Frequency space correlation between REITs and capital market indices

Abstract. Several studies have examined real estate investment trust (REIT) co-movement with stocks or bonds using traditional time domain based methods, such as linear regression or correlation. Results of these studies have produced inconsistent statistical model parameters. The erratic behavior of the models may have resulted from the different time periods in the studies, the REITs included in a study or the market indices. Another factor contributing to the variation of the models comes from the compression of cyclical information over a study's time period by time domain based techniques. Cross-spectral analysis provides a frequency space method of examining the coherency (ie., frequency space correlation) between two time series across all frequencies. This article contains an examination of the coherency between REITs and stock market indices and REITs and U.S. Treasury debt indices for the period 1989-95. Results of the coherency spectra show significant co-movement between REITs and stock market indices, while debt instruments show very few frequencies with significant coherency. Furthermore, phase spectra provide evidence of contemporaneous movement between REITs and stock indices at all frequencies.

Introduction

Modern portfolio theory suggests that investments should be allocated over different asset classes to minimize a portfolio's unsystematic risk. Thus, a properly diversified portfolio would contain only systematic, i.e., market risk. Since real estate represents a separate asset class, researchers have developed risk-minimizing models for allocating portions of a portfolio's wealth to real estate. These models require time series returns for real estate that can be compared to time series returns for other asset classes such as equity or debt instruments. Typically, investigators used the NCREIF indices as a proxy for equity returns generated by real estate holdings. Yet, these indices have been criticized for incompleteness and biasness. Hence, results from asset allocation models using these indices may be questionable.

Recently, institutional investors have been exchanging their ownership interests in real estate for equity positions in real estate investment trusts (REITs). Shifting their real estate holdings to REITs reduces the liquidity and management risks associated with direct ownership in real estate and allows investors to benefit from the dividend cash flow generated by most REITs. During the past decade investors have observed a proliferation of public stock offerings in REITs. These public offerings have provided a new outlet for real estate companies to raise capital for the development, acquisition and management of real estate assets. Since many REITs specialize in particular types of properties, (e.g., apartment REITS, office REITs, retail REITS, etc.,) investors can construct diversified REIT portfolios containing investments across many different types of real estate.

Although previous research has shown that returns from direct ownership of real estate perform differently than stocks and bonds, thus helping to diversify a mixed-asset portfolio, a question remains as to whether REITs provide a similar advantage. REIT pricing must demonstrate different behavior than stocks or bonds in order to hedge a mixed-asset portfolio against unsystematic risk. To understand how REITs fit into a mixed-asset portfolio, it is imperative to investigate the co-movement of REIT pricing with other asset classes. Several researchers have used time domain models, such as CAPM, APT, ARMA and correlation, to examine the co-movement between REITs, stocks and bonds. Results of these studies have been mixed, with some showing similarities between REITs and stocks and others between REITs and bonds. Often these studies are sensitive to the time period studied and the indices or REITs included in the study. To be effective these statistical analyzes require regular time invariant cycles (Knif, Pynnonen and Luoma, 1995) or a linear relationship between the asset time series. Unlike spectral analysis, time domain analyzes do not provide frequency specific information because they aggregate statistical parameters, such as mean, variance and covariance over all frequencies (Erol and Balkan, 1991). The following pedagogical example demonstrates a case where spectral (frequency) analysis yields different information than a classical time domain based model. Consider the two time series plotted in Exhibit 1. Selecting either series as a dependent variable and the other as an independent variable in a linear regression model produces the following estimates of the univariate regression line: Y Intercept, beta^sub o^ = .1286, coefficient of determination, R^sup 2^ = .0087, slope of the regression line, (beta^sub 1^, = -.0277 and t-Stat = 0.5.

These values represent insignificant estimates in the time domain of the linear relationship between the two series. Yet, a frequency space analysis would show that the coherency between the two series equals one at all frequencies. The only difference detected in frequency space would be a phase shift of ninety degrees. The authors constructed the first series using a number of cosine functions and the second series with an identical number of sine functions containing the same amplitudes and frequencies. Since the sine function is identical to the cosine function except for a ninety degree (II/2 radians) phase shift, the frequency space correlation (coherency) between the two series equals one at all frequencies. Hence, this example shows that frequency space analysis can produce radically different results than time domain analysis.

The research in this article employs frequency domain models that focus on the cyclical relationships of REIT pricing to other asset indices. Cross spectral analyzes, which produce coherency and phase spectra, disclose leads and lags between two return series that may not be apparent using time domain based models. When correlations between two signals vary across time, cross spectral analysis provides a method of detecting hidden relationships between the two series by transforming the signals into their cyclical components.

Literature Review

The following summarizes past research that used time domain-based models for examining REIT pricing relative to stocks and bonds. Typically, the researchers focused on whether REITs demonstrated abnormal returns compared to the stock market or whether market yields on debt instruments influenced REIT pricing. Also, some studies examined the inflation hedging characteristics of REITs compared to other assets. Current literature does not contain a frequency space study of REIT comovement with capital markets.

Gyourko and Linneman (1988) used a modified CAPM to compare quarterly REIT returns with the S&P 500 and bonds for the period 1972-86. Results of their analysis showed significant positive correlations between REITs, stocks and bonds. Sagalyn (1990) constructed a quarterly index from survivor REITs that spanned 1973 to 1987. Sagalyn's CAPM analyzes showed a lower coefficient of determination (R^sup 2^ = .27) between the S&P 500 and REITs during high growth periods when compared to low growth periods (R^sup 2^ = .648). Results of a Chow test showed a significant difference between regression beta^sub s^ calculated for high and low growth periods. Although researchers (e.g., Han and Liang, 1995) have criticized Sagalyn's study for survivorship bias, the analysis provided evidence of the relationship between the S&P 500 and REITs.

Han and Liang (1995) recognized the inconsistencies in the REIT samples used in the above articles, the time frame of the studies and the market benchmarks for performance. The authors investigated REIT performance relative to a broader market index and examined the selection bias associated with using only survivor REITs in an index. Unlike previous studies, Han and Liang used the return on savings accounts as their risk free rate and the returns on the equally-weighted CRSP Stock Index as a market benchmark because most REITs are small capitalization issues. Since the authors speculated that using survivor REITs represented an ex post sampling bias, they constructed their monthly time series of REIT returns from all REITs trading on the markets. Results of their CAPM analysis showed significant co-movement between REIT returns and the CRSP Index. Based on comparing the regression coefficients of determination, R^sup 2^, the CRSP index produced better explanatory power than the S&P 500. Also, the R^sup 2^ for a multivariate linear regression model containing both the CRSP Index and S&P 500 as independent variables proved inferior compared to the univariate CAPM containing only the CRSP Index.

The CAPM analyzes in the above studies identified significant co-movement between REITs and stock indices. In addition, results of these studies suggested that REITs and small stock returns are more correlated than REITs and large stock returns. Also, regression coefficients were not stable across time periods, indicating that macroeconomic conditions may affect REITs differently than general stock indices.

Park, Mullineaux and Chew (1990) developed a multiple linear regression model where Treasury bills (TBILL) represented anticipated inflation and the difference between the CPI and the Treasury bill equaled unanticipated inflation. Results of the authors examination of NAREIT's total return series for REITS from 1972-86 showed that REITs did not provide a hedge against anticipated or unanticipated inflation. From comparing the assets' regression parameters, Park et al. reported similarities between the hedging characteristics of stocks and REITs against inflation that supports the theory of co-movement between REITs and stock indices.

Liu and Mei (1992) developed a model for predicting the risk premium associated with REITs. Their vector autoregressive forecasting model consisted of five independent variables: a dummy variable set to one for observations from January, the yield on Treasury bills, the yield difference between Treasury bills and AAA corporate bonds, the dividend yield on stocks and a real estate capitalization rate published by the American Council of Life Insurance. The authors regressed the dependent variable, risk premium for REITs, against lagged values of the independent variables as demonstrated in the following equation:

Significant regression coefficients for the January effect, Treasury bills and the capitalization rate resulted from this analysis. The coefficient of determination (R^sup 2^) equaled .175, which the authors cited as evidence that a significant percentage of the risk premium for REITs could be predicted one period in advance.

An article by Myer and Webb (1993) investigated the distributional and time series characteristics of REITs compared to other assets. The authors constructed a quarterly equally weighted index of equity REIT returns from data contained on the CRSP tapes to examine the relationship between REITS, the Russell-NCREIF index and the S&P 500. The first eight autocorrelation parameters (p^sub 1^, through p^sub 8^ contained insignificant values. Based on the results of a Box-Pierce-Ljuge Portmanteau Q tests for white noise (i.e., random noise) the researchers concluded that the time series represented white noise. The researchers constructed a vector autoregressive model between the REIT return series and the S&P 500 and the REIT return series and the Russell-NCREIF. Results of a Granger causality test demonstrated that lagged values of the S&P 500 failed to explain the return on REITs. Granger causality tests using CRSP value-weighted and equally-weighted stock indices produced similar results.

The above literature suggests that REITs may co-move with stock market indices, such as the S&P 500. Recent articles, such as Han and Liang (1995), present convincing data that REIT returns follow indices constructed from small company stocks. If REIT prices move in the same direction as broad-based stock indices, then inclusion of REITs into a mixed-asset portfolio may not provide the same diversification benefits that direct ownership in real estate provide to a portfolio.

Data

In 1995, the National Association of Real Estate Investment Trusts (NAREIT) listed a total of 178 publicly traded equity REITs with a total market capitalization of approximately $60 billion. Roughly 110 of these REITs trade on the New York Stock Exchange (NYSE), with the remaining traded on the American Stock Exchange (AMEX) or the NASDAQ exchange. Because REITs typically focus on a specific real estate market,1 such as retail or apartment, they can be classified into subcategories based on the property types within their portfolio. Although NAREIT publishes a monthly index for REIT returns containing several subclassifications based on REIT types and dividends, they do not maintain a daily return index. Because a long-term daily REIT index does not exist, this study computes a price-weighted index for equity REITs for the period January 1,1989 to December 31, 1994. In addition, the study includes price-weighted indices for office, apartment, retail, health and mixed asset REIT subcategories. Each index contains as many companies as possible (see Exhibit 11). The overall index consists of an aggregate of the sub-category indices. A REIT must meet the following criteria to be included into an index:

The REIT's initial public offering (IPO) must be one quarter prior to inclusion in an index.

The REIT must have a minimum capitalization of $20 million as of January, 1989.

The REIT must have a majority of its assets in equity ownership of real estate.

Failure to meet one of these criteria causes a REIT to be excluded from the index. Once a REIT enters the index it remains until it merges with another REIT or discontinues trading on the exchanges. Thus, the index equates to a buy and hold strategy.

This investigation uses daily pricing of futures contracts on market indices as proxies for market prices. Since futures contracts converge to the market price at maturity, they demonstrate similar price movements as spot prices. More importantly, futures contracts represent a liquid asset that can be easily purchased by managers to balance their portfolios' market exposure. This study investigates a range of the debt term structure by including futures contracts on three different maturities of Treasury debt: Treasury bill contracts traded on the International Monetary Market (IMM) of the Chicago Mercantile Exchange (CME), ten-year Treasury note contracts traded on the Chicago Board of Trade (CBT) and Treasury bond contracts traded on the CBT. Also, this investigation compares REIT pricing to the pricing of three stock futures contracts: the Value Line Index traded on the Kansas City Board of Trade (KBOT), the NYSE composite index and the CME's S&P 500 futures contract.

Long term time series studies of futures' pricing present problems to researchers because of the relatively short life of futures contracts. Typically, trading volume on a contract remains low until the final three to six months of the contract's life. After a contract matures, the researcher faces a discontinuity in the pricing series when they continue with the next nearest contract to maturity. To resolve this dilemma researchers (Herbst, Kare and Caples, 1989) employed Pelletier's (1983) weighted average method for computing perpetual contract prices that equate to ninety-day forward rates. The data used in this research also uses perpetual futures contracts to facilitate examining the long time series of futures contracts.

Methodology

Results

Exhibit 2 lists the standard time domain correlation matrix between the 1989 to 1994 REIT indices and the perpetual futures contracts. Only the office REIT index showed negative correlation, which occurred with both the REIT indices and perpetual futures contracts. The Value Line Index had the highest correlation with the total REIT index. The health REIT index correlated highest with the NYSE composite index and the S&P 500. Although, the apartment index showed the highest correlation with the health REIT index (87%), it still correlated higher with the Value Line Index.

The authors computed the first difference of the log of the data series to obtain stationary time series that approximate daily percentage return for the indices and perpetual futures contracts. Spectra of these stationary time series were computed using cosine (Hanning) data windows of 260 data points that overlapped by 130 data points. Overlap processing (Bendat and Piersol, 1986) counteracts the increase in spectral estimate variability that results from applying the cosine window to reduce leakage between frequencies. The above window parameters produced spectral estimates at discrete Fourier frequencies:

The Nyquist sampling rate of one day (Delta t = 1) resulted in a Nyquist frequency of 1/2Deltat = .5 cycles per day.

According to Table 8.1 in Koopman (1974), the coherency estimates for a Hanning window contain 2.67(N/M) degrees of freedom, where N is the length of the data series and M is the window length. Since the windows overlapped, N and M must be adjusted by the overlap factor of 130 data points, thus N equals 1,560 - 130 and M equals 260 - 130, which results in 29 degrees of freedom for each coherency estimate. Table A9.6 in Koopman gives the critical values for testing:

Koopman's table lists .39 as the critical value (p

Exhibits 3-10 list the frequencies that contain statistically significant coherence between an input signal (i.e., one of the perpetual futures contracts and an output signal). A negative phase indicates that the output signal lags the input signal (i.e., a REIT index lags a perpetual futures contract whenever the phase was negative). Treasury bill perpetual futures contracts fail to show significant coherency with any of the REIT indices, thus they do not appear in any of the exhibits.

Exhibits 3, 4 and 5 show that the total REIT index has the greatest number of periods containing significant coherencies with the perpetual futures contracts. Also, with the exception of the health REIT index, the Value Line Index shows the highest number of periods containing significant coherencies with the REIT indices as evidenced by Exhibits 4, 6, 8, 9 and 10. According to the results in Exhibit 7, the NYSE perpetual futures contain the greatest number of significant coherent periods with the health REIT index. The Treasury perpetual futures demonstrates fewer significant coherency values with the REITs than the stock perpetual futures. Also, Exhibits 5, 7, 8, 9 and 10 show that most of the significant coherencies between the REIT indices and a Treasury perpetual futures occur in periods longer than thirty-two days and were out of phase.

Also, the width of a phase angle's confidence interval depends on the coherency level between the input and output signals. An increase in coherency between the signals results in a narrower confidence interval for phase. Consequently, many of the periods with the highest coherency have phase angles that differ from zero (p

Findings and Conclusions

This study investigated equity REIT co-movement with capital market indices. Unlike past REIT research, the authors use spectral analysis to examine stock and futures price returns in frequency space. We identify common cycles between REITs and market indices that may not be observable using traditional time domain based models. The results show stronger co-movements in price between REITs and stocks than REITs and Treasury instruments.

The exhibits show the periods containing significant coherency between the REIT indices and the perpetual futures contracts. Stock market pricing appears to influence the movement of REIT prices more than the Treasury instruments because they contain more periods with significant coherency and larger coherency values. This supports the findings of Kuhle, Walther and Wurtzebach (1986), Gyourko and Kiem (1992) and Liu and Mei (1992). Further, the Value Line Index, which represents a broader stock market index than the S&P 500 and the NYSE composite indices, produced the strongest coherency with all the REIT indices, except the health REIT index. This supports the Han and Liang (1995) findings that REIT returns, which are relatively low capitalization stocks, should be compared to small stock returns.

Treasury instruments produce the fewest periods containing significant coherency with the REIT indices. The absence of significant coherency between the Treasury bill perpetual futures and the REIT indices indicate that they have little short-term or long-term influence on investors' pricing of REITs. However, the consistent occurrence of low frequency coherency between longer term debt instruments and the REIT indices imply that investors used a longer horizon to adjust REIT pricing to correspond with changes in interest rates on middle and long term debt instruments. Furthermore, the phase spectra show that REIT prices lag changes in interest rates by two to three weeks. The low frequency correlation and lags between debt and REITs may account for the results published by Gyourko and Linneman (1988) and Sagalyn (1990) that found significant co-movement in REIT pricing and interest rates.

In addition, the coherency values provide information on the relationship between REIT pricing and real estate returns. Martin and Cook (1991) use perfect market theory to argue that investors price REIT stocks based on the value of their underlying assets (i.e., equity in real estate holdings). Thus, REITs should represent a proxy for real estate returns. Since appraisal theory uses discount rates based on the term structure for estimating real estate values, REIT stock prices should co-move with interest rates if investors value REITs based on their real estate holdings. Yet both the time domain correlation in Exhibit 2 and the frequency based coherency in Exhibits 3-10 show that Treasury debt instruments are only weakly correlated with REIT pricing. Consequently, the coherency results support Myer and Webb's (1993) hypothesis that overall stock market movement has a greater effect on REIT pricing because REITs trade on the major exchanges. Certainly, some component of REIT pricing results from the value of the underlying real estate assets, but the high frequency input of the stock market signal makes it difficult to observe the real estate component of the REIT pricing signal without further filtering the data.

Still, the negative correlation between the pricing of the office index and the futures contracts (Exhibit 2) suggests counter-cyclical properties of the office REIT. Yet, the coherency spectrum (Exhibit 6) results in very few frequencies containing significant coherency. Although the phase spectrum shows a lag between the office index and the S&P 500 Index in the lower frequencies, they are not sufficiently negative (i.e., a -180 degree phase angle shift) to indicate significant counter-cyclical behavior from the office REITs. However, the poor performance of the office index appears to reflect the higher vacancy rates and poor performance of office building revenues during the study period. Thus, the severe conditions that exist in the office real estate markets during the study period appear to dominate investor pricing of REIT stocks.

The coherency spectra between stocks and REITs indicates that REITs share common cycles with stocks, while the phase spectra between stocks and REITs shows that neither signal lags the other. Hence, in a mixed stock portfolio, REITs would not provide an anti-cyclical hedge against inflation, which has been a characteristic attributed to real estate holdings in a mixed asset portfolio and supports the findings of Park, Mullineaux and Chew (1990) that REITs do not provide a hedge against inflation.

In summary, we use spectral analysis to discover common periodicities between REITs and financial futures contracts. Based on the evidence in the coherency spectra, stock indices have a dominate influence on REIT price movement in frequencies with a period of six weeks or less. The Treasury debt instruments appeared to contain partial influence over REIT pricing in frequencies with a period of thirteen weeks or more. Also, small company stock index futures share more common harmonics with REITs than the larger company stock index futures. Although two of the REIT indices (health and office) appeared to be influenced by market conditions specific to their industry, the strong co-movement between REITs and stock indices implies that REITs do not provide a hedge against inflation.

Notes

NAREIT lists only twenty-two diversified equity REITs.

2 Koopman's table uses a value, n, that equals one half the effective degrees of freedom. In this case n = .5 * 29.5 or approximately 15.

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Peter Oppenheimer* Terry V Grissom**

*Department of Financial Information Systems, Columbus State University, Columbus, GA 31907-5645 or oppenheimer_pete @colstate.edu.

**Department of Real Estate, Georgia State University, Atlanta, GA 30302-4020 redajz@langate.gsu.edu.

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