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Weekly Updates[DELTA][X.sub.t] = [delta][X.sub.t-1] + [m.summation over (i=1)] [[beta].sub.i][DELTA][X.sub.t-i] + [[epsilon].sub.t], (1)
[DELTA][X.sub.t] = [alpha] + [delta][X.sub.t-1] + [m.summation over (i=1)] [[beta].sub.i][DELTA][X.sub.t-i] + [[epsilon].sub.t], (2)
[DELTA][X.sub.t] = [alpha] + [[beta].sub.t] + [delta][X.sub.t-1] + [m.summation over (i=1)] [[beta].sub.i][DELTA][X.sub.t-i] + [[epsilon].sub.t]. (3)
The symbol of [alpha] is an intercept and the product of [beta] and t is a deterministic trend. Equation 1 contains no intercept and trend; this means that X is a stationary time series with a zero mean if the null hypothesis is rejected. In the same way, equation 2 comprises an intercept but no trend; this means that X is a stationary time series with a non zero mean. Equation 3 includes an intercept and a trend; this means that X is a stationary time series around a deterministic trend.
Table 1 shows the unit root test results of eight capital cities, using the ADF unit root test and the PP unit root test. The null hypothesis of non-stationarity was performed at the 1% and 5% significance levels. There are three different null hypotheses of the time series processes in this test: process as a random walk, process as a random walk with drift, and process as a random walk with drift around a deterministic trend. They are shown in Table 1 respectively: no trend and intercept, intercept without trend and, intercept and trend. The results shows that eight capital cities' house price index data series are not stationary at the level form but stationary after the first difference at the 1% and 5% significance levels. That is, all the eight data series are I(1) which denotes that the time series is integrated at the first difference level.
5. CONSTRUCTING THE VECTOR AUTOREGRESSION MODEL
Unfortunately, the VAR model using the price indices as variables directly, does not satisfy the stability condition; due to the house price indices probably not being stationary in level form. If the VAR model does not satisfy the stability condition, certain results such as impulse response standard errors are not valid (Lutkepohl, 1993; Greene, 2000). This will lead to an invalid conclusion. In this case, the house price indices series after first difference are used to construct the VAR model.
There are at least two advantages when using the first difference data series to explain the impulse response function. Firstly, it focuses more on the increase or decrease trend rather than the actual house prices change. Because the first difference data series is the increase or decrease between every two consecutive quarters, a strengthening or weakening of the trend will be detected by the impulse response function. Secondly, it captures more information on the shocks of house prices, because the first difference data shows the changes in the past two quarters while the level data shows the changes in one quarter in impulse response function. In this section, a regional housing market affecting the others means that the movement trend change in a market could affect the trend change in the others. The symbols such as 'D(Adelaide)' in tables or figures stand for the first difference series.
5.1. Selection of optimal lag
One of the biggest and common practical problems in the VAR model is to select the optimal lagged term. One of the common and simple approaches in selecting optimal lag length is to reestimate a VAR model, reducing lag length from a large lag term until 0. In each of these models, the smallest value of the Akaike information criterion and the Schwarz criterion are used to select the optimal lag length (Grasa, 1989; DeJong et al., 1992; Maddala and Kim, 1998; Gujarati, 2003). Using VAR estimates, the optimal lag length can be determined by comparing the Akaike information criterion (AIC) and the Schwarz criterion (SC) (Grasa, 1989). Moreover, the judgement of the optimal lag length should still take other factors into account: for example, autocorrelation, heteroskedasticity, possible ARCH effects and normality and normality of the residuals (Asteriou, 2005). In this study, 5 criteria: Sequential modified LR test statistics (LR), Final prediction error (FPE), Akaike information criterion (AIC), Schwarz criterion (SC) and Hannan-Quinn information criterion (HQ), which have been introduced by Lutkepohl (1993) were inspected. Similarly, the smallest value of these 5 criteria points to the optimal lag length.
Table 2 shows the results of VAR lag order selection criterion. The first left hand column shows the lag orders from 0 to 4. The LR, FPE, AIC, SC and HQ are the 5 criteria mentioned above. The numbers with an asterisk are the smallest value in each of criteria. Before selecting the lag length, two situations should be identified. Firstly, too short a lag length in the VAR may not capture the dynamic behaviour of the variables (Chen and Patel, 1998), so the optimal lag length would be selected by the smallest lag shown under the criteria. Secondly, DeJong et al. (1992) point out that too long a lag length will distort the data and lead to a decrease in power. Based on the results, one lag which is considered as one quarter is selected in the VAR model; that is VAR (1).
5.2. Test for the stability of the VAR model
Once the VAR model is constructed, the stability of the model should be verified. If the VAR model does not satisfy the stability condition, certain results such as impulse response standard errors are not valid. Stability is achieved if the characteristic roots of the matrix of coefficients have a modulus of less than one. Table 3 shows the results of the roots of the characteristic polynomial. The results show that all roots are less than 1 and no root lies outside the unit circle. It indicates that the VAR(1) model satisfies the stability condition. So the results of the impulse response function deriving from the VAR(1) are valid in our study.
6. IMPULSE RESPONSES AMONG REGIONAL HOUSING MARKETS
One of the key elements of the VAR model is the impulse response analysis. It presents the dynamic effect of each exogenous variable response to the individual unitary impulse from other variables. The IRF can explain the current and lagged effect over time of shocks in the error term. It estimates the sensitivity of one variable to the change in another. The impulse response function (IRF) derived from the VAR model is used to trace out the response of one variable to the shocks in the error term of another variable. The IRF can explain the current and lagged effect over time of shocks in the error term.
6.1. Impulse response of regional house prices
Figure 2 shows the impulse response results of the eight capital cities' housing markets individually. It traces out the response of each regional housing market to the shocks in the error terms of other markets. There are eight curved lines in each figure. Seven lines in the eight starting from zero in time 1 explain the impulse response of one housing market to the other seven markets. The impulse response of the seven markets is assumed as zero in the first quarter and these seven markets are assumed to receive a one positive unit standard deviation shock from external markets in the first quarter. The eighth line explains the response of one market to its past shock. The X axis shows the quarters and the Y shows the shock in the movement trend. The positive symbol does not mean an increase in house price. It means an increase in movement trend is strengthened or a decrease in movement trend is weakened. In the same way, the negative symbol means an increase in trend is weakened or a decrease in trend is strengthened. In short, a positive symbol means a favourable effect on house prices growth and a negative symbol means an adverse effect. In addition, the value shown in the figure indicates a change on the house prices movement trend.
Figure 2 shows that, all of the capital cities are impacted more from themselves than the exogenous factors. Canberra and Hobart received a stronger impact (positive) from the past performance of themselves, while Darwin has a negative impact on itself after the fourth period. Each of the five housing markets in Sydney, Melbourne, Adelaide, Brisbane, and Perth received more influence from the Canberra or Hobart markets than others. The influences on the five markets from Darwin are negative. Two conditions are applied to judge the epicentre, which aggregates Australian housing market in this study. The first one is that the most important influence is from the past performance of the market itself. The second one is that this market should transfer more impact than other markets. In this case, it can be concluded that the main epicentres in the Australian housing market are Canberra and Hobart. Canberra is the key engine of the area of Adelaide, Brisbane, Canberra, Melbourne and Sydney while Hobart is another key engine of the area of Hobart and Perth. However, the Darwin housing market is more independent which can also be detected in the diffusion pattern.
[FIGURE 2 OMITTED]
As the one of most import housing markets in Australia, Sydney does not exert its power as expected. The influence from Sydney is weaker than from Canberra and Hobart. This obviously supports the results in Figure 2 that Sydney does not affect other markets directly. It is surprising that Sydney is not the main epicentre in the aggregate Australian market. Similar findings can be seen in the study by Stevenson (2004) which showed how Dublin is not the most influential market in the Irish housing market. The Rural areas' market was identified as the 'surrogate' of the Dublin market and Dublin affected the provincial markets through its surrogate. If so, Canberra would be seen as the surrogate of Sydney, because the impulse response results show that Canberra market influenced the Sydney market more than Sydney market itself does. Moreover, Canberra and Hobart deliver more influence (positive) than the others.
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