Comovements among U.S. state housing prices: Evidence from fractional cointegration

Economic Modelling 29 (2012) 936–942

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Comovements among U.S. state housing prices: Evidence from fractional cointegration Carlos Pestana Barros a,⁎, Luis A. Gil-Alana b, James E. Payne c a b c

Technical University of Lisbon, Lisbon, Portugal University of Navarra, Faculty of Economics, Edificio Biblioteca, Entrada Este, E-31080 Pamplona, Spain University of South Florida Polytechnic, 3433 Winter Lake Road, Lakeland, FL 33803, USA

a r t i c l e

i n f o

Article history: Accepted 7 February 2012 JEL classification: C22 Keywords: U.S. state housing prices Persistence Long memory Fractional cointegration

a b s t r a c t This study investigates the relationship between U.S. state housing prices and overall U.S. housing prices as well as the relationship among state housing prices using fractional integration and cointegration techniques. The results based on parametric and semiparametric estimators reveal that some states contain unit roots though we fail to find cointegrating relations between U.S. states housing prices and the overall U.S. housing prices as well as among state housing prices. The results raise doubts regarding the long-run convergence in U.S. state housing prices and the presence of the ripple effect. © 2012 Elsevier B.V. All rights reserved.

1. Introduction This study examines the comovements among U.S. housing prices at the state level along with the relation of each state's housing price to the overall U.S. housing price. In light of the recent housing price bubble and its collapse due to the financial crisis, it is interesting to analyze to what extent regional and national shocks are transmitted across states. The transmission of shocks across regional housing markets, known as the rippled effect, will result in housing prices moving together in the long-run (Meen, 1999). The underlying economic factors associated with the ripple effect include migration patterns, home equity conversion, spatial arbitrage, as well as the correlated movement in the spatial patterns of the determinants of house prices. Indeed, the ripple effect implies that the ratio of regional house prices to national house prices should be stationarity, but also comovements among the regional prices should exist. On the other hand, as pointed out by Johnes and Hyclak (1994) and Drake (1995), the absence of stationarity (or long-run convergence) has implications for labour market mobility as well as financial capital mobility to effectively arbitrage house price differentials. Unlike previous studies, which adopt standard (integer) cointegration analysis, this study extends this literature by employing fractional cointegration to infer whether there is a long-run equilibrium relationship first between state level housing prices and the overall

⁎ Corresponding author. E-mail addresses: [email protected] (C.P. Barros), [email protected] (J.E. Payne). 0264-9993/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2012.02.006

U.S. housing price, and then among the state housing prices in the U.S. Specifically, standard cointegration analysis allows only integer values for the memory parameters, say d (individual series) and b (cointegrating regression), and tests for the presence of cointegration depending on the existence of unit roots in the cointegrating relationships. On the other hand, fractional cointegration is a more general framework which allows the memory parameters to take fractional values as positive real numbers. Moreover, allowing the orders of integration to be real values provides for a richer degree of flexibility in the dynamic specification of the time series, not achieved when using the classical approaches based on integer degrees of differentiation, i.e. d = 1 for the individual time series and b = 0 for the cointegrating regression. The flexibility in the dynamic specification of housing prices through fractional integration and cointegration techniques provides for the possibility of long-memory (persistence) behavior which characterizes housing prices. Persistence measures the extent to which short term shocks in current market conditions lead to permanent future changes (Caporale and Gil-Alana, 2004). A shock represents an event which takes place at a particular point in the time series, and it is not confined to the point at which it occurs. A shock is known to have a temporary or short term effect, if after a number of periods the time series returns back to its original performance level. On the other hand, a shock is known to have a persistent or long term impact if its short run impact is carried forward to set a new trend in performance. For instance, in the case of a unit root, shocks will be permanent and the time series will be highly persistent.

C.P. Barros et al. / Economic Modelling 29 (2012) 936–942

As discussed by Case and Shiller (1989), Tirtiroglu (1992), Clapp and Tirtiroglu (1994), Gatzlaff and Tirtiroglu (1995), and Cho (1996), the empirical evidence raises doubts with respect to the efficiency of the housing market and prices to adjust in the wake of new information. Furthermore, Holmes et al. (2011) and Apergis and Payne (forthcoming) also note that regional housing markets may not respond at the same time to a common shock as regional sensitivities to demand and supply factors vary across regions. Differences in such regional factors as migration patterns, spatial arbitrage, per capita income, availability of mortgage financing, labour mobility, rental housing market, demographic composition, weather, degree of urbanization, among other socio-economic factors, impact the degree of responsiveness of regional housing markets to shocks. 2. Brief literature overview The majority of the literature on the ripple effect (i.e., long-run convergence) has focused on the regional housing markets within the U.K. using a variety of econometric approaches to detect comovements between regional housing prices. Rosenthal (1986) utilized cross-spectral analysis of regional house price transactions to cast doubt on the convergence hypothesis. Giussani and Hadjimatheou (1991), MacDonald and Taylor (1993), Alexander and Barrow (1994), and Ashworth and Parker (1997) employ standard cointegration analysis to infer the long-run equilibrium relationship between U.K. regional housing prices yielding mixed results. Drake (1995) uses time varying parameter estimation to test for convergence between regional house prices to find regional differences in U.K. house price movements. Cook (2003) undertakes asymmetric unit root tests to examine the stationarity of the U.K. regional/national house price ratios. Cook and Thomas (2003) utilize non-parametric tests and business cycle dating procedures to present evidence supportive of the ripple effect. In another study of U.K. regional house prices, Cook (2005) employs a threshold autoregressive cointegration model to find asymmetric adjustment to equilibrium. Holmes and Grimes (2008) combine principal components with unit root testing to examine the long-run relationships in U.K. regional house prices to show that regional house prices exhibit strong convergence in the long-run. Tests of long-run convergence have also been investigated in a number of other countries as well. Within a vector error correction model Stevenson (2004) lends support for the ripple effect from Dublin to other regions of the Irish housing market. Oikarinen (2006) utilizes a vector error correction model to reveal ripple effects from the Helsinki metropolitan area to the regional centers than to peripheral areas in Finland. Luo et al. (2007) employ cointegration analysis to identify ripple effects across Australia's capital cities. Larraz-Iribas and Alfaro-Navarro (2008) analyze the cointegrated relationship among housing prices in Spanish regions with evidence of convergence among regions. Burger and Rensburg (2008) examine the stationarity of metropolitan housing prices in South Africa to find segmented housing markets with varying levels of convergence. Chien (2010) tests the stationarity of regional/national house price ratios for cities within Taiwan to show ripple effects for each city in Taiwan except in the case of Taipei City. In the case of the U.S., a number of studies have been undertaken. Pollahowski and Ray (1997) investigate the spatial and temporal behavior of house price dynamics for the nine U.S. census regions as well as metropolitan areas to find a relationship between spatial prices, but no difference in price diffusion patterns between neighboring and non-neighboring divisions. Zohrabyan et al. (2008) utilize cointegration techniques to reveal cointegrated relationships among U.S. regions led by regions influential in financial and economic aspects. Clark and Coggin (2009) employ structural time series models in the examination of U.S. regional house price convergence to render mixed results with respect to convergence behavior. Gupta and Miller

937

(forthcoming) use a battery of cointegration and causality test to infer the causal relationship between housing prices in Los Angeles, Las Vegas, and Phoenix to find housing prices in Los Angeles causes housing prices in Las Vegas directly and Phoenix indirectly. Using a similar methodological approach, Gupta and Miller (2012) examine eight South California MSAs to find a long-run relationship among MSA housing prices. Kuethe and Pede (2011) undertake a spatial–temporal analysis of the states in the western U.S. to provide evidence of convergence among these states. Holmes et al. (2011) examine U.S. state housing pries using a probabilities test statistic for convergence with the results supportive of convergence with the speed of adjustment inversely related to distance between states. Barros et al. (2011) examine the stationarity of state/national house price ratios in the U.S. using fractional integration with results indicating a great deal of variation in the stationarity properties and long memory behavior of state housing prices. Payne (forthcoming) uses autoregressive distributed lag models to examine the long-run relationship between U.S. regional housing prices to find cointegration across regions. Apergis and Payne (forthcoming) utilize clustering and club convergence procedures to identify three convergence clubs with respect to U.S. state housing prices. The rest of the study is organized as follows: Section 2 briefly surveys the literature. Section 3 describes the methodology used in the study. Section 4 presents the data and the main results. Section 5 provides concluding remarks. 3. Hypotheses and methodology of fractional cointegration The methodology employed is based on the concept of long memory or long range dependence to examine several hypotheses related to housing prices. First, given the differences in the sensitivity of demand and supply factors within housing markets across states, we test whether (i) housing prices in U.S. states are persistent. Second, in light of the ripple effect and long-run convergence, housing prices should move together in the long-run, we test whether (ii) comovements exist between individual U.S. state housing prices and the overall U.S. housing price and (iii) comovements exist between housing prices across U.S. states. Given a covariance stationary process {xt, t = 0, ±1, … }, with autocovariance function E(xt − Ext)(xt-j − Ext) = γj, according to McLeod and Hipel (1978), xt displays the property of long memory if

limT→∞

T   X   γ j 

ð1Þ

j¼−T

is infinite. An alternative definition, based on the frequency domain, is the following: Suppose that xt has an absolutely continuous spectral distribution function, so that it has a spectral density function, denoted by f(λ), and defined as f ðλÞ ¼

∞ 1 X γ cosλj; 2π j¼−∞ j

−πbλ≤π:

ð2Þ

Then, xt displays the property of long memory if the spectral density function has a pole at some frequency λ in the interval [0, π]. Most of the empirical literature has concentrated on the case where the singularity or pole in the spectrum occurs at the smallest (zero) frequency. This is the standard case of I(d) models of the form: ð1−LÞd xt ¼ ut ;

t ¼ 0; 1; …;

ð3Þ

where L is the lag operator (Lxt = xt-1) and ut is I(0) defined as a covariance stationary process with spectral density function that is positive and bounded at any frequency.

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Thus, the process ut is short memory and maybe a stationary and invertible ARMA sequence, when its autocovariances decay exponentially; however, it could decay at a much slower rate than exponentially (in fact, hyperbolically) if d is positive. When d = 0 in Eq. (3), xt = ut, and xt is said to be “weakly autocorrelated” as opposed to the case of “strongly autocorrelated” if d > 0. Moreover, if 0 b d b 0.5, xt is still covariance stationary, but its lag-j autocovariance γj decreases very slowly, at the rate of j 2d-1 as j → ∞, and so the γj are absolutely non-summable. We say then that xt has long memory given that f(λ) is unbounded at the origin. Also, as d in Eq. (3) increases beyond 0.5 and through 1 (the unit root case), xt can be viewed as becoming “more nonstationary” in the sense, for example, that the variance of the partial sums increases in magnitude. Processes of the form given by (3) with positive non-integer d are called fractionally integrated, and when ut is ARMA(p, q), xt has been called a fractionally ARIMA (or ARFIMA) model. This type of model provides a higher degree of flexibility in modelling low frequency dynamics which is not achieved by non-fractional ARIMA models. Engle and Granger (1987) suggested that, if two processes xt and yt are both I(d), then it is generally true that for a certain scalar a ≠ 0, a linear combination wt = yt − axt, will also be I(d), although it is possible that wt be I(b) with b b d. This is the concept of cointegration, which Engle and Granger (1987) adapted from Granger (1981) and Granger and Weiss (1983). Given two real numbers d, b, the components of the vector zt are said to be cointegrated of order d, b: (i) all the components of zt are I(d), (ii) there exists a vector α ≠ 0 such that st = α'zt ~ I(b), with b b d. Here, α and st are called the cointegrating vector and error, respectively. 1 This prompts consideration of an extension of Phillips' (1991) triangular system, which for a very simple bivariate case is: yt ¼ νxt þ u1t ð−bÞ;

ð4Þ

xt ¼ u2t ð−dÞ;

ð5Þ

for t = 0, ±1, …, where for any vector or scalar sequence wt, and any c, we introduce the notation wt(c) = (1 − L) cwt. ut = (u1t, u2t) T is a bivariate zero mean covariance stationary I(0) unobservable process and ν ≠ 0, b b d. Under (4) and (5), xt is I(d), as is yt by construction, while the cointegrating error yt − νxt is I(b). Models (4) and (5) reduces to the bivariate version of Phillips' (1991) triangular form when b = 0 and d = 1, which is one of the most popular models displaying standard cointegration considered in both the empirical and theoretical literature. Moreover, this model allows greater flexibility in representing equilibrium relationships between economic variables than the traditional integer (d = 1, b = 0) prescription. The methodology employed to examine the hypothesis of fractional cointegration is based on three steps: 3.1. Step 1 We first estimate individually the orders of integration of the time series. Here, we employ parametric and semiparametric methods. In the parametric context, we use a method proposed by Robinson (1994) which is basically a Lagrange Multiplier (LM) test that uses the Whittle function in the frequency domain. This approach tests the null hypothesis, H o : d ¼ do ;

ð6Þ

1 Even considering only integer orders of integration, a more general definition of cointegration than the one given by Engle and Granger (1987) is possible, allowing for a multivariate process with components having different orders of integration. Nevertheless, in this study, we focus exclusively on bivariate relationships and a necessary condition is that the two series display the same integration order.

for any real value do, in a model given by (1) where xt can be the errors in a regression model of the form: yt ¼ βT zt þ xt ;

ð7Þ

t ¼ 1; 2; …;

where yt is the observed time series, β is a (kx1) vector of unknown coefficients and zt is a set of deterministic terms that might include an intercept (i.e., zt = 1), an intercept with a linear time trend (zt = (1, t) T), or any other type of deterministic processes. Robinson (1994) has shown that, under certain very mild regularity conditions, the LM-based statistic ^r →dtb Nð0; 1Þas T→∞; where “ →dtb “ stands for convergence in distribution, and this limit behavior holds independently of the regressors zt used in Eq. (7) and the specific model for the I(0) disturbances ut in Eq. (3). The functional form of this procedure can be found in any of the numerous empirical applications based on his tests (see, e.g., Gil-Alana and Robinson, 1997; Gil-Alana and Henry, 2003; Cunado et al., 2005, among others). As in other standard large-sample testing situations, Wald and LR test statistics against fractional alternatives will have the same null and limit theory as the LM test of Robinson (1994). In fact, Lobato and Velasco (2007) essentially employ such a Wald testing procedure, though clearly this method requires an efficient estimate of d. In this sense the LM procedure of Robinson (1994) seems computationally simpler. 2 With respect to the semiparametric approaches, there exist two main branches, those based on the Whittle function and others that use log-periodogram-type estimators. We employ both of them. First, we use a “local” Whittle estimator in the frequency domain (Robinson, 1995a), using a band of frequencies that degenerates to zero. This estimator is implicitly defined by: d^ ¼ arg mind C ðdÞ ¼

! m 1X logC ðdÞ−2d logλs ; m s¼1

m 1X 2d Iðλ Þλ ; m s¼1 s s

λs ¼

2πs ; T

1 m þ →0; m T

ð8Þ

where I(λs) is the periodogram of the raw time series, xt, given by:  2  T 1 X iλs t  xt e  ; Iðλs Þ ¼   2πT  t¼1

ð9Þ

and d ∈ (− 0.5, 0.5). Under finiteness of the fourth moment and other mild conditions, Robinson (1995a) proved that:  pffiffiffiffiffi ^ m d−d →dtb Nð0; 1=4Þas T→∞;

ð10Þ

where d⁎ is the true value of d. This estimator is robust to a certain degree of conditional heteroscedasticity (Robinson and Henry, 1999) and is more efficient than other semi-parametric competitors. 3 Finally, we use the log-periodogram estimator of Robinson (1995b), which is defined as: d^ ðlÞ ¼

m  X

   aj −a logI λj =Sl ;

ð11Þ

j¼lþ1

2 Other parametric approaches like Sowell's (1992) maximum likelihood in the time domain and Beran's (1995) method were also employed and the results were substantially the same as those reported in this study. 3 There exist further refinements of this procedure, e.g., Velasco (1999), Velasco and Robinson (2000), Phillips and Shimotsu (2004), Shimotsu and Phillips (2005), and Abadir et al. (2007), though these methods require additional user-chosen parameters, and the estimates of d may then be very sensitive to the choice of these parameters.

C.P. Barros et al. / Economic Modelling 29 (2012) 936–942

as a whole is used in the analysis. The data were obtained from the St. Louis Federal Reserve database, FRED II, with a base period 1980:1 = 100. First, we estimate the order of integration in each of the individual series, using the three approaches described in the Step 1 in Section 3. For the parametric approach (Robinson, 1994) we choose a model with an intercept and a linear time trend, removing the components insignificantly different from zero, and using a seasonal AR(1) process for the disturbance term ut. The estimated values of d are displayed in the second column in Table 1. The third and fourth columns in this

where    p 1 X 2 λj ; a ¼ a u þ εt ; aj ¼ − log 4 sin m−l j¼1 j t−j 2 Sl ¼

939

m  2 X 2πj ; aj −a ; λj ¼ T j¼lþ1

and 0 ≤ l b m b n. 4 3.2. Step 2 We test the homogeneity of the orders of integration in the bivariate systems (i.e., Ho: dx = dy), where dx and dy are the orders of integration of the two individual time series, by using an adaptation of Robinson and Yajima (2002) statistic T^ xy to log-periodogram estimation. The statistic is: T^ xy

  m1=2 d^ x −d^ y ¼   1=2 1 ^ = G ^ G ^ 1−G þ hðT Þ; xy

2

xx

ð12Þ

yy

^ is the (xy) th element of where h(T) > 0 and G xy          m X −1 −1 ^¼ 1 Re ΑΛ^ λj I λj Λ^ λj ; Λ^ λj G m j¼1

 iπd^ =2 −d^ iπ d^ =2 −d^ ¼ diag e x λ x ; e y λ y ; with a standard limit normal distribution. 3.3. Step 3 In the third step, we perform the Hausman test for no cointegration set forth by Marinucci and Robinson (2001) comparing the estimate d^ x of d with the more efficient bivariate one of Robinson x

(1995a), which uses the information that dx = dy = d*. Marinucci and Robinson (2001) show that  2 2 H is ¼8s d^  −d^ i →dtb χ 1

1 s as þ →0; s T

ð13Þ

with i = x, y, and where s b [T/2] is a bandwidth parameter, analogous to m introduced earlier; d^ are univariate estimates of the parent sei

ries, and d^  is a restricted estimate obtained in the bivariate context under the assumption that dx = dy. In particular, s P

d^  ¼ −

j¼1

0

^ −1 Y v 12 Ω j j

0 ^ −1 1 P v2 2 12 Ω 2 j

s

;

ð14Þ

j¼1

with Yj = [log Ixx(λj), log Iyy(λj)] T, and vj ¼ logj− 1s

s P

log j: The limit-

j¼1

ing distribution above is presented heuristically, but the authors argue that it seems sufficiently convincing for the test to warrant serious consideration. 4. Data and results Quarterly data from 1975:1 to 2010:7 for the house price index from the Federal Housing Finance Agency of each state and the U.S. 4 Extensions of this method can be found in Moulines and Soulier (1999), Velasco (2000), Phillips and Shimotsu (2002), and Andrews and Guggenberger (2003).

Table 1 Summary of the results in terms of the estimated values of d. States

Parametric

Whittle semiparametric

Log-Periodogram

AK AL AR AZ CA CO CT DE FL GA HI ID IL IN IA KS KY LA ME MD MA MI MN MS MO MT NE NV NH NJ NM NY NC ND OH OK OR PA RI SC SD TN TX VT UT VA WA WV WI WY US

0.944 0.897 1.022 1.342 1.768 1.294 1.390 0.901 1.021 1.081 0.758 0.893 1.189 1.126 0.943 1.114 1.206 1.219 0.666 1.425 1.404 1.248 1.334 0.708 0.912 0.866 0.944 1.387 1.074 1.440 1.124 1.189 1.165 0.676 1.332 1.242 1.183 1.151 1.346 0.935 0.421 0.942 1.156 0.445 1.278 1.333 1.439 0.539 1.125 1.139 1.478

1.094 1.500 1.176 1.500 1.500 1.500 1.500 1.500 1.500 1.500 1.319 1.500 1.500 1.500 1.426 1.500 1.480 1.500 1.228 1.500 1.500 1.500 1.500 1.456 1.500 1.500 1.007 0.940 0.965 0.961 1.010 0.964 0.986 1.022 0.997 1.040 1.002 1.002 1.500 1.500 0.738 1.500 1.500 1.382 1.500 1.500 1.500 1.253 1.435 1.500 1.500

1.115 1.292 1.398 1.445 1.648 1.372 1.728 1.243 1.545 1.726 1.141 1.410 1.192 1.809 1.334 1.768 1.196 1.850 1.330 1.558 1.875 1.651 1.537 1.295 1.441 1.817 1.006 0.936 0.947 0.945 1.005 0.950 0.985 1.023 0.996 1.036 0.998 0.998 1.639 1.571 0.522 1.823 1.833 1.115 1.506 1.372 1.170 1.339 1.176 1.522 1.816

In bold, the cases with evidence of unit roots, i.e., d = 1. State abbreviations: AK (Alaska), AL (Alabama), AR (Arkansas), AZ (Arizona), CA (California), CO (Colorado), CT (Connecticut), DE (Delaware), FL (Florida), GA (Georgia), HI (Hawaii), ID (Idaho), IL (Illinois), IN (Indiana), IA (Iowa), KS (Kansas), KY (Kentucky), LA (Louisiana), ME (Maine), MD (Maryland), MA (Massachusetts), MI (Michigan), MN (Minnesota), MS (Mississippi), MO (Missouri), MT (Montana), NE (Nebraska), NV (Nevada), NH (New Hampshire), NJ (New Jersey), NM (New Mexico), NY (New York), NC (North Carolina), ND (North Dakota), OH (Ohio), OK (Oklahoma), OR (Oregon), PA (Pennsylvania), RI (Rhode Island), SC (South Carolina), SD (South Dakota), TN (Tennessee), TX (Texas), VT (Vermont), UT (Utah), VA (Virginia), WA (Washington), WV (West Virginia), WI (Wisconsin), WY (Wyoming), US (United States).

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C.P. Barros et al. / Economic Modelling 29 (2012) 936–942

Table 2 Tests of the homogeneity condition for the orders of integration.

AK NE NH NM NY NC OR PA

AK

NE

NH

NM

NY

NC

OR

PA

xxx xxx xxx xxx xxx xxx xxx xxx

0.511⁎ xxx xxx xxx xxx xxx xxx xxx

0.788⁎ 1.720⁎⁎ xxx xxx xxx xxx xxx xxx

0.517⁎ 0.066⁎ − 1.559⁎ xxx xxx xxx xxx xxx

0.774⁎ 1.796⁎⁎ − 0.192⁎ 1.695⁎⁎ xxx xxx xxx xxx

0.609⁎ 1.754⁎⁎

0.371⁎ − 1.291⁎ − 1.819⁎⁎ − 1.286⁎ − 1.838⁎⁎ − 1.727⁎⁎ xxx xxx

0.548⁎ 0.459⁎ − 1.792⁎⁎ 0.320⁎ − 2.152 − 0.970⁎ 1.291⁎

− 1.401⁎ 1.203⁎ − 1.585⁎ xxx xxx xxx

xxx

* and ** means non-rejection of the null hypothesis of equal orders of integration at the 5% and 10% levels, respectively.

table refer to the estimated values of d using the Whittle and the logperiodogram-type (semiparametric) estimators of d, respectively. 5 The results in Table 1 indicate the potential existence of unit roots in the following 31 states: Alaska, Alabama, Arkansas, Delaware, Florida, Georgia, Idaho, Illinois, Indiana, Iowa, Kansas, Missouri, Nebraska, Nevada, New Hampshire, New Jersey, New Mexico, New York, North Carolina, North Dakota, Ohio, Oklahoma, Oregon, Pennsylvania, South Carolina, Tennessee, Texas, Vermont, Washington, Wisconsin, and Wyoming. However, of these 31 states, we only found conclusive evidence of unit roots based on the three methods presented for eight states, which are Alaska, Nebraska, New Hampshire, New Mexico, New York, North Carolina, Oregon, and Pennsylvania. In the other cases, we find at least one case where the unit root is rejected. On the other hand, the estimated value of d for the overall U.S. housing price, displayed in the last row of the table, indicates that this value is clearly greater than 1: 1.478 with the parametric method, 1.500 with the Whittle semiparametric approach, 6 and 1.816 with the logperiodogram method. Therefore, we can rule out any long run equilibrium relationship between the overall U.S. state house price and the eight states in which unit roots may be present due to the different orders of integration. Nevertheless, within these eight states, we can still find a long run equilibrium relationship, and this is undertaken in the Step 2. In Step 2 we test for homogeneity across the eight states where we found strong evidence in favor of unit roots by using the Robinson and Yajima (2002) procedure. Table 2 reports the results of the test for homogeneity in the orders of integration. We notice that we are unable to reject the hypothesis of equal orders of integration for the eight states examined, which is not surprising given the strong support in favor of the I(1) case obtained in Step 1. In light of the results in Table 2, the Hausman test for cointegration is employed (Step 3 in the methodology). Following Marinucci and Robinson (2001), we test the null hypothesis of no cointegration against fractional cointegration. Table 3 displays the results which indicate that the null hypothesis of no cointegration cannot be rejected in any case. In fact, we observe that the orders of integration in the cointegrating regressions are equal to or slightly below 1, rejecting the hypothesis of cointegration of any order. Therefore, according to the results, there is no long-run equilibrium relationship between U.S. state housing prices. 5. Concluding remarks This study examines the long-run equilibrium relationship between state housing prices and overall U.S. housing price as well as the relationship among state housing prices. We examine the 5

nonstationarity of housing prices by conducting tests based on fractional integration and cointegration. The results can be summarized as follows: we find strong evidence in favor of unit roots in only eight states: Alaska, Nebraska, New Hampshire, New Mexico, New York, North Carolina, Oregon, and Pennsylvania. In the remaining U.S. states, we reject the unit root hypothesis as well as for the overall U.S. housing price. In the majority of these cases the rejections are in favor of alternatives with orders of integration which are above 1, implying that housing prices are highly persistent. Focusing on the relationship between individual state housing prices and the overall U.S. housing price, the results do not support comovements between state housing prices and the overall U.S. housing price, at least for the states where the unit root cannot be rejected. The reason is that the housing prices for these states exhibit an order of integration which is substantially different from the overall U.S. housing price. Finally, focusing on the eight states where the unit root hypothesis cannot be rejected, we examine the possibility of comovements among the housing prices of these states. The results fail to support cointegration, which implies that state housing prices move relatively independent of one another. These results indicate that housing prices are highly persistent and therefore shocks in current market conditions lead to permanent future changes. The results also suggest that housing prices across states are influenced more by state-specific economic and demographic factors than national factors. Table 3 Test the null hypothesis of no cointegration against the alternative of fractional cointegration. Hx

AK

NE

NH

NM

NY

NC

OR

PA

AK

xxx

0.013 0.902 1.102

NE

xxx

xxx

0.159 1.554 1.074 0.089 0.077 0.976

NH

xxx

xxx

xxx

0.015 0.910 1.102 0.0005 0.0002 1.003 0.074 0.087 0.975

NM

xxx

xxx

xxx

xxx

0.134 1.561 1.077 0.071 0.079 0.979 0.0008 0.000001 0.950 0.069 0.076 0.979

NY

xxx

xxx

xxx

xxx

xxx

0.041 1.145 1.094 0.010 0.010 0.996 0.037 0.032 0.967 0.009 0.009 0.995 0.038 0.021 0.970

NC

xxx

xxx

xxx

xxx

xxx

xxx

0.022 0.390 1.099 0.002 0.118 1.000 0.060 0.390 0.973 0.002 0.122 0.999 0.062 0.351 0.976 0.004 0.184 0.993

OR

xxx

xxx

xxx

xxx

xxx

xxx

xxx

PA

xxx

xxx

xxx

xxx

xxx

xxx

xxx

0.056 0.823 1.090 0.019 0.003 0.992 0.024 0.117 0.963 0.018 0.004 0.992 0.025 0.096 0.967 0.0003 0.021 0.983 0.215 0.008 0.989 xxx

Hy d⁎

0.5

In these two cases, we choose as bandwidth number m = (T) and for the logperiodogram estimator l = 1. Other values produced essentially the same results. 6 Many Whittle semiparametric estimates in the table display the value of 1.500. This is due to the fact that the estimates were obtained based on the first differenced data, adding then 1 to obtain the proper estimate, and noting that the method computes d in the range (− 0.5, 0.5), the value of 1.500 indicate that the order of integration may be higher than that number.

Χ12(5%) = 3.84. The first two values refer to the test statistics for Hx and Hy respectively. The third value is the estimation of d⁎.

C.P. Barros et al. / Economic Modelling 29 (2012) 936–942

The results of this study contribute to the literature on the ripple effect and long-run convergence in regional housing prices by providing more accurate evidence on measuring persistence in U.S. housing prices through the use of fractional integration and cointegration techniques. Models based on fractional integration and cointegration are more general than the classical models based on integer degrees of differentiation and thus allow for a much richer degree of flexibility in the dynamic specification of housing prices. The persistence behavior exhibited by state housing prices confirms previous studies related to the efficiency of the housing market and prices in the adjustment process in response to new information. The results also highlight the importance of identifying the differences in regional sensitivities to demand and supply factors across regions that may contribute to such persistence behavior. Furthermore, as noted by Grenadier (1995) and Gu (2002), inefficiencies in the housing market may be attributed to the non-standardized nature of housing, transaction costs, demand uncertainty, adjustment costs, and construction lags which contribute to the observed persistence. Another consideration is the potential nonlinear behavior of housing prices associated with boom and bust housing cycle experienced in the U.S. (Kim and Bhattacharya, 2009; Miles, 2008). With respect to the policy implications of our findings, policymakers at the national level through fiscal and/or monetary policy actions should be cognizant of the regional differences in housing markets and the varying response across regions to policy actions as our results suggest that housing prices are influenced more by state-specific economic and demographic factors. On the other hand, state and local policymakers need to understand the unique demand and supply factors driving regional housing markets and design the appropriate policies to ensure stable housing prices over time. Furthermore, the methodological approach utilized in this study can be extended to the case of U.S. housing markets at the metropolitan level as well as to regional housing markets in other countries in the identification of long-memory behavior and the influence of such behavior on the ripple effect and long-run convergence in regional housing prices. In addition, future research should be focused on the potential presence of structural breaks and seasonality, which is especially relevant in the context of fractional integration given these issues are related (Diebold and Inoue, 2001; Granger and Hyung, 2004).

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