# Leads and Lags — Dynamic Effects

Dynamic promotion effects can also be explicitly modelled by incorporating variables that capture lead and lag effects, i.e.:

$$S_t=f(X_{t+k}), \, S_t=f(X_{t-k}).$$

Commonly used in marketing, the Geometric Distributed Lag (GL) model has a functional form that is structurally similar to that used for stock variables, i.e.:

$$S_t=α+β(1-\lambda)\sum_{l=0}^{\infty}\lambda^l X_{t-l} + \delta_t,\,where\,\delta_t\,is\,the\,disturbance\,term.$$

This relation, which is nonlinear, may be converted to a linear estimation model by applying the Koyck transformation:

$$S_t=α+β(1-\lambda)X_t + β(1-\lambda)\sum_{l=1}^{\infty}\lambda^l X_{t-l} + \delta_t$$ $$-\lambda S_{t-1}=-\lambda α - β(1-\lambda)\sum_{l=1}^{\infty}\lambda^l X_{t-l} - \lambda\delta_{t-1}$$ $$S_t-\lambda S_{t-1}=(1-\lambda)α+β(1-\lambda)X_t+(𝛿_t-\lambda 𝛿_{t-1})$$ $$S_t=β_0+β_1X_t+\lambda S_{t-1} + v_t.$$

Where vt=𝛿t−λ𝛿t-1, β0=(1−λ)α and β1=(1−λ)β.

In general, for multiple marketing mix variables, the Koyck model becomes:

$$S_t=β_0+\sum_{k=1}^K β_k X_{kt}+\lambda S_{t-1} + v_t$$

This model however captures only monotonically decaying carryover effects that do not have a hump. Moreover, estimating the carryovers is tricky when there are multiple independent variables, each with its own carryover effect. Even so the model may provide a fairly good approximation of the underlying response function.

The Autoregressive Distributed Lag Model (ADL) which contains an autoregressive component for sales and a moving average distributed lag component for the mix variables is a general model that captures all types of carryover effects.

$$S_t=β_0+\sum_{p=1}^P \lambda_p S_{t-p} + \sum_{k=1}^K\sum_{q=1}^Q β_{kq} X_{k,t-q}+v_t$$

The rate at which the carryover effects peak and decay is controlled by λ, and the number of peaks and their heights is controlled by β.

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