Diagnosing and Modeling Autocorrelation

You can diagnose autocorrelation with an autocorrelation plot created with the ARIMA procedure.

ods graphics on;
ods select ChiSqAuto SeriesACFPlot SeriesPACFPlot;
proc arima data=Chemical plots(only)=series(acf pacf);
   identify var = xt;
run;
quit;

Refer to SAS/ETS User's Guide for details on the ARIMA procedure. The output, shown in Figure 17.191 and Figure 17.192, indicates that the data are highly autocorrelated with a lag 1 autocorrelation of 0.83.

Figure 17.191: Autocorrelation Check for Chemical Data

Individual Measurements Chart

The ARIMA Procedure

Autocorrelation Check for White Noise
To Lag Chi-Square DF Pr > ChiSq Autocorrelations
6 228.15 6 <.0001 0.830 0.718 0.619 0.512 0.426 0.381
12 315.34 12 <.0001 0.360 0.364 0.380 0.347 0.348 0.354
18 406.76 18 <.0001 0.349 0.371 0.348 0.353 0.368 0.341
24 442.15 24 <.0001 0.303 0.261 0.230 0.184 0.141 0.098


Figure 17.192: Autocorrelation Plots for Chemical Data

Autocorrelation Plots for Chemical Data
External File:images/ac2outb1.png


The partial autocorrelation plot in Figure 17.192 suggests that the data can be modeled with a first-order autoregressive model, commonly referred to as an AR(1) model.

\[  \tilde{x}_{t} \equiv x_{t} - \mu = \phi _{0} + \phi _{1} \tilde{x}_{t-1} + \epsilon _{t}  \]

You can fit this model with the ARIMA procedure. The results in Figure 17.193 show that the equation of the fitted model is $\tilde{x}_{t} = 13.05 + 0.847 \tilde{x}_{t-1}$.

ods select ParameterEstimates;
proc arima data=Chemical;
   identify var=xt;
   estimate p=1 method=ml;
run;

Figure 17.193: Fitted AR(1) Model

Individual Measurements Chart

The ARIMA Procedure

Maximum Likelihood Estimation
Parameter Estimate Standard Error t Value Approx
Pr > |t|
Lag
MU 85.28375 2.32973 36.61 <.0001 0
AR1,1 0.84694 0.05221 16.22 <.0001 1