Functional Form, Partial Effects, and Omitted-Variable Bias

1. Partial effects

For a model \(Y=g(X;\beta)+u\) with \(E[u\mid X]=0\), the partial effect of a continuous regressor \(X_j\) is

In the linear model it is constant and equal to \(\beta_j\). With nonlinear terms it depends on \(x\), and we report either the partial effect at a representative value (PEA) or the average partial effect \(APE_j=E[\theta_j(X)]\), estimated by \(n^{-1}\sum_i\theta_j(X_i;\widehat\beta)\).

2. Logarithms

Exact percentage effects. In the log-level model, a one-unit change in \(x\) changes \(Y\) by exactly \(100(e^{\beta_1}-1)\)%. The approximation \(100\beta_1\) is accurate for \(|\beta_1|<0.1\). For a dummy variable \(D\) in \(\log Y=\beta_0+\delta D+\dots\), the percentage difference is \(100(e^{\delta}-1)\). With \(\delta=0.4\), that is 49%, not 40%.

Retransformation. A model for \(E[\log Y\mid X]\) is not a model for \(E[Y\mid X]\). By Jensen's inequality, \(\exp(X'\widehat\beta)\) underpredicts \(E[Y\mid X]\). If \(u\) is independent of \(X\), \(E[Y\mid X]=\exp(X'\beta)E[e^u]\), and Duan's smearing estimator \(n^{-1}\sum_ie^{\widehat u_i}\) estimates the correction. If \(u\) is heteroskedastic, log-linearized OLS is inconsistent even for elasticities. Then estimate \(E[Y\mid X]=\exp(X'\beta)\) directly by Poisson pseudo-maximum likelihood (Santos Silva and Tenreyro, 2006), which also handles zeros in \(Y\).

3. Polynomials and interactions

Quadratics. In \(Y=\beta_0+\beta_1x+\beta_2x^2+u\), the partial effect is \(\beta_1+2\beta_2x\). If \(\beta_2<0\), it turns negative at \(x^*=-\beta_1/(2\beta_2)\). A turning point is only meaningful if it lies well inside the support of the data. Its standard error follows from the delta method with gradient

Interactions. In \(Y=\beta_0+\beta_1x+\beta_2z+\beta_3xz+u\), the effect of \(x\) is \(\beta_1+\beta_3z\). Hence:

• \(\beta_1\) is the effect of \(x\) at \(z=0\), which may be outside the data. Centering \(z\) at its mean makes \(\beta_1\) the effect at the average \(z\).
• A test of \(\beta_1=0\) is not a test of "no effect of \(x\)". The hypothesis \(H_0:\beta_1=\beta_3=0\) is.
• The standard error of the effect at \(z_0\) is \(\sqrt{\operatorname{Var}(\widehat\beta_1)+z_0^2\operatorname{Var}(\widehat\beta_3)+2z_0\operatorname{Cov}(\widehat\beta_1,\widehat\beta_3)}\).

Dummy variables. With an intercept and \(G\) mutually exclusive categories, include \(G-1\) dummies; including all \(G\) creates perfect collinearity (the dummy-variable trap). Each coefficient is a difference relative to the omitted base category. A full interaction of a group dummy with all regressors is equivalent to estimating separate regressions by group. The Chow test is the \(F\) test of the interaction coefficients.

4. Omitted-variable bias

Suppose the long regression is \(Y=X_1'\beta_1+X_2'\beta_2+u\) with \(E[Xu]=0\), but we estimate the short regression of \(Y\) on \(X_1\) only.

\(\Gamma\) collects the coefficients of the auxiliary regressions of each omitted variable on the included ones. With one included and one omitted regressor (plus intercepts), the bias is

Two facts that are often missed.

1. There is no bias if the omitted variable is irrelevant (\(\beta_2=0\)) or uncorrelated with the included regressors (\(\Gamma=0\)). Randomized experiments work through the second channel.
2. Adding a control changes the coefficient of interest by exactly \(\widehat\Gamma\widehat\beta_2\). Coefficient movement when controls are added is therefore informative about selection on observables. Oster (2019) formalizes how far one can extrapolate from it to selection on unobservables.

5. Measurement error

Classical error in the regressor. Let the true model be \(Y=\beta_0+\beta_1x^*+u\), but we observe \(x=x^*+v\), with \(v\) independent of \(x^*\) and \(u\), mean zero, and variance \(\sigma_v^2\). Then \(Y=\beta_0+\beta_1x+(u-\beta_1v)\), and the composite error is correlated with \(x\). OLS satisfies

The factor \(\lambda\) is the reliability ratio: the estimate is attenuated toward zero. With other controls, the attenuation worsens because the controls absorb part of the true signal in \(x^*\) but none of the noise: replace \(\sigma^2_{x^*}\) by the residual variance of \(x^*\) after partialling out the controls. Fixed-effects and first-difference estimators can therefore amplify attenuation (Lecture 7).

Error in the dependent variable. If \(y=y^*+w\) with \(w\) uncorrelated with \(X\), OLS remains consistent; only precision falls. Non-classical error, for example misreporting correlated with the true value, can bias in either direction.

Remedies. Use an instrument such as a second, independent measurement (Lecture 5), a validation sample, or bounds based on assumed reliability ratios.

6. Bad controls

More controls are not always better. A bad control is a variable that is itself an outcome of the regressor of interest. Example: when estimating the effect of a college degree on earnings, controlling for occupation conditions on a channel through which the degree works. It also introduces selection bias, because people in the same occupation with and without degrees differ in unobserved ways. In the language of causal graphs, conditioning on a mediator removes part of the effect, and conditioning on a collider opens a spurious path. Rule: good controls are determined before the treatment (Angrist and Pischke, 2009, §3.2.3).

7. Specification tests

RESET (Ramsey, 1969). Add \(\widehat y^2,\widehat y^3\) to the regression and test their joint significance with a robust Wald test. Rejection indicates that the conditional mean is nonlinear in a way the specification misses. RESET is not a test for omitted variables in general: an omitted variable that is linearly related to the included ones leaves no trace in the fitted values.

Model choice. Between non-nested specifications (for example, level versus log of \(Y\)), \(R^2\) values are not comparable when the dependent variables differ. Compare on a common scale, use out-of-sample prediction error, or use economic theory. A specification search conducted on the same data used for inference invalidates nominal p-values. Report the search, or hold out data.

import numpy as np
rng = np.random.default_rng(3)
n = 5_000
A = rng.normal(size=n)                              # ability
S = 12 + 2 * (0.2 * A + rng.normal(size=n))         # schooling, correlated with ability
logw = 0.5 + 0.08 * S + 0.10 * A + rng.normal(0, 0.3, n)

def ols(y, *cols):
    X = np.column_stack([np.ones(len(y)), *cols])
    return np.linalg.lstsq(X, y, rcond=None)[0]

long_ = ols(logw, S, A)
short = ols(logw, S)
gamma = ols(A, S)
print(short[1], long_[1] + gamma[1] * long_[2])      # identical: the OVB identity

S_obs = S + rng.normal(0, 1, n)                     # add classical measurement error
lam = S.var() / S_obs.var()
print(ols(logw, S_obs, A)[1], "reliability ratio", round(lam, 3))

import numpy as np, wooldridge as wd, statsmodels.formula.api as smf
df = wd.data("wage2")
ctrl = "exper + tenure + married + south + urban + black"
short = smf.ols(f"np.log(wage) ~ educ + {ctrl}", data=df).fit()
# TODO: long regression adds IQ; auxiliary regression IQ ~ educ + ctrl

import numpy as np, wooldridge as wd, statsmodels.formula.api as smf
df = wd.data("wage2")
ctrl = "exper + tenure + married + south + urban + black"
short = smf.ols(f"np.log(wage) ~ educ + {ctrl}", data=df).fit()
long_ = smf.ols(f"np.log(wage) ~ educ + IQ + {ctrl}", data=df).fit()
aux = smf.ols(f"IQ ~ educ + {ctrl}", data=df).fit()
bS, bL, bIQ, gam = short.params["educ"], long_.params["educ"], long_.params["IQ"], aux.params["educ"]
print(f"short-regression return to educ: {bS:.5f}")
print(f"long-regression return to educ:  {bL:.5f}")
print(f"IQ coefficient {bIQ:.5f};  auxiliary slope of IQ on educ {gam:.4f}")
print(f"bL + gamma * bIQ = {bL + gam * bIQ:.5f}   (identity holds: {abs(bL + gam*bIQ - bS) < 1e-10})")
print(f"omitted-variable bias = {gam * bIQ:.5f}, i.e. {100 * gam * bIQ / bS:.1f}% of the short estimate")

short-regression return to educ: 0.06543
long-regression return to educ:  0.05441
IQ coefficient 0.00356;  auxiliary slope of IQ on educ 3.0963
bL + gamma * bIQ = 0.06543   (identity holds: True)
omitted-variable bias = 0.01102, i.e. 16.8% of the short estimate

import numpy as np, wooldridge as wd, statsmodels.formula.api as smf
df = wd.data("wage1")
base = "np.log(wage) ~ female + educ + exper + I(exper**2) + tenure"
r1 = smf.ols(base, data=df).fit(cov_type="HC1")
# TODO: exact gap 100*(exp(d)-1); interaction model; RESET with fitted values squared and cubed

import numpy as np, wooldridge as wd, statsmodels.formula.api as smf
df = wd.data("wage1")
base = "np.log(wage) ~ female + educ + exper + I(exper**2) + tenure"
r1 = smf.ols(base, data=df).fit(cov_type="HC1")
d, sd = r1.params["female"], r1.bse["female"]
print(f"female coefficient {d:.4f}: approx gap {100*d:.1f}%, exact gap {100*(np.exp(d)-1):.1f}% (se {100*np.exp(d)*sd:.1f} pp)")

r2 = smf.ols(base + " + female:educ", data=df).fit(cov_type="HC1")
print(f"female:educ = {r2.params['female:educ']:.4f} (robust se {r2.bse['female:educ']:.4f}, p = {r2.pvalues['female:educ']:.3f})")
for e in (12, 16):
    t = r2.t_test(f"female + {e}*female:educ = 0")
    print(f"  log gap at educ = {e}: {float(np.squeeze(t.effect)):.4f} (se {float(np.squeeze(t.sd)):.4f})")

df["yh"] = smf.ols(base, data=df).fit().fittedvalues
r3 = smf.ols(base + " + I(yh**2) + I(yh**3)", data=df).fit(cov_type="HC1")
w = r3.wald_test("I(yh ** 2) = 0, I(yh ** 3) = 0", scalar=True)
print(f"RESET (robust): chi2(2) = {w.statistic:.3f}, p = {w.pvalue:.3f}")

female coefficient -0.2979: approx gap -29.8%, exact gap -25.8% (se 2.7 pp)
female:educ = -0.0040 (robust se 0.0148, p = 0.787)
  log gap at educ = 12: -0.2958 (se 0.0367)
  log gap at educ = 16: -0.3118 (se 0.0662)
RESET (robust): chi2(2) = 25.470, p = 0.000