6.
Process or Product Monitoring and Control
6.4. Introduction to Time Series Analysis 6.4.4. Univariate Time Series Models


Series F 
We analyze the series F data set
in
Box, Jenkins, and Reinsel, 1994. A plot of the 70 raw
data points is shown below.


Model Identification 
We compute the
autocorrelation function
(ACF) of the data for the first 35 lags to determine the
type of model to fit to the data. We list the numeric results and
plot the ACF (along with 95 % confidence limits) versus the lag number.
Lag ACF 0 1.000000000 1 0.389878319 2 0.304394082 3 0.165554717 4 0.070719321 5 0.097039288 6 0.047057692 7 0.035373112 8 0.043458199 9 0.004796162 10 0.014393137 11 0.109917200 12 0.068778492 13 0.148034489 14 0.035768581 15 0.006677806 16 0.173004275 17 0.111342583 18 0.019970791 19 0.047349722 20 0.016136806 21 0.022279561 22 0.078710582 23 0.009577413 24 0.073114034 25 0.019503289 26 0.041465024 27 0.022134370 28 0.088887299 29 0.016247148 30 0.003946351 31 0.004584069 32 0.024782198 33 0.025905040 34 0.062879966 35 0.026101117 

Model Fitting 
We fit an AR(2) model to the data.
$$ X_t = \delta + \phi_1 X_{t1} + \phi_2 X_{t2} + A_t $$
The model fitting results are shown below.
Source Estimate Standard Error    φ_{1} 0.3198 0.1202 φ_{2} 0.1797 0.1202 δ = 51.1286 Residual standard deviation = 10.9599 Test randomness of residuals: Standardized Runs Statistic Z = 0.4887, pvalue = 0.625 

Forecasting 
Using our AR(2) model, we forcast values six time periods into the future.
Period Prediction Standard Error 71 60.6405 10.9479 72 43.0317 11.4941 73 55.4274 11.9015 74 48.2987 12.0108 75 52.8061 12.0585 76 50.0835 12.0751The "historical" data and forecasted values (with 90 % confidence limits) are shown in the graph below. 