diff --git a/CHANGELOG.rst b/CHANGELOG.rst
index 95b6811b..839f2248 100644
--- a/CHANGELOG.rst
+++ b/CHANGELOG.rst
@@ -7,6 +7,14 @@
 Changelog
 =========
 
+1.5.1 - 2021-07-22
+------------------
+
+**Bug fix:**
+
+* Have the :meth:`linear_predictor` and :meth:`predict` methods of :class:`~quantcore.glm.GeneralizedLinearRegressor` and :class:`~quantcore.glm.GeneralizedLinearRegressorCV`
+  honor the offset when ``alpha`` is ``None``.
+
 1.5.0 - 2021-07-15
 ------------------
 
@@ -21,7 +29,6 @@ Changelog
 * Don't list ``sparse_dot_mkl`` as a runtime requirement from the conda recipe.
 * The minimal ``numpy`` pin should be dependent on the ``numpy`` version in ``host`` and not fixed to ``1.16``.
 
-
 1.4.3 - 2021-06-25
 ------------------
 
diff --git a/src/quantcore/glm/_glm.py b/src/quantcore/glm/_glm.py
index 02dc21be..5f2a560b 100644
--- a/src/quantcore/glm/_glm.py
+++ b/src/quantcore/glm/_glm.py
@@ -1158,6 +1158,8 @@ def linear_predictor(
 
         if alpha_index is None:
             xb = X @ self.coef_ + self.intercept_
+            if offset is not None:
+                xb += offset
         elif np.isscalar(alpha_index):  # `None` doesn't qualify
             xb = X @ self.coef_path_[alpha_index] + self.intercept_path_[alpha_index]
             if offset is not None:
diff --git a/tests/glm/test_glm.py b/tests/glm/test_glm.py
index fd50ac12..0f1f4864 100644
--- a/tests/glm/test_glm.py
+++ b/tests/glm/test_glm.py
@@ -1624,7 +1624,7 @@ def test_alpha_search(regression_data):
 def test_predict_scalar(regression_data, alpha, alpha_index):
 
     X, y = regression_data
-    offset = np.zeros_like(y)
+    offset = np.ones_like(y)
 
     estimator = GeneralizedLinearRegressor(alpha=[0.5, 0.75], alpha_search=True)
     estimator.fit(X, y)
@@ -1632,7 +1632,7 @@ def test_predict_scalar(regression_data, alpha, alpha_index):
     target = estimator.predict(X, alpha_index=alpha_index)
 
     candidate = estimator.predict(X, alpha=alpha, offset=offset)
-    np.testing.assert_allclose(candidate, target)
+    np.testing.assert_allclose(candidate, target + 1)
 
 
 @pytest.mark.parametrize(
@@ -1642,7 +1642,7 @@ def test_predict_scalar(regression_data, alpha, alpha_index):
 def test_predict_list(regression_data, alpha, alpha_index):
 
     X, y = regression_data
-    offset = np.zeros_like(y)
+    offset = np.ones_like(y)
 
     estimator = GeneralizedLinearRegressor(alpha=[0.5, 0.75], alpha_search=True)
     estimator.fit(X, y)
@@ -1656,10 +1656,10 @@ def test_predict_list(regression_data, alpha, alpha_index):
     )
 
     candidate = estimator.predict(X, alpha=alpha, offset=offset)
-    np.testing.assert_allclose(candidate, target)
+    np.testing.assert_allclose(candidate, target + 1)
 
     candidate = estimator.predict(X, alpha_index=alpha_index, offset=offset)
-    np.testing.assert_allclose(candidate, target)
+    np.testing.assert_allclose(candidate, target + 1)
 
 
 def test_predict_error(regression_data):