and in many others, each observation contains the complete address for each property. Software packages allow researchers to map addresses to latitude and longitude coordinates; this process is commonly referred to as geocoding. The software package used here is the ggmap package in R. The package ggmap matches up to 2,500 addresses over a 24-hour span at no cost. We suggest a hybrid model that uses both level and differenced sale prices. By differencing, any time-invariant price effects specific to property are removed from both the regression and error term. This procedure is similar to matching methods found in Deng, McMillen & Sing (2012), McMillen (2012) and the pseudo repeat-sales method outlined in Guo, Zheng, Geltner & Liu (Working Paper). Our method builds on their work and exploits unobserved price effects common to properties within a location without specifying an explicit matching algorithm. 1 Pace, Barry & Sirmans (1998) provide a survey of spatial modeling.

4 To summarize, the proposed method 1) utilizes all observations in a differenced sale price framework, 2) produces a house price index with increased precision, 3) exploits spatial correlation in error terms, and 4) requires no more data than is required for a repeat-sales regression. Methodology Models for House Prices and Changes in House Prices We first describe a model for house prices, the resulting model for differenced house prices and our proposed method. The model for house price follows Quigley (1995). Sale price is represented as the product of a price index and a flow of services,, where is the price index and is the flow of services. The price index is assumed to be time variant. The log sale price for the house in the location is given by 1 Here, is the log sale price, is the price index for time period, is a flow of services associated with the house, and is a random error term uncorrelated with or. Because the flow of services is not directly observable. We model as as a linear combination of observable and unobservable attributes. Flow of services is given by 2 The vector is an observable vector of structural attributes, and is a vector of attribute loadings. The term captures all flows of services that are associated with unobserved factors. We assume can be decomposed as 3 The term represents a property-specific flow of services due to unobserved attributes specific to property. The term represents a location-specific flow of services from unobserved factors that are common to all properties located at location. Here, the are assumed iid across properties, and the are iid across location. With this model for the flow of services, the sale price can be written using 1 and 2 as

5 4 Equation 4 is a hedonic model that can be used to estimate a price index. The hedonic model in 4 expresses house prices as a linear function of a time index and observable attributes. This equation for price levels is used to develop a model for price changes as in the RSR. The hedonic model in 4 also plays a direct role in other hybrid methods such as Case & Quigley (1991), Hill, Knight & Sirmans (1997). In matrix form, we can write the vector of prices as 4 Here, is an 1 vector of prices, is a 1 matrix of time dummies, is a 1 1 vector of time coefficeints, is a matrix of hedonic variables including a column of 1s and is a 1 vector of error terms. Using 2 and 4 the is a composite error term is given by 5 Unmeasured attributes are property and location-specific and can be both quantitative and qualitative. Location attributes include school quality, subdivision amenities, distance to commercial areas, etc. The distinction we make between property-specific and location-specific attributes is that property-specific attributes are specific to property while location attributes are common to all properties in location. Location should be thought of as a flow of services common to multiple properties located in close proximity to each other. Further, the flow of services form location is in contrast to the flow of services associated with a specific parcel due to its location within a subdivision. For example, two neighboring houses might have identical commutes to downtown but different backyard views. In this setting, the backyard view is captured by while the commute is captured by. We assume that unobserved attributes do not change over time; however, the analysis can be modified to accommodate dynamic, unobserved attributes. For example, we can write model either of the unobserved attributes as having temporal correlation. Quigley (1995) assumes that

6 is constant. Hill, Knight & Sirmans (1999) assume that follows a first-order autocorrelation, AR(1), process. Other authors have further found that the error term follows a stationary, autoregressive process. Unreported results prove that our conclusions do not change when accounting for an autoregressive structure in. The RSR uses differenced sale prices. Given a set of observations, there are a total of repeat-sales. A repeat-sale is defined as a sale for which there exists a previous sale of the same house. For each of these observations, we can difference the log sale prices. Using the above equations, differenced sale prices can be expressed as 6 6 A common assumption is that attributes do not change over time resulting in which results in 6. Alternatively, we can assume that the expected change in property attributes is 0. In matrix form, the vector of price changes in 6 or 6 as 7 7 Here, is an 1 vector of differences prices, is a 1 matrix of time dummies, is a matrix of differenced hedonic variables excluding a column of 1s and is a 1 vector of differenced error terms. For the repeat sale given in 6, the row elements of contain a +1 in column 1, a -1 in column 1 and 0s elsewhere. Likewise, the rows in contain the. The RSR uses only the differenced sale prices from 6. While the assumption that hedonic variables do not change precludes the need to collect property attributes, a serious drawback is that the researcher must discard observations. With this in mind, Case and Quigley (1991) suggest a hybrid approach. This method simultaneously estimates equation 4 for the unpaired observations and equation 6 for the repeat-sales with the constraint that be equal across equations.

7 In the spirit of this hybrid model, we seek an estimation approach that incorporates all observations and controls for unobserved factors. This motivation is important for several reasons. First, early studies did not have access to large data sets and instead used data sets with sales from single condominium buildings or subdivisions that implicitly controlled for location. With recent advances in data collection, storage and availability, data sets can now include hundreds of thousands of transactions from a single metro area with heterogeneous locations. Because the heterogeneity is spatial in nature, it is possible that regression estimates can be improved. Second, there is a wide variety of attributes from one data set to the next. Our study uses assessor s office data from both King County, Washington and Maricopa County, Arizona. While the number of bedrooms is included in the King County data, bedrooms is missing in the Maricopa data. Further, while both data sets include structure square footage, only the Maricopa County data includes the lot square footage. For this reason, we seek a model that mitigates the reliance on observable property attributes as in the RSR. Although real estate data sets vary from one to the next, it is a given that each data set will include both sale price and some form of property identification. In most cases, properties are identified by address or latitude and longitude. Using a parsimonious data set that includes only sale price and address, a RSR can estimate a house price index. We seek to provide alternative methods for such parsimonious data that does not discard observations as the RSR does. Proposed Nearest-Neighbor Estimator For the remaining unpaired observations in the data, we propose difference observations using a variant of nearest neighbor methods. Our approach is as follows: for an unpaired observation of house sold at time, first collect all unpaired observations sold at any time period. Next, calculate the distance between house and these other properties. Define the house nearest to property in this set as. Property is the nearest, previously sold, unpaired neighbor to property. Price and observable attributes for house are used to model differences in sale prices as 8

8 The pairs from equation 8 can be used to augment the repeat-sales used in the RSR. This augmentation is similar to the hybrid methods augmenting repeat sales with unpaired sale prices in level form. In matrix form, we can write the vector of price changes in 8 as 9 Here, is an 1 vector of differences prices, is a 1 matrix of time dummies, is a matrix of differenced hedonic variables excluding a column of 1s, is a 1 vector of differenced error terms and is a 1 vector of differenced property fixed-effects. For the differenced sale prices given in 9, the row elements of contain a +1 in column 1, a -1 in column 1 and 0s elsewhere. Likewise, the rows in contain. AS mentioned above, we use sales from. Because of this, we cannot pair any unpaired sales from 1. Therefore, sales from 1 enter into the regression in level form as in Here, equation 9 contains sale prices, hedonic variables and errors from 1 in level form. Our proposed estimator is a least-squares problem. We stack differenced observations from equations 7, 9 and 10 as 11 The estimator in 11 requires observable property attributes. However, we also specify another estimator that does not require any observable property attributes. We assume 0 or alternatively 0. With this assumption and dropping the sales in 10, the model we estimate is now 12

9 The terms and represent errors associated with the 0 expected attribute differences across properties. The model in 12 requires only time of sale and address, as does the RSR. Thus, there is no onus on the researcher to collect more data than is required by the RSR. Competing Models As mentioned above, there have been many proposed methods for estimating a property price index. In this section, we sketch several competing models to which we compare to our own. The first model is a variant of the hybrid model. We estimate 13 Here, is a vector of prices for the unpaired observations, is the 1 submatrix created from the unpaired observations in 4, is the submatrix created from the unpaired observations in 4 and is the 1 vector created from the unpaired observations in 4. Because our estimator is based on the flow of services associated with location, we also consider a fixed-effects model that controls for location. Using zip code fixed effects, we estimate 14 Where is an 1 matrix of fixed effects for each zip code 1,, and is a 1 vector of location fixed-effects. Although 14 might appear to effectively control for unobserved location effects, there are two issues that the researcher must keep in mid. First, the choice of location grouping is debatable as properties can be categorized according to location using zip code, school district or census block. This is important because, many school districts with different qualities might be located within a single zip code. Second, large sample properties of fixed effects estimator typically rely on while holding the number of regressors,, constant. However, in practice can be quite large and the resulting assumption of a fixed number of regressors is not valid. This is especially true if the researcher defines location at a granular level, necessitating a large value of. Because we exploit spatial correlation, a model of spatial correlation in errors is also considered. For a review of nearest neighbors estimator and spatial errors, see (X). We parameterize

10 correlation in the of 4 using a spatial weighting matrix. In particular, we use the spatial error model (SEM) and assume prices are generated by Where 0 1 is a spatial correlation term, is a block spatial weighting matrix of nearest neighbors and is a vector of normally distributed variables with mean 0 and variance. As our proposed method uses a single nearest neighbor, the matrix is also created also using the single nearest neighbors. Lastly, we account for potential serial correlation in the. We assume is dynamic and follows an AR(1) process where 16 Here, 1 1, is an iid random variable with mean 0 and variance. Using the error process in 16, we estimate 4 using a two-step feasible GLS procedure. We first estimate 4 and calculate the estimated residuals,. Next we use residuals from the paired observations and create differenced,. Our estimator for is found by taking expectations of 16 after scaling we have 1 17 This scaling method is similar to that in Quigley (1995). The AR parameter can be estimated using GMM on 15 where an estimate of can be found as the mean-squared error of the residuals from 4. Data The data comes from two sources. The first source is the King County assessor s office in of Washington state. This data set contains all transactions for real property in King County beginning 1982 and ending June, The second data set comes from the Maricopa County assessor s office in Arizona. This data set contains all transactions for real property in Maricopa County beginning 1982 and ending December, For computational reasons, we truncate

11 both data sets before January 1, This truncation does not alter conclusions in any considerable manner. The King County data set was chosen because it is available for download at no cost using the assessor s website 2. The Maricopa County website was obtained using the private firm Ion Data Express while the authors were at Arizona State University. Due to financial considerations, it is not possible to compare our method using any other county-wide data sets. However, King County and Maricopa County include the cities of Seattle and Phoenix, respectively, so the results are most-likely applicable to counties with large metropolitan areas. Gary & Barrett (2009) and Liu, Nowak & Rosenthal (2014) use data from Maricopa County. Palmquist ( ) and Cunningham (2007) examine King County. Summary statistics for King County are displayed in Table 1. The average sale price in King County was $413,370 over this 13-year period while the average house size was 2,060 square feet. The average age of a house sold was 36 years. The median sale date was indicating that half of the houses in the sample were sold before August of The Maricopa County summary statistics are reported in Table 2. The average house sold in Maricopa County had fewer square feet and sold for less than the average house in King County. However, the houses sold in Maricopa County were much newer; the average age was only 15 years, while half of the houses sold were less than 9 years old. As mentioned above, the RSR discards observations. Table 3 displays the total number of sales and repeat-sales for each county. Discarding the unpaired transactions would discard roughly 57% of the observations in Maricopa County and 63% of the observations in King County. Results Main Findings The primary statics of interest are standard errors for time coefficients,. Figures 1a and 1b display the time coefficients for King County. Figure 1a displays the time coefficients for those models that include property attribtues in the regression. The figure shows that prices in King County were increasing from the beginning of the sample until peaking in the summer of Following the peak, prices began to fall until reaching a bottom in the first quarter of

12 Figure 1b displays the time coefficients for the parsimonious models. These models are parsimonious in that they only require sale price and property address in order to estimate the index. The peak in King County prices was approximately equal to 0.61 or approximately 61% above prices in the base period of Q1, The trough was approximately 0.15; thus, prices bottomed out at levels that were above Q1, 2000 prices. In Maricopa County, the time coefficient index peaked at roughly 0.81 and bottomed out at roughly Figures 2a and 2b display the time coefficients for Maricopa County. The Maricopa County price series peaks in Q2 of Following the peak, prices declined until reaching a low in Q1 of In summary, house price movements in Maricopa county led house price movements in King County. However, the peak in King County was larger than the peak in Maricopa County and the trough in Maricopa County was lower than the trough in King County. Tables 4 and 5 report these standard errors for the various models considered and Figures 3a to 4b graph the standard errors. Figures 3a and 3b are for King County and Figures 4a and 4b are from Maricopa County. Figure 3a shows that the FGLS model that uses the proposed nearestneighbor estimator has smaller standard errors than any competing model requiring property attributes. Figure 3b shows that the proposed nearest-neighbor estimator has smaller standard errors than the traditional RSR. This relative performance is important because both the RSR and our proposed method require no more data than sale prices and property address. Thus, increased performance in the proposed method comes at no additional data collection cost to the researcher but instead a more exhaustive incorporation of data on hand. The results for Maricopa County are similar to results for King County. Figures 4a and 4b display the same pattern. Standard errors are smaller for the proposed nearest-neighbor estimator than any other competing estimator. In addition, the parsimonious estimators in Figure 4b show that the estimator in 12 produces smaller standard errors than the RSR. Localized Property Effects The proposed nearest-neighbor estimator assumes that there are unobserved location effects as in 3. In order to provide evidence that the unobserved error term does include a location effect, we preform an F-test for the joint-significance of fixed effects in 14. Specifically, we test the null hypothesis that all of the zip code coefficients are equal to 0, : 0. Tests using F- statistic methods can be performed using the data in Tables 6 and 7.

13 For Maricopa County, The F-statistic from the change in the sum of squared residuals is distributed as 139, The value for the F-statistic is 1,521.1 while the critical value for the 0.1% level is 1.3; thus, there is strong evidence that location effects are significant in pricing after controlling for hedonic attributes. For King County, the F-statistic is and is also highly significant. In addition to the fixed effects model, we also test for spatial correlation using the spatial model in 15. We find that 0.19 in King County and 0.31 in Maricopa County. Using a likelihood ratio test, we strongly reject the null hypothesis that 0 for both data sets. Thus, the spatial correlation model supports the idea that error terms for properties located near one another are spatially correlated. Old and New Houses We are also interested in how our proposed method performs when the sample is bifurcated into old and new properties. A property is defined as an old property if its construction year is after the median construction year for its respective data set. A property that is not an old property is a new property. Tables 1 and 2 display the cutoffs used: the cutoff for old houses is 1969 for King County and 1996 for Maricopa County. Standard errors for old and new houses are displayed in Figures 5a and 5b for King County. The proposed nearest-neighbors method has smaller standard errors than the competing models. Other competing models are omitted for graphing clarity, but our conclusions are not affected by their omission. A similar patter for old and new houses is also apparent in Maricopa County. Figures 7a and 7b display the standard errors for the time coefficients. Figure 7a shows that the results for old houses are the same as for King County. Figure 7b shows that the standard errors for new houses are smaller when using the proposed nearest-neighbor method. Figure 7b also shows that the NNG model that uses property attributes shows standard errors that rise following the peak of the housing bubble. However, the standard errors associated with the NNT model decrease following the peak and are everywhere else lower than the standard errors of the RSR or hedonic model. Small and Big Houses

14 In addition to comparing the results for a subset of the population based on age, we also consider bifurcating the sample based on square footage. We group houses into big and small groups. A big houses is defined as a property with square footage larger than the median square footage for all houses built in the same year. We create this group by first sorting houses into bins according to their construction year. Next, we calculate the median and define any houses with a square footage larger than the median square footage in that year as big. Standard errors for old and new houses are displayed in Figures 6a and 6b for King County. The proposed nearest-neighbors method has smaller standard errors than the competing models. A similar patter for old and new houses is also apparent in Maricopa County. Figures 7a and 7b display the standard errors for the time coefficients. Figure 7a shows that the results for old houses are the same as for King County. Conclusion Lenders, property owners and policy makers will always want to value the real estate they own. Because real estate is infrequently traded and heterogeneous in nature, it is necessary to develop methods for price indexes that operate alongside these features. The method we have proposed produces smaller prediction intervals for the price index than other competing models we consider. In part, this is because we remove nuisance parameters by differencing yet do not sacrifice sample size. We have documented the performance using one private data set and one public data set; each contains sales from large metropolitan areas. In our data, the standard errors from our methods are almost two-thirds the standard errors for the repeat-sales index. The method we discuss takes advantage of common features associated with location. Exploiting these features does not require the researcher to collect any more data than if he were to compute a repeat sales index. Although motivated by spatial methods, our method is easy to implement and does not require any computation outside of pairwise distance calculations. Further, because one version of our procedure does not require hedonic variables, it can be applied to data sets without accurate or many hedonic variables. In addition, because our data set does not discard observations as in the repeat-sales method, our method is also potentially applicable to cities with fewer transactions.

15 Table 1: Descriptive Statistics of the King County Data Set Mean Median Standard Deviation Min Max SALEPRICE SQFT AGE SALE DATE BEDROOMS CONSTRUCTION YEAR MULTISTORY GARAGE BASEMENT HALF BATHS FULL BATHS Table 1 displays the descriptive statistics for the Seattle data. SALEPRICE is the sale price measured in $1,000s. SQFT is the square footage measured in 1,000s. AGE is the number of years between the first recorded construction date for the property and the sale year. SALEDATE is the year of sale. BEDROOMS is the number of bedrooms. CONSTRUCTION YEAR is the year of construction. MULTISTORY is an indicator equal to 1 if the property has two or more stories. GARAGE is an indicator equal to 1 if the property has a garage. BASEMENT is an indicator equal to 1 if the property has a basement. HALFBATHS is the number of half baths. FULLBATHS is the number of full baths.

16 Table 2: Descriptive Statistics of the Maricopa County Data Set Mean Median Standard Deviation Min Max SALEPRICE SQFT LOT SQFT AGE SALE DATE BATH FIXTURES CONSTRUCTION YEAR GARAGE CARPORT PATIOS POOL AC Table 2 displays the descriptive statistics for the Phoenix data. SALEPRICE is the sale price measured in $1,000s. SQFT is the square footage measured in 1,000s. LOT SQFT is the lot square footage measured in 1,000s. AGE is the number of years between the first recorded construction date for the property and the sale year. SALEDATE is the year of sale. BATH FIXTURES is the number of fixtures in the house. A bathroom with a sink, shower and bathtub would have 3 fixtures. CONSTRUCTION YEAR is the year of construction. GARAGE is an indicator equal to 1 if the property has a garage. CARPORT is an indicator equal to 1 if the property has a carport. PATIOS is the number of patios. POOL is an indicator variable equal to 1 if the property has a pool. AC is an indicator variable equal to 1 if the property has central air.

17 Table 3: Counts For King and Maricopa County Maricopa County King County SALES 719, ,474 REPEAT SALES 312, ,681 ZIP CODES PARCELS 406, ,793 REPEAT SALES / SALES SALES / ZIP CODE 5, , PARCELS / ZIP CODE 2, , Table 3 displays the counts by sale in the two data sets. SALES is the total number of sales. REPEAT SALES is the total number of sales where there exists a previous same-house sale. ZIP CODES is the total number of zip codes. PARCELS is the total number of unique parcels.

18 Table 4: Standard Errors for King County Time Coefficients Date AR HED NNO NNG NNT HYB SPA ZIP RSR

19 Table 4 displays the standard errors for the time coefficients for the regression models. AR: A GLS estimation assuming an error process as in 15 HED: A baseline hedonic model as in equation 4 NNO: An ordinary least-squares regression for the model in 11 NNG: A FGLS regression for the model in 11 NNT: A FGLS regression for the model in 12 HYB: A hybrid model in 13 SPA: A spatial weighting model as in 15 ZIP: A fixed-effects model for zip codes as in 14 RSR: The repeat-sales regression as in equation 7

20 Table 5: Standard Errors for Maricopa County Time Coefficients AR HED NNO NNG NNT HYB SPA ZIP RSR

21 Table 5 displays the standard errors for the time coefficients for the regression models. AR: A GLS estimation assuming an error process as in 15 HED: A baseline hedonic model as in equation 4 NNO: An ordinary least-squares regression for the model in 11 NNG: A FGLS regression for the model in 11 NNT: A FGLS regression for the model in 12 HYB: A hybrid model in 13 SPA: A spatial weighting model as in 15 ZIP: A fixed-effects model for zip codes as in 14 RSR: The repeat-sales regression as in equation 7

22 Table 6: Hedonic Estimates for King County AR HED NNO NNG HYB SPA ZIP SQFT AGE BASEMENT BEDROOMS FULLBATHS GARAGE HALFBATHS MULTISTORY SSE 78,598 78,597 98,345 98,349 76,344 76,344 78,265 MSE R N 367, , , , , , ,474 DF 367, , , , , , ,335 Table 6 displays the standard errors for the hedonic coefficients for the regression models. SQFT is the square footage measured in 1,000s. AGE is the number of years between the first recorded construction date for the property and the sale year. BASEMENT is an indicator equal to 1 if the property has a basement. BEDROOMS is the number of bedrooms. FULLBATHS is the number of full baths. GARAGE is an indicator equal to 1 if the property has a garage. HALFBATHS is the number of half baths. MULTISTORY is an indicator equal to 1 if the property has two or more stories. SSE is the total sum of squared errors. MSE is the mean squared error. R2 is the r-squared. N is the total number of observations. DF is the total degrees of freedom. Standard errors are below. All variables are significant at the 1% confidence level. AR: A GLS estimation assuming an error process as in 15

23 HED: A baseline hedonic model as in equation 4 NNO: An ordinary least-squares regression for the model in 11 NNG: A FGLS regression for the model in 11 HYB: A hybrid model in 13 SPA: A spatial weighting model as in 15 ZIP: A fixed-effects model for zip codes as in 14

24 Table 7: Regression Estimates for Maricopa County AR HED NNO NNG HYB SPA ZIP SQFT LOT SQFT AGE BATH FIXTURES CARPORT GARAGE POOL PATIOS AC SSE 104, ,253 92,919 92,948 96,730 96,730 74,445 S R N 719, , , , , , ,127 DF 719, , , , , , ,935 Table 7 displays the standard errors for the hedonic coefficients for the regression models. SQFT is the square footage measured in 1,000s. LOT SQFT is the lot square footage measured in 1,000s. AGE is the number of years between the first recorded construction date for the property and the sale year. BATH FIXTURES is the number of fixtures in the house. A bathroom with a sink, shower and bathtub would have 3 fixtures. CARPORT is an indicator equal to 1 if the property has a carport. GARAGE is an indicator equal to 1 if the property has a garage. POOL is an indicator variable equal to 1 if the property has a pool. PATIOS is the number of patios. AC is an indicator variable equal to 1 if the property has central air. Standard errors are below. All variables are significant at the 1% confidence level.

25 AR: A GLS estimation assuming an error process as in 15 HED: A baseline hedonic model as in equation 4 NNO: An ordinary least-squares regression for the model in 11 NNG: A FGLS regression for the model in 11 HYB: A hybrid model in 13 SPA: A spatial weighting model as in 15 ZIP: A fixed-effects model for zip codes as in 14

26 Figure 1a: Attribute Price Indexes for King County 0.7 Time Coefficients for House Price Index King County, Washington 0.6 Index: 2000 Q1 = Year and Quarter AR HED NNG HYB SPA ZIP Figure 1a displays the time coefficient estimates for the regression models that require property attributes. AR: A GLS estimation assuming an error process as in 15 HED: A baseline hedonic model as in equation 4 NNG: A FGLS regression for the model in 11 HYB: A hybrid model in 13 SPA: A spatial weighting model as in 15 ZIP: A fixed-effects model for zip codes as in 14

27 Figure 1b: Parsimonious Price Indexes for King County 0.7 Time Coefficients for House Price Index King County, Washington 0.6 Index: 2000 Q1 = Year and Quarter NNT RSR Figure 1b displays the time coefficient estimates for the time coefficients for the regression models that do not require property attributes. NNT: A FGLS regression for the model in 12 RSR: The repeat-sales regression as in equation 7

28 Figure 2a: Attribute Price Indexes for Maricopa County 1 Time Coefficients for House Price Index Maricopa County, Arizona Index: 2000 Q1 = Year and Quarter AR HED NNG HYB SPA ZIP Figure 2a displays the time coefficient estimates for the regression models that require property attributes. AR: A GLS estimation assuming an error process as in 15 HED: A baseline hedonic model as in equation 4 NNG: A FGLS regression for the model in 11 HYB: A hybrid model in 13 SPA: A spatial weighting model as in 15 ZIP: A fixed-effects model for zip codes as in 14

29 Figure 2b: Parsimonious Price Indexes for Maricopa County 1 Time Coefficients for House Price Index Maricpoa County, Arizona 0.8 Index: 2000 Q1 = Year and Quarter NNT RSR Figure 2b displays the time coefficient estimates for the time coefficients for the regression models that do not require property attributes. NNT: A FGLS regression for the model in 12 RSR: The repeat-sales regression as in equation 7

30 Figure 3a: Attribute Index Standard Errors, King County Index: 2000 Q1 = Standard Errors for House Price Index King County, Washington Year and Quarter AR HED NNG HYB SPA ZIP Figure 3a displays the standard errors for the regression models that require property attributes. AR: A GLS estimation assuming an error process as in 15 HED: A baseline hedonic model as in equation 4 NNG: A FGLS regression for the model in 11 HYB: A hybrid model in 13 SPA: A spatial weighting model as in 15 ZIP: A fixed-effects model for zip codes as in 14

31 Figure 3b: Parsimonious Index Standard Errors, King County Index: 2000 Q1 = Standard Errors for House Price Index King County, Washington Year and Quarter NNT RSR Figure 3b displays the standard errors for the time coefficients for the regression models that do not require property attributes. NNT: A FGLS regression for the model in 12 RSR: The repeat-sales regression as in equation 7

32 Figure 4a: Attribute Index Standard Errors, Maricopa County Standard Errors for House Price Index Maricopa County, Arizona Index: 2000 Q1 = Year and Quarter AR HED NNG HYB SPA ZIP Figure 4a displays the standard errors for the regression models that require property attributes. AR: A GLS estimation assuming an error process as in 15 HED: A baseline hedonic model as in equation 4 NNG: A FGLS regression for the model in 11 HYB: A hybrid model in 13 SPA: A spatial weighting model as in 15 ZIP: A fixed-effects model for zip codes as in 14

33 Figure 4b: Parsimonious Index Standard Errors, Maricopa County Standard Errors for House Price Index Maricpoa County, Arizona Index: 2000 Q1 = Year and Quarter NNT RSR Figure 4b displays the standard errors for the time coefficients for the regression models that do not require property attributes. NNT: A FGLS regression for the model in 12 RSR: The repeat-sales regression as in equation 7

34 Figure 5a: Old House Standard Errors, King County Standard Errors for Old Houses, King County Standard Error Year and Quarter HED NNG NNT RSR Figure 5a displays the standard errors for time coefficients for houses built after the median construction year in King County. AR: A GLS estimation assuming an error process as in 15 HED: A baseline hedonic model as in equation 4 NNG: A FGLS regression for the model in 11 NNT: A FGLS regression for the model in 12 RSR: The repeat-sales regression as in equation 7

35 Figure 5b: New House Standard Errors, King County Standard Errors for New Houses, King County Standard Error Year and Quarter HED NNG NNT RSR Figure 5b displays the standard errors for time coefficients for houses built before the median construction year in King County. AR: A GLS estimation assuming an error process as in 15 HED: A baseline hedonic model as in equation 4 NNG: A FGLS regression for the model in 11 NNT: A FGLS regression for the model in 12 RSR: The repeat-sales regression as in equation 7