6. What survives demeaning?
(a) Years of schooling. Pooled OLS can estimate it. Worker FE cannot: the variable equals its own worker mean in every year, so demeaning turns it into a column of zeros. Adding year effects changes nothing.
(b) The national unemployment rate. Pooled OLS and worker FE can both estimate it, because it varies over time and the within-worker deviations are non-zero. Adding year effects removes it, since the variable takes a single value per year and is perfectly collinear with the year dummies.
(c) Union membership. All three specifications can estimate it, provided some workers actually change status during the sample. With worker FE the coefficient is identified only by those switchers.
(d) Tenure. Pooled OLS and worker FE can estimate it. Under worker and year effects it is identified only if tenure varies across workers within the same year, which requires job changes or different start dates. If tenure rose by exactly one for every worker in every year, its within-worker deviation would be a pure function of \(t\) and therefore collinear with the year effects.
(e) Because a regressor that takes one value per year lies in the space spanned by the year dummies. The year effects absorb every common time shock, so no independent variation is left to attribute to that variable.
(f) It is true that worker fixed effects remove the influence of schooling, and of every other permanent worker characteristic, from the estimation of the remaining coefficients: those coefficients are no longer contaminated by time-invariant confounding. It is not true that the return to schooling has been estimated. Schooling has no within-worker variation here, so its own effect is absorbed into \(\alpha_i\) and cannot be separated from it.
7. Why keep the intercept in a first-difference regression?
(a) Writing the model at \(t\) and \(t-1\) and subtracting,
\[
y_{it}-y_{i,t-1}
=(\alpha_i-\alpha_i)+g\bigl(t-(t-1)\bigr)
+\beta(x_{it}-x_{i,t-1})+(u_{it}-u_{i,t-1}),
\]
so
\[
\Delta y_{it}=g+\beta\Delta x_{it}+\Delta u_{it}.
\]
The unit effect disappears, and the common trend survives as the constant.
(b) Regressing \(\Delta y\) on \(\Delta x\) through the origin imposes \(g=0\): it assumes there is no common linear trend in the levels. If \(g\neq0\) and \(E(\Delta x)\neq0\), the omitted constant is an omitted regressor that is correlated with \(\Delta x\) in the sample, and the usual omitted-variable-bias reasoning applies: \(\hat\beta\) is pulled in the direction of \(\operatorname{sign}(g)\times\operatorname{sign}(E(\Delta x))\). A general rise in the outcome is then misattributed to a regressor that also happens to be rising.
(c) With \(y_{it}=\alpha_i+g_it+\beta x_{it}+u_{it}\), first differencing gives
\[
\Delta y_{it}=g_i+\beta\Delta x_{it}+\Delta u_{it}.
\]
A unit-specific constant remains, so first differencing removes the level effect \(\alpha_i\) but not the trend \(g_i\). Differencing a second time removes it, as does including unit-specific linear trends in the levels or FE specification. Either way, the identifying variation becomes changes in the growth of \(x\), which is demanding of the data.
(d) A remaining \(g_i\) that is correlated with \(\Delta x_{it}\) makes the transformed regressor correlated with the transformed error, which biases \(\hat\beta\). Clustering changes only the estimated variance of \(\hat\beta\) around whatever it converges to. This is the identification-versus-inference distinction: a standard-error choice cannot repair a misspecified conditional mean.