diff --git a/linearmodels/panel/results.py b/linearmodels/panel/results.py
index c85531cee9..962b5f3736 100644
--- a/linearmodels/panel/results.py
+++ b/linearmodels/panel/results.py
@@ -647,11 +647,19 @@ def pesaran_cd(self) -> InapplicableTestStatistic | NormalTestStatistic:
         Under the null, the statistic is asymptotically standard normal.
 
         The shocks are the model's ``idiosyncratic`` values, so the weights of
-        a weighted model are not applied to them, and the shocks of
-        ``FirstDifferenceOLS`` are in levels rather than the residuals of the
-        differenced equation. Since the correlations are computed over time,
-        the test assumes that the shocks are not serially correlated, and it
-        over-rejects if they are persistent.
+        a weighted model are not applied to them. The exception is
+        ``FirstDifferenceOLS``, which uses ``resids``, the residuals of the
+        differenced equation, as ``plm::pcdtest`` does for a pooled regression
+        of the differenced data. Its ``idiosyncratic`` values are in levels,
+        and are persistent when the errors are, for example a random walk,
+        which is when first differences are used. Since the correlations are
+        computed over time, the test assumes that the shocks are not serially
+        correlated, and it over-rejects if they are persistent. Conversely, if
+        the errors in levels are serially uncorrelated, the residuals of
+        ``FirstDifferenceOLS`` are negatively autocorrelated (an MA(1) with
+        autocorrelation -0.5) and the test over-rejects, with a size of about
+        11% at the nominal 5% level in large panels. The test is then better
+        applied to a ``PanelOLS`` model with entity effects.
 
         The test is not valid for every model and an InapplicableTestStatistic
         is returned for the following.
@@ -692,10 +700,15 @@ def pesaran_cd(self) -> InapplicableTestStatistic | NormalTestStatistic:
            problem in testing for remaining cross-section correlation.
            Journal of Business & Economic Statistics, 40(3), 1191-1203.
         """
+        from linearmodels.panel.model import FirstDifferenceOLS  # noqa: PLC0415
+
         reason = self._pesaran_cd_inapplicable()
         if reason is not None:
             return InapplicableTestStatistic(reason=reason, name="Pesaran CD test")
-        eps = self.idiosyncratic.iloc[:, 0]
+        if isinstance(self.model, FirstDifferenceOLS):
+            eps = self.resids
+        else:
+            eps = self.idiosyncratic.iloc[:, 0]
         wide = eps.unstack(level=0)
         wide = wide.loc[:, wide.notnull().any(axis=0)]
         nentity = wide.shape[1]
diff --git a/linearmodels/tests/panel/results/pesaran-cd-fd-panels.csv b/linearmodels/tests/panel/results/pesaran-cd-fd-panels.csv
new file mode 100644
index 0000000000..b02cdd03b8
--- /dev/null
+++ b/linearmodels/tests/panel/results/pesaran-cd-fd-panels.csv
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diff --git a/linearmodels/tests/panel/results/pesaran-cd-fd-reference.R b/linearmodels/tests/panel/results/pesaran-cd-fd-reference.R
new file mode 100644
index 0000000000..d36d4772dd
--- /dev/null
+++ b/linearmodels/tests/panel/results/pesaran-cd-fd-reference.R
@@ -0,0 +1,202 @@
+# Reference values for PanelResults.pesaran_cd of FirstDifferenceOLS, which
+# computes the Pesaran CD statistic from the residuals of the differenced
+# equation. The script simulates the panels, so that nothing is shared with
+# linearmodels, and uses the R package plm.
+#
+# Run from this directory:   Rscript pesaran-cd-fd-reference.R
+# Requires:                  plm (tested with 2.6.7 on R 4.6.1)
+# Output:                    pesaran-cd-fd-panels.csv     the simulated panels
+#                            pesaran-cd-fd-reference.csv  the reference values
+#                            (both are read by ../test_results_pesaran_cd.py)
+#
+# Definition (Pesaran 2021, Empirical Economics 60(1), 13-50):
+#
+#   e_it   residual of the estimated equation, for the periods in which the
+#          entity has a residual
+#   T_ij   number of periods in which e_it and e_jt are both observed
+#   rho_ij sample correlation of e_it and e_jt over those T_ij periods
+#   CD   = sum_{i<j} sqrt(T_ij) rho_ij / sqrt(M)
+#
+# where the sum, and M the number of its terms, are over the pairs with T_ij >= 2
+# and a defined correlation. CD is asymptotically standard normal under the
+# null of cross-sectional independence, so the p-value is 2 (1 - Phi(|CD|)).
+# This is what plm::pcdtest(test = "cd") computes for the residuals of a panel
+# model. The script computes it from the definition, in two ways, and checks
+# that both equal pcdtest.
+#
+# The residuals are those of the first-difference estimator: the regression
+# of the first difference of y on the first difference of x, without a
+# constant. A difference exists for the period t if the entity is observed in
+# the periods t and t - 1, and its residual is labelled t.
+#
+# The model = "fd" of plm differences the observations of an entity that follow
+# each other in the data, so that it also differences across a period that is
+# missing for the entity. FirstDifferenceOLS differences adjacent periods only.
+# The differences here are therefore those of diff() of plm for panel series,
+# which respects the time index, followed by a pooled regression of the
+# differences. The pcdtest of plm cannot be used on an fd model ("argument 1 is
+# not a vector"), but it can be used on this regression. If no entity is
+# missing a period inside its observed range (case "unbalanced-window") the
+# differences are those of model = "fd", and the script checks that the
+# estimates agree.
+#
+# Cases. There are N entities and T periods (0, ..., T - 1) and
+#
+#   y = 1 + 0.5 x + a_i + e_it,   x = standard normal + 0.3 f_t,
+#
+# where a_i, f_t and the shocks e_it are standard normal, except where noted:
+#   balanced-null          independent shocks
+#   balanced-factor        e_it = u_it + 0.3 l_i f_t, with l_i = 1 + 0.5 N(0, 1)
+#   balanced-random-walk   the shocks are independent random walks; their
+#                          first differences, which this estimator uses, are
+#                          independent
+#   unbalanced-window      each entity is observed from a first period in 0-3
+#                          to a last period in T - 4 to T - 1
+#   unbalanced-gaps        like balanced-factor, with each observation missing
+#                          with probability 0.3 (an entity needs at least 3
+#                          observations to be kept)
+#   unbalanced-sparse      each observation is missing with probability 0.5, so
+#                          that most pairs of entities have fewer than two
+#                          differences in common
+# cd_levels is the statistic of the series in levels, y - x b, where b is
+# the estimate. It is what pesaran_cd used for FirstDifferenceOLS before it used
+# the residuals of the differenced equation, and is only here to show how much
+# the choice matters.
+
+suppressMessages(library(plm))
+
+cat(R.version.string, "and plm", as.character(packageVersion("plm")), "\n")
+set.seed(20261012)
+
+# CSV files with "\n" line endings on every platform
+write_csv <- function(x, path, ...) {
+  con <- file(path, open = "wb")
+  on.exit(close(con))
+  write.csv(x, con, row.names = FALSE, eol = "\n", ...)
+}
+
+simulate <- function(case, n, nt, errors = "iid", loading = 0, observed = "all") {
+  common <- rnorm(nt)
+  load <- 1 + 0.5 * rnorm(n)
+  x <- matrix(rnorm(n * nt), n, nt) + 0.3 * matrix(rep(common, each = n), n, nt)
+  shock <- matrix(rnorm(n * nt), n, nt)
+  if (errors == "random-walk") shock <- t(apply(shock, 1, cumsum))
+  shock <- shock + loading * outer(load, common)
+  effect <- matrix(rep(rnorm(n), nt), n, nt)
+  y <- 1 + 0.5 * x + effect + shock
+  keep <- matrix(TRUE, n, nt)
+  if (observed == "window") {
+    first <- 1 + floor(runif(n) * 4)
+    last <- nt - floor(runif(n) * 4)
+    keep <- outer(seq_len(n), seq_len(nt), function(i, t) t >= first[i] & t <= last[i])
+  } else if (is.numeric(observed)) {
+    keep <- matrix(runif(n * nt) >= observed, n, nt)
+    keep <- keep & rowSums(keep) >= 3
+  }
+  d <- data.frame(case = case,
+                  entity = rep(seq_len(n) - 1L, times = nt),
+                  time = rep(seq_len(nt) - 1L, each = n),
+                  y = signif(as.vector(y), 8), x = signif(as.vector(x), 8))
+  d <- d[as.vector(keep), ]
+  d[order(d$entity, d$time), ]
+}
+
+panels <- rbind(
+  simulate("balanced-null", 40, 8),
+  simulate("balanced-factor", 40, 8, loading = 0.3),
+  simulate("balanced-random-walk", 40, 8, errors = "random-walk"),
+  simulate("unbalanced-window", 40, 10, observed = "window"),
+  simulate("unbalanced-gaps", 40, 10, loading = 0.3, observed = 0.3),
+  simulate("unbalanced-sparse", 40, 8, observed = 0.5)
+)
+write_csv(panels, "pesaran-cd-fd-panels.csv")
+# Everything below uses the file, so that the reference is that of the data
+# that the tests read
+panels <- read.csv("pesaran-cd-fd-panels.csv", stringsAsFactors = FALSE)
+
+## ------------------------------------------------------------- CD test ---
+# The CD statistic of the residuals e of the entities and periods (entity, time),
+# from the N x N matrices of the correlations and of the numbers of common periods
+cd_matrix <- function(entity, time, e) {
+  ent <- sort(unique(entity))
+  tim <- sort(unique(time))
+  m <- matrix(NA_real_, length(tim), length(ent))
+  m[cbind(match(time, tim), match(entity, ent))] <- e
+  tij <- crossprod(!is.na(m))
+  rho <- suppressWarnings(cor(m, use = "pairwise.complete.obs"))
+  pair <- lower.tri(rho) & tij >= 2 & is.finite(rho)
+  list(stat = sum(sqrt(tij[pair]) * rho[pair]) / sqrt(sum(pair)),
+       npairs = sum(pair))
+}
+
+# The same with a loop over the pairs, which uses the common periods of each pair
+cd_loop <- function(entity, time, e) {
+  series <- split(setNames(e, time), entity)
+  total <- 0
+  npairs <- 0
+  for (i in seq_len(length(series) - 1)) {
+    for (j in (i + 1):length(series)) {
+      both <- intersect(names(series[[i]]), names(series[[j]]))
+      if (length(both) < 2) next
+      rho <- suppressWarnings(cor(series[[i]][both], series[[j]][both]))
+      if (!is.finite(rho)) next
+      total <- total + sqrt(length(both)) * rho
+      npairs <- npairs + 1
+    }
+  }
+  list(stat = total / sqrt(npairs), npairs = npairs)
+}
+
+agree <- function(a, b, what, rtol = 1e-10) {
+  if (abs(a - b) > rtol * max(1, abs(b))) {
+    stop(sprintf("%s: %.15g vs %.15g", what, a, b))
+  }
+}
+
+## --------------------------------------------------------- the cases ----
+rows <- list()
+for (case in unique(panels$case)) {
+  d <- panels[panels$case == case, c("entity", "time", "y", "x")]
+  pd <- pdata.frame(d, index = c("entity", "time"))
+
+  # Differences of adjacent periods, and the regression of the differences
+  fd <- data.frame(entity = d$entity, time = d$time,
+                   dy = as.numeric(diff(pd$y)), dx = as.numeric(diff(pd$x)))
+  fd <- fd[!is.na(fd$dy), ]
+  fit <- plm(dy ~ dx - 1, data = pdata.frame(fd, index = c("entity", "time")),
+             model = "pooling")
+  e <- as.numeric(residuals(fit))
+  agree(sum(e * fd$dx), 0, "the residuals are orthogonal to dx")
+
+  by_hand <- cd_matrix(fd$entity, fd$time, e)
+  looped <- cd_loop(fd$entity, fd$time, e)
+  agree(by_hand$stat, looped$stat, paste(case, "matrix and loop"), 1e-12)
+  stopifnot(by_hand$npairs == looped$npairs)
+  # plm warns that it omits the pairs without two common periods, as the
+  # definition does
+  pcd <- suppressWarnings(pcdtest(fit, test = "cd"))
+  agree(by_hand$stat, unname(pcd$statistic), paste(case, "pcdtest of plm"), 1e-10)
+
+  # The levels, with the estimate of the differenced equation
+  levels <- cd_matrix(d$entity, d$time, d$y - unname(coef(fit)) * d$x)
+
+  consecutive <- all(tapply(d$time, d$entity, function(t) all(diff(t) == 1)))
+  if (consecutive) {
+    plm_fd <- plm(y ~ x - 1, data = pd, model = "fd")
+    agree(unname(coef(plm_fd)), unname(coef(fit)), paste(case, "model = fd of plm"), 1e-10)
+    stopifnot(length(residuals(plm_fd)) == nrow(fd))
+    cat(sprintf("%-22s model = 'fd' of plm has this coefficient and %d residuals\n",
+                case, nrow(fd)))
+  }
+
+  rows[[case]] <- data.frame(case = case, nobs = nrow(fd),
+                             coef = unname(coef(fit)), npairs = by_hand$npairs,
+                             cd_stat = by_hand$stat,
+                             cd_pvalue = 2 * pnorm(-abs(by_hand$stat)),
+                             cd_levels = levels$stat)
+  cat(sprintf("%-22s nobs %4d  pairs %4d  CD %9.6f  p %.6g  levels %9.6f\n", case,
+              nrow(fd), by_hand$npairs, by_hand$stat,
+              2 * pnorm(-abs(by_hand$stat)), levels$stat))
+}
+write_csv(do.call(rbind, rows), "pesaran-cd-fd-reference.csv", quote = FALSE)
+cat("wrote pesaran-cd-fd-panels.csv and pesaran-cd-fd-reference.csv\n")
diff --git a/linearmodels/tests/panel/results/pesaran-cd-fd-reference.csv b/linearmodels/tests/panel/results/pesaran-cd-fd-reference.csv
new file mode 100644
index 0000000000..23f9d92dc9
--- /dev/null
+++ b/linearmodels/tests/panel/results/pesaran-cd-fd-reference.csv
@@ -0,0 +1,7 @@
+case,nobs,coef,npairs,cd_stat,cd_pvalue,cd_levels
+balanced-null,280,0.496938833909092,780,-0.366978247478736,0.713635243727115,-0.699092395728204
+balanced-factor,280,0.620586914236625,780,4.97305388448626,6.59063321869154e-07,6.67457138202798
+balanced-random-walk,280,0.53582473502824,780,-0.639887390382865,0.522245812297977,-1.27791497537843
+unbalanced-window,234,0.612643943922457,780,-0.581545549620823,0.560872828552816,-0.655132520804181
+unbalanced-gaps,172,0.471570161846001,477,4.915933423038,8.83604250838333e-07,6.27580589857232
+unbalanced-sparse,76,0.487123421933064,92,-0.783691374882634,0.433221224719038,0.729178940387188
diff --git a/linearmodels/tests/panel/test_results.py b/linearmodels/tests/panel/test_results.py
index cfce9ce4a2..d24b2d0450 100644
--- a/linearmodels/tests/panel/test_results.py
+++ b/linearmodels/tests/panel/test_results.py
@@ -241,7 +241,12 @@ def test_pesaran_cd_all_estimators(estimator):
     res = models[estimator]().fit()
     cd = res.pesaran_cd
     assert isinstance(cd, NormalTestStatistic)
-    expected = brute_force_pesaran_cd(res.idiosyncratic)
+    if estimator == "first-difference":
+        # The residuals of the differenced equation, not the shocks in levels
+        shocks = res.resids.to_frame()
+    else:
+        shocks = res.idiosyncratic
+    expected = brute_force_pesaran_cd(shocks)
     assert_allclose(cd.stat, expected, rtol=1e-10)
     assert_allclose(cd.pval, 2 * stats.norm.sf(abs(expected)), rtol=1e-10)
 
diff --git a/linearmodels/tests/panel/test_results_pesaran_cd.py b/linearmodels/tests/panel/test_results_pesaran_cd.py
index a4961f6abb..13637c99e1 100644
--- a/linearmodels/tests/panel/test_results_pesaran_cd.py
+++ b/linearmodels/tests/panel/test_results_pesaran_cd.py
@@ -1,8 +1,14 @@
 """
-Tests of PanelResults.pesaran_cd: which models the test is applicable to.
+Tests of PanelResults.pesaran_cd: which models the test is applicable to, and
+the residuals that FirstDifferenceOLS uses.
+
+The reference values of FirstDifferenceOLS in results/pesaran-cd-fd-reference.csv
+were produced by results/pesaran-cd-fd-reference.R, which computes the statistic
+from the residuals of the differenced equation using the R package plm.
 """
 
 from itertools import combinations
+import os
 
 import numpy as np
 from numpy.testing import assert_allclose
@@ -24,6 +30,8 @@
     NormalTestStatistic,
 )
 
+RESULTS = os.path.join(os.path.split(os.path.abspath(__file__))[0], "results")
+
 
 def cd_panel(seed=0, nentity=30, nperiod=8, loading=0.0):
     """Panel with independent errors, or a common factor if loading > 0"""
@@ -91,6 +99,10 @@ def cd_models(frame):
         ),
         "random": (lambda: RandomEffects(y, xc), True),
         "first-difference": (lambda: FirstDifferenceOLS(y, x), True),
+        "first-difference-weighted": (
+            lambda: FirstDifferenceOLS(y, x, weights=weights),
+            True,
+        ),
         # Time-invariant groups of entities, and groups unrelated to the periods
         "other-group": (lambda: PanelOLS(y, xc, other_effects=entity % 3), True),
         "entity-other-random": (
@@ -156,7 +168,13 @@ def test_pesaran_cd_applicability(name):
     cd = res.pesaran_cd
     if applicable:
         assert isinstance(cd, NormalTestStatistic)
-        expected = brute_force_cd(res.idiosyncratic)
+        if name.startswith("first-difference"):
+            # The residuals of the differenced equation, not the shocks in
+            # levels, and the weights are not applied to them
+            shocks = res.resids.to_frame()
+        else:
+            shocks = res.idiosyncratic
+        expected = brute_force_cd(shocks)
         assert_allclose(cd.stat, expected, rtol=1e-10)
         assert_allclose(cd.pval, 2 * stats.norm.sf(abs(expected)), rtol=1e-10)
         return
@@ -240,12 +258,63 @@ def test_pesaran_cd_time_invariant_other_effects_unbalanced():
     assert_allclose(cd.stat, brute_force_cd(res.idiosyncratic), rtol=1e-10)
 
 
-def test_pesaran_cd_first_difference_uses_levels():
-    # The shocks are in levels and not the residuals of the differenced equation
-    frame = cd_panel(seed=3, nentity=40, nperiod=8)
+def test_pesaran_cd_first_difference_closed_form():
+    # With N entities that are observed in T periods there are T - 1 residuals
+    # of the differenced equation for each, so that
+    # CD = sqrt(2 (T - 1) / (N (N - 1))) * the sum of the N (N - 1) / 2 correlations
+    nentity, nperiod = 40, 8
+    frame = cd_panel(seed=3, nentity=nentity, nperiod=nperiod)
+    y = frame.y.to_numpy().reshape(nentity, nperiod)
+    x = frame.x.to_numpy().reshape(nentity, nperiod)
+    dy, dx = np.diff(y, axis=1), np.diff(x, axis=1)
+    beta = np.sum(dx * dy) / np.sum(dx * dx)
+    corr = np.corrcoef(dy - beta * dx)
+    pairs = corr[np.triu_indices(nentity, k=1)]
+    expected = np.sqrt(2 * (nperiod - 1) / (nentity * (nentity - 1))) * pairs.sum()
+
     res = FirstDifferenceOLS(frame.y, frame[["x"]]).fit()
+    assert_allclose(res.params.iloc[0], beta, rtol=1e-10)
     cd = res.pesaran_cd
     assert isinstance(cd, NormalTestStatistic)
-    assert_allclose(cd.stat, brute_force_cd(res.idiosyncratic), rtol=1e-10)
-    differenced = brute_force_cd(res.resids.to_frame())
-    assert not np.isclose(cd.stat, differenced, rtol=1e-3)
+    assert_allclose(cd.stat, expected, rtol=1e-10)
+    assert_allclose(cd.pval, 2 * stats.norm.sf(abs(expected)), rtol=1e-10)
+    # It is not the statistic of the shocks in levels
+    assert not np.isclose(cd.stat, brute_force_cd(res.idiosyncratic), rtol=1e-3)
+
+
+def test_pesaran_cd_first_difference_two_periods():
+    # There is one difference for each entity, so that no pair of entities has
+    # two overlapping residuals. The shocks in levels have two for every pair
+    frame = cd_panel(seed=2, nentity=10, nperiod=2)
+    res = FirstDifferenceOLS(frame.y, frame[["x"]]).fit()
+    assert res.resids.shape[0] == 10
+    cd = res.pesaran_cd
+    assert isinstance(cd, InapplicableTestStatistic)
+    assert "two or more overlapping observations" in str(cd)
+
+
+FD_PANELS = pd.read_csv(os.path.join(RESULTS, "pesaran-cd-fd-panels.csv"))
+FD_REFERENCE = pd.read_csv(os.path.join(RESULTS, "pesaran-cd-fd-reference.csv"))
+
+
+@pytest.mark.parametrize(
+    "row",
+    [pytest.param(row, id=row.case) for row in FD_REFERENCE.itertuples(index=False)],
+)
+def test_pesaran_cd_first_difference_plm(row):
+    panel = FD_PANELS.loc[FD_PANELS.case == row.case].set_index(["entity", "time"])
+    res = FirstDifferenceOLS(panel.y, panel[["x"]]).fit()
+    # The reference uses the same differences and the same estimate
+    assert res.nobs == row.nobs
+    assert_allclose(res.params.iloc[0], row.coef, rtol=1e-8)
+    # Pairs of entities with at least two residuals at the same times
+    seen = res.resids.unstack(level=0).notnull().to_numpy(dtype=int)
+    assert np.tril(seen.T @ seen >= 2, k=-1).sum() == row.npairs
+
+    cd = res.pesaran_cd
+    assert isinstance(cd, NormalTestStatistic)
+    assert_allclose(cd.stat, row.cd_stat, rtol=1e-8)
+    assert_allclose(cd.pval, row.cd_pvalue, rtol=1e-8)
+    # The shocks in levels give the reference of the levels, which is not the same
+    assert_allclose(brute_force_cd(res.idiosyncratic), row.cd_levels, rtol=1e-8)
+    assert not np.isclose(cd.stat, row.cd_levels, rtol=1e-2)