Transfer Function Model

Summary

Transfer function (TF) models relate a marketing output series $y_t$ (sales) to one or more input series $x_t$ (advertising, price) while accounting for the autocorrelated noise structure. They combine the ARIMA framework (Ch.6) with regression, generalizing the ADL model to include arbitrary lag structures and correlated errors.

Transfer Function Framework

Transfer Function Model (Single Input)

The general single-input TF model (impulse response form):

$$y_t={\alpha }_0+(v_0+v_{1}L+v_2{L}^2+\cdots )x_t+n_t\text{\hspace{1em}(Eq 7.7)}$$

or compactly:

$$y_t={\alpha }_0+V(L)x_t+n_t\text{\hspace{1em}(Eq 7.8)}$$

where:

• $v_j$: impulse response weights (effect of $x$ on $y$ at lag $j$)
• $n_t$: added noise process, typically modeled as ARMA($p$,$q$)
• $V(L)=\omega (L)/\delta (L)\cdot L^b$: rational polynomial in $L$ with $b$-period delay

When $r=0$ (no denominator), $V(L)$ is a finite polynomial; when $r>0$, it is an infinite series (like the Koyck model).

Two-Input TF Model

$$y_t={\alpha }_0+\frac{{\omega }_1(L)}{{\delta }_1(L)}{L}^{b_1}{x}_{1t}+\frac{{\omega }_2(L)}{{\delta }_2(L)}{L}^{b_2}{x}_{2t}+\frac{\Theta (L)}{\Phi (L)}{w}_t\text{\hspace{1em}(Eq 7.14)}$$

Impulse response form: $y_t={\alpha }_0+V_1(L)x_{1t}+V_2(L)x_{2t}+n_t$ (Eq 7.15)

Prewhitening Identification (Box-Jenkins Method)

Three-Step Prewhitening

Step 1: Find the ARMA model for input $x_t$ (prewhiten):

$$\hat{\Phi }(L)x_t=\hat{\Theta }(L){\hat{w}}_t\text{\hspace{1em}(Eq 7.9)}$$ $${\hat{w}}_t=\hat{\Theta }(L{)}^{-1}\hat{\Phi }(L)x_t\text{\hspace{1em}(Eq 7.10)}$$

Step 2: Apply the same filter to output $y_t$:

$${\hat{\beta }}_t=\hat{\Theta }(L{)}^{-1}\hat{\Phi }(L)y_t\text{\hspace{1em}(Eq 7.11)}$$

Step 3: Compute CCF (cross-correlation function) between ${\hat{w}}_t$ and ${\hat{\beta }}_t$:

$${\hat{v}}_j=\frac{s_{\hat{\beta }}}{s_{\hat{w}}}{r}_{\hat{w}\hat{\beta }}(j)\text{\hspace{1em}(Eq 7.12)}$$

The CCF pattern directly reveals the impulse response weights $v_j$ and identifies the TF order $(r,s,b)$.

CCF Patterns and Transfer Function Shapes

Figure 7-1 patterns (from the book):

CCF Pattern	Transfer Function	Marketing Example
Single spike at $j=0$, negative at $j=1$	$({\omega }_0+{\omega }_{1}L)x_t$, ${\omega }_1<0$	Dynamic effects of dealing (promotion with stockpiling)
Monotone decay starting at $j=0$	${\omega }_0/(1-\delta L)$, $0<\delta <1$	Monotonic advertising carryover
Spike then decay	$({\omega }_0+{\omega }_{1}L+{\omega }_2{L}^2)/(1-\delta L)$	Advertising buildup then decay

Multi-Input Identification (Liu-Hanssens Method)

The prewhitening approach becomes cumbersome with many inputs (each requires separate prewhitening). The Liu-Hanssens (1982) direct-lag regression approach:

1. Estimate a long-lag OLS regression (Eq 7.16):

$$y_t={\alpha }_0+\sum _{k=0}^{K_1}{v}_{1k}{x}_{1,t-k}+\sum _{k=0}^{K_2}{v}_{2k}{x}_{2,t-k}+u_t$$

2. Examine pattern of OLS coefficients ${\hat{v}}_{jk}$ to identify cutoffs vs. dying-out patterns

3. If inputs are highly autocorrelated (AR-dominated), apply a common filter to reduce collinearity

Problem 1 (collinearity): if $x_t$ is AR(1) with $\varphi =0.9$, adjacent lags are $\rho =0.9$ correlated. Remedy: filter out the AR component with a common filter before OLS. Problem 2 (non-white residuals): use GLS — estimate ARMA structure from OLS residuals, transform, re-estimate.

Intervention Analysis

Intervention Analysis (Box-Tiao 1975)

Qualitative events (advertising copy changes, competitor entry, regulation) that cannot be quantified as continuous variables are modeled as dummy variable inputs.

Pulse intervention (temporary):

$$D_{\text{pulse},t}=\{\begin{array}{ll}0& t<k\\ 1& k\le t\le k+l\\ 0& t>k+l\end{array}$$

Step intervention (permanent):

$$D_{\text{step},t}=\{\begin{array}{ll}0& t<k\\ 1& t\ge k\end{array}$$

Note: $(1-L)D_{\text{step},t}=D_{\text{pulse},t}$ (Eq 7.20) — the first difference of a step is a pulse.

General pulse intervention TF model (Eq 7.19):

$$Y_t={\alpha }_0+\frac{\omega (L)}{\delta (L)}{L}^b{D}_{\text{pulse},t}+\frac{\Theta (L)}{\Phi (L)}{w}_t$$

Intervention Scenarios (Figure 7-2)

Four canonical response patterns to pulse input + step input:

TF	Pulse Response	Step Response	Example
$y_t={\alpha }_0+{\omega }_0{x}_t$	Immediate, temporary	Level shift	Static sales response
$y_t={\alpha }_0+\frac{{\omega }_0}{1-\delta L}{x}_t$	Gradual decay	Trending up	Advertising carryover
$y_t={\alpha }_0+\frac{{\omega }_0}{1-L}{x}_t$	Permanent	Linear trend	Sustained competitive advantage
$y_t={\alpha }_0+({\omega }_0+{\omega }_{1}L)x_t$	Temporary blip	Blip then back	Promotion with stockpiling

Diagnostic Checking for TF Models

Two residual checks:

1. ACF of residuals should be flat (no autocorrelation) — adjust $\Phi (L)$/$\Theta (L)$ if not
2. CCF of residuals with prewhitened input should be flat — a spike at lag $k=2$ indicates a lag-2 response was omitted from the TF

Cross-Links

• ARIMA background: Single Marketing Time Series
• Koyck model (special case): Carryover Effects and Distributed Lags
• Multi-input extension: Multivariate Persistence and Cointegration
• Intervention analysis in causal inference context: Bayesian Structural Time-Series Model
• ADL model: Design of Dynamic Response Models
• Causal ordering of inputs: Empirical Causal Ordering — determining whether $x_t$ leads or lags $y_t$ before specifying the TF delay $b$