have equity reits experienced periodically collapsing bubbles? · 2013. 9. 12. · literature on...

Have Equity REITs Experienced Periodically Collapsing Bubbles?

James E. Payne Professor and Chair Department of Economics Illinois State University Normal, IL 61790-4200 [email protected] 309-438-8588

and

George A. Waters Assistant Professor Department of Economics Illinois State University Normal, IL 61790-4200 [email protected] 309-438-7301 Final Version Accepted November 2005

Abstract: This paper uses the momentum threshold autoregressive (MTAR) model and the residuals-augmented Dickey-Fuller (RADF) test to examine the possibility of Evans’ (1991) periodically collapsing bubbles in the equity REIT market. The results are mixed. The MTAR model indicates that overall real equity REIT prices and dividends are cointegrated with asymmetric adjustment towards the long-run equilibrium. However, the estimated coefficients of the MTAR model do not indicate the presence of periodically collapsing bubbles. Adjustment in the standard cointegration tests of bubbles for skewness and excess kurtosis via the RADF test fails to reject the null hypothesis of no cointegration, leaving the possibility of periodically collapsing bubbles. The MTAR and RADF results with respect to equity REIT sub-sectors are mixed. Lodging is the only sub-sector in which both the MTAR and RADF results support periodically collapsing bubbles. Moreover, market fundamentals proxied by two alternative measures of capacity utilization do not explain either real equity REIT prices or dividends.

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Have Equity REITs Experienced Periodically Collapsing Bubbles?

I. Introduction

Within the standard present value model, stock prices are determined by the discounted

values of expected future dividends. However, asset prices that are in excess of what is viewed

as the asset’s fundamental value have been interpreted as speculative bubbles. A class of

speculative bubbles known as rational bubbles, do not violate the rational expectations

hypothesis and are consistent with the efficient markets hypothesis. Investors recognize the

overvaluation; however, investors are compensated with excess positive returns for the risk of a

bubble collapsing. Such rational bubbles are due to self-fulfilling expectations that can break the

connection between prices and dividends over the short term.

With respect to the REIT market there are a number of reasons to explore the possibility

of bubble formation in REIT prices. First, the empirical evidence indicates that REITs are

integrated with the stock market and share common risk factors.1 In light of the growing

literature on speculative bubbles with respect to the stock market, it is a natural extension to test

for bubble-like behavior in the REIT markets.2 Second, there is some evidence for the presence

of speculative bubbles in the housing market, but this issue has not been studied from the

standpoint of commercial real estate.3 Third, when prices continue to increase beyond

fundamental values there is an increase in short selling, a signal of overvaluation in the market.

In the case of REIT markets, Li and Yung (2004) argue that REIT markets are not liquid enough

to support such short selling as a means to signal overvaluation in the market and the formation

of a bubble. Fourth, due to informational problems and market inefficiency, there is an

underpricing of REIT seasoned equity offerings [Howe and Shilling, 1998 and Ghosh et al,

3

2000] which serves as a deterrent to the issuance of seasoned equity offerings to capture market

overvaluation (Wang et al, 1995).4

Recently, Jirasakuldech et al (2005) test for the presence of rational speculative bubbles

in the equity REIT market over the period 1973:01 to 2003:12 along with the sub-periods

1973:01 to 1991:10 and 1991:11 to 2003:12 with the results indicating the absence of rational

bubbles. Specifically, following the approach of Diba and Grossman (1984, 1988a,b),

Jirasakuldech et al (2005) implement the standard unit root and cointegration tests of equity

REIT returns and macro fundamentals along with tests of duration dependence in equity REIT

returns.5 Evidence against the presence of bubbles is supported if REIT prices and macro

fundamental variables are respectively integrated of order one and the existence of a

cointegrating vector between REIT prices and the macro fundamental variables. Moreover,

evidence against the presence of bubbles is supported if REIT prices do not exhibit negative

duration dependence.

However, Evans (1991) argues that this standard approach will not be able to detect a

class of periodically collapsing rational bubbles. For example, the sudden collapse of a bubble

may be mistaken by standard cointegration tests for mean reversion, resulting in a bias towards

rejection of the null hypothesis of no cointegration. In the case of equity REITs, the approach

followed by Jirasakuldech et al (2005) implicitly assumes the bubble component follows a linear

process whereas the bubble specification of Evans can follow a non-linear process. The task of

this study is to extend the recent work of Jirasakuldech et al (2005) by investigating whether or

not periodically collapsing bubbles exist within the equity REIT market. The existence of

periodically collapsing bubbles is examined using the momentum threshold autoregressive

(MTAR) model advanced by Enders and Granger (1998) and Enders and Siklos (2001) and the

4

residuals-augmented Dickey-Fuller (RADF) test by Taylor and Peel (1998). The MTAR model

will capture the possible asymmetries in the adjustment towards the long-run equilibrium,

particularly the sharp falls in the asset price after the price has reached a certain threshold

relative to dividends. The RADF test corrects the standard cointegration tests for the skewness

and excess kurtosis that may arise in the presence of periodically collapsing bubbles. The results

are mixed as to whether periodically collapsing bubbles exist in the equity REIT market.

The paper is organized as follows. Section II provides the theoretical framework for

periodically collapsing bubbles. Section III presents the empirical methodologies, the data and

empirical results. Section IV investigates the robustness of the results and the role of market

fundamentals with respect to equity REIT sub-sectors. Concluding remarks are presented in

Section V.

II. Theoretical Framework of Periodically Collapsing Bubbles

Solutions to the present value model represent both fundamental and bubble solutions for

asset prices. Evans (1991) provides a specific form for generating periodically collapsing

bubbles that might not be detected by simple unit root analysis.6 The asset price Pt at time t

depends on the expectation at time t of next period’s price Pt and dividend Dt such that

11t ttt DPEP (1)

where the discount factor is 0

5

variables a la McCallum (1983, 1997)) condition on (2), then the bubble term must be zero, B t =

0. In this case, the asset price Pt is determined solely by expected future dividends,

corresponding to the fundamental solution in the asset pricing literature.

While some have argued for the non-existence of bubbles on the basis of the conditions

mentioned above, much work has been advanced recently on ways to detect whether asset prices

are determined by dividends alone. Evans (1991) discusses a class of bubbles that would not be

detected by simple cointegration techniques. Such periodically collapsing bubbles may be

represented as follows.

,11

1

ttt vBB if tB (3a)

1111 tttt vBB , if tB . (3b) The parameters in the above equations satisfy and 0 < . The stochastic process

vt is iid and has conditional expectation Etvt+1 = 1, which ensures that a bubble will not switch

sign. The term t is a Bernoulli process that takes the value 1 with probability and the value 0

with probability 1-. Equation (3a) represents the phase when the bubble grows at mean rate

, but equation (3b) shows that if the bubble exceeds the threshold , it explodes at mean rate

However, this phase does not last indefinitely as the bubble collapses with probability 1-

each period.

Evan’s model of a bubble incorporates three important characteristics, Bt satisfies the

martingale property EtBt+1 = Bt, bubbles cannot be detected by simply examining

cointegration between asset prices and dividends, and bubbles that are initially positive stay

positive. The last observation that bubbles stay positive fits an assumption that, while not

universally accepted, is commonly made in the literature. In this paper we test for positive

bubbles, without denying the possibility of negative bubbles.

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III. Data, Methodology, and Results

Monthly data on the prices and dividends for equity REITs were obtained from the

National Association of Real Estate Investment Trusts (NAREIT) for the period 1972:01 to

2005:03. The original data has been converted to natural logarithms. Following Jirasakuldech

et al (2005), we recognize the structural shift in the REIT markets in 1991 in testing for unit

roots in equity REIT prices and dividends, respectively. The hypothesized structural shift can be

attributed to the following: (1) the REIT market became more dominated by large institutional

investors along with increased market liquidity in the post 1992 period [Damodaran and Liu,

1993; Wang et al, 1995; Below et al, 1996; Chan et al, 1998; and Chan et al, 2003]; (2) analyst

and media coverage of the REIT markets increased, thereby increasing the transparency of the

market throughout the 1990s [Gentry et al, 2003 and Chui et al, 2003]; and (3) the creation of the

umbrella partnership REIT organization structure in 1991 allowed for more flexibility in

purchasing property; however, the lack of transparency has made valuation more difficult

[Damodaran et al, 1997 and Ling and Ryngaert, 1997].

For prices and dividends to be cointegrated, representing a long run connection between

the two, they must be integrated of the same order. We begin by testing the null hypothesis of a

unit root in real equity REIT prices, tp and real equity REIT dividends, td , incorporating the

possibility of structural breaks using Perron’s (1989) unit root test.

titk

i itbtycyTDDTtDUy 11)( (4)

where tt py or td ; )(1 bTtDU is a post-break constant dummy variable; t is a linear time

trend; tTtDT b )(1 is a post-break slope dummy variable; )1(1)( bb TtTD is the break

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dummy variable; and t are white noise error terms.7 The null hypothesis of a unit root is given

by 1 . Table 1 reports the results of the unit root tests allowing for a structural break in

1991:11 as suggested by Jirasakuldech et al (2005). Neither real equity REIT prices nor

dividends yield statistically significant coefficients on the respective dummy variables.

Moreover, the results are unable to reject the null hypothesis of a unit root in either real equity

REIT prices or dividends.

Given the respective real equity REIT prices and dividends contain a unit root, the

momentum threshold autoregressive (MTAR) model advanced by Enders and Granger (1998)

and Enders and Siklos (2001) and the residuals-augmented Dickey-Fuller (RADF) test set forth

by Taylor and Peel (1998) are estimated in order to capture the dynamics of periodically

collapsing bubbles.8

MTAR Approach:

The following cointegration equation representing the relationship between real equity

REIT prices, tp , and dividends, td , is estimated.

ttt dp (5)

Indeed, if real equity REIT prices and dividends are integrated of order one, the residuals from

the cointegration equation, t , should be stationary in levels. The cointegration test with the

possibility of asymmetric adjustment is undertaken by the following regression of the residuals.

titp

i itttttvII 11211 )1( (6)

Heaviside indicator function, tI , is represented by:

1

1

ˆ0ˆ1

t

tt if

ifI (7)

8

where the threshold is the value that minimizes the residual sum of squares.9 The MTAR

model allows the speed and direction of adjustment, represented by and, to depend on the

previous period’s change in 1t . This model is especially valuable when the adjustment is

believed to exhibit more momentum in one direction than the other, as in the case of collapsing

bubbles. The null hypothesis of no cointegration is tested by the restriction,

Indeed, if real equity REIT prices, tp , and dividends, td , are cointegrated, the null hypothesis of

symmetry is tested by the restriction,. If the estimated coefficient, , is statistically

significant and negative and larger in absolute terms relative to the estimated coefficient, ,

there is evidence of a sharp correction when prices have risen above a certain threshold relative

to dividends. Therefore, if the null hypothesis of symmetric adjustment is rejected and 2 < 1 ,

we conclude that periodically collapsing bubbles are present in REIT prices.

The cointegration and momentum threshold tests are reported in Panel A of Table 2.

Real equity REIT prices and dividends appear cointegrated as evident from the significant ADF

test statistic (CR in Panel A of Table 2). According to the Diba and Grossman approach the

presence of a cointegrated relationship between real equity REIT prices and dividends can be

interpreted as evidence against the presence of speculative bubbles in the REIT market. To

address the possible asymmetries in the adjustment towards the long-run equilibrium relationship

between real equity REIT prices and dividends that might occur in the presence of periodically

collapsing bubbles, the MTAR model is explored. Note from equations (6) and (7), the MTAR

specification provides point estimates of 1 and 2 . As in the case of the Engle-Granger

cointegration tests, the null hypothesis of no cointegration, , is rejected.

Furthermore, the null hypothesis of symmetry,, is rejected; however, the point estimates,

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1 and 2 , do not satisfy the condition 2 < 1 . Therefore, the results from the MTAR

approach do not provide evidence in support of periodically collapsing bubbles in the equity

REIT market. However, this model requires the very restrictive assumption that the threshold

for the collapse is constant throughout the sample.

RADF Approach:

The argument of Diba and Grossman (1984, 1988a,b) and Jirasakuldech et al (2005) that

cointegration between price and dividends is evidence against bubbles assumes a linear process

for the growth of tB , implying normality in the residuals. However, a preliminary test on the

residuals from equation (5) shows both skewness and excess kurtosis in the residuals, which

could be caused by the presence of periodically collapsing bubbles (see Panel B of Table 2).

Taylor and Peel (1998) present Monte Carlo evidence to suggest that periodically collapsing

bubbles generate skewness and excess kurtosis in stock prices. Following the work of Im

(1996), Taylor and Peel (1998) argue that the presence of skewness and excess kurtosis can be

used to obtain a more efficient estimator in the cointegration tests of bubbles. Standard tests of

cointegration examine the stationarity of the residuals from equation (5) as follows:

ttt u 1 (8)

The null hypothesis of no cointegration is stated as 0 while the alternative hypothesis of a

stationary residual is 0 . Taylor and Peel (1998) argue that correcting the least squares

estimate in equation (8) for skewness and excess kurtosis provides a more efficient estimator of

and improves the ability to detect periodically collapsing bubbles.

Taylor and Peel (1998) suggest the following two-step estimator in the construction of

the residuals-augmented Dickey-Fuller (RADF) test of the null hypothesis of no cointegration.10

First, regress the first difference of the residuals of the cointegrating equation on their lagged

10

level (see equation 8 above) and use the new residuals, tû , and the estimated variance, 2̂ , to

construct the vector, )]ˆˆ(),ˆˆ3ˆ[(ˆ 2223 tttt uuuw . Next, re-estimate equation (8) with the

addition of the vector, tŵ , which corrects the estimate of for skewness and excess kurtosis of

the residuals as follows:

tttt w ˆ1 (9)

where t is white noise. The key test statistic is, )ˆ(/ˆ** VCR , where

*̂ is the estimator

in equation (9).11 Taylor and Peel (1998), Sarno and Taylor (2003), as well as Capelle-Blancard

and Raymond (2004) conduct Monte Carlo Studies to construct critical values and analyze the

power of this test against alternatives. In particular, Taylor and Peel (1998) show that the

adjustment for skewness and excess kurtosis has superior power over standard cointegration tests

to correctly reject a mean-reverting error model as a bubble. Panel B of Table 2 reports the

results of the RADF test on equity REITs. The value of CR is -0.79, far below the 10 percent

critical value, hence the null hypothesis of no cointegration (i.e. presence of a bubble

component) cannot be rejected. The results of the RADF test leaves room for the possibility of

periodically collapsing bubbles in the equity REIT market.

IV. Further Inspection of Equity REIT Sub-Sectors

Given the mixed results associated with equity REITs from the MTAR and RADF

specifications, a further inspection of equity REIT sub-sectors for robustness is warranted.

While the evidence on periodically collapsing bubbles is mixed for the overall equity REIT

market, it is possible that various sub-sectors may exhibit periodically collapsing bubbles.12 The

analysis begins with testing the null hypothesis of a unit root in the respective real equity REIT

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prices and dividends along with two proxies for market fundamentals using the standard

augmented Dickey-Fuller (ADF, 1979) and Phillips-Perron (PP, 1988) unit root tests. Panels A

and B of Table 3 reports the results of the ADF and PP unit root tests for real equity prices and

dividends for overall equity REITs along with the equity REIT sub-sectors: apartments,

industrial, lodging, manufactured homes, office, and regional malls. Panel C of Table 3 reports

the corresponding unit root tests for two measures of capacity utilization, the National

Association of Purchasing Managers tnapm index and the actual capacity utilization rate tcu ,

which will be used as proxies for market fundamentals in explaining real equity REIT prices and

dividends. The unit root results indicate that the respective real equity prices and dividends

along with the capacity utilization measures are stationary after first-differencing, so there is the

possibility of cointegration between these variables. Next, tests for periodically collapsing

bubbles are performed for the overall equity REITs and the equity REIT sub-sectors using the

MTAR (equations 6 through 7) and RADF (equations 8 and 9) specifications and reported in

Table 4.

The Engle-Granger (1987) cointegration tests reveal that real equity REIT prices and

dividends are cointegrated at the 10 percent significance level for overall equity REITs and the

equity REIT sub-sector, regional malls (CR in Panel A of Table 4). The presence of

cointegration provides evidence of a connection between prices and dividends and the absence of

a bubble component, but, as we have noted, cointegration tests may not detect periodically

collapsing bubbles. The MTAR testing strategy, as shown in (6) and (7), has superior power

relative to the alternative assumption of symmetric adjustment associated with standard Engle-

Granger tests (Enders and Siklos, 2001). Thus, the null hypothesis of no cointegration,

, is tested allowing for the possibility of asymmetric adjustment. The column labeled

12

FC in Table 4 displays the test statistics associated with the MTAR adjustment in the

cointegration tests. It appears that the null hypothesis of no cointegration, associated

with the MTAR specification is rejected for the equity REIT sub-sectors, apartments, industrial,

lodging, and office.13 Thus, for those equity REIT sub-sectors in which the null hypothesis of no

cointegration is rejected using the MTAR specification, the null hypothesis of symmetry,

is tested.14 Indeed, the null hypothesis of symmetry is rejected for the equity REIT sub-

sectors, apartments, industrial, lodging, and office, indicating that prices above (or below) a

certain threshold, τ, exhibit differing speeds of adjustment toward the long-run equilibrium (i.e.

cointegrating) relationship between real equity REIT prices and dividends. However, upon

further inspection of the point estimates, 1 and 2 , the condition, 2 < 1 , for the presence of

periodically collapsing bubbles is not satisfied. Thus, it appears from the MTAR specifications

over the period, 1994:01 to 2005:03, that only the equity REIT sub-sector, lodging, exhibits

behavior consistent with periodically collapsing bubbles.

Panel B of Table 4 reports the results of the RADF test on the overall equity REIT market

and equity REIT sub-sectors. In every case the value of CR is far below the 10 percent critical

value, hence the null hypothesis of no cointegration (i.e. presence of a bubble component) cannot

be rejected. The results of the RADF test leave room for the possibility of periodically

collapsing bubbles in both the overall equity REIT market as well as the equity REIT sub-

sectors.

In light of the common criticism that bubble-like behavior may be attributed to changes

in market fundamentals, as opposed to self-fulfilling expectations as specified in section II, the

influence of such fundamentals are incorporated in the relationship between real equity prices

and dividends. Two alternative measures are used as proxies for market fundamentals: the

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National Association of Purchasing Managers Index, a leading indicator of capacity utilization,

and the actual capacity utilization rate. The rationale for using either measure is that as firms

increase their capacity utilization, there is an increase in the demand for space. The increase

demand for space leads to a reduction in vacancy rates and upward pressure on rents which will

affect the respective REIT markets.15

The analysis proceeds by first establishing whether or not the respective proxies for

market fundamentals share a long-run cointegrating relationship with the respective real equity

REIT prices and dividends. Appendix A presents the results of the Johansen-Juselius (1990)

multivariate cointegration procedure to test the possible long-run relationships between the

respective real equity REIT prices, dividends, and the alternative measures of capacity

utilization, using the maximum eigenvalue and trace tests. As shown in Panels A through G of

Appendix A, the maximum eigenvalue and trace tests for cointegration proceed sequentially

from the first hypothesis of no cointegration to increasing numbers of cointegrating vectors.

More specifically, the maximum eigenvalue test max is based on the null hypothesis that the

number of cointegrating vectors is r against the alternative r + 1 cointegrating vectors. The trace

tests trace is based on the null hypothesis that the number of cointegrating vectors is less than

or equal to r against a general alternative. The results displayed in Panels A through G of

Appendix A fail to reject the null hypothesis of zero cointegrating vectors based on either the

maximum eigenvalue or trace tests. These results indicate that neither the NAPM index nor the

capacity utilization rate shares a common stochastic trend with real equity REIT prices and

dividends (i.e. not cointegrated).

Given the absence of a long-run relationship as indicated by the Johansen-Juselius

cointegration tests, the short-run dynamics are explored in Panels A through G of Appendix B

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with the estimation of vector autoregressive models for the respective real equity REIT prices

and dividends using either the NAPM index or the capacity utilization rate. While there is

evidence of autoregressive behavior in many of the variables and the occasional feedback

between prices and dividends, neither the NAPM index nor the capacity utilization rate have

predictive power, as evident from their partial-F-statistics, in explaining either real equity REIT

prices or dividends. Thus, the results with respect to periodically collapsing bubbles cannot be

attributed to changing market fundamentals as proxied by the NAPM index or the capacity

utilization rate.

V. Concluding Remarks

This study extends the recent work of Jirasakuldech et al (2005) on rational speculative

bubbles in the equity REIT market by exploring the possibility of periodically collapsing

bubbles. Detection of periodically collapsing bubbles in asset markets, such as those in Evans

(1991), requires econometric tests beyond the standard cointegration tests proposed by Diba and

Grossman (1984, 1988a,b). Two approaches are undertaken to discern the whether periodically

collapsing bubbles are present in the equity REIT market: the momentum autoregressive

threshold (MTAR) model of Enders and Granger (1998) and Enders and Siklos (2001) and the

residuals-augmented Dickey-Fuller (RADF) test by Taylor and Peel (1998). The MTAR model

will capture the possible asymmetries in the adjustment towards the long-run equilibrium,

particularly the sharp falls in the asset price after the price has reached a certain threshold

relative to dividends. The RADF test corrects the standard cointegration tests for the skewness

and excess kurtosis that may arise in the presence of periodically collapsing bubbles. Over the

period 1972:01 to 2005:03, the MTAR results indicate that while overall real equity REIT prices

15

and dividends are cointegrated and exhibit asymmetries, the point estimates do not indicate the

presence of periodically collapsing bubbles in the equity REIT market. On the other hand, the

RADF results suggest the possibility of periodically collapsing bubbles over this period.

However, within the sub-period, 1994:01 to 2005:03, the MTAR approach provides

support for periodically collapsing bubbles for only the equity REIT sub-sector, lodging. The

RADF results fail to reject the null hypothesis of no cointegration in the overall equity REIT

market and equity REIT sub-sectors, hence the possibility of periodically collapsing bubbles.

Moreover, to account for the possibility that the relationship between real equity REIT prices and

dividends have been driven by market fundamentals, Johansen-Juseilus cointegration tests are

performed to infer the long-run relationship between real equity REIT prices, dividends, and two

alternative measures of market fundamentals, the National Association of Purchasing Managers

Index and the actual capacity utilization rate. Across equity REIT sub-sectors, the cointegration

results indicate the absence of a long-run relationship between real equity REIT prices,

dividends, and market fundamentals. In addition, vector autoregressive models for the

respective real equity REIT prices, dividends, and proxies for market fundamentals are estimated

to capture the short-run dynamics among these variables. It appears from the results that

changing market fundamentals proxied by two alternative measures of capacity utilization do not

explain either real equity REIT prices or dividends.

In summary, the results are mixed as to whether periodically collapsing bubbles exist in

the equity REIT market. An interesting extension would be to examine the connection

empirically between REITs and housing prices in the context of bubble formation.

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Notes

1. See Myer and Webb (1992), Li and Wang (1995), Peterson and Hsieh (1997), Okunev and Wilson (1997), Ling and Naranjo (1999), Okunev et al (2000), and Glascock et al (2000) for studies on the relationship between REITs and the general stock market.. 2. See Diba and Grossman (1988a,b), Camerer (1989), Dezbaksh and Demirguc-Kunt (1990), Evans (1991), Topol (1991), Taylor and Peel (1998), Bohl (2003), Capelle-Blancard and Raymond (2004), and Harman and Zuehlke (2004) for studies on speculative bubbles in the general stock market. 3. See Case and Shiller (1990), Kim and Shuh (1993), Grenadier (1995), Abraham and Hendershott (1996), Clayton (1997), Bjorklund and Soderberg (1999), Hendershott (2000), and Roche and McQinn (2001) for studies on speculative bubbles in real estate markets. 4. Jirasakuldech et al (2005) also mention the latter three points with respect to bubble formation. 5. Harman and Zeuhlke (2004) discuss several shortcomings of using duration dependence tests to detect bubbles. In particular, duration dependence is sensitive to the method of correcting for discrete observation of continuous duration, the use of value-weighted versus equal-weighted portfolios, and the frequency of the observed data. 6. Charemza and Deadmen (1995) specify a more general model of periodically collapsing bubbles. 7. Equity REIT prices and dividends are converted into real terms by deflating by the consumer price index obtained from the St. Louis Federal Reserve Bank. 8. The use of the MTAR model parallels the approach undertaken by Bohl (2003) and the RADF test has been used in studies by Taylor and Peel (1998), Sarno and Taylor (1999, 2003), and Capelle-Blancard and Raymond (2004) for stock markets. 9. While the Chan (1993) method is typically used for choosing the threshold parameter, , the idea of eliminating extreme values when testing for bubble behavior seems counter intuitive. Specifically, Chan (1993) requires sorting the estimated residuals in ascending order, eliminating 15 percent of the largest and smallest values. The threshold parameter that yields the lowest sum of squared errors from the remaining 70 percent of the residuals is used in the MTAR model. 10. See Im (1996) and Taylor and Peel (1998) for the theoretical underpinnings of the two-step approach of the RADF test. Note that the basis of this test draws from the work of Im (1996) on residuals-augmented least squares estimators. 11. The covariance matrix of *̂ , )ˆ( *V , is estimated by 1ˆ

2* )~~()ˆ( XMXV wA where

17

2235

23

6246

44

44

244

235

443

23

6246

2322

)4()96)(()()3()4)(3(2)96(

A

and i denotes the ith central moment of tu . X

~ is the vector of the lagged series of centered

residuals and the idempotent matrix, wM ˆ , is given by WWWWIM Tw~)~~(~ 1ˆ

where TI is the

identity matrix and W~ is the matrix of the centered residuals of tŵ . 12. Upon the recommendation of an anonymous referee tests for periodically collapsing bubbles were examined for equity REIT sub-sectors. The equity REIT sub-sector data provided by the referee covered the period 1994:01 to 2005:03. The respective prices and dividends were converted into real terms by deflating by the consumer price index. 13. Note that for overall equity REITs along with the equity REIT sub-sectors, manufactured homes and regional malls, one fails to reject the null hypothesis of no cointegration (i.e. FC column of Table 4). Thus, as stated by Enders and Sikos (2001, p. 170) there is no need to pursue the tests of symmetry (i.e. FA column of Table 4). 14. Given the relatively short time horizon, the power of the cointegration tests are questionable (Hakkio and Rush, 1991 as well as Kremers et al, 1992). 15. We appreciate the suggestion of an anonymous referee to examine whether the NAPM index and/or the capacity utilization rate affect real equity REIT prices and dividends.

18

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22

Table 1 Perron Unit Root Tests Real Equity REIT Prices and Dividends 1972:01 to 2005:03 Panel A: Real Equity REIT Prices, tp DU t DT )( bTD 1tp k

2.RAdj )36(Q 0.175 -0.012 2.2E-5 4.5E-5 -0.018 0.964 1 0.946 32.60 (0.06)a (0.02) (3.8E-5) (8.1E-5) (0.04) (0.01) [0.63] Panel B: Real Equity REIT Dividends, td DU t DT )( bTD 1td k

2.RAdj )36(Q 0.121 -4.0E-4 6.2E-6 -3.5E-6 -0.003 0.974 4 0.961 43.40 (0.06)a (0.02) (2.7E-5) (5.5E-5) (0.03) (0.01) [0.19] Notes: Standard errors are denoted in parentheses and probability values in brackets. k is the number of augmented lags of the first-differences of tp and td , respectively. )36(Q denotes the Ljung-Box Q-statistic distributed as 236 . Critical values to test the null hypothesis of a unit root, 1 , is drawn from Table VI.B p. 1377 of Perron (1989) for 60. as follows: 1% -4.88, 5% -4.24, and 10% -3.95. For equity REIT prices the t-statistic for the null hypothesis, 1 , is -3.6 and for equity REIT dividends the t-statistic for the null hypothesis, 1 , is -2.6.

23

Table 2 MTAR and RADF Results

1972:01 to 2005:03 Panel A: Momentum Threshold Autoregressive Model Results

CR τ ρ1 ρ2 FC FA Q(5) k Equity -3.39b 0.31 0.05 -0.07 12.70a 10.10a 1.20 4 (1.74)c (-4.11)a {0.95} Panel B: Residuals –Augmented Dickey-Fuller Results

Skewness Kurtosis JB CRτ Q(5) Equity -0.001 3.56 5.23 -0.79 3.75 {0.07}c {0.59} Notes: CR is the Dickey-Fuller test statistic applied to the residuals from the cointegration equation (5) under the null hypothesis of no cointegration with critical values: a(1%) -3.73, b(5%) -3.17, and c(10%) -2.91 (Engle and Granger, 1987). CRτ is the residuals-augmented Dickey-Fuller test statistic applied to the residuals from the cointegrating equation (5) under the null hypothesis of no cointegration with critical values: a(1%) -3.98, b(5%) -3.44, and c(10%) -3.13 (Capelle-Blancard and Raymond, 2004). is the estimated threshold. 1 and 2 are the estimated parameters from the MTAR specification. Standard errors denoted by [ ], t-statistics denoted by ( ), and probability values in { } where a(1%), b(5%), and b(10%). FC represents the F-statistic corresponding to the null hypothesis of no cointegration (i.e. 021 ) with critical values provided by Enders and Siklos (2001, Table 5, p. 172, n = 250 and four lags): a(1%) 8.47, b(5%) 6.32, and c(10%) 5.32. FA represents the F-statistic corresponding to the null hypothesis of symmetry (i.e. 21 ) using the standard F distribution with critical values a(1%) 4.61 and b(5%) 3.00. k is the number of lags in equation (6). Q(5) denotes the Ljung-Box Q-statistic at 5 lags.

24

Table 3 Unit Root Tests 1994:01 to 2005:03 Panel A: Overall Equity REITs

tp tp td td ADF -1.58 -12.19a -2.55 -9.85a PP -1.67 -12.18a -2.99 -11.75a Panel B: Equity REIT Sub-Sectors Apartments tp tp td td ADF -3.04 -13.54a -0.79 -14.39a PP -3.12 -13.54a -1.48 -15.16a Industrial tp tp td td ADF -1.76 -13.95a -2.64 -6.87a PP -2.19 -13.88a -3.09 -29.32a Lodging tp tp td td ADF -1.79 -9.33a -2.47 -11.13a PP -1.79 -9.33a -2.45 -11.15a Manufactured Homes tp tp td td ADF -2.52 -12.05a -1.71 -4.15a PP -2.62 -12.05a -1.50 -15.41a Office tp tp td td ADF -1.90 -12.25a -1.53 -5.11a PP -1.98 -12.24a -2.78 -15.91a Regional Malls tp tp td td ADF -0.76 -13.87a -0.57 -13.27a PP -0.94 -13.75a -0.64 -13.35a Panel C: Measures of Capacity Utilization: National Association of Purchasing Managers Index, tnapm , and Capacity Utilization Rate, tcu tnapm tnapm tcu tcu ADF -2.43 -11.70a -1.66 -4.09a PP -2.57 -11.70a -1.55 -11.91a Notes: Lag length selection for the ADF unit root tests is based on Akaike’s information criterion while the PP unit root tests is based on Newey-West bandwidth using Bartlett kernel. ADF and PP unit root tests include constant and linear trend terms with the following critical values: a(1%) -4.03, b(5%) -3.44, and c(10%) -3.15.

25

Table 4 MTAR and RADF Results

1994:01 to 2005:03 Panel A: Momentum Threshold Autoregressive Model Results

CR τ ρ1 ρ2 FC FA Q(5) k Equity -2.94c 0.071 0.12 -0.06 2.21 0.86 4 (1.09) (-1.84)c {0.99} Apartments -2.03 0.041 -0.10 -0.12 5.67c 4.68b 0.49 4 (-0.12) (-2.67)a {0.99} Industrial -1.55 0.010 0.03 -0.13 6.59b 4.07b 0.63 4 (0.66) (-2.69)a {0.99} Lodging -2.41 0.149 -0.86 -0.05 15.67a 9.85a 1.78 4 (-4.20)a (-1.77)c {0.88} Manufactured Homes -2.69 0.017 -0.15 -0.05 1.54 0.61 4 (-2.59)a (-0.92) {0.99} Office -2.88 0.043 0.09 - 0.12 5.82c 6.08a 0.64 4 (1.11) (-3.25)a {0.99} Regional Malls -3.02c 0.002 -0.27 -0.21 4.07 4.84 4 (-0.40) (-3.42)a {0.44}

26

Table 4 (Continued) MTAR and RADF Results 1994:01 to 2005:03 Panel B: Residuals –Augmented Dickey-Fuller Results

Skewness Kurtosis JB CRτ Q(5) Equity 0.65 2.54 10.58 -0.41 5.15 {0.01}a {0.40} Apartments 0.48 3.16 5.27 -0.83 5.57 {0.07}a {0.35} Industrial 1.06 3.39 26.32 -0.49 8.82 {0.00}a {0.12} Lodging -0.18 2.45 2.48 -1.04 9.49 {0.29} {0.09}c Manufactured Homes -0.45 2.66 5.23 -1.28 2.35 {0.07}c {0.80} Office 0.36 3.06 2.87 -0.94 10.76 {0.24} {0.06}c Regional Malls 0.12 2.50 1.73 -1.85 4.74 {0.42} {0.45} Notes: CR is the Dickey-Fuller test statistic applied to the residuals from the cointegration equation (5) under the null hypothesis of no cointegration with critical values: a(1%) -3.73, b(5%) -3.17, and c(10%) -2.91 (Engle and Granger, 1987). CRτ is the residuals-augmented Dickey-Fuller test statistic applied to the residuals from the cointegrating equation (5) under the null hypothesis of no cointegration with critical values: a(1%) -4.36, b(5%) -3.61, and c(10%) -3.32 (Sarno and Taylor, 2003). is the estimated threshold. 1 and 2 are the estimated parameters from the MTAR specification. Standard errors denoted by [ ], t-statistics denoted by ( ), and probability values in { } where a(1%), b(5%), and b(10%). FC represents the F-statistic corresponding to the null hypothesis of no cointegration (i.e. 021 ) with critical values provided by Enders and Siklos (2001, Table 5, p. 172, n = 100 and four lags): a(1%) 8.91, b(5%) 6.56, and c(10%) 5.52. FA represents the F-statistic corresponding to the null hypothesis of symmetry (i.e. 21 ) using the standard F distribution with critical values a(1%) 4.79 and b(5%) 3.07. k is the number of lags in equation (6). Q(5) denotes the Ljung-Box Q-statistic at 5 lags.

27

Appendix A Johansen-Juselius Cointegration Tests Equity REIT Sub-Sectors 1994:01-2005:03

Panel A: Equity

Variables: tp , td , tnapm Null Alt. Statistic 95%CV 99%CV

max r = 0 r = 1 10.74 20.97 25.52 r ≤ 1 r = 2 6.74 14.07 18.63 r ≤ 2 r = 3 2.69 3.76 6.65

trace r = 0 r ≥ 1 20.17 29.68 35.65 r ≤ 1 r ≥ 2 9.44 15.41 20.04 r ≤ 2 r ≥ 3 2.29 3.76 6.65

Variables: tp , td , tcu Null Alt. Statistic 95%CV 99%CV

max r = 0 r = 1 13.51 20.97 25.52 r ≤ 1 r = 2 7.37 14.07 18.63 r ≤ 2 r = 3 1.76 3.76 6.65

trace r = 0 r ≥ 1 22.64 29.68 35.65 r ≤ 1 r ≥ 2 9.13 15.41 20.04 r ≤ 2 r ≥ 3 1.76 3.76 6.65 Panel B: Apartments


max r = 0 r = 1 11.19 20.97 25.52 r ≤ 1 r = 2 3.79 14.07 18.63 r ≤ 2 r = 3 2.70 3.76 6.65

trace r = 0 r ≥ 1 17.68 29.68 35.65 r ≤ 1 r ≥ 2 6.49 15.41 20.04 r ≤ 2 r ≥ 3 2.70 3.76 6.65

28

Appendix A (continued) Johansen-Juselius Cointegration Tests Equity REIT Sub-Sectors

1994:01-2005:03 Variables: tp , td , tcu

Null Alt. Statistic 95%CV 99%CV max r = 0 r = 1 18.88 20.97 25.52

r ≤ 1 r = 2 3.78 14.07 18.63 r ≤ 2 r = 3 2.99 3.76 6.65

trace r = 0 r ≥ 1 18.88 29.68 35.65 r ≤ 1 r ≥ 2 3.78 15.41 20.04 r ≤ 2 r ≥ 3 2.99 3.76 6.65 Panel C: Industrial


max r = 0 r = 1 11.60 20.97 25.52 r ≤ 1 r = 2 7.38 14.07 18.63 r ≤ 2 r = 3 0.46 3.76 6.65

trace r = 0 r ≥ 1 19.44 29.68 35.65 r ≤ 1 r ≥ 2 7.84 15.41 20.04 r ≤ 2 r ≥ 3 0.46 3.76 6.65


max r = 0 r = 1 8.12 20.97 25.52 r ≤ 1 r = 2 4.63 14.07 18.63 r ≤ 2 r = 3 1.58 3.76 6.65

trace r = 0 r ≥ 1 14.33 29.68 35.65 r ≤ 1 r ≥ 2 6.21 15.41 20.04 r ≤ 2 r ≥ 3 1.58 3.76 6.65

29


1994:01-2005:03 Panel D: Lodging


max r = 0 r = 1 13.95 20.97 25.52 r ≤ 1 r = 2 4.60 14.07 18.63 r ≤ 2 r = 3 0.67 3.76 6.65

trace r = 0 r ≥ 1 19.23 29.68 35.65 r ≤ 1 r ≥ 2 5.27 15.41 20.04 r ≤ 2 r ≥ 3 0.67 3.76 6.65


max r = 0 r = 1 17.70 20.97 25.52 r ≤ 1 r = 2 7.16 14.07 18.63 r ≤ 2 r = 3 1.73 3.76 6.65

trace r = 0 r ≥ 1 26.60 29.68 35.65 r ≤ 1 r ≥ 2 8.89 15.41 20.04 r ≤ 2 r ≥ 3 1.73 3.76 6.65 Panel E: Manufactured Homes


max r = 0 r = 1 16.94 20.97 25.52 r ≤ 1 r = 2 10.56 14.07 18.63 r ≤ 2 r = 3 2.07 3.76 6.65

trace r = 0 r ≥ 1 29.57 29.68 35.65 r ≤ 1 r ≥ 2 12.63 15.41 20.04 r ≤ 2 r ≥ 3 2.07 3.76 6.65

30


1994:01-2005:03


max r = 0 r = 1 7.60 20.97 25.52 r ≤ 1 r = 2 3.95 14.07 18.63 r ≤ 2 r = 3 3.50 3.76 6.65

trace r = 0 r ≥ 1 15.04 29.68 35.65 r ≤ 1 r ≥ 2 7.44 15.41 20.04 r ≤ 2 r ≥ 3 3.50 3.76 6.65 Panel F: Office


max r = 0 r = 1 11.33 20.97 25.52 r ≤ 1 r = 2 6.22 14.07 18.63 r ≤ 2 r = 3 2.21 3.76 6.65

trace r = 0 r ≥ 1 19.76 29.68 35.65 r ≤ 1 r ≥ 2 8.43 15.41 20.04 r ≤ 2 r ≥ 3 2.21 3.76 6.65


max r = 0 r = 1 18.99 20.97 25.52 r ≤ 1 r = 2 2.62 14.07 18.63 r ≤ 2 r = 3 2.18 3.76 6.65

trace r = 0 r ≥ 1 23.79 29.68 35.65 r ≤ 1 r ≥ 2 4.80 15.41 20.04 r ≤ 2 r ≥ 3 2.18 3.76 6.65

31


1994:01-2005:03 Panel G: Regional Malls


max r = 0 r = 1 16.30 20.97 25.52 r ≤ 1 r = 2 8.31 14.07 18.63 r ≤ 2 r = 3 0.06 3.76 6.65

trace r = 0 r ≥ 1 24.67 29.68 35.65 r ≤ 1 r ≥ 2 8.37 15.41 20.04 r ≤ 2 r ≥ 3 0.06 3.76 6.65


max r = 0 r = 1 18.82 20.97 25.52 r ≤ 1 r = 2 6.22 14.07 18.63 r ≤ 2 r = 3 2.03 3.76 6.65

trace r = 0 r ≥ 1 27.08 29.68 35.65 r ≤ 1 r ≥ 2 8.26 15.41 20.04 r ≤ 2 r ≥ 3 2.03 3.76 6.65 Notes: Critical values for the cointegration tests were obtained from Osterwald-Lenum (1992). Lag lengths were determined by Akaike’s information criterion. The Johansen-Juselius cointegration tests were performed with stochastic trends (Johansen, 1995 p.80-84).

32

Appendix B Granger Causality Tests Equity REIT Sub-Sectors and Capacity Utilization Measures 1994:01-2005:03 Panel A: Equity

Independent Variables Dependent Variable tp td tnapm F-statistic

2R k LM(4)

tp 0.36 0.33 0.48 0.43 0.010 1 3.84

td 0.18 0.12 0.18 0.15 0.004 (0.92)

tnapm 1.36 0.23 0.18 0.54 0.013

Independent Variables Dependent Variable tp td tcu F-statistic

2R k LM(4)

tp 0.77 2.50c 0.55 1.40 0.095 3 6.23

td 2.59c 2.24c 0.92 1.81c 0.119 (0.72)

tcu 2.05 2.08 6.98a 3.76a 0.219

Panel B: Apartments


2R k LM(4)

tp 2.96c 2.49 0.68 2.21c 0.049 1 7.41

td 0.01 5.60b 0.06 1.94 0.043 (0.59)

tnapm 0.55 0.02 0.13 0.22 0.005


2R k LM(4)

tp 3.47c 2.41 0.57 2.17c 0.048 1 10.21

td 0.01 6.22b 2.49 2.79b 0.061 (0.33)

tcu 0.84 1.24 0.24 0.77 0.018 Panel C: Industrial


2R k LM(4)

tp 5.06b 0.04 0.00 1.75 0.039 1 12.76

td 0.44 0.01 0.01 0.15 0.004 (0.17)

tnapm 0.56 0.51 0.13 0.37 0.008

33

Appendix B (Continued) Granger Causality Tests Equity REIT Sub-Sectors and Capacity Utilization Measures 1994:01-2005:03


2R k LM(4)

tp 1.37 2.34b 1.52 1.82b 0.195 5 10.37

td 0.45 4.52a 0.58 1.96b 0.206 (0.32)

tcu 1.35 1.99c 3.92a 2.39a 0.241

Panel D: Lodging


2R k LM(4)

tp 1.55 1.13 1.83 1.14 0.132 5 8.92

td 6.54a 0.20 1.59 3.64a 0.326 (0.45)

tnapm 2.31b 1.08 0.68 1.71c 0.185

Independent Variables

Dependent Variable tp td tcu F-statistic 2R k LM(4)

tp 1.03 0.52 0.38 0.74 0.070 4 10.24

td 9.68a 0.58 1.16 4.22a 0.302 (0.33)

tcu 1.94 0.56 5.52a 2.69a 0.216

Panel E: Manufactured Homes


2R k LM(4)

tp 0.63 3.27b 0.13 1.51 0.101 3 8.26

td 0.30 4.30a 0.77 1.98b 0.128 (0.51)

tnapm 0.85 0.34 0.65 0.60 0.042


2R k LM(4)

tp 0.83 2.73b 1.33 1.95c 0.127 3 5.74

td 0.23 4.29a 1.77 2.36b 0.149 (0.77)

tcu 0.41 0.65 6.29a 2.71a 0.168

34

Appendix B (Continued) Granger Causality Tests Equity REIT Sub-Sectors and Capacity Utilization Measures 1994:01-2005:03 Panel F: Office


2R k LM(4)

tp 0.59 1.86 0.19 0.88 0.020 1 4.20

td 0.01 6.79a 0.16 2.36c 0.006 (0.90)

tnapm 0.09 0.87 0.13 0.35 0.008


2R k LM(4)

tp 0.78 1.74 0.81 1.09 0.024 1 4.73

td 0.01 6.83a 0.72 2.56c 0.056 (0.86)

tcu 1.26 0.05 0.21 0.49 0.011 Panel G: Regional Malls


2R k LM(4)

tp 3.75c 0.65 1.77 2.07c 0.046 1 10.06

td 0.16 2.50 0.03 0.87 0.020 (0.35)

tnapm 2.01 1.53 0.16 1.32 0.030


2R k LM(4)

tp 1.45 0.85 1.82 1.45 0.161 5 8.23

td 0.60 0.92 0.30 0.60 0.074 (0.51)

tcu 1.79 1.21 4.03a 2.35a 0.238

Notes: Partial F-statistics are denoted under the respective independent variables. F-statistic represents the overall F-statistic for the equation. Lag lengths, k , determined by Akaike’s information criterion. The multivariate Lagrange multiplier test for serial correlation at 4 lags is denoted by LM(4) with probability values in parentheses.

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