BUG: Correct exog forecast

Author: bashtage

arch/tests/univariate/test_forecast.py

@@ -1205,3 +1205,22 @@ def test_forecast_simulation_horizon_1():
     res = mod.fit(first_obs=0, last_obs=98)
     res.forecast(start=1, x=x, method="simulation", simulations=2)
     res.forecast(start=1, x=x, method="simulation", simulations=1)
+
+
+def test_forecast_start():
+    rg = np.random.default_rng(0)
+    y = rg.standard_normal(10)
+    x = pd.DataFrame(rg.standard_normal((10, 1)), columns=["x"])
+    mod = ARX(y, x=x, lags=3)
+    res = mod.fit(first_obs=0, last_obs=98)
+    fcast = res.forecast(start=2, x=x)
+    fcast2 = res.forecast(start=2, method="simulation", simulations=1, x=x)
+    assert_allclose(fcast.mean, fcast2.mean)
+
+    c, p1, p2, p3, b, _ = res.params
+    oos = np.full((8, 1), np.nan)
+    for i in range(2, 9):
+        oos[i - 2, 0] = (
+            c + p1 * y[i] + p2 * y[i - 1] + p3 * y[i - 2] + b * x.iloc[i + 1, 0]
+        )
+    assert_allclose(fcast.mean, oos)

arch/univariate/mean.py

@@ -918,8 +918,12 @@ def _reformat_forecast_x(
                 f"the original sample size ({self._y.shape[0]}) or the number of "
                 f"forecasts ({self._y.shape[0] - start})."
             )
-        if arr.shape[1] > (self._y.shape[0] - start):
-            arr = arr[:, start:]
+        if arr.shape[1] == self._y.shape[0]:
+            arr = arr[:, start:].copy()
+            arr[:, :-1, :] = arr[:, 1:, :]
+            # Nan fill last value because true out-of-sample not available
+            # When using x same size as original y
+            arr[:, -1:, :] = np.nan
         return arr
 
     def forecast(