## 4 Case Example: Australian Public-Private Partnership (PPP) Schools
### Forecasting Analysis in \(t=25\) and Discussion  
To validate a time-based performance curve, a long-term PPP school performance is predicted. An autoregressive integrated moving average model is used to forecast the actual quarterly service payments in future, based on historical data. It is a famous model in time-series analysis in statistics and econometrics and fitted to time series data either to better understand the data or to predict future points in the series, and has been applied in some cases where data show evidence of non-stationarity. This model considers the evolving variable of interest is regressed on its own lagged values and indicates the regression error is actually a linear combination of error terms whose values occurred contemporaneously and at various times in the past. It has the following general form: where \(X_{t}\) is the actual quarterly service payments in past years in this study, \(L\) is the 'backward operator' with the quarterly service payments, i.e., \(L\cdot X_{t}~{}=~{}X_{t-1}\), and \(L^{i}\) means L with power \(i\). \(\varphi_{i}\) and \(\theta_{i}\) are fixed parameters needed to be estimated. \(\varepsilon_{t}\) is the measurement error with zero mean. When the time series \(X_{t}\) is non-stationary, a differencing step (corresponding to the 'integrated' part of the model) is used to get a stationary process \(X_{t}^{*}\) by the first Equation in (8). \[\left\{ \begin{aligned} & X_{t}^{*}~{}=~{}(1-L)^{d}\cdot X_{t}\\ &\left(1-\sum\limits_{i~{}=~{}1}^{p}\varphi_{i}L^{i}\right)X_{t}^{* }~{}=~{}\left(1+\sum\limits_{i~{}=~{}1}^{q}\theta_{i}L^{i}\right)\varepsilon _{t}\end{aligned} \right.\, \tag{8}\]  
The parameter \(d\) is determined by a unit root test with the least order coefficient of less than 0.05 in its \(P\) value, rejecting the null hypothesis that the process is non-stationary. Then the second equation in (6) is to handle the process \(X_{t}^{*}\). \(p\) is determined by the time-lag with the highest order coefficient of less than 0.05 in its \(p\)-value, rejecting the null hypothesis when the null hypothesis is assumed to be true. This allows a 5% probability of a Type I error to reject the null hypothesis when it is true. The weighting and strength between lagged and current periods are inferred from \(\varphi_{1}\), \(\varphi_{2}\),..., \(\varphi_{i}\), similar as \(\theta_{1}\), \(\theta_{2}\),..., \(\theta_{i}\). The procedures for deducing the parameter \(d\),\(p\),\(q\) of the autoregressive integrated moving average model in this study is briefly described using the following steps: determining the value of \(d\) by unit root test, determining the value of \(p\), \(\varphi_{1}\), \(\varphi_{2}\),..., \(\varphi_{i}\), and \(\theta_{1}\), \(\theta_{2}\),..., \(\theta_{i}\) by regression and the maximum loglikelihood estimate method, and identification of inconsistencies between regressed values and actual data. The mean value of \(A(t)\) is derived from history data of the first 9 years and forecasted using this model, please see Figure 7a. To predict the end-user's perceived service quality \(P(t)\) in future, a power-law curve is fitted using current data (t = 9) under the hypothesis that: when t = 0, the end user's satisfaction is 7, thus in the ideal situation, the end-user is satisfied with the new PPP school project at the earliest beginning of operations, and when the T goes to infinite, the satisfaction score is 3, thus the bottom line for the end-user is 3, meaning finally no availability for all the services; please see Figure 7b. Based on this synthetic projection, the end-user's perceived service quality in the year 25 is forecasted as \(\mu_{P,25}\ =\ 5.203\), \(\sigma_{P,25}\ =\ 0.356\). Meanwhile, \(\mu_{A,25}\ =\ 4.871\) according to the ARIMA method, and \(\sigma_{A,25}\) is assumed as the same with that in t = 9, i.e., \(\sigma_{A,25}\ =\ 0.014\).