Single Marketing Time Series

Summary

Chapter 6 applies Box-Jenkins ARIMA methodology to individual marketing time series (sales, advertising, price). The goal is to characterize the within-series stochastic structure before modeling inter-series relationships (Ch.7). Covers stationarity, ACF/PACF diagnostics, AR/MA/ARMA/ARIMA model families, and Box-Cox transformation.

Two-Step Extraction Framework

Every marketing time series $Z_t$ can be decomposed:

1. Remove deterministic components: trend, seasonality, cyclicality, heteroscedasticity
2. Model stochastic component: the residual $z_t$ as ARMA

Stationarity

A time series $\{z_t\}$ is weakly stationary (covariance-stationary) if:

$$E(z_t)=\mu \forall t\text{\hspace{1em}(Eq 6.5)}$$ $$\text{Var}(z_t)={\sigma }_z^2\forall t\text{\hspace{1em}(Eq 6.6)}$$ $$\text{Cov}(z_t,z_{t+k})={\gamma }_k\text{depends only on lag }k$$

Non-stationarity (unit root) requires differencing before ARMA modeling.

Deterministic Components

Four types that must be removed before modeling the stochastic component:

Component	Type	Treatment
Trend	Linear or nonlinear	Differencing or detrending
Seasonality	Regular, periodic	Seasonal dummies or seasonal differencing
Cyclicality	Business cycle	Differencing at cycle frequency
Heteroscedasticity	Variance changes with level	Box-Cox transformation

Yule’s Linear Filter / ARMA Representation

Linear Filter

Any stationary ARMA process is a linear filter of white noise:

$$z_t=\Psi (L)w_t=\sum _{j=0}^{\infty }{\psi }_j{w}_{t-j}\text{\hspace{1em}(Eq 6.10)}$$

where $w_t\sim \text{WN}(0,{\sigma }_w^2)$ and $\Psi (L)=1+{\psi }_{1}L+{\psi }_2{L}^2+\cdots$ is a (possibly infinite) lag polynomial.

The general ARMA form (Eq 6.11):

$$\Phi (L)z_t=\Theta (L)w_t$$ $$\Phi (L)=1-{\varphi }_{1}L-\cdots -{\varphi }_p{L}^p\text{(AR polynomial)}$$ $$\Theta (L)=1-{\theta }_{1}L-\cdots -{\theta }_q{L}^q\text{(MA polynomial)}$$

ACF and PACF

Sample ACF and PACF

Autocorrelation function (ACF):

$$r_k=\frac{{\sum }_{t=k+1}^T(z_t-\bar{z})(z_{t-k}-\bar{z})}{{\sum }_{t=1}^T(z_t-\bar{z}{)}^2}\text{\hspace{1em}(Eq 6.27)}$$

Approximate variance of $r_k$ (Bartlett):

$$\text{Var}(r_k)\approx \frac{1}{T}\left(1+2\sum _{v=1}^{k-1}{\rho }_v^2\right)\text{\hspace{1em}(Eq 6.28)}$$

Partial autocorrelation function (PACF): ${\varphi }_{kk}$ = the coefficient on $z_{t-k}$ in an AR($k$) regression on $z_t,z_{t-1},\dots ,z_{t-k}$. Captures the direct effect at lag $k$, removing effects of intermediate lags.

AR(p) Models

AR(1) Model

$$z_t={\alpha }_0+\varphi z_{t-1}+w_t$$

Stationarity: $|\varphi |<1$ (root of $1-\varphi L=0$ must exceed 1 in absolute value)

ACF: ${\rho }_k={\varphi }^k$ — exponential decay (Eq 6.40)

PACF: Cuts off after lag 1 — ${\varphi }_{11}=\varphi$, ${\varphi }_{kk}=0$ for $k>1$ (Eq 6.42)

For $\varphi >0$: positive, monotone decay. For $\varphi <0$: oscillating decay.

MA(q) Models

MA(1) Model

$$z_t=\mu +w_t-{\theta }_1{w}_{t-1}$$

Invertibility: $|{\theta }_1|<1$ (root of $1-{\theta }_{1}L=0$ outside unit circle)

ACF: Single spike at lag 1 only:

$${\rho }_1=\frac{-{\theta }_1}{1+{\theta }_1^2},{\rho }_k=0\text{ for }k>1$$

PACF: Dies out geometrically (infinite AR representation)

ACF/PACF Diagnostic Summary (Table 6-2)

Model	ACF	PACF
AR($p$)	Decays to zero geometrically	Cuts off after lag $p$
MA($q$)	Cuts off after lag $q$	Decays to zero
ARMA($p$,$q$)	Decays after lag $q$	Decays after lag $p$
White noise	No significant spikes	No significant spikes
Unit root (I(1))	Very slow, linear decay	Spike near 1.0 at lag 1

Box-Jenkins Identification Procedure (Figure 6-4)

Three-Step Box-Jenkins Method

Step 1: Identification

• Plot series; apply Box-Cox if variance non-constant
• Difference until stationary (ADF unit root test)
• Examine ACF/PACF to determine tentative $p$, $q$

Step 2: Estimation

• Estimate ARMA parameters by conditional or exact MLE
• Check standard errors and parameter significance

Step 3: Diagnostic Checking

• Residuals should be white noise: Ljung-Box Q test
• ACF of residuals should show no pattern
• If inadequate: return to Step 1 with revised order

ARIMA(p, d, q)

ARIMA

Regular differencing to achieve stationarity (order $d$):

$$z_t=(1-L{)}^d{Z}_t\text{\hspace{1em}(Eq 6.76)}$$

For $d=1$ (most common): $z_t=Z_t-Z_{t-1}$ (first difference = period-over-period change)

Seasonal ARIMA (Box-Jenkins seasonal model) adds seasonal differencing:

$$z_t=(1-L^s{)}^{d_s}{Z}_t\text{\hspace{1em}(Eq 6.79)}$$

where $s$ is the seasonal period (12 for monthly, 52 for weekly). Full model:

$$\Phi (L){\Phi }_s(L^s)(1-L{)}^d(1-L^s{)}^{d_s}{Z}_t=\Theta (L){\Theta }_s(L^s)w_t$$

Box-Cox Variance Stabilization

$$Z_t^{\prime }=\frac{(Z_t+c{)}^{\lambda }-1}{\lambda }\text{\hspace{1em}(Eq 6.74)}$$

Common choices: $\lambda =0.5$ (square root), $\lambda \to 0$ (log), $\lambda =1$ (no transform). Choose $\lambda$ by profile MLE or visual inspection of the variance-mean plot.

Marketing Applications of ARIMA

• Sales forecasting: pure ARIMA for baseline forecast without explanatory variables
• Prewhitening input series before transfer function identification (Ch.7)
• Unit root testing: determine if sales is I(0) or I(1) before using in ECM
• Seasonality modeling: seasonal ARIMA for weekly scanner data

Cross-Links

• Transfer function extension: Transfer Function Model
• VAR and cointegration: Multivariate Persistence and Cointegration
• Carryover and ADL: Carryover Effects and Distributed Lags
• Causal ordering from prewhitened series: Empirical Causal Ordering
• Parameter estimation methods for ARIMA models: Parameter Estimation in Market Response
• Bayesian state-space alternative to ARIMA: Bayesian Structural Time-Series Model
• Model selection (AIC/BIC for order selection): Model Selection and Exploratory Analysis

See Also

• State-Space Models and the Kalman Filter - Overview — ARIMA as a special-case state-space model