The General and Special Models for Demand Forecasting

Work note

These are the equations that should be starting point for any forecating effort. The general model is rarely used. Instead pick one of the special models.

General Model

Schematically, the general model looks like this:

This is the equation for a “raw” regression:

\[(1) \quad \Delta y_t= \alpha+ \sum_{k \in \mathcal{LF}} \beta_k(L)\,\Delta x_{k,t}+ \sum_{j \in \mathcal{HF}} \gamma_j(L)\,\Delta w_{j,t}+ u_t,\quad u_t \sim \text{ARIMA(p,d,q)}\]

We often prefer to run a de-trended, stationary model in differences which has a slight modification in the timeseries part:

\[(2) \quad \Delta y_t= \alpha+ \sum_{k \in \mathcal{LF}} \beta_k(L)\,\Delta x_{k,t}+ \sum_{j \in \mathcal{HF}} \gamma_j(L)\,w_{j,t}+ u_t,\quad u_t \sim \text{ARMA(p,q)}\]

(L) means that a lag term may be included. LF = Low frequency (long-term influences on demand) such as population growth; HF = High frequency (short-term influences on demand) such as unemployment. Price often shows up as both LF and HF with HF usually more important.

Note the subtle variations: (1) has Δw while (2) has w since it is already differenced. (1) has ARIMA(p,d,q) while (2) has ARMA(p,q) since d disappears when differencing.

Note that LF or HF coefficients may be calculated in a separate model and elasticities then set as static.

These are ARIMAX equations, but with a clear distinction between long-term and short-term independent variables and the timeseries component.

Special Models

All conceivable forecasting needs are covered by 5 subsets of the general model. This is MECE. [It is possible to set up all kinds variants to this (such as including the timeseries component as dummies in the S/T model), but those are rearrangements, not different models.]

A. Long-term regression without lag effects and timeseries component

This is useful for strategic forecasting and cross-sectional analyses. Works with pooling. 3-10 years horizon.

B. Long-term regression with timeseries component

This is useful for strategic forecasting with a timeseries component. It cannot easily be pooled. 3-10 years horizon.

C. Short-term regression with timeseries component

This is a typical ARIMAX or ARMAX case for operational forecasting 1 to 12 months out.

D. Short-term regression with long-term drivers inpact and with timeseries component

This is the most complex model. Seldom used, but important. It allows for combination of short-term and long-term demand drivers. The long-term drivers are usually estimated with A. or B. above, and then “grafted” on as predetermined static coefficients. Also for operational forecasting 1-12 months out.

E. Ultra-short-term regression with only timeseries component

For tactical forecasting less than 4 weeks out.

Once one of these basic equations has been modelled and understood, more complex models may be pursued such as ECM / ARDL.