REVISITING THE ANOMALOUS RELATIONSHIP BETWEEN INFLATION AND REAL ESTATE INVESTMENT TRUST RETURNS IN PRESENCE OF STRUCTURAL BREAKS: EMPIRICAL EVIDENCE FROM THE USA AND THE UK

In this paper we have re-investigated the frequently observed anomalous negative relationship between inflation (INF) and real estate investment trust (REIT) returns for two most important economies viz., the USA and the UK by addressing two aspects of misspecification: Inappropriate functional form and omission of relevant variable. We have found that the anomalous relationship between REIT and INF appear to proxy for the significant effect of relative price variability (RPV) on REIT returns (REITR) in both the countries. Further, it is evidenced that the effect of RPV on REITR is not stable over time in case of the USA while in the UK there is no structural change in the relationship.


INTRODUCTION
A number of papers have attempted to explain the anomalous relationship between real estate investment trust returns (REITR) and inflation (INF). The relationship is anomalous in the sense that the effect of INF has often been found to be negative though it was expected to be positive from theoretical consideration. Such an expectation was derived due to existence of positive relationship between residential real estate returns and INF, and real estate investment trust (REIT) being a securitized form of this traditional real estate investment, it was also expected to have positive association with INF.
In this paper our aim is to resolve this anomalous relationship considering two important issues which the previous studies have failed to capture. The idea is that the anomalous nature of this relation is essentially due to what, in general econometrics terminology, can be called "misspecification" of the underlying model. These two aspects of misspecification that appears relevant in this context are inappropriate functional form of the model and omission of relevant variables in the existing studies. Keeping this in mind we have considered firstly the non-linear modeling approach where structural break is explicitly considered not only for the variables concerned but also in the relationship involving the variables by using Qu and Perron (2007) methodology. Secondly, the inclusion of a relevant variable namely, relative price variability (RPV) other than REITR and INF. The inclusion of RPV has provided some new insights which have explained the existing anomalous relationship as it affects INF directly as well as asset returns through the change in real economic activity. Barrow (1976) and Cukierman (1982) both have argued that increased RPV has negative impact on economic production which happens due to misallocation of resources caused by the increased RPV.
For our study, monthly data of the USA and the UK, covering the period from January 1990 to December 2014 andJanuary 1996 This Journal is licensed under a Creative Commons Attribution 4.0 International License to December 2014, respectively, have been used. The starting date of the UK data is on January 1996 as most of the individual price series which are required for the construction of RPV are available only after 1996. In this paper, we have found empirically in both the countries that the existing negative relationship between REITR and INF appears to proxy for the significant effect of RPV on the REITR and INF as well. Moreover, our results indicate that the relationship involving all the variables are not stable over the entire sample period for the USA while in case of the UK there is no significant structural change in the underlying relationship. The time points of the structural changes obtained for the USA are, however, found to coincide with the period of the Global Financial Crisis.
The paper is organized as follows: Section 2 provides extensive review of literature on this anomalous relationship. The details about data and methodology used in this study are discussed in the Section 3. Next Section presents the estimation results. The paper ends with some concluding remarks in Section 5.

LITERATURE REVIEW
As discussed in the introduction section, our paper has contributed to an extensive body of literature studying on the anomalous negative relationship between REITR and INF. Extant literatures tends to suggest that REITR have negative relationship with INF while few studies, such as Chen and Tzang (1988), Liang et al. (1998), and have indicated that REIT possesses some INF hedging properties which establishes the positive relationship between REITR and INF. Chen and Tzang (1988) documented that REIT has some ability to hedge expected component of INF and found that REITR are closely related to interest rates. Liang et al. (1998) ruled out the possibility that a stock market-induced proxy effect is the cause for the apparent lack of relationship between REITR and INF, and found that REITR are positively related to temporary or permanent components of INF measures. Chatrath and Liang (1998) have empirically found the long-run co-movement between REITs and INF. However, most of the empirical evidences tend to suggest the opposite i.e., REITR have negative relationship with INF. For instances, Goebel and Kim (1989), Park et al. (1990), and Adrangi et al. (2004) have found the similar negative relationship between REITR and INF. Chan et al. (1990) analyzed monthly returns on equity REIT that were traded on major stock exchanges over the period of 1973-1987, and concluded that returns from REIT is not a hedge against unexpected INF. Liu et al. (1997) examined whether real estate securities continue to act as perverse INF hedges from a global perspective in countries like Australia, France, South Africa, Switzerland, the UK, and the USA. With few exceptions, the results were found to be consistent with negative INF hedging ability of REITR. The characteristic of perverse hedging ability is quite common in the literature of the stock market as well. In a study, Darrat and Glascock (1989) Simpson et al., 2007;Chang et al., 2011;Pierdzioch et al., 2018).

About the Data
In this section, we first discuss about the details on the relevant aspects of the data sets. The standard descriptive statistics values of these data sets are also presented and discussed. The sample period for all the time series used in this study ranges from January 1990 to December 2014 for the USA and January 1996 to December 2014 for the UK. Monthly data of the equity REITR for the USA has been taken from the National Association of REIT (NAREIT) REIT Handbook, whereas it has been taken from Data Stream for the UK. The price series used to construct a RPV measure called RPV t , and as described below, involves the seasonally adjusted price indices of the components of the consumer price index (CPI) at the item/product level. As summarized in Table 1, the resulting series for both the USA and the UK which are available for 38 and 40 product categories, respectively, have been taken from CEIC data source. For the purpose of computation of INF rate, data on seasonally adjusted CPI for all items are required. This data have been obtained from the Federal Reserve Bank at St. Louis for both the countries. RPV is most often constructed by the weighted average of sub-aggregate INF series using the standard deviation (s.d.).
The primary measure of INF used here is the monthly logdifference of the seasonally adjusted CPI.  Table 2 shows the summary statistics of the three variables for both the countries under investigation, viz., REITR, RPV, and INF. Note that REITR in both the countries have the highest s.d. among the three variables, followed by RPV. The skewness value for RPV is highest in both the countries while the value is lowest for REITR 1 Given the nature of index data, the RPV measure adopted here should be read as relative inflation variability. In this paper, however, we have followed the tradition in the literature and referred to this measure as RPV.

Summary Statistics
Another common formulation for RPV is the coefficient of variation (c.v. All the variables in both the countries have very high kurtosis values, and hence as seen from the J-B test statistic values, the null hypothesis of normality is rejected. It is evident from these

Methodology
Both the statistics and econometrics literature have a good deal of work on inferences on structural changes with unknown break dates, most of which are specifically designed for the case of a single change. 2 The issue of multiple structural changes has received more attention recently, and this is in the context of single regression model only (see, Bai and Perron, 1998;2003). However, work concerning structural changes in the context of a system of equations is only very recent and very few in number, 2 See, Perron (2006), for an extensive review. and the important ones are Perron (1998), Hansen (2003), and Qu and Perron (2007).
The main advantages of Qu and Perron (2007) methodology is that it provides a comprehensive treatment of issues related to estimation, inference, and computation with multiple structural changes that occur at unknown points in linear multivariate regression models that include vector autoregressive (VAR) model, certain linear panel data models, and seemingly unrelated regression (SUR) model. Changes can occur in the parameters of the conditional mean, the covariance matrix of errors, or both, and the distribution of the regressors can also be allowed to change across regimes. It may be noted that for this methodology it is not required to assume that the regressors are independent of the errors at all leads and lags in presence of heteroskedasticity and/ or autocorrelation. Let the variables of interest Y t =(Y 1,t Y 2,t …Y n,t )' be an (n×1) vector at time point t. The general model considered by Qu and Perron (2007) is as follows.
where Y t =(Y 1,t Y 2,t …Y n,t )' for n equations and T observations. The total number of structural changes in the system is m and the break dates are denoted by the (1×m) vector T=(T 1 T 2 …T m ), taking into account that T 0 =1 and T m+1 =T. The subscript j indexes a regime where (j=1,2,…,m+1), subscript t indexes the temporal observation (t=1,2,…,T), and i indexes the i th equation where i=1,2,…,n to which a scalar dependent variable Y i,t is associated. The number of regressors is q and x t is the (q×1) vector which includes the regressors from all the equations i.e., x t =(x 1,t x 2,t .x q,t )', and φ j is the set of parameters in the model for the j th regime. The selection matrix is denoted by S in the above equation, which involves elements that take the values 0 and 1, and thus indicate which regressors appear in each equation. When using a vector autoregressive model, we have x t =(1 y 1,t−1 y 1,t−2… y 1,t−q y 2,t−1 y 2,t−2 y 2,t−q y 3,t−1 y 3,t−2 y 3,t−q) ', which simply contains the lagged dependent variables including intercept term, and here will be an identity matrix.
This general framework of VAR is adopted for the purpose of studying structural breaks in the relationships involved in this study, and to estimate the parameters thereafter for different regimes separately based on the Qu-Perron test. In our case, Y t now consists of RPV, INF, and REITR i.e., Y t =(RPV t , INF t , REIT)'. The quasi maximum likelihood method is used to estimate the above model. Qu and Perron (2007) have proposed a number of test statistics for identifying multiple break points, and these are stated below. i. The sup F T (k) test i.e., a sup F-type test of the null hypothesis of no structural break versus the alternative of a fixed number of breaks (k). ii. The double maximum test, denoted as UD max test and WD max test, from consideration of having equal weighting scheme and unequal weighting scheme where weights depend on the number of regressors and the significance level of the tests. For these two tests, the alternative hypothesis is that the number of breaks is unknown, but up to some specified maximum 3 . iii. The sup F T (l+1|l) test i.e., a sequential test of the null hypothesis of l breaks versus the alternative of (l+1) breaks with the starting value of l being 1.
It should be quite obvious that size and power of these tests are important issues for final testing conclusions. Similar to Bai and Perron (1998;2003), Qu and Perron have suggested the following useful strategy. First the UD max test and the WD max test 3 The methodology of these two tests is same as that in Bai and Perron (1998).  Figure 2: Time series plots of real estate investment trust returns, relative price variability, and inflation for the UK are used to find if at least one break is present. If these indicate the presence of at least one break, then the number of breaks can be decided based upon the sequential examination of the sup F T (l+1|l) statistic which is constructed using global minimizers for the break dates. While applying these tests, we set the value of the trimming parameter to 0.15. Since the focus of this study is on the stability of the relationship among the variables of interest, we restrict our attention to tests for changes in the regression coefficients only. Once the tests for structural breaks have been carried out, the subsequent estimation of the relations involving these variables for each regime is done by VAR model whose explicit form is as follows:

EMPIRICAL RESULTS
We first report the results of the augmented Dickey and Fuller (1979) (ADF) test which has been applied to find if all the variables are stationary. This step is necessary as the usual unrestricted vector autoregressive model requires all the variables to be stationary. The optimum lag length for the ADF test has been chosen by the Schwarz information criteria (SIC).
It can be concluded from the test results, presented in Table 3, that all the variables are stationary at 1% level of significance for both the countries. In addition, the structural stability of each of the variables during the sample period has been examined by carrying out the Bai and Perron (1998;2003a) multiple structural breaks test for both the countries. This test requires using UD max test, WD max test and then sup F T (l+1|l) test, as described earlier.
It is evident from the results of the tests presented in Table 3 that at least one structural break is present in all the series of both the countries. The WD max test statistic values are found to be higher than the critical value of 12.81 at 5% level of significance for INF series in the USA, and for all other series the values of this test statistic indicate significance at 1% level. Hence, the null hypothesis of "no break" is rejected in favor of alternative of "upto one break." To detect further if there is more than one structural break, the sequential break test has been performed. This test also suggests that all the three series are structurally unstable. In fact, looking at Sup F T (3|2) test statistic values, it is evident that there are two breaks in all the series of the USA and RPV series of the UK. In case of RPV series in the USA, the estimated break dates are 1993:M08 Since all the series are found to be structurally unstable, any study of the relationship involving these variables cannot be taken to be of the fixed coefficient kind. These findings of structural stability of all the series provides justification for our approach of considering VAR model allowing for multiple structural breaks.
We now report the results of the Qu and Perron (2007) test for detecting breaks in the system of equations involving these three variables. The values of this test statistic are given in Table 4. It is clear that the test results indicate the presence of structural breaks in the relationship involving REITR, RPV and INF for the USA while in case of the UK there is no structural change in the underlying relationship.
In case of the USA, the test statistic value of the WD max test is found to be 42.01, which is significant at 1% level of significance, and hence the null hypothesis of "no break" is rejected in favor of the alternative of "up to one break" in this system of equations.
To detect further if there is more than one structural break, the sequential break test was carried out. By looking at the relevant entry of the table, we note that the test statistic value of Sup F T (2|1)) is 53.57, which is significant at 1% level. So the test rejects the null hypothesis of "one break" in favor of "two breaks." However, the sequential test for detecting more than two breaks i.e., Sup F T (3|2) test statistic yields that the underlying null hypothesis of "two breaks" cannot be rejected in favor of "three breaks." Hence, no further test is required. Finally, the break points for the USA have been estimated following the procedure of Qu-Perron, and these are found to be May 2005 andMay 2009. In what follows we attempt at providing plausible economic explanations for the findings on the break dates for both the countries. For the USA, the first break date has been found to be close to the middle of the year 2005, which coincide with the period of bubble in the real estate market. In that period real estate price peaked its high, causing high fluctuations in all the series. The occurrence of second break in middle of 2009 can be attributed to the severe recession in the US economy which occurred as a result of busting of this bubble, causing huge fluctuations in those series again. In case of the UK, as already stated, there is no evidence of any structural break in the relationship involving these variables though it is obvious from the test results in Table 3 that each of the individual series has one or more structural breaks. This may be due to the existence of "co-breaking" in the variables which is defined as the cancellation of structural shifts across linear combinations of variables (see for details, Hendry and Mizon, 1998;and Clements and Hendry, 1999). However, it is important to note that the effect of RPV on REIT in both the countries is contrasting in nature. In case of the USA, this effect is positive and the coefficient values are 0.9 and 2.10 in first and third regimes respectively. Eaton (1980) has argued that RPV may have positive effect on the return of an asset if the elasticity of demand of this asset and marginal propensity to consumption from the returns of this asset are large. Since returns from real estate asset have some impact on the level of future consumption (see, for instance, Brayton and Tinsley, 1996), it is expected to have positive relationship between RPV and REITR. In contrast, in the UK this effect is negative and the coefficient value is −2.54. The negative effect of RPV on REITR is due to the adverse effect of RPV −0.04 "*" and "**" indicate significance at 5% and 1% levels, respectively on economic production which in turn lowers the returns from REIT. The negative effect of RPV on economic production is due to misallocation of resources caused by the increased RPV (see for instance, Barrow 1976;and Cukierman, 1982). Looking at the results in Table 5 it is evidenced that significant negative relations between INF and RPV exists in both the USA and the UK. Despite the existence of a large body of empirical studies reporting positive relationship (see, for example, Parks, 1978;Lach and Tsiddon 1992;Parsley and Wei, 1996;Debelle and Lamant, 1997), a number of studies have supported a negative relationship between RPV and INF. For instance, Reinsdorf (1994) found this relationship to be negative During 1980s in the USA. Fielding and Mizen (2000) and Silver and Ioannidis (2001) also reported the same for several European countries. They have argued that this result is consistent with the fact that the law of one price tends to hold more strongly with higher INF. In other words, if firms make adjustment to prices towards desired levels during inflationary process then price dispersion may fall. In that case RPV will be negatively related to INF. Further, Rogers and Jenkins (1995) and Engel and Rogers (1998) support the hypothesis that there are frictions to the price setting process, justifying a negative relationship between price variability and INF.

CONCLUSIONS
In this paper, we have explored the effects of RPV and INF on REITR and re-examined the spurious relationship between REITR and INF. The evidence shows that the anomalous negative relation between REITR and INF appear to proxy for the effectiveness of RPV on REITR. We have also found that the effect of RPV on REITR is positive and different across the different regimes in the USA while it has remained the same in the UK over the entire sample period. In other words, we have found multiple structural changes in the relationship involving REIT, RPV and INF in the USA, whereas in the UK, there is no such structural change in this relationship. It is also important to note that RPV and INF is negatively related in both the countries.
Our findings have important policy implications. For instance, our finding of increased RPV having positive effect on REITR in the USA combined with the observation by Kaul and Seyhun (1990), viz., that RPV affects stock returns negatively, suggests that investors can diversify their portfolios and maximize their returns by investing more on REIT market than on any other stock market.