13.1 Endogenous regressors
A regressor is endogenous if it is correlated with the disturbance, and exogenous otherwise. In the model \(Y = X\beta + U\), the least squares estimator satisfies \[ b = \beta + \Bigl (\frac 1nX'X\Bigr )^{-1}\frac 1nX'U \xrightarrow {p} \beta + Q^{-1}\gamma , \qquad Q = \plim \tfrac 1nX'X, \quad \gamma = \plim \tfrac 1nX'U , \] by the argument of Theorem 4.4.1. If any element of \(\gamma \) is non-zero, \(b\) is inconsistent, and in general every coefficient is affected, not just that of the endogenous regressor. Three mechanisms produce endogeneity.
Omitted variables
If \(Y = X\beta + W\delta + \varepsilon \) and \(W\) is omitted, the disturbance is \(W\delta + \varepsilon \), correlated with \(X\) whenever \(W\) is. This is Theorem 8.1.1 in the language of endogeneity. The classic example is ability in an earnings equation: it raises earnings and is correlated with schooling, so the least squares return to schooling is biased upwards.
Measurement error
Theorem 13.1.1. Let \(Y_i = \alpha + \beta X_i^* + \varepsilon _i\), but suppose that \(X_i^*\) is observed with error, as \(X_i = X_i^* + m_i\), where \(m_i\) has mean zero and variance \(\sigma _m^2\) and is uncorrelated with \(X_i^*\) and \(\varepsilon _i\). Then the least squares slope of \(Y\) on \(X\) satisfies \[ \plim b = \lambda \beta , \qquad \lambda = \frac {\sigma _*^2}{\sigma _*^2 + \sigma _m^2} , \] where \(\sigma _*^2 = \Var (X^*)\). The estimate is biased towards zero (attenuation), by the factor \(\lambda \), the reliability of \(X\).
Proof. In terms of the observed \(X\), \(Y_i = \alpha + \beta X_i + (\varepsilon _i - \beta m_i)\), and the disturbance \(\varepsilon _i - \beta m_i\) is correlated with \(X_i\): \(\Cov (X_i, \varepsilon _i - \beta m_i) = -\beta \sigma _m^2\). Hence \[ \plim b = \frac {\Cov (X, Y)}{\Var (X)} = \frac {\beta \Cov (X^* + m, X^*)}{\Var (X^* + m)} = \frac {\beta \sigma _*^2}{\sigma _*^2 + \sigma _m^2} . \] □
Error in the dependent variable is harmless by comparison: it adds to the disturbance and inflates the variance, but leaves least squares consistent. With several regressors, measurement error in one of them biases its own coefficient towards zero and contaminates the others through their correlation with it.
Simultaneity
Haavelmo’s (1943) example is the simplest. Consumption depends on income, \(C_t = \alpha + \beta Y_t + u_t\), and income is consumption plus investment, \(Y_t = C_t + I_t\), where investment is determined outside the model. Solving the two equations, \[ Y_t = \frac {\alpha + I_t + u_t}{1 - \beta } , \] so a positive consumption shock raises income: \(\Cov (Y_t, u_t) = \sigma _u^2/(1 - \beta )\).
Theorem 13.1.2. In Haavelmo’s model, with \(I_t\) uncorrelated with \(u_t\), the least squares estimate of the marginal propensity to consume satisfies \[ \plim b = \beta + (1 - \beta )\frac {\sigma _u^2}{\sigma _I^2 + \sigma _u^2} > \beta . \]
Proof. \(\Var (Y) = (\sigma _I^2 + \sigma _u^2)/(1 - \beta )^2\) and \(\Cov (Y, u) = \sigma _u^2/(1 - \beta )\), so \(\plim b - \beta = \Cov (Y, u)/\Var (Y) = (1 - \beta )\sigma _u^2/(\sigma _I^2 + \sigma _u^2)\). □
The bias is upward, and it is worse when consumption shocks are large relative to the variation in investment. Simultaneity is the general problem of the next section; here we need only its symptom, a regressor correlated with the disturbance.
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