情境式问题设计

Scenario-Based Problem: Instrumental Variables Identification and Endogeneity

Research question Estimate the causal effect of years of schooling (S) on log annual earnings (Y) in a cross-sectional sample of N = 5,000 individuals born 1980–1985. The baseline structural model is: Y_i = β S_i + X_i′γ + u_i, where X includes age, gender, race, parental education, and region fixed effects. Schooling S_i may be endogenous due to unobserved ability and measurement error, implying Cov(S_i, u_i) ≠ 0.

Candidate instruments

• Z1: Indicator equal to 1 if there was a 4-year college within 10 km of the individual’s residence at age 17 (following the proximity strategy in Card 1995).
• Z2: Count of 2- and 4-year colleges within 25 km at age 17.

Both Z1 and Z2 are measured historically (at age 17), prior to observed schooling and earnings. Researchers argue that proximity affects schooling decisions through lower costs of attendance but has no direct effect on adult earnings, conditional on X and region fixed effects.

Selected estimation results (controlling for X and region fixed effects)

• OLS (Y on S, X): β̂_OLS = 0.074; SE = 0.006.
• First stage with Z1 only: S on Z1 and X
  • π̂1 = 0.350; SE = 0.080; t-stat = 4.375; first-stage F on Z1 = 19.14.
• Reduced form with Z1 only: Y on Z1 and X
  • ρ̂1 = 0.035; SE = 0.012.
• 2SLS with Z1 and X:
  • β̂_IV (implied by Wald ratio) = 0.035 / 0.350 = 0.100.
• First stage with Z1 and Z2 jointly: S on Z1, Z2, and X
  • Joint F for instruments = 23.4; partial R^2 of instruments = 0.020.
• Overidentification (2SLS with Z1 and Z2): Hansen J = 1.80, df = 1, p = 0.18.
• Endogeneity (Hausman test comparing OLS vs. 2SLS with Z1): χ^2(1) = 4.60, p = 0.032.

Assumptions to evaluate

• Relevance: Cov(Zk, S | X) ≠ 0 for k ∈ {1, 2}.
• Exclusion: Cov(Zk, u | X) = 0.
• Monotonicity (for LATE interpretation): For any two values z′ > z of the binary proximity instrument Z1, S_i(z′) ≥ S_i(z) for all i (no “defiers”) (Imbens and Angrist 1994).

Tasks

1. Identification logic (short answer)

• State precisely why OLS is potentially inconsistent in this setting. Frame your answer in terms of omitted variables and measurement error. Then state the moment condition that justifies IV estimation with Z1 (and with Z1, Z2 jointly).

2. Instrument selection (multiple choice; select all that apply and justify) Which statements are correct? A. Z1 is relevant if the first-stage F-statistic exceeds conventional weak-IV thresholds. B. Z1 is valid if its coefficient is statistically significant in the first stage. C. Z1 is valid if, conditional on X and region fixed effects, it affects Y only through S. D. Z2 is redundant if Z1 is already strong in the first stage; adding Z2 cannot improve identification. E. If both Z1 and Z2 are valid, 2SLS identifies β consistently and the J test should not reject asymptotically.

3. First-stage diagnostics (calculation + interpretation)

• Using the first-stage with Z1 only, compute and interpret the first-stage F-statistic. Based on the Staiger–Stock rule-of-thumb (F ≳ 10), assess whether weak instrument concerns are likely to be severe. Briefly note limitations of the rule-of-thumb and when Stock–Yogo critical values would be preferable.

4. Wald/2SLS estimation (calculation)

• Using the reduced form and first stage with Z1 only, compute the Wald estimator for β. Report the point estimate and interpret it as an approximate percentage effect on earnings per additional year of schooling.

5. Comparing OLS and IV (short answer)

• Explain why β̂_IV > β̂_OLS in these results. Discuss at least two mechanisms consistent with this pattern (e.g., classical measurement error in S, or negative selection of marginal students whose schooling decisions are more sensitive to proximity).

6. Endogeneity test (interpretation)

• Interpret the reported Hausman test (χ^2(1) = 4.60, p = 0.032). What does this imply about the consistency of OLS and the necessity of IV in this application? State the null and alternative hypotheses clearly.

7. Overidentification (interpretation; short answer)

• Interpret the Hansen J statistic (1.80, df = 1, p = 0.18) obtained when using Z1 and Z2 jointly. What does this test assess? Why does a failure to reject not prove instrument validity? Under what circumstances can the J test have low power?

8. Exclusion restriction analysis (applied reasoning)

• Critically evaluate the exclusion restriction for Z1 and Z2 in this setting. Discuss at least two plausible threats (e.g., local labor market opportunities, urban amenities correlated with earnings). Propose empirical strategies to probe these threats (e.g., richer geographic controls, pre-trend falsification with outcomes measured before schooling completion, covariate balance tests, or using historical college openings).

9. LATE interpretation (conceptual)

• Define the population for which β̂_IV identifies a Local Average Treatment Effect when Z1 is used. Characterize “compliers” in this context and discuss external validity: to whom does this estimate generalize, and to whom might it not?

10. Weak instruments—what if? (short answer)

• Suppose in an alternative subsample the first-stage F falls to 5. Explain the consequences for bias and inference in 2SLS. Name at least two methods that provide more reliable inference under weak instruments (e.g., LIML, Fuller, Anderson–Rubin test, conditional likelihood ratio). Cite the relevant literature.

11. Alternative instrument critique (applied reasoning)

• A colleague proposes using the local unemployment rate at age 17 as an instrument for schooling. Analyze whether it likely satisfies relevance and exclusion. If exclusion is doubtful, suggest design modifications or alternative strategies.

Answer format

• Provide numerical answers where requested (with brief interpretation).
• For conceptual questions, use concise, evidence-based reasoning tied to the identification assumptions and reported diagnostics.
• When referencing diagnostics, explicitly connect conclusions to the reported statistics and their sampling uncertainty.

References

• Angrist, J. D., and A. B. Krueger. 1991. Does Compulsory School Attendance Affect Schooling and Earnings? Quarterly Journal of Economics 106(4): 979–1014.
• Angrist, J. D., and J.-S. Pischke. 2009. Mostly Harmless Econometrics: An Empiricist’s Companion. Princeton University Press.
• Card, D. 1995. Using Geographic Variation in College Proximity to Estimate the Return to Schooling. In Aspects of Labour Market Behaviour: Essays in Honour of John Vanderkamp, edited by L. N. Christofides et al., 201–222. University of Toronto Press.
• Hansen, L. P. 1982. Large Sample Properties of Generalized Method of Moments Estimators. Econometrica 50(4): 1029–1054.
• Hausman, J. A. 1978. Specification Tests in Econometrics. Econometrica 46(6): 1251–1271.
• Imbens, G. W., and J. D. Angrist. 1994. Identification and Estimation of Local Average Treatment Effects. Econometrica 62(2): 467–475.
• Sargan, J. D. 1958. The Estimation of Economic Relationships Using Instrumental Variables. Econometrica 26(3): 393–415.
• Staiger, D., and J. H. Stock. 1997. Instrumental Variables Regression with Weak Instruments. Econometrica 65(3): 557–586.
• Stock, J. H., and M. Yogo. 2005. Testing for Weak Instruments in Linear IV Regression. In Identification and Inference for Econometric Models: Essays in Honor of Thomas Rothenberg, edited by D. W. K. Andrews and J. H. Stock, 80–108. Cambridge University Press.

情境式问题：销售团队培训—季度目标拆解与客户拜访优先级

情境背景 一家B2B SaaS厂商（面向中大型企业的流程自动化平台）进入新季度（Q）。你作为华东大区销售经理，需在首两周内完成季度目标拆解，并制定前四周的客户拜访优先级与排程。公司对季度业绩的结构性要求如下：

• 大区季度营收目标：人民币 6,000,000 元（其中新签客户至少占比 70%，存量扩展不超过 30%）。
• 机会阶段到成交的历史转化率（以过去 8 个季度为基准）：采购谈判 70%；方案报价 45%；方案评估 25%；需求探索 10%。
• 各阶段平均剩余销售周期（从当前阶段至签单）：采购谈判 20 天；方案报价 35 天；方案评估 60 天；需求探索 90 天。本季度剩余天数：84 天。
• 管理假设（用于测算与排序）：对于A层级战略客户，若获得跨部门高层联动，可在不改变方案范围的情况下将剩余周期缩短最多 20%；对于标注“需线下”的账户，若当周无面对面拜访则无法推进至下阶段。
• 资源约束：团队4名AE；每名AE每周可安排最多6次线下拜访，团队合计每周不超过24次；每次拜访含路程在内计作1个拜访名额；出差城市间当周往返不超过1次/人。

账户与机会清单（当前时点）

• A1 Alpha制造（新签）：阶段=方案报价；预计合同额=1.2M；干系人覆盖=60/100；采购窗口=第70天截止；竞争强度=高；客户层级=A；需线下=是；距离=单程2小时；最近接触=7天前。
• A2 Beta物流（扩展）：阶段=采购谈判；预计合同额=0.8M；覆盖=85；窗口=第40天；竞争=低；层级=B；需线下=否；距离=0.5小时；最近接触=5天。
• A3 Gamma科创（新签）：阶段=方案评估；预计合同额=1.5M；覆盖=50；窗口=第90天；竞争=中；层级=A；需线下=是；距离=1小时；最近接触=21天。
• A4 Delta医药（新签）：阶段=需求探索；预计合同额=2.0M；覆盖=20；窗口=第110天；竞争=中；层级=A；需线下=是；距离=3小时；最近接触=14天。
• A5 Epsilon零售（扩展）：阶段=方案报价；预计合同额=0.6M；覆盖=70；窗口=第60天；竞争=中；层级=C；需线下=否；距离=0.5小时；最近接触=10天。
• A6 Zeta汽车（新签）：阶段=方案报价；预计合同额=0.9M；覆盖=40；窗口=第50天；竞争=高；层级=A；需线下=是；距离=2.5小时；最近接触=9天。
• A7 Eta能源（新签）：阶段=方案评估；预计合同额=1.1M；覆盖=55；窗口=第80天；竞争=低；层级=B；需线下=否；距离=1小时；最近接触=12天。
• A8 Theta金融（扩展）：阶段=采购谈判；预计合同额=1.0M；覆盖=90；窗口=第30天；竞争=低；层级=A；需线下=是；距离=0.3小时；最近接触=3天。
• A9 Iota教育（新签）：阶段=需求探索；预计合同额=0.7M；覆盖=30；窗口=第75天；竞争=低；层级=C；需线下=否；距离=1小时；最近接触=20天。
• A10 Kappa食品（新签）：阶段=方案报价；预计合同额=0.5M；覆盖=65；窗口=第42天；竞争=中；层级=B；需线下=是；距离=1.5小时；最近接触=6天。

提示性方法（用于作答时采用的证据基础与工具）

• 目标拆解：以阶段到签单的历史转化率与剩余销售周期为约束，先估算当前管道的加权可兑现额与缺口，再将缺口分解至月、周、个人层级（Jordan & Vazzana, 2011/2012；Johnston & Marshall, 2016）。
• 拜访优先级：建议采用多属性期望价值法，综合考虑本季度可兑现概率、交易规模、时间约束（采购窗口与剩余周期）、客户层级权重与关系强度等，形成可比的优先级得分（Keeney, 1992；Zoltners, Sinha, & Lorimer, 2009）。
• 结构性配比：对扩展类机会可引入客户终身价值与留存贡献的权重，以避免短期目标与长期价值错配（Gupta & Lehmann, 2005）。对新签与扩展的比例需同时满足公司结构性要求。
• 账户分层与覆盖：对A层级账户要求更高的决策者覆盖与节奏密度，优先分配线下资源（Zoltners et al., 2009）。
• 历史行为度量：可参考RFM思想将“最近接触、接触频次、金额”内生到评分模型，以校准推进难度与时效性（Blattberg, Kim, & Neslin, 2008）。

任务与问题

1. 目标缺口测算与拆解

• 计算当前机会池对本季度的加权可兑现额，并与6,000,000元目标对比，明确总缺口以及新签（需达≥4,200,000元）与扩展（≤1,800,000元）的各自缺口。
• 考虑阶段剩余周期与采购窗口，判断哪些机会在不加速条件下可在本季度兑现；在“最多20%周期加速仅适用于A层级”的前提下，复核可兑现清单并重算缺口。
• 将缺口分解为前四周的周度达成路径，给出团队与个人（4名AE）层面的周目标（包括：需推进至下阶段的机会数量/金额、新生成或再资格化的机会金额），并说明与历史转化率之间的逻辑一致性。

2. 拜访优先级建模与选择

• 构建一套明确的优先级评分公式，至少包含以下维度及其权重：本季度成交概率（结合阶段转化率与时间约束计算）、预计合同额、客户层级（A/B/C权重）、采购窗口紧迫度、是否需线下、干系人覆盖度。请给出每个维度的定量化方法与权重设定依据，并引用相应文献支持你的建模选择。
• 基于评分结果，遴选出“首两周必须安排线下拜访”的前六个账户，并给出排序与理由。说明你的排序如何最大化本季度的期望营收，同时兼顾公司对新签与扩展的结构性配比。

3. 两周拜访排程与资源约束

• 在“团队每周最多24次线下拜访、当周城市间往返每人不超过1次、需线下账户若无拜访则不推进”的约束下，制定前两周的线下拜访排程（到账户-频次层级）。说明排程如何支持关键机会的阶段推进，以满足第1题拆解出的周目标。
• 对每个被排期的账户，明确本次拜访的推进假设（如：从评估推进至报价、从报价推进至采购谈判）与预期影响（对成交概率与周期的量化影响假设），并说明假设的证据基础。

4. 敏感性与风险对策

• 进行一次单因素敏感性分析：若整体阶段转化率下滑10%或关键A层级账户无法实现20%周期加速，你的优先级序列与排程需要如何调整，方能尽量保持结构性目标与季度达成的可行性？
• 提出一项增量管道策略，用于在第3-4周内补充可在本季度兑现的“短周期扩展”机会，并说明其对整体目标的边际贡献。

作答要求与评分要点

• 计算正确性与逻辑一致性：转化率、期望价值与时间约束的结合需自洽，并与拆解的周目标匹配。
• 模型透明度与可复用性：优先级评分方法需清晰、可复核、可在后续周度滚动更新。
• 资源—产出匹配：拜访排程需在资源约束内最大化当季期望回报，同时满足新签/扩展比例要求。
• 证据与引用：关键建模选择须有文献支持，并按下列参考文献格式进行规范引用。

参考文献

• Blattberg, R. C., Kim, B.-D., & Neslin, S. A. (2008). Database marketing: Analyzing and managing customers. Springer.
• Gupta, S., & Lehmann, D. R. (2005). Managing customers as investments: The strategic value of customers in the long run. Wharton School Publishing.
• Johnston, M. W., & Marshall, G. W. (2016). Sales force management: Leadership, innovation, technology (12th ed.). Routledge.
• Jordan, J. A., & Vazzana, M. (2011/2012). Cracking the sales management code: The secrets to measuring and managing sales performance. McGraw-Hill.
• Keeney, R. L. (1992). Value-focused thinking: A path to creative decisionmaking. Harvard University Press.
• Zoltners, A. A., Sinha, P., & Lorimer, S. E. (2009). Building a winning sales force: Powerful strategies for driving high performance. AMACOM.