Unit Roots and Structural Change

Abstract

This paper employs time-series analysis to investigate the price dynamics of the house price indices included in the S&P/Case–Shiller Composite10 index and the validity of the ‘ripple effect’, following the approach outlined by Meen (1999). More specifically, the paper first considers the time-series properties of the capital gain from the sale of houses. That is, it examines whether shocks to the capital gain series produce permanent or transitory changes. In general, the findings lack uniformity and depend upon the assumptions imposed by the testing procedures. Secondly, it considers the time-series properties of the ratio of regional house price indices to the Composite10 index. That is, it examines whether shocks to these house price ratios exhibit trend reversion. The tests of this ‘ripple effect’ also display conflicting evidence.

1. Introduction

The behaviour of regional house prices constitutes an important area of research, which emerged in recent years, in part, because of the boom and bust cycles undergone by many local housing markets. Most analysts attribute the collapse of house prices in recent years as triggering the financial crisis that led to the significant recession in the US (and world) economy. The analysis of the run-up and collapse of house prices in the past decade requires a careful investigation of the characteristics of house price time-series.¹

Economic data frequently exhibit stochastic trend (i.e. non-stationarity and unit-root processes) (Nelson and Plosser, 1982; Juselius, 2009) and, a priori, no reason exists to exclude house prices from containing stochastic trend. Stochastic trend imposes important characteristics on how theoretical models can explain economic reality. That is, stochastic trends imply that economic series wander around with no tendency to revert to some mean value. Moreover, stochastic trends also render standard statistical inferences invalid. In this paper, we employ a battery of unit-root tests to analyse two separate, but intertwined, issues of the housing markets. First, we consider whether the rate of capital gain from the sale of houses exhibits trend-reverting movement. Testing formally for deterministic versus stochastic trends² is important for the applied time-series analysis of housing markets for several reasons. One, the two processes imply different dynamics (Rudebusch, 1993), especially for forecasting analyses. Two, unit-root tests provide the starting-point for cointegration analysis, Granger-causality tests and impulse response functions. Three, unit-root tests shed light on the basic hypothesis about asset prices, weak-form efficiency. Weak-form efficiency requires that we cannot use the history of an asset price to predict future changes in any meaningful manner. In an efficient market, asset prices fully incorporate all relevant information and, hence, the rate of capital gain displays unpredictable (unit-root) behaviour. If the capital gain from house price movements does not contain a unit root, then we can predict future capital gain changes by the historical sequence of its past changes.

Meen (1999) and Peterson et al. (2002), using standard unit-root tests, find that the UK national house price series follows a unit-root process. More recently, however, Cook and Vougas (2009) show that the use of a more sophisticated testing methodology can reverse findings derived using the conventional unit-root approach. Cook and Vougas (2009), using the smooth transition momentum-threshold autoregressive (ST-MTAR) test of Leybourne et al. (1998), confirm the stationarity characteristic of house price changes but find that house prices exhibit structural change. Using quarterly data from 1975 to 1996 from the 50 US states, Muñoz (2003) finds unit roots in house price changes, using the Dickey–Fuller generalised least squares (DF–GLS) test (Elliott et al., 1996). Meen (2002) compares the time-series behaviour of house prices in the US and the UK. Using quarterly data from 1976 to 1999 for the US and from 1969 to 1999 for the UK, Meen (2002) conducts both augmented Dickey–Fuller (ADF) and Phillips–Perron (PP) unit-root tests on the level of house prices and finds that in both countries house prices follow a difference stationary process. That is, house prices are integrated of order one, or I(1). By implication, the rate of capital gain from the sale of houses should prove integrated of order zero, or I(0), since the rate of capital gain approximately equals the logarithmic difference in the house price between months. Given the recent boom and bust of the housing markets in the US, compelling reasons exist to investigate further the behaviour of house prices in the US.

Two extensions deserve consideration— non-linear unit-root tests and linear unit-root tests with structural breaks. Conventional unit-root tests, which assume structural stability and linear adjustment, can interpret departures from linearity and structural instabilities as permanent stochastic disturbances. We control for two sources of non-linearities in the dynamics of house prices when applying unit-root tests. First, non-linearities can exist in the form of threshold effects, whereby the price dynamics follow a non-stationary process at some threshold, but follow a stationary process outside the threshold (Teräsvirta, 1994). Kapetanios et al. (2003) propose a non-linear unit-root test, which permits a stable dynamic process with an inherently non-linear adjustment caused by market frictions and transaction costs, and show that the non-linear test proves more powerful than the standard unit-root tests. Secondly, non-linearities can also exist when the economic series suffer from structural changes. Bierens (1997) suggests modelling these changes as broken deterministic time trends, which produces a non-linear deterministic trend. Alternatively, Lumsdaine and Papell (1997) and Lee and Strazicich (2003) propose tests that directly incorporate structural changes. Much research argues that the presence of structural breaks distorts the results of conventional unit-root tests (Perron, 1989, 1997).

Lee et al. (2006) find that accurate forecasting and empirical verification of theories can depend critically on understanding the appropriate nature of structural change in time-series data. Consequently, we analyse the time-series characteristics of the rate of capital gain from the sale of houses, checking for unit roots both with and without structural breaks. When considering structural breaks, we implement the two endogenous structural break models developed by Lumsdaine and Papell (1997) and Lee and Strazicich (2003).The use of tests that allow for the possible presence of structural breaks possesses at least two advantages. First, it protects against test results that, in the linear framework, are biased towards non-rejection, as suggested by Perron (1989). Secondly, since this procedure can, unlike the non-linear tests, identify when structural breaks occur, it can provide valuable information about whether the break associates with a particular government policy, economic crises, war, regime shifts or other factors.

The tests that assume structural constancy and linear adjustment serve as a comparison for the effect of adding endogenous breaks into the test procedure. Researchers have not applied linear unit-root tests with two structural breaks or non-linear unit-root tests to US metropolitan rate of capital gain from the sale of houses. Compared with the Lumsdaine–Papell tests, the Lee–Strazicich unit-root tests incorporate the endogenous breaks also in the null. That is, the endogenous two-break unit-root test of Lumsdaine and Papell (1997) assumes no structural breaks under the null. As Lee and Strazicich (2003) emphasise, rejection of the null does not necessarily imply rejection of a unit root per se, but may imply rejection of a unit root without break. Similarly, the alternative does not necessarily imply trend-stationarity with breaks, but may indicate a unit root with breaks. Lee and Strazicich (2003) propose an endogenous two-break minimum Lagrange multiplier (LM) unit-root test that allows for breaks under both the null and alternative hypotheses. As a result, rejection of the null unambiguously implies broken-trend stationarity.

The key to understanding the issues relates to the critical values that the research must generate through Monte Carlo simulations. The larger the breaks in the trend, the further the critical values computed under no and trend breaks diverge from each other (Lee and Strazicich, 2003, p. 1082). In other words, to determine unambiguously if the time-series in question achieves broken-trend stationarity, researchers must include the breaks in the trend in the null hypothesis.

Secondly, we consider the house price diffusion effect or ‘ripple effect’ and address the issue of convergence/divergence in US metropolitan housing markets. UK housing experts identify a ‘ripple effect’ of house prices that begins in the south-east UK and proceeds towards the north-west. Economic theory and intuition suggest that different regional house prices should not move together. House prices depend mostly on local housing market supply and demand factors, which can differ substantially between regions due to differences in regional economic and demographic environments. Yet, a variety of empirical studies present extensive evidence on the so-called ripple effect, the interregional or spatial transmission of shocks in house prices.

Meen (1999) describes four different theories that may explain the ripple effect— migration, equity conversion, spatial arbitrage and exogenous shocks with different timing of spatial effects. Migration could cause house price ripples, if households relocate in response to changes in the spatial distribution in house prices. House prices need not equalise among regions because long-lasting differences exist in regional or metropolitan area fixed endowments (for example, climate) or scale economies (Haurin, 1980). An exogenous shock to a region, however, may disrupt local house price levels, causing migration (Haurin and Haurin, 1988). Migration spreads the effect of the shock throughout a region or country, causing a spatial ripple of house price change. Changes in house prices change homeowners’ equity (Stein, 1995). An increase in equity relaxes downpayment constraints, permitting additional mobility. In contrast, falling nominal house prices reduce equity and constrain mobility. The spatial diffusion of house prices proves a manifestation of arbitrage mitigated by search costs or by the diffusion of news throughout a region. Pollakowski and Ray (1997) test whether house price changes in one region predict price changes in other regions using a vector autoregressive (VAR) model. Their work builds on Tirtiroglu (1992) and Clapp and Tirtiroglu (1994), who find that excess returns to houses in a sub-market diffuse to other sub-markets of the same MSA. Pollakowski and Ray (1997) find statistically significant cross-price effects at the regional level, but no sensible economic pattern to their results exists. This purely spatial approach implicitly argues that the transmission mechanism flows across space, not across economically similar housing sub-markets. Finally, Meen assumes that all regions react to shocks with different speeds. House prices change first in the fastest-reacting region, followed by price changes in slower-reacting areas. Thus, price ripples occur, although no transmission mechanism exists. Meen (1999) develops a time-series econometric test of this hypothesis using UK regional data. He finds evidence supporting the claim of different response rates. In the long run, house prices tend to return to the same pre-shock relative values.

The ripple effect implies that the long-run convergence of house prices occurs and requires that deviations of regional prices from the national price are stationary (Meen, 1999). The ripple effect hypothesis receives little support in the US economy, although exceptions do exist. For example, most analyses relate to a given geographical housing market, such as a metropolitan area (Tirtiroglu, 1992; Clapp and Tirtiroglu, 1994). More recent evidence across census regions also exists, which may reflect the fourth of Meen’s explanations (Pollakowski and Ray, 1997; Meen, 2002). Finally, Gupta and Miller (2010b, 2010a) find evidence of house price linkages between the eight Southern California metropolitan statistical areas (MSAs) and between Los Angeles, Las Vegas and Phoenix.

Several econometric approaches to examine the ripple effect exist in the empirical literature. A number of studies investigate the links between regional housing markets using Granger causality (Giussani and Hadjimatheou, 1991; MacDonald and Taylor, 1993; Alexander and Barrow, 1994; Berg, 2002; Peterson et al., 2002; Gupta and Miller, 2010a, 2010b) and the Engle–Granger two-step or Johansen cointegration procedures (MacDonald and Taylor, 1993; Alexander and Barrow, 1994; Munro and Tu, 1996; Ashworth and Parker, 1997; Luo et al., 2007).³ Other approaches include Kalman filtering (Drake, 1995), cross-correlation matrices (Peterson et al., 2002) and unit-root methods (Meen, 1999; Cook, 2003, 2005a, 2005b). Still other methods compare the long-run equilibrium relationships between house prices and ‘fundamentals’ (Alexander and Barrow, 1994; Malpezzi, 1999; Case and Shiller, 2003).⁴ More recently, researchers have studied the ripple effect using non-parametric methods (Cook and Thomas, 2003), principal component analysis (Holmes and Grimes, 2008), panel unit-root tests (Holmes, 2007), spatial versions of autoregression and Granger causality models (Kuethe and Pede, 2009), as well as spatial and temporal diffusion models (Holly et al., 2010).⁵

Following the analysis in Meen (1999), we cast the issue as a univariate unit-root problem. That is, we consider the time-series characteristics of the ratios of the metropolitan house price indices to the national house price index in the US. This approach provides an advantage over spatial methods in that it focuses on stochastic convergence (in the sense of Carlino and Mills, 1993) and, rather than examining how house prices in a regional market respond to house price changes emanating from contiguous housing markets, this approach examines regional housing markets in relation to the US. Furthermore, this approach is unique in that it directly models up to two endogenous structural breaks in the series. Spatial dependence models emphasise short-run adjustment processes and provide behavioural explanations of the spatial patterns, but do not provide formal test results about the nature of the long-term trends in the housing markets. Meen (1999) emphasises that the diffusion of changes in house prices implies a long-run constancy in the ratio of regional house prices to the national house price. Alternatively, the ratio of regional house prices to the national house price exhibits stationarity under the ripple-effect hypothesis, reverting to an underlying trend value. This represents an additional reason why researchers need to understand the nature of the shocks to the housing market. If the ripple effect exists, then a given price shock in a metropolitan area may produce permanent or transitory implications for house prices in other metropolitan areas, depending on the unit-root characteristics of the data. Meen (1999), using the ADF unit-root test, fails to find significant evidence of stationarity in the house price ratios for the UK. Conversely, Cook (2003) detects overwhelming convergence in a number of regions in the UK, using an asymmetric unit-root test. Cook (2005b) detects stationarity by jointly applying the DF–GLS test (Elliott et al., 1996) and the Kwaitkowski–Phillips–Schmidt–Shin (KPSS) stationarity test (Kwiatkowski et al. 1992).

The rest of the paper is structured as follows. Section 2 discusses the data and method. Section 3 reports results of linear and non-linear unit-root tests of the rate of capital gain from the sale of houses in 10 US metropolitan areas under alternative assumptions regarding linearity of the model or structural constancy in the deterministic components of the trend. We find that the integration characteristics of the rate of capital gains differ markedly across alternative assumptions. Section 4 reports the empirical results of the analysis of the ripple effect in the US. We show that the assumptions of linearity and structural constancy significantly affect the time-series characteristics of the ratios of the metropolitan house price indices to the national house price index in the US. Section 5 concludes.

2. Data and Method

This section considers the data and method of analysis. We briefly describe seven unit-root tests used in the analysis of the dynamics of the capital gain on the sale of houses and the ripple effect. We illustrate the methods using the capital gain series. The first five include linear and non-linear tests that assume structural stability in the time-series pattern of the data. The last two, instead, test the unit-root hypothesis under the assumption of two structural breaks.

2.1 Data

We extract the data utilised in this paper from the S&P/Case–Shiller Home Price Indices (HPI) database and include seasonally adjusted monthly house price indices for the metropolitan statistical areas (MSAs) measured by the S&P/Case–Shiller HPI Composite10 index: Boston, Chicago, Denver, Las Vegas, Los Angeles, Miami, New York, San Diego, San Francisco and Washington, DC. The sample period of each series equals monthly data from January 1987 through April 2009, 268 observations.

House price indices possess well-known issues surrounding their validity or appropriateness, reflecting mainly the non-fungible nature of housing. The S&P/Case–Shiller HPI data possess several advantages over the Federal Housing Finance Agency (FHFA) (formerly Office of Federal Housing Enterprise Oversight) house price indices, ordinarily used in the literature (Deng and Quigley, 2008; Himmelberg et al., 2005, among others). These two indices measure house prices quite differently. For instance, the S&P/Case–Shiller house price index includes foreclosed houses, while the FHFA indices do not. Consequently, the S&P/Case–Shiller house price index shows a larger decline in national and metropolitan house prices than the FHFA indices. Both the FHFA and S&P/Case–Shiller HPI use the weighted repeated-sales methodology.⁶ The FHFA indices, however, exhibit more limitations than the S&P/Case–Shiller indices. First, the FHFA indices (at the MSA level) appear quarterly, while the S&P/Case–Shiller indices appear monthly. Monthly data provide a better opportunity to model the house price and the rate of capital gain from the sale of houses in a shorter time-interval. Secondly, the S&P/Case–Shiller indices only include actual house transactions and do not include, like the FHFA indices, refinance appraisals, which produce ‘appraisal smoothing bias’ for the rate of capital gains measurement (Geltner, 1989; Edelstein and Quan, 2006). Thirdly, the FHFA indices only incorporate Fannie Mae and Freddie Mac conforming mortgages, which concentrate at the lower end of prices in the housing markets. Finally, the Chicago Mercantile Exchange (CME) uses the S&P/Case–Shiller indices for housing futures and options. These derivatives enable investors to take positions on the movement of the Composite10 index and any of the indices that compose it. The CME housing futures contracts trade on the CME Globex electronic trading platform and the options on futures trade on the trading floor in an open outcry style. Although trading volumes for the CME housing futures and options remain relatively thin, the analysis of the S&P/Case–Shiller HPI also provides practical implications for the homebuilding industry (Jud and Winkler, 2008).⁷

We construct the time-series of the rate of capital gain from the sale of houses indices, y_t, defined as follows

y_{t} = p_{t} - p_{t - 1}

(1)

where, p_t refers to the natural logarithm of the house price index at time t.⁸

2.2 The Unit-root Model

The long-standing debate on housing market efficiency (Case and Shiller, 1989) connects intimately to the question of unit roots in the house price data. In its simplest form, housing market efficiency requires that today’s house price provides the best prediction of tomorrow’s price. In other words, the house price series conforms to a random-walk or non-stationary process, while the capital gain series, which we can approximate with the logarithmic difference in the house price, conforms to a stationary process. Assuming that the capital gain series y_t follows an autoregressive process with two structural breaks, we can write

y_{t} = α + β t + θ D U_{1 t} + γ D T_{1 t} + ω D U_{2 t} + ψ D T_{2 t} + ρ y_{t - 1} + ε_{t}

(2)

where, the variable t is a time trend; ϵ _tand is an error term. DU₁ _t and DU₂ _t are indicator variables for the breaks in the intercept, occurring at times TB₁ and TB₂ respectively, while DT₁ _t and DT₂ _t are indicator variables for the breaks in the time trend occurring at times TB₁ and TB₂ espectively. TB₁ and TB₂ denote the dates of the two structural breaks. The values of the coefficients α, β and ρ determine the basic character of the time-series. The parameter α represents ‘drift’ (i.e. a fixed movement in each time-period), while the parameter β represents the effect of a linear time trend. The most important parameter for determining the character of the series, however, is ρ. Subtracting y_t−1 from both sides of equation (2) and rearranging the model generates the following

Δ y_{t} = α + β t + θ D U_{1 t} + γ D T_{1 t} + ω D U_{2 t} + ψ D T_{2 t} + (ρ - 1) y_{t - 1} + ε_{t}

(3)

where, Δ is the difference operator.

If ρ < 1, then (ρ − 1) < 0 and the change in the capital gain Δy_t depends on the capital gain at t-1. This denotes a lack of efficiency. Such a series is called mean- or trend-reverting (β = 0, or β ≠ 0 respectively) and enables the researcher to forecast future capital gains from past capital gains. Any house price shock that pushes the capital gain price away from its trend will eventually dissipate. By contrast, if ρ = 1 then (ρ − 1) = 0, a change in the capital gain in any period simply consists of the drift and trend component (if any) plus a random change ε_t. Thus, researchers cannot forecast future capital gains from past capital gains and the market is (weak-form) efficient. Such a series is termed a random walk (with trend and/or drift). Any shocks will permanently affect the price and no mean- or trend-reversion tendency exists. The time-series already described may exhibit either stationarity (if ρ < 1) or non-stationarity (if ρ = 1). We can test for (weak-form) market efficiency by testing for the value of ρ —that is, by testing whether the series possesses a unit root.

If the house price time-series is non-stationary, then the issue emerges as to the time-series characteristics of the rate of capital gain on the sale of houses. That is, y_t approximates the rate of capital gain, excluding the implicit consumption benefits from living in the house. If p_t is an I(1) process, then y_t is an I(0) process, by definition. Conversely, if y_t is an I(1) process, then p_t must behave as an I(2) process. The dynamics of I(2) processes exhibit more complex structure than the dynamics of I(1) processes (Haldrup, 1998). In both cases, the shocks possess permanent effects, but housing prices that conform to I(2) processes are driven by different permanent shocks that housing prices that conform to I(1) processes.⁹ Researchers, therefore, need to assess the empirical reliability of the unit-root hypothesis.

Conventional unit-root tests such as the ADF and PP tests lose power dramatically against stationary alternatives with a low-order, moving-average (MA) process: a characterisation that fits well to all the rate of capital gains on S&P/Case–Shiller HPI. We use four more efficient procedures to test the null hypothesis that each series contains a unit root. First, the generalised least squares (GLS) version of the Dickey–Fuller (DF) test due to Elliott et al. (1996) (DF–GLS) exhibits superior power to the ADF test (i.e. is more likely to reject the unit-root hypothesis against the stationary alternative hypothesis when the alternative is true). Secondly, we use the point-optimal, unit-root test developed by Elliott et al. (1996) (ERS-P_T). Finally, Ng and Perron (2001) developed modified versions of the PP test (NP-MZ_t ) and of the ERS point-optimal test of Elliott et al. (1996), (NP-MP_T), both of which exhibit excellent size and power characteristics.

The DF–GLS of Elliott et al. (1996) is essentially an ADF test, except that they transform the data via a generalised least squares (GLS) regression prior to performing the test. They perform the test in two steps. First, they de-trend (de-mean) the data using the GLS approach. Secondly, they use an ADF test to test for a unit root. For a detailed discussion of the DF–GLS test, see Stock and Watson (2010, pp. 644–647).

The DF–GLS test that allows for a linear time trend relies on the following regression

Δ y_{t}^{d} = α y_{t - 1}^{d} + \sum_{j = 1}^{p} β_{j} Δ y_{t - j}^{d} + v_{t}

(4)

where, $y_{t}^{d}$ equals the de-trended (de-meaned) capital-gain series and v_t equals an error term.

The DF–GLS statistic equals the t-ratio testing H₀: α = 0 against H₁: α and < 0. In addition to the DF–GLS test, Elliott et al. (1996) compute a second unit-root test, the so-called point-optimal test. They derive the power envelope and maximise the power for a given alternative hypothesis (point-optimal test) against the background that no uniformly most powerful unit-root test exists. The test statistic that consistently asymptotically satisfies this condition is

P_{T} = [S S R (\bar{a}) - S S R (1)] / f_{0}

(5)

The point-optimal test involves the compu-tation of the sum of squared residuals $S S R (\bar{a}) = \sum_{i = 1}^{t} {\hat{η}}_{t}^{2} (\bar{a}) .$ The null hypothesis for the point optimal test is α = 1 and the alternative hypothesis is $α = \bar{a} .$ The test statistic equals

P_{T} = [S S R (\bar{a}) - S S R (1)] / f_{0}

(6)

where, f₀ estimates the residual spectrum at frequency zero. SSR(a) equals the sum of the squared residuals of a quasi-differenced OLS regression, given the alternative hypothesis $\bar{a} .$ The Akaike information criterion (AIC) determines the lag length in f₀.

Ng and Perron (2001) construct four test statistics that use the GLS de-trended data $y_{t}^{d} .$ Two of these statistics modify the statistic of Phillips and Perron (1988) and point-optimal statistics of Elliott et al. (1996). Letting $κ = \sum_{t = 2}^{T} (y_{t - 1}^{d})^{2} / T^{2},$ we can write the GLS modified statistics as follows

N P - M Z_{t} = (κ / f_{0})^{1 / 2} (T^{- 1} (y_{T}^{d})^{2} - f_{0}) / 2 κ

(7)

and

N P - M P_{T} = ({\bar{a}}^{2} κ - \bar{a} T^{- 1} (y_{T}^{d})^{2}) / f_{0}

(8)

in the de-meaned case where $\bar{a} = - 7$

N P - M P_{T} = ({\bar{a}}^{2} κ + (1 - \bar{a}) T^{- 1} (y_{T}^{d})^{2}) / f_{0}

(9)

in the de-trended case where $\bar{a} = - 13.5$

For each test, we include a constant and a time trend and estimate the residual spectrum at frequency zero using the GLS de-trended autoregressive spectral density estimator. We determine the lag length for the test regressions by the AIC procedure assuming the maximum lag k = 12.

2.3 Non-linear Unit-root Tests

Linear unit-root tests assume that a symmetrical adjustment process exists. A number of studies provide empirical evidence for non-linear dynamics for unit-root testing procedures (Caner and Hansen, 2001; Shin and Lee, 2001; Kapetanios et al., 2003). Taylor (2001) indicates that the power of linear unit-root tests is poor, if the series follows a non-linear threshold process. To accommodate the possibility of a non-linear dynamics of house prices, we employ the non-linear test of Kapetanios et al. (2003), which tests for a unit root against a non-linear stationary process based on an exponential smooth transition autoregressive (ESTAR) process. The use of this test can clearly and directly advance the discussion of whether house prices follow a stable dynamic process with an inherently non-linear adjustment caused by market frictions and transaction costs.

Kapetanios et al. (2003) extend the standard ADF test and introduce a new unit-root test to test for a linear unit root against an alternative of non-linear stationary exponential smooth transition autoregressive process. Kapetanios et al. (2003) propose the following univariate STAR model

Δ y_{t} = γ y_{t - 1} + [1 - \exp (- θ y_{t - 1}^{2})] + ε_{t}

(10)

where, $[1 - \exp (- θ y_{t - 1}^{2})]$ is the exponential transition function and θ ≥ 0.

The test focuses on the parameter θ, which equals zero under the null and is positive under the alternative. Since γ is not identified under the joint null hypothesis of linearity and a unit root, testing the null hypothesis of H₀ : θ = 0 against the alternative hypothesis of H₁ : θ > 0 is not feasible. Thus, Kapetanios et al. (2003) reparameterise equation (10) using a Taylor series approximation to obtain

Δ y_{t}^{d} = δ (y_{t - 1}^{d})^{3} + e_{t}

(11)

Δ y_{t}^{d} = δ (y_{t - 1}^{d})^{3} + \sum_{j = 1}^{q} ρ_{j} Δ y_{t - j}^{d} + e_{t}

(12)

where, $y_{t}^{d}$ equals the de-trended (de-meaned) capital-gain series and q equals the maximum autoregressive lag order to eliminate serially correlated errors.

Equations (11) and (12) correspond to the Dickey Fuller and the augmented Dickey–Fuller regressions respectively, differing only in that the lagged level of $y_{t}^{d}$ is raised to the power of 3 rather than the power of 1. In both equations (11) and (12), we test the null hypothesis of H₀: δ = 0 against the alternative of H₀: δ > 0 by using familiar t ratio obtained for δ. Since the asymptotic distribution of t is not standard normal, asymptotic critical values of the test statistic are tabulated by Kapetanios et al. (2003). We determine the lag length for the test regressions by the AIC procedure assuming the maximum lag k = 12.

2.4 Unit-root Tests with Two Structural Breaks

One major drawback of linear unit-root tests exists. In all such tests, we implicitly assume that we correctly specify the deterministic trend. The non-linear unit-root test allows for structural change in a smooth process. Yet, a superficial visual inspection of the rate of capital gain from the sale of houses series suggests the presence of potential structural breaks, which reflect shocks rather than smooth change. For example, an economic series that conforms to a stationary process around a fixed mean, which undergoes a one-time shift, will appear to conform to a non-stationary process, unless one incorporates the shift in the mean. Following the seminal work of Perron (1989), we recognise that the presence of structural change can substantially reduce the power of unit-root tests. Zivot and Andrews (1992) propose a unit-root test that allows for an endogenous structural break. More recently, Lumsdaine and Papell (1997) propose a sequential ADF-type unit-root test that allows for two shifts in the deterministic trend at two distinct unknown dates. For significant breaks, the test proves more powerful than unit-root tests allowing for only one structural break. The Lumsdaine and Papell (1997) modified version of the ADF test extends the Zivot and Andrews (1992) test as follows

Δ y_{t} = μ + β t + θ D U_{1 t} + γ D T_{1 t} + ω D U_{2 t} + ψ D T_{2 t} + α y_{t - 1} + \sum_{i = 1}^{k} c_{i} Δ y_{t - i} + ε_{t}

(13)

We include the lagged terms Δy_{t − i} to correct for serial correlation. Lumsdaine and Papell (1997) call this model CC in analogy to model C of Zivot and Andrews (1992), since it allows breaks both in the intercept and the slope of the trend function.

Lumsdaine and Papell (1997) describe the estimation procedure in more detail. We reject the null hypothesis of a unit root in favour of broken trend-stationarity, if significantly differs from zero. Since the asymptotic distribution of t is not standard normal, Lumsdaine and Papell (1997) provide asymptotic critical values of the test statistics. We determine the lag length for the test regressions by the AIC procedure, assuming the maximum lag k = 12.

The minimum LM unit-root test proposed by Lee and Strazicich (2003) allows for breaks under both the null and the alternative hypotheses in a consistent manner. According to the minimum LM principle, a unit-root test statistic comes from the following regression

Δ y_{t}^{d} = δ^{'} Δ Z_{t} + ϕ y_{t - 1}^{d} + \sum_{i = 1}^{k} γ_{i} Δ y_{t - i}^{d} + μ_{t}

(14)

where, the de-trended capital-gain series $y_{t}^{d}$ is defined as follows: $y_{t}^{d} = y_{t} - {\tilde{ψ}}_{x} - Z_{t} \tilde{δ},$ t = 2, …, T; $\tilde{δ}$ equal the coefficients in the regression of Δy_t onto ΔZ_t; ${\tilde{ψ}}_{x}$ equals $y_{1} - Z_{t} \tilde{δ},$ where y₁ and Z₁ correspond to the first observations of y_t and Z_t respectively. The lagged terms $Δ y_{t - i}^{d}$ are included to correct for serial correlation.

Considering structural breaks in both the intercept and the slope of the trend function (model C)

Δ Z_{t} = [1, t, D U_{1 t}, D U_{2 t}, D T_{1 t}, D T_{2 t}]

where, as in Lumsdaine and Papell (1997), DU_1t and DU_2t are indicator variables for the intercept changes in the trend function occurring at times TB₁ and TB₂ respectively; and DT_1t and DT_2t are indicator variables for the slope changes in the trend function occurring at times TB₁ and TB₂ respectively. We test the null hypothesis of a unit root in equation (14) that ϕ = 0 with a t-ratio. Lee and Strazicich (2003) provide the critical values, which depend on the location of the breaks.

3. Empirical Results

Table 1 reports the results of linear and non-linear unit-root tests with a constant and trend without structural breaks. This table presents overwhelming evidence in favour of a unit root in all series for the linear tests, including the Composite10 index, as each test reaches the same conclusion regarding each time-series. The non-linear test, however, does reject the null hypothesis of a unit root at the 5 per cent level for Los Angeles and San Francisco.¹⁰ This suggests that, for Los Angeles and San Francisco, the trend-reverting characteristics of the rate of capital gains could follow a non-linear path.¹¹

Table 1.

Unit-root tests without structural breaks for rate of capital gains on S&P/ Case–Shiller HPI

Series	DF–GLS	ERS-P_T	NP-MZ_t	NP-MP_T	KSS
Boston	−1.121	33.823	−1.337	23.572	−2.791
Chicago	−0.861	144.588	−1.747	14.612	0.686
Denver	−0.546	37.795	−1.153	33.943	−2.072
Las Vegas	−1.682	62.126	−1.137	22.079	−0.308
Los Angeles	−1.981	34.984	−1.209	26.913	−3.525*
Miami	−1.492	90.103	−0.651	45.888	−1.597
New York	−1.106	85.789	−0.456	81.262	−2.003
San Diego	−2.092	34.659	−1.231	28.207	−2.502
San Francisco	−1.847	44.349	−1.085	38.167	−3.773*
Washington, DC	−2.335	46.825	−1.166	33.241	−2.481
Composite10	−1.334	68.348	−1.069	32.464	−2.889

Notes: The test critical values with constant and trend equal the following

(1) DF–GLS Elliott–Rothenberg–Stock: −3.465(1 per cent level), −2.919 (5 per cent level) and −2.612 (10 per cent level) (Elliott et al., 1996, Table 1). (2) ERS-P_T Elliott–Rothenberg–Stock: 4.019 (1 per cent level), 5.646 (5 per cent level) and 6.870 (10 per cent level) (Elliott et al., 1996, Table 1). (3) NP-MZ_t Ng–Perron: −3.420 (1 per cent level), −2.910 (5 per cent level) and −2.620 (10 per cent level) (Ng and Perron, 2001, Table 1). (4) NP–MP_T Ng–Perron MP_T: 4.030 (1 per cent level), 5.480 (5 per cent level) and 6.670 (10 per cent level) (Ng and Perron, 2001, Table 1). (5) KSS, Kapetanios–Shin-Snell: −3.93 (1-percnt level); −3.40 (5 per cent level); and −3.13 (10 per cent level) (Kapetanios et al., 2003, Table 1). * denotes rejection of the null hypothesis at the 5 per cent significance level.

Table 2 reports the empirical results of the Lumsdaine–Papell test using model CC, which allows for two breaks in the constant and the trend. Overall, by allowing for two breaks, we cannot reject the unit-root hypothesis in favour of the (broken) trend-stationary alternative for 10 of the 11 series. We can reject the null only for the metropolitan area of Las Vegas. This implies that shocks to the capital gains in housing markets other than Las Vegas are permanent in nature and not trend-reverting. Overall, these findings, while illustrating the importance of allowing for breaks in the slope and the intercept of the trend function, do not produce results too dissimilar from the tests that assume structural constancy. Ben-David et al. (2003) point out that allowing for additional breaks does not necessarily produce more rejections of the unit-root hypothesis, because the critical value increases in absolute value when we include more breaks. The results in Table 2, however, indicate that allowing breaks in both the intercept and the slope of the trend function proves important. The break dates themselves are of interest. We determine the significance of the breaks using the conventional t-statistic. For over half of the series, including the Composite10 index, the first break occurs in 1991, or the second in 2003. For San Diego, however, the first break is not significant. For Boston, Los Angeles, New York and the Composite10 indices, the changes in the intercept and the slope of the trend function prove significant in both breaks. For the remaining series, only either the intercept or the slope of the trend function tests significant.

Table 2.

Lumsdaine–Papell and Lee–Strazicich minimum LM two-break, unit-root tests for rate of capital gains on S&P/Case–Shiller HPI

	Lumsdaine-Papell		Lee-Strazicich
	TB₁ TB₂	y_{t − 1}	TB₁ TB₂	y^d_{t − 1}
Boston	April 1991	−0.598	March 1991	−0.521
	March 2000	(−6.06)	October 2002	(−5.39)
Chicago	March 1999	−0.511	January 1994	−0.298
	August 2005	(−4.58)	August 2006	(−4.26)
Denver	June 1994	−0.497	January 1991	−0.348
	May 2001	(−4.81)	March 2001	(−5.60)
Las Vegas	September 1991	−0.703*	August 2003	−0.678*
	November 2003	(−7.14)	October 2005	(−6.57)
Los Angeles	August 1993	−0.207	June 1990	−0.287
	June 2003	(−5.34)	September 2005	(−5.33)
Miami	July 1996	−0.054	June 1992	−0.374*
	January 2005	(−5.83)	January 2007	(−6.66)
New York	April 1991	−0.477	December 1990	−0.337
	December 2005	(−6.03)	March 2006	(−5.53)
San Diego	July 1991	−0.278	March 1990	−0.447*
	June 2003	(−4.46)	April 2004	(−5.78)
San Francisco	April 1996	−0.209	March 1995	−0.268
	June 2003	(−4.77)	February 2007	(−5.15)
Washington, DC	April 1991	−0.308	December 1990	−0.283
	July 2003	(−5.16)	November 2005	(−5.41)
Composite10	March 1991	−0.208	December 1990	−0.271
	June 2003	(−5.48)	November 2005	(−5.41)

Notes: The numbers in parenthesis equal the t-statistics for the estimated coefficients. TB₁ and TB₂ equal the break dates and the coefficient of y _t−1tests for the unit root. Critical values for the coefficients on the dummy variables follow the standard normal distribution. The critical values for Lumsdaine–Papell, from Ben-David et al. (2003, Table 3), equal −7.19 (1 per cent level), −6.75(5 per cent level) and −6.48 (10 per cent level). The critical values for the unit-root test, tabulated in Lee and Strazicich (2003, Table 2), depend upon the location of the breaks. For λ₁ = 0.2 and λ₂ = 0.8, the critical values equal, respectively, −6.32 (1 per cent level), −5.71 (5 per cent level) and −5.33 (10 per cent level). * denotes rejection of the null hypothesis at the 5 per cent significance level.

As already noted, a potential problem of the Lumsdaine–Papell unit-root test exists because typically the derivation of the critical values assumes no breaks under the null hypothesis. This assumption may lead to conclude incorrectly that rejection of the null is evidence of trend stationarity when, in fact, the series is difference-stationary with breaks (Lee and Strazicich, 2001, 2003). To avoid this potential problem, Lee and Strazicich (2003) propose a minimum LM unit-root test that allows for two endogenously determined breaks in the level and trend.

The minimum LM unit-root test of Lee and Strazicich (2003) incorporates structural breaks under the null hypothesis and rejection of the minimum LM test null hypothesis provides genuine evidence of stationarity. In addition, the results of Lee and Strazicich (2003) show that the minimum LM test possesses greater power than the test of Lumsdaine and Papell (1997).

Table 2 also reports the results of the Lee–Strazicich unit-root test based on model C, which allows for two breaks in the constant and the trend. The findings overturn most of the previously presented results suggesting that the rates of capital gain from the sale of houses are non-stationary and provide significant evidence in favour of segmented trend stationarity for the majority of the rates of capital gains series. We can reject the unit-root hypothesis at the 10 per cent level for Boston, Denver, Los Angeles, New York, Washington, DC, and the Composite10 indices; at the 5 per cent level for San Diego and at the 1 per cent level for Las Vegas and Miami. For Chicago and San Francisco, however, we cannot reject the unit-root hypothesis.¹² The Kapetanios–Shin–Snell (KSS) tests reported in Table 1 reject the unit-root hypothesis at the 5 per cent level for Los Angeles and San Francisco. Thus, combining the findings of the non-linear KSS unit-root tests with the results of the Lee–Strazicich tests with two structural breaks leads to the rejection of the unit-root null hypothesis for every city, except Chicago, and the Composite10 index.

The two date breaks that minimise the LM statistics provide suggestive information. As in the Lumsdaine–Papell test, the significance of the breaks is determined using a conventional t-statistic. No a priori reason exists to expect the break dates estimated by the Lee–Strazicich and Lumsdaine–Papell procedures to coincide. These break dates, however, generally fall close to each other. The recession of the early 1990s roughly coincides with the first break in both the Lumsdaine–Papell and Lee–Strazicich procedures for most series. The only exception, Las Vegas, does not exhibit a break in the 1990s under the Lee–Strazicich procedure. The second break, instead, is clustered in the first half of the current decade for all series, which corresponds to the dramatic rise in house prices nationally.

4. The ‘Ripple Effect’

The ‘ripple effect’ or interregional transmission of house price shocks emerged in studies of the UK housing markets (Meen, 1999; Cook, 2003, 2005a, 2005b; and Holmes and Grimes, 2008). In recent papers on the predictability of US house prices, Gupta and Miller (2010a, 2010b) present evidence of a regional ‘ripple effect’ (i.e. forecasting house prices in one metropolitan area improve by including house prices in nearby metropolitan areas) for the eight Southern California MSAs as well as Los Angeles, Las Vegas and Phoenix.

This section concerns the time-series characteristics of the ratios of the metropolitan house price indices to the national house price index in the US.¹³ Define the metropolitan house price ratio as follows

d_{t} = p_{t} - m_{t}

(15)

where, m_t refers to the natural logarithm of the Composite10 index, t = 1, 2, … T.¹⁴ Assume that d_t comes from a first-order autoregressive with two structural breaks process as follows

d_{t} = ς + ξ t + θ D U_{1 t} + γ D T_{1 t} + ω D U_{2 t} + ψ D T_{2 t} + ρ d_{t - 1} + ε_{t}

(16)

Subtracting d_{t − 1} from both sides of equation (16) and rearranging the model generates the familiar ADF regression

Δ d_{t} = ς + ξ t + θ D U_{1 t} + γ D T_{1 t} + ω D U_{2 t} + ψ D T_{2 t} + τ d_{t - 1} + ε_{t}

(17)

where, τ = ρ − 1. Acceptance of the null hypothesis (H₀: ρ =1) means that d_t is a non-stationary series, whereas rejection of the null means that dt is stationary (i.e. a ripple effect exists in the sense of Meen, 1999).

Table 3 reports the results of applying the conventional linear unit-root tests as well as the non-linear unit-root test previously used in the paper, which do not allow for structural breaks, but allow for a constant and a trend. The findings sometimes prove inconsistent. For example, the ERS test, as well as the two Ng–Perron tests, provide evidence of stationarity for the Chicago price ratio. The DF–GLS test, however, fails to reject the unit root. Conversely, the DF–GLS test finds stationarity for the Denver and Miami price ratios, but the remaining three linear tests do not. For all remaining price-ratio series, each of the four linear test statistics consistently rejects linear stationarity. The non-linear unit-root test, however, rejects the unit-root hypothesis at the 5 per cent level in five cases—Boston, Denver, Miami, New York and San Diego.¹⁵

Table 3.

Unit-root tests for house price ratios without structural breaks

Series	DF–GLS	ERS-P_T	NP-MZ_t	NP-MP_T	KSS
Boston	−2.381	155.422	−0.264	115.352	−4.939*
Chicago	−0.173	0.980*	−8.158*	0.702*	−1.505
Denver	−3.816*	328.374	−0.377	302.091	−4.201*
Las Vegas	−1.820	62.973	−0.213	48.792	−0.465
Los Angeles	−1.932	60.903	−0.811	60.101	−2.334
Miami	−2.976	81.878	−0.551	66.347	−3.617*
New York	−1.631	250.408	0.771	125.258	−4.487*
San Diego	−2.073	109.664	−0.007	76.051	−5.440*
San Francisco	−0.911	170.831	0.868	81.349	−2.919
Washington, DC	−1.351	27.606	−1.214	30.687	−2.105

Notes: See Table 1. * denotes rejection of the null hypothesis at the 5 per cent significant level.

Table 4 reports the results of applying the Lumsdaine–Papell two-break, unit-root test for the house price ratios based on model CC, which assumes a constant and a trend. The results of the Lumsdaine–Papell test consistently fail to reject the null of non-stationarity of the price ratios for all metropolitan areas. Consequently, these empirical results fail to support the ripple effect for each metropolitan area. In other words, the housing markets are segmented and shocks to the house prices of each city do not ‘ripple out’ across the nation.

Table 4.

Lumsdaine–Papell and Lee–Strazicich minimum lm two-break, unit-root test for house price ratios

	Lumsdaine-Papell		Lee-Strazicich
Series	TB₁ TB₂	d_{t − 1}	TB₁ TB₂	d^d_{t − 1}
Boston	March 2003	−0.0173	December 1994	−0.0244*
	February 2005	(−5.68)	April 2004	(−5.81)
Chicago	February 1996	−0.0357	Jun. 1994	−0.0299
	December 2003	(−4.38)	May. 2003	(−3.95)
Denver	September 1998	−0.0166	February 1995	−0.0143
	December 2004	(−5.55)	Aug. 2003	(−5.22)
Las Vegas	February 1994	−0.0283	Nov. 1991	−0.0202
	February 2006	(−3.96)	December 2003	(−3.57)
Los Angeles	March 1998	−0.0124	February 1994	−0.0185
	October 2004	(−4.86)	October 2004	(−4.84)
Miami	July 1992	−0.0230	Aug. 1994	−0.0271
	July 2004	(−4.96)	May 2004	(−5.38)
New York	January 2000	−0.0182	October 1992	−0.0335*
	May 2003	(−5.39)	October 2004	(−6.09)
San Diego	July 1994	−0.0327	Jun. 1996	−0.0332
	May 2003	(−5.10)	March 2003	(−5.19)
San Francisco	April 1999	−0.0272	October 1992	−0.0362
	January 2006	(−4.71)	July 1999	(−4.96)
Washington, DC	February 1994	−0.0489	January 1993	−0.0397
	January 1999	(−5.08)	Jul. 2000	(−3.64)

Notes: See Table 2. The critical values for Lumsdaine–Papell, from Ben-David et al. (2003, Table 3), equal −7.19 (1 per cent level), −6.75 (5 per cent level) and −6.48 (10 per cent level). The critical values for the unit-root test, tabulated in Lee and Strazicich (2003, Table 2), depend upon the location of the breaks. For λ₁ = 0.2 and λ₂ = 0.8 the critical values equal, respectively, −6.32 (1 per cent level), −5.71 (5 per cent level) and −5.33 (10 per cent level). * denotes rejection of the null hypothesis at the 5 per cent significance level.

Table 4 also reports the results of applying the Lee–Strazicich two-break, unit-root test for the house price ratios based on model C, which assumes a constant and a trend. In contrast to the Lumsdaine–Papell tests, the Lee–Strazicich tests reject the null hypothesis of a unit root with two structural breaks at the 5 per cent level for Boston and New York. In turn, the non-linear KSS tests reject the null hypothesis of a unit root for Denver, Miami and San Diego (in the constant only specification), in addition to Boston and New York. These findings reject the notion of segmentation of the US housing market and suggest that convergence and the ripple effect are not UK-specific phenomena. The findings provide new insight about the dynamics of the US housing market. The house price shocks stemming from Boston, Denver, Miami, New York and San Diego ‘ripple out’ and significantly influence house price changes in the US. The dynamics of the ripple effect also provide important consequences for the US economy. Housing comprises a large share of assets for many households (Alexander and Barrow, 1994; Holmes and Grimes, 2008) and house prices affect labour mobility as well as migration, although the relationships are weak because most households move from one region to another not only for house price differences but also for other factors (job opportunities, etc.). In addition, the ability to predict house prices correctly in one region of the US may improve, if we consider the significant effect of other regional house prices.

5. Conclusions

This paper contributes to the literature on the long-run behaviour of the house prices and the ‘ripple effect’ in the US by addressing the issue of non-stationarity using an empirical approach not previously considered in the literature. US house prices provide an interesting demonstration of non-linearities and structural change. We address these issues by applying Lumsdaine–Papell and Lee–Strazicich two-break unit-root tests and the KSS non-linear unit-root test. The two structural-break tests are dissimilar. Lumsdaine and Papell (1997) take into account the existence of segmented trend stationarity under the alternative hypothesis, while Lee and Strazicich (2003) include the segmented trend hypothesis under the null as well the alternative hypothesis. The Lumsdaine–Papell test rejects the null of a unit root only for one of the eleven series (Las Vegas). The Lee–Strazicich test finds broken-trend stationarity for three of eleven series (Las Vegas, Miami, San Diego). The KSS test finds non-linear stationarity for two of the eleven series (Los Angeles and San Francisco). This contrasts with the results of the standard linear tests that reject stationarity in all 11 series.

Although the two structural-break tests provide contradictory findings, both tests indicate that structural breaks exist in the rate of capital gain from the sale of houses. No common significant structural breaks exist that characterise all rate of capital gains series, but the estimated breaks roughly cluster around two periods: the recession of the early 1990s and the first half of the current decade, which experiences low interest rate policies of the FED, the housing bubble and significant sub-prime lending activity.

The tests used offer further insights about the ‘ripple effect’. Following Meen (1999) and Cook (2003, 2005a, 2005b), we examine the ratio of the S&P/Case–Shiller 10 metropolitan price indices to a national house price index (Composite10). The Lumsdaine–Papell test fails to reject the null of a unit root in for all 10 series of price ratios. Conversely, the Lee–Strazicich test finds broken-trend stationarity for two of the ten series (Boston and New York), while the KSS tests finds non-linear stationarity for five of the ten series (Boston, Denver, Miami, New York and San Diego). These findings provide partial evidence that rejects the notion of segmentation of the US housing market and confirms the findings of previous research (for example, MacDonald and Taylor, 1993), which indicate that house price changes in the east and west coast metropolitan areas exert a significant influence on house price changes in the rest of the US.

Footnotes

Notes

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