The dataset for this study was obtained from Cabanatuan City Health Office, Immunization Department. The dataset consists of the number of infant per month receiving the measles immunization for 60 month period from January 2014 to December 2018 in Cabanatuan City, Nueva Ecija, Philippines.

Autoregressive Integrated Moving Average model is popular and widely used stochastic time series model. It is a combination of three statistical models. An ARIMA model is denoted as an ARIMA model (p, d, q), where p is the number of autoregressive terms, d is the degree of differencing involve, and q is the number of moving-average terms [11] [12] [13]. This study follows the Box and Jenkins methodology [14], which composed of four main steps as shown in Figure 1.

1) Model Identification. In this step, the order of respective time series model is determined, the order of differencing “d”, the order of AR process which is “p” and order of MA process is “q”. This mainly leads to the use of graphical procedures such as plotting the series, the ACF and PACF.

2) Estimation of Model Parameters. Involves the estimation of the parameters of the different models using previously available data and proceeds to the first selection of models using the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) values.

3) Diagnostic checks (model checking). This step determines whether the model(s) specified and estimated is valid. Thus, the residual values are normally distributed, no significant periodicity is present in the residual, and the residual series is uncorrelated. The process will start again if this step is not fulfilled.

4) Forecasting. In this step, the model is now ready to be tested on real data to forecast future values of immunization coverage.

The first stage in developing an ARIMA model is to determine if the series is stationary. The graph below (Figure 2) depicts that the series has a large variance. To reduce the variance, the “raw data” was transformed into its natural logarithmic equivalent. The graph of the log (immunization) is shown below.

Figure 2 and Figure 3 show that the mean of the series is a function of time

(i.e., the mean of the series decreases as time increases). Also, there are some points in time where the variances are smaller and larger (i.e., the variances are not constant over time). Visually, both sequence charts show that the monthly measles immunization data is non-stationary.

Augmented Dickey-Fuller (ADF) test was conducted to test the hypothesis statistically that the natural log-transformed series has a unit root or not. The series is non-stationary if there is a unit root and the series is stationary if there is no unit root. Results of the ADF test is summarized in Table 1(a) below. The results of the three test equations are inconsistent, an indication that the series is non-stationary. The finding using the ADF test is consistent of the sequence charts above.

Differencing the natural log-transformed series can transform the series from non-stationary series to stationary. The graph below shows that the “first difference of the log-transformed series” is stationary (Figure 4).

The ADF test for the first difference of the natural log-transformed data confirms that a stationary series can be achieved after the first differencing.

The ACF and PACF of the first difference natural log-transformed series are shown below. The height of bars in the ACF represents the autocorrelation coefficients. The two parallel lines represent the confidence interval. All bars fall inside the confidence interval, indicating that the autocorrelations are statistically non-significant at 5% level. This means that an ARIMA with Moving Average component of order 0 can be tentatively considered, that is, MA (0) (Figure 5).

The height of bars in the PACF represents the partial autocorrelation coefficients. Though the partial autocorrelation coefficients fall within the confidence level, that is, the partial autocorrelation coefficients are statistically nonsignificant at 5%, some are larger than the others. The PACF suggests an ARIMA model with AR of order 0 can be considered. AR of orders 1, 4, 8, and 9 can also

Note: No unit root indicates a stationary series. To be stationary, the three test equations should show no unit root.

be considered. In short, ARIMA with AR (0), AR (1), AR (4), AR (8), and AR (9) can be considered.

Based on the results of the ACF and PACF, the following ARIMA Models can be considered:

ARIMA Model 1: ARIMA (0, 1, 0)

ARIMA Model 2: ARIMA (1, 1, 0)

ARIMA Model 3: ARIMA (4, 1, 0)

ARIMA Model 4: ARIMA (8, 1, 0)

ARIMA Model 5: ARIMA (9, 1, 0)

After the estimation of the parameters, the Root Mean Square Errors (RMSE) and Normalized BIC are summarized as shown in Table 2. The smaller the RMSE and Normalized BIC, the better the model. In this study, the normalized BIC is used because its value is not a function of the number of parameters.

Using the Normalized BIC as the basis for the selection of the best ARIMA Model, model #1, the ARIMA (0, 1, 0), emerged as the best model with the smallest Normalized BIC of 8.673. The estimates of the model parameters generated by the IBM SPSS Forecasting module are presented below. In equation form, the ARIMA (0, 1, 0) can be written as: DLog(Immunization) = −0.009 (Figure 6).

ARIMA (0, 1, 0) was used to forecast the monthly measles immunization coverage for the next 36 months from January 2018 to December 2020. Figure 10 shows the actual data (blue line) and fitted/forecasted values (red line) using the

equation. Table 3 shows forecast values of immunization coverage from January 2018 to December 2021 using ARIMA (0, 1, 0).