An application of the ARIMA model to real-estate prices in Hong Kong

Transcription

1 Journal of Property Finance 8,2 152 Journal of Property Finance Vol. 8 No. 2, 1997, pp MCB University Press, X An application of the ARIMA model to real-estate prices in Hong Kong Raymond Y.C. Tse Department of Architecture, The University of Hong Kong Introduction The foundation for fundamental analysis is supply and demand for the commodity in question. Fundamental analysis involves modelling the supplydemand relationships in a market. Analysis of investments, including commodity futures, can be labelled as fundamental or technical. Supply and demand are far more difficult to analyse and model than generally realized. To provide useful results, the analyst cannot deal only with static equilibrium conditions for a commodity but must incorporate dynamic influences also. If markets were perfectly efficient, the practice of either fundamental or technical analysis as a predictive tool would be in vain. No one could hope to earn consistently higher returns than a naïve investor could do. This research attempts to identify the relationships within the system of real estate market being modelled. Using the ARIMA model, these relationships become equations in an interlinked system of equations. While some property analysts use fundamental analysis to try to determine the direction of the market trend, it seems that few can apply technical analysis for timing their entry and exit from market positions. In general, fundamental analysis is useful to gauge the long-term market factors, whereas technical analysis is important for dealing with shorter-term influences. Forecasting the real estate market is neither an easy task to accomplish nor susceptible to naïve, mechanistic approaches. Nevertheless, any forecasting approach that consistently provides better odds than those from tossing a coin in making the correct investment decision should merit careful examination. In fact, technical analysis has been developed and applied to finance for many years, and this research will apply the Box-Jenkins methodology ARIMA model to the study of Hong Kong s real estate prices. In a time series, it is important to identify the data series in the following processes: (a) are the data random?; (b) do the data have a trend?; (c) model identification and (d) testing for model adequacy. If a series is random, the correlation between successive values in a time series is close to zero. However, if the observations of time series are statistically dependent on or related to one another, then the Box-Jenkins (ARIMA) methodology is appropriate. By looking at autocorrelation coefficients for time lags of more than one period, one can determine additional information on how values of a given time series are related. This method produces forecasts that are likely to be more accurate than the forecasts

2 produced by other approaches. The ARIMA models have also proved to be excellent short-term forecasting models for a wide variety of time series because short-term factors are expected to change slowly. The simpler autoregressive and moving average models are actually special cases of the ARIMA classes. Moving averages are popular for determining turning points, that is when a market trend changes direction. The basic concept behind application of moving averages is that, when a price series crosses the correct moving average of itself, the price series will continue in the direction of the crossing. Moving averages are also useful for filtering the effects of cycles of known period in data. The simple models can contain either autoregressive or moving average components but not both. A mixed autoregressive and moving average model with both components is known as an ARIMA model. This paper is the first to apply such a technical model in studying the patterns of real estate prices in Hong Kong. The most important practicality of the research is to show how to determine the cyclical turning points in a real-estate price series. The model helps to track the direction of changes in the real-estate prices. An application of the ARIMA model 153 The data The data for this study are drawn from two sources: Property Review (various issues), the Hong Kong Rating and Valuation Department and Hong Kong Monthly Digest of Statistics (various issues). As the residential property market in Hong Kong has been subject to government interventions, we will restrict our attention to office and industrial properties. The main disadvantage of the data is, however, the level of aggregation. The composite quarterly index for a certain type of premises is simply compiled by calculating a weighted average of the index for a property class or grade. Thus, the price indices can only be regarded as providing a broad indication of the price trend. Aggregation tends to result in an information loss. Moreover, there is an aggregate error arising from the use of a common regression coefficient. As a consequence, the forecasts obtained from disaggregated data are better in terms of mean square error than those obtained from the aggregates. However, it is in any case difficult to identify the most appropriate proxy for the price index in the real estate market, since this heterogeneous sector includes different types and classes of building, and demand for them is generated across all sectors of the economy. Despite data limitations, there is some evidence in favour of a simple averaging of the available data as a reasonable method of forming forecasts. Above all Hong Kong s properties are more homogeneous since multi-storey development has remained predominant in the real estate market. It at least has the merit of being a comprehensive measure. While ideal data on which to base a study of future price trends are not available, an accurate indication could be furnished. The price indices are based on an analysis of prices paid for completed flats as recorded in Sale and Purchase agreements. The indices are based on a rateable value for measuring price changes with quality kept constant. Quarterly data for the period 1980Q1 to 1995Q2 which contains 59

3 Journal of Property Finance 8,2 154 observations, are employed in the ARIMA model. A 50-sample observations is adequate for ARIMA analysis (Holden et al., 1990). The ARIMA model is essentially an approach to economic forecasting based on time-series data. However, the ARIMA model requires the use of stationary time-series data (Dickey and Fuller, 1981; Granger and Newbold, 1974; Tse, 1996). Under current practice, developing such data requires that the observed data series should be tested for unit roots. The tests for unit roots are also known as Dickey-Fuller (DF) and augmented Dickey-Fuller (ADF) tests. Typically, the ADF test is based on the following formulation: where Y t = Y t Y t 1, µ is a drift term and T is the time trend with the null hypothesis H 0 : α = 0 and its alternative hypothesis H 1 : α 0, N is the number of lags necessary to obtain white noise and u t is the error term. The simpler Dickey-Fuller (DF) test removes the summation term. However, the implied t- statistic is not the student t distribution, instead it is generated from Monte Carlo simulations (Engle and Granger, 1987, 1991). Note that failing to reject H 0 implies that the time series is non-stationary. Generally, many kinds of non-stationarity are present in time series data. A non-stationary time series is one for which the parameters are functions of time, and thus one for which its mean, variance, and so on, change over time. A time series is referred to as stationary when it contains no growth or decline and as non-stationary when a trend is present. When a time series is non-stationary, autocorrelations will dominate the pattern. To model the non-trend patterns in the series, trends must be removed before further analysis can take place. The most popular approach is to carry out consecutive differencing on the series concerned to achieve stationarity and then fit the ARIMA model to them. In fact, many time series can be made stationary by replacing the original data points with their first differences; that is, the differences between successive observations. Since there is a strong upward trend in the series throughout the period, we must therefore consider how to transform the data to remove this trend. It seems sensible to deflate the data by Consumer Price Index in order to generate a constant-price series. If the series is not deflated, it may have to be differenced one more time to obtain stationarity. However, overdifferencing may introduce unnecessary correlations into the model and cause information loss. Official data are available up to the first quarter of 1996, but we only use quarterly data in the period 1980Q1-1995Q2. The estimated equations will be used to forecast for the next three quarters. There are two reasons for using quarterly data: first, there are not enough annual data for performing an ARIMA analysis, and second, annual data will average out aberrations that tend to distort the estimation results in any given short span of time (there are no monthly data as such in Hong Kong). Inflation rate in quarter j of time t (P tj ) is computed as (1)

4 P tj = (CPI tj CPI (t 1)j )/CPI (t 1)j, where CPI tj refers to the Consumer Price Index- A in quarter j of time t. Thus, real prices are discounted by the factor 1/(1 + P tj ). Let ROFF and RIND represent the real values of office and industrial property prices respectively. Table I reports the DF and ADF test statistics on the ROFF and RIND in equation (1) omitting minus signs for simplicity. In Table I, the null hypothesis of unit root for the ROFF in level form with and without time trend, is rejected at all conventional levels of significance, but not rejected for RIND (at the 0.05 level) when the calculated ADF test statistics associated with the numerical coefficients of the variables are compared with their critical values, as given in Engle and Granger (1987). It seems reasonable to carry out the analysis assuming that the series of ROFF and RIND in level terms are non-stationary. This implies that the properties of the series do not satisfy the usual assumptions of econometric theory of constant mean and variance that, conversely, evolve with time. The two series were first-differenced and the unitroot tests re-run; in this form, the DF and ADF test statistics reject the hypothesis of a unit root at all conventional levels of significance, suggesting that the first differences of all the series under consideration are stationary. Thus, the ARIMA tests were carried out in first differences of the data. An application of the ARIMA model 155 The methodology The ARIMA technique does not assume any particular pattern in the historical data of the series to be forecast. The applications of an ARIMA model are well documented in Barras (1983), Box and Jenkins (1976), Chow and Choy (1993), Cleary and Levenbach (1982), Hanke and Reitsch (1986), Herbst (1992), and Nazem (1988) for example. An ARIMA model uses an iterative approach of identifying a possible useful model from a general class of models. Another tool for identification of stationarity in an ARIMA model is the ordinary and partial autocorrelation. Non-stationarity may be present if the values plotted in the correlogram do not diminish at large lags. When the original series or correlogram exhibits non-stationarity, successive differencing is carried out Level First differences Variables Trend No trend Trend No trend ROFF t 1 DF-0 lag * 5.92* ADF-1 lag * 3.87* ADF-4 lags RIND t 1 DF-0 lag * 6.23* ADF-1 lag ADF-4 lags 4.02** Notes: * and ** indicate significance at the 1% and 5% levels. The critical values of DF statistics are: 3.5 with trend and 2.93 without trend, and 4.15 with trend and 3.58 without trend at the 5% and 1% levels of significance respectively Table I. Results of unit-root tests

5 Journal of Property Finance 8,2 156 until the correlogram of the differenced series dies out reasonably rapidly. There is a convenient notational device for expressing an ARIMA operation which relates a dependent variable to lagged terms of itself and to lagged error terms. The general form of the ARIMA model can be written as: (2) where B represents the backshift operator such that BY t = Y t 1, Y t is the value of the time series observation at time t, ε t is a series of random shocks which are assumed to be independently, normally distributed with zero mean and variance and d represents the order of difference. If a series is stationary, then d = 0. In equation (2), φ(b) is a polynomial of order p in the backshift operator B, which is defined as: (3) Similarly, φ(b) is defined to be a polynomial of order q in B, such that: The conditions for the existence of the model require (i) stationarity, that is the statistical equilibrium about a fixed mean and (ii) invertibility, which guarantees uniqueness of representation. While one may not be aware of the appropriate order of the autoregressive process to fit to a time series, we examine the time series of the property markets by taking differences of order 1 of the data. Once a tentative model has been selected, the parameters for that model must be estimated. The first differences of the two series are given by: (5) (6) Now, we shall proceed to estimating such a model. The ARIMA model is generated iteratively by using the RATS program package until convergence is obtained. The estimated parameters are shown as follows: Office Property ARIMA(2,1,1) Industrial Property ARIMA(2,1,1) As seen, the lag structures of the two series are similar. Using equations (7) and (8), the values of ROFF and RIND without error-correction terms (WET) which are, respectively, ROFF^ and RIND^ can be estimated and compared with their actual values (see Figures 1 and 2). The lag structure of the equations suggests that the cyclical effects generated in the past information are transmitted (4) (7) (8)

6 endogenously to current prices through the lagged variables. The error-correction terms represent the random noise that tends to obscure the true, fundamental movements of the market. It should be noted that there are seasonal factors in a time series, which manifest themselves in the sample autocorrelation function. Recognition of seasonality is important because it provides information about regularity in the series that help make a forecast. Figure 1. Office property price Figure 2. Industrial property price

7 Journal of Property Finance 8,2 158 We can identify seasonality by observing regular peaks in the autocorrelation function, even if seasonal peaks cannot be discerned in the time series itself (Pindyck and Rubinfeld, 1991). Estimation and forecasting Generally, a comparison between the original time series and the estimated model provides a measure of the model s ability to explain the variability in the original time series data. If the model is specified correctly, the residuals ε t should resemble a white noise. For a large displacements m, the residual autocorrelation r k (k = 1,, l) are supposed to be uncorrelated random variables, normally distributed as N(0, 1/n) (Pindyck and Rubinfeld, 1991). The model can be checked for adequacy by doing a diagnostic chi-square test, known as the Box-Pierce test, on the autocorrelations of the residuals. The test statistic due to Box and Pierce (1970) and modified by Ljung and Box (1978) which is widely used today, is given by the formula: where n is the number of observations used to fit the model, and l is the number of autocorrelations included in the test, which is usually taken to be 6, 12 or 18, m is the number of parameters estimated, and (l m) equals degree of freedom. This computed chi-square value can be compared against the corresponding chi-square value from the chi-square table at a certain level of significance. If the chi-square statistic does not exceed the threshold value, it can be concluded that no recognizable pattern is left in the residual series and the model is not rejected. The estimated values of the diagnostic Q-statistics for different lags are summarized in Table II. Comparing the diagnostic Q statistics against the theoretical chi-square values with corresponding degrees of freedom, it can be concluded that none of these chi-square statistics is significant at the 0.1 level. Thus the result suggests that there is no further pattern left in the residual series of the estimated models. From the modelling results described above, office and industrial property prices in Hong Kong can be fitted into an ARIMA model. In the office and industrial property sectors, it is user demand, rather than investment demand, which is the dominant influence on real estate price trends, whereas in the housing sector, investment demand appears to exert the strongest influence. If investment demand is the dominant factor, speculators play an important role (9) Lag Office Industrial df Critical value (10%) Table II. The diagnostic chi-square test: Q-statistics

8 in the market. Their continuous buying and selling activities increase market liquidity and transactions, although the housing market is subject to government interventions. However, if user demand is the dominant factor, a tendency to cyclical fluctuations in development will be created, corresponding to alternating phases of relative undersupply and oversupply in the user market. Thus, exogenously determined economic factors, such as construction costs, the variation in user activity, and the terms and availability of credit will all act on the inherent cyclical tendency, either reinforcing or dampening the price trend according to circumstances (Barras, 1983). The changes in prices can result from gradual shifts in demand and supply, and the interaction of these economic factors is illustrated by the strong cyclical trend observed in the office and industrial property prices. In a perfectly efficient market, price at all times reflects the consensus of value determined by sellers and buyers acting on their assessments of all pertinent information. Thus, any new information will cause prices to change quickly until a new consensus of value is reached, too quickly for investors to profit from the news. The deviations from the path of real estate prices are as likely to be positive as they are to be negative. However, the term market efficiency must be hedged to some extent since it is unlikely that any market could be so completely efficient that all prices adjust instantaneously to new information. Above all, real estate markets are supposed to be of a lower degree of liquidity than financial markets. Our forecasts will be based on the assumption that the real estate markets are less than perfectly efficient. It follows that the forecasts should be tempered by the observation that mass psychology can influence market price behaviour in ways that can be traced by an ARIMA model. The forecasting method can provide an indication of shortterm market direction, a sense of whether or not the movement will be small or large, and a warning well ahead of turning points supplementary to investment strategy. The approach adopted is to assume that a causal model will help in understanding behaviour and so will produce reasonable forecasts. With further assumptions it is possible to make forecasts of future real-estate prices. Suppose the estimation model is: Xt = f(x t 1, X t 2,, X t L ) + εt θε t 1, for the given values of X = {X 1, X 2,, X T ). The problem is to forecast X t for t > T. Since the expected value of the residual term should be zero: E(εt θε t 1 ) = 0, we assume initially that εˆt+1 = θε T. Therefore the estimated values εˆt+2, εˆt+3 can be obtained. The forecasts: {Xˆt, > T} can be generated as: An application of the ARIMA model 159

9 Journal of Property Finance 8,2 160 The corresponding predicted values of office and industrial property prices, respectively, in the periods: 1995Q3 1996Q1 are illustrated in Figures 1 and 2. In order to verify whether these forecasts were profitable in practice, the predicted values can be compared with actual figures. The forecasts which yield a decrease of 18.3 per cent and 24.6 per cent in office and industrial property prices during the period 1995Q3 1996Q1 are close to the actual figures. Moreover, the direction of price changes is as expected. Although the chi-square diagnostic tests do not indicate problems of misspecification, tests for structural breaks (using Chow test and Hendry s forecasting test) cannot be carried out in this research owing to data limitations. Note that a sample of not less than 50 observations is required for an ARIMA analysis. However, it should be noted that forecasts from an ARIMA model are less susceptible to structural breaks. It is because the ARIMA procedure is an adaptive process which can be started up at any point in a series. It recomputes the regression while adding an observation to the sample. This means that the regression will be updated, once the observation at time t + 1 is available. Clements and Hendry (1996) argue that some models, such as the ARIMA modelling, may offer greater protection against unforeseen structural breaks than others. Moreover, in the presence of structural breaks, which occurred at the time of forecasting, time-series models in differences tend to outperform econometric models. An adaptive forecasting procedure like the ARIMA model is expected to perform well in one-to-four-step-ahead forecasts. One of the objectives of the ARIMA analysis includes improving the accuracy of forecasts in real-estate prices. To measure the deviations of ROFF^ and RIND^ from their actual values, we can compute the root mean square error (RMSE) of the naïve forecast: δ= [( n ) ( Z Z ˆ)], where Z = {ROFF i, RIND i }. Table III shows that the estimated equation of the industrial property prices has a better fit compared with the office property prices. For (n 3) observations, the forecast accuracy can also be measured by the Theil U-statistic which is defined as: n i= = t+ T t+ T t+ T t U [ ( Z Z ˆ ) / ( Z Z ˆ )]. The U-statistic computes the ratio of the RMSE of the model forecasts to the RMSE of naïve, no-change forecasts (Dua and Ray, 1995). This statistic implies that: (1) U = 1, the model s forecasts match, on average, the naïve forecasts; (2) U < 1, the model s forecasts outperform the naïve forecasts; (3) U > 1, the naïve forecasts outperform the model forecasts. Thus, a model producing a lower average U-statistic has a relatively better forecasting accuracy. Table III reports the U-statistics for the office and

10 δ µ a σ µ/σ µ g /σ g Office 80/1-82/ /1-85/ /1-88/ /1-91/ /1-95/ Average U Industrial 80/1-82/ /1-85/ /1-88/ /1-91/ /1-95/ Average U Notes: The risk-adjusted price performance µ/σ = {1/(ΣZ 2 i /(nµ2 ) 1)} µ g = Σz i /(n 1) and σ g = {Σ(z i µ g ) 2/(n 1)}, where z i = log(z i /Z i 1 ) a First quarter of 1980 = 100 Table III. Average performance of real estate prices industrial properties. It indicates that the industrial-property model provides a slightly better performance relative to the office-property model. Furthermore, the forecasting performance can be gauged by comparing the RMSE (δ) with the mean value µ: we obtain δ/µ of 12.4 per cent for office and 7.5 per cent for industrial property. Accordingly, we suggest that a turning point is likely to occur when the reversed change in prices is greater than δ/µ. Using this criterion, the results correctly indicate that the movements of the office and industrial property prices changed directions in To compare the average performances of the property prices, we compute the mean values (µ) and standard errors (σ) of their nominal prices with moving time periods (Table III). Thus, µ measures the average price performance, and σ measures the associated risks. The risk-adjusted price performance which is equal to µ/σ, is better in office than industrial property in the period 1980Q1-1982Q4. Moreover, the office property performs better than industrial investments in terms of return and risk in that period. However, in the longer term, it turns out that industrial property is better than office investments. As shown in Table III, the risk-adjusted price performance of industrial property is 1.76, which is significantly much higher than that of office property at When the mean value of rate of price change (µ g ) is compared with its deviation (σ g ) in the whole period 1980Q1-1995Q2, the risk-adjusted price change (µ g /σ g ) of the industrial property is 0.35, which is also significantly better than that of office at 0.26.

11 Journal of Property Finance 8,2 162 Conclusions The technical analysis presented in this paper provides sufficient evidence in support of the adequacy of the estimated models for office and industrial properties in Hong Kong. In fact, the behaviour of real estate prices has long been a matter of interest. Technical analysis was built on the belief that human mass behaviour over the centuries has exhibited a tendency to move in trends (Flood and Hodrick, 1990). Such psychological factors are probably considered to fuel shorter-term cycles in financial and futures markets. This paper concentrates on an analysis that attempts to fit real estate prices in Hong Kong into the ARIMA model. This paper shows how the office and industrial property prices in Hong Kong can be fitted into the ARIMA equation. The core of the ARIMA model is premissed on the fact that the market price at any time is revealed by the pattern of prior price movements. While fundamental analysis suffers from the problems of model building, technical analysis suffers from problems related to pattern validation. Changes in fundamental factors can of course alter prices. However, as far as the property market is concerned, these changes are unlikely to cause gradual shifts in consumer tastes and preferences or conversion to substitute or complementary products. It should be noted that technical analysis is inadequate at signalling market turning points in the longer term. Property price trends may fade because of fundamental economic causes, effectively reducing the predictive power of the technical analysis. Thus it has already become an accepted practice that ARIMA models and econometric models can be combined to give an improved forecast. While econometric methods are not always more accurate than time-series models, they help in understanding causal relationships between variables and can provide evidence of the validity of economic theory. Moreover, all forecasting methods require the use of qualitative judgement. Simple methods based on sound judgement frequently give better forecasts than complex and sophisticated methods (Holden et al., 1990). Thus, a point forecast is of limited use and tends to be associated with uncertainty (Granger, 1996). More attention could be paid to how forecasts are to be used, since the purpose of forecasting is to help in investment decisions. We do not suggest an ARIMA methodology is used to provide a forecasting model directly without resort to other procedures. If forecasts from a technical and behavioural model can be combined to give an improved forecast, this suggests that each of these models is inadequate if used independently. A better forecast might arise from a union of the two models. However, the investor may wish to incorporate forecasts from an ARIMA model into his investment strategy, to help with timing. References Barras, R. (1983), A simple theoretical model of the office-development cycle, Environment and Planning, Vol. 15 No. 6, pp Box, G.E.P. and Jenkins, G.M. (1976), Time Series Analysis, Forecasting and Control, Holden-Day, Oakland, CA.

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