Flexible Functional Forms

Summary

Flexible functional forms provide second-order approximations to an arbitrary unknown response function without committing to a specific parametric shape. They are particularly useful when the shape of the marketing response is uncertain and the data should drive the specification.

Motivation

Standard functional forms (linear, log-log, ADBUDG) impose strong a priori shape restrictions. When theory is ambiguous about whether response is concave, S-shaped, or convex, a flexible form lets the data determine the local curvature.

The Translog (Transcendental Logarithmic) Form

Translog

A second-order Taylor expansion in logs around the sample mean:

$$\ln Q={\alpha }_0+\sum _j{\beta }_j\ln X_j+\frac{1}{2}\sum _j\sum _k{\gamma }_{jk}\ln X_j\cdot \ln X_k+ϵ$$

The translog nests:

• Log-log (power): when all ${\gamma }_{jk}=0$
• Interaction effects between instruments: ${\gamma }_{jk}\ne 0$

Symmetry restriction: ${\gamma }_{jk}={\gamma }_{kj}$ (curvature matrix is symmetric)

Own-price elasticity: ${\eta }_j={\beta }_j+{\sum }_k{\gamma }_{jk}\ln X_k$ (non-constant, varies with marketing levels)

The Box-Cox Transformation

Box-Cox Power Transformation

The Box-Cox transformation:

$$Q^{(\lambda )}=\frac{Q^{\lambda }-1}{\lambda },\lambda \to 0\Rightarrow \ln Q$$

Applied to both the dependent and independent variables with potentially different $\lambda$:

$$Q^{({\lambda }_0)}=\alpha +\sum _j{\beta }_j{X}_j^{({\lambda }_j)}+ϵ$$

MLE over the $\lambda$ parameters yields a data-driven choice of transformation. If ${\hat{\lambda }}_0\approx 0$, log transformation of $Q$ is appropriate. If ${\hat{\lambda }}_j\approx 0$, log transformation of $X_j$ is appropriate.

See also Box-Cox for variance stabilization in ARIMA — Single Marketing Time Series.

Locally Weighted Regression (LOESS)

For purely nonparametric estimation of the response function $Q=f(X)$ without any parametric restriction:

• Fit a weighted local polynomial at each evaluation point $x_0$
• Weight observations by proximity: $w_i=K((X_i-x_0)/h)$ with bandwidth $h$
• The estimated $\hat{f}(x_0)$ traces the nonparametric response

Tradeoff: high flexibility but low interpretability and poor out-of-sample performance. Useful as a diagnostic check against parametric forms.

Spline Models

Regression Spline

Splines fit piecewise polynomials with continuity constraints at knots ${\tau }_1,\dots ,{\tau }_K$:

$$Q=\sum _{j=0}^p{\beta }_j{X}^j+\sum _{k=1}^K{\delta }_k(X-{\tau }_k{)}_{+}^p+ϵ$$

where $(X-{\tau }_k{)}_{+}=\max (0,X-{\tau }_k)$. The knot positions can be pre-specified or estimated.

Marketing application: different response regimes at low, medium, and high advertising levels (captures both S-shape and saturation in separate segments).

Comparison of Flexible Forms

Form	Parameters	Globally valid?	Marketing Use
Translog	$\sim J^2/2$	No (local approx)	Multi-instrument response
Box-Cox	$J+1$ ($\lambda$‘s)	Yes (nests linear & log)	Form selection
LOESS	Nonparametric	Locally only	Diagnostic
Spline	$p+K+1$	By segment	Threshold + saturation

Cross-Links

• Standard parametric forms: Functional Forms in Marketing
• Specification testing: Model Testing and Specification
• Model selection: Model Selection and Exploratory Analysis
• Nonparametric Bayes: Nonparametric Models Overview