# install.packages(c("wooldridge", "stargazer"))
library(wooldridge) # provides the wage2 dataset
library(stargazer) # side-by-side regression tablesAn estimator \(\hat{\theta}\) is a rule (a formula applied to sample data) used to guess an unknown population parameter \(\theta\) — for us, the true regression coefficient \(\beta\). Because it depends on a random sample, \(\hat{\theta}\) is itself a random variable with a sampling distribution.
The bias of the estimator is the difference between its expected value (the average estimate we would get across infinitely many random samples) and the true parameter:
\[ \text{Bias}(\hat{\theta}) \;=\; E[\hat{\theta}] - \theta . \]
An estimator is unbiased when \(E[\hat{\theta}] = \theta\), i.e. the bias is zero — on average it hits the target, even if any single sample is off. It is biased when \(E[\hat{\theta}] \neq \theta\); the estimates are systematically centered away from the truth. Bias is about the center of the sampling distribution, not its spread. That distinction matters here: a biased estimator does not get better just because the spread shrinks.
A bigger sample does NOT fix omitted variable bias. OVB is a bias problem, not a precision problem. Adding rows (observations) shrinks the standard errors and narrows the sampling distribution, so the estimate becomes more precise — but it becomes more precise around the wrong number. As \(n \to \infty\) the biased estimator converges to \(\beta + \text{bias}\), not to \(\beta\). In fact a large sample makes things worse in a sense: you get a tight, confident, and wrong answer. (This is the lighter-and-lung-cancer / smoking example from class.)
Adding the right variable CAN fix it — but “more variables” means new columns (the omitted variable itself, or a good proxy for it), not more rows. If you can actually measure and include the confounder, the bias disappears because the two OVB conditions can no longer both hold. When the confounder is unmeasurable, you instead change the research design (experiment, diff-in-diff, RDD, IV, fixed effects) to get around it. Note that adding irrelevant variables does nothing for bias (and only costs you degrees of freedom) — see the bonus questions.
I use wage2 from the
wooldridge package: 935 working men from the U.S. National
Longitudinal Survey. Key variables:
wage — monthly earnings (dollars); I use
lwage = log(wage)educ — years of educationIQ — IQ score (a proxy for cognitive
ability)age — age in yearssibs — number of siblingsThis is a different dataset and a different question from the class examples (gender/tenure wages and test-scores/STR).
data(wage2)
wage2$lwage <- log(wage2$wage)
stargazer(wage2[, c("wage", "educ", "IQ", "age", "sibs")],
type = "text", title = "Summary Statistics")##
## Summary Statistics
## ========================================
## Statistic N Mean St. Dev. Min Max
## ----------------------------------------
## wage 935 957.945 404.361 115 3,078
## educ 935 13.468 2.197 9 18
## IQ 935 101.282 15.053 50 145
## age 935 33.080 3.108 28 38
## sibs 935 2.941 2.306 0 14
## ----------------------------------------We want the causal return to education. Wages depend on both schooling and underlying ability; ability is rewarded in the labor market and helps people get more schooling. The correct model therefore includes ability (proxied by IQ):
\[ \log(wage)_i \;=\; \beta_0 \;+\; \beta_1\, educ_i \;+\; \beta_2\, IQ_i \;+\; u_i \]
My key independent variable of interest is
educ (\(\beta_1\)
= the return to a year of schooling).
full_model <- lm(lwage ~ educ + IQ, data = wage2)
summary(full_model)##
## Call:
## lm(formula = lwage ~ educ + IQ, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.01601 -0.24367 0.03359 0.27960 1.23783
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.6582876 0.0962408 58.793 < 2e-16 ***
## educ 0.0391199 0.0068382 5.721 1.43e-08 ***
## IQ 0.0058631 0.0009979 5.875 5.87e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.3933 on 932 degrees of freedom
## Multiple R-squared: 0.1297, Adjusted R-squared: 0.1278
## F-statistic: 69.42 on 2 and 932 DF, p-value: < 2.2e-16In practice we rarely have a clean ability measure, so a
researcher would be forced to drop it. Omitting IQ gives
the short model:
\[ \log(wage)_i \;=\; \alpha_0 \;+\; \alpha_1\, educ_i \;+\; v_i \]
short_model <- lm(lwage ~ educ, data = wage2)
summary(short_model)##
## Call:
## lm(formula = lwage ~ educ, data = wage2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.94620 -0.24832 0.03507 0.27440 1.28106
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.973062 0.081374 73.40 <2e-16 ***
## educ 0.059839 0.005963 10.04 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4003 on 933 degrees of freedom
## Multiple R-squared: 0.09742, Adjusted R-squared: 0.09645
## F-statistic: 100.7 on 1 and 933 DF, p-value: < 2.2e-16For omitted variable bias to occur, both conditions
must hold. With educ as the key regressor \(A\) and the omitted IQ as
\(B\):
Condition I — The omitted variable is correlated with the key
regressor. Here: is IQ correlated with
educ? More able people tend to acquire more schooling, so
we expect a positive correlation.
cor.test(wage2$IQ, wage2$educ)##
## Pearson's product-moment correlation
##
## data: wage2$IQ and wage2$educ
## t = 18.385, df = 933, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.4670255 0.5612531
## sample estimates:
## cor
## 0.515697The correlation is about +0.52 and highly significant (\(p < 2.2\times10^{-16}\)), so Condition I is met (positive).
Condition II — The omitted variable is a determinant of the
outcome \(Y\). Here: does
IQ affect lwage? Ability is rewarded in the
labor market, so we expect a positive effect. Both the raw correlation
and the significant, positive \(\beta_2\) in the full model support
this.
cor.test(wage2$IQ, wage2$lwage)##
## Pearson's product-moment correlation
##
## data: wage2$IQ and wage2$lwage
## t = 10.13, df = 933, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## 0.2558383 0.3714047
## sample estimates:
## cor
## 0.3147877The correlation is about +0.31 and highly significant (\(p < 2.2\times10^{-16}\)), and in the full model \(\hat\beta_2 \approx 0.0059\) (\(p<0.001\)). Condition II is met (positive).
Both conditions hold, so omitting IQ biases the
coefficient on educ.
Full model: \(Y = \beta_0 + \beta_1 A + \beta_2 B + u\); short model: \(Y = \alpha_0 + \alpha_1 A + v\). The OVB formula gives
\[ \alpha_1 \;=\; \beta_1 \;+\; \underbrace{\beta_2 \cdot \delta_1}_{\text{bias}}, \qquad \text{where } B = \delta_0 + \delta_1 A + e . \]
The sign of the bias is the sign of \(\beta_2\) (effect of \(B\) on \(Y\)) times the sign of \(\delta_1\) (correlation of \(A\) and \(B\)):
| \(A,B\) positively correlated | \(A,B\) negatively correlated | |
|---|---|---|
| \(B\) has positive effect on \(Y\) | Positive bias | Negative bias |
| \(B\) has negative effect on \(Y\) | Negative bias | Positive bias |
In our case educ and IQ are
positively correlated (\(+0.52\)) and IQ has a
positive effect on lwage (\(\hat\beta_2 > 0\)). That is the
top-left cell → positive bias. So the short-model
return to education is biased upward: it overstates the
true return.
# delta1: slope of IQ on educ
aux <- lm(IQ ~ educ, data = wage2)
delta1 <- coef(aux)["educ"]
beta1 <- coef(full_model)["educ"] # true-ish return to educ
beta2 <- coef(full_model)["IQ"] # effect of IQ on lwage
alpha1 <- coef(short_model)["educ"] # biased return to educ
c(beta1_full = beta1, bias = beta2 * delta1,
predicted_short = beta1 + beta2 * delta1, actual_short = alpha1)## beta1_full.educ bias.IQ predicted_short.educ
## 0.03911990 0.02071931 0.05983921
## actual_short.educ
## 0.05983921The predicted short coefficient (\(\beta_1 + \beta_2\delta_1\)) reproduces the actual short coefficient almost exactly (≈ 0.0598), confirming the formula.
stargazer(full_model, short_model,
type = "text",
column.labels = c("Full (correct)", "Short (omits IQ)"),
covariate.labels = c("Education", "IQ", "Constant"),
title = "Return to Education: Full vs. Short Model") Dependent variable:
------------------------------------------------
lwage
Full (correct) Short (omits IQ)
(1) (2) | Education 0.039*** 0.060*** (0.007) (0.006) |
| IQ 0.006*** (0.001) |
| Constant 5.658*** 5.973*** (0.096) (0.081) |
Observations 935 935
R2 0.130 0.097
Adjusted R2 0.128 0.096
Residual Std. Error 0.393 (df = 932) 0.400 (df = 933)
F Statistic 69.419*** (df = 2; 932) 100.700*** (df = 1; 933)
====================================================================
Note: p<0.1; p<0.05; p<0.01
educ ≈
0.039 (about a 3.9% wage gain per year of school).educ ≈
0.060 (about a 6.0% gain).The short coefficient is larger, exactly the positive (upward) bias the OVB formula predicted. Confirmed.
In the short model, educ is doing two jobs at once. It
carries its own genuine effect on wages plus it stands
in for the ability it is correlated with but which we failed to control
for. More-educated workers also tend to be higher-ability workers, and
ability independently raises pay. So part of the wage premium that
really comes from ability gets wrongly credited to schooling.
educ “absorbs” ability’s effect, and because both links are
positive (able people get more schooling, and ability pays), the credit
pushes the coefficient up. Control for ability and that borrowed credit
is handed back to IQ, so the education coefficient falls to
its true, smaller value.
The two OVB conditions are jointly necessary. Break either one and the omitted variable causes essentially no bias — the key coefficient barely moves. Each bonus below breaks exactly one condition.