diff --git a/aeon/forecasting/__init__.py b/aeon/forecasting/__init__.py
index 0b134857dd..7d7cb7e6c4 100644
--- a/aeon/forecasting/__init__.py
+++ b/aeon/forecasting/__init__.py
@@ -1,13 +1,15 @@
 """Forecasters."""
 
 __all__ = [
-    "NaiveForecaster",
     "BaseForecaster",
+    "NaiveForecaster",
     "RegressionForecaster",
     "ETSForecaster",
     "TVPForecaster",
+    "ARIMA",
 ]
 
+from aeon.forecasting._arima import ARIMA
 from aeon.forecasting._ets import ETSForecaster
 from aeon.forecasting._naive import NaiveForecaster
 from aeon.forecasting._regression import RegressionForecaster
diff --git a/aeon/forecasting/_arima.py b/aeon/forecasting/_arima.py
new file mode 100644
index 0000000000..6f08af4325
--- /dev/null
+++ b/aeon/forecasting/_arima.py
@@ -0,0 +1,301 @@
+"""ARIMA.
+
+An implementation of the ARIMA forecasting algorithm.
+"""
+
+__maintainer__ = ["alexbanwell1", "TonyBagnall"]
+__all__ = ["ARIMA"]
+
+import numpy as np
+from numba import njit
+
+from aeon.forecasting._extract_paras import _extract_arma_params
+from aeon.forecasting._nelder_mead import nelder_mead
+from aeon.forecasting.base import BaseForecaster
+
+
+class ARIMA(BaseForecaster):
+    """AutoRegressive Integrated Moving Average (ARIMA) forecaster.
+
+    ARIMA with fixed model structure and fitted parameters found with an
+    nelder mead optimizer to minimise the AIC.
+
+    Parameters
+    ----------
+    p : int, default=1,
+        Autoregressive (p) order of the ARIMA model
+    d : int, default=0,
+        Differencing (d) order of the ARIMA model
+    q : int, default=1,
+        Moving average (q) order of the ARIMA model
+    use_constant : bool = False,
+        Presence of a constant/intercept term in the model.
+    iterations : int, default = 200
+        Maximum number of iterations to use in the Nelder-Mead parameter search.
+
+    Attributes
+    ----------
+    residuals_ : np.ndarray
+        Residual errors from the fitted model.
+    aic_ : float
+        Akaike Information Criterion for the fitted model.
+    c_ : float, default = 0
+        Intercept term.
+    phi_ : np.ndarray
+        Coefficients for autoregressive terms (length p).
+    theta_ : np.ndarray
+        Coefficients for moving average terms (length q).
+
+    References
+    ----------
+    .. [1] R. J. Hyndman and G. Athanasopoulos,
+       Forecasting: Principles and Practice. OTexts, 2014.
+       https://otexts.com/fpp3/
+    """
+
+    _tags = {
+        "capability:horizon": False,  # cannot fit to a horizon other than 1
+    }
+
+    def __init__(
+        self,
+        p: int = 1,
+        d: int = 0,
+        q: int = 1,
+        use_constant: bool = False,
+        iterations: int = 200,
+    ):
+        self.p = p
+        self.d = d
+        self.q = q
+        self.use_constant = use_constant
+        self.iterations = iterations
+        self.phi_ = 0
+        self.theta_ = 0
+        self.c_ = 0
+        self._series = []
+        self._differenced_series = []
+        self.residuals_ = []
+        self.fitted_values_ = []
+        self.aic_ = 0
+        self._model = []
+        self._parameters = []
+        super().__init__(horizon=1, axis=1)
+
+    def _fit(self, y, exog=None):
+        """Fit ARIMA forecaster to series y to predict one ahead using y.
+
+        Parameters
+        ----------
+        y : np.ndarray
+            A time series on which to learn a forecaster to predict horizon ahead
+        exog : np.ndarray, default =None
+            Not allowed for this forecaster
+
+        Returns
+        -------
+        self
+            Fitted ARIMA.
+        """
+        self._series = np.array(y.squeeze(), dtype=np.float64)
+        # Model is an array of the (c,p,q)
+        self._model = np.array(
+            (1 if self.use_constant else 0, self.p, self.q), dtype=np.int32
+        )
+        self._differenced_series = np.diff(self._series, n=self.d)
+        # Nelder Mead returns the parameters in a single array
+        (self._parameters, self.aic_) = nelder_mead(
+            0,
+            np.sum(self._model[:3]),
+            self._differenced_series,
+            self._model,
+            max_iter=self.iterations,
+        )
+        #
+        (self.aic_, self.residuals_, self.fitted_values_) = _arima_model(
+            self._parameters,
+            self._differenced_series,
+            self._model,
+        )
+        formatted_params = _extract_arma_params(
+            self._parameters, self._model
+        )  # Extract
+        # parameters
+        differenced_forecast = self.fitted_values_[-1]
+
+        if self.d == 0:
+            forecast_value = differenced_forecast
+        elif self.d == 1:
+            forecast_value = differenced_forecast + self._series[-1]
+        else:  # for d > 1, iteratively undifference
+            forecast_value = differenced_forecast
+            last_vals = self._series[-self.d :]
+            for _ in range(self.d):
+                forecast_value += last_vals[-1] - last_vals[-2]
+                # Shift values to avoid appending to list (efficient)
+                last_vals = np.roll(last_vals, -1)
+                last_vals[-1] = forecast_value  # Extract the parameter values
+        self.forecast_ = forecast_value
+        if self.use_constant:
+            self.c_ = formatted_params[0][0]
+        self.phi_ = formatted_params[1][: self.p]
+        self.theta_ = formatted_params[2][: self.q]
+
+        return self
+
+    def _predict(self, y, exog=None):
+        """
+        Predict the next step ahead for y.
+
+        Parameters
+        ----------
+        y : np.ndarray, default = None
+            A time series to predict the value of. y can be independent of the series
+            seen in fit.
+        exog : np.ndarray, default =None
+            Optional exogenous time series data assumed to be aligned with y
+
+        Returns
+        -------
+        float
+            Prediction 1 step ahead of the last value in y.
+        """
+        y = y.squeeze()
+        p, q, d = self.p, self.q, self.d
+        phi, theta = self.phi_, self.theta_
+        c = 0.0
+        if self.use_constant:
+            c = self.c_
+
+        # Apply differencing
+        if d > 0:
+            if len(y) <= d:
+                raise ValueError("Series too short for differencing.")
+            y_diff = np.diff(y, n=d)
+        else:
+            y_diff = y
+
+        n = len(y_diff)
+        if n < max(p, q):
+            raise ValueError("Series too short for ARMA(p,q) with given order.")
+
+        # Estimate in-sample residuals using model (fixed parameters)
+        residuals = np.zeros(n)
+        for t in range(max(p, q), n):
+            ar_part = np.dot(phi, y_diff[t - np.arange(1, p + 1)]) if p > 0 else 0.0
+            ma_part = (
+                np.dot(theta, residuals[t - np.arange(1, q + 1)]) if q > 0 else 0.0
+            )
+            pred = c + ar_part + ma_part
+            residuals[t] = y_diff[t] - pred
+
+        # Use most recent p values of y_diff and q values of residuals to forecast t+1
+        ar_forecast = np.dot(phi, y_diff[-np.arange(1, p + 1)]) if p > 0 else 0.0
+        ma_forecast = np.dot(theta, residuals[-np.arange(1, q + 1)]) if q > 0 else 0.0
+
+        forecast_diff = c + ar_forecast + ma_forecast
+
+        # Undifference the forecast
+        if d == 0:
+            return forecast_diff
+        elif d == 1:
+            return forecast_diff + y[-1]
+        else:
+            return forecast_diff + np.sum(y[-d:])
+
+    def _forecast(self, y, exog=None):
+        """Forecast one ahead for time series y."""
+        self.fit(y, exog)
+        return float(self.forecast_)
+
+    def iterative_forecast(self, y, prediction_horizon):
+        self.fit(y)
+        n = len(self._differenced_series)
+        p, q = self.p, self.q
+        phi, theta = self.phi_, self.theta_
+        h = prediction_horizon
+        c = 0.0
+        if self.use_constant:
+            c = self.c_
+
+        # Start with a copy of the original series and residuals
+        residuals = np.zeros(len(self.residuals_) + h)
+        residuals[: len(self.residuals_)] = self.residuals_
+        forecast_series = np.zeros(n + h)
+        forecast_series[:n] = self._differenced_series
+        for i in range(h):
+            # Get most recent p values (lags)
+            t = n + i
+            ar_term = 0.0
+            if p > 0:
+                ar_term = np.dot(phi, forecast_series[t - np.arange(1, p + 1)])
+            # Get most recent q residuals (lags)
+            ma_term = 0.0
+            if q > 0:
+                ma_term = np.dot(theta, residuals[t - np.arange(1, q + 1)])
+            next_value = c + ar_term + ma_term
+            # Append prediction and a zero residual (placeholder)
+            forecast_series[n + i] = next_value
+            # Can't compute real residual during prediction, leave as zero
+
+        # Correct differencing using forecast values
+        y_forecast_diff = forecast_series[n : n + h]
+        d = self.d
+        if d == 0:
+            return y_forecast_diff
+        else:  # Correct undifferencing
+            # Start with last d values from original y
+            undiff = list(self._series[-d:])
+            for i in range(h):
+                # Take the last d values and sum them
+                reconstructed = y_forecast_diff[i] + sum(undiff[-d:])
+                undiff.append(reconstructed)
+            return np.array(undiff[d:])
+
+
+@njit(cache=True, fastmath=True)
+def _aic(residuals, num_params):
+    """Calculate the log-likelihood of a model."""
+    variance = np.mean(residuals**2)
+    likelihood = len(residuals) * (np.log(2 * np.pi) + np.log(variance) + 1)
+    return likelihood + 2 * num_params
+
+
+# Define the ARIMA(p, d, q) likelihood function
+@njit(cache=True, fastmath=True)
+def _arima_model(params, data, model):
+    """Calculate the log-likelihood of an ARIMA model given the parameters."""
+    formatted_params = _extract_arma_params(params, model)  # Extract parameters
+
+    # Initialize residuals
+    n = len(data)
+    num_predictions = n + 1
+    residuals = np.zeros(num_predictions - 1)
+    fitted_values = np.zeros(num_predictions)
+    # Leave first max(p,q) residuals and fitted as zero.
+    for t in range(max(model[1], model[2]), num_predictions):
+        fitted_values[t] = _in_sample_forecast(
+            data, model, t, formatted_params, residuals
+        )
+        if t != num_predictions - 1:
+            # Only calculate residuals for the predictions we have data for
+            residuals[t] = data[t] - fitted_values[t]
+    return _aic(residuals, len(params)), residuals, fitted_values
+
+
+@njit(cache=True, fastmath=True)
+def _in_sample_forecast(data, model, t, formatted_params, residuals):
+    """Efficient ARMA one-step forecast at time t for fitted model."""
+    p = model[1]
+    q = model[2]
+    c = formatted_params[0][0] if model[0] else 0.0
+
+    ar_term = 0.0
+    for j in range(min(p, t)):
+        ar_term += formatted_params[1, j] * data[t - j - 1]
+
+    ma_term = 0.0
+    for j in range(min(q, t)):
+        ma_term += formatted_params[2, j] * residuals[t - j - 1]
+
+    return c + ar_term + ma_term
diff --git a/aeon/forecasting/_extract_paras.py b/aeon/forecasting/_extract_paras.py
new file mode 100644
index 0000000000..2e38a3e4dd
--- /dev/null
+++ b/aeon/forecasting/_extract_paras.py
@@ -0,0 +1,27 @@
+"""ARIMA Utility Function."""
+
+import numpy as np
+from numba import njit
+
+
+@njit(cache=True, fastmath=True)
+def _extract_arma_params(params, model):
+    """Extract ARIMA parameters from the parameter vector."""
+    n_parts = len(model)
+    starts = np.zeros(n_parts, dtype=np.int32)
+    for i in range(1, n_parts):
+        starts[i] = starts[i - 1] + model[i - 1]
+
+    max_len = np.max(model)
+    result = np.empty((n_parts, max_len), dtype=params.dtype)
+    for i in range(n_parts):
+        for j in range(max_len):
+            result[i, j] = np.nan
+
+    for i in range(n_parts):
+        length = model[i]
+        start = starts[i]
+        for j in range(length):
+            result[i, j] = params[start + j]
+
+    return result
diff --git a/aeon/forecasting/_hypo_tests.py b/aeon/forecasting/_hypo_tests.py
new file mode 100644
index 0000000000..198a4472a8
--- /dev/null
+++ b/aeon/forecasting/_hypo_tests.py
@@ -0,0 +1,94 @@
+import numpy as np
+
+
+def kpss_test(y, regression="c", lags=None):  # Test if time series is stationary
+    """
+    Perform the KPSS (Kwiatkowski-Phillips-Schmidt-Shin) test for stationarity.
+
+    The KPSS test evaluates the null hypothesis that a time series is
+    (trend or level) stationary against the alternative of a unit root
+    (non-stationarity). It can test for either stationarity around a
+    constant (level stationarity) or arounda deterministic trend
+    (trend stationarity).
+
+    Parameters
+    ----------
+    y : array-like
+        Time series data to test for stationarity.
+    regression : str, default="c"
+        Indicates the null hypothesis for stationarity:
+        - "c"  : Stationary around a constant (level stationarity)
+        - "ct" : Stationary around a constant and linear trend (trend stationarity)
+    lags : int or None, optional
+        Number of lags to use for the
+        HAC (heteroskedasticity and autocorrelation consistent) variance estimator.
+        If None, defaults to sqrt(n), where n is the sample size.
+
+    Returns
+    -------
+    kpss_stat : float
+        The KPSS test statistic.
+    stationary : bool
+        True if the series is judged stationary at the 5% significance level
+        (i.e., test statistic is below the critical value); False otherwise.
+
+    Notes
+    -----
+    - Uses asymptotic 5% critical values from Kwiatkowski et al. (1992): 0.463 for level
+      stationarity, 0.146 for trend stationarity.
+    - Returns True for stationary if the test statistic is below the 5% critical value.
+
+    References
+    ----------
+    Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992).
+    "Testing the null hypothesis of stationarity against the alternative
+    of a unit root."
+    Journal of Econometrics, 54(1–3), 159–178.
+    https://doi.org/10.1016/0304-4076(92)90104-Y
+    """
+    y = np.asarray(y)
+    n = len(y)
+
+    # Step 1: Fit regression model to estimate residuals
+    if regression == "c":  # Constant
+        X = np.ones((n, 1))
+    elif regression == "ct":  # Constant + Trend
+        X = np.column_stack((np.ones(n), np.arange(1, n + 1)))
+    else:
+        raise ValueError("regression must be 'c' or 'ct'")
+
+    beta = np.linalg.lstsq(X, y, rcond=None)[0]  # Estimate regression coefficients
+    residuals = y - X @ beta  # Get residuals (u_t)
+
+    # Step 2: Compute cumulative sum of residuals (S_t)
+    S_t = np.cumsum(residuals)
+
+    # Step 3: Estimate long-run variance (HAC variance)
+    if lags is None:
+        # lags = int(12 * (n / 100)**(1/4)) # Default statsmodels lag length
+        lags = int(np.sqrt(n))  # Default lag length
+
+    gamma_0 = np.sum(residuals**2) / (n - X.shape[1])  # Lag-0 autocovariance
+    gamma = [np.sum(residuals[k:] * residuals[:-k]) / n for k in range(1, lags + 1)]
+
+    # Bartlett weights
+    weights = [1 - (k / (lags + 1)) for k in range(1, lags + 1)]
+
+    # Long-run variance
+    sigma_squared = gamma_0 + 2 * np.sum([w * g for w, g in zip(weights, gamma)])
+
+    # Step 4: Calculate the KPSS statistic
+    kpss_stat = np.sum(S_t**2) / (n**2 * sigma_squared)
+
+    # 5% critical values for KPSS test
+    if regression == "ct":
+        # p. 162 Kwiatkowski et al. (1992): y_t = beta * t + r_t + e_t,
+        # where beta is the trend, r_t a random walk and e_t a stationary
+        # error term.
+        crit = 0.146
+    else:  # hypo == "c"
+        # special case of the model above, where beta = 0 (so the null
+        # hypothesis is that the data is stationary around r_0).
+        crit = 0.463
+
+    return kpss_stat, kpss_stat < crit
diff --git a/aeon/forecasting/_loss_functions.py b/aeon/forecasting/_loss_functions.py
new file mode 100644
index 0000000000..c88211b645
--- /dev/null
+++ b/aeon/forecasting/_loss_functions.py
@@ -0,0 +1,40 @@
+"""Loss functions for optimiser."""
+
+import numpy as np
+from numba import njit
+
+from aeon.forecasting._extract_paras import _extract_arma_params
+
+LOG_2PI = 1.8378770664093453
+
+
+@njit(cache=True, fastmath=True)
+def _arima_fit(params, data, model):
+    """Calculate the AIC of an ARIMA model given the parameters."""
+    formatted_params = _extract_arma_params(params, model)  # Extract parameters
+
+    # Initialize residuals
+    n = len(data)
+    residuals = np.zeros(n)
+    c = formatted_params[0][0] if model[0] else 0
+    p = model[1]
+    q = model[2]
+    for t in range(n):
+        ar_term = 0.0
+        max_ar = min(p, t)
+        for j in range(max_ar):
+            ar_term += formatted_params[1, j] * data[t - j - 1]
+        ma_term = 0.0
+        max_ma = min(q, t)
+        for j in range(max_ma):
+            ma_term += formatted_params[2, j] * residuals[t - j - 1]
+        y_hat = c + ar_term + ma_term
+        residuals[t] = data[t] - y_hat
+    sse = 0.0
+    start = max(p, q)
+    for i in range(start, n):
+        sse += residuals[i] * residuals[i]
+    variance = sse / (n - start)
+    likelihood = (n - start) * (LOG_2PI + np.log(variance) + 1.0)
+    k = len(params)
+    return likelihood + 2 * k
diff --git a/aeon/forecasting/_nelder_mead.py b/aeon/forecasting/_nelder_mead.py
new file mode 100644
index 0000000000..f617aa5727
--- /dev/null
+++ b/aeon/forecasting/_nelder_mead.py
@@ -0,0 +1,130 @@
+"""Optimisation algorithms for automatic parameter tuning."""
+
+import numpy as np
+from numba import njit
+
+from aeon.forecasting._loss_functions import _arima_fit
+
+
+@njit(cache=True, fastmath=True)
+def dispatch_loss(fn_id, params, data, model):
+    if fn_id == 0:
+        return _arima_fit(params, data, model)
+    else:
+        raise ValueError("Unknown loss function ID")
+
+
+@njit(cache=True, fastmath=True)
+def nelder_mead(
+    loss_id,
+    num_params,
+    data,
+    model,
+    tol=1e-6,
+    max_iter=500,
+):
+    """
+    Perform optimisation using the Nelder–Mead simplex algorithm.
+
+    This function minimises a given loss (objective) function using the Nelder–Mead
+    algorithm, a derivative-free method that iteratively refines a simplex of candidate
+    solutions. The implementation supports unconstrained minimisation of functions
+    with a fixed number of parameters.
+
+    Parameters
+    ----------
+    loss_id : int
+        ID for loss function to optimise, used by dispatch_loss.
+    num_params : int
+        The number of parameters (dimensions) in the optimisation problem.
+    data : np.ndarray
+        The input data used by the loss function. The shape and content depend on the
+        specific loss function being minimised.
+    model : np.ndarray
+        The model or context in which the loss function operates. This could be any
+        other object that the `loss_function` requires to compute its value.
+        The exact type and structure of `model` should be compatible with the
+        `loss_function`.
+    tol : float, optional (default=1e-6)
+        Tolerance for convergence. The algorithm stops when the maximum difference
+        between function values at simplex vertices is less than `tol`.
+    max_iter : int, optional (default=500)
+        Maximum number of iterations to perform.
+
+    Returns
+    -------
+    best_params : np.ndarray, shape (`num_params`,)
+        The parameter vector that minimises the loss function.
+    best_value : float
+        The value of the loss function at the optimal parameter vector.
+
+    Notes
+    -----
+    - The initial simplex is constructed by setting each parameter to 0.5,
+    with one additional point per dimension at 0.6 for that dimension.
+    - This implementation does not support constraints or bounds on the parameters.
+    - The algorithm does not guarantee finding a global minimum.
+
+    References
+    ----------
+    .. [1] Nelder, J. A. and Mead, R. (1965).
+       A Simplex Method for Function Minimization.
+       The Computer Journal, 7(4), 308–313.
+       https://doi.org/10.1093/comjnl/7.4.308
+    """
+    points = np.full((num_params + 1, num_params), 0.5)
+    for i in range(num_params):
+        points[i + 1][i] = 0.6
+    values = np.empty(len(points), dtype=np.float64)
+    for i in range(len(points)):
+        values[i] = dispatch_loss(loss_id, points[i].copy(), data, model)
+    for i in range(max_iter):
+        # Order simplex by function values
+        order = np.argsort(values)
+        points = points[order]
+        values = values[order]
+
+        # Centroid of the best n points
+        centre_point = points[:-1].sum(axis=0) / len(points[:-1])
+
+        # Reflection
+        # centre + distance between centre and largest value
+        reflected_point = centre_point + (centre_point - points[-1])
+        reflected_value = dispatch_loss(0, reflected_point, data, model)
+        # if between best and second best, use reflected value
+        if len(values) > 1 and values[0] <= reflected_value < values[-2]:
+            points[-1] = reflected_point
+            values[-1] = reflected_value
+            continue
+        # Expansion
+        # Otherwise if it is better than the best value
+        if reflected_value < values[0]:
+            expanded_point = centre_point + 2 * (reflected_point - centre_point)
+            expanded_value = dispatch_loss(0, expanded_point, data, model)
+            # if less than reflected value use expanded, otherwise go back to reflected
+            if expanded_value < reflected_value:
+                points[-1] = expanded_point
+                values[-1] = expanded_value
+            else:
+                points[-1] = reflected_point
+                values[-1] = reflected_value
+            continue
+        # Contraction
+        # Otherwise if reflection is worse than all current values
+        contracted_point = centre_point - 0.5 * (centre_point - points[-1])
+        contracted_value = dispatch_loss(0, contracted_point, data, model)
+        # If contraction is better use that otherwise move to shrinkage
+        if contracted_value < values[-1]:
+            points[-1] = contracted_point
+            values[-1] = contracted_value
+            continue
+
+        # Shrinkage
+        for i in range(1, len(points)):
+            points[i] = points[0] - 0.5 * (points[0] - points[i])
+            values[i] = dispatch_loss(0, points[i], data, model)
+
+        # Convergence check
+        if np.max(np.abs(values - values[0])) < tol:
+            break
+    return points[0], values[0]
diff --git a/aeon/forecasting/_seasonality.py b/aeon/forecasting/_seasonality.py
new file mode 100644
index 0000000000..356b1a40d2
--- /dev/null
+++ b/aeon/forecasting/_seasonality.py
@@ -0,0 +1,101 @@
+"""Seasonality Tools.
+
+Includes autocorrelation function (ACF) and seasonal period estimation.
+"""
+
+import numpy as np
+from numba import njit
+
+
+@njit(cache=True, fastmath=True)
+def acf(X, max_lag):
+    """
+    Compute the sample autocorrelation function (ACF) of a time series.
+
+    Up to a specified maximum lag.
+
+    The autocorrelation at lag k is defined as the Pearson correlation
+    coefficient between the series and a lagged version of itself.
+    If both segments at a given lag have zero variance, the function
+    returns 1 for that lag. If only one segment has zero variance,
+    the function returns 0.
+
+    Parameters
+    ----------
+    X : array-like, shape (n_samples,)
+        The input time series data.
+    max_lag : int
+        The maximum lag (number of steps) for which to
+        compute the autocorrelation.
+
+    Returns
+    -------
+    acf_values : np.ndarray, shape (max_lag,)
+        The autocorrelation values for lags 1 through `max_lag`.
+
+    Notes
+    -----
+    The function handles cases where the lagged segments have zero
+    variance to avoid division by zero.
+    The returned values correspond to
+    lags 1, 2, ..., `max_lag` (not including lag 0).
+    """
+    length = len(X)
+    X_t = np.zeros(max_lag, dtype=float)
+    for lag in range(1, max_lag + 1):
+        lag_length = length - lag
+        x1 = X[:-lag]
+        x2 = X[lag:]
+        s1 = np.sum(x1)
+        s2 = np.sum(x2)
+        m1 = s1 / lag_length
+        m2 = s2 / lag_length
+        ss1 = np.sum(x1 * x1)
+        ss2 = np.sum(x2 * x2)
+        v1 = ss1 - s1 * m1
+        v2 = ss2 - s2 * m2
+        v1_is_zero, v2_is_zero = v1 <= 1e-9, v2 <= 1e-9
+        if v1_is_zero and v2_is_zero:  # Both zero variance,
+            # so must be 100% correlated
+            X_t[lag - 1] = 1
+        elif v1_is_zero or v2_is_zero:  # One zero variance
+            # the other not
+            X_t[lag - 1] = 0
+        else:
+            X_t[lag - 1] = np.sum((x1 - m1) * (x2 - m2)) / np.sqrt(v1 * v2)
+    return X_t
+
+
+@njit(cache=True, fastmath=True)
+def calc_seasonal_period(data):
+    """
+    Estimate the seasonal period of a time series using autocorrelation analysis.
+
+    This function computes the autocorrelation function (ACF) of
+    the input series up to lag 24. It then identifies peaks in the
+    ACF above the mean value, treating the first such peak
+    as the estimated seasonal period. If no peak is found,
+    a period of 1 is returned.
+
+    Parameters
+    ----------
+    data : array-like, shape (n_samples,)
+        The input time series data.
+
+    Returns
+    -------
+    period : int
+        The estimated seasonal period (lag) of the series. Returns 1 if no significant
+        peak is detected in the autocorrelation.
+    """
+    lags = acf(data, 24)
+    lags = np.concatenate((np.array([1.0]), lags))
+    peaks = []
+    mean_lags = np.mean(lags)
+    for i in range(1, len(lags) - 1):  # Skip the first (lag 0) and last elements
+        if lags[i] >= lags[i - 1] and lags[i] >= lags[i + 1] and lags[i] > mean_lags:
+            peaks.append(i)
+    if not peaks:
+        return 1
+    else:
+        return peaks[0]
diff --git a/aeon/forecasting/tests/test_arima.py b/aeon/forecasting/tests/test_arima.py
new file mode 100644
index 0000000000..73d99d7b50
--- /dev/null
+++ b/aeon/forecasting/tests/test_arima.py
@@ -0,0 +1,113 @@
+"""Test the ARIMA forecaster."""
+
+import numpy as np
+import pytest
+
+from aeon.forecasting import ARIMA
+
+y = np.array(
+    [112, 118, 132, 129, 121, 135, 148, 148, 136, 119, 104, 118], dtype=np.float64
+)
+
+
+def test_arima_zero_orders():
+    """Test ARIMA(0,0,0) which should return the mean if constant is used."""
+    model = ARIMA(p=0, d=0, q=0, use_constant=True)
+    model.fit(y)
+    forecast = model.predict(y)
+    assert np.isfinite(forecast)
+    assert abs(forecast - np.mean(y)) < 10
+
+
+@pytest.mark.parametrize(
+    "p, d, q, use_constant",
+    [
+        (1, 0, 1, True),  # basic ARIMA
+        (2, 1, 1, False),  # no constant
+        (1, 2, 1, True),  # higher-order differencing
+    ],
+)
+def test_arima_fit_and_predict_variants(p, d, q, use_constant):
+    """Test ARIMA fit and predict for various (p,d,q) and use_constant settings."""
+    model = ARIMA(p=p, d=d, q=q, use_constant=use_constant)
+    model.fit(y)
+    forecast = model.forecast_
+    assert isinstance(forecast, float)
+    assert np.isfinite(forecast)
+
+
+def test_arima_iterative_forecast():
+    """Test multi-step forecasting using iterative_forecast method."""
+    model = ARIMA(p=1, d=1, q=1)
+    horizon = 3
+    preds = model.iterative_forecast(y, prediction_horizon=horizon)
+    assert preds.shape == (horizon,)
+    assert np.all(np.isfinite(preds))
+
+
+@pytest.mark.parametrize(
+    "y_input, error_match",
+    [
+        (np.array([1.0, 2.0]), "Series too short for differencing"),
+        (np.array([1.0, 2.0, 3.0]), "Series too short for ARMA"),
+    ],
+)
+def test_arima_too_short_series_errors(y_input, error_match):
+    """Test errors raised for too short input series."""
+    model = ARIMA(p=3, d=2, q=3)
+    model.fit(y)
+    with pytest.raises(ValueError, match=error_match):
+        model._predict(y_input)
+
+
+def test_forecast_attribute_set():
+    """Test that calling _forecast sets the internal forecast_ attribute."""
+    model = ARIMA(p=1, d=1, q=1)
+    forecast = model._forecast(y)
+    assert hasattr(model, "forecast_")
+    assert np.isclose(forecast, model.forecast_)
+
+
+def test_iterative_forecast_with_d2():
+    """Test iterative forecast output shape and validity with d=2."""
+    model = ARIMA(p=1, d=2, q=1)
+    preds = model.iterative_forecast(y, prediction_horizon=5)
+    assert preds.shape == (5,)
+    assert np.all(np.isfinite(preds))
+
+
+@pytest.mark.parametrize(
+    "p, d, q, use_constant, expected_forecast",
+    [
+        (1, 0, 1, False, 118.47506756),  # precomputed from known ARIMA implementation
+        (2, 1, 1, False, 209.1099231455),  # precomputed
+        (3, 0, 0, True, 137.47368045155),  # precomputed
+    ],
+)
+def test_arima_fixed_paras(p, d, q, use_constant, expected_forecast):
+    """Test ARIMA fit/predict accuracy against known forecasts.
+
+    expected values calculated with values fitted by Nelder-Mead:
+
+    1. phi = [0.99497524] theta [0.0691515]
+    2. phi = [ 0.02898788 -0.4330671 ] theta [1.26699252]
+    3. phi = [ 0.19202414  0.05207654 -0.07367897] theta [], constant 105.970867164
+
+    """
+    model = ARIMA(p=p, d=d, q=q, use_constant=use_constant)
+    model.fit(y)
+    forecast = model.forecast_
+    assert isinstance(forecast, float)
+    assert np.isfinite(forecast)
+    assert np.isclose(forecast, expected_forecast, atol=1e-6)
+
+
+def test_arima_known_output():
+    """Test ARIMA for fixed parameters.
+
+    Test ARMIMA with forecast generated externally.
+    """
+    model = ARIMA(p=1, d=0, q=1)
+    model.fit(y)
+    f = model.forecast_
+    assert np.isclose(118.47506756, f)