Error Correction Models of MSA Housing Supply Elasticities

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Error Correction Models of MSA Housing Supply Elasticities

Transcript Of Error Correction Models of MSA Housing Supply Elasticities

2nd Draft: January 19, 2014
Error Correction Models of MSA Housing “Supply” Elasticities: Implications for Price Recovery
William C. Wheaton Department of Economics
Center for Real Estate MIT*
Cambridge, Mass 02139 [email protected]
Serguei Chervachidze Capital Markets Economist CBRE Econometric Advisers [email protected]
Gleb Nechayev Senior Economist CBRE Econometric Advisers [email protected]
The authors are indebted to, the MIT Center for Real Estate, and CBRE. They remain responsible for all results and conclusions derived there from. * Corresponding Author
JEL Code: R. Keywords: Housing

ABSTRACT MSA-level estimates of a housing supply schedule must offer a solution to the twin problems of simultaneity and stationarity that plague the time series data for local housing prices and stock. An Error Correction Model (ECM) is shown to provide a solution to stationarity, but not simultaneity. A Vector Error Correction Model (VECM) is suggested to handle both the stationarity and endogeneity problems. Such models also nicely distinguish between (very) long run elasticities and a variety of short term impacts. We estimate these models separately for 68 US MSA using quarterly data on housing prices and residential construction permits since 1980. The results provide long run supply elasticity estimates for each market that are better bounded than previous panel-based attempts and also correspond with much conventional thought. We find these elasticities are well explained by geographic and regulatory barriers, and that inelastic markets exhibit greater price volatility over the last two decades. Using the models’ short run dynamics we make several forecasts of prices over the next decade. In current dollars, some MSA will still not recover to recent peak (2007) house price levels by 2022, while others should exceed it by as much as 70%.

I. Introduction
There has been a revival of interest in the supply schedule that characterizes urban housing markets. It has been argued that variation in the elasticity of this schedule may explain the wide differences observed across US metropolitan statistical areas (MSA) in house price levels, their growth and also volatility [(Campbell-Davis-Martin (2009), Capozza-Hendershott-Mack (2004), Cannon-Miller-Pandher (2006), Glaeser-GyourkoSaiz (2005), Paciorek (2013), Davidoff (2013)]. It has also been argued that housing supply elasticities impact the allocation of labor across MSAs and can affect aggregate productivity [Nieuwerburgh-Weil (2009), Eckhout-Pinheiro-Schmidheiny (2010)]. Thus there is much interest in the possible “determinants” of local housing supply elasticities, be they related to intrinsic geographic land supply [Saiz (2010)], local land use regulations [Gyourko-Saiz-Summers (2008)], or the organizational structure of the homebuilding industry [Somerville (1999)]. Despite all this interest, much of the discussion is conducted in an empirical vacuum, for there are no individual time series estimates of long or short run housing supply elasticities in San Francisco versus Dallas or any of the other 300 US MSAs. The literature review section that follows will make this more apparent.
One purpose of this study therefore is to apply modern time series analysis to quarterly data on MSA constant dollar (repeat sale) house prices, new housing construction and the growth of the stock – covering 1980:1 through 2012:2. We do this individually for each of the largest 68 MSA. At the start, we find that (real) house price levels and stock are not stationary - in all but a few MSA. This calls into question much of the early empirical research on this topic. Both series, however, are stationary in differences, or I(1). Most importantly house prices and stock (measured in levels) are cointegrated (again in all but a handful of markets). This suggests that an error-correction methodology is the proper framework for understanding the relationship between these two variables – at least at the MSA level.
Within this framework we estimate a single equation Error Correction Model (ECM) in which stock determines prices, and then a Vector Error Correction Model (VECM) in which both variables are simultaneously determined as well as determining. The cointegrating vector between the two variables yields a consistent estimate of the long

run elasticity of supply and across our 68 MSA the results are extremely plausible: ranging from .2 to 3.1. The short run dynamic impacts estimated through both models are also significant however and provide us with an alternative short-run measure of housing supply elasticity. With the ECM we ask what recovery in prices – from their current (2012:2) levels - will be needed to support anticipated new supply over the next decade. To allow for the possibility that demographic aging will slow long run growth of housing demand from its historic trend, we set the anticipated new supply at 80% of that built in previous decades. With the VECM we simply do an out-of-sample forecast of both stock and price for the next 10 years. The ratio of the stock/price forecasts then provides a measure of the effective short run elasticity to be observed over the next decade.
We find great consistency between these supply elasticities, with correlations in the .9 range. We also find that they relate statistically to those “determinants” of supply elasticity recently hypothesized in the literature. Larger cities, with geographic land constraints, that have high scores on a survey of regulatory barriers have significantly lower supply elasticities – as we have measured them.
We then investigate the role of supply elasticity in an MSA’s price experiences over the 2001-2012 period characterized by the housing “boom” and “bust”. Consistent with the conventional thinking that this period was marked by housing demand shocks, markets with more inelastic supply exhibit both greater price increases over 2001-2007, and greater price declines from 2007 to 2012.
Finally, we use the short run forecasts generated by the models to assess the degree of housing price growth and eventual price recovery that is likely across our 68 MSA, beginning in 2012:3. We find that markets with inelastic supply will experience faster price growth between 2012 and 2022, thus exhibiting similar dynamics to the prior decade. That said inelastic markets generally have no greater tendency to recover or exceed their 2007 “bubble” peak levels – in constant dollars. We suspecte that this finding is due to the fact that areas with inelastic supply generally experience slower demand growth than areas with more elastic supply.

In the next section we review the empirical literature on housing supply and its determinants. Section III reviews the standard model of land development that underlies much of the supply elasticity discussion, while Section IV then examines the data series available on prices, construction and the stock – testing for stationarity and cointegration. Section V lays out our ECM and VECM models, along with some alternative estimation strategies. Section VI presents results from estimating the various models, and displays their long and short run implied elasticities. In section VII we examine whether these elasticities line up with often used supply “instruments”. Section VIII examines the relationship between these elasticity estimates and market behavior from 2001 through 2012, while Section IX examines the magnitude and patterns in housing price recovery over the next decade that is likely across our 68 MSA.
II. Housing Supply Literature
There is an early literature on housing supply [Alberts (1962), Burns-Grebbler (1982)] that looks at housing construction as a “business” in which the level of price relative to some opportunity cost (including credit) drives housing “investment” or unit flows (permits, starts or completions). The most recent elaboration of this approach is Topel-Rosen (1988) in which great attention is given to the expectation mechanism for future prices. Following Abel-Blanchard (1986), DiPasquale-Wheaton (1992, 1994) add to this approach with the notion that expected price levels determine a “desired” stock towards which the actual stock adjusts slowly with new investment. In this approach the existing stock is a critical additional variable (to prices) in explaining new investment. Much of this early literature is summarized in Blackley (1999), who notes that it contains little or no application of true time series analysis. For example the series are rarely tested for stationarity or cointegration, and regressions contain mixed I(0) and I(1) series.
Mayer-Somerville (2000) redefines the entire discussion of supply elasticity and links it more explicitly to land development. In their framework the key ingredient necessary for cities to grow through new construction is the development of additional land. Following a long and voluminous literature on urban spatial models - this requires higher land values which are simply a residual from housing prices. Thus like DiPasqualeWheaton the underlying “supply” relationship is between housing price levels and housing

stock, rather than price levels and housing flows. They also argue that any empirically estimated relationship between house price levels and stock is likely to be miss specified as both variables are I(1) and not I(0). They present estimates of a relationship between changes in both housing stock and price that they argue is reflective of that between housing price and stock levels. We return to this specification issue in the next section.
All of the literature above uses US national data to estimate some supposed national supply elasticity. But if the supply elasticity is really about land development, then surely there should be significant variation across cities of different sizes, with different geographies, transportation systems and regulatory processes. The availability of widespread MSA-level housing price indices has prompted a series of more recent analysis of differences in price movements across MSA. Capozza-Hendershott-Mack (2004), Cannon-Miller-Pandher (2006) and Campbell-Davis-Martin (2009) all examine the movement in prices to see if there is any relationship between price appreciation and price volatility across (respectively) MSA, ZIP codes and US census regions. None of these studies explicitly include study of housing supply or stock.
Harter-Drieman (2004) opens up a new line of inquiry by using panel data analysis from 1980 to 1998 to compare 49 MSA. She develops a VECM model relating prices to income (rather than stock or households) and does not test for cointegration between these two variables. By allowing for a market specific constant term in the long run cointegrating relationship, she is able to calculate a price response to an income shock that is unique to each market. Then assuming a common price elasticity of demand, she is able calculate an implicit estimate of each market’s supply elasticity. Not surprising, the result is a very narrow range of implicit supply elasticities; for example, she finds that no markets are inelastic.
Saiz (2010) adopts a similar panel approach, but with limited time series: there are only 3 decadal changes (observations) for each MSA. He develops an estimating equation wherein price changes are predicted using household (housing stock) changes, fixed effects, and two variables that logically would impact a supply elasticity: the Wharton Land Use Regulatory Index (WLURI) [Gyourko-Saiz-Summers (2008)] and a newly constructed measure of geographic land unavailability. By interacting these variables with

household changes he again is able to calculate an implicit supply elasticity for each market – yielding estimates of between .6 and 5.0.
Over this time span there is only one attempt to separately estimate a supply elasticity uniquely by market: Green-Malpezzi-Mayo (2011). Their paper simply presents the elasticity resulting from a series of simple bivariate regressions with no statistical specification tests, and no resulting statistics. The elasticities vary widely (-.3 to 29.0) and half are reported as insignificant. A second stage cross-section regression explaining the elasticities contains 9 variables, most of which likely to be endogenous to a housing supply process. The approach of this paper is in principle similar to Green-MalpezziMayo, but with far greater series detail and testing. Such testing suggests the need for a very different model – one following an error correction framework.
III. Long-Run models of Local Housing Supply.
As discussed above, newer models of housing supply are actually models of land development and city expansion – rather than of “investment” in housing. In this literature, monocentric land use models yield relationships between equilibrium city size (housing stock) and the difference between central housing prices and edge housing prices1. This latter can be thought of as a crude “average” city price. This relationship begins with the set of variable definitions in (1) below.

B : traveldistancefrom the urban center toedgein a circularcity.

T : capitalized cost of travelinga unit distance(fixed acrosslocations)

L :lot size of house (fixed acrosslocation)

K : Capitalcost of a house (fixed acrosslocation)

ra : price of ubiquitousagricultural land outside the circle

P0 , PB : house Prices at urban center and edge

S : stock of housing (population)

A : fraction of land (at all locations)that is developable


1 Here we use the most simple of monocentric models where transport costs are constant and exogenous, and where land consumption is similarly. The voluminous literature on such models often has land consumption determined by agent utility and prices, while congestion can make transport costs non-linear and endogenous.


The equations below derive a set of equilibrium relationships between these variables. Equation (2) requires that house price differences between the center and edge equal the capitalized value of traveling from the edge to center. This is a spatial equilibrium condition. Equation (3) links edge prices to replacement costs using capital and agricultural land (both exogenous), while (4) links the urban border with total housing stock. Combining equations, we get the result in (5) where equilibrium average house prices (the difference between center and edge prices) depend positively on the size of the housing stock. While (5) expresses price as a function of stock, it is equally true that to provide a larger stock of urban land prices must rise and hence stock is a function of price; in other words equilibrium implies joint causality. Expression (5) has often been interpreted as an inverse housing “supply” schedule [Saiz (2010), Mayer and Somerville (2000)]. Expression (5) illustrates that the long term relationship between house prices and stock may also depend on land availability and the efficiency of the city transportation system (Saiz, 2010).

P0  PB  TB


PB  K  Lra


B  ( SA)1/ 2


P0  PB  T (SA )1/ 2


Economists also have modeled a competitive system of cities [Weil and Nieuwerburgh-Weil (2009), Eckhout-Pinheiro-Schmidheiny (2010)]. In these models a “national” population selects where to reside among cities. Jobs or employment do likewise. In addition to productivity and amenity considerations, these choices are influenced in no small part by house prices. Thus through the migration of population and the relocation of employment there can arise a parallel (but negative) “demand” relationship D(…) between prices and stock. Ceteris paribus, lower prices eventually attract an expanding population and job base (E). In equilibrium this base must equal the stock (S). Expressions (6) and (7) can be combined with (5) to complete any full model of a system of cities.


E  D(P0 ) , D' 0




In recent years monocentric models have given away to “polycentric” theory wherein a city has multiple centers or locations of economic activity. Each such center however behaves like a “sub-center” – following similar conditions as in (2)-(5). The main difference from the monocentric framework is that some sub centers compete with each other over land, rather than just with agricultural uses. Some theory of “agglomeration” is also necessary to determine whether the city is composed of many small sub centers or a few very large ones. With endogenous employment location, this family of models does not always yield the same comparative static results as with single centered city models. For example, in monocentric models higher transportation costs raise land rent and generate more dense cities. In polycentric models, higher transportion costs lead to greater employment dispersal and little or no increase in land rents [McMillen-Smith (2003), Helsley-Sullivan (1991)]. The empirical implication of these models is that often they have almost infinite long run housing (land) supply. With a single fixed center, increasing the supply of land requires greater commuting and hence land rent. With multiple endogenous centers more land can be had by expanding the number of centers with little attendant increase in commuting since the dominant pattern of commutes is within a center and not between centers. Hence land rents also need not increase.
Finally, it is important to remember that (1) – (7) inform us only about long run equilibrium relationships and provide very little guidance in terms of short term dynamics. Short run dynamics are often thought to involve the capital portion of housing rather than land: the difficulty of constructing new structures and their durability once built. GlaeserGyourko (2005) for example convincingly show that even over decade long intervals the derivative of house prices with respect to stock is asymmetric: falling prices do little to shrink the stock, but rising prices are needed to expand it. Put differently, the identity between S and E in (6) rarely holds over short run intervals, when the stock of housing adjusts slowly.


While this discussion is certainly not new, it does highlight a range of important issues that any empirical study of local housing supply elasticities must address. The first such issue is that estimation must be flexible enough to accommodate a full range of potential long run elasticity values, matching the variation anticipated from polycentric as opposed to monocentric urban models. Secondly, the estimation should allow for short run dynamic effects that may be distinct from long run equilibrium impacts. For example, if the stock increases from a true “shock” then prices may fall in the short run rather than rise along the long run schedule. Finally, stock and prices are jointly determined in equilibrium and so there is a high likelihood of simultaneity between the series Using local economic data as an instrument to resolve this simultaneity is certainly not valid in the long run. Given that structural identification is difficult, non-structural macroeconomic time series analysis may offer more viable solutions.
IV. Empirical Tests of MSA House Price and Housing Stock Series
The data used in this analysis consists of two time series for each of 68 US metropolitan areas, at quarterly frequency, covering 1980:1 through 2012:2. The first is the Federal Housing Finance Agency’s (FHFA) all-transactions house price index (HPI) based on repeat transactions involving conventional mortgages purchased or securitized by Fannie Mae and Freddie Mac. The second is a series of total housing stock, starting with the 2010 Decennial Census and adding (for post 2010) or subtracting (for pre 2010) housing permits each quarter. It should be noted that this estimated stock series will not produce values for 1980, 1990 or 2000 stock that match the Decennial Census unit counts for those years, due largely to the effects of demolitions and undercounting. It would be possible to calculate a 3-decade average quarterly stock adjustment for each MSA and apply this to the estimated stock series, but such scalar adjustment would not impact the statistical results, although it could alter slightly the estimated elasticities.
To test for stationarity we undertake augmented Dickey-Fuller tests (ADF) of house price and stock in levels, using 4 and then 8 lags. For house prices, we can reject the