3.1. The Data
The data set consists of quarterly observations for the period 2001:Q1–2018:Q4 for residential building costs of low-rise and high-rise buildings in New Zealand, and two explanatory variables including housing prices and work volume. The training sample is from 2001:Q1 to 2014:Q4, a total of 56 observations. The validation sample used the remaining 16 observations, from 2015:Q1 to 2018:Q4. The cost data is the approximate cost per building type per square meter in New Zealand.
Note that, this study categorised residential building costs into two major segments, namely, low-rise building and high-rise building costs. The low-rise residential building is believed to be the most representative of the general house type prevailing in New Zealand, while high-rise residential buildings are apartments or retirement villages. LBC and HBC are the building cost indexes of residential low-rise and high-rise buildings, respectively, which are obtained from the QV cost builder that is widely used in the construction industry of New Zealand. Cost estimation at the early stage is highly based on the use and availability of historical cost data and information [
42].
As a measure of house prices (HP), the study employed house price index is quarterly data and spans from 2001:Q1 to 2018:Q4 obtained from Reserve Bank of New Zealand (RBNZ). The annual number of building consents issued is the major indicator of new building projects in New Zealand. The number of building consents is a significant indicator of construction work volume which can be used as a proxy of construction work volume in the present study. Building consent series (BC) indicates the quarterly number of building consents awarded in New Zealand officially reported by Statistics of New Zealand, from 2001:Q1 to 2018:Q4. A complete set of descriptive statistics is available upon request.
3.2. ARIMA Model
The ARIMA model is a combination of the autoregressive (AR) model, differencing of time series, and moving average model (MA). The modelling process of the ARIMA technique was proposed by [
43]. The model ARIMA (p,d,q)(P,D,Q)
L can be expressed in Equation (1). The basic steps of applying an ARIMA model include identification, estimation, diagnosis checking, and forecasting. The examination of the autocorrelation function (ACF) and partial autocorrelation function (PACF) to gain insights into the characteristics of the time series is the first stage of the approach. The series is differenced to attain stationarity. The ACF and PACF plots of the stationary series are examined to identify an appropriate model. At the estimation and diagnostic checking stages, the portmanteau test is used to evaluate the identified model. The process is repeated until a reliable model is obtained. Finally, the selected model is used to yield out-of-sample forecasts.
where
B is the backshift operator;
L is the number of seasons in a year (
L = 4 for quarterly data and
L = 12 for monthly data);
is a constant term;
is a random shock;
is a non-seasonal autoregressive parameter;
is a seasonal autoregressive parameter;
is a non-seasonal moving average parameter,
is a seasonal moving average parameter.
3.3. Bivariate Transfer Function Model
The transfer function approach is a time series model that involves more than one time series and explains explicitly the dynamic characteristics of the process [
44]. The general form of the function can be written in Equation (2).
where
where
is the numerator polynomial,
is the denominator polynomial, and
is the explanatory variable,
is the independent noise term generated by an ARIMA process. However, if the time series exhibits non-stationarity, an appropriate degree of differencing should be applied to the series to achieve stationarity. In reality, stationary dependent and explanatory variables are necessary to develop a transfer function model. As a result, the transfer function model can be more generally expressed in Equation (6).
where
where
D and
d are orders of regular and seasonal differencing that transform non-stationary
to stationary
;
D′ and
d′ are orders of regular and seasonal differencing that transform non-stationary
to stationary
;
l is the period of seasonality;
is an ARIMA process that transfer noise term
to white noise
.
According to [
43], a comprehensive procedure to develop a transfer function model is displayed. The cross-correlation function (CCF) was introduced to identify the model orders such as delay order b, numerator polynomial order s, denominator polynomial order r. However, the spurious effects usually appear due to the autocorrelation in the input series. Thus, a pre-whiten filter is introduced to alleviate this spurious effect. The main principle of the pre-whiten filter transfers the input variable to a white noise process. In other words, the autocorrelation in the input variable series was removed.
The first step in pre-whitening is to identify and fit an ARIMA model to the stationary input data . An appropriate ARIMA model for the input data might be written in Equation (9).
Rewriting Equation (6), the residuals are given in Equation (10).
After pre-whitening the input data, the next step is to pre-whiten the stationary output data . This is to filter the output data through the same ARIMA model with the same model parameters fitted to the input data. In other words, is substituted by in Equation (10) to obtain the filtering equation for the pre-whitened output, as shown in Equation (11).
The next step is to compute the cross-correlation function between the pre-whitened input and the pre-whiten output . It should be emphasised here that all the cross-correlation for negative should be insignificant. Since the model assumed that the input variable has effects on the output variable, no feedback effects exist. The first significant spike lag indicates a time delay order b. Then the damping pattern of the cross-correlation function can be used to identify the denominator polynomial order r. If the CCF fades out in an exponential decay pattern, this suggests r = 1. If the CCF damp down in an oscillating pattern, this indicates r = 2. While the numerator polynomial order s is equal to the number of lags that exist between the first spike in the CCF and the beginning of the clear damping down pattern. For example, if the first spike starts at lag 2 and the CCF exhibits decay after lag 4, this suggests s = 4 − 2 = 2. At this point, it should be noted that the study does not attempt to search the right values of the model orders. The models are only tentatively identified, further diagnostics checking should be performed to obtain adequate models.
The transfer function model parameters are estimated based on the identified model orders at this step. After the model has been identified, the error term should be checked. If significant autocorrelation exists in the error series, an ARIMA model should identify and fit to the error series and then be incorporated into the overall model in Equation (2). The model can be identified by examining the auto-correlation function (ACF) and partial auto-correlation function (PACF) of the error. In this study, SPSS 23 was used to develop the transfer function models. The parameter estimates and standard errors suggest whether the parameters should be omitted from the model.
For diagnostics checking, the cross-correlation between the residuals from the developed transfer function model and the residuals of the ARIMA model that used to pre-whiten the input variable. If there is no significant spike that exists in the CCF, indicating there is no significant cross-correlation left between the input and output variables, the transfer function model is correctly identified. If that is not the case, this indicates that some information still exists in the output variable that can be explained by the input variable. Moreover, the autocorrelation function (ACF) of the residuals of the transfer function model should also be checked. If the model is appropriately identified, there should be no significant autocorrelation in the residuals.
3.4. Multivariate Transfer Function Model
The transfer function model can be further expanded to include multivariate, explaining variables [
25]. If there are two or more independent variables, the cross-correlation to identify a transfer function relating the dependent variable to each independent variable. The model development process follows the strategy described by [
43].
So, in this study, the two time series (input X
1t and X
2t) to estimate the dependent variable of another time series (output Y
t) is applied. This is done by modelling a linear system, which takes the form in Equation (12).
where
and
are, respectively, the polynomials of the s and r orders;
and
can be identified by examining the cross-correlation between pre-whitened
and
;
and
can be identified by examining the cross-correlation between pre-whitened
and
. The transfer function in this study used the house price and work volume for the primary series (
and
).