特性重要性分析描述

Why price sensitivity matters in models

1. Definition and core measures

• Price sensitivity quantifies how outcomes change when price changes. The standard metric is price elasticity of demand:
  • Elasticity ε = dQ/Q ÷ dP/P = d ln(Q)/d ln(P). Typically ε < 0.
  • Own-price elasticity: response of a product’s demand to its own price.
  • Cross-price elasticity: response of demand to competitors’ or substitutes’ prices.
• Willingness to pay (WTP): the maximum price at which a customer would purchase; can be estimated at individual or segment levels.

2. Why it is critical

• Revenue and profit optimization: The optimal monopoly markup satisfies (P − c)/P = −1/ε (Lerner index). Without ε, pricing decisions are guesswork.
• Forecast accuracy: Demand forecasts that ignore price variation misattribute effects to seasonality, advertising, or trend, degrading accuracy and biasing planning.
• Marketing ROI and MMM: Price and promotion influence both volume and baseline; omitting or mis-specifying price inflates the apparent effect of media or promotions.
• Customer models: Conversion, churn, and CLV are price-sensitive. Optimizing for conversion alone can destroy margin; models must capture trade-offs between volume and profit.
• Assortment and cannibalization: Cross-price effects determine substitution patterns when prices move or items are added/removed.
• Dynamic pricing and experimentation: Reliable price response curves guide exploration, guardrails, and on-policy learning.

3. Modeling price response: practical choices

• Outcome and link:
  • Sales count: Poisson/NB (with zero inflation if needed), or log-linear regression on ln(Q).
  • Purchase probability: logistic/probit; use care to convert to revenue/margin targets.
  • Revenue or profit: direct modeling is useful when margin varies.
• Functional form:
  • Log-log: ln(Q) = α + β ln(P) + …; β is elasticity (β < 0).
  • Nonlinear/semiparametric: splines, gradient boosting, GAMs; add monotonic constraints to enforce non-increasing demand with price.
  • Discrete choice: MNL, nested or mixed logit for assortment and cross-price elasticities; captures substitution and random coefficients for heterogeneity.
• Heterogeneity:
  • Segment-level or hierarchical Bayes/random effects for product-, store-, and customer-level variation in ε.
  • Interaction terms: price × income, channel, promo, season, inventory, device, region.
• Cross-effects:
  • Include competitors’ prices, relative price (own minus competitor), price rank within category, and attribute-based similarity to approximate substitution when competitor data are sparse.
• Data issues and censoring:
  • Zero sales and stock-outs imply censored demand; consider Tobit, zero-inflated models, or explicit stock-out controls.
  • Represent discounts correctly: use net price after coupons/rebates; separate list price, promo depth, and deal type (feature/display).

4. Identification and endogeneity

• Problem: Price is set in response to expected demand, promotions, and competition, creating simultaneity. Naive regressions bias ε toward zero.
• Strategies:
  • Randomized pricing experiments or geo experiments with stratification and guardrails.
  • Panel fixed effects with within-entity variation and rich controls for time, seasonality, and promotions.
  • Instrumental variables/control functions using cost shifters (wholesale costs, exchange rates, input costs), tax/fee changes, or supply shocks that affect price but not demand directly.
  • Difference-in-differences or regression discontinuity around exogenous policy or threshold changes.
  • Double/debiased ML or causal forests to estimate ATE/heterogeneous treatment effects when using high-dimensional controls.
• Diagnostics: First-stage strength for IV, overidentification tests, falsification checks (pre-trends), stability of ε across specifications.

5. Feature engineering for price

• Absolute and relative measures: price level, ln(price), delta vs prior week, percent discount, price endings (e.g., .99), price relative to category median or competitor.
• Promotion decomposition: depth, type (BOGO, feature, display), interaction with price.
• Temporal: time since last price change, price volatility, lagged price to capture reference price effects.
• Availability and constraints: stock indicators, shipping fees, taxes, bundle pricing, and minimum advertised price rules.

6. Validation and diagnostics

• Sign and magnitude checks: own-price elasticity should be negative; plausible ranges by category (e.g., necessities often −0.2 to −1.0, discretionary goods commonly more elastic).
• Local vs global elasticity: report ε at relevant price points; plot demand curves and marginal revenue.
• Counterfactual backtesting: simulate historical price changes; compare predicted vs realized revenue/volume.
• Cross-price consistency: substitution matrices should be positive semidefinite and respect share constraints in choice models.
• Stability: evaluate across time, regions, and segments; check for Simpson’s paradox when pooling.
• Monotonicity: ensure predicted demand does not increase with price unless explained by confounders (e.g., Veblen effects, limited cases).

7. From estimation to decisions

• Profit targeting: Optimize P to maximize (P − c) × Q(P) subject to business constraints (price ladders, MAP, inventory, fairness, and legal requirements).
• Multi-product optimization: Account for cannibalization, category profit, and constraints via quadratic/convex optimization or simulation over choice model utilities.
• Dynamic pricing: Use contextual bandits or reinforcement learning with safety constraints; seed with causal estimates; continually re-estimate ε with new data.
• Segmentation and personalization: Use heterogeneous ε/WTP for targeted offers; ensure compliance and avoid discriminatory pricing where prohibited.
• A/B testing and monitoring: Monitor revenue lift, profit lift, and customer outcomes; use off-policy evaluation (IPW/DR) when necessary.

8. Common pitfalls

• Endogeneity ignored: leads to underestimating true elasticity in magnitude.
• Mis-measured price: excluding coupons, rebates, shipping, or taxes skews ε.
• Promo collinearity: discounts coincide with advertising and seasonality; separate effects carefully.
• Inventory confounding: stock-outs cap observed sales; treat as censoring, not zero demand.
• Sparse variation: insufficient price movement per product/store yields unstable ε; use hierarchical pooling.
• Objective mismatch: optimizing conversion instead of profit yields suboptimal prices.

Summary Price sensitivity is foundational to accurate forecasting and economically sound decision-making. Correctly specifying, identifying, and validating price effects enables reliable demand estimates, robust profit optimization, and effective marketing and assortment decisions. Ignoring price sensitivity, or modeling it naively, systematically biases results and erodes business performance.

折扣力度（割引率・割引額）は、需要予測・価格最適化・販促効果測定・LTV推定において中核となる説明変数です。適切に組み込まないと、弾力性の歪み、チャネル効果の誤帰属、最適割引の誤設計につながります。以下にモデル内での重要性と実装上の要点を整理します。

1. 推定バイアスの防止

• 価格弾力性の歪み: 実務では「ネット価格＝定価−割引」と「割引指標」の両方の管理が必要です。割引を欠落させると、需要の変動を季節性や広告に誤って帰属し、弾力性やMMMの係数が偏ります。
• 同時性・内生性: 割引は需要ショック（在庫過多、天候、競合価格）に反応して設定されることが多く、通常の回帰では上流ショックが残差に漏れて係数が内生化します。回避には以下を用います。
  • 固定効果（店舗×週、SKU×月など）で未観測の恒常要因を吸収
  • 計画ベースのIV（事前に確定した販促カレンダー、仕入れコスト、割引予算の締め日）による2SLS
  • DID/合成コントロールで時変の共通ショックを差分除去
  • 需要式の同時方程式化（価格・割引の決定式と需要式の同時推定）

2. 非線形・閾値・相互作用の表現

• 非線形な反応: 割引深度に対する反応は逓減や閾値（例: 10%、30%で段差）を示しやすい。GAM/スプライン、区分線形、単調性制約（monotonic）で形状を明示的に学習。
• 相互作用: 陳列（display）、チラシ/メール告知、広告GRP、季節、チャネル（EC/実店舗）、在庫水準と強い交互作用。相互作用項や分層ランダム係数（階層ベイズ、混合効果モデル）で異質性を表現。

3. 行動・タイミング効果

• 前倒し購買・キャリーオーバー: 大幅割引は一時的に需要を引き上げる一方、将来需要を先食い。DLNM/分布ラグや購買時点のハザードモデルで短期・遅行効果を分離。
• 参照価格・相対割引: 消費者は通常価格や競合価格を基準に反応する。絶対額/率だけでなく「参照価格との差」「カテゴリ内相対割引指数」を明示する。

4. 施策評価・最適化への直接的インパクト

• 因果効果推定: 割引深度は連続処置量。DR-learner/Generalized Random Forest/causal forest（continuous treatment）でCATE/個客別最適割引を推定。A/Bでは多水準因子計画で用量反応曲線を取得。
• 利益最大化: 利潤 π(d) = (p(d) − c)·q(d) − 販促コスト。限界利益が0になる深度が最適。推定した q(d) の導関数から最適dを導出し、在庫・キャパ・価格ルール（下限/上限）制約下で最適化。

5. 特徴量設計（推奨）

• 水準・相対尺度:
  • 割引率（%）、割引額、ネット価格、対参照価格差、カテゴリ/競合比の相対割引指数
  • ヒストリカル平均からの偏差（そのSKUの「いつもより安いか」）
• 形状・安定化:
  • 価格はlog変換、割引率は0–1でクリップ、極端値はウィンズライズ
  • 閾値バケット（0、(0,10]、(10,20]、…）とスプラインの併用
• コンテキスト:
  • 陳列/告知ダミー、広告接触（adstock/飽和変換）、在庫可用性/欠品フラグ、季節・祝日、天候、店舗/地域・週固定効果
• データ品質:
  • 返品・値引きの重複記録統合、税込税抜の統一、無料配布・セット/BOGOの正規化、時刻粒度の整合

6. モデル化の選択肢

• 需要予測: log-線形/Poisson/NB（階層ベイズ推奨）、パネル固定効果、交差価格効果（クロスエラスティシティ）
• MMM: 価格/割引を独立チャネルとして入れ、飽和・adstockは広告、割引は非線形逓増/逓減を別形状で。季節・カレンダー・供給制約を明示的に制御
• 離反/LTV: マージン低下と継続率への影響を同時にモデル化（構造方程式、joint modeling）。過度割引がLTVを毀損するセグメントをCATEで特定
• 動的価格: 在庫・需要不確実性下の収益最大化（コンテキスト付きバンディット、動的計画）。安全制約とガードレール（価格下限・頻度上限）を設定

7. 検証・健全性チェック

• 共線性: 価格と割引のVIF/条件数を監視（必要ならネット価格のみ＋参照差分に再パラメータ化）
• 擬似効果検出: プレトレンド/プラセボ期間での効果が0に近いか
• IVの妥当性: 弱IV検定（F統計）、過同定検定を実施
• オフポリシー評価: IPW/DR推定で割引ポリシー変更の期待収益を評価
• キャンペーン単位の時系列・地理ホールドアウトで再現性を確認

失敗パターンの回避

• 割引を入れないMMM/需要モデル: 広告や季節の係数が過大化
• 欠品を無視: 割引中の供給制約で需要潜在を過小推定
• ターゲティング内生性を無視: 割引の負の係数（「割引すると売れない」）という誤結論

結論 割引力度は、価格反応の推定精度、因果アトリビューション、利益最適化に直結する一次の説明変数です。非線形・内生性・異質性・在庫制約を前提とした特徴量設計と因果的推定枠組みで扱うことで、需要予測と最適割引政策の精度が大きく向上します。