• Group Name: Insights
  • Group Members:
  1. Arn Joseph Napinas (A0178485L)
  2. Akwila Jeremiah Suria (A0137701M)
  3. Jayanthi D/O Duraisamy (A0178542X)
  4. Shi Peigong (A0178286M)
  5. Yang Chia Lieh (A0178500J)
  6. Lee Geok Joo (A0178392R)


This study analyzes the dynamic relationship between an input and output time series using conventional transfer function model. The following sections of this report will describe the data source and data records, identify the leading indicator (by pre-whitening), and the necessary analysis and diagnostic check to make an appropriate conclusion between the input and output time series.

Data Preparation

Data Preparation

Quarter: YYYY-Quarter
– rates of re-entry into employment in a quarter for retrenched residents
– Statistics on re-entry into employment allows government to gauge how well retrenched workers are able to secure a new job
– Percentages are expressed as a value over 100, i.e. “100” represents 100%
– Job Vacancy to Unemployed Person Ratio
– calculated by taking the ratio of the estimates of the total number of job vacancies for the whole economy to the total number of unemployed persons
– Ratio

A total of 65 records are used in this study, which starts from year 2002 first quarter until year 2018 first quarter.

Data Source:

Develop Transfer Function Model

1. Time series plot

In Figure 1, a set of 65 pairs of simultaneous observations of reentry-rate and jb_to_ue are plotted in a time series manner with respective quarters of each year. They denote the input by x and the output by y, respectively. As it is difficult to assess the delayed dynamic relationship between reentry_rate and jb_to_ue in the early stage, an assumption is made that reentry_rate is the output variable, y and jb_to_ue is the input variable, x. This assumed relationship will be verified and confirmed during the series pre-whitening stage.

figure 1

2. Pre-Whitening

If the input is autocorrelated, the effect of any change in the input itself will take some time to play out. Therefore, the subsequent effects of the input on the output tend to linger around. A spurious effect of changes in y may appear to have led to the change in x. To alleviate the spurious effect in the estimated cross correlation function caused by the autocorrelation in the input, pre-whitening method is applied to make the input look like white noise. Below is the summary of pre-whitening steps.

Figure 2.pngStep 1:
The first step in pre-whitening involves identifying and fitting a time series model to the input data x. In Time series plot section, it is assumed that jb_to_ue is the input variable, x. The autocorrelation and partial autocorrelation of input data x is shown in Figure 2. The ACF dies down quickly and the PACF cut off after the lag 1. These two plots, when combined, indicate that an appropriate model might be a first order autoregressive AR(1) Model.

figure 3.pngStep 2:
The residuals after fitting the AR(1) model should look like “white noise” as shown in Figure 3. Figure 4 shows the plot of ACF and PACF for the residuals which has no significant autocorrelation at higher lags and deem the model to be appropriate.
figure 4.pngStep 3:
After pre-whitening the input, the next step is to pre-whitening the output, y. Figure 5 and Figure 6 show the residuals plot, ACF and PACF plot after fitting the AR(1) model into the output.

figure 5.pngStep 4:
From the flow chart, the next step in the pre-whitening process is to compute the cross-correlation function between the pre-whitened input and output. The correlation function is shown in Figure 7.
In Figure 7, it provides a picture of input-output relationship. Also, it shows that the cross correlations for lag 2 and 3 are significant and it indicates that there is a time delay of about 2-time units in the system and a dynamic relationship stretching over 2 time periods that initially build up and then fades out. Thus, the leading indicator is jb_to_ue which prove that the assumption in Step 1 is correct.

figure 6Step 5:
The approximate transfer function weights for the data in Figure 7. Clearly, the first nonzero weight starts at lag 2, indicating that the delay is 2 times units, that is b=2. For the nonzero weights, one plausible interpretation is that the weights exhibit exponential decay after lag 2 which suggests that s = 2-2 = 0. The exponential decay after lag 2 could be due to a single exponential decay term or sum of more than one exponential decay terms. This suggests that r = 1 or 2. To build the transfer function model, fit r with value of either 1 or 2 to obtain the optimal model.

3. Transfer function Model

With all parameters obtained from above process, the transfer function model is generated as shown in Figure 8 and Figure 9:

a) Transfer Function Model with r=1:

figure 7.pngb) Transfer Function Model with r=2:

figure 8

4. Diagnostic Check

In diagnostic check, residuals check and parameters check are performed.
For the residual check, from the ACF and PACF plots of the residuals for transfer function model 1 and 2 in Figure 10 and Figure 11, it can see that there is no autocorrelation left in the residuals.

figure 9.png

figure 10.pngFor the parameter check, in it shows that in transfer function model 1, Figure 8, the first and second parameters are not significant and in the transfer function model 2, Figure 9, second parameter is not significant. As both models has insignificant terms, the built transfer model is not optimal. It is recommended that alternative regression strategies (e.g. linear transfer function modeling strategy for multiple input model, where total_paid_hours in the same dataset can be included as an input variable) be considered.


In this study, the relationship between the reentry_rate (rates of re-entry into employment in a quarter for retrenched residents) and jb_to_ue (Job Vacancy to Unemployed Person Ratio) was modeled using Box-Jenkins approach. This study uses AR(1) pre-whitening process to “de-correlate” the input and output series, and based on these pre-whitened series to build a transfer function. However, during the diagnostic check, it is found that not all the parameters are significant, which lead refitting the data with various ARIMA model for the pre-whitening process. After several iterations, it seems that there is no better ARIMA model to fit the input data and generate the transfer function. Therefore, this result may suggest that there is no dynamic delayed relationship between reentry_rate and jb_to_ue. Besides, it is suspected that the failure to build an appropriate transfer function could be due to insufficient data set for the analysis, since there are only 65 observations in the study data set.