1. Exploratory Data Analysis
1.4. EDA Case Studies
1.4.2. Case Studies
1.4.2.3. Random Walk

1.4.2.3.3.

Develop A Better Model

Lag Plot Suggests Better Model

Since the underlying assumptions did not hold, we need to develop a better model.

The lag plot showed a distinct linear pattern. Given the definition of the lag plot, Y_i versus Y_i-1, a good candidate model is a model of the form

Fit Output

A linear fit of this model generated the following output.

 LEAST SQUARES MULTILINEAR FIT
       SAMPLE SIZE N       =      499
       NUMBER OF VARIABLES =        1
       NO REPLICATION CASE
  
  
               PARAMETER ESTIMATES           (APPROX. ST. DEV.)    T VALUE
        1  A0                  0.501650E-01   (0.2417E-01)        2.075
        2  A1       YIM1       0.987087       (0.6313E-02)        156.4
  
       RESIDUAL    STANDARD DEVIATION =        0.2931194
       RESIDUAL    DEGREES OF FREEDOM =         497

The slope parameter, A1, has a t value of 156.4 which is statistically significant. Also, the residual standard deviation is 0.29. This can be compared to the standard deviation shown in the summary table, which is 2.08. That is, the fit to the autoregressive model has reduced the variability by a factor of 7.

Time Series Model

This model is an example of a time series model. More extensive discussion of time series is given in the Process Monitoring chapter.