Solutions Tutorial 5

Retrieval practice

  1. Pooled OLS, ordinary RE and Mundlak can estimate a time-invariant coefficient. For Mundlak, a causal interpretation requires the remaining individual effect \(c_i\) to be unrelated to the time-invariant regressor conditional on the person means and other controls.
  2. It subtracts a fraction \(\theta\) of each individual’s mean from every variable; \(\theta=0\) gives pooled OLS and \(\theta=1\) gives FE.
  3. It tests \(H_0:\sigma_\alpha^2=0\); rejection says only that a persistent individual component exists for GLS to model, not that \(\alpha_i\) is uncorrelated with the regressors.
  4. It compares the FE and RE estimates of the coefficients both estimators identify; the classical statistic additionally requires RE to be efficient under the null, so it is unreliable under heteroskedasticity or serial correlation.
  5. It adds each individual’s means of the time-varying regressors; their joint test assesses the ordinary RE restriction within that specification and permits person-clustered inference.

Solutions

Question 1: What variation identifies the union premium?

Part 1: Within and between variation

The variation calculation gives:

Variable Between-person SD Average within-person SD
educ 1.748 0.000
union 0.329 0.195
married 0.377 0.258

FE can identify the coefficients on union and married, because both change within at least some people. It cannot identify the coefficient on educ: for every person, \(\text{educ}_{it}-\overline{\text{educ}}_i=0\) in every year.

Part 2: Comparing the estimates

The estimates are:

Estimator union coefficient educ coefficient
Between 0.2707 0.0946
Pooled OLS 0.1801 0.0994
Random effects 0.1074 0.1012
Fixed effects 0.0821 not identified
Mundlak (CRE) 0.0821 0.0946

Part 3: Ordering the union coefficients

The union estimates are between \(>\) pooled OLS \(>\) ordinary RE \(>\) FE \(=\) Mundlak. The between estimator uses only cross-person differences. Pooled OLS mixes those differences with within-person changes. Ordinary RE partially demeans each person’s observations. FE fully demeans them. Mundlak includes all the person means, separating the within and between slopes and reproducing FE’s union coefficient in this balanced panel.

Part 4: Interpreting schooling

Education has no within-person variation, so FE cannot estimate its coefficient. Mundlak estimates it by comparing people with different schooling, conditional on their time-varying regressor means and other controls. The coefficient, 0.0946, corresponds to about \(100(e^{0.0946}-1)=9.9\%\) higher wages per additional year of schooling.

A causal interpretation requires the remaining person effect \(c_i\) to be unrelated to education conditional on those controls. If ability still differs by schooling level after conditioning on the means, the coefficient contains ability bias. Similar ordinary RE and Mundlak schooling coefficients do not establish exogeneity.

Part 5: Clustered standard errors

For pooled OLS, the default standard error on union is 0.0171 and the person-clustered one is 0.0276. The coefficient remains 0.1801 in both cases. Clustering changes the estimated sampling uncertainty, not the OLS minimization problem. The default calculation incorrectly treats eight observations on the same man as independent even though they share \(\alpha_i\).

Code 
library(dplyr)
library(fixest)
library(haven)
library(lmtest)
library(plm)

wagepan <- read_dta(
  "https://empirical-economics.netlify.app/tutorials/datafiles/WAGEPAN.DTA"
)

# Between and within variation.
bind_rows(lapply(c("educ", "union", "married"), function(v) {
  by_person <- wagepan |>
    group_by(nr) |>
    summarise(person_mean = mean(.data[[v]]),
              within_sd = sd(.data[[v]]), .groups = "drop")
  data.frame(
    variable = v,
    between_sd = sd(by_person$person_mean),
    average_within_sd = mean(by_person$within_sd)
  )
}))

f_all <- lwage ~ union + educ + black + hisp + exper + expersq + married
tv <- c("union", "exper", "expersq", "married")
model_vars <- c("lwage", "nr", "year", tv, "educ", "black", "hisp")
wagepan <- wagepan |>
  filter(if_all(all_of(model_vars), ~ !is.na(.x)))

pooled  <- plm(f_all, wagepan, index = c("nr", "year"), model = "pooling")
re      <- plm(f_all, wagepan, index = c("nr", "year"), model = "random")
fe      <- plm(f_all, wagepan, index = c("nr", "year"), model = "within")
wagepan_cre <- wagepan |>
  group_by(nr) |>
  mutate(across(all_of(tv), mean, .names = "mean_{.col}")) |>
  ungroup()
cre <- feols(
  lwage ~ union + exper + expersq + married + educ + black + hisp +
    mean_union + mean_exper + mean_expersq + mean_married,
  data = wagepan_cre, vcov = ~nr
)

# On a balanced panel, OLS on the person means is the between estimator.
wagepan_means <- wagepan |>
  group_by(nr) |>
  summarise(across(all_of(all.vars(f_all)[-1]), mean),
            lwage = mean(lwage), .groups = "drop")
between <- lm(f_all, data = wagepan_means)

# The coefficients.
lapply(list(Between = between, Pooled = pooled, RE = re, FE = fe,
            Mundlak = cre), function(m) coef(m)[c("union", "educ")])

# The standard errors to report.
V_between <- sandwich::vcovHC(between, type = "HC0")
V_pooled  <- vcovHC(pooled, method = "arellano", type = "HC0",
                    cluster = "group")
V_re      <- vcovHC(re, method = "arellano", type = "HC0",
                    cluster = "group")
V_fe      <- vcovHC(fe, method = "arellano", type = "HC0",
                    cluster = "group")

coeftest(between, V_between)
coeftest(re, V_re)
coeftest(fe, V_fe)
coeftest(cre, vcov. = vcov(cre))
coeftest(pooled)            # default standard errors
coeftest(pooled, V_pooled)  # person-clustered standard errors
Code 
import pandas as pd
from linearmodels.panel import BetweenOLS, PanelOLS, PooledOLS, RandomEffects

df = pd.read_stata(
    "https://empirical-economics.netlify.app/tutorials/datafiles/WAGEPAN.DTA"
)

variation = []
for v in ["educ", "union", "married"]:
    by_person = df.groupby("nr")[v]
    variation.append({
        "variable": v,
        "between_sd": by_person.mean().std(),
        "average_within_sd": by_person.std().mean(),
    })
print(pd.DataFrame(variation))

panel = df.set_index(["nr", "year"])
y = panel["lwage"]
all_vars = ["union", "educ", "black", "hisp",
            "exper", "expersq", "married"]
tv_vars = ["union", "exper", "expersq", "married"]
df = df.dropna(subset=["lwage", "nr", "year"] + all_vars).copy()
panel = df.set_index(["nr", "year"])
y = panel["lwage"]
X = panel[all_vars].assign(const=1.0)
X_tv = panel[tv_vars].assign(const=1.0)

between = BetweenOLS(y, X).fit(cov_type="robust")
pooled = PooledOLS(y, X).fit(
    cov_type="clustered", cluster_entity=True
)
re = RandomEffects(y, X).fit(
    cov_type="clustered", cluster_entity=True
)
fe = PanelOLS(y, X_tv, entity_effects=True).fit(
    cov_type="clustered", cluster_entity=True
)
means = panel[tv_vars].groupby(level="nr").transform("mean").add_prefix("mean_")
cre = PooledOLS(y, X.join(means)).fit(
    cov_type="clustered", cluster_entity=True
)

for name, model in {
    "Between": between, "Pooled": pooled, "RE": re, "FE": fe,
    "Mundlak": cre
}.items():
    print(name, model.params.reindex(["union", "educ"]))

pooled_default = PooledOLS(y, X).fit()
print(pooled_default.std_errors[["union", "educ"]])
print(pooled.std_errors[["union", "educ"]])

Question 2: Where does random effects sit?

Substitution gives

\[ \hat\theta =1-\sqrt{\frac{0.1235}{0.1235+8(0.1053)}} \approx 0.6425. \]

RE therefore subtracts about 64% of each person’s mean. It lies substantially closer to FE than to pooled OLS, but it does not remove the between-person comparison completely.

For a time-invariant regressor, \(\bar z_i=z_i\), so quasi-demeaning gives

\[ z_i-\theta\bar z_i=(1-\theta)z_i. \]

This retains variation when \(0<\theta<1\). At the FE endpoint, \(\theta=1\), the transformed regressor is zero and its coefficient is not identified.

If \(\sigma_\alpha^2\to0\), then \(\theta\to0\): there is no persistent component to remove, and RE becomes pooled OLS. If \(T\to\infty\) while \(\sigma_\alpha^2>0\), then \(\theta\to1\): quasi-demeaning approaches full demeaning and RE approaches FE.

A \(\theta\) close to one does not repair correlation between \(\alpha_i\) and a regressor. For every finite \(T\), RE retains some between variation. If that variation is endogenous, the RE estimator remains inconsistent; being numerically close to FE is not an identification argument.

Question 3: What do the specification tests actually establish?

Part 1: The LM test

The Breusch–Pagan test gives \(\chi^2(1)=3216.7\) and \(p<0.001\). Its null is \(H_0:\sigma_\alpha^2=0\). We reject: the same person’s errors share a persistent component, so RE’s GLS transformation has something to model and can improve efficiency relative to pooled OLS. The test does not examine \(\operatorname{Cov}(\alpha_i,x_{it})\), so it establishes the credibility of neither estimator.

Part 2: The Hausman test

The classical Hausman comparison gives \(\chi^2(4)=250.26\) and \(p<0.001\). educ, black and hisp must be excluded because FE cannot identify them; Hausman can compare only coefficients estimated by both models. Under

\[ H_0:E(\alpha_i\mid x_{i1},\ldots,x_{iT})=0, \]

FE and RE are both consistent and should converge to the same slopes. The decisive rejection indicates a systematic difference and points to correlation between the individual effect and at least one regressor.

The classical covariance-difference formula relies on RE being efficient under \(H_0\). Cluster-robust covariance matrices do not preserve the required efficiency ordering, so their difference need not be positive definite and can even produce a negative statistic. Clustered inference belongs in the Mundlak test, not inside the classical Hausman formula.

Part 3: The Mundlak regression and joint test

In the Mundlak regression, the coefficients on the time-varying regressors equal the FE coefficients (to the displayed precision):

Regressor FE Mundlak
union 0.0821 0.0821
exper 0.1168 0.1168
expersq -0.0043 -0.0043
married 0.0453 0.0453

Part 4: Interpretation

The person means are jointly significant with person-clustered inference (\(p<0.001\)), so the ordinary RE restriction is rejected. FE and Mundlak estimate the union coefficient from within-person changes under strict exogeneity. The Mundlak test does not test strict exogeneity or whether the remaining individual effect is unrelated to education.

Mundlak estimates a conditional between-person schooling association of 0.0946 log points, about 9.9% per additional year. A causal interpretation needs the additional assumption stated in Question 1. The joint rejection neither establishes nor rules out that assumption.

Code 
library(dplyr)
library(fixest)
library(haven)
library(plm)

wagepan <- read_dta(
  "https://empirical-economics.netlify.app/tutorials/datafiles/WAGEPAN.DTA"
)
f_all <- lwage ~ union + educ + black + hisp + exper + expersq + married
pooled <- plm(f_all, wagepan, index = c("nr", "year"), model = "pooling")

# H0: sigma_alpha^2 = 0.
plmtest(pooled, type = "bp")

# Classical Hausman: common, time-varying coefficients and classical vcovs.
f_tv <- lwage ~ union + exper + expersq + married
fe_tv <- plm(f_tv, wagepan, index = c("nr", "year"), model = "within")
re_tv <- plm(f_tv, wagepan, index = c("nr", "year"), model = "random")
phtest(fe_tv, re_tv)

# Mundlak / correlated random effects with person-clustered inference.
tv <- c("union", "exper", "expersq", "married")
wagepan_cre <- wagepan |>
  group_by(nr) |>
  mutate(across(all_of(tv), mean, .names = "mean_{.col}")) |>
  ungroup()

cre <- feols(
  lwage ~ union + exper + expersq + married + educ + black + hisp +
    mean_union + mean_exper + mean_expersq + mean_married,
  data = wagepan_cre, vcov = ~nr
)
fe_clustered <- feols(
  lwage ~ union + exper + expersq + married | nr,
  data = wagepan, vcov = ~nr
)

coef(cre)[tv]
coef(fe_clustered)[tv]
wald(cre, "mean_")
Code 
import numpy as np
import pandas as pd
from linearmodels.panel import PanelOLS, PooledOLS, RandomEffects
from scipy import stats

df = pd.read_stata(
    "https://empirical-economics.netlify.app/tutorials/datafiles/WAGEPAN.DTA"
)
panel = df.set_index(["nr", "year"])
y = panel["lwage"]
all_vars = ["union", "educ", "black", "hisp",
            "exper", "expersq", "married"]
tv = ["union", "exper", "expersq", "married"]

# Breusch--Pagan LM statistic for this balanced panel.
pooled = PooledOLS(y, panel[all_vars].assign(const=1.0)).fit()
resid = pooled.resids.rename("e").reset_index()
N, T = resid["nr"].nunique(), resid["year"].nunique()
numerator = resid.groupby("nr")["e"].sum().pow(2).sum()
denominator = resid["e"].pow(2).sum()
LM = N * T / (2 * (T - 1)) * (numerator / denominator - 1) ** 2
print(f"LM = {LM:.2f}, p = {stats.chi2.sf(LM, 1):.3g}")

# Classical Hausman: do not use cluster-robust vcovs in this formula.
X_tv = panel[tv].assign(const=1.0)
fe_u = PanelOLS(y, X_tv, entity_effects=True).fit()
re_u = RandomEffects(y, X_tv).fit()
d = fe_u.params[tv] - re_u.params[tv]
V = fe_u.cov.loc[tv, tv] - re_u.cov.loc[tv, tv]
H = float(d @ np.linalg.inv(V) @ d)
print(f"H = {H:.2f}, p = {stats.chi2.sf(H, len(tv)):.3g}")

# Mundlak regression and a clustered joint Wald test.
means = df.groupby("nr")[tv].transform("mean").add_prefix("mean_")
cre_df = pd.concat([df, means], axis=1).set_index(["nr", "year"])
rhs = tv + ["educ", "black", "hisp"] + list(means.columns)
cre = PooledOLS(
    cre_df["lwage"], cre_df[rhs].assign(const=1.0)
).fit(cov_type="clustered", cluster_entity=True)

fe_c = PanelOLS(y, X_tv, entity_effects=True).fit(
    cov_type="clustered", cluster_entity=True
)
print(pd.concat(
    [fe_c.params[tv], cre.params[tv]],
    axis=1, keys=["FE", "Mundlak"]
))

R = np.zeros((len(means.columns), len(cre.params)))
for row, name in enumerate(means.columns):
    R[row, list(cre.params.index).index(name)] = 1
print(cre.wald_test(R, np.zeros(len(means.columns))))

Question 4: Does the error process favour FE or FD?

If \(u_{it}\) is serially independent with constant variance, adjacent differences share \(-u_{i,t-1}\) with opposite signs:

\[ \operatorname{Corr}(\Delta u_{it},\Delta u_{i,t-1})=-\frac12. \]

If the level error is a random walk, \(u_{it}=u_{i,t-1}+e_{it}\), then \(\Delta u_{it}=e_{it}\) and the benchmark correlation is zero.

The interval \([-0.438,-0.354]\) contains neither \(-0.5\) nor zero. Thus, the level errors look like neither serially independent noise nor a random walk. The diagnostic concerns the relative efficiency of FE and FD under alternative error processes; it does not change their common identifying logic. Both remove \(\alpha_i\) and both require strict exogeneity of the idiosyncratic error in the static model.

The point estimates can differ because FE uses deviations from the full person mean whereas FD weights adjacent changes, and because a finite sample produces sampling variation. A large gap can also be a warning about a misspecified functional form, measurement error, or failure of strict exogeneity. It is not by itself proof that one estimator is consistent and the other is not.

Both models should use person-clustered standard errors because remaining disturbances can still be heteroskedastic and serially correlated within a person. The simple FE and FD transformations do not guarantee independent transformed errors.

Question 5: Make a model choice, not a test choice

Firms with stronger management or a more trusting workplace culture may both adopt remote work earlier and achieve higher productivity. Then remote-work adoption is correlated with \(\alpha_i\). Pooled OLS and ordinary RE retain the contaminated between-firm comparison and are inconsistent. FE and FD remove \(\alpha_i\) and remain credible under strict exogeneity.

The three tests answer different questions:

  • The LM rejection says \(\sigma_\alpha^2>0\), so there is a persistent firm component and RE may be more efficient than pooled OLS if their shared exogeneity assumption holds.
  • The classical Hausman non-rejection says that this particular test did not detect a difference. It does not prove orthogonality, may have low power, and relies on the classical efficiency assumptions.
  • The clustered Mundlak rejection directly detects correlation between the firm means of the regressors and the outcome, using inference appropriate for within-firm dependence. It is evidence against ordinary RE.

These statements are compatible: a firm effect can exist, the classical test can fail to detect its correlation with adoption, and the clustered Mundlak test can detect it.

For the remote-work coefficient, FE and FD both remove the firm effect and are more credible than pooled OLS or RE. The FD-residual correlation being close to zero is the random-walk benchmark, so it favours FD for efficiency; FE should still be reported as a robustness comparison. This efficiency choice is conditional on having already chosen a within-firm design for credibility.

Neither FE nor FD can identify the coefficient on FounderLed_i, because that variable has no within-firm changes. An RE coefficient is not an automatic substitute: it uses between-firm variation under the very orthogonality restriction rejected by the Mundlak test. Estimating a credible founder-led effect requires an additional source of identifying variation or a different research design.

The researcher should cluster by firm, the level at which policy adoption varies and within which errors may be dependent over time. Clustering can make standard errors valid under within-firm dependence. It never changes a coefficient and cannot repair omitted-variable bias.

Self-study

Question 6: Reading \(\theta\) off the variance components

1. Using \(\theta=1-\sqrt{\sigma_u^2/(\sigma_u^2+T\sigma_\alpha^2)}\):

Panel \(T\hat\sigma_\alpha^2\) \(\hat\sigma_u^2+T\hat\sigma_\alpha^2\) \(\hat\theta\)
A 0.06 0.56 \(1-\sqrt{0.893}=0.055\)
B 4.50 4.65 \(1-\sqrt{0.032}=0.820\)
C 0.40 0.60 \(1-\sqrt{0.333}=0.423\)

2. Panel A has \(\hat\theta\approx0.06\), so RE removes almost nothing of each individual’s mean and sits essentially on top of pooled OLS. Panel B has \(\hat\theta\approx0.82\) and is the closest to FE. Panel C sits between the two. The pattern is intuitive: \(\theta\) rises with the share of the total variance attributable to the individual effect and with the number of periods.

3. No. Consistency is not continuous in \(\theta\). RE is consistent only under \(E(\alpha_i\mid x_{i1},\ldots,x_{iT},z_i)=0\). If that condition fails, RE remains inconsistent for every \(\theta<1\), because a fraction \(1-\theta\) of the contaminated between-individual variation is still used. A large \(\theta\) shrinks the asymptotic bias relative to pooled OLS, but only the complete removal at \(\theta=1\), which is FE, eliminates it.

4. The agreement is mechanical rather than evidential. With \(\hat\theta\approx0.06\) the RE transformation subtracts about six per cent of each person’s mean, so RE and pooled OLS are using almost exactly the same variation and are numerically bound to be close. Two estimators that share a comparison cannot be used to check whether that comparison is contaminated. The informative comparison is with FE, which uses only within-person variation, or better a Mundlak regression with person-clustered standard errors that tests the restriction directly.

Question 7: Audit the claims

1. Incorrect. Failing to reject is not evidence that the null is true. With little within-person variation the test has low power, and the classical statistic itself presumes that RE is efficient under the null, which fails under heteroskedasticity or serial correlation. A Mundlak test with clustered standard errors, together with an argument about what sits inside \(\alpha_i\), is the more informative route.

2. Overstated. FE is more robust because it allows \(\alpha_i\) to be correlated with the regressors, but it cannot identify the coefficient on any time-invariant regressor and it throws away the between-individual comparison. When a regressor barely moves within individuals, the FE estimate can be very imprecise. If the RE restriction is genuinely defensible, RE uses more of the information in the data.

3. Confused about what efficiency means. RE is more efficient than FE only when the RE assumptions hold. A smaller standard error around an inconsistent estimate is worse than a larger standard error around a consistent one. Efficiency is a conditional property, not a reason to prefer an estimator whose identifying assumption is in doubt, and once standard errors are clustered by individual the RE standard errors need not be smaller anyway.

4. Partly correct. FE and FD cannot estimate it: education has no within-person variation, so it is absorbed by \(\alpha_i\). Pooled OLS and RE do report a coefficient, but only under \(E(\alpha_i\mid x,z)=0\), which is exactly what is doubted when ability is inside \(\alpha_i\). Mundlak estimates a conditional between-person schooling association. A causal interpretation still requires the remaining individual effect to be unrelated to education conditional on the included controls. The absence of within variation does not prevent numerical estimation, but it prevents using within-person comparisons to justify that coefficient.

5. Incorrect. The LM test’s null is \(\sigma_\alpha^2=0\). Rejection says only that a persistent individual component exists, so pooled OLS is inefficient and RE has something to model. The FE-versus-RE decision turns on whether \(\alpha_i\) is correlated with the regressors, which the LM test does not examine.

Takeaways

What should you be able to do now?

  1. Say which variation pooled OLS, the between estimator, ordinary RE, FE, Mundlak and FD each use, and which coefficients each of them can identify.
  2. Calculate \(\theta\) from the estimated variance components and read off where RE sits between pooled OLS and FE.
  3. Estimate Mundlak alongside the other panel models in R or Python and compare the coefficients with individual-clustered inference.
  4. Run the LM, Hausman, and Mundlak diagnostics and state precisely what each does and does not establish.
  5. Justify an estimator from the economics of what may lie inside \(\alpha_i\), and state the extra assumption needed for a causal time-invariant coefficient.