Case ARMA: Modeling Item Movement by Using ARMA Process

After learning two classes of models, i.e., AR (p) and MA(q), for tracking movements of grocery items, Jo has decided to re-analyze movements of major categories using the new tools. She has hired an assistant with a BA degree whom she plans to train to become a data analyst. She tells Jane, the assistant, to develop ACF and PACF for each series. Jo can then suggest plausible models for each series.

Jo however soon is presented with a series with the following correlational characteristics: both ACF and PACF do not appear truncated.

1. Variable & Timeplot

Variable: Weekly Movement of Ft. Free Italian Dressing -- 16 oz.

Y = natural log transformation

2. Correlational Characteristics

ACF PACF

Jo calls Professor Victoria Foresight, who recommends Jo to mix AR and MA models using however only small orders. Professor Foresight reminds Jo that the key idea is to keep the number of parameters low. In below we show Jo’s modeling exercise for the data.

3-1: Jo fits MA (q) models for q = 1, 2, 3 and summarizes the results in a table

q / Stand. Error / Constant / q1 / q2 / q3
1 / .35293 / 4.1591 / .413037
2 / .33367 / 4.1628 / .488331 / .31557
3 / .32981 / 4.1643 / .47547 / .37180 / .18011*

Jo seems to be able to keep improving by increasing the order. This could lead to a model with a high order, which Jo should avoid. (* not significant at 5%)

3-2:.Jo fits an ARMA(1, 1) and ARMA(1, 2)

p / q / Stand. Error / Constant / f1 / f2 / q1 / q2
1 / 1 / .31827 / 4.16769 / .85344 / -.43351
2 / 1 / .31545 / 4.07251 / 1.28099 / -.32242 / -.84262
1 / 2 / .31888 / 4.14963 / .91606 / -.50302 / -.11344*

It appears that ARMA (2, 1) works best for the data. This exercise teaches Jo the power of working with a wide range of models.

Appendix

1. Eviews procedure for fitting ARMA models: ARMA(2,2)


2. Modeling Guide

Stay at low p and low q, while exploring the trees shown below using AFC, PACF and testing significance of the parameters. Select one with a lowest Standard Error, (or AIC or SBC).

1