Empirical Modeling of Social Science Theory

The workhorse of empirical social science. You specify a linear relationship: y = Xβ + ε. The coefficients β represent how Y changes with each X (holding others constant). OLS minimizes the sum of squared residuals — geometrically, it fits a hyperplane that minimizes vertical distance to the data.

经验社会科学的主要工具。你指定线性关系：y = Xβ + ε。系数 β 表示 Y 与每个 X 的变化（固定其他变量）。OLS 最小化残差平方和——从几何上讲，它拟合一个超平面，使到数据的垂直距离最小。

The Gauss-Markov Theorem: Under five classical assumptions, OLS produces the Best Linear Unbiased Estimator (BLUE) — no other linear estimator is more efficient.

The Five Classical Assumptions:

1. Linearity: The true relationship is linear in parameters (y = Xβ + ε).
2. Exogeneity: E(ε|X) = 0 — errors are uncorrelated with regressors.
3. No Multicollinearity: X'X is invertible (no perfect linear dependence among regressors).
4. Homoscedasticity: Var(ε|X) = σ² — errors have constant variance.
5. No Autocorrelation: Cov(εᵢ, εⱼ|X) = 0 for i ≠ j — errors are uncorrelated with each other.

Gauss-Markov 定理：在五个经典假设下，OLS 产生最佳线性无偏估计量 (BLUE)——没有其他线性估计量更有效。

五个经典假设：

1. 线性性：真实关系在参数中是线性的 (y = Xβ + ε)。
2. 外生性：E(ε|X) = 0 — 误差与回归变量不相关。
3. 无多重共线性：X'X 是可逆的（回归变量间无完全线性依赖）。
4. 同方差性：Var(ε|X) = σ² — 误差有常数方差。
5. 无自相关：Cov(εᵢ, εⱼ|X) = 0 对于 i ≠ j — 误差相互不相关。

Three Important Intuitions:

(1) Omitted Variable Bias (OVB): If you omit variable Z that both affects Y and correlates with X, then plim(b₁) = β₁ + β₃·(Cov(X,Z)/Var(X)) — your coefficient absorbs Z's effect, scaled by their correlation. The direction and magnitude of bias depends on the signs and magnitudes of these correlations.

(2) Measurement Error & Attenuation Bias: If X is measured with error (X* = X + u), then the observed regressor X* has added noise. The coefficient on X* becomes b ≈ β·(Var(X)/(Var(X) + Var(u))) — biased toward zero. This is called attenuation bias, and it's especially problematic for concluding "no effect" when in fact there's just too much noise.

(3) Simultaneity Bias: If X and Y cause each other, then the error ε in the Y equation is correlated with X (because Y appears in the X equation's error). This creates bias in both directions of causality, and the direction/magnitude depends on the system parameters.

三个重要的直觉：

(1) 遗漏变量偏差 (OVB)：如果你遗漏变量 Z 既影响 Y 又与 X 相关，那么 plim(b₁) = β₁ + β₃·(Cov(X,Z)/Var(X)) — 你的系数吸收 Z 的效应，按它们的相关性缩放。偏差的方向和幅度取决于这些相关性的符号和幅度。

(2) 测量误差与衰减偏差：如果 X 用误差测量 (X* = X + u)，那么观测到的回归变量 X* 有额外噪声。X* 上的系数变成 b ≈ β·(Var(X)/(Var(X) + Var(u))) — 向零有偏。这称为衰减偏差，对于结论"无效应"特别有问题，而实际上只是噪声太多。

(3) 同时性偏差：如果 X 和 Y 相互影响，那么 Y 方程中的误差 ε 与 X 相关（因为 Y 出现在 X 方程的误差中）。这导致因果关系两个方向的偏差，方向/幅度取决于系统参数。

OLS assumes errors are independent with constant variance. Reality is messier: errors might be correlated across time (autocorrelation) or across groups (clustering). GLS down-weights observations with more noise and accounts for error correlations. The idea: if you know something is measured with more error, don't trust it as much.

OLS 假设误差独立且方差恒定。现实更复杂：误差可能随时间相关（自相关）或跨组相关（聚集）。GLS 降低噪声较大的观测的权重并说明误差相关性。思想：如果你知道某物测量噪声较大，就不要那么相信它。

Specific Cases:

Heteroscedasticity (unequal variances): If Ω is diagonal but not σ²I, use Weighted Least Squares (WLS). Weight each observation inversely by its variance: observations with higher variance get lower weight.

Autocorrelation (time series): If Ω has off-diagonal elements due to temporal correlation, use methods like Cochrane-Orcutt or Prais-Winsten that iteratively estimate the correlation coefficient ρ and transform the data.

Clustering (grouped data): If Ω has blocks of correlation within groups, use Feasible GLS (FGLS) that estimates Ω from the data, or simply report cluster-robust standard errors (down-weight within-cluster variance).

特殊情况：

异方差（不等方差）：如果 Ω 是对角线但不是 σ²I，使用加权最小二乘法 (WLS)。通过方差的倒数加权每个观测：方差更高的观测获得较低权重。

自相关（时间序列）：如果 Ω 因时间相关性具有非对角线元素，使用 Cochrane-Orcutt 或 Prais-Winsten 等方法，迭代估计相关系数 ρ 并转换数据。

聚集（分组数据）：如果 Ω 在组内具有相关性块，使用可行 GLS (FGLS) 从数据估计 Ω，或简单地报告聚集稳健标准误（降低聚集内方差）。

When your outcome is not continuous (binary: yes/no, count: 0, 1, 2, ..., or categorical: A, B, C, D), OLS is inappropriate. Instead, you specify a likelihood function based on the theoretical distribution of your outcome, then find the parameters that maximize it.

当你的结果不是连续的（二元：是/否，计数：0、1、2 等，或分类：A、B、C、D），OLS 不合适。相反，你基于结果的理论分布指定似然函数，然后找到最大化它的参数。

How MLE Works Conceptually: You propose a model where observations come from a probability distribution, parameterized by θ. Each observed data point has some probability under this model. The likelihood is the joint probability of observing all the data, given θ. MLE finds the θ that makes the observed data most probable. We maximize the log-likelihood (equivalent but numerically stable) using numerical optimization (Newton-Raphson, BFGS, etc.).

The Log-Likelihood Trick: The log-likelihood is ℓ(θ) = Σᵢ log f(yᵢ | xᵢ, θ). Why use logs? (1) Products become sums, easier to compute; (2) log is monotonic, so maximizing ℓ is equivalent to maximizing L; (3) avoids numerical underflow when likelihoods are very small.

Standard Errors & Information Matrix: The information matrix I(θ) ≈ -∂²ℓ/∂θ² tells you the curvature of the likelihood surface. Sharper curvature = more certainty about θ. Standard errors are roughly proportional to the inverse of the information matrix: SE(θ̂) ≈ √(Var(θ̂)), where Var(θ̂) ≈ I(θ̂)⁻¹.

MLE 工作的概念方式：你提出一个模型，其中观测来自概率分布，由 θ 参数化。每个观测数据点在此模型下有某个概率。似然是观测所有数据的联合概率，给定 θ。MLE 找到使观测数据最可能的 θ。我们使用数值优化（Newton-Raphson、BFGS 等）最大化对数似然（等价但数值稳定）。

对数似然技巧：对数似然是 ℓ(θ) = Σᵢ log f(yᵢ | xᵢ, θ)。为什么使用对数？(1) 乘积变成和，更容易计算；(2) 对数是单调的，所以最大化 ℓ 等同于最大化 L；(3) 避免当似然非常小时的数值下溢。

标准误与信息矩阵：信息矩阵 I(θ) ≈ -∂²ℓ/∂θ² 告诉你似然表面的曲率。更锐利的曲率 = 关于 θ 更多的确定性。标准误大致与信息矩阵的逆成比例：SE(θ̂) ≈ √(Var(θ̂))，其中 Var(θ̂) ≈ I(θ̂)⁻¹。

Common Models:

Logit (binary outcomes): P(y=1|x) = Λ(xβ) = 1/(1 + e^{-xβ}). The latent variable interpretation: assume y* = xβ + ε where ε ~ Logistic, and observe y = 1 if y* > 0, else 0. The logistic CDF becomes Λ.

Probit (binary outcomes): P(y=1|x) = Φ(xβ), where Φ is the standard normal CDF. Similar to logit but with normal errors instead of logistic. Probit and logit give very similar predictions; choose based on theory or convention.

Poisson (count outcomes): E(y|x) = λ(x) = e^{xβ}. Assumes y follows a Poisson distribution with mean λ. Good for counts (0, 1, 2, ...) where variance might be small. If variance >> mean (overdispersion), use Negative Binomial instead.

Negative Binomial (overdispersed counts): Extends Poisson by allowing var(y) ≠ E(y). Uses an additional dispersion parameter α.

常见模型：

Logit（二元结果）： P(y=1|x) = Λ(xβ) = 1/(1 + e^{-xβ})。潜在变量解释：假设 y* = xβ + ε 其中 ε ~ Logistic，观测 y = 1 如果 y* > 0，否则 0。逻辑 CDF 变成 Λ。

Probit（二元结果）： P(y=1|x) = Φ(xβ)，其中 Φ 是标准正态 CDF。类似于 logit 但具有正态误差而不是逻辑误差。Probit 和 logit 给出非常相似的预测；根据理论或惯例选择。

Poisson（计数结果）： E(y|x) = λ(x) = e^{xβ}。假设 y 遵循具有平均值 λ 的 Poisson 分布。适合计数 (0, 1, 2, ...)，方差可能很小。如果方差 >> 平均值（过度离散），使用负二项分布。

负二项分布（过度离散计数）：通过允许 var(y) ≠ E(y) 扩展 Poisson。使用额外的离散参数 α。

The simplest way: add a term that multiplies two variables. If you think "the effect of X on Y depends on Z," you write:

The marginal effect of X on Y is now: ∂y/∂x = β₁ + β₃z. It depends on z! When z increases, so does X's effect on Y (if β₃ > 0).

最简单的方式：加一个乘以两个变量的项。如果你认为"X 对 Y 的影响取决于 Z"，你写：

X 对 Y 的边际效应现在是：∂y/∂x = β₁ + β₃z。它取决于 z！当 z 增加时，X 对 Y 的影响也增加（如果 β₃ > 0）。

Three Things You Must Report for Interactions:

1. The Marginal Effect: ∂y/∂x = β₁ + β₃z (this is what the effect actually is, not just β₁)
2. Its Standard Error: SE(∂y/∂x) — varies with z, so report it for meaningful values of z (e.g., mean, ±1 SD)
3. The Range of Significance: For what values of z is the effect statistically significant? (Solve: β₁ + β₃z ≈ 1.96·SE for the bounds)

Common Mistake: Looking at the coefficient β₃ on the interaction term alone is meaningless. β₃ tells you how the effect of X changes with Z, but you need β₁ too to understand the effect at baseline. Similarly, ignoring β₁ and β₂ (the "constitutive terms") leads to wrong interpretations.

Brambor, Clark & Golder (2006) Rules: (1) Always include the main terms (β₁x and β₂z); (2) Interpret effects using the marginal effect formula; (3) Plot the relationship for visual clarity; (4) Test whether the interaction is significant using a hypothesis test, not just looking at β₃'s p-value.

你必须为交互报告三件事：

1. 边际效应： ∂y/∂x = β₁ + β₃z（这是效应实际上是什么，不仅仅是 β₁）
2. 其标准误： SE(∂y/∂x) — 随 z 变化，所以为 z 的有意义值报告它（例如，均值、±1 SD）
3. 显著性范围：对于 z 的哪些值，效应在统计上显著？（求解：β₁ + β₃z ≈ 1.96·SE 对于边界）

常见错误：单独查看交互项的系数 β₃ 毫无意义。β₃ 告诉你 X 的效应随 Z 如何变化，但你也需要 β₁ 来理解基线处的效应。同样，忽略 β₁ 和 β₂（"构成项"）导致错误解释。

Brambor、Clark & Golder (2006) 规则： (1) 总是包含主项 (β₁x 和 β₂z)；(2) 使用边际效应公式解释效应；(3) 为清晰起见绘制关系；(4) 使用假设检验测试交互是否显著，不仅仅是查看 β₃ 的 p 值。

Linear interactions work great for continuous outcomes. But what if your model is inherently nonlinear? For instance, a demand function might be multiplicative: Q = A·P^β·I^γ (quantity depends on price and income). Or a production function: Y = A·K^α·L^β. You can't just multiply the variables — you need nonlinear least squares or maximum likelihood to estimate the exponents directly.

线性交互适用于连续结果。但如果你的模型本质上是非线性的呢？例如，需求函数可能是乘法的：Q = A·P^β·I^γ（数量取决于价格和收入）。或生产函数：Y = A·K^α·L^β。你不能只是乘以变量——你需要非线性最小二乘法或最大似然法来直接估计指数。

Additive vs. Non-Additive Error Structures: A linear model has an additive error: y = Xβ + ε. A nonlinear model might be multiplicative: y = f(X, β)·ε (errors scale with fitted values). Or purely nonlinear: y = f(X, β) + ε. The error structure matters for likelihood specification and standard errors.

How NLS Works: You specify a nonlinear function f(X, β) and minimize the sum of squared residuals: S(β) = Σ(yᵢ - f(Xᵢ, β))². Unlike OLS, there's no closed-form solution — you use iterative algorithms: Grid search + gradient descent (Newton-Raphson, BFGS). Start with multiple initial guesses because the objective function is not convex — you might get stuck in local optima.

Multiple Local Optima Problem: The function f(X, β) might have many combinations of β that fit the data nearly equally well. To handle this: (1) try many starting values; (2) plot the likelihood surface if possible; (3) use grid search to narrow the region, then refine; (4) check second-order conditions (Hessian) to verify you found a global optimum.

可加与非可加错误结构：线性模型有可加误差：y = Xβ + ε。非线性模型可能是乘法的：y = f(X, β)·ε（误差按拟合值缩放）。或纯非线性：y = f(X, β) + ε。错误结构对似然规范和标准误很重要。

NLS 工作原理：你指定非线性函数 f(X, β) 并最小化残差平方和：S(β) = Σ(yᵢ - f(Xᵢ, β))²。与 OLS 不同，没有闭式解——你使用迭代算法：网格搜索+梯度下降（Newton-Raphson、BFGS）。从多个初始猜测开始，因为目标函数不是凸的——你可能被困在局部最优值中。

多个局部最优值问题：函数 f(X, β) 可能有许多 β 组合几乎同样好地适应数据。处理方法：(1) 尝试许多起始值；(2) 如果可能，绘制似然表面；(3) 使用网格搜索缩小区域，然后细化；(4) 检查二阶条件（Hessian）以验证你找到了全局最优值。

Students are nested in classrooms, classrooms in schools, schools in districts. Observations are not independent — students in the same classroom are more alike than students in different classrooms. Standard regression ignores this clustering. Multilevel models explicitly account for it, allowing both fixed effects (intercepts and slopes that differ by group) and random effects (intercepts and slopes that are drawn from a distribution).

学生嵌套在教室，教室嵌套在学校，学校嵌套在地区。观测值不独立——同一教室的学生比不同教室的学生更相似。标准回归忽视了这种聚集。多层次模型显式考虑了它，允许固定效应（因群体不同的截距和斜率）和随机效应（从分布中抽取的截距和斜率）。

Intraclass Correlation Coefficient (ICC): How much variance is between groups vs. within groups? ICC = τ/(τ + σ²), where τ is between-group variance and σ² is within-group variance. If ICC = 0, observations are completely independent; if ICC = 1, all variation is between groups. Rule of thumb: If ICC > 0.05, you should use multilevel modeling. If ICC > 0.10, ignoring it leads to very misleading standard errors.

Random Intercept Model: yᵢⱼ = β₀ + u₀ⱼ + β₁xᵢⱼ + εᵢⱼ, where u₀ⱼ ~ N(0, τ) is the group-level deviation. Each group j has its own intercept (β₀ + u₀ⱼ), but the slope β₁ is the same across groups. Useful when groups differ in baseline but respond similarly to predictors.

Random Slope Model: yᵢⱼ = β₀ + u₀ⱼ + (β₁ + u₁ⱼ)xᵢⱼ + εᵢⱼ. Both intercept and slope vary by group. The slope u₁ⱼ captures group-specific responses to x. More complex but needed when theory suggests the effect of x differs by group.

Shrinkage/Partial Pooling: Multilevel models borrow information across groups. A small group (e.g., a school with only 5 students) gets its estimate "pulled" toward the grand mean, weighted by the ICC. More extreme estimates for small groups are down-weighted because they're likely just noise. This is more principled than listwise averaging within each group.

类内相关系数 (ICC)：有多少方差在组之间 vs. 组内？ICC = τ/(τ + σ²)，其中 τ 是组间方差，σ² 是组内方差。如果 ICC = 0，观测完全独立；如果 ICC = 1，所有变异在组间。经验法则：如果 ICC > 0.05，你应该使用多层次建模。如果 ICC > 0.10，忽视它导致非常误导的标准误。

随机截距模型： yᵢⱼ = β₀ + u₀ⱼ + β₁xᵢⱼ + εᵢⱼ，其中 u₀ⱼ ~ N(0, τ) 是组级偏差。每个组 j 有其自己的截距 (β₀ + u₀ⱼ)，但斜率 β₁ 在组间相同。当组在基线上不同但对预测器的反应相似时很有用。

随机斜率模型： yᵢⱼ = β₀ + u₀ⱼ + (β₁ + u₁ⱼ)xᵢⱼ + εᵢⱼ。截距和斜率都按组变化。斜率 u₁ⱼ 捕捉组特定的对 x 的反应。更复杂但当理论提示 x 的效应因组而异时需要。

收缩/部分汇总：多层次模型在组间借用信息。一个小组（例如，只有 5 个学生的学校）其估计被"拉向"整体均值，按 ICC 加权。小组的更极端估计被降权，因为它们可能只是噪声。这比在每个组内列表平均更原则性。

Lagged Dependent Variable (LDV): The simplest approach — include y_(t-1) as a predictor of y_t. Captures persistence (some things continue because they were already happening). But LDV models are controversial: the lagged y is endogenous if errors are correlated over time, requiring IV estimation.

Autoregressive Distributed Lag (ADL): Include both lagged values of the dependent variable AND lagged values of regressors: y_t = β₀ + β₁y_(t-1) + γ₁x_t + γ₂x_(t-1) + ε_t. Richer dynamics — how long does an X shock take to affect Y? Compute impulse responses by simulating the system forward.

Error Correction Model (ECM): If X and Y are cointegrated (they share a long-run equilibrium relationship), an ECM lets you model both short-run adjustment and long-run equilibrium. Variables that drift together will eventually correct back toward their equilibrium.

滞后因变量 (LDV)：最简单的方法——包含 y_(t-1) 作为 y_t 的预测器。捕捉持久性（某些事情继续因为它们已经在发生）。但 LDV 模型是有争议的：如果误差随时间相关，滞后的 y 是内生的，需要 IV 估计。

自回归分布滞后 (ADL)：同时包含因变量的滞后值和回归变量的滞后值：y_t = β₀ + β₁y_(t-1) + γ₁x_t + γ₂x_(t-1) + ε_t。更丰富的动态——X 冲击需要多长时间才能影响 Y？通过向前模拟系统计算脉冲响应。

误差修正模型 (ECM)：如果 X 和 Y 协整（它们共享长期均衡关系），ECM 让你同时建模短期调整和长期均衡。一起漂移的变量最终会向其均衡调整回来。

Unit Roots & Stationarity: A series is stationary if it has a constant mean and variance, and autocorrelations decay over time. A unit-root process (I(1)) is nonstationary — it's a random walk where shocks permanently affect the level. Spurious Regression Problem: If you regress two independent random walks on each other, you'll find a "significant" relationship just by chance. The solution: test for unit roots (Augmented Dickey-Fuller test), difference the data if nonstationary, or use cointegration methods.

Cointegration: Two I(1) series are cointegrated if some linear combination of them is I(0). Think of it as: the variables wander together like "a drunk and her dog" — they don't move independently, but they don't have a fixed relationship either. An ECM models this relationship.

Long-Run Multiplier in ADL: In an ADL(1,1): y_t = β₀ + β₁y_(t-1) + γ₁x_t + γ₂x_(t-1) + ε_t, the long-run effect of X on Y is the "long-run multiplier": (γ₁ + γ₂)/(1 - β₁). This is the total impact after all adjustments have played out. The immediate impact is γ₁, but future lags add γ₂, and the lagged y_(t-1) perpetuates the effect dynamically.

单位根与平稳性：如果序列有常数均值和方差，自相关随时间衰减，则该序列是平稳的。单位根过程 (I(1)) 是非平稳的——这是一个随机游走，其中冲击永久影响水平。虚假回归问题：如果你对两个独立的随机游走进行回归，你只会凭机会找到"显著"关系。解决方案：测试单位根（增强 Dickey-Fuller 检验），如果非平稳差分数据，或使用协整方法。

协整：两个 I(1) 序列是协整的，如果它们的某个线性组合是 I(0)。把它看作：变量一起漂移，像"一个醉汉和她的狗"——它们不独立移动，但也没有固定关系。ECM 建模这个关系。

ADL 中的长期乘数：在 ADL(1,1)：y_t = β₀ + β₁y_(t-1) + γ₁x_t + γ₂x_(t-1) + ε_t 中，X 对 Y 的长期效应是"长期乘数"：(γ₁ + γ₂)/(1 - β₁)。这是所有调整都已发生后的总体影响。立即影响是 γ₁，但未来滞后添加 γ₂，滞后 y_(t-1) 动态地延续效应。

Just as past events affect the present, nearby places affect each other. Countries' trade policies influence neighbors. Riots spread geographically. Pollution drifts across borders. Spatial econometrics lets you model these spillovers.

就如过去事件影响现在一样，附近的地方相互影响。国家的贸易政策影响邻近国家。骚乱地理上传播。污染跨越边界漂移。空间计量经济学让你建模这些溢出效应。

The Spatial Weight Matrix W: Define which units are "neighbors." Common approaches: (1) Contiguity: W_ij = 1 if i and j share a border, 0 otherwise; (2) Inverse distance: W_ij = 1/distance_ij, down-weighting far units; (3) k-nearest neighbors: connect each unit to its k nearest neighbors. Row-standardize W so each row sums to 1 for interpretability.

Spatial Autoregressive Model (SAR): y = ρWy + Xβ + ε. The outcome y depends on X (as in OLS), but also on neighbors' outcomes Wy, scaled by ρ (spatial coefficient). If ρ = 0, no spatial dependence. If ρ > 0, neighbors' high outcomes pull up your outcome. This is like a spatial analog of a lagged-y model in time series.

Spatial Error Model (SEM): y = Xβ + ε, where ε = λWε + u. The errors (not the outcomes themselves) are spatially correlated. This arises when unmeasured confounders are spatially clustered. Less commonly used than SAR but sometimes more appropriate theoretically.

Moran's I Test: Detects spatial autocorrelation in the residuals. I = (n/(W sum)) · (ε'Wε / ε'ε). If I is significantly different from 0, you have spatial dependence. Always run this test on OLS residuals to check whether spatial methods are needed.

Spatiotemporal Models (STAR, STADL): Combine both time and space: y_it = ρ₁W_sy_(t-1) + ρ₂ y_i(t-1) + Xβ + ε. This says: your outcome depends on neighbors last period, your own lag last period, and contemporary X. More complex but captures both spatial spillovers and temporal dynamics simultaneously.

空间权重矩阵 W：定义哪些单位是"邻近"。常见方法：(1) 邻接性：W_ij = 1 如果 i 和 j 共享边界，否则 0；(2) 逆距离：W_ij = 1/distance_ij，降低远单位的权重；(3) k-最近邻：连接每个单位到其 k 个最近邻。行标准化 W 使每行求和为 1，便于解释。

空间自回归模型 (SAR)： y = ρWy + Xβ + ε。结果 y 依赖于 X（如 OLS），但也依赖于邻近结果 Wy，按 ρ（空间系数）缩放。如果 ρ = 0，无空间依赖。如果 ρ > 0，邻近高结果拉高你的结果。这就像时间序列中滞后 y 模型的空间类似物。

空间误差模型 (SEM)： y = Xβ + ε，其中 ε = λWε + u。误差（不是结果本身）是空间相关的。这发生在未测量的混淆因子空间聚集时。比 SAR 使用较少，但有时在理论上更合适。

Moran 的 I 检验：检测残差中的空间自相关。I = (n/(W sum)) · (ε'Wε / ε'ε)。如果 I 显著不同于 0，你有空间依赖。始终在 OLS 残差上运行此检验以检查是否需要空间方法。

时空模型 (STAR、STADL)：同时结合时间和空间：y_it = ρ₁W_sy_(t-1) + ρ₂ y_i(t-1) + Xβ + ε。这说：你的结果取决于上一期邻近，上一期你自己的滞后，和当代 X。更复杂但同时捕捉空间溢出和时间动态。

The Problem: If X and Y cause each other simultaneously, OLS is biased. Example: higher wages might increase education demand (Y on X), but more education also increases wages (X on Y). Which direction is the effect? Without external information, you can't identify them separately (as Challenge 04 showed).

The Solution: Instrumental Variables. Find a variable Z that affects X but does NOT directly affect Y (an "instrument"). Then use Z to predict X, and use those predictions (not X itself) in your regression. This breaks the simultaneity because Z affects Y only through X.

Two Conditions for a Valid Instrument: (1) Relevance: Cov(Z, X) ≠ 0 (Z must be correlated with X). (2) Exclusion Restriction: Cov(Z, ε) = 0 (Z must be uncorrelated with the error in the Y equation, i.e., affect Y only through X).

Two-Stage Least Squares (2SLS): Stage 1: regress X on Z to get predicted values X̂ = Z(Z'Z)⁻¹Z'X. Stage 2: regress Y on X̂ to get β̂₂SLS = (X̂'X̂)⁻¹X̂'y. The coefficient on X̂ is consistent and asymptotically normal under the two conditions above.

Weak Instruments Problem: If Z is weakly correlated with X (low R² in first stage), the 2SLS estimates become very noisy. Stock & Yogo recommend: if first-stage F < 10, your instruments are "weak" and inference can be severely distorted. Report first-stage F-statistics always.

Over-identification Test (Sargan/Hansen J-test): If you have more instruments than endogenous variables (over-identified), you can test whether the excluded instruments are truly exogenous. This tests the exclusion restrictions. If J-test rejects, some instruments may be invalid.

Three-Stage Least Squares (3SLS): When you have multiple equations simultaneously, 3SLS estimates them jointly, accounting for correlations in errors across equations. More efficient than estimating each equation by 2SLS separately.

Generalized Method of Moments (GMM): A more flexible framework that works with general moment conditions E[g(y, X, θ)] = 0. Handles endogeneity, dynamics (lagged dependent variables), and weak instruments better than 2SLS. Requires choosing how to weight different moment conditions (via weighting matrix).

问题：如果 X 和 Y 同时相互影响，OLS 有偏。例子：更高工资可能增加对教育的需求（Y 在 X 上），但更多教育也增加工资（X 在 Y 上）。哪个方向是效应？没有外部信息，你不能单独识别它们（如挑战 04 所示）。

解决方案：工具变量。找一个变量 Z 影响 X 但不直接影响 Y（一个"工具"）。然后用 Z 预测 X，用这些预测（不是 X 本身）在你的回归中。这打破了同时性，因为 Z 只通过 X 影响 Y。

有效工具的两个条件： (1) 相关性：Cov(Z, X) ≠ 0（Z 必须与 X 相关）。(2) 排除限制：Cov(Z, ε) = 0（Z 必须与 Y 方程的误差不相关，即仅通过 X 影响 Y）。

两阶段最小二乘法 (2SLS)：阶段 1：对 Z 回归 X 得到预测值 X̂ = Z(Z'Z)⁻¹Z'X。阶段 2：对 X̂ 回归 Y 得到 β̂₂SLS = (X̂'X̂)⁻¹X̂'y。X̂ 上的系数在上述两个条件下是一致的且渐近正态的。

弱工具问题：如果 Z 与 X 的相关性弱（第一阶段 R² 低），2SLS 估计变得非常噪声。Stock & Yogo 建议：如果第一阶段 F < 10，你的工具是"弱"的，推断可能严重扭曲。总是报告第一阶段 F 统计量。

过度识别检验（Sargan/Hansen J 检验）：如果你有比内生变量更多的工具（过度识别），你可以测试被排除的工具是否真正外生。这测试排除限制。如果 J 检验拒绝，某些工具可能无效。

三阶段最小二乘法 (3SLS)：当你有多个方程同时时，3SLS 联合估计它们，考虑方程间误差中的相关性。比单独为每个方程估计 2SLS 更有效。

广义矩量法 (GMM)：一个更灵活的框架，使用一般矩条件 E[g(y, X, θ)] = 0。比 2SLS 更好地处理内生性、动态性（滞后因变量）和弱工具。需要选择如何加权不同的矩条件（通过加权矩阵）。

When you have multiple time series that cause each other, VAR is ideal. Instead of trying to figure out which variable causes which, VAR treats all variables symmetrically: each variable is regressed on its own lags and the lags of all other variables. The system captures feedback loops.

当你有多个相互影响的时间序列时，VAR 是理想的。与其试图弄清楚哪个变量导致哪个，VAR 对所有变量平等对待：每个变量对其自身滞后和所有其他变量的滞后回归。该系统捕捉反馈环。

VAR Specification: With M variables and p lags, the reduced-form VAR(p) is: Y_t = A₁Y_(t-1) + A₂Y_(t-2) + ... + ApY_(t-p) + ε_t, where Y_t is M×1, each A_i is M×M, and ε_t ~ N(0, Σ). Estimate each equation by OLS (feasible because regressors are the same across equations).

Impulse Response Functions (IRFs): Trace the effect of a one-unit shock to one variable through time. Example: Federal Reserve raises interest rates by 100 basis points. How does GDP respond? Inflation? Unemployment? IRFs show the time path of responses over the next 8-12 quarters. Computed by the Moving Average representation of the VAR.

Forecast Error Variance Decomposition (FEVD): At a given horizon (e.g., 8 quarters ahead), how much of the forecast error for GDP is explained by shocks to interest rates vs. other variables? FEVD tells you the relative importance of different shocks for forecasting each variable.

Granger Causality: X "Granger-causes" Y if past values of X help predict Y better than Y's own past alone. Test: regress Y on its own lags, then add lags of X and test if they're jointly significant (F-test). Note: Granger causality is predictive precedence, not true causality — it's controversial because reverse causality can produce spurious Granger causality.

Structural VAR (SVAR) & Identification: The reduced-form VAR has contemporaneous correlations in ε_t (errors across equations are correlated). To interpret IRFs as causal, you need to identify the structural shocks. Common approach: Choleski decomposition — assume a recursive causal structure among variables (e.g., Fed doesn't respond to current inflation, only lagged inflation; inflation doesn't respond to current unemployment, only lagged). This orders the variables and identifies the shocks.

VAR 规范：有 M 个变量和 p 个滞后，约化形式 VAR(p) 是：Y_t = A₁Y_(t-1) + A₂Y_(t-2) + ... + ApY_(t-p) + ε_t，其中 Y_t 是 M×1，每个 A_i 是 M×M，ε_t ~ N(0, Σ)。通过 OLS 估计每个方程（可行因为回归变量在方程间相同）。

脉冲反应函数 (IRF)：追踪一个变量的一单位冲击通过时间的效应。例子：联邦储备加息 100 个基点。GDP 如何反应？通胀？失业？IRF 显示未来 8-12 季度的响应时间路径。通过 VAR 的动态平均表示计算。

预测误差方差分解 (FEVD)：在给定地平线上（例如，8 季度前），GDP 预测误差有多少由利率冲击解释 vs. 其他变量？FEVD 告诉你不同冲击对预测每个变量的相对重要性。

Granger 因果性：X "Granger 导致"Y 如果 X 的过去值比 Y 仅自身过去更好地预测 Y。测试：对 Y 仅其自身滞后回归，然后添加 X 的滞后并测试它们是否联合显著（F 检验）。注意：Granger 因果是预测优先级，不是真正的因果——有争议因为反向因果可能产生虚假的 Granger 因果。

结构 VAR (SVAR) 和识别：约化形式 VAR 在 ε_t 中有当代相关性（方程间的误差相关）。要解释 IRF 为因果，你需要识别结构冲击。常见方法：Choleski 分解——假设变量间的递归因果结构（例如，联邦不对当前通胀反应，仅对滞后通胀；通胀不对当前失业反应，仅对滞后失业）。这排序变量并识别冲击。

Here's a crucial insight: coefficients are impulses, not responses. In a system where variables cause each other, the full causal response includes feedback through the system.

Imagine a Keynesian system: I (investment) causes Y (output), and Y causes C (consumption). If I increases by 1 dollar:

• Directly: Y increases by some amount (say, 1.5 times the investment — the multiplier)
• Indirectly: Higher Y causes higher C, which causes even higher Y (feedback loop)
• The ultimate response is larger than the direct effect

这里有一个关键的见解：系数是冲击，不是反应。在变量相互影响的系统中，完全因果反应包括通过系统的反馈。

想象一个凯恩斯制系统：I（投资）导致 Y（产出），Y 导致 C（消费）。如果 I 增加 1 美元：

• 直接：Y 增加某个数量（比如投资的 1.5 倍——乘数）
• 间接：更高的 Y 导致更高的 C，这导致更高的 Y（反馈环）
• 最终反应大于直接效应

The Multiplier Matrix Formula: Consider a simultaneous system: Ay = x, where A is M×M matrix of contemporaneous coefficients, y is the M×1 endogenous variables, x is the exogenous shocks. The solution is:

The matrix A⁻¹ is the "impact multiplier" — it converts exogenous shocks into equilibrium responses. For dynamic systems with lags, you must compute (I - A₁L - A₂L² - ...)⁻¹ where L is the lag operator, then evaluate at different horizons to get impact, interim, and long-run multipliers.

Why Not Just Look at Coefficients? Suppose in a trade model, when China increases tariffs by 1%, US tariffs also increase (retaliation). But that 1% increase in US tariffs affects US GDP, which affects demand for imports, which feeds back to China. The total US GDP response is much larger than any single coefficient. You must trace all these feedback loops.

Distinction: Impact vs. Long-Run Multipliers: Impact multiplier: immediate effect (same period). Interim multiplier: cumulative effect over k periods. Long-run multiplier: final equilibrium after all adjustments. Example: fiscal stimulus → immediate GDP boost, but inflation builds → interest rates rise → private investment crowded out → long-run boost smaller than impact.

Static vs. Dynamic Multipliers: A static multiplier compares two equilibria (before vs. after tariff change). A dynamic multiplier traces the time path of adjustment (quarter-by-quarter). For policy analysis, you usually want the dynamic multiplier — how does the economy actually transition, not just where it ends up?

where A is the matrix of direct effects and (I - A)⁻¹ is the "multiplier matrix" — it translates impulses into full responses through the system.

乘数矩阵公式：考虑一个同时系统：Ay = x，其中 A 是 M×M 当代系数矩阵，y 是 M×1 内生变量，x 是外生冲击。解是：

矩阵 A⁻¹ 是"冲击乘数"——它将外生冲击转化为均衡反应。对于具有滞后的动态系统，你必须计算 (I - A₁L - A₂L² - ...)⁻¹，其中 L 是滞后算子，然后在不同地平线评估以获得冲击、临时和长期乘数。

为什么不仅查看系数？假设在贸易模型中，当中国将关税增加 1% 时，美国关税也增加（报复）。但美国关税的 1% 增加影响美国 GDP，这影响进口需求，反馈给中国。美国 GDP 的总反应比任何单一系数大得多。你必须追踪所有这些反馈环。

区分：冲击 vs. 长期乘数：冲击乘数：立即效应（同一时期）。临时乘数：k 个时期内的累积效应。长期乘数：所有调整后的最终均衡。例子：财政刺激 → 立即 GDP 提升，但通胀建立 → 利率上升 → 私人投资挤出 → 长期提升小于冲击。

静态 vs. 动态乘数：静态乘数比较两个均衡（前 vs. 后关税变化）。动态乘数追踪调整的时间路径（逐季度）。对于政策分析，你通常想要动态乘数——经济如何实际过渡，而不仅仅是最终位置。

其中 A 是直接效应矩阵，(I - A)⁻¹ 是"乘数矩阵"——它通过系统将冲击转化为完整反应。