Multivariate Persistence and Cointegration

Summary

When marketing variables are jointly evolving (nonstationary), VAR models capture the full dynamic system including six channels of total impact (contemporaneous, carryover, purchase reinforcement, feedback, decision rules, competitive reaction). Cointegration tests for long-run equilibrium between evolving variables; error-correction models (ECM) incorporate that equilibrium into short-run dynamics.

VARMA and VAR Models

VARMA Model

The Vector ARMA model for a $k\times 1$ vector of stationary observations ${\mathbf{z}}_t$ (Eq 7.21):

$$\mathbf{\Phi }(L){\mathbf{z}}_t={\mathbf{\alpha }}_0+\mathbf{\Theta }(L){\mathbf{w}}_t$$

where:

• $\mathbf{\Phi }(L)=\mathbf{I}-{\mathbf{\varphi }}_{1}L-\cdots -{\mathbf{\varphi }}_p{L}^p$ ($k\times k$ AR matrix polynomial)
• $\mathbf{\Theta }(L)=\mathbf{I}-{\mathbf{\theta }}_{1}L-\cdots -{\mathbf{\theta }}_q{L}^q$ ($k\times k$ MA matrix polynomial)
• ${\mathbf{w}}_t\sim \text{i.i.d. MVN}(0,\mathbf{\Sigma })$

VAR Model

The Vector Autoregressive model — VARMA with $\mathbf{\Theta }(L)=\mathbf{I}$ (Eq 7.22):

$$\mathbf{\Phi }(L){\mathbf{z}}_t={\mathbf{\alpha }}_0+{\mathbf{w}}_t$$

Advantages over VARMA: (1) straightforward OLS estimation equation-by-equation, (2) no nonlinear MA terms, (3) easily extends to cointegrated systems.

Bivariate Sales-Advertising VAR

$$\left[\begin{array}{c}{Q}_t\\ AD{V}_t\end{array}\right]=\left[\begin{array}{cc}{\pi }_{11}^1& {\pi }_{12}^1\\ {\pi }_{21}^1& {\pi }_{22}^1\end{array}\right]\left[\begin{array}{c}{Q}_{t-1}\\ AD{V}_{t-1}\end{array}\right]+\cdots +\left[\begin{array}{cc}{\pi }_{11}^I& {\pi }_{12}^I\\ {\pi }_{21}^I& {\pi }_{22}^I\end{array}\right]\left[\begin{array}{c}{Q}_{t-I}\\ AD{V}_{t-I}\end{array}\right]+\left[\begin{array}{c}{w}_{Q,t}\\ w_{ADV,t}\end{array}\right]\text{\hspace{1em}(Eq 7.23)}$$

Order $I$ determined by AIC/BIC. Off-diagonal elements ${\pi }_{12}^i$ (advertising’s effect on sales at lag $i$) and ${\pi }_{21}^i$ (sales feedback to advertising) capture the full dynamic structure.

Six Channels of Total Impact (Dekimpe & Hanssens 1995a)

A marketing shock propagates through:

Channel	Mechanism
Contemporaneous	Direct same-period effect
Carryover	Past advertising influences current sales via goodwill
Purchase reinforcement	New buyers from ad increase repeat purchase base
Feedback	Higher sales → higher budgets → more advertising
Decision rules	Firm’s spending patterns (momentum) affect future spending
Competitive reactions	Rivals match or counter, dampening/amplifying net effect

The VAR captures all six simultaneously without imposing causal structure a priori.

Multivariate Persistence

Impulse Response and Persistence

Writing the VAR as an infinite-order VMA:

$$\left[\begin{array}{c}{Q}_t\\ AD{V}_t\end{array}\right]=\mathbf{I}\left[\begin{array}{c}{w}_{Q,t}\\ w_{ADV,t}\end{array}\right]+{\mathbf{\alpha }}^1\left[\begin{array}{c}{w}_{Q,t-1}\\ w_{ADV,t-1}\end{array}\right]+{\mathbf{\alpha }}^2\left[\begin{array}{c}{w}_{Q,t-2}\\ w_{ADV,t-2}\end{array}\right]+\cdots \text{\hspace{1em}(Eq 7.24)}$$

${\alpha }_{12}^k$: impact on $Q_t$ of a one-unit advertising shock $k$ periods ago.

Multivariate persistence = ${\lim }_{k\to \infty }{\alpha }_{12}^k$

• Zero persistence (stationary system): impulse response decays to zero, brand returns to pre-shock baseline
• Non-zero persistence (evolving system): shock has a permanent effect — see “hysteresis” scenario

Four Strategic Scenarios (Dekimpe & Hanssens 1999)

Market Performance	Marketing Mix	Scenario	Implication
Stationary I(0)	Stationary I(0)	Business as usual	Temporary marketing effects only
Stationary I(0)	Evolving I(1)	Escalation	Marketing spending battles; no long-run gain
Evolving I(1)	Stationary I(0)	Hysteresis	Temporary marketing creates permanent sales change
Evolving I(1)	Evolving I(1)	Co-evolution	Long-run equilibrium; ECM required

Cointegration

Cointegration (Engle-Granger 1987)

Two I(1) series $Y_t$ and $X_t$ are cointegrated if there exists ${\beta }_1$ such that:

$$e_t=Y_t-{\beta }_0-{\beta }_1{X}_t\text{\hspace{1em}(Eq 7.26)}$$

is stationary I(0). Cointegration means a long-run equilibrium relationship exists between the two evolving series — they cannot drift arbitrarily far apart.

Engle-Granger test: (1) OLS of $Y_t$ on $X_t$; (2) ADF unit root test on residuals ${\hat{e}}_t$ (with different critical values)

Johansen FIML test (Eq 7.27-7.28): full-information maximum likelihood; handles multiple cointegrating vectors. Starting from VAR($k$):

$${\tilde{\mathbf{X}}}_t=\mathbf{c}+{\mathbf{\Pi }}_1{\tilde{\mathbf{X}}}_{t-1}+\cdots +{\mathbf{\Pi }}_k{\tilde{\mathbf{X}}}_{t-k}+{\mathbf{w}}_t$$

The number of cointegrating vectors = rank of ${\mathbf{\Gamma }}_k=-{\mathbf{I}}_N+{\mathbf{\Pi }}_1+\cdots +{\mathbf{\Pi }}_k$ ($=\mathbf{w}{\mathbf{\beta }}^{\prime }$).

Error-Correction Model (ECM)

Error-Correction Model

If $Y_t$ and $X_t$ are cointegrated with equilibrium error $e_t=Y_t-{\beta }_0-{\beta }_1{X}_t$, the Engle-Granger ECM is (Eq 7.29):

$$\Delta Y_t={\alpha }_0+{\alpha }_1\Delta Y_{t-1}+{\alpha }_2\Delta X_{t-1}+{\alpha }_3{e}_{t-1}+u_t$$

where ${\alpha }_3<0$ is the speed of adjustment parameter. When $Y_{t-1}$ is too high relative to the equilibrium, ${\alpha }_3{e}_{t-1}$ pulls it back down in period $t$.

Marketing interpretation: if sales are unusually high given current advertising, the ECM predicts sales will fall back toward the long-run advertising-supported level.

Incorporating ECM in transfer function models significantly improves long-horizon forecasting accuracy (Hanssens 1998: 63% improvement over univariate models).

Key Empirical Findings (Table 7-1)

Study	Period	Variable	Key Finding
Dekimpe & Hanssens (1995b)	—	Sales/market share	Sales series mostly evolving; market share mostly stationary
Bronnenberg et al. (2000)	Weekly	Market share	Distribution coverage drives long-run share
Dekimpe et al. (1999)	Weekly	Industry sales	Little evidence of long-run promotional effects in FPCG
Hanssens (1998)	Monthly	Factory orders	Factory orders and retail sales cointegrated; ECM forecasts 63% better
Hanssens & Ouyang (2000)	Monthly	Sales	Hysteresis: long-run profit-maximizing rules differ from short-run

Cross-Links

• Transfer function model (single equation): Transfer Function Model
• ARIMA foundation: Single Marketing Time Series
• Reaction functions within VAR: Reaction Functions and Competitive Dynamics
• Causal ordering: Empirical Causal Ordering
• Cointegration in econometrics: Differences-in-Differences
• Bayesian structural time series: Bayesian Structural Time-Series Model