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A Frequency Domain Analysis of Common Cycles in and Related Sectors

Author Peijie Wang

Abstract This study examines cycles and common cycles in property and related sectors in the frequency domain. The findings indicate that property shares common cycles with a number of economic sectors and, in particular, with those sectors that are the user markets of property, and lags behind in business cycle phases. Property has large coherence at most frequencies with most economic sectors, but seems to have large discrepancy with them in the cycles at the annual frequency. The property market swings more severely than the economy as a whole. However, fluctuations in the property market are considered moderate relative to those in the housing market.

Introduction

This study examines cycles and common cycles in the property market and the economy using an econometric approach in the frequency domain. The common cycle is one type of common factors that have attracted much attention in contemporary econometric modeling. The other common factors include, prominently, the common trend and cointegration, which focus on the long-run comovement between two or more time series. While there have been several studies of common factor analysis involving property and other economic and financial variables, they exclusively adopt the cointegration procedure, and are predominantly on the cointegration relationship between direct property investment and indirect property investment. The latter is usually represented by Investment Trusts (REITs) in the United States and property company shares in the United Kingdom. Examples of such research can be found in Lizieri and Satchell (1997) and Wang, Lizieri and Matysiak (1997), among others. Currently, none have studied common cycles in property and other sectors in the economy in a modern business cycle framework, incorporating contemporary econometric modeling strategies. Lack of empirical research on common cycles of property and other sectors in the economy arises from the fact that it is hard to quantify or even qualitatively confirm the characteristics of the interaction using the traditional time domain method, which has motivated this study adopting a non-conventional approach in

JRER ͉ Vol. 25 ͉ No. 3 – 2003 326 ͉ Wang the frequency domain to discover common cycle features and patterns of interaction between property and economic sectors. The existing literature has highlighted the state of present research that recognizes a close link between property and the economy but is yet to go further to identify the patterns of association between them. Examining the price-income relationship, Bjorklund and Soderberg (1999) suggest that Swedish property market cycles may have been partly driven by a speculative bubble during the 1980s. The results of Wang (2000), derived from the analysis of capital value and rent relationships, suggest that there are no bubbles in the office, retail and aggregate property cycles in the U.K., but the existence of bubbles in the industrial property market cannot be ruled out. Pyhrr, Roulac and Born (1999) review extensively the literature and research relevant to property cycles. In their discussion of macroeconomic relevance of cycles, they summarize and make comments on research on cycles in the national economy, national economy linked to and macro property cycles. The study presents fundamental cycle concept and recognizes the relevance of phases in property supply and demand cycles, but stops short of establishing a link with the national economy and providing empirical results. The demand cycle leads the supply cycle, and the rate is found to be the best indicator of the phase of the cycle. Dokko, Edelstein, Lacayo and Lee (1999) develop a property cycle model linking property value and net operating income. Although there are no economic fundamentals, other than value and income, involved in their statistical equations, the relationship between value and income is the kind for the fundamentals. They claim that twenty office markets, that exhibit different cyclical behavior, may be represented by their three-parameter econometric specification, with the three parameters being for the value variable, the change in value and a time trend. Varied approaches are adopted in the investigation of property cycle behavior, e.g., Wheaton (1999) applies a theoretical stock-flow model incorporating agents’ expectations to demonstrate various cyclical features, and Grenadier (1995) examines the prolonged cycles, or persistence, in property markets. The study by Grissom and DeLisle (1999) is truly linked to the national economy and the financial market, using standard time domain regression with relevant augmentation. Included in the macro-financial market analysis are variables of GNP, the interest rate, unanticipated inflation, tax shelter and capital gains, all of them being contemporary, neither leading nor lagging property returns. Dividing the entire time period into several time segmentations, the role of these variables in explaining property returns vary. Further analysis of the results indicates that these variables and the relationships help distinguish property cycle stages and the relationships are stable in identifying cyclical changes. The most significant variables over the entire period to have an influence on property returns have been identified as changes in GNP and the interest rate, which reflects anticipated inflation. The sign of the coefficient for GNP is, reasonably, always positive. While the sign of the coefficient for the interest rate alters during different time segmentations, it is always the same as the coefficient for unanticipated inflation. Common Cycles in Property and Related Sectors ͉ 327

Clayton (1996) shows that the risk premium on Canadian property varies over time and is strongly related to general economic conditions. The study adopts the appraisal-based Morguard Property Index and Russell Canadian Property Index (MRCPI) and the indirect property investment on the Toronto Stock Exchange (TSE 300 real estate index) in the empirical investigation where the indexes are unsmoothed prior to statistical estimation. Using a VAR that includes the total return, income return and net operating income level of MRCPI and the TSE 300 real estate index, the study suggests that time variation in property risk is partly predictable, and thus can help forecast future movements in values. In an international setting, Renaud (1997), prompted by the phenomenal effect of the globalization of financial markets on property markets around the world, documents the international and domestic factors that contributed to this strong global property cycle. The above analysis of the literature indicates that extensive efforts have been made in recent years to examine property performance in association with the economy and financial markets. However, while a few do so in establishing some kinds of links between property performance and macroeconomic variables and using the latter to help explain the former, a fair portion of the studies are still confined within the property market itself. Nevertheless, the concepts and methodologies of these studies can be claimed to be those used in modern economic research based on fundamental relationships between economic variables, with the relationship between income and value being the most commonly referred one. All the statistical procedures used are time domain regressions, which, as will be discussed later, are not powerful when the variables are featured by cycles and, in a multivariate setting, by common cycles and phases. Subsequently, the results from the above-mentioned studies have little to offer with regard to the cycle components in the variables, which does not appear to be particularly encouraging when the study is intended to focus on cycles and common cycles. The findings of the above studies on property cycles are generally better explained and as expected when they are confined to the property sector itself than in a multi-sectoral setting. These findings have the following strands: (1) how cycles or fluctuations develop and whether they are bound by the price-income relationship; (2) what are the phase relations in demand cycles and supply cycles; (3) whether there exist identifiable stages of property cycles; and (4) whether macroeconomic variables help explain property cycles and performance. The first two sets of investigations and associated findings, though multivariate, involve only property variables that represent the fundamental economic relationship as in the former, and the market mechanism and process as in the latter. The third set of investigations is specific applications of business cycle phases to property, typically including peak, trough, declining and recovery and their evolution processes. Only the fourth set of investigations links property returns and cycles to macroeconomic variables and attempts to explain the former with the help of the latter. The present study goes beyond establishing a link between property and the economy. It attempts to identify common cycle characteristics and patterns in the

JRER ͉ Vol. 25 ͉ No. 3 – 2003 328 ͉ Wang interaction between property and the real sectors of the economy, covering the whole spectrum of short, medium and long cycles and the phase relations. Moreover, the article pays attention to the specific relationship of individual economic sectors with property in theoretical analysis and empirical investigation, since the roles of the economic sectors as the user market or supplier differ substantially. The empirical investigation is further empowered by the frequency domain method to achieve the set objectives effectively. Overall, this empirical study contributes to the literature through opening up a new channel of research to gain knowledge in such important aspects of property performance that is either overlooked or unable to be quantified previously and, therefore, advances our understanding about the property market and its position in the economy. Common cycle analysis is, in a sense, an extension of common trend analysis. Common cycles differ from common trends in that the phase matters in the former; whereas there is no role for it in the latter. With common trends and cointegration analysis, one of the can be stated as: if xt is cointegrated with yt, then xt is also cointegrated with ytϪ1. However, this does not hold in common cycle analysis. It is possible that xt and yt have no common cycle but xt and ytϪ1 share a common cycle. This nature is important because usually economic fluctuations and movement are not in the same phase. In particular, property exhibits noticeable cyclical characteristics but may not be in the same phase as aggregate economic fluctuations. Therefore, considering phases in common cycle analysis would have profound implications in property research. This can be more effectively achieved adopting analysis in the frequency domain, or spectral analysis, as the usual time domain methods are empirically difficult to implement for these types of common cycles. In fact, spectral analysis is particularly useful and easy to understand regarding cycles and their phases. Common cycle analysis is important to property in that property is featured by cyclical behavior that exhibits phenomenal fluctuations. Moreover, fluctuations in property originate not only in the property market itself, but also in some sectors in the economy. There are interactions between property and the economy— property is influenced by and influences other economic and financial sectors, in one way or another. Common cycle analysis of property, therefore, attempts to identify the patterns of cyclical movement in the property market, and establish how these patterns fit into the business cycle in the economy. This article is organized as follows. The next section presents the rationale of common cycle analysis between property and the real sectors of the economy. It evaluates the expected interactions between property and the economic sectors with possibly predictable outcome in a common cycle analysis framework, and shows how the approach of this empirical study may enhance our understanding of property cycles in a national economy. It is followed by a brief discussion of the frequency domain approach to the analysis of common cycles, in comparison with the generally adopted common cycle concept in the time domain. Then the article progresses to examine common cycles in property and related sectors and makes an empirical inquiry into issues on common cycles in the U.K. property Common Cycles in Property and Related Sectors ͉ 329

market and the economy, and reports estimation results and findings that augment our knowledge and the empirical literature. The final section is the conclusion.

͉ Rationale of Common Cycle Analysis for Property and the Economy In recent years, a few researchers have adopted analysis of direct property investment performance in relation to indirect property investment on the stock market. Although the results challenge the early perception or judgment that the performance of property has little to do with that of other financial market investments, and thus cannot be explained by or predicted via the latter, the results are somewhat mixed. This type of approach is motivated by the search for an easy way to describe and explain property performance and property market behavior in the absence of reliable transaction-driven market data. However, analysis has been restricted to stock market investment. One of the reasons behind this type of analysis is that property companies are one of the major players in the property market. They are actively engaged in property development, investment and trading. The activities of large companies usually cover all of these three aspects, while the small ones are mostly involved in trading only. For example, Land Securities, British Land and MEPC groups all have a diverse portfolio of offices, shopping centers, out of town retail parks, food superstores, industrial and warehouse buildings in the U.K., invest in all kinds of commercial properties, and are additionally involved in property trading and development. As property companies, as described above, develop, invest in and deal with properties, property company shares are therefore considered surrogate for property. They also provide a channel for investment in property for the investors, who either lack large financial resources or management skills, to invest in property directly. To invest in property company shares allows people to ‘‘buy’’ property piecemeal. In addition, property company shares are highly liquid, and can normally be traded many times in a single day; in contrast to real property trading that can take several months to complete a transaction. Due to these factors, property company shares are financial vehicles providing a medium for indirect investment in property. This indirect investment, while enjoying liquidity and divisibility, should reflect the performance of the underlying property in some ways. Nevertheless, although the value of a property company’s shares bear some relationship to the value of the property it owns and the income it derives from property operations, the link is not so straightforward. First, property company shares are stock market investment. They will fluctuate on the market, depending on the value of property owned; but the stock market will also tend to discount expected economic and financial events before they happen, while valuations may lag the events. Therefore, property company shares may lead property in time sequence. Nevertheless, the two sharply dissimilar types of investments inevitably exhibit differential performance. This is particularly true in timing of returns and

JRER ͉ Vol. 25 ͉ No. 3 – 2003 330 ͉ Wang cash flows. This means that the two investments have little short-term resemblance, as reported in a number of studies. Second, property companies are leveraged, and the effect of leverage is higher volatility in property company shares. Third, dividend policy tends to induce additional volatility with regard to uncertainty in the timing and amount of cash flows, as observed empirically. Fourth, tax regimes also affect the profit and performance. While property company shareholders are double taxed on the one hand, they do enjoy a corporate tax shield on the other due to financial leverage that is typically high in property companies. The combined effects could benefit or disfavor investors, depending on the individual companies and investor utilities. So it is a natural extension of empirical research to look beyond indirect property investment on the stock market and consider property’s interaction with the activity in the real sectors of the economy. In the following, the interaction that may exist between property and the economy is briefly analyzed, with specific reference to cycles and common cycles. Construction is by all means relevant to property development. However, the relationship may not be clear using conventional statistical methods, due to timing differences. It is known that there are time lags or leads in construction output with reference to property activity, but the lag or lead may not be the same in different cycles (e.g., a semi-annual cycle and an annual cycle). This blurs a probable relationship linking property to construction. It is expected that construction output would adjust and respond to property performance. Nevertheless, due to the complexity in the building process, how and how long it may take for construction output to respond to property market performance is an empirical matter. In contrast to construction, the other major sectors of the economy in this study are the users of property. It is expected that these sectors would lead property in business cycles one way or another when property adjusts to the demand of these sectors. One of the largest user markets of commercial property is the services sector, so there is an imbedded connection between services and property that might also be strong. Since both the services sector and commercial property for services, retail property and office property, are comparatively easier to adjust in the economy, a lead/lag relation, if it exists, would be short and take place at the short cycles (high frequencies) end. The link between property and the production/manufacturing sector is through the latter’s use of industrial property. Relative to the services sector and retail and office property, the players involved here are illiquid and inflexible to make adjustment in the short-term. Consequently, the production/manufacturing sector’s lead over property would be longer than that of the services sector and in the medium cycles range. From the viewpoint of economic fundamentals and economic agents’ adjustment to changes or expected changes in the fundamentals, most economic sectors would be expected to share a larger coherence in lower frequency cycles (longer run) as Common Cycles in Property and Related Sectors ͉ 331

they tend to move together in the long-run, as well as in higher frequency cycles (shorter run) as they response to the change in and the news about the fundamentals. I n the middle range, sectors may depart from each other as they have yet to be bound by economic fundamentals in their respective adjustment to the change in and the news about the fundamentals. There are, nonetheless, variations and leads/lags among sectors in such cyclical movements and adjustments that reveal the characteristics and patterns required in specific and individual empirical studies—the present study is one among them with specific relevance to common cycles involving property. The advantages of estimation in the frequency domain are that the characteristics and patterns discussed above are not readily identifiable by time domain regression analysis. Prior to presenting and discussing the frequency domain approach to common cycles, it is worthwhile having some basic sense of this approach and the rationale for its adoption in this empirical study. Consider two time series of economic activities that consist of the components of the quarterly cycle, the annual cycle and the bi-annual cycle. If the two time series have the annual cycle in the same phase, i.e., without lead or lag, but the first time series has a one-quarter lead over the second in the quarterly cycle and has a three-quarter lag over the second in the bi-annual cycle, what results would be expected in a traditional regression analysis in the time domain? Probably none of the regression coefficients at lag zero, one or three is significant. Even if one or several coefficients have been estimated, traditional time domain regression only tells, for example, that a change in the first time series is caused by the change in the second time series three periods earlier. It, however, does not tell the characteristics of the association between the two time series. Consequently, the coincident link in the annual cycle, and the lead/lag relations in the quarterly cycle and the bi-annual cycle can be overlooked, leading to a possibly wrong conclusion. The frequency domain approach in this study attempts to identify such relations, which is especially effective for research in cyclical fluctuations featured by the property market. ͉ The Frequency Domain Approach Spectral analysis, or studies in the frequency domain, is one of the unconventional subjects in time series econometrics. Analysis in the frequency domain does not bring in new or additional information, it is simply an alternative method with which information is observed, processed and abstracted. This is sometimes helpful. Depending on the characteristics of the issues, analysis in one domain may be more powerful than in the other. For example, cycles are better and more explicitly observed and represented in the frequency domain. Correlations in the time domain and cross spectra in the frequency domain deal with the relationship between two time series from different perspectives and have defined links.

Common Cycles in the Time Domain

The success and popularity of research in common trends and cointegration have generated much research interest in other types of common factors, common cycle

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analysis being one of them. Recent work on modeling common cycles can be found in Engle and Kozicki (1993), Vahid and Engle (1993, a,b), Lippi and Reichlin (1994), Forni and Reichlin (1995), Gallo and Kempf (1995), Engle and Issler (1995) and Wang (1999). The idea of common cycles bears a remarkable similarity to that of common trends and cointegration. But, while one lag/lead (phase-shifting) in one time series does not change common trend and cointegration relations, it does affect the way in which a combination of cycles

behaves. In the simplest case, if time series xt and yt have a common trend, a linear combination of xt and ytϪ1 will have a common trend as well (c.f., Engle and Granger, 1987). But this is not the case for common cycles. Indeed, it is because the phase plays an important role in common cycles. Nevertheless, phases cannot be exactly replicated by leads/lags in the time domain, so the task of modeling and analyzing common cycles has to be in the frequency domain1 empirically.

Common Cycles in the Frequency Domain: Phases and Coherence

Next, the concept of cross spectra, coherence and phases, based on the Fourier transform in the frequency domain. The basics of the Fourier transform is provided in Appendix 1. The cross spectrum of the two time series is:

NϪ1 ϭ ͸ ␶ Ϫj␶ (2␲k/N) hX,YX(k) Cov ,Y ( )e , (1) ␶ϭϪ(NϪ1)

␶ ϭ ⌬ Ϫ ␮ ⌬ Ϫ ␮ ␮ ϭ ⌬ ␮ ϭ ⌬ where CovX,Y ( ) E{( Xt X)( YtϪ␶ Y)}, X E{ Xt} and Y E{ Yt}. ␶ Cov( ) is in general not an even function, so hX,Y (k) is in general a complex number:

2␲k 2␲k h (k) ϭ c(k) cosͩͪ␶ ϩ jq(k) sin ͩͪ␶ . (2) X,Y NN

Unlike the univariate Fourier transform where the imaginary part is zero, the cross spectrum has both magnitude and phase as follows:

m(k) ϭ ͙c22(k) ϩ q (k) (3) Common Cycles in Property and Related Sectors ͉ 333

and

q(k) p(k) ϭ tanϪ1 . (4) c(k)

Equations (3) and (4) are called magnitude spectrum and phase spectrum ␶ respectively. It can be seen, from the above analysis, that if CovX,Y ( )isaneven function, then the phase spectrum is zero (i.e., there is no overall lead of series

Xt over series Yt, and vice versa). With Equations (3) and (4), the cross spectrum can also be expressed as:

ϭ jp(k) hX,Y (k) m(k)e , (5)

so that both magnitude and phase are shown explicitly. Another measure of the closeness of two time series is coherence, defined, in a very similar way to the correlation coefficient, as:

ϭ hX,Y (k) CohX,Y (k) 1/2 1/2 . (6) hX,XY(k)h ,Y (k)

If a comparison of the measures in the frequency domain is made with those in the time domain, then the cross spectrum of Equation (2) corresponds to the covariance in the time domain, which is not standardized; the coherence as with Equation (6) corresponds to the correlation in the time domain, which is standardized by the square roots of the two time series’ spectra and the two time series’ standard deviations respectively; and the phase of Equation (4) addresses leads and lags. The closeness of two time series is straightforwardly observed with the standardized measures of coherence, together with the phase measure, which was adopted in this study. Typical lead/lag relations are shown in Exhibit A1 in Appendix 2 with interpretations.

͉ Frequency Domain Analysis and Presentation of Cycles and Common Cycles in Property and the Economy A range of economic and financial activities were selected to investigate common cycles shared by property and the economy. These include the GDP sectors, the money supply, the leading indicators, unemployment and the price. The GDP sectors include the aggregate GDP itself and its three major components:

JRER ͉ Vol. 25 ͉ No. 3 – 2003 334 ͉ Wang construction output (CO), industrial production (PDN) and the services sector (SVC). The agricultural sector is excluded, as the Jones Lang Wootten (JLW) Index,2 which is used in this study, covers virtually no farmland. In addition, there is manufacturing (MNG), as it is one of the most important components of the production sector, influenced by general economic conditions rather than other non-economic events (oil is one of the examples, utilities another). The money supply variable is the narrowly defined money M0.3 The unemployment variable is the unemployment rate (UER). There are leading indicators of the long lead (LL), short lead (SL), coincident (CC) and lag indicators (LG); and the coincident indicator is used in the empirical study. The house price is represented by the Nationwide Building Society house price index4 (NTW). Property is represented by the JLW Total Return Index (JLW). All of these data are quarterly, running from the second quarter of 1977 to the second quarter of 1993 with sixty-five observations. The data set can form thirty-two quarterly alternations, eight annual alternations and two alternations at a four-year frequency, so it performs reasonably in frequency domain analysis. As the JLW Index is used throughout this study to represent property performance, it is helpful to introduce some institutional background and the construction of the index. The JLW Property Index was launched in 1977 and it is the longest property index available in the U.K.5 The properties in the JLW Index are drawn from twenty different funds, none of which account for more than 20% of its overall portfolio. These are funds that JLW values, advises or manages. The Index consisted of 179 properties as in March 1998, of which 49% are offices, 31% retail, 19% industrial, and the remaining are farms and miscellaneous. The value of these properties was £457 million and that of the funds was £3.01 billion as of March 31, 1998. The JLW Index is appraisal- or valuation-based; therefore, there is a smoothing problem. The JLW Index is unsmoothed6 before it is applied to empirical investigations.

Tests on Common Cycles in the Time Domain

The results from common cycle tests are reported in Exhibit 1, using the unsmoothed JLW series to represent property investment. The instrumental ␣ ⌬ variable (IV) method is used to estimate the coefficient in a relation that ( y1t Ϫ ␣⌬ ⌬ Ϫ ␣⌬ y2t) has no cycles. This means ( y1t y2t) is the white noise residual and ⌬ ⌬ has no serial correlation with lagged y1t and y2t, and with the cointegration Ϫ residual at t 1ify1t and y2t are cointegrated. The coefficient from direct ⌬ ⌬ ⌬ regression of y1t on y2t would be biased, as y2t is correlated with the current period innovation or residual. The instruments used are the first and second lags of the JLW series and the other variable in the pair, plus the first lag of the cointegration vector if there is a cointegration relation. Two-stage least squares is a similar method. ␣ ⌬ There are four statistics reported. , the coefficient of the other variable ( y2t)in the regression (the coefficient of JLW is set to one); a significant ␣ suggests a Common Cycles in Property and Related Sectors ͉ 335

Exhibit 1 ͉ Common Cycle Tests Using the IV Method

Series ␣ F-Test ␹ 2 Test Q

GDP 2.4395 2.8848 8.8887 23.8026 (0.0087) (0.0222) (0.0029) (0.0939) CO Ϫ0.0814 3.6716 9.5803 42.8476 (0.6448) (0.0062) (0.0020) (0.0003) NTW 0.7103 1.0086 3.0500 19.7933 (0.0001) (0.4217) (0.0807) (0.1800) SVC 2.5033 0.7650 2.2982 15.6771 (0.0000) (0.5791) (0.1295) (0.4757) PDN 1.0038 2.0996 4.5010 18.5485 (0.0465) (0.0795) (0.0339) (0.2928) MNG 0.9347 1.5394 0.2551 19.7070 (0.0151) (0.1931) (0.6135) (0.2337) UER 0.0010 3.3809 9.0456 41.9067 (0.0306) (0.0152) (0.0026) (0.0004) M0 0.7684 1.9423 5.1160 29.4691 (0.0095) (0.1024) (0.0237) (0.0210) CC 0.0009 2.9531 4.1912 34.3338 (0.2604) (0.0278) (0.0406) (0.0049)

Note: Standard errors are in parentheses.

relation or correlation, though may not necessarily a common cycle relation, exists ⌬ ␹ 2 between property and y2t. Both F-test and statistics are used to check the existence of common cycles, or the cancellation of cyclical components, which is suggested by the insignificant test statistic. The combined series is also examined against the serial correlation with the Ljung-Box Q statistic—no correlation with the lagged variables is equivalent to no serial correlation in the combined series itself. In Exhibit 1, JLW’s common cycle relationship is clearly found to be with the house price, the services sector and the manufacturing sector, with very low insignificant levels for F, ␹ 2 and Q, and a very significant ␣. Property seems to be more cyclical (i.e., the magnitude of its cycles are larger than the service sector with ␣ being 2.5033). Put another way, the magnitude of cycles in property is about 2.5 times that of the cycles in the services sector. But the cyclical fluctuations in property are less than those in the housing market suggested by the ␣ coefficient of 0.7103. The magnitude is about the same for property and the manufacturing sector. The existence of common cycles between property and the money supply and between property and total production is marginally confirmed. In the case of total production, the Ljung-Box Q statistic is the criterion, but recall

JRER ͉ Vol. 25 ͉ No. 3 – 2003 336 ͉ Wang that the PDN series is much more white than the MNG series, partly due to the aggregation, this result should be viewed with caution. With the money supply, only the F-statistic marginally accepts the existence of common cycles, and property is relatively less cyclical than the money supply variable. There is no common cycle relationship found between property and the GDP series. This is not to rule out the cyclical co-movement of property with GDP, because the aggregation in GDP has reduced or phased out the fluctuations in the GDP index in general, and the GDP series is rather white in this given short period in particular. In a sense, investigations at the sectoral level are helpful, not only in the sectoral analysis itself, but also in inferring implications for some economic aggregates. Quite beyond imagination, if not surprisingly, property shares no common cycles with the construction sector, although the existence of common trends or long-run co-movement between them is so evident. The series of coincident leading indicator, GDP and total production, lacking a common cycle relationship with property, have a very clear correlation with property. One should notice that, theoretically and empirically, the conditions for common cycles are rather less possible to meet than those for common trends or cointegration, as the former requires that the components in two series are proportional at every frequency of their cycles; whereas in the latter, the proportionality requirement merely applies to the zero frequency component plus some elements very close to zero frequency (in fact it is these elements that would decide a cointegration relation, otherwise two I(1) series would always be cointegrated). Therefore, while many economic time series variables have common trends and are cointegrated, not so many have common cycles. Put another way, there are many different paths to reach a certain level of activity, any the path different from a pure random walk path are cycles or fluctuations. So, there could be many different cycle patterns; even two time series are bound to move together in their levels.

Frequency Domain Analysis of Common Cycles

In Exhibit 2, there is about 80% coherence in most of the cyclical components from quarterly to semi-annual cycles, but the cycles around the annual frequency have little in common in the two series. Property and GDP are in the same phase over nearly the whole spectrum except property lags in the annual cycle. As the phase statistic is not on a straight line in Exhibit 2a, they are not purely coincident nor phase-shift with a single order, so time domain techniques are not effective. There would be many statistics for different leads in the time domain, few of them would be statistically significant individually, while a possible joint significant statistic does not tell anything about individual leads and how big the lead is at each cycle. It has been noticed that the time domain analysis on common cycles between property and GDP suggests that such characteristics probably exist. The coherence of property with construction is about 0.5, with no regular phase patterns, as shown in Exhibit 2b. The higher coherence happens with the quarterly Common Cycles in Property and Related Sectors ͉ 337

Exhibit 2 ͉ Coherence and Phase: Property with Other Variables

(a) (b)

(c) (d)

cycle that seems to be in the same phase, and the annual cycle with rather large leads of property over construction. Overall, no single order phase shift (and even two or three combinations) can describe their phase relations, resulting in a straightforward rejection of any common features in cycles in the time domain. In fact, property and construction do share some common cyclical movements or fluctuations. Property and housing, as portrayed in Exhibit 2c, have higher coherence at the lower frequencies, close to 0.8 at around the bi-annual cycle, and the value is about 0.5 at the higher frequencies around the quarterly cycle. They lack coherence at the annual frequency, and large phase difference also occurs at this frequency. Observing Exhibit 2d, the services sector and property have a large coherence of about 0.8 at both the higher and lower ends of the frequencies and the two series are in the same phase at these frequencies. The services sector leads property (compared with Exhibit 2 bÐd) in the semi-annual and annual cycles to the extent of one to two quarters, but the value of their corresponding coherence is relatively small. As seen in Exhibit 2e and f, the patterns for JLW with the production and manufacturing sectors are similar, and again, property and these sectors are in the same phase at the lower and higher ends of the spectrum, and the production

JRER ͉ Vol. 25 ͉ No. 3 – 2003 338 ͉ Wang

Exhibit 2 ͉ (continued) Coherence and Phase: Property with Other Variables

(e) (f)

(g) (h)

sector and the manufacturing sector have a lead of three to four quarters over property at the semi-annual, annual and bi-annual frequencies. As the services and production sectors constitute a large part of GDP, property’s coherence and phase relations with these sectors should be reflected in the relations with GDP as well. In fact, property’s phase relations with GDP are reasonably consistent with the mixture of these sectors, and so is the coherence. One interesting and common feature being found in this empirical study is that property seems to have large discrepancy with almost all the other sectors in the cycles at the annual frequency, with less coherence and larger phase leads/lags. Property’s coherence with the unemployment rate, shown in Exhibit 2g, is found largely at quarterly or semi-annual frequencies, beyond that, there is little in common. Property seems to lag behind unemployment for about a third of a complete phase, amounting to a month with the quarterly data; this is by no means quite accurate when the quarterly data have to be used. A large phase difference in the semi-annual cycle does not count much as the coherence at that point is so small. With the coincident leading indicator, property appears to be in the same Common Cycles in Property and Related Sectors ͉ 339

phase, as revealed by Exhibit 2h, except in the very low frequency cycles (annual and lower). But the phase lag to the vertical axis just accounts for a very small portion, when the spectrum is considered to be continuous, not discrete. Since these two series are (regarded) stationary in their original form, a big difference with all other series emerges at the zero frequency: the coherence is about zero. In Exhibit 2aÐf, the coherence at the zero frequency (the unit root circle) is about their cointegration vector (1, ␤), with ␤ being close to the coherence at the zero frequency. Taking JLW and NTW, for example, if coherence and phase are analyzed with the variables in levels instead of differences, then the coherence is about 0.9 (the value at the zero frequency in Exhibit 2d) and the phase is zero over the whole horizontal axis. This also holds when the pair is JLW and the lagged NTW, and simply means that if one series is cointegrated with another series, then it is also cointegrated with the lagged series. Therefore, phase does not play a role in cointegration but it matters in common cycle analysis. This comparison may not be appropriate but does point out that cointegration, or the verifying of a cointegration relationship, is not always necessary and rarely reveals something, at least in many studies carried out in the last decade.

͉ Conclusion In this study, cycles and common cycles are examined in property and related economic sectors. Common cycle analysis on its own is an extension of common trend analysis. But common cycles differ from common trends in that the phase matters in the former, therefore analysis is more complicated, and sometimes, rather difficult. Nevertheless, on the methodology side, the empirical investigation in this study has been reinforced by the frequency domain analysis method, which is more effective in presentation when phases are concerned. On the empirical property research side, the study has contributed to the existing literature by extending common trend analysis to common cycle analysis involving property on the one hand, and extending property cycle analysis to common cycle analysis in property and related sectors on the other hand. The study has identified common cycle characteristics and patterns in the interaction between property and the related economic sectors, covering the whole spectrum of short, medium and long cycles and the phase relations. It opens up a new channel of research in our continuing search for knowledge and understanding about property markets. Due to these distinctive features, this study is able to examine property cycles more effectively and in a fundamental way in relation to the real sectors of the economy, compared with previous research. Consequently, the findings of this study enhance our knowledge in property cycles and contribute to the literature in four aspects. First, while recognizing the importance of value-income relationships in property markets in the existing literature, the present study envisages that fluctuations in both value/price and income are subject to the

JRER ͉ Vol. 25 ͉ No. 3 – 2003 340 ͉ Wang influence of the macroeconomic environment in which the property sector operates. Therefore, the association between property and the related economic sectors in their common cycle interactions is a practically more relevant matter than the value-income relationship within the property sector and should be examined in detail. Second, while recognizing the relevance of demand, supply and their phase differences in property markets and property cycles in the existing literature, the present study investigates the activity of those sectors that constitute the user market and the supplier of property in a common cycle analysis framework, instead of investigating the demand and supply themselves. In other words, the causes to, and the sources of, demand and supply cycles that eventually surface as property demand and supply at a later stage are examined. Third, while recognizing the usefulness of stages or phases in property cycles and their evolution paths in the existing literature, the present study scrutinizes phase leads/ lags between property and the related economic sector in short, medium and long cycles, instead of inspecting the stages or phases and the evolution path of the property cycle itself. This is particularly relevant and useful in property development and related investment activity in the real world. Finally, while recognizing the significance of establishing a link between property and the economy in the existing literature, the present study goes further to identify common cycle characteristics and patterns in the interaction between property and the real sectors of the economy and pays attention to the specific relationship of individual economic sectors with property in theoretical analysis and empirical investigation. It has been found that property fits into the business cycle well and has the long- run comovement with most parts of the economy. It is because property is not a purely financial market investment. It is mainly an investment in production, trading, work and storage spaces and capacity from the point of view of most companies or industries. Property has large coherence at most frequencies with most economic sectors and the economy as a whole and is in the same phase at these frequencies with the latter. But, property seems to have a large discrepancy with almost all the other sectors in the cycles at the annual frequency, with less coherence and larger phase leads/lags. Though time domain analysis does not have the capability to reveal these features and thus is unable to lend support to these results, it does have an additional finding: due to the reason that amounts of available property cannot be increased or reduced quickly and easily, the magnitude in property cycles is larger than that in the business cycle for the economy as a whole and for some sectors. In particular, property shares common cycles with the residential property sector, the services sector and manufacturing sector. Combining the analyses in the frequency domain and the time domain, the findings suggest that adjustments in the property market are more sluggish than those in the economy in general, and in the services sector, the most liquid part of the economy, in particular; while they are actively compared with the residential property sector, which could be attributed to the existence of an indirect investment market for commercial property that reduces the cycles in the direct property investment market. Common Cycles in Property and Related Sectors ͉ 341

The study has the following implications. First, the findings suggest that prediction of cycles in the property market could be improved by analyzing cycles in other related sectors, as property and the other sectors in the economy share common cycles. The advantages of a joint analysis of property and the economy are the use of more information from other sectors, and the recognition of the underlying mechanism driving cycles and common cycles. This practice may help mitigate the fluctuations and the magnitudes of cycles in property. Second, it is beneficial to watch the performance of those sectors that are the user market of property and, in particular, the movement of their underlying cycle components, so the patterns of interaction that helps present a sensible outlook, instead of blurred picture, can be observed. Third, the development of an indirect investment vehicle may help smooth out fluctuations in a related sector, as the empirical findings have suggested with regard to the commercial and residential property markets. Although the commercial property market exhibits more significant cyclical behavior than most part of the economy, it fluctuates less severely than the residential property market, due partly to the existence of an indirect investment market for commercial property. The above findings and implications will benefit practitioners as well. It has already been recognized in the previous research that studies of property performance should not be confined in the property market itself. What are more relevant currently are that studies of property performance should also go beyond of establishing a link between property and the economy, and further identify common cycle characteristics and patterns in the interaction between property and the individual and respective sectors of the economy, paying attention to short, medium and long cycles and the phase relations. As the results of the study suggest that property lags the services sector and the production/manufacturing sector to varied degrees in different cycle components, it would be beneficial for practitioners to scrutinize the performance of these sectors in their property investment decision-making processes and the timing, though the task is not always easy. Moreover, practitioners in other areas may also take the advantages. For those in the construction sector, they would be better positioned if they inspect the specific interaction between property and construction prudently. For those who use property, they may benefit from their extended intelligence in property- related sectors to make sensible and timely moves in their business adventures that, nevertheless, requires additional knowledge in the working of the economy.

͉ Appendix 1 ͉ The Fourier Transform and Spectra Only the discrete Fourier transform is presented here. For a discrete time series Ä(n) with N observations, the discrete Fourier transform (DFT) is:

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NϪ1 F(k) ϭ ͸ Ä(n)eϪjn(2␲k)/N ), (A1) nϭ0 and the inverse discrete Fourier transform (IDFT) is:

1 NϪ1 Ä(n) ϭ ͸ F(k)e jk(2␲n / N ), (A2) N kϭ0 with ⌬␻ ϭ 2␲/N, F(k) ϭ F(k⌬␻) ϭ F(2␲k/N). That is, time domain series can be expressed with different frequency components. Equation (A1) is the energy spectrum. In the case of stochastic processes, the Fourier transform is concerned with the power spectrum or the power spectral density function (which can be simply called spectral density function when there is no confusion).7 The spectral ⌬ ϭ Ϫ ϭ density function of a discrete random process Xt Xt XtϪ1 (t 1,...N) is:

NϪ1 h(k) ϭ ͸ R(␶)eϪj␶ (2␶k / N ), (A3) ␶ϭϪ(NϪ1)

␶ ⌬ ␶ ϭ ⌬ Ϫ ␮ ⌬ where R( ) is the autocovariance function of Xt, i.e., R( ) E{( Xt )( XtϪ␶ Ϫ ␮ ␮ ϭ ⌬ )} and E{ Xt}. The inverse Fourier transform of Equation (A3) is:

1 NϪ1 R(␶) ϭ ͸ h(k)e jk(2␲␶ / N ). (A4) N kϭϪ(NϪ1)

Setting ␶ ϭ 0 in Equation (A4) produces:

1 NϪ1 ϭ ⌬ 2 ϭ ͸ jk(2␲␶ / N ) R(0) E{( Xt)} h(k)e . (A5) N kϭϪ(NϪ1)

It is the mean squared value of the process and has the meaning of power of the process, so equation (A3) is called the power spectrum. Equation (A1), in contrast, is the energy spectrum as it has the features of electrical current or voltage. R(␶) usually takes real values and is an even function, i.e., R(Ϫ␶) ϭ R(␶). Accordingly, the spectral density function can be written as: Common Cycles in Property and Related Sectors ͉ 343

NϪ1 2␲␶k ϭ ␴ 2 ϩ ͸ ␶ ͩͪ h(k) X 2 R( ) cos (A6) ␶ϭ1 N

͉ Appendix 2 Phase Lead/Lag Relations In Exhibit A1 four examples are drawn to show typical leads ranging from one phase lead to four phase lead, using one time series (GDP in difference) against its own lags. The left hand axis is for coherence and the right hand axis for phase. Coherence is symmetric about the vertical axis and phase is symmetric about the origin point, therefore only the right half of the graph is displayed. The horizontal axis ranges from zero to ␲Ä (the other half from Ϫ␲Ä to zero) with 2␲Ä being one cycle (one quarter in this case). At the half point of the horizontal axis the frequency is lower at Ä/2 and a complete cycle is in two quarters, at the frequency of Ä/4 the cycle is annual, and so on. A value of one for coherence at a particular

Exhibit A1 ͉ Coherence and Phase—An Example (GDP with lagged GDP)

(a) (b)

(c) (d)

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point means the two series are altogether in common at that frequency or cycle; if coherence is one over the whole spectrum then the two series are common at all frequencies or cycles. If there is no phase lead/lag, the line for phase would be zero over the whole range in the horizontal axis. If there is a strict one phase (period) lead at every frequency, then it would be a linear line as depicted in Exhibit A1 a. A value of one means an 180Њ (␲) lead in half cycle (as the other half is not displayed). A time series has perfect coherence with itself so the value of coherence is one over the whole spectrum. The phase statistic is one (180Њ) lead for the quarterly frequency in half quarter, equivalent to 360Њ in one quarter; at the half point on the horizontal axis, the phase is 0.5 (90Њ) lead for the semi- annual frequency that is also one quarter (1/4ϫ2quartersϫ2). So, as long as the line for phase statistic is linear, every frequency component will have the same lead; if the line is not linear, then some frequency components would have longer or shorter leads than others, and would not be a purely, say, one phase lead. Exhibit A1 b is for two phase lead, its coherence is the same, the phase is twice as big as in Exhibit A1 a,asϪ1 is the same as ϩ1 (360Њ ϩ (Ϫ180Њ ϭ 180Њ)), and the negative part of the phase line can be continued upwards at the point (0.5f, 1). In Exhibit A1 c, the phase is three times as big as that in Exhibit A1 a, and in Exhibit A1 d, four times as big. As can be seen, the coherence becomes more distorted, as the series is not ideal (from Ϫϱ to ϩϱin time).

͉ Endnotes 1 Or in other transformations, e.g., the Laplas transform and the z transform. 2 It is now the Jones Lang LaSalle Property Index. 3 M0, the narrowly defined money, is chosen as the money supply variable in this study. The reasons for using M0 instead of M4, the broad money supply, are empirical. There was a big break in the M4 series in the fourth quarter of 1981, caused by the switch between the old banking sector and the new monetary sector. In July 1989, Abbey National’s conversion to a public limited company caused minor breaks in the M0 series and major breaks in the M4 series. Although the first breaks in the fourth quarter of 1981 were removed from the changes in M4, the removal of the breaks in the changes in M4 resulted in as much distortion as the retaining of the breaks in M4 levels. Besides these breaks, the M0 and M4 series had a similar pattern. Beyond the concern in breaks, M0 is more liquid and more public sensitive in representing the demand factors, separated from the supply factors or real factors. 4 Although the Halifax Building Society (converted to a bank in 1998) House Price Index has the widest coverage in the U.K., its quarterly index only started in 1983. 5 The other major U.K. property index, IPD (the Investment Data Bank), although with wider coverage, did not start compiling the monthly index until 1987; it was annual before 1987. 6 Blundell and Ward (1987), Firstenberg, Ross and Zisler (1988) and Ross and Zisler (1991) were the earliest studies that raised the issue of smoothing in appraisal-based property indices and proposed approaches to correcting such indices. More recent Common Cycles in Property and Related Sectors ͉ 345

research includes Giaccotto and Clapp (1992), Geltner (1993), Shilling (1993), Barkham and Geltner (1995) and Wang (1998), to mention a few. The early research on the issue, concerned by a substantially lower standard deviation in property return indices relative to that in the returns on other financial market investments, adopted an approach, which assumes a random walk in the property return process, to correcting or unsmoothing the indices. However, more recent studies have pointed out that, while there exists smoothing in property return indices, the property return process does not necessarily follow a random walk. Therefore, on the one hand, the valuation-based property index should be corrected or unsmoothed to get the true standard deviation in its return or the true risk associated with property investment. On the other hand, the index should not be ‘‘fully’’ unsmoothed, assuming a random walk process in property returns, which would have exaggerated the standard deviation in the returns on property. Almost all recent studies have recognised and accepted the stance that valuation-based property indices should be unsmoothed to a right extent. The difficulty and difference are, then, how to decide an unsmoothing factor reasonably. This study adopts a smoothing factor ␣ ϭ 0.6241 from Wang (1998), which is an empirical study on the same JLW Index. The fully unsmoothing procedures can be found in most early studies, e.g., Blundell and Ward (1987) and Firstenberg, Ross and Zisler (1988). 7 This can also be the product of the Fourier transform and it’s conjugate. ͉ References Barkham, R. D. and D. Geltner, Price Discovery in American and British Property Markets, Journal of the American Real Estate and Urban Economics Association, 1995, 23, 21Ð44. Bjorklund, K. and B. Soderberg, Property Cycles, Speculative Bubbles and the Gross Income Multiplier, Journal of Real Estate Research, 1999, 18, 151Ð74. Blundell, G. F. and C. W. R. Ward, Property Portfolio Allocation: A Multi-Factor Model, Land Development Studies, 1987, 4, 145Ð56. Clayton, J., Market Fundamentals, Risk and the Canadian Property Cycle: Implications for Property Valuation and Investment Decisions, Journal of Real Estate Research, 1996, 12, 347Ð67. Dokko, Y., R. H. Edelstein, A. J. Lacayo and D. C. Lee, Real Estate Income and Value Cycles: A Model of Market Dynamics, Journal of Real Estate Research, 1999, 18, 69Ð95. Engle, R. F. and C. W. J. Granger, Co-integration and Error Correction Representation, Estimation, and Testing, Econometrica, 1987, 55, 251Ð67. Engle, R. F. and J. V. Issler, Estimating Common Sectoral Cycles, Journal of Monetary Economics, 1995, 35, 83Ð113. Engle, R. F. and S. Kozicki, Testing for Common Features, Journal of Business and Economic Statistics, 11, 1993, 369Ð95. Firstenberg, P. M., S. A. Ross and R. C. Zisler, Real estate: The whole story, Journal of Portfolio Management, 1988, Spring, 22Ð34. Forni, M. and L. Reichlin, Dynamic Common Factors in Large Cross-Sections, CEPR Discussion Paper No 1285, 1995. Gallo, G. M. and H. Kempf, Cointegration, Codependence and Economic Fluctuations, European University Institute, Working paper, ECO No 95/27, 1995.

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Geltner, D. M., Temporal Aggregation in Real Estate Return Indices, Journal of the American Real Estate and Urban Economics Association, 1993, 21, 141Ð66. Giaccotto, C. and J. Clapp, Appraisal-based Real Estate Returns under Alternative Market Regimes, Journal of the American Real Estate and Urban Economics Association, 1992, 20, 1Ð24. Grenadier, S. R., The Persistence in Real Estate Cycles, Journal of Real Estate Finance and Economics, 1995, 10, 95Ð119. Grissom, T. and J. R. DeLisle, A Multiple Index Analysis of Real Estate Cycles and Structural Change, Journal of Real Estate Research, 1999, 18, 97Ð129. Jones Lang Wootton, The JLW Index—Explanatory Notes, Jones Lang Wootton Consulting and Research, London, 1991. Lippi, M. and L. Reichlin, Common and Uncommon Trends and Cycles, European Economic Review, 1994, 38, 624Ð35. Lizieri, C. and S. Satchell, Interaction between Property and Equity Markets: An Investigation of Linkages in the U.K. 1972Ð1992, Journal of Real Estate Finance and Economics, 1997, 15, 11Ð26. Pyhrr, S. A., S. E. Roulac and W. L. Born, Real Estate Cycles and Their Strategic Implications for Investors and Portfolio Managers in the Global Economy, Journal of Real Estate Research, 1999, 18, 7Ð68. Renaud, B., The 1985 to 1994 Global Real Estate Cycle: An Overview, Journal of Real Estate Literature, 1997, 5, 13Ð44. Ross, S. A. and R. C. Zisler, Risk and Return in Real Estate, Journal of Real Estate Finance and Economics, 1991, 4, 175Ð90. Shilling, J. D., Measurement Error in FRC/NCREIF Returns on Real Estate, Southern Economic Journal, 1993, 60, 210Ð19. Vahid, F. and R. F. Engle, Common Trends and Common Cycles, Journal of Applied Econometrics, 1993a, 8, 341Ð60. ——., Non-synchronous Common Cycles, UCSD Economics Discussion Paper, No. 93-55, 1993b. Wang, P., Unsmoothing Property Returns with the Implied Cointegration Relationship, Journal of Property Valuation and Investment, 1998, 16, 358Ð68. ——., Phase-Shifting Common Cycles and Common Trends, Communications in Statistics Theory and Methods, 1999, 28, 1311Ð29. ——., Market Efficiency and Rationality in Property Investment, Journal of Real Estate Finance and Economics, 2000, 21, 185Ð202. Wang, P., C. Lizieri and G. Matysiak, Information Asymmetry, Long-run Relationship and Price Discovery in Property Investment Markets, European Journal of Finance, 1997, 3, 262Ð75. Wheaton, W. C., Real Estate ‘‘Cycles’’: Some Fundamentals, , 1999, 27, 209Ð30.

Peijie Wang, City University Business School, London EC2Y 8HB, UK or p.wang-1@ city.ac.uk.