The focus of the empirical section of this paper is on Market 5 in Maricopa County, AZ, for properties that sold post-2011 during a recovery period when the economy experienced a positive shock to demand.Footnote 22 The geographic boundaries of Market 5 (Fig. 3) are defined by the Maricopa County assessor to include sales of higher priced single-family homes (SFR), averaging $504,517 compared to a Maricopa County overall mean sale price of $253,028. As discussed above, we selected Market 5 because it contains active markets for vacant land and new home sales as well as single family residential properties. This allows us to compare the three general methods for land valuation to our option value method. Our choice of market 5 favors the land residual method because new construction can be substituted for teardowns in most of the market.
Figure 3 shows a map of Maricopa County, and demonstrates that Market 5 is linked to downtown Phoenix by major roads. A typical Market 5 location has a 10–15-mile drive to the downtown area, 10–15 minutes in average traffic according to Google maps. Figure 4 shows neighborhoods (as defined by the Maricopa County assessor) in Market 5. It is noteworthy that there is a grid pattern of streets, as well as many golf courses, schools, mountains and water amenities. After examining Google Maps it is apparent that the areas where there are no transactions are often locations with mountains, parks, and golf courses offering amenities to bordering residential areas. Market 5 clearly benefits by ease of access to amenities as well as access to downtown.
Using a GIS we add layers of data on distances to downtown and to roads by type of road, schools, parks and water bodies. Most houses are within 1.4 miles of a primary road; the mean of 0.92 miles indicates a substantial majority closer than one mile. Distances to secondary roads, parks and water show substantial variation. Using an overlay of topographical maps, we add a dummy variable (high elevation dummy) which has values of 1 for high elevation locations, otherwise zero. About 5% of Market 5 houses are at high elevations, whereas many Maricopa County markets have no houses at high elevation.
A data filtering process was applied to the entire dataset for Maricopa County, to yield the final data that we use in our analysis. To mitigate possible confounding factors from the financial crisis and obtain conservative estimates of the relevance of option values, we focus on the sale years for the period 2012–2018. The filtering process is described in Table 2.
In Table 2, SFR structures in Market 5 average about 31 years old, with 25% being greater than 38 years at the time of sale (2012–2018).
Over two-thirds of sales have a swimming pool, an important amenity in the warm, dry climate. Market 5 does not have many golf communities (about 15%). Locations on cul-de-sacs and greenbelts are desirable amenities available for a small percentage of sales (Table 3).
Table 4 indicates a great deal of variation in structure size (improved area), with the average around 2400 sf. Most are single story or split-level ranch houses as indicated by the first-floor area (not shown in Table 4), which in many cases is the same as total square footage, which in many cases is the same as total square footage.
We applied the cost approach described above to value each SFR structure: the result is the “replace cost” variable. This variable starts with the characteristics of each SFR sale and estimates the cost to build a new structure with the same characteristics in the year of sale (“sale year”), then subtracts an estimate of depreciation to arrive at the depreciated cost of a new structure, the estimate of structure value according to the land residual method. Because Market 5 has thousands of sales of vacant land and new construction, older housing must compete with newer. A buyer considering an older house will be able to compare prices, location and characteristics to those of a newer house, or buy vacant land and buy a custom-built house. We expect significant substitution between new construction and older existing houses. This is important: we do not expect dramatic differences among the land valuation methods in Market 5. Any differences we do find will illustrate conservative differences, i.e., that should be accentuated in markets with limited substitution between new construction and older properties.
Market 5 is useful for comparing valuation methods because there are substantial differences in locations of SFR, vacant land and new construction within Market 5. Table 5 shows number of transactions by each of the 28 neighborhoods with boundaries defined by the Maricopa County assessor. Table 5 shows that there are at least 125 SFR sales in each neighborhood and at least one vacant or new construction sale except for 5015 which has no vacant sales but an active new construction market. A handful of neighborhoods have less than 10 transactions in vacant land and in new construction: a buyer wanting to locate in those neighborhoods would have little alternative to an older SFR.
Table 6 characterizes neighborhoods using the ratio of vacant and new construction sales to all SFR sales, where we define new construction as structures less than 16 years old at the time of sale. Clearly many neighborhoods have substantial vacant and/or new construction options for buyers. Neighborhoods with active vacant land and new construction are likely to have substantial tract development. In these, the value of land at the point of construction, but not necessarily over the ensuing 15 years, should be well-measured by the land residual method. The eight neighborhoods without a lot of alternative to existing SFR should have most land values determined by irreversibility.
An alternative to the choice between new and existing SFR is to buy a property (typically a small, old house), tear it down and rebuild. The Maricopa County assessor has identified 75 sales for teardown (Table 7). More than half the teardowns are in two neighborhoods, 5004 and 5005, and almost all of the teardown sales took place in 3 years, 2016–2018. Moreover, the median teardown prices are high compared to the sales of all SFR (see Table 4).
The teardown numbers in Table 7 are highly consistent with the option exercise as predicted by real options theory. Options are exercised at the high contact point, where the price of a newly constructed house at that location (i.e., the price of the underlying asset) has risen permanently to a level that triggers options exercise. The high-contact point must be high enough to justify sacrificing the value of the option which is extinguished after exercise and the value of the existing structure, plus teardown and new construction costs. Ben Bernanke famously described the requirement of a permanently (as believed by the option holder) high price with the regret principle: the person exercising wants some assurance that she will not regret the decision if there is a negative shock to the market. (Note that the prices of teardowns in neighborhoods 5004 and 5005, shown in Table 7, are substantially higher than vacant land in Appendix 2.) This is because the high contact point is far above the NPV = 0 point. In the housing market this implies that option exercise is highly concentrated in time and space, and this is the case in Market 5 (Table 7). This characteristic of option exercise has been documented in other markets by Dye and McMillen (2007) and by Munneke and Womack (2020).
Regression Results and Land Ratios over Time
Table 8 presents hedonic regression coefficients starting with a baseline hedonic valuation model where the dependent variable is SFR sales prices, model (1). Land residual estimates of land values are the dependent variables for models (2) and (3); models (4)–(7) are designed to test the land residual model by regressing sales prices on various combinations of land residual variables and hedonic variables. Model (5) includes all the variables in any of the other models: i.e., it is the unrestricted model which nests models (1), (4), (6) and (7). We chose to estimate models linear in improved area and structure age because nonlinear regressions (not shown) support the linear restriction and results are easier to interpret.Footnote 23 We estimate models in levels because the land residual model is additive in levels, not logs, and some land residual values are negative. We enter land square feet linearly and as a square root to account for “excess acreage”; the value of a building lot per square foot declines if lot size is above the amount needed to build with a normal yard. The square root specification performed well when tested against nonlinear specifications.
All coefficients in the baseline hedonic model (1) have the expected signs. Larger interior area increases structure value at the rate of $122,400 for each additional thousand square feet of floor area: note that sales prices are in hundreds of thousands of dollars and interior area is in thousands of square feet. Property age has the expected negative sign with value decreasing at the rate of $4300 per year from a base in 2012 of $407,400. The presence of a pool adds about 5% to value and construction quality increases value at an increasing rate. Recall from Table 4, that the vast majority of structures have an average quality rating of 4. Properties with an average rating are not worth significantly more than the nearly 5000 properties with rating of 3. Taken together, these sales form a base value typical of market 5: higher quality ratings are exceptional with quality rating 6 (only 302 sales) worth nearly $200,000 more than average. We conclude that the market validates assessor construction quality estimates. This is important because construction quality figured prominently in our estimates (based on cost manuals) of the cost to rebuild structures. In models (2) and (3) the dependent variables are land residual values: i.e., estimates from Eq. (3), sales price minus depreciated construction costs. Model (2) explains land values with all the location variables in our dataset, providing estimated (“hat”) values for models (4) and (5); in addition, model (2) is used for out-of-sample analysis. Significant coefficients in model (2) are the same signs as coefficients in model (1), and magnitudes are similar except for a few of the distance variables and the lot area variables. This means that the land residual method is capturing many of the characteristics of land value that are relevant to market pricing, but with somewhat different weights on several characteristics. The square root of lot size has a much larger coefficient in model (2) compared to model (1) but marginal valuations are virtually identical for the two models, ranging from $14 per square foot to $4 per square foot over a relevant range of lot sizes (6000 to 27,000 square feet).
Structural characteristics are included in model (3) to examine their relationship with land residuals. Land residual theory would predict that land and structure are two separate components of property value with structure properly valued by depreciated construction costs. If this were the case, then we would expect zero coefficients on structure size unless our construction cost estimates were incorrect, or structure is substituted for land. The near-zero coefficients on construction quality and pool dummies suggest that the market values these factors in about the same way as we included them in construction costs: i.e., the land residual model and our cost estimates are jointly supported.
Interpretation of the large positive coefficient on interior area and the $27,000 reduction in land value per year of age is difficult. The signs of these two variables are opposite those that might follow from a purely mechanical relationship: more square footage (higher age) adds to (subtracts from) estimates of structure cost, meaning that they have the opposite mechanical influence on land residual values which are estimated using Eq. (3). It is highly unlikely that we underestimated the influence of size and age enough to account for the signs observed in model (3) because the cost manual provided for large influence of these variables. We conclude that these two large, significant coefficients provide evidence of a problem with the land residual method: for a typical property in the rising market studied, marginal structure value per square foot is undervalued and the amount of structure depreciation per year of age is greater than the cost method would indicate. These marginal effects are consistent with option value theory and with the simple example of option value in a rising market, Table 1.
Model (4) tests the additive separability assumption of the land residual model. If the assumptions are correct, and structure costs correctly estimated then the two should add up to the predicted sales price. Instead, marginal land residual values are overestimated by about $9200 per thousand dollars increase in the land residual, and structure values are underestimated by $26,100. For large houses with cost estimates in the $300,000 to $400,000 range these results can be interpreted to mean that the market value is roughly $100,000 higher than the cost estimate. It may be objected that the discrepancy is due to errors in our cost estimates, but as pointed out above, this is a problem common to all land residual models, and we have a much more detailed and plausible method for estimating structure cost than the previous literature.
Model (5) is the unrestricted model which contains all explanatory variables in any other model. It is included to provide for nested tests of differences in model fit. Model (5) double counts the effects of many variables. This results in a negative sign on lot size because land residuals already account for lot size.
Models (6) and (7) are hedonic models supplemented with depreciated structure cost estimates. Our estimate based on a cost manual includes many variables that are not in the hedonic. We estimate the cost of a finished basement, a garage (attached or detached), additional square footage (e.g., outbuildings) and sports courts (costs vary with size). Also, cost of a second story is estimated separately from the first story. Therefore, we expect structure costs to add information to the hedonic variables, and we find that the R-squared is higher than the baseline hedonic. Model (7) is the same as (6) except that structural characteristics are omitted. Consistent with model (4), replacement cost substantially underestimates the marginal economic value of the structure.
Evaluation of Alternatives to the Simple Land Residual Model
This section evaluates nested models in and out of sample in order to determine if alternatives to the land residual model add explanatory and predictive power to the baseline hedonic.
We use RMSE (last line of Table 8) to compare in sample because the R-squares for models (2) and (3) are not comparable to the other models whereas all RMSEs are calculated based on the variance of sales price compared to predictions of sales price given parameter restrictions.Footnote 24
Not surprisingly, the lowest RMSE is the unrestricted model (5) in which other models are nested. A likelihood ratio test on model (1) versus (5) produces a strongly significant chi-squared statistic, 717 (p-value = 0.0000): i.e., the addition of the land residual variables adds explanatory power. Similar comparison of models (5) and (6) shows that only one of the two land residual variables, construction costs is needed as an addition to the baseline hedonic: there is no chi-squared value because the two models differ only by the redundancy of the land residual variable.
Importantly, the land residual model standard in the literature, model (4), performs poorly in-sample. The additive separability restrictions it requires produce a very large chi-squared statistic, 2461 (p value = 0.0000). A particularly interesting comparison is model (6) vs model (3) which differ by the restriction of the parameter on depreciated cost to equal one in model (3). The RMSE for model (3) is .002 higher than model (6), chi-square = 100.41, significant at less than the 1% level.
Table 9, panel A replicates all these in-sample tests using a leave-one-out framework; results are consistent. We conclude that the additive separability restrictions imposed by the land residual model are not supported by the data. But the addition of structure replacement costs to the standard hedonic model produces the best results in and out of sample. This supports our very detailed use of a cost manual to estimate construction costs, and it shows that construction costs are valued in the housing market.
Are Model Results Consistent with Predictions from Option Value and Land Residual Theories?
Table 9 adds variables to Table 8 models in order to test the option value and land residual theories. The simple model discussed above says that, in a rising market, structure value declines towards zero for small old structures. The model shows that the economic value of these small old structures is substantially less than predicted by land residual methods: see the numerical example in Table 1.
Table 9, panel B tests this by adding a dummy variable for small old structures interacted with the relevant valuation variable from each model. As predicted by option value theory, the marginal value of interior square footage is reduced dramatically for these structures, from $1258 for an additional thousand square feet to $633, a 50% value reduction. Panel C interacts the small old dummy with one for the moderate to high option value neighborhoods identified in Table 10. The result is a larger reduction in valuation to $588, as implied by theory. The large additions to the constant term (1.245 and 1.388) can be interpreted as higher land valuation for these properties.Footnote 25
The interactions of small-old dummies with “replace cost” in models (4)–(7), panels B and C, all strongly confirm the predictions of option value theory given that house prices are increasing in the market.Footnote 26 The decrease in marginal valuation of additional cost for small old houses is between 25% and 67% of the valuation for other houses. Larger decreases in absolute dollars and as a percentage are observed for small old houses in moderate to high option value neighborhoods, panel C.
We tested nonlinear models similar to those in panels B and C. When interior area and replace cost variables were included as quadratics, conclusions are similar, but interpretation is obscured by the squared variables. When quantile dummies (an approach to allow for nonlinearities) are substituted for property age, interior area and replace cost, most of the significance of the small-old dummy goes away. This is not surprising since lack of significance means that quantiles on age and structure size or cost capture the effect predicted by the model: interaction terms are not required. Changes over time in land values and land shares.
Panel A of Table 10 presents changes in price indices from 2012 to 2018 for models (1) and (2), Table 8 and for a third model using “replace cost” as the dependent variable. The increase in construction costs (model (3)) is the rate of increase from the cost manual we used, weighted by the ages and characteristics of structures in our market 5 sample.
Two problems with the land residual method are apparent in Table 10. First, land shares appear very high when compared to shares of around 20% in Ahlfeldt and McMillen (2020). This is due to the undervaluation of structures in a rising market which increases the economic value of both structure and land, whereas the land residual method applies a depreciation rate to construction cost growth: I.e., structure value is said to increase at less than construction costs despite a 47% increase in property values! Table 8 provided evidence for this with the 1.26 coefficient on construction costs, model (4). If we increase structure value in the first line of panel A by 26%, reducing the land residual by about $54,000, then we get a more plausible land share of 36%.
The second problem is the very high rate of increase in land share, from 49.1% to 60.7%, an increase of nearly 24% in only six years. The 78% increase in land values follows from the slow growth in depreciated construction costs (11.3%) when subtracted from rapidly increasing house prices (up by 47.3%). The land residual model does little to constrain the increase in land share in a market with increasing house prices as illustrated over a longer time in Fig. 1.
Land Value Shares over Time: Option Value Compared to Land Residual Assumptions
We use option value to construct land value shares (Table 10 panel B) by first classifying neighborhoods according to the amount of option value and then using option value concepts based on Eqs. (4) and (5) to estimate land value ratios and evolution of ratios over time for each neighborhood type. Our strategy is motivated by our objective of showing that application of option value concepts to land valuation might produce land value ratios dramatically different than the land residual method, even in market #5 which has a lot of new construction and large numbers of vacant land sales that can be substituted for existing housing. We compare to land residual estimates that we calculate and those produced by the FHFA (Davis et al., 2019).
Using results in Tables 6 and 7, we identified four types of neighborhoods:
Neighborhoods with high concentrations of teardown sales in 2016–2018. Table 7 which shows that the 75 teardown sales are concentrated in the last 3 years of the time period and that options exercise is concentrated in neighborhoods 5003, 5004 and 5005. These three neighborhoods are classified as in neighborhood type (A). Note that this concentration of options exercise in time and space is as predicted by option value theory, where the high contact point holds when expected future implicit rental income is unusually high (See Helms, 2003; Munneke & Womack, 2020). For this category, we assume that land values evolve based on the evolution of vacant land prices divided by prices of existing SFR properties from the baseline hedonic model.
Neighborhoods that are characterized by tract development or a high percentage of new construction sales. There are 14 neighborhoods in this category, neighborhood type (B). We assume no option value in these neighborhoods: for comparison we assume that land shares evolve as predicted by the land residual model, Table 10 panel A.
Neighborhoods with high vacant land percentages but not high rates of new construction. There are 3 neighborhoods in this category. Their codes are 5008, 5022 and 5028. We exclude these neighborhoods from our option value analysis since many vacant properties have not reached the point of option exercise. Including them does not substantially influence our conclusions.
Neighborhoods with high percentages of existing housing sales: i.e., neighborhoods that do not fall into any of the above categories. There are 8 neighborhoods in this category, neighborhood type (C), completing an exhaustive classification of the 28 neighborhoods. Option value theory predicts that land shares are increasing slowly due to annual structure depreciation in this category. Construction costs are not a factor for this type of property as indicated by Table 1 and the above discussion of theory.
To finish implementing our strategy, we obtain land value ratios in 2012 based on land residual theory using median prices for land and new houses for each neighborhood type. We think these ratios are too high because residual assumptions undervalue most existing structures through the assumption that structure and land evolve independently and by ignoring capital land substitution. Ahlfeldt and McMillen (2020) use sales of lots linked to subsequent sales of new SFR properties to arrive at much lower ratios. Nevertheless, we use land residual theory for initial ratios because our focus is on changes over time in these ratios, not their level.
Option value theory says that each property starts with a ratio given by the land residual method. Over subsequent years this ratio increases slowly until depreciation and, more importantly, functional obsolescence increases the ratio toward one; a ratio = 1 is achieved at the high contact point when it becomes economically feasible to tear down and rebuild to highest and best use. We increase land values in type A neighborhoods at the rate of vacant land price changes and divide by prices in each year for new houses (less than 16 years old).
Results indexed to base 100 in 2012 show that land value shares for high option value neighborhoods increased by about 11% over six years compared to nearly 24% under land residual assumptions. There is more variability in neighborhood type (A), consistent with the evolution of vacant land prices in Fig. 2. The neighborhoods classified as existing with little option value, neighborhood type (C), have low growth in the land ratio due to depreciation only, as required by option value theory.Footnote 27 This 2% growth in land shares is dramatically less than 24% for neighborhoods where we use the land residual assumptions. Neighborhoods with little option value (type C) represents the vast majority of urban neighborhoods which have aging housing and little new construction or redevelopment, suggesting that the 2% to 24% comparison represents a general pattern in markets with increasing house prices. These large differences over only six years suggest that the hockey stick pattern over time found by Knoll et al. (2017) is largely an artifact of land residual assumptions.
Generalization to the U.S.: FHFA Land Residual Estimates of Land Share over Time
The last column of Table 10, panel B presents land share data from the FHFA (Davis et al., 2019), a study that provides similar information for numerous MSAs representing a large share of the U.S. housing market.Footnote 28 They apply the land residual method only to structures built within 10 years of sale, reasoning that these new properties should follow the land residual theory. But we find that FHA land shares increase even more rapidly than in neighborhoods with a lot of new construction, 27% vs 24%. Option value theory says that the problem with FHFA numbers is that they do not allow new structure values to increase with property values over the first 10 years, and they do not allow for building larger houses as land becomes more expensive (capital land substitution).Footnote 29
It may be objected that this conclusion follows from our assumption that land ratios in the 8 neighborhoods dominated by older existing SFR are constant except for depreciation. The objection is correct, but is it an objection or just a comment? The assumption reflects the reality of irreversibility – land and structure are a bundled good – and this applies to the majority of the built environment in most urban areas. Moreover, we have provided considerable empirical support for the assumption in Fig. 2, the temporal and spatial clustering of market 5 transactions in Tables 5 through 7 and the coefficients in Tables 8 and 9. Most importantly, we began with a simple thought experiment: what typically happens to urban structure and land values when the only change within a fully built-up neighborhood is an unexpected, permanent shock to demand? Land residual assumptions do not provide a plausible answer when the value of a large structure is compared to the otherwise identical small structure.
Do we contradict ourselves with our theory which nests the land residual Eq. (1) within option value theory, but then presenting evidence that land residuals give incorrect patterns over time even for houses built within 10 or 15 years of sale? No, because land residual theory holds at the moment of construction, not after changes such as the boom in house prices in Maricopa county, 2012–2018. To correctly apply option value theory, the analyst needs construction costs for each property at the time it was built. The resulting land value ratio changes over time as predicted by irreversibility, not with the ratio of other newly built properties in the neighborhood: land and structure function as a bundled good after construction whereas land residual theory severs the connection between land value and the structure that provides the source of residual value to land.