ForecastingconstructionindustrydemandpriceandproductivityinSingaporetheBoxJenkinsapproach1.pdf
Introduction
In the construction industry, regression techniques have been used often to model and forecast construc- tion variables such as demand and price owing to their relative simplicity in both concept and application. Goh (1999) evaluated the accuracy of the multiple regres- sion approach in forecasting sectoral construction demand in Singapore. In the UK, Akintoye and Skitmore (1994) produced regression models to predict private-sector construction demand. Tang et al. (1990) also applied regression analysis to project values of total construction activities for the Thai construction industry. In the USA, Killingsworth (1990) developed a demand forecasting model based on a regression analysis of the leading indicators of industrial con- struction. Regression models, using leading indicators of construction workload, were also built to forecast
UK contractors’ total new orders (Oshobajo and Fellows, 1989). For construction cost prediction, linear and loglinear regression methods were explored for public school buildings in Jordan (Al-Momani, 1996). Flanagan and Norman (1983) applied simple linear and curvilinear regressions to obtain extrapolated fore- casts of tender price, while McCaffer et al. (1983) developed a regression model to predict construction price movements. The feasibility of econometric cost modelling was examined by Bowen (1982) as an alter- native estimating approach for reinforced concrete frame structures.
In essence, regression techniques are concerned with the modelling of relationships among variables, and they quantify how a response variable is related to a set of explanatory variables. The strength of a causal model as a forecasting tool is that it allows the impact of various alternative inputs to be evaluated. However, the dif� culty of this approach is that it requires infor- mation on several variables in addition to the variable
Construction Management and Economics (2000) 18, 607–618
Forecasting construction industry demand, price and productivity in Singapore: the Box–Jenkins approach
GOH BEE HUA* AND T EO HO PIN
School of Building and Real Estate, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260
Received 12 July 1999; accepted 13 January 2000
In academic research, the traditional Box–Jenkins approach is widely acknowledged as a benchmark technique for univariate methods because of its structured modelling basis and acceptable forecasting performance. This study examines the versatility of this approach by applying it to analyse and forecast three distinct variables of the construction industry, namely, tender price, construction demand and productivity, based on case studies of Singapore. In order to assess the adequacy of the Box–Jenkins approach to construction industry forecasting, the models derived are evaluated on their predictive accuracy based on out-of-sample forecasts. Two measures of accuracy are adopted, the root mean-square-error (RMSE) and the mean absolute percentage error (MAPE). The conclusive � ndings of the study include: (1) the prediction RMSE of all three models is consistently smaller than the model’s standard error, implying the models’ good predictive performance; (2) the prediction MAPE of all three models consistently falls within the general acceptable limit of 10%; and (3) among the three models, the most accurate is the demand model which has the lowest MAPE, followed by the price model and the productivity model.
Keywords: Box–Jenkins approach, forecasting, construction demand, tender price, productivity, accuracy
* Author for correspondence. e-mail: [email protected]
Construction Management and Economics
ISSN 0144–6193 print/ISSN 1466-433X online © 2000 Taylor & Francis Ltd
http://www.tandf.co.uk/journals
that is being predicted. By contrast, a univariate model predicts future values of a time series solely on the basis of its historical values. It treats the system as a black box and makes no attempt to discover the factors affecting its behaviour. The time-series model always assumes that some pattern or combination of patterns is recurring over time. Hence, by identifying and extrapolating that pattern, forecasts for subsequent time periods can be obtained. However, one limitation of such simpler methods is that they are suitable for making only short term forecasts. It is also disadvan- taged by the black box characteristics, whereby explanatory capabilities are lacking, and hence it is not suitable for applications where explanation of reasoning is critical.
Nevertheless, empirical studies dealing with fore- casting accuracy (Slovic, 1972; Armstrong, 1978; and Makridakis and Hibon, 1979) have shown that ‘more complex or statistically sophisticated methods are not necessarily more accurate than simpler methods’ (Makridakis et al., 1982, p. 112). Greene (1997, p. 823) notes that ‘researchers have observed that the large simultaneous equations macroeconomic models constructed in the 1960s frequently have poorer forecasting performance than fairly simple, univariate time-series models based on just a few parameters and compact speci� cations’. Similarly, in construction- related studies, researchers have found that a higher level of accuracy can be achieved by using time-series methods as they are based upon a stochastic rather than deterministic approach (McCaffer et al., 1983; Fellows, 1987; Taylor and Bowen, 1987; and Goh, 1998). As a new approach, Wang and Mei (1998) presented a time-series model for forecasting construc- tion cost indices in Taiwan.
Univariate time-series forecasting techniques range from simple smoothing methods such as moving aver- ages and exponential smoothing to more advanced techniques such as decomposition and the Box–Jenkins approach. The simpler techniques are often applied in practice in situations where clearly it is not worth the time or the money to develop and apply more sophisticated ones, for instance, preparing daily fore- casts for a number of different variables. However, in academic research, the traditional Box–Jenkins approach is used widely as a benchmark technique for univariate methods because of its structured modelling basis and acceptable forecasting performance. For immediate and short term forecasts, ‘the Box–Jenkins method is probably the most accurate of the time-series models’ (O’Donovan, 1983, p. 9). Taylor and Bowen (1987), and Fellows (1987, 1991) have found the tech- nique to be satisfactory in modelling construction tender price movements as it permits judgment, models the changing process adequately and provides a
class of models of stationary stochastic processes. Its strength lies in its generality, as it can handle virtually any time-series data, partly owing to its strong theo- retical foundations and partly to its success in empir- ical comparisons which have found it to be as accurate as many complex econometric models (Wheelwright and Makridakis, 1985). Abdelhamid and Everett (1999) found this technique, in particular, useful in analysing construction productivity data, as order and dependence are the exact two properties that are found in data collected for productivity improvement � eld experiments. The versatility of the Box–Jenkins approach is examined in this paper, where it is used to analyse and forecast three distinct variables of the construction industry, namely, price, demand and productivity, based on case studies of Singapore.
Objectives and scope
This paper constitutes a culmination of empirical studies on the application of an advanced univariate time-series technique to model and forecast construc- tion demand, tender price and industry-level produc- tivity in Singapore. More speci� cally, these studies, or rather the objectives of this paper, include the use of the Box–Jenkins approach to analyse and forecast: 1. public industrial building tender prices (case study I); 2. residential construction demand (case study II); and 3. construction productivity at industry level (case study III). These case studies were undertaken consec- utively over a period of six years, each serving a different purpose to cater to speci� c needs of the Singapore construction industry. For case study I, the intention of developing a tender price index forecasting model for public industrial buildings in the early 1990s was in anticipation of a rise in demand for such devel- opments to support a booming manufacturing industry. This was in line with Singapore’s new economic thrust ‘to adopt a global strategy and become a competitive total business centre’ (Jurong Town Corporation, 1988, p. 2). For case study II, a model to produce accurate demand forecasts was needed in the mid-1990s to cater to a rapidly growing residen- tial sector where it had a share of 32–50% in the total value of construction contracts awarded between 1986 and 1994 (CIDB, 1994). Case study III was under- taken most recently, as there has been an urgent need to look into ways of improving industry productivity as Singapore enters into the next millennium. If productivity levels can be predicted with reasonable accuracy, future levels of competitiveness in the industry could be raised through proper planning of its workload and better use of its limited resources to achieve national targets.
608 Goh and Teo
Time-series data
For case study I, the Singapore public industrial building tender price index series, constructed by Goh (1992), was used as the time-series data set. The SBEM Modal House approach (Goh and Teo, 1993) to index construction was adapted from the concepts of the New Zealand Institute of Valuers Modal House approach (NZIV, 1967a, b, 1972). Figure 1 shows a plot of the quarterly compiled index series from the � rst quarter of 1980 to the � rst quarter of 1992, giving a total of 49 data points. With the intention of utilizing as many data points as possible to build the forecasting model, the � rst 47 quarters formed the modelling data set, while the remaining 2 quarters were used to test the accuracy of the model for out-of-sample forecasts.
For case study II, ‘gross � oor area (GFA) of devel- opments commenced’ was used to represent demand for residential construction. A study on construction statistics in the UK has singled out its usefulness as a forward indicator of demand (MPBW, 1968). Alternative series, such as the ‘gross � xed capital formation’ and ‘value of contracts awarded’, were considered, but deemed less suitable because the former is an output rather than a demand measure, while the latter had data starting from only 1986. A graphical plot of the time series from the � rst quarter of 1975 to the � rst quarter of 1994, comprising a total of 77 data points, is shown in Figure 2. The � rst 72 quarters formed the modelling data set and the remaining 5 quarters were used for model testing.
For case study III, ‘value added per worker’ was chosen as the industry-level productivity measure for two reasons. First, a longer time series was required for modelling purposes and the alternative measure, ‘square metres of built-up area per man-day’, had data starting from only 1991. Second, the output and input measures, ‘GDP at 1990 market prices, by industry: Construction’, and ‘Employed persons aged 15 years and above, by industry: Construction’, respectively, used for its computation were published national statis- tics compiled by the Department of Statistics, which could be obtained readily. Data for these two series were abstracted from statistical yearbooks from 1975 to 1996 and comprising a total of 22 data points. Based on the same intention of utilizing more data for model- ling, the � rst 20 periods were used for modelling, with the last two periods for testing. Figure 3 shows a plot of the time series for ‘value added per worker’.
The Box–Jenkins approach
The general methodology suggested by Box and Jenkins (1976) for applying autoregressive-integrated-
moving-average (ARIMA) models to time-series analysis, forecasting and control has come to be known as the Box–Jenkins approach for time-series analysis and forecasting. It is a systematic approach for iden- tifying characteristics of a time series such as station- arity and seasonality, and eventually it provides a class of models of stationary stochastic processes. It relies on an iterative approach for identifying a possible useful model from a general class of models. The chosen model is then checked against the historical data to see whether it describes the series accurately. The model presents a good � t if the residuals between the forecasting model and the historical data are small, randomly distributed and independent. If the speci� ed model is not satisfactory, the process is repeated by using a different model that is designed to improve on the original one. This process is repeated until a satis- factory model has been identi� ed.
Forecasting tender price, demand and productivity 609
Figure 1 Time series of public industrial building tender price index in Singapore (1980 Q1 = 100)
Figure 2 Time series of gross � oor area (GFA in sq. m) of residential developments commenced in Singapore
This approach divides the forecasting problem into three stages. Initially, a general class of forecasting models is postulated. Stage 1 involves identifying a speci� c model that can be tentatively entertained as one that best suits the characteristics of the time series. This model is � tted to the historical data in stage 2, and a check is carried out to determine whether it is adequate. If the tentative model is not satisfactory, the approach returns to stage 1, where an alternative model is identi� ed. Once an adequate model is determined, it will be used to generate forecasts for the variable concerned in stage 3.
The stages of identi� cation, estimation and diag- nostic checking, and application can be carried out via statistical analysis software, such as SPSS for Windows or SAS for PC. However, skilled user intervention is required at the stage of identifying the best-� t model. Also substantial knowledge in time-series analysis is necessary for this approach, such as analysing the time series for stationarity, seasonality, and identifying and specifying the form of the time-series model using the autocorrelations and partial autocorrelations. Once the model has been identi� ed, � tting and forecasting are automatic.
Case study I: public industrial building tender price index forecasting in Singapore
Autoregressive-integrated-moving-average (ARIMA) models are designed for stationary time series. A stationary series is one whose basic statistical proper- ties, such as the mean or variance remain constant over time. To build ARIMA models, � rst periodic varia- tions and systematic changes in these properties must be identi� ed and removed. One method of checking for stationarity is the use of the autocorrelation func- tion (ACF). Typically, the ACFs for nonstationary data are signi� cantly different from zero for the � rst several time lags. Figure 4 plots the pattern of the autocorre- lations of the time series for ‘public industrial building tender price index’. It was con� rmed that the time series was nonstationary in the mean since the ACF died down very slowly. The series was transformed into one that is stationary by the method of ‘differencing’. It involved the process of taking � rst differences of the series and creating a new time series of successive differences. The results of the � rst differences of lag 1 are shown in Figure 5. The Box–Ljung test statistic was used to test the null hypothesis that the residuals of the differenced series are random. Based on the ‘Prob’ values given in Figure 5, the null hypothesis was not rejected at the 5% signi� cance level for almost all the lags. The ACF plot also resembled ‘white noise’, that is, all of the autocorrelations were near zero,
con� rming that the differenced series was ready for model � tting.
During model � tting, the behaviour of the ACF and the partial ACF (PACF) served as the key to model identi� cation as shown in Table 1. The simple corre- lation coef� cient r between Yt and Yt-1 was derived by means of the following equation:
n
S t=2
(Yt – Y ]
)(Yt–1 –Y ]
) rYtYt–1 = —————————n
S t=1
(Yt – Y ] )2
where Y ]
is the mean value of Y; and n is the number of time periods. Similarly, the autocorrelations for 1, 2, 3, 4, ..., k time
lags can be found and denoted by rk, as
n –k
S t =1
(Yt – Y ]
)(Yt–k –Y ]
) rk = —————————n
S t =1
(Yt – Y ]
)2
where D(q) means the function drops off to 0 after lag q; D(p) means the function drops off to 0 after lag p; T means the function tails off exponentially; and 0 means the function is 0 at all nonzero lags.
The ACF and PACF plots of the differenced series did not give a clear indication of an autoregressive (AR) or moving average (MA) model. On the one hand, the ACF seemed to indicate an MA model, but on the other, the PACF was consistent with an AR model. However, based on the study by Fellows (1987) which found that low-order AR models of differenced series provide the best-� t models for tender price indices, a tentative AR model was chosen.
At the estimation stage, the tentative AR(1) model was � tted to the differenced series and the model para- meters were estimated. Table 2 provides the results of the AR(1) model � t. It was found that the parameter
610 Goh and Teo
Figure 3 Time series of value added per worker (in S$) in Singapore
estimate for the constant term, was statistically insig- ni� cant at the 10% level and, hence, was removed from the model. The AR(1) model was re� tted but without a constant term and the results are given in Table 3. It was con� rmed that the removal of the constant term did not affect the signi� cance of the parameter estimate of the AR(1) term.
Diagnostic checking involved either checking for random residuals or, by ‘over� tting’, to determine the adequacy of the � tted AR(1) model without a constant term. In this case, both procedures were conducted to test for consistency of result. First, the residuals were checked for randomness by observing the pattern of the autocorrelations, together with their respective Box–Ljung test statistics. As shown in Figure 6, all the autocorrelations were highly insigni� cant, indicating that the residuals were random and the � tted model was adequate. Second, the series was ‘over� tted’ with a higher order AR model. The parameter estimate for the AR(2) term was found to be insigni� cant at the 10% level (t-ratio = –0.280). Hence both procedures have con� rmed that an AR(1) model best � ts the data series for Singapore’s public industrial building tender price index.
Then the equation for the best-� t model, ARIMA (1,1,0), can be expressed as
Yt = Yt–1 + f 1Yt–1 – f 1Yt–2 + et
where Yt is the variable to be forecast at current time t; Yt–1 and Yt–2 are past values of Yt at time lags 1 and 2, respectively; f 1 is the � rst-order autoregression coef� cient; and et is the random error term. From the analysis, the parameter estimate of the AR(1) term, f 1, was found to be –0.3864. The forecasts generated for the validation period (1991 Q4–1992 Q1) are given in Table 4.
Case study II: residential construction demand forecasting in Singapore
The data series for ‘GFA of developments commenced’ was � rst checked for stationarity. The pattern of the autocorrelations is plotted in Figure 7. The ACF con� rmed that the data series was nonstationary in the mean since it died down very slowly. Once again, ‘differencing’ was performed to transform the series
Forecasting tender price, demand and productivity 611
Figure 4 ACF of ‘public industrial building tender price index’
Table 1 Summary of model identi� cationa
MA (q) AR (p) ARMA (p,q) White noise
ACF D(q) T T 0 PACF T D(p) T 0
a Source: Brocklebank and Dickey (1986).
Autocorrelations: TPI
Auto- Stand. Lag Corr. Err. –1 –.75 –.5 –.25 0 .25 .5 .75 1 Box-Ljung Prob.
+----+----+----+----+----+----+----+----+ 1 .812 .141 . I*****.********* * 33.026 .000 2 .726 .140 . I*****.******** * 59.971 .000 3 .604 .138 . I*****.***** * 79.062 .000 4 .489 .137 . I****.**** * 91.894 .000 5 .433 .135 . I****.**** 102.154 .000 6 .335 .133 . I****.** 108.445 .000 7 .222 .132 . I****. 111.291 .000 8 .156 .130 . I*** . 112.725 .000 9 .050 .128 . I* . 112.876 .000
10 –.005 .127 . * . 112.878 .000 11 –.074 .125 . *I . 113.229 .000 12 –.112 .123 . **I . 114.061 .000 13 –.164 .122 . ***I . 115.889 .000 14 –.187 .120 . ****I . 118.342 .000 15 –.210 .118 . ****I . 121.505 .000 16 –.213 .116 . ****I . 124.888 .000
Plot Symbols: Autocorrelations * Two Strand Error Limits .
Total Cases: 47 Computable � rst lags: 46
612 Goh and Teo
Figure 5 ACF and PACF of � rst differences of ‘public industrial building tender price index’
Autocorrelations: TPI
Transformations: difference (1)
Auto- Stand. Lag Corr. Err. –1 –.75 –.5 –.25 0 .25 .5 .75 1 Box-Ljung Prob.
+----+----+----+----+----+----+----+----+ 1 –.416 .143 **. *****I . 8.490 .004 2 .090 .141 . I** . 8.901 .012 3 .102 .140 . I** . 9.439 .024 4 –.020 .138 . * . 9.460 .051 5 –.009 .136 . * . 9.465 .092 6 .090 .135 . I** . 9.915 .128 7 –.168 .133 . ***I . 11.505 .118 8 .213 .131 . I****. 14.142 .078 9 –.154 .129 . ***I . 15.562 .077
10 .079 .128 . I** . 15.947 .101 11 –.016 .126 . * . 15.963 .143 12 –.008 .124 . * . 15.967 .193 13 –.082 .122 . **I . 16.413 .228 14 .066 .120 . I* . 16.710 .272 15 –.165 .118 . ***I . 18.641 .230 16 .212 .117 . I****. 21.959 .145
Plot Symbols: Autocorrelations * Two Strand Error Limits .
Total Cases: 47 Computable � rst lags after differencing: 45
Partial Autocorrelations: TPI
Transformations: difference (1)
Pr–Aut- Stand. Lag Corr. Err. –1 –.75 –.5 –.25 0 .25 .5 .75 1
+----+----+----+----+----+----+----+----+ 1 –.416 .147 **. *****I . 2 –.100 .147 . **I . 3 .125 .147 . I** . 4 .101 .147 . I** . 5 .015 .147 . * . 6 .081 .147 . I** . 7 –.133 .147 . ***I . 8 .108 .147 . I** . 9 –.042 .147 . *I .
10 .032 .147 . I* . 11 –.005 .147 . * . 12 –.009 .147 . * . 13 –.105 .147 . **I . 14 –.045 .147 . *I . 15 –.134 .147 . ***I . 16 .117 .147 . I** .
Plot Symbols: Autocorrelations * Two Strand Error Limits .
Total Cases: 47 Computable � rst lags after differencing: 45
into a stationary one. The ACF of the differenced series generally resembled ‘white noise’, except for the � rst few lags, and model � tting was initiated.
At the model identi� cation stage, it was observed that the ACF tailed off exponentially while the PACF dropped off to zero after lag 1. Hence, a tentative AR(1) model was chosen and � tted to the differenced series. The results of the model � t are provided in Table 5, which indicates that the parameter estimate of the constant term is insigni� cant at the 10% level. Based on this observation, the constant term was dropped and the results of the re� tting are given in Table 6, con� rming that the parameter estimate of the AR(1) term is still highly signi� cant.
For diagnostic checking, the residuals were exam- ined for randomness to test for goodness of � t of the AR(1) model without a constant term. The ACF plot of the residuals indicated that all the autocorrelations were highly insigni� cant, and this was re-con� rmed by
Forecasting tender price, demand and productivity 613
Table 2 Results of the AR(1) model � t with a constant term
Parameter Coef� cient u SE (u ) t-ratio Approx. estimate prob.
AR(1) –0.4172 0.136 –3.059 0.0038 Constant 2.8406 1.778 1.597 0.1173
Table 3 Results of the AR(1) model � t without a constant term
Parameter Coef� cient u SE (u ) t-ratio Approx. estimate prob.
AR(1) –0.3864 0.1368 –2.824 0.0071
Figure 6 Autocorrelations of residuals of the � tted ARIMA (1,1,0) model
Table 4 Details of forecasts generated by the ARIMA (1,1,0) model
Period Actual Forecast Error Lower 95% Upper 95% values A values F (A–F) con� dence limit con� dence limit
1991 Q4 224 236 –12 201 270 1992 Q1 240 228 12 194 262
Auto- Stand. Lag Corr. Err. –1 –.75 –.5 –.25 0 .25 .5 .75 1 Box-Ljung Prob.
+----+----+----+----+----+----+----+----+ 1 –.072 .143 . *I . .253 .615 2 –.024 .141 . * . .283 .868 3 .176 .140 . I**** . 1.878 .598 4 .022 .138 . * . 1.904 .753 5 .021 .136 . * . 1.928 .859 6 .045 .135 . I* . 2.041 .916 7 –.090 .133 . **I . 2.498 .927 8 .150 .131 . I*** . 3.808 .874 9 –.075 .129 . **I . 4.146 .902
10 .034 .128 . I* . 4.217 .937 11 .015 .126 . * . 4.231 .963 12 –.059 .124 . *I . 4.457 .974 13 –.081 .122 . **I . 4.896 .977 14 –.025 .120 . *I . 4.940 .987 15 –.100 .188 . **I . 5.646 .985 16 .135 .117 . I*** . 6.994 .973
Plot Symbols: Autocorrelations * Two Strand Error Limits .
Total Cases: 47 Computable � rst lags: 45
the Box–Ljung test statistics generated for each lag. The best-� t univariate time-series forecasting model
for Singapore’s residential construction demand has been identi� ed as ARIMA(1,1,0) and the equation can be expressed as
Yt = Yt–1 + f 1Yt–1 – f 1Yt–2 + et
where Yt is the variable to be forecast at current time t; Yt–1 and Yt–2 are past values of Yt at time lags 1 and 2, respectively; f 1 is the � rst-order autoregression coef- � cient; and et is the random error term. For this study,
the parameter estimate of the AR(1) term f 1 was found to be 0.7279 in the analysis. The forecasts generated for the validation period (1993 Q1–1994 Q1) are given in Table 7.
Case study III: construction industry productivity forecasting in Singapore
To study the characteristics of the time series ‘value added per worker’ for Singapore’s construction indus- try productivity the ACF plot was generated as shown in Figure 8. It was con� rmed that the data series was nonstationary in the mean since the ACF drops off exponentially to zero. When � rst differences of the series were taken it was suspected that the series had not achieved stationarity, since the ACF plot did not behave much differently from the earlier plot. Essentially, the autocorrelations did not drop to zero rapidly, suggesting that nonstationarity was still present in the � rst-order differenced data series. In this instance, it was necessary to take � rst differences of the � rst-order differenced series. The ACF of the second- order differenced series indicated stationarity and, hence, was ready for model � tting.
At the model identi� cation stage, the behaviour of the ACF and the PACF indicated a tentative MA(1)
614 Goh and Teo
Table 5 Results of the AR(1) model � t with a constant term
Parameter Coef� cient u SE (u ) t-ratio Approx. estimate prob.
AR(1) 0.7237 0.0812 8.914 0.0000 Constant 18.0252 30.731 0.587 0.5594
Table 6 Results of the AR(1) model � t without a constant term
Parameter Coef� cient u SE (u ) t-ratio Approx. estimate prob.
AR(1) 0.7279 0.0803 9.063 0.0000
Figure 7 ACF of ‘GFA of residential developments commenced’
Autocorrelations: DEMAND Demand
Auto- Stand. Lag Corr. Err. –1 –.75 –.5 –.25 0 .25 .5 .75 1 Box-Ljung Prob.
+----+----+----+----+----+----+----+----+ 1 .955 .115 . ý****.************* * 68.446 .000 2 .873 .115 . ý****.*********** * 126.432 .000 3 .873 ,115 . ý****.********* * 172.432 .000 4 .655 .113 . ý****.******* * 205.684 .000 5 .547 .112 . ý***.****** * 229.435 .000 6 .445 .111 . ý***.**** * 245.438 .000 7 .348 .110 . ý***.*** 255.345 .000 8 .249 .110 . ý***.* 260.502 .000 9 .144 .109 . ý***. 262.268 .000
10 .038 .108 . ý* . 262.390 .000 11 –.065 .107 . *ý . 262.758 .000 12 –.157 .106 .***ý . 264.949 .000 13 –.242 .105 *.***ý . 270.234 .000 14 –.328 .104 ***.***ý . 280.105 .000 15 –.418 .103 ****.***ý . 296.412 .000 16 –.507 .103 ******.*** ý . 320.891 .000
Plot Symbols: Autocorrelations * Two Strand Error Limits .
Total Cases: 72 Computable � rst lags: 71
model. The PACF tailed off exponentially while the ACF dropped off to zero after lag 1. Table 8 gives the results of the MA(1) model � t. The MA(1) parameter estimate was highly signi� cant at the 10% level, implying that the correct model might indeed be a � rst- order moving average. The constant term was not necessary, in this case, since the process of differencing would have transformed the data to have a series mean about zero.
Diagnostic checking was carried out by observing the pattern of the autocorrelations of the residuals of the MA(1) model. The ACF plot, together with their respective Box–Ljung test statistic, indicated that all the autocorrelations were highly insigni� cant and, hence, the residuals were random and the � tted
model was adequate. The series was also ‘over� tted’ with an MA(2) model and the results are given in Table 9. They show clearly that the parameter estimates for MA1 and MA2 are insigni� cant at the 10% level. The best-� t model was con� rmed to be an ARIMA (0,2,1) and the derivation of the model’s equation is detailed below.
By using the backward shift operator B, in the form
BYt = Yt–1
a second-order difference is represented as
Y 0 t = (1 – B) 2Yt
If the second-order difference, Y0t, forms an MA(1) model, then the appropriate equation is
Forecasting tender price, demand and productivity 615
Table 7 Details of forecasts generated by the ARIMA (1,1,0) model
Period Actual Forecast Error Lower 95% Upper 95% values A values F (A–F) con� dence limit con� dence limit
1993 Q1 1638 1642 –4 1500 1785 1993 Q2 1681 1622 59 1479 1764 1993 Q3 1724 1712 12 1569 1854 1993 Q4 1752 1755 –3 1612 1898 1994 Q1 1760 1772 –12 1630 1915
Figure 8 ACF of ‘value added per worker’
Autocorrelations: VADDPWK
Auto- Stand. Lag Corr. Err. –1 –.75 –.5 –.25 0 .25 .5 .75 1 Box-Ljung Prob.
+----+----+----+----+----+----+----+----+ 1 .617 .208 . ý*******.*** * 8.826 .003 2 .135 .202 . ý*** . 9.268 .010 3 –.312 .197 . ******ý . 11.780 .008 4 –.506 .191 **.*******ý . 18.831 .001 5 –.395 .185 *.******ý . 23.410 .000 6 –.246 .178 . *****ý . 25.305 .000 7 –.073 .172 . *ý . 25.484 .001 8 .107 .165 . ý** . 25.906 .001 9 .247 .158 . ý*****. 28.347 .001
10 .261 .151 . ý*****. 31.342 .001 11 .093 .143 . ý** . 31.762 .001 12 –.084 .135 . **ý . 32.148 .001 13 –.289 .126 *.****ý . 37.399 .000 14 –.298 .117 *.****ý . 43.909 .000 15 –.133 .107 .***ý . 45.475 .000 16 .021 .095 . * . 45.525 .000
Plot Symbols: Autocorrelations * Two Strand Error Limits .
Total Cases: 22 Computable � rst lags: 19
(1 – B)2 Yt = (1 – u 1B)et
and substituting B into the above equation gives
Yt = 2Yt–1 – Yt–2 + et – u 1et–1
Here Yt is the variable to be forecast at current time t; Yt–1 and Yt–2 are past values of Yt at time lags 1 and 2, respectively; u 1 is the � rst-order moving average coef� cient; and et and et–1 are the random error term at current time t and lag 1, respectively. For this study, the parameter estimate of the MA(1) term u 1 was found to be 0.8134 in the analysis. The forecasts generated for the validation period (1995–1996) are given in Table 10.
Evaluation of forecasting accuracy
The forecasts generated by the models were evaluated on two measures of accuracy, namely, the root-mean- square error (RMSE) and the mean absolute percentage error (MAPE). The characteristics of these measures have been elaborated in other studies (Akintoye and Skitmore, 1994; Fitzgerald and Akintoye, 1995). In brief, these measures can be explained as follows. Root-mean-square error
RMSE = ! n
S t=1
———
e2t / n
where et = At – Pt, et is the forecast error at current time t; At is the actual value at current time t; and Pt
is the predicted value at current time t. Mean absolute percentage error
1 |et|MAPE = – n
S t=1
— 3 100 n At
These two measures were used because the prediction RMSE could be evaluated against the model RMSE, while the magnitude of the prediction MAPE could be gauged by the general acceptable limit of 10%. Based on the models’ out-of-sample forecasts, the computed values for these measures are given in Table 11.
Discussion of � ndings
There are basically two types of forecast, the point forecast and the prediction interval forecast (Newbold and Bos, 1990). The three models have been used to generate the two types of forecast, as shown in Tables 4, 7 and 10, and their predictive performance has been evaluated on the point forecast by applying the RMSE and MAPE. Essentially, a prediction interval forecast is an interval of values that is calculated based on a speci� ed level of con� dence that the actual value will be contained in the interval. For example, in case study I the prediction interval for 1991 Q4 implies that there is 95% con� dence level that the actual tender price index for Singapore’s public industrial projects will fall between 201 and 270.
Based on the point forecasts, the evaluation of the models’ forecasting accuracy has produced two posi- tive results. For the three models, the prediction RMSE is consistently smaller than the model’s stan- dard error (or RMSE) and also the prediction MAPE is consistently within the acceptable limit of 10%. Generally speaking, assessing the predictive ability of models based on out-of-sample forecasts is considered as an objective and stringent test. This is because the data set used to build the model is distinct from that used to test its accuracy.
In this instance, both absolute and relative measures of accuracy have been used in the evaluation. The stan- dard error is generated automatically as a descriptive statistic of a model and often it is useful to compare it with the prediction’s RMSE. As such, it serves as
616 Goh and Teo
Table 10 Details of forecasts generated by the ARIMA (0,2,1) model
Period Actual Forecast Error Lower 95% Upper 95% values A values F (A–F) con� dence limit con� dence limit
1995 66598 67505 –907 50715 84295 1996 78111 70755 7356 44700 96811
Table 8 Results of the MA(1) model � t
Parameter Coef� cient u SE (u ) t-ratio Approx. estimate prob.
MA1 0.8134 0.2112 3.8519 0.0013
Table 9 Results of the MA(2) model � t
Parameter Coef� cient u SE (u ) t-ratio Approx. estimate prob.
MA1 0.7071 0.5224 1.3536 0.1947 MA2 0.2361 0.3465 0.6813 0.5055
an absolute measure of accuracy for each model. On the other hand, the relative measure, MAPE, provides a basis for comparing among different models, as well as against the general acceptable limit. For example, it is observed that among the three models, the one generated in case study II to predict residential construction demand is the most accurate as its prediction MAPE is the lowest (1.07%). However, all three models have high predictive ability since their prediction MAPE is within the 10% acceptable limit.
Conclusion
This paper presents an empirical study of the use of a benchmark univariate time-series forecasting tech- nique, the Box–Jenkins approach, to model and forecast three distinct construction industry variables, price, demand and productivity. The derived models have been used to generate out-of-sample forecasts to provide a basis for assessing the predictive performance of the models and, hence, adequacy of the approach. The results of the evaluation can be summarized as follows. 1. The prediction RMSE of all three models is consistently smaller than the model’s standard error, implying the models’ good predictive performance. 2. The prediction MAPE of all three models consis- tently falls within the general acceptable limit of 10%. Among the three models, the most accurate is the demand model (MAPE = 1.07%), followed by the price model (MAPE = 5.18%) and the productivity model (MAPE = 5.39%).
Despite the favourable results of the study, it has to be highlighted that the derived models are suitable for producing only short-term forecasts, that is, one to � ve periods ahead. In this respect, the choice to build a complex econometric model or a simple univariate model has to be based on considerations such as time, data and cost. In any case, even if conditions permit the building of a larger model, the need for a simpler and easier-to-use model is still justi� able, since it would serve as a benchmark or control model to check on the larger model’s performance.
Broadly speaking, the purpose of building an indus- try-level forecasting model is to treat its predictions as targets for the industry to plan its future workload and
use of limited resources, such as land, � nancial capital and manpower. In essence, the tender price index fore- casting model allows the industry to assess current against future prevailing price levels in order to achieve higher price competitiveness by either lowering costs or increasing productivity. Having a construction demand forecasting model would assist the industry in antici- pating any future changes and constraints, with the objective of maintaining a healthy balance between demand and supply of construction items such as resi- dential buildings. The ability to predict industry pro- ductivity enables future levels of competitiveness in the industry to be raised through increasing value added in construction or reducing manpower utilization.
Although the Box–Jenkins technique has been proved in the study of Singapore to be a reliable time- series forecasting approach to construction industry forecasting, its inherent shortcomings should not be ignored. Recent studies in empirical macroeconomics involving nonstationary and trending variables, such as income, consumption and exchange rates, have explored more interesting and appropriate methods of trend analysis. Granger causality analysis has the ability to test for signi� cant short run causal relationship between two trending variables, while a technique known as ‘cointegration’ analyses the long run rela- tionship of such variables. Essentially, these new techniques have combined the principles of causality and time series to produce a more comprehensive approach to analysing related trending variables.
A report by the Royal Institution of Chartered Surveyors (RICS, 1996) has presented a review of the application of non-traditional forecasting techniques and, in the process, highlighted the works of Barras and Ferguson (1987a, b). Their progressive studies have culminated in the development of a theoretical model that incorporates an error correction term into a transfer function model to capture both short run dynamics and long run behaviour of the private prop- erty market. In concurrence, the RICS undertook a pilot study to examine the use of a vector error correc- tion model (VECM), incorporating a vector of six variables, to forecast construction orders and output. Despite promising results from the VECM approach, it was acknowledged that the major dif� culty had been in the interpretation of the coef� cients of the derived equations. To assess the performance of the VECM, it was suggested that a comparison be made with some traditional structural equation models. This is, in fact, necessary since there has to be a formalized way of evaluating state-of-the-art techniques against conven- tional ones which serve as appropriate benchmarks. Further adoption of such new techniques would have to rely on more conclusive � ndings in future studies of construction industry forecasting.
Forecasting tender price, demand and productivity 617
Table 11 Measures of accuracy for out-of-sample forecasts
Model Prediction Prediction RMSE RMSE MAPE
Case study I 17.27 12.00 5.18 Case study II 73.79 27.55 1.07 Case study III 7955.47 5240.87 5.39
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