17. Philosophy and Method

Unlike asset pricing and macroeconomics, corporate finance does not have a single dominating framework. Modigliani-Miller (1958) is the frictionless benchmark to compare against, not the framework actually used; it organizes the corporate finance literature as: every study poses a question that breaks the MM assumptions in some practical way that makes corporate finance research meaningful.

Therefore corporate finance research is more driven by questions than economic models. All sorts of techniques could answer an interesting question, of which some are reduced-form empirical tools; such tools are useful as long as they answer important questions even without a structural model. Typical corporate finance research estimates parameters not directly tied to economic models, such as:

• the effect of an exogenous cash flow shock on investment decisions;
• the effect of higher leverage on enterprise value;
• the effect of management compensation on a firm's risk-taking behavior;
• the effect of credit availability on consumption.

Reduced-form empirical research doesn't need economic assumptions for modeling, and understands a parameter/effect in a statistical sense rather than an economic sense. But a good empirical corporate finance paper is often motivated by structural theoretical models rather than just identifying a causal effect. Empirical and theoretical works inspire each other: theory draws on cleanly identified causal effects to write a model; empirical work draws insights from theory to know which causal effects are interesting to test.

Although corporate finance has no dominating framework, reduced-form empirical studies do. The causal-analysis framework of Angrist and Pischke (2014) has become increasingly common.

17.2.1 Notation. Treatment of individual \(i\): treated \(D_i=1\), not treated \(D_i=0\). Potential outcomes: if treated \(Y_{1i}\equiv Y_i\mid(D_i=1)\), if not \(Y_{0i}\equiv Y_i\mid(D_i=0)\). Sample average \(\text{Avg}_n[X_i]\), population average \(\mathbb{E}[X_i]\).

17.2.2 Treatment effect and selection bias.

• Average treatment effect (ATE): population \(\mathbb{E}[Y_{1i}]-\mathbb{E}[Y_{0i}]\).
• ATT (on the treated): population \(\mathbb{E}[Y_{1i}\mid D_i=1]-\mathbb{E}[Y_{0i}\mid D_i=1]\).
• ATU (on the untreated): population \(\mathbb{E}[Y_{1i}\mid D_i=0]-\mathbb{E}[Y_{0i}\mid D_i=0]\).

But for individual \(i\) we cannot observe both \(Y_{1i}\) and \(Y_{0i}\), so none of these is directly observable. Instead we use the observable \(\text{Avg}_n[Y_{1i}\mid D_i=1]-\text{Avg}_n[Y_{0i}\mid D_i=0]\). Suppose we want ATT \(=\kappa\):

$$\text{Avg}_n[Y_{1i}\mid D_i{=}1]-\text{Avg}_n[Y_{0i}\mid D_i{=}0]=\kappa+\underbrace{\text{Avg}_n[Y_{0i}\mid D_i{=}1]-\text{Avg}_n[Y_{0i}\mid D_i{=}0]}_{\text{selection bias}} \tag{17.1}$$

The selection bias is unobserved (\(\text{Avg}_n[Y_{0i}\mid D_i{=}1]\) is counterfactual). It could be non-zero, so the researcher must think about its sign, the key being what underlying variation drives both treatment and outcome. Example: \(D_i=1\) means a firm recently changed its CEO, \(D_i=0\) no change, outcome \(Y_i\) = ROE; the selection bias could be negative since bad performance causes both lower ROE and a management change.

Randomization. If treatment is randomly assigned, \(\mathbb{E}[Y_{0i}\mid D_i{=}1]=\mathbb{E}[Y_{0i}\mid D_i{=}0]\), the selection bias in (17.1) is zero, and \(\kappa\) is consistently estimated by the observable \(\kappa=\text{Avg}_n[Y_{1i}\mid D_i{=}1]-\text{Avg}_n[Y_{0i}\mid D_i{=}0]\). But randomized treatment is almost impractical for corporate finance.

Regression with controls. Suppose the true model is \(Y_i=\alpha+\beta_1 X_{1i}+\beta_2 X_{2i}+\varepsilon_i\), \(\text{Cov}(X_{1i},\varepsilon_i)=\text{Cov}(X_{2i},\varepsilon_i)=0\), and we want \(\beta_1\). It is identified by

$$\beta_1^{\text{Reg}}=\frac{\text{Cov}(Y_i,\tilde X_{1i})}{\text{Var}(\tilde X_{1i})} \tag{17.2}$$

where \(\tilde X_{1i}\) is the residual of regressing \(X_{1i}\) on \(X_{2i}\) (\(X_{1i}=\pi_0+\pi_1 X_{2i}+\tilde X_{1i}\), \(\mathbb{E}[\tilde X_{1i}]=0\), \(\text{Cov}(\tilde X_{1i},X_{2i})=0\)). The point: identifying \(\beta_1\) means orthogonalizing \(X_1\)'s effect from \(X_2\)'s. If we only care about \(\beta_1\), this is legitimate; we can add controls to "shut down" unwanted channels of causality. But if we care about the synthesized effect of \(X_1\) on \(Y\) through all channels, no controls should be added. The core question: what underlying sources of variation remain to drive the variable of interest after taking out the sources added as controls.

Fixed effects regression. The true model

$$Y_i=\alpha+\beta X_{ij}+\sum_{j=1}^J\gamma_j\cdot\text{Group}_{ji}+\varepsilon_{ij} \tag{17.3}$$

(\(\text{Group}_{ji}=1\) if \(i\) belongs to group \(j\), controlling for unobserved fixed effects). \(\beta\) is identified by \(\beta^{\text{FE}}=\frac{\text{Cov}(Y_i,\tilde X_{ij})}{\text{Var}(\tilde X_{ij})}\) (17.4) with \(\tilde X_{ij}=X_{ij}-\bar X_j\), so

$$\beta^{\text{FE}}=\frac{\text{Cov}(Y_i,X_{ij}-\bar X_j)}{\text{Var}(X_{ij}-\bar X_j)} \tag{17.5}$$

the within estimator. Fixed effects solve inconsistency only when we believe the true model is (17.3); with multiple possible fixed effects, present results both with and without FE, and with different FE, to uncover the economic story and show robustness. For example, the table below (Mian-Sufi 2009) shows the effect of productivity growth on mortgage credit growth varies with the level of fixed effects (region / division / state / county):

Omitted variable bias (OVB). The "long" model \(Y_i=\alpha^l+\beta^l X_{1i}+\gamma X_{2i}+\varepsilon_i^l\) (17.6) and the "short" model \(Y_i=\alpha^s+\beta^s X_{1i}+\varepsilon_i^s\). Then

$$\beta^s=\beta^l+\pi_1\gamma \tag{17.7}$$

where \(\pi_1\) comes from \(X_{2i}=\pi_0+\pi_1 X_{1i}+u_i\), and OVB \(=\pi_1\gamma\) (not necessarily 0). Adding controls solves OVB, but the controls might also mess up \(X_{1i}\) (even near-perfect collinearity).

For the model \(Y=\mathbf X'\boldsymbol\beta+u\) (\(\mathbb{E}[\mathbf X u]=0\), \(\mathbb{E}[\mathbf{XX}']<\infty\) invertible), the OLS estimator's limiting distribution is \(\sqrt n(\hat{\boldsymbol\beta}_n-\boldsymbol\beta)\xrightarrow{d}\mathcal N(\mathbf 0,\boldsymbol\Omega)\), with \(\boldsymbol\Omega=\mathbb{E}[\mathbf{XX}']^{-1}\text{Var}(\mathbf X u)\mathbb{E}[\mathbf{XX}']^{-1}\).

• Homoskedasticity \(\text{Var}(u\mid\mathbf X)=\sigma^2\): \(\boldsymbol\Omega=\sigma^2\mathbb{E}[\mathbf{XX}']^{-1}\), consistently estimated by

$$\left(\frac1n\sum_{i=1}^n\mathbf X_i\mathbf X_i'\right)^{-1}\hat\sigma_n^2 \tag{17.8}$$

• Heteroskedasticity \(\text{Var}(u\mid\mathbf X)=\sigma^2(\mathbf X)\): estimated by

$$\left(\frac1n\sum\mathbf X_i\mathbf X_i'\right)^{-1}\left(\frac1n\sum\mathbf X_i\mathbf X_i'\hat u_i^2\right)\left(\frac1n\sum\mathbf X_i\mathbf X_i'\right)^{-1}$$

the (unclustered) heteroskedasticity-robust variance-covariance estimator.

• Clustering (within-group correlation):

$$\begin{cases}\text{Cov}(u_m,u_n\mid\mathbf X_m,\mathbf X_n)=0 & m,n\text{ different groups}\\[2pt] \text{Cov}(u_m,u_n\mid\mathbf X_m,\mathbf X_n)\ne0 & m,n\text{ same group}\end{cases} \tag{17.9}$$

With \(J\) groups, the OLS estimator \(\hat{\boldsymbol\beta}_n=\boldsymbol\beta+\left(\frac1n\sum\mathbf X_i\mathbf X_i'\right)^{-1}\left[\frac1n\sum\mathbf X_i u_i\right]\) (17.10). By (17.9) cross-group covariances are zero,

$$\text{Var}\left(\frac1n\sum\mathbf X_i u_i\right)=\frac1{n^2}\sum_{j=1}^J\text{Var}\left(\sum_{i\in\text{Group }j}\mathbf X_{ij}u_{ij}\right) \tag{17.11}$$

so the variance-covariance is the cluster-robust estimator

$$\left(\sum\mathbf X_i\mathbf X_i'\right)^{-1}\left(\sum_{j=1}^J\hat{\boldsymbol\mu}_j\hat{\boldsymbol\mu}_j'\right)\left(\sum\mathbf X_i\mathbf X_i'\right)^{-1},\qquad \hat{\boldsymbol\mu}_j=\sum_{i\in\text{Group }j}\mathbf X_{ij}\hat u_{ij} \tag{17.12}$$

Heteroskedasticity and clustering are two different things: heteroskedasticity is about the structure of the distribution of \(\mathbf X\) and \(u\) (nothing about sample correlation); clustering is about the sample correlation of conditional errors (nothing about the variance of \(u\) as a function of \(\mathbf X\)). The cluster-robust estimator is already heteroskedasticity-robust (it doesn't make the homoskedasticity simplification of (17.8)).

Classical measurement error. True model \(Y_i=\alpha+\beta X_i^*+\varepsilon_i\), \(X_i^*\) unobservable, \(\text{Cov}(\varepsilon_i,X_i^*)=0\). Replace with a mismeasured proxy \(X_i=X_i^*+e_i\) (\(\text{Cov}(X_i^*,e_i)=0\), \(\text{Cov}(\varepsilon_i,e_i)=0\), \(e_i\) the classical measurement error). The OLS regression \(Y_i=\alpha+\beta^{\text{OLS}}X_i+\epsilon_i\) gives

$$\beta^{\text{OLS}}=\underbrace{\frac{\text{Var}(X_i^*)}{\text{Var}(X_i^*)+\text{Var}(e_i)}}_{\text{attenuation bias}}\beta \tag{17.13}$$

biased towards 0 (attenuation bias), proven in (17.14).

IV solves classical measurement error. An instrument \(Z_i\) with relevance \(\text{Cov}(X_i^*,Z_i)\ne0\) and exclusion \(\text{Cov}(\varepsilon_i,Z_i)=\text{Cov}(e_i,Z_i)=0\). The IV estimator

$$\beta^{\text{IV}}=\frac{\text{Cov}(Y_i,Z_i)}{\text{Cov}(X_i,Z_i)} \tag{17.15}$$

identifies \(\beta\).

IV solves OVB. True \(Y_i=\alpha+\beta_1 X_{1i}+\gamma X_{2i}+\varepsilon_i\), but we only estimate \(Y_i=\alpha+\beta_1^{\text{OVB}}X_{1i}+\eta_i\) (\(\eta_i=\gamma X_{2i}+\varepsilon_i\)), with \(\beta_1^{\text{OVB}}\ne\beta_1\). With an instrument \(Z_i\) satisfying relevance \(\text{Cov}(X_{1i},Z_i)\ne0\) and exclusion \(\text{Cov}(\eta_i,Z_i)=0\) (\(Z\) affects \(Y\) only through \(X_1\)), then

$$\beta_1^{\text{IV}}=\frac{\text{Cov}(Y_i,Z_i)}{\text{Cov}(X_{1i},Z_i)} \tag{17.16}$$

identifies \(\beta_1\) (proven in (17.17)).

Suppose the true model is \(Y_i=\alpha+\beta X_i+\boldsymbol\gamma\cdot\mathbf V_i+\varepsilon_i\) (\(\boldsymbol\gamma,\mathbf V_i\) vectors), and we want \(\beta\). If \(Z_i\) satisfies relevance and exclusion, two steps obtain the true \(\beta\):

• Stage 1: regress \(X\) on \(Z\), \(X_i=\mu+\phi Z_i+\epsilon_i\), obtain \(\mu^{2SLS},\phi^{2SLS}\), construct \(\hat X_i=\mu^{2SLS}+\phi^{2SLS}Z_i\).
• Stage 2: regress \(Y\) on \(\hat X\), \(Y_i=\xi+\lambda\hat X_i+e_i\), then \(\lambda^{2SLS}=\beta\).

Benefits of 2SLS over IV: more flexibility. It accommodates additional controls in both stages (which can differ across stages); the first stage allows multiple instruments with optimal weights to optimally construct \(\hat X\) for the second stage.

Example: Shue-Townsend (2017) study the effect of executive option grants on risk-taking. Difficulty: omitted variables may affect both option granting and risk-taking (riskier firms tend to use more equity compensation). They exploit that executive compensation has fixed cycles: the instrument is a dummy \(I_{ijt}^{\text{PredictedFY}}\) identifying the predicted first year in each cycle of firm \(i\) (industry \(j\), year \(t\)), determined by the length of the last cycle. The predicted-first-year dummy is pinned down by the fixed cycle length, not co-moving with omitted variables → exclusion satisfied; compensation is adjusted in the first year of each cycle, correlated with the option grant → relevance. First stage \(\Delta O_{ijt}=\beta_0+\beta_1 I_{ijt}^{\text{PredictedFY}}+\gamma_t+\psi_j+\mu_{ijt}\) (\(\gamma_t\) year FE, \(\psi_j\) industry FE); second stage \(\Delta y_{ijt}=\delta_0+\delta_1\Delta\hat O_{ijt}+\gamma_t+\psi_j+e_{ijt}\). Result: more option grants increase the firm's stock volatility (the manager raises risk to maximize his own value after receiving more options). Limitation of 2SLS: it identifies a local average treatment effect (LATE), only for the subgroup of compliers, so external validity is hard to justify.

Regression discontinuity (RD). We want the effect of \(X\) on \(Y\), but omitted variables drive both. Suppose a continuous running variable \(R\) relates to \(X\) discontinuously at some point \(R=r\), i.e. \(\lim_{R\downarrow r}X(R)-\lim_{R\uparrow r}X(R)>0\). Just left and right of \(R=r\), \(R\) moves up infinitesimally, the other omitted variables don't move but \(X\) does, so computing

$$\lim_{R\downarrow r}Y(R)-\lim_{R\uparrow r}Y(R)$$

identifies the variation in \(Y\) contributed purely by \(X\).

Difference-in-differences (DiD). Two groups: treatment and control. The treatment group receives a single shock at \(t=\tau\); the common-trend assumption — both groups have the same time trend, parallel before and after the treatment. The DiD regression

$$Y_{it}=\alpha+\beta\,\text{TREAT}_i+\gamma\,\text{POST}_t+\delta\,\text{TREAT}_i\times\text{POST}_t+\varepsilon_{it}$$

\(\text{TREAT}_i\) identifies the treatment group, \(\text{POST}_t\) the post-treatment periods (\(\text{POST}_t>0\) for \(t\ge\tau\)), and the parameter of interest is the interaction coefficient \(\delta\). Example: Richardson-Troost (2009) study the effect of central-bank liquidity on bank failures: they use that Caldwell and Company collapsed on Nov 7, 1930, with part under the St. Louis Fed (no bailout) and part under the Atlanta Fed (bailout); assuming the two parts have the same time trend, taking the difference of the two across-time differences estimates the effect of bailing out on bank failures (e.g. suspension, liquidation).