Parameter Estimation in Market Response

Summary

Chapter 5 covers the estimation toolkit for market response models: OLS and its 8 assumptions, GLS/FGLS for autocorrelated errors, SUR for multi-equation systems, 2SLS/3SLS for simultaneous equations, and Bayesian approaches including hierarchical Bayes (HB) and Empirical Bayes (EB). Also covers IV estimation for endogenous regressors.

Variable Classification

Variable Types

• Exogenous: determined outside the model (macro variables, season)
• Predetermined: includes lagged endogenous variables; not correlated with current errors
• Current endogenous: jointly determined with the dependent variable in the same period (e.g., advertising when spending reacts to same-period sales signals)

Simultaneity arises when marketing instruments are current endogenous — OLS is biased and inconsistent.

OLS

OLS Estimator

For the linear model $\mathbf{q}=\mathbf{X}\mathbf{\beta }+\mathbf{u}$:

$${\hat{\mathbf{\beta }}}_{\text{OLS}}=({\mathbf{X}}^{\prime }\mathbf{X}{)}^{-1}{\mathbf{X}}^{\prime }\mathbf{q}\text{\hspace{1em}(Eq 5.5)}$$ $$\text{Var}(\hat{\mathbf{\beta }})={\sigma }^2({\mathbf{X}}^{\prime }\mathbf{X}{)}^{-1}\text{\hspace{1em}(Eq 5.6)}$$

BLUE (Best Linear Unbiased Estimator) under all 8 Gauss-Markov assumptions (Table 5-1):

1. Linearity
2. Fixed $\mathbf{X}$ (or independence of $\mathbf{X}$ and $\mathbf{u}$)
3. Full rank of $\mathbf{X}$
4. $E(\mathbf{u})=0$
5. $E(\mathbf{u}{\mathbf{u}}^{\prime })={\sigma }^2\mathbf{I}$ (homoscedasticity + no autocorrelation)
6. No multicollinearity
7. Correct functional form
8. No simultaneity bias

Generalized Least Squares (GLS/FGLS)

When errors are autocorrelated or heteroscedastic ($E(\mathbf{u}{\mathbf{u}}^{\prime })={\sigma }^2\mathbf{\Omega }$, $\mathbf{\Omega }\ne \mathbf{I}$):

$${\hat{\mathbf{\beta }}}_{\text{GLS}}=({\mathbf{X}}^{\prime }{\mathbf{\Omega }}^{-1}\mathbf{X}{)}^{-1}{\mathbf{X}}^{\prime }{\mathbf{\Omega }}^{-1}\mathbf{q}$$

When $\mathbf{\Omega }$ is unknown, Feasible GLS (FGLS) substitutes a consistent estimate $\hat{\mathbf{\Omega }}$. In practice: estimate the error ARMA structure from OLS residuals, then transform data by the estimated filter.

Seemingly Unrelated Regressions (SUR)

For a system of $M$ equations (e.g., sales equations for multiple brands) with correlated errors across equations:

$${\mathbf{q}}_m={\mathbf{X}}_m{\mathbf{\beta }}_m+{\mathbf{u}}_m,m=1,\dots ,M$$ $$E(u_{mt}{u}_{m^{\prime }t})={\sigma }_{m{m}^{\prime }}$$

SUR (Zellner 1962) is more efficient than OLS equation-by-equation when:

1. Errors are correlated across equations
2. Regressors differ across equations

SUR is the standard estimator for MCI/MNL market share systems (see Market Share Models) and multi-brand advertising effects.

Simultaneous System and 2SLS/3SLS

Structural System

The full simultaneous system in matrix form:

$$\mathbf{Y\Gamma }+\mathbf{XB}=\mathbf{U}\text{\hspace{1em}(Eq 5.4)}$$

where $\mathbf{Y}$ = current endogenous variables, $\mathbf{X}$ = predetermined variables, $\mathbf{\Gamma }$ and $\mathbf{B}$ = structural coefficient matrices.

2SLS (Two-Stage Least Squares): instrument endogenous RHS variables with exogenous variables. First stage: regress endogenous $\mathbf{Y}$ on $\mathbf{X}$; second stage: substitute fitted values $\hat{\mathbf{Y}}$ in structural equation. Consistent but less efficient than 3SLS.

3SLS: adds SUR cross-equation correlation to 2SLS. Full system efficiency when the model is correctly specified.

ILS (Indirect Least Squares): OLS on the reduced form, then back-solves for structural parameters. Exact identification only.

See Instrumental Variables for the IV estimator in a causal inference context.

Bayesian Estimation

Bayesian Posterior

The Bayesian approach updates a prior distribution $p(\mathbf{\beta },\sigma )$ with the likelihood $p(\mathbf{q}|\mathbf{\beta },\sigma )$:

$$p(\mathbf{\beta },\sigma |\mathbf{q})\propto p(\mathbf{q}|\mathbf{\beta },\sigma )\cdot p(\mathbf{\beta },\sigma )\text{\hspace{1em}(Eq 5.14)}$$

With a noninformative prior $p(\mathbf{\beta },\sigma )\propto 1/\sigma$ (Eq 5.15):

$$[\mathbf{\beta }|\mathbf{q}]\sim N({\hat{\mathbf{\beta }}}_{\text{OLS}},{\sigma }^2({\mathbf{X}}^{\prime }\mathbf{X}{)}^{-1})\text{\hspace{1em}(Eq 5.27)}$$

The posterior mean equals the OLS estimate; Bayesian and frequentist results coincide under diffuse priors.

Hierarchical Bayes (HB) Shrinkage

HB Shrinkage Estimator

When parameters vary across brands/markets (random coefficients), the HB estimator (Eq 5.30) shrinks individual estimates toward the grand mean:

$${\hat{\mathbf{\beta }}}_i^{\text{HB}}={\mathbf{W}}_i{\hat{\mathbf{\beta }}}_i+(\mathbf{I}-{\mathbf{W}}_i)\bar{\mathbf{\beta }}$$

where ${\mathbf{W}}_i$ is a matrix that downweights individual estimates with high sampling variance and upweights the pooled mean. This is a Stein-like shrinkage rule that dominates OLS in mean squared error when there are $\ge 3$ parameters.

Empirical Bayes (EB): estimates the hyperparameters $(\bar{\mathbf{\beta }},\mathbf{\Sigma })$ from data rather than specifying them a priori. More automated but ignores uncertainty in hyperparameters.

Related to Hierarchical Linear Models and the partial-pooling perspective in Partial Pooling as Multiple Comparisons Correction.

Estimation Decision Tree (Figure 5-2)

Single equation?
├── No autocorrelation, homoscedastic → OLS
├── Autocorrelated errors → GLS/FGLS
└── Endogenous regressors → IV / 2SLS

Multiple equations?
├── No cross-equation correlation → OLS equation-by-equation
├── Correlated errors, different X → SUR
└── Simultaneous system → 2SLS or 3SLS

Hausman Test for Endogeneity

Hausman Test

To test whether a regressor $X$ is endogenous (correlated with $u$):

1. Regress $X$ on all exogenous variables $\mathbf{Z}$; save residuals $\hat{v}$
2. Include $\hat{v}$ in the structural equation alongside $X$
3. If the coefficient on $\hat{v}$ is significant (Wald statistic), $X$ is endogenous — use IV/2SLS

This is also called the regression-based Hausman-Wu test.

Related to Instrumental Variables (MHE context).

Cross-Links

• Functional forms requiring nonlinear estimation: Functional Forms in Marketing
• ADL / Koyck estimation issues (MA errors): Carryover Effects and Distributed Lags
• Testing and diagnostics: Model Testing and Specification
• Bayesian workflow: Bayesian Workflow - Overview
• IV estimator theory: Instrumental Variables
• SUR for market share systems: Market Share Models