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ARFIMA

ARFIMA\((p, d, q)\) Estimator.

Models time series data featuring long-memory processes through fractional integration, where the difference parameter \(d\) is non-integer, solving the general equation \(\phi(L)(1-L)^dy_t=\theta(L)w_t\)

Source code in src\arfima\core.py 
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class ARFIMA:
    r"""
    ARFIMA$(p, d, q)$ Estimator.

    Models time series data featuring long-memory processes through fractional 
    integration, where the difference parameter $d$ is non-integer, solving the 
    general equation $\phi(L)(1-L)^dy_t=\theta(L)w_t$ 
    """
    def __init__(self):
        self.p : int = 0
        self.q : int = 0
        self.d : float = 0
        self.ar_coeffs = []
        self.ma_coeffs = []
        self.in_sample : NDArray[np.float64] = np.array([], dtype=np.float64)
        self.is_fitted = False

    def __repr__(self):
        return f"ARFIMA(p={self.p}, d={self.d}, q={self.q})"

    def fit(self, time_series:NDArray[np.float64], p_list:list[int], q_list:list[int], method:str="L-BFGS-B",
            criterion:str='aic', returns=True):
        r"""
            Evaluates combinations of AR and MA lags to minimize the selected information criterion. 
            Upon convergence, calculates the inverse Hessian matrix to derive standard errors, 
            t-statistics, and p-values for statistical inference.

            Parameters
            ----------
            time_series : NDArray
                The 1D time series data object to fit.
            p_list : list[int]
                A list of autoregressive (AR) lag orders to test (e.g., [0, 1, 2]).
            q_list : list[int]
                A list of moving average (MA) lag orders to test (e.g., [0, 1]).
            method : str, default="L-BFGS-B"
                The SciPy optimization algorithm to use. Options:

                * "L-BFGS-B" (Default, highly recommended for bounded parameters)
                * "TNC"
                * "SLSQP"
                * "Powell"
            criterion : str, default="aic"
                The information criterion used to penalize model complexity. Options:

                * "aic" : Akaike Information Criterion
                * "bic" : Bayesian Information Criterion
                * "hqic" : Hannan-Quinn Information Criterion
                * "aicc" : Corrected AIC
            """
        self.in_sample = np.asarray(time_series,dtype=float)
        best_ic_score = np.inf
        best_params = None
        n = len(time_series)

        ic_formulas = {
            'aic': lambda n, k, neg_ll: 2 * k + 2 * neg_ll,
            'bic': lambda n, k, neg_ll: k * np.log(n) + 2 * neg_ll,
            'hqic': lambda n, k, neg_ll: 2 * k * np.log(np.log(n)) + 2 * neg_ll,
            'aicc': lambda n, k, neg_ll: (2 * k + 2 * neg_ll) + (2 * k**2 + 2 * k) / (n - k - 1) if (n - k - 1) > 0 else np.inf
        }

        criterion = criterion.lower()

        if criterion not in ic_formulas:
            raise ValueError(f"Invalid Information Criteria {criterion}. Choose from {", ".join(ic_formulas.keys())}")

        for p in p_list:
            for q in q_list:

                d_seed = _gph_estimator(self.in_sample)
                initial_guess = [d_seed] + [0.0]*p + [0.0]*q
                bounds = [(-0.49,0.49)] + [(-0.99,0.99)]*(p+q)

                res = minimize(fun=_score_function,
                               x0=initial_guess,
                               args=(time_series, p),
                               method=method,
                               bounds=bounds)

                if res.success:
                    k = p+q+1
                    negative_likelihood = res.fun
                    ic_score = ic_formulas[criterion](n, k, negative_likelihood)

                    if ic_score < best_ic_score:
                        best_ic_score = ic_score
                        self.p = p
                        self.q = q
                        best_params = res.x
                else:
                    print(f"Failed to fit ARFIMA({p}, d, {q}): {res.message}") 

        if best_params is not None:

                    final_bounds = [(-0.49, 0.49)] + [(-0.99, 0.99)] * (self.p + self.q)

                    final_result = minimize(
                        fun=_score_function, 
                        x0=best_params, 
                        args=(time_series, self.p), 
                        method='L-BFGS-B', 
                        bounds=final_bounds
                    )

                    best_params = final_result.x
                    self.d = best_params[0]
                    self.ar_coeffs = best_params[1 : self.p + 1] if self.p > 0 else []
                    self.ma_coeffs = best_params[self.p + 1 :] if self.q > 0 else []
                    self.is_fitted = True


                    inv_hessian = final_result.hess_inv.todense()
                    variances = np.diag(inv_hessian)
                    self.std_errors = np.sqrt(np.abs(variances))
                    self.t_stats = best_params / self.std_errors 
                    self.p_values = 2 * stats.norm.sf(np.abs(self.t_stats))
                    print(f"\nFinal Fit Complete: ARFIMA({self.p}, {self.d:.4f}, {self.q})")
        if returns:
            return self

    def summary(self, returns:bool=True):
        """
        Prints a formatted statistical regression table for the fitted model.

        Outputs the estimated parameters (fractional differencing $d$, AR, and MA coefficients), 
        along with their corresponding standard errors, t-statistics, and two-tailed p-values 
        derived from the inverse Hessian matrix.

        Returns
        -------
        None
            This method prints directly to the console and does not return any objects.

        Prints
        ------
        str
            A formatted ASCII table containing the model structure and statistical inference metrics.
        """
        if not self.is_fitted:
            print("Model is not fitted yet.")
            return

        print("\n" + "="*50)
        print(f"{'ARFIMA MODEL SUMMARY':^50}")
        print("="*50)
        print(f"Model Structure: ARFIMA({self.p}, d, {self.q})")
        print("-" * 50)

        print(f"{'Parameter':<10} | {'Coef':>8} | {'Std Err':>8} | {'t-stat':>8} | {'P>|t|':>8}")
        print("-" * 50)              

        def print_row(name, index):
            coef = getattr(self, 'd') if name == 'd' else (self.ar_coeffs[index-1] if 'AR' in name else self.ma_coeffs[index-1 - self.p])
            actual_index = 0 if name == 'd' else index

            print(f"{name:<10} | {coef:>8.4f} | {self.std_errors[actual_index]:>8.4f} | {self.t_stats[actual_index]:>8.4f} | {self.p_values[actual_index]:>8.4f}")

        print_row('d', 0)

        for i in range(1, self.p + 1):
            print_row(f'AR({i})', i)

        for j in range(1, self.q + 1):
            print_row(f'MA({j})', self.p + j)

        print("="*50 + "\n")

        if returns:
            return self

    def predict(self, steps:int=0):
        """
        Forecasts future values of the time series out-of-sample.

        Mathematically projects the series forward by assuming all future 
        random shocks ($w_t$) are exactly zero. The projected shocks are then 
        fractionally integrated using $-d$ to restore the original scale of the data.

        Parameters
        ----------
        steps : int, default=0
            The number of future time periods to forecast.

        Returns
        -------
        np.ndarray
            A 1D array of length `steps` containing the out-of-sample forecasted values.

        Raises
        ------
        ValueError
            If the method is called before the model has been successfully fitted.
        """
        if not self.is_fitted:
            raise ValueError("ARFIMA not fitted yet")

        y_history = _frac_diff(self.in_sample, self.d)
        w_history = _fast_residuals(y_history, self.ar_coeffs, self.ma_coeffs)

        w_future = np.zeros(steps)
        w_full = np.concatenate((w_history, w_future))

        y_full = _arma_generate(w_full, self.ar_coeffs, self.ma_coeffs)
        integrated_series = _frac_diff(y_full, -self.d)

        return integrated_series[-steps:]

    def predict_in_sample(self):
        """
        Generates in-sample fitted values for the historical data.

        Reconstructs the model's historical fit by calculating the exact historical 
        residuals via linear filtering, and subtracting those residuals 
        from the raw input data.

        Returns
        -------
        np.ndarray
            A 1D array of fitted values, equal in length to the input time series.

        Raises
        ------
        ValueError
            If the method is called before the model has been successfully fitted.
        """
        if not self.is_fitted:
            raise ValueError("ARFIMA not fitted yet")


        ar_poly = [1.0] + [-phi for phi in self.ar_coeffs]
        ma_poly = [1.0] + list(self.ma_coeffs)

        y_history = _frac_diff(self.in_sample, self.d)

        w_history = np.asarray(signal.lfilter(ar_poly, ma_poly, y_history))

        fitted_values = self.in_sample - w_history

        return fitted_values

    @property
    def residuals(self)-> np.ndarray:
        if not self.is_fitted:
            raise ValueError("Model must be fitted first.")
        y_history = _frac_diff(self.in_sample, self.d)
        return _fast_residuals(y_history, self.ar_coeffs, self.ma_coeffs)

fit(time_series, p_list, q_list, method='L-BFGS-B', criterion='aic', returns=True)

Evaluates combinations of AR and MA lags to minimize the selected information criterion. Upon convergence, calculates the inverse Hessian matrix to derive standard errors, t-statistics, and p-values for statistical inference.

Parameters

time_series : NDArray The 1D time series data object to fit. p_list : list[int] A list of autoregressive (AR) lag orders to test (e.g., [0, 1, 2]). q_list : list[int] A list of moving average (MA) lag orders to test (e.g., [0, 1]). method : str, default="L-BFGS-B" The SciPy optimization algorithm to use. Options:

* "L-BFGS-B" (Default, highly recommended for bounded parameters)
* "TNC"
* "SLSQP"
* "Powell"

criterion : str, default="aic" The information criterion used to penalize model complexity. Options:

* "aic" : Akaike Information Criterion
* "bic" : Bayesian Information Criterion
* "hqic" : Hannan-Quinn Information Criterion
* "aicc" : Corrected AIC
Source code in src\arfima\core.py 
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def fit(self, time_series:NDArray[np.float64], p_list:list[int], q_list:list[int], method:str="L-BFGS-B",
        criterion:str='aic', returns=True):
    r"""
        Evaluates combinations of AR and MA lags to minimize the selected information criterion. 
        Upon convergence, calculates the inverse Hessian matrix to derive standard errors, 
        t-statistics, and p-values for statistical inference.

        Parameters
        ----------
        time_series : NDArray
            The 1D time series data object to fit.
        p_list : list[int]
            A list of autoregressive (AR) lag orders to test (e.g., [0, 1, 2]).
        q_list : list[int]
            A list of moving average (MA) lag orders to test (e.g., [0, 1]).
        method : str, default="L-BFGS-B"
            The SciPy optimization algorithm to use. Options:

            * "L-BFGS-B" (Default, highly recommended for bounded parameters)
            * "TNC"
            * "SLSQP"
            * "Powell"
        criterion : str, default="aic"
            The information criterion used to penalize model complexity. Options:

            * "aic" : Akaike Information Criterion
            * "bic" : Bayesian Information Criterion
            * "hqic" : Hannan-Quinn Information Criterion
            * "aicc" : Corrected AIC
        """
    self.in_sample = np.asarray(time_series,dtype=float)
    best_ic_score = np.inf
    best_params = None
    n = len(time_series)

    ic_formulas = {
        'aic': lambda n, k, neg_ll: 2 * k + 2 * neg_ll,
        'bic': lambda n, k, neg_ll: k * np.log(n) + 2 * neg_ll,
        'hqic': lambda n, k, neg_ll: 2 * k * np.log(np.log(n)) + 2 * neg_ll,
        'aicc': lambda n, k, neg_ll: (2 * k + 2 * neg_ll) + (2 * k**2 + 2 * k) / (n - k - 1) if (n - k - 1) > 0 else np.inf
    }

    criterion = criterion.lower()

    if criterion not in ic_formulas:
        raise ValueError(f"Invalid Information Criteria {criterion}. Choose from {", ".join(ic_formulas.keys())}")

    for p in p_list:
        for q in q_list:

            d_seed = _gph_estimator(self.in_sample)
            initial_guess = [d_seed] + [0.0]*p + [0.0]*q
            bounds = [(-0.49,0.49)] + [(-0.99,0.99)]*(p+q)

            res = minimize(fun=_score_function,
                           x0=initial_guess,
                           args=(time_series, p),
                           method=method,
                           bounds=bounds)

            if res.success:
                k = p+q+1
                negative_likelihood = res.fun
                ic_score = ic_formulas[criterion](n, k, negative_likelihood)

                if ic_score < best_ic_score:
                    best_ic_score = ic_score
                    self.p = p
                    self.q = q
                    best_params = res.x
            else:
                print(f"Failed to fit ARFIMA({p}, d, {q}): {res.message}") 

    if best_params is not None:

                final_bounds = [(-0.49, 0.49)] + [(-0.99, 0.99)] * (self.p + self.q)

                final_result = minimize(
                    fun=_score_function, 
                    x0=best_params, 
                    args=(time_series, self.p), 
                    method='L-BFGS-B', 
                    bounds=final_bounds
                )

                best_params = final_result.x
                self.d = best_params[0]
                self.ar_coeffs = best_params[1 : self.p + 1] if self.p > 0 else []
                self.ma_coeffs = best_params[self.p + 1 :] if self.q > 0 else []
                self.is_fitted = True


                inv_hessian = final_result.hess_inv.todense()
                variances = np.diag(inv_hessian)
                self.std_errors = np.sqrt(np.abs(variances))
                self.t_stats = best_params / self.std_errors 
                self.p_values = 2 * stats.norm.sf(np.abs(self.t_stats))
                print(f"\nFinal Fit Complete: ARFIMA({self.p}, {self.d:.4f}, {self.q})")
    if returns:
        return self

predict(steps=0)

Forecasts future values of the time series out-of-sample.

Mathematically projects the series forward by assuming all future random shocks (\(w_t\)) are exactly zero. The projected shocks are then fractionally integrated using \(-d\) to restore the original scale of the data.

Parameters

steps : int, default=0 The number of future time periods to forecast.

Returns

np.ndarray A 1D array of length steps containing the out-of-sample forecasted values.

Raises

ValueError If the method is called before the model has been successfully fitted.

Source code in src\arfima\core.py 
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def predict(self, steps:int=0):
    """
    Forecasts future values of the time series out-of-sample.

    Mathematically projects the series forward by assuming all future 
    random shocks ($w_t$) are exactly zero. The projected shocks are then 
    fractionally integrated using $-d$ to restore the original scale of the data.

    Parameters
    ----------
    steps : int, default=0
        The number of future time periods to forecast.

    Returns
    -------
    np.ndarray
        A 1D array of length `steps` containing the out-of-sample forecasted values.

    Raises
    ------
    ValueError
        If the method is called before the model has been successfully fitted.
    """
    if not self.is_fitted:
        raise ValueError("ARFIMA not fitted yet")

    y_history = _frac_diff(self.in_sample, self.d)
    w_history = _fast_residuals(y_history, self.ar_coeffs, self.ma_coeffs)

    w_future = np.zeros(steps)
    w_full = np.concatenate((w_history, w_future))

    y_full = _arma_generate(w_full, self.ar_coeffs, self.ma_coeffs)
    integrated_series = _frac_diff(y_full, -self.d)

    return integrated_series[-steps:]

predict_in_sample()

Generates in-sample fitted values for the historical data.

Reconstructs the model's historical fit by calculating the exact historical residuals via linear filtering, and subtracting those residuals from the raw input data.

Returns

np.ndarray A 1D array of fitted values, equal in length to the input time series.

Raises

ValueError If the method is called before the model has been successfully fitted.

Source code in src\arfima\core.py 
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def predict_in_sample(self):
    """
    Generates in-sample fitted values for the historical data.

    Reconstructs the model's historical fit by calculating the exact historical 
    residuals via linear filtering, and subtracting those residuals 
    from the raw input data.

    Returns
    -------
    np.ndarray
        A 1D array of fitted values, equal in length to the input time series.

    Raises
    ------
    ValueError
        If the method is called before the model has been successfully fitted.
    """
    if not self.is_fitted:
        raise ValueError("ARFIMA not fitted yet")


    ar_poly = [1.0] + [-phi for phi in self.ar_coeffs]
    ma_poly = [1.0] + list(self.ma_coeffs)

    y_history = _frac_diff(self.in_sample, self.d)

    w_history = np.asarray(signal.lfilter(ar_poly, ma_poly, y_history))

    fitted_values = self.in_sample - w_history

    return fitted_values

summary(returns=True)

Prints a formatted statistical regression table for the fitted model.

Outputs the estimated parameters (fractional differencing \(d\), AR, and MA coefficients), along with their corresponding standard errors, t-statistics, and two-tailed p-values derived from the inverse Hessian matrix.

Returns

None This method prints directly to the console and does not return any objects.

Prints

str A formatted ASCII table containing the model structure and statistical inference metrics.

Source code in src\arfima\core.py 
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def summary(self, returns:bool=True):
    """
    Prints a formatted statistical regression table for the fitted model.

    Outputs the estimated parameters (fractional differencing $d$, AR, and MA coefficients), 
    along with their corresponding standard errors, t-statistics, and two-tailed p-values 
    derived from the inverse Hessian matrix.

    Returns
    -------
    None
        This method prints directly to the console and does not return any objects.

    Prints
    ------
    str
        A formatted ASCII table containing the model structure and statistical inference metrics.
    """
    if not self.is_fitted:
        print("Model is not fitted yet.")
        return

    print("\n" + "="*50)
    print(f"{'ARFIMA MODEL SUMMARY':^50}")
    print("="*50)
    print(f"Model Structure: ARFIMA({self.p}, d, {self.q})")
    print("-" * 50)

    print(f"{'Parameter':<10} | {'Coef':>8} | {'Std Err':>8} | {'t-stat':>8} | {'P>|t|':>8}")
    print("-" * 50)              

    def print_row(name, index):
        coef = getattr(self, 'd') if name == 'd' else (self.ar_coeffs[index-1] if 'AR' in name else self.ma_coeffs[index-1 - self.p])
        actual_index = 0 if name == 'd' else index

        print(f"{name:<10} | {coef:>8.4f} | {self.std_errors[actual_index]:>8.4f} | {self.t_stats[actual_index]:>8.4f} | {self.p_values[actual_index]:>8.4f}")

    print_row('d', 0)

    for i in range(1, self.p + 1):
        print_row(f'AR({i})', i)

    for j in range(1, self.q + 1):
        print_row(f'MA({j})', self.p + j)

    print("="*50 + "\n")

    if returns:
        return self

FIGARCH

Source code in src\arfima\core.py 
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class FIGARCH:
    def __init__(self):
        self.p : int = 0
        self.q : int = 0
        self.d : float = 0.0
        self.omega : float = 0.0
        self.arch_coeffs = []
        self.garch_coeffs = []
        self.is_fitted = False

    def fit(self, squared_residuals: NDArray[np.float64], p_list: list[int], q_list: list[int], method: str="L-BFGS-B", 
            criterion:str='aic'):
        r"""
        Fits the optimal FIGARCH structure using Maximum Likelihood Estimation.

        Parameters
        ----------
        squared_residuals : NDArray[np.float64]
            The squared white noise shocks ($\epsilon_t^2$) extracted from the mean equation.
        p_list : list[int]
            List of ARCH lag orders to test.
        q_list : list[int]
            List of GARCH lag orders to test.
        method : str, default="L-BFGS-B"
            The SciPy optimization algorithm to use. Options:

            * "L-BFGS-B" (Default, highly recommended for bounded parameters)
            * "TNC"
            * "SLSQP"
            * "Powell"
        criterion : str, default="aic"
            The information criterion used to penalize model complexity. Options:

            * "aic" : Akaike Information Criterion
            * "bic" : Bayesian Information Criterion
            * "hqic" : Hannan-Quinn Information Criterion
            * "aicc" : Corrected AIC
        """
        self.in_sample_variance = np.asarray(squared_residuals, dtype=float)
        best_params = None

        omega_guess = np.var(self.in_sample_variance)

        best_ic_score = np.inf
        n = len(self.in_sample_variance)

        ic_formulas = {
            'aic': lambda n, k, neg_ll: 2 * k + 2 * neg_ll,
            'bic': lambda n, k, neg_ll: k * np.log(n) + 2 * neg_ll,
            'hqic': lambda n, k, neg_ll: 2 * k * np.log(np.log(n)) + 2 * neg_ll,
            'aicc': lambda n, k, neg_ll: (2 * k + 2 * neg_ll) + (2 * k**2 + 2 * k) / (n - k - 1) if (n - k - 1) > 0 else np.inf
        }

        criterion = criterion.lower()
        d_v_seed = _gph_estimator(self.in_sample_variance)

        for p in p_list:
            for q in q_list:


                initial_guess = [omega_guess, d_v_seed] + [0.05] * p + [0.8 / q if q > 0 else 0.0] * q


                bounds = [(1e-6, None), (0.0, 1.0)] + [(0.0, 0.99)] * p + [(0.0, 0.99)] * q

                res = minimize(
                    fun=_figarch_score_function,
                    x0=initial_guess,
                    args=(self.in_sample_variance, p, q),
                    method=method,
                    bounds=bounds
                )

                if res.success:
                    k = p + q + 2
                    negative_likelihood = res.fun
                    ic_score = ic_formulas[criterion](n, k, negative_likelihood)

                    if ic_score < best_ic_score:
                        best_ic_score = ic_score
                        self.p = p
                        self.q = q
                        best_params = res.x

                elif not res.success:
                    print(f"Failed to fit FIGARCH({p}, d_v, {q}): {res.message}")

        if best_params is not None:
            final_bounds = [(1e-6, None), (0.0, 1.0)] + [(0.0, 0.99)] * self.p + [(0.0, 0.99)] * self.q

            final_result = minimize(
                fun=_figarch_score_function,
                x0=best_params,
                args=(self.in_sample_variance, self.p, self.q),
                method='L-BFGS-B',
                bounds=final_bounds
            )

            best_params = final_result.x
            self.omega = best_params[0]
            self.d_v = best_params[1]
            self.arch_coeffs = best_params[2 : self.p + 2] if self.p > 0 else []
            self.garch_coeffs = best_params[self.p + 2 :] if self.q > 0 else []
            self.is_fitted = True


            inv_hessian = final_result.hess_inv.todense()
            variances = np.diag(inv_hessian)
            self.std_errors = np.sqrt(np.abs(variances))
            self.t_stats = best_params / self.std_errors 
            self.p_values = 2 * stats.norm.sf(np.abs(self.t_stats))

            print(f"\nFinal Fit Complete: FIGARCH({self.p}, {self.d_v:.4f}, {self.q})")

        return self
    def summary(self, returns:bool=True):
        r"""
        Prints a formatted statistical regression table for the fitted FIGARCH model.

        Outputs the estimated parameters (baseline variance $\omega$, fractional 
        variance differencing $d_v$, ARCH, and GARCH coefficients), along with 
        their corresponding standard errors, t-statistics, and two-tailed p-values 
        derived from the inverse Hessian matrix.
        """
        if not self.is_fitted:
            print("Model is not fitted yet.")
            return

        print("\n" + "="*50)
        print(f"{'FIGARCH MODEL SUMMARY':^50}")
        print("="*50)
        print(f"Model Structure: FIGARCH({self.p}, d_v, {self.q})")
        print("-" * 50)

        print(f"{'Parameter':<10} | {'Coef':>8} | {'Std Err':>8} | {'t-stat':>8} | {'P>|t|':>8}")
        print("-" * 50)              

        def print_row(name, actual_index, coef):
            std_err = self.std_errors[actual_index]
            t_stat = self.t_stats[actual_index]
            p_val = self.p_values[actual_index]
            print(f"{name:<10} | {coef:>8.4f} | {std_err:>8.4f} | {t_stat:>8.4f} | {p_val:>8.4f}")


        print_row('omega', 0, self.omega)
        print_row('d_v', 1, self.d_v)


        for i in range(1, self.p + 1):
            print_row(f'ARCH({i})', i + 1, self.arch_coeffs[i-1])


        for j in range(1, self.q + 1):
            print_row(f'GARCH({j})', self.p + 1 + j, self.garch_coeffs[j-1])

        print("="*50 + "\n")

        if returns:
            return self

    def predict_in_sample(self):
        r"""
        Generates in-sample conditional variances for the historical data.

        Reconstructs the model's historical volatility fit by calculating the 
        ARCH($\infty$) decay weights and convoluting them with the actual 
        historical squared shocks.

        Returns
        -------
        np.ndarray
            A 1D array of conditional variances ($\sigma_t^2$), equal in length 
            to the input time series.

        Raises
        ------
        ValueError
            If the method is called before the model has been successfully fitted.
        """
        if not self.is_fitted:
            raise ValueError("FIGARCH not fitted yet")

        n = len(self.in_sample_variance)


        weights = _figarch_weights(self.d_v, self.arch_coeffs, self.garch_coeffs, n)


        beta_sum = np.sum(self.garch_coeffs) if self.q > 0 else 0.0
        omega_star = self.omega / (1.0 - beta_sum)


        conditional_variance = _dynamic_variance(self.in_sample_variance, omega_star, weights)

        return conditional_variance

    def predict(self, steps: int = 0):
        r"""
        Forecasts future conditional variance out-of-sample.

        Mathematically projects volatility forward iteratively. Because future 
        squared shocks ($\epsilon_{t+k}^2$) are unobservable, the model substitutes 
        them with their expected value, which is the forecasted conditional 
        variance ($\sigma_{t+k}^2$).

        Parameters
        ----------
        steps : int, default=0
            The number of future time periods to forecast.

        Returns
        -------
        np.ndarray
            A 1D array of length `steps` containing the forecasted conditional variances.

        Raises
        ------
        ValueError
            If the method is called before the model has been successfully fitted.
        """
        if not self.is_fitted:
            raise ValueError("FIGARCH not fitted yet")

        if steps <= 0:
            return np.array([])

        n = len(self.in_sample_variance)

        beta_sum = np.sum(self.garch_coeffs) if self.q > 0 else 0.0
        omega_star = self.omega / (1.0 - beta_sum)


        extended_shocks = np.copy(self.in_sample_variance)
        forecasted_variance = np.zeros(steps)


        for k in range(steps):
            current_len = n + k


            weights = _figarch_weights(self.d_v, self.arch_coeffs, self.garch_coeffs, current_len)
            current_variance_path = _dynamic_variance(extended_shocks, omega_star, weights)
            next_variance = current_variance_path[-1]
            forecasted_variance[k] = next_variance

            extended_shocks = np.append(extended_shocks, next_variance)

        return forecasted_variance

fit(squared_residuals, p_list, q_list, method='L-BFGS-B', criterion='aic')

Fits the optimal FIGARCH structure using Maximum Likelihood Estimation.

Parameters

squared_residuals : NDArray[np.float64] The squared white noise shocks (\(\epsilon_t^2\)) extracted from the mean equation. p_list : list[int] List of ARCH lag orders to test. q_list : list[int] List of GARCH lag orders to test. method : str, default="L-BFGS-B" The SciPy optimization algorithm to use. Options:

* "L-BFGS-B" (Default, highly recommended for bounded parameters)
* "TNC"
* "SLSQP"
* "Powell"

criterion : str, default="aic" The information criterion used to penalize model complexity. Options:

* "aic" : Akaike Information Criterion
* "bic" : Bayesian Information Criterion
* "hqic" : Hannan-Quinn Information Criterion
* "aicc" : Corrected AIC
Source code in src\arfima\core.py 
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def fit(self, squared_residuals: NDArray[np.float64], p_list: list[int], q_list: list[int], method: str="L-BFGS-B", 
        criterion:str='aic'):
    r"""
    Fits the optimal FIGARCH structure using Maximum Likelihood Estimation.

    Parameters
    ----------
    squared_residuals : NDArray[np.float64]
        The squared white noise shocks ($\epsilon_t^2$) extracted from the mean equation.
    p_list : list[int]
        List of ARCH lag orders to test.
    q_list : list[int]
        List of GARCH lag orders to test.
    method : str, default="L-BFGS-B"
        The SciPy optimization algorithm to use. Options:

        * "L-BFGS-B" (Default, highly recommended for bounded parameters)
        * "TNC"
        * "SLSQP"
        * "Powell"
    criterion : str, default="aic"
        The information criterion used to penalize model complexity. Options:

        * "aic" : Akaike Information Criterion
        * "bic" : Bayesian Information Criterion
        * "hqic" : Hannan-Quinn Information Criterion
        * "aicc" : Corrected AIC
    """
    self.in_sample_variance = np.asarray(squared_residuals, dtype=float)
    best_params = None

    omega_guess = np.var(self.in_sample_variance)

    best_ic_score = np.inf
    n = len(self.in_sample_variance)

    ic_formulas = {
        'aic': lambda n, k, neg_ll: 2 * k + 2 * neg_ll,
        'bic': lambda n, k, neg_ll: k * np.log(n) + 2 * neg_ll,
        'hqic': lambda n, k, neg_ll: 2 * k * np.log(np.log(n)) + 2 * neg_ll,
        'aicc': lambda n, k, neg_ll: (2 * k + 2 * neg_ll) + (2 * k**2 + 2 * k) / (n - k - 1) if (n - k - 1) > 0 else np.inf
    }

    criterion = criterion.lower()
    d_v_seed = _gph_estimator(self.in_sample_variance)

    for p in p_list:
        for q in q_list:


            initial_guess = [omega_guess, d_v_seed] + [0.05] * p + [0.8 / q if q > 0 else 0.0] * q


            bounds = [(1e-6, None), (0.0, 1.0)] + [(0.0, 0.99)] * p + [(0.0, 0.99)] * q

            res = minimize(
                fun=_figarch_score_function,
                x0=initial_guess,
                args=(self.in_sample_variance, p, q),
                method=method,
                bounds=bounds
            )

            if res.success:
                k = p + q + 2
                negative_likelihood = res.fun
                ic_score = ic_formulas[criterion](n, k, negative_likelihood)

                if ic_score < best_ic_score:
                    best_ic_score = ic_score
                    self.p = p
                    self.q = q
                    best_params = res.x

            elif not res.success:
                print(f"Failed to fit FIGARCH({p}, d_v, {q}): {res.message}")

    if best_params is not None:
        final_bounds = [(1e-6, None), (0.0, 1.0)] + [(0.0, 0.99)] * self.p + [(0.0, 0.99)] * self.q

        final_result = minimize(
            fun=_figarch_score_function,
            x0=best_params,
            args=(self.in_sample_variance, self.p, self.q),
            method='L-BFGS-B',
            bounds=final_bounds
        )

        best_params = final_result.x
        self.omega = best_params[0]
        self.d_v = best_params[1]
        self.arch_coeffs = best_params[2 : self.p + 2] if self.p > 0 else []
        self.garch_coeffs = best_params[self.p + 2 :] if self.q > 0 else []
        self.is_fitted = True


        inv_hessian = final_result.hess_inv.todense()
        variances = np.diag(inv_hessian)
        self.std_errors = np.sqrt(np.abs(variances))
        self.t_stats = best_params / self.std_errors 
        self.p_values = 2 * stats.norm.sf(np.abs(self.t_stats))

        print(f"\nFinal Fit Complete: FIGARCH({self.p}, {self.d_v:.4f}, {self.q})")

    return self

predict(steps=0)

Forecasts future conditional variance out-of-sample.

Mathematically projects volatility forward iteratively. Because future squared shocks (\(\epsilon_{t+k}^2\)) are unobservable, the model substitutes them with their expected value, which is the forecasted conditional variance (\(\sigma_{t+k}^2\)).

Parameters

steps : int, default=0 The number of future time periods to forecast.

Returns

np.ndarray A 1D array of length steps containing the forecasted conditional variances.

Raises

ValueError If the method is called before the model has been successfully fitted.

Source code in src\arfima\core.py 
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def predict(self, steps: int = 0):
    r"""
    Forecasts future conditional variance out-of-sample.

    Mathematically projects volatility forward iteratively. Because future 
    squared shocks ($\epsilon_{t+k}^2$) are unobservable, the model substitutes 
    them with their expected value, which is the forecasted conditional 
    variance ($\sigma_{t+k}^2$).

    Parameters
    ----------
    steps : int, default=0
        The number of future time periods to forecast.

    Returns
    -------
    np.ndarray
        A 1D array of length `steps` containing the forecasted conditional variances.

    Raises
    ------
    ValueError
        If the method is called before the model has been successfully fitted.
    """
    if not self.is_fitted:
        raise ValueError("FIGARCH not fitted yet")

    if steps <= 0:
        return np.array([])

    n = len(self.in_sample_variance)

    beta_sum = np.sum(self.garch_coeffs) if self.q > 0 else 0.0
    omega_star = self.omega / (1.0 - beta_sum)


    extended_shocks = np.copy(self.in_sample_variance)
    forecasted_variance = np.zeros(steps)


    for k in range(steps):
        current_len = n + k


        weights = _figarch_weights(self.d_v, self.arch_coeffs, self.garch_coeffs, current_len)
        current_variance_path = _dynamic_variance(extended_shocks, omega_star, weights)
        next_variance = current_variance_path[-1]
        forecasted_variance[k] = next_variance

        extended_shocks = np.append(extended_shocks, next_variance)

    return forecasted_variance

predict_in_sample()

Generates in-sample conditional variances for the historical data.

Reconstructs the model's historical volatility fit by calculating the ARCH(\(\infty\)) decay weights and convoluting them with the actual historical squared shocks.

Returns

np.ndarray A 1D array of conditional variances (\(\sigma_t^2\)), equal in length to the input time series.

Raises

ValueError If the method is called before the model has been successfully fitted.

Source code in src\arfima\core.py 
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def predict_in_sample(self):
    r"""
    Generates in-sample conditional variances for the historical data.

    Reconstructs the model's historical volatility fit by calculating the 
    ARCH($\infty$) decay weights and convoluting them with the actual 
    historical squared shocks.

    Returns
    -------
    np.ndarray
        A 1D array of conditional variances ($\sigma_t^2$), equal in length 
        to the input time series.

    Raises
    ------
    ValueError
        If the method is called before the model has been successfully fitted.
    """
    if not self.is_fitted:
        raise ValueError("FIGARCH not fitted yet")

    n = len(self.in_sample_variance)


    weights = _figarch_weights(self.d_v, self.arch_coeffs, self.garch_coeffs, n)


    beta_sum = np.sum(self.garch_coeffs) if self.q > 0 else 0.0
    omega_star = self.omega / (1.0 - beta_sum)


    conditional_variance = _dynamic_variance(self.in_sample_variance, omega_star, weights)

    return conditional_variance

summary(returns=True)

Prints a formatted statistical regression table for the fitted FIGARCH model.

Outputs the estimated parameters (baseline variance \(\omega\), fractional variance differencing \(d_v\), ARCH, and GARCH coefficients), along with their corresponding standard errors, t-statistics, and two-tailed p-values derived from the inverse Hessian matrix.

Source code in src\arfima\core.py 
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def summary(self, returns:bool=True):
    r"""
    Prints a formatted statistical regression table for the fitted FIGARCH model.

    Outputs the estimated parameters (baseline variance $\omega$, fractional 
    variance differencing $d_v$, ARCH, and GARCH coefficients), along with 
    their corresponding standard errors, t-statistics, and two-tailed p-values 
    derived from the inverse Hessian matrix.
    """
    if not self.is_fitted:
        print("Model is not fitted yet.")
        return

    print("\n" + "="*50)
    print(f"{'FIGARCH MODEL SUMMARY':^50}")
    print("="*50)
    print(f"Model Structure: FIGARCH({self.p}, d_v, {self.q})")
    print("-" * 50)

    print(f"{'Parameter':<10} | {'Coef':>8} | {'Std Err':>8} | {'t-stat':>8} | {'P>|t|':>8}")
    print("-" * 50)              

    def print_row(name, actual_index, coef):
        std_err = self.std_errors[actual_index]
        t_stat = self.t_stats[actual_index]
        p_val = self.p_values[actual_index]
        print(f"{name:<10} | {coef:>8.4f} | {std_err:>8.4f} | {t_stat:>8.4f} | {p_val:>8.4f}")


    print_row('omega', 0, self.omega)
    print_row('d_v', 1, self.d_v)


    for i in range(1, self.p + 1):
        print_row(f'ARCH({i})', i + 1, self.arch_coeffs[i-1])


    for j in range(1, self.q + 1):
        print_row(f'GARCH({j})', self.p + 1 + j, self.garch_coeffs[j-1])

    print("="*50 + "\n")

    if returns:
        return self

_arma_generate(shocks, ar_coeffs, ma_coeffs)

Generates time series data \(y_t\) from given residuals \(w_t\).

Acts as the inverse of _fast_residuals by solving \((1 - \phi)y_t = (1 + \theta L)w_t\) for \(y_t\).

Parameters

shocks : NDArray[np.float64] The array of residuals. ar_coeffs : list[float] The autoregressive coefficients (\(\phi\)). ma_coeffs : list[float] The moving average coefficients (\(\theta\)).

Returns

np.ndarray A 1D array of generated ARMA values.

Source code in src\arfima\core.py 
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def _arma_generate(shocks: NDArray[np.float64], ar_coeffs: list[float], ma_coeffs: list[float]):
    r"""
    Generates time series data $y_t$ from given residuals $w_t$.

    Acts as the inverse of `_fast_residuals` by solving $(1 - \phi)y_t = (1 + \theta L)w_t$ for $y_t$.

    Parameters
    ----------
    shocks : NDArray[np.float64]
        The array of residuals.
    ar_coeffs : list[float]
        The autoregressive coefficients ($\phi$).
    ma_coeffs : list[float]
        The moving average coefficients ($\theta$).

    Returns
    -------
    np.ndarray
        A 1D array of generated ARMA values.
    """
    a = [1.0] + [-i for i in ar_coeffs]  
    b = [1.0] + list(ma_coeffs)          
    return np.asarray(signal.lfilter(b, a, shocks))

_dynamic_variance(squared_shocks, omega, weights)

Calculates the conditional variance array (\(\sigma_t^2\)) using FFT convolution.

Applies the pre-calculated ARCH(\(\infty\)) decay weights to the historical squared shocks via Fast Fourier Transform convolution to efficiently construct the dynamic variance path. Applies a strict mathematical floor to prevent variance from dropping below zero.

Parameters

squared_shocks : NDArray[np.float64] The array of squared residuals (\(\epsilon_t^2\)) extracted from the mean equation. omega : float The adjusted baseline variance constant (\(\omega^*\)). weights : NDArray[np.float64] The array of ARCH(\(\infty\)) decay weights.

Returns

np.ndarray A 1D array of conditional variances (\(\sigma_t^2\)) equal in length to the input shocks.

Source code in src\arfima\core.py 
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def _dynamic_variance(squared_shocks: NDArray[np.float64], omega: float, weights: NDArray[np.float64]):
    """
    Calculates the conditional variance array ($\sigma_t^2$) using FFT convolution.

    Applies the pre-calculated ARCH($\infty$) decay weights to the historical 
    squared shocks via Fast Fourier Transform convolution to efficiently construct 
    the dynamic variance path. Applies a strict mathematical floor to prevent 
    variance from dropping below zero.

    Parameters
    ----------
    squared_shocks : NDArray[np.float64]
        The array of squared residuals ($\epsilon_t^2$) extracted from the mean equation.
    omega : float
        The adjusted baseline variance constant ($\omega^*$).
    weights : NDArray[np.float64]
        The array of ARCH($\infty$) decay weights.

    Returns
    -------
    np.ndarray
        A 1D array of conditional variances ($\sigma_t^2$) equal in length to the input shocks.
    """
    n = len(squared_shocks)

    arch_effect = signal.fftconvolve(squared_shocks, weights, mode="full")[:n]
    cond_variance = omega + arch_effect

    return np.maximum(cond_variance, 1e-6)

_fast_residuals(diff_data, ar_coeffs, ma_coeffs)

Returns the residuals \(w_t\) from differenced data using linear filtering.

Solves the ARMA equation \((1 - \phi)y_t = (1 + \theta)w_t\) for \(w_t\).

Parameters

diff_data : NDArray[np.float64] The fractionally differenced time series (\(y_t\)). ar_coeffs : list[float] The autoregressive coefficients (\(\phi\)). ma_coeffs : list[float] The moving average coefficients (\(\theta\)).

Returns

np.ndarray A 1D array of residuals.

Source code in src\arfima\core.py 
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def _fast_residuals(diff_data:NDArray[np.float64], ar_coeffs:list[float], ma_coeffs:list[float]):
    r"""
    Returns the residuals $w_t$ from differenced data using linear filtering.

    Solves the ARMA equation $(1 - \phi)y_t = (1 + \theta)w_t$ for $w_t$.

    Parameters
    ----------
    diff_data : NDArray[np.float64]
        The fractionally differenced time series ($y_t$).
    ar_coeffs : list[float]
        The autoregressive coefficients ($\phi$).
    ma_coeffs : list[float]
        The moving average coefficients ($\theta$).

    Returns
    -------
    np.ndarray
        A 1D array of residuals.
    """

    b = [1.0] + [-i for i in ar_coeffs]
    a = [1.0] + list(ma_coeffs)

    residuals = signal.lfilter(b, a, diff_data)

    return np.asarray(residuals)

_figarch_score_function(params, squared_shocks, p, q)

Evaluates the Gaussian negative log-likelihood for dynamic volatility.

Extracts the FIGARCH parameters, constructs the conditional variance path (\(\sigma_t^2\)), and calculates the likelihood score used by the optimizer. Implements strict penalty boundaries to reject non-stationary GARCH parameter combinations or invalid variance arrays.

Solves: \(-\ln(L) = 0.5 * \sum[ ln(\sigma_t^2) + (\epsilon_t^2 / \sigma_t^2)]\)

Parameters

params : list[float] The optimization parameters containing omega, \(d_v\), followed by ARCH and GARCH coefficients. squared_shocks : NDArray[np.float64] The 1D array of squared residuals (\(\epsilon_t^2\)). p : int The ARCH lag order. q : int The GARCH lag order.

Returns

float The negative log-likelihood scalar, or a penalty value (\(1^{10}\)) if bounds are violated.

Source code in src\arfima\core.py 
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def _figarch_score_function(params: list[float], squared_shocks: NDArray[np.float64], p: int, q: int):
    """
    Evaluates the Gaussian negative log-likelihood for dynamic volatility.

    Extracts the FIGARCH parameters, constructs the conditional variance path 
    ($\sigma_t^2$), and calculates the likelihood score used by the optimizer. 
    Implements strict penalty boundaries to reject non-stationary GARCH 
    parameter combinations or invalid variance arrays.

    Solves: $-\ln(L) = 0.5 * \sum[ ln(\sigma_t^2) + (\epsilon_t^2 / \sigma_t^2)]$

    Parameters
    ----------
    params : list[float]
        The optimization parameters containing omega, $d_v$, followed by ARCH and GARCH coefficients.
    squared_shocks : NDArray[np.float64]
        The 1D array of squared residuals ($\epsilon_t^2$).
    p : int
        The ARCH lag order.
    q : int
        The GARCH lag order.

    Returns
    -------
    float
        The negative log-likelihood scalar, or a penalty value ($1^{10}$) if bounds are violated.
    """
    omega = float(params[0])
    d_v = float(params[1])
    arch_coeffs = params[2 : p + 2]
    garch_coeffs = params[p + 2 :]

    n = len(squared_shocks)

    weights = _figarch_weights(d_v, arch_coeffs, garch_coeffs, n)

    beta_sum = np.sum(garch_coeffs)
    if beta_sum >= 1.0:
        return 1e10 

    omega_star = omega / (1.0 - beta_sum)

    cond_variance = _dynamic_variance(squared_shocks, omega_star, weights)

    log_likelihood = 0.5 * np.sum(np.log(cond_variance) + (squared_shocks / cond_variance))

    if np.isnan(log_likelihood) or np.isinf(log_likelihood):
        return 1e10

    return log_likelihood

_figarch_weights(d_v, arch_coeffs, garch_coeffs, n)

Computes the ARCH(infinity) representation weights for the FIGARCH process.

Expands the fractional differencing polynomial \((1-L)^d_v\) up to lag n and filters it through the ARCH and GARCH polynomials to derive the decay weights \(\lambda_i\) for the conditional variance equation: \(\lambda(L) = 1 - [\phi(L) / \beta(L)] * (1-L)^{d_v}\).

Parameters

\(d_v\) : float The fractional integration parameter for volatility. arch_coeffs : list[float] The autoregressive conditional heteroskedasticity (ARCH) coefficients. garch_coeffs : list[float] The generalized autoregressive conditional heteroskedasticity (GARCH) coefficients. n : int The length of the time series (the truncation point for the infinite expansion).

Returns

np.ndarray A 1D array of length n containing the ARCH(\(\infty\)) decay weights.

Source code in src\arfima\core.py 
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def _figarch_weights(d_v: float, arch_coeffs: list[float], garch_coeffs: list[float], n: int):
    r"""
    Computes the ARCH(infinity) representation weights for the FIGARCH process.

    Expands the fractional differencing polynomial $(1-L)^d_v$ up to lag n 
    and filters it through the ARCH and GARCH polynomials to derive the 
    decay weights $\lambda_i$ for the conditional variance equation:
    $\lambda(L) = 1 - [\phi(L) / \beta(L)] * (1-L)^{d_v}$.

    Parameters
    ----------
    $d_v$ : float
        The fractional integration parameter for volatility.
    arch_coeffs : list[float]
        The autoregressive conditional heteroskedasticity (ARCH) coefficients.
    garch_coeffs : list[float]
        The generalized autoregressive conditional heteroskedasticity (GARCH) coefficients.
    n : int
        The length of the time series (the truncation point for the infinite expansion).

    Returns
    -------
    np.ndarray
        A 1D array of length n containing the ARCH($\infty$) decay weights.
    """
    frac_weights = np.zeros(n)
    frac_weights[0] = 1.0
    for i in range(1, n):
        frac_weights[i] = ((i - 1 - d_v) / i) * frac_weights[i - 1]

    phi = [1.0] + [-phi_i for phi_i in arch_coeffs]
    beta = [1.0] + [-beta_i for beta_i in garch_coeffs]

    core_expansion = signal.lfilter(phi, beta, frac_weights)

    lambda_weights = -core_expansion
    lambda_weights[0] = 0.0  

    return lambda_weights

_frac_diff(time_series, d)

Applies the fractional differencing operator \((1-L)^d\) using Fast Fourier Transform (FFT).

Parameters

time_series : NDArray[np.float64] The 1D time series array to difference. d : float The fractional integration parameter. Pass \(-d\) for inverse differencing (integration).

Returns

np.ndarray A 1D array of the fractionally differenced time series.

Source code in src\arfima\core.py 
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def _frac_diff(time_series:NDArray[np.float64], d:float):
    """
    Applies the fractional differencing operator $(1-L)^d$ using Fast Fourier Transform (FFT).

    Parameters
    ----------
    time_series : NDArray[np.float64]
        The 1D time series array to difference.
    d : float
        The fractional integration parameter. Pass $-d$ for inverse differencing (integration).

    Returns
    -------
    np.ndarray
        A 1D array of the fractionally differenced time series.
    """
    w = [1.0]
    n = len(time_series)
    for i in range(1,n):
        w_i = ((i-1-d) / i)*w[-1]
        w.append(w_i)

    conv_result = signal.fftconvolve(time_series, w, mode="full")

    return conv_result[:n]

_gph_estimator(time_series, bandwidth_power=0.5)

Approximates d(fractional differencing parameter) using the Geweke and Porter Hudak(GPH) semiparametric estimator.

Parameters

time_series: NDArray[np.float64] | np.ndarray 1D time series array bandwidth_power: float (default = 0.5) determines the number of periodogram frequencies included in the regression. Ideally between 0.5 and 0.7

Returns

float returns the estimated fractional difference parameter d. d is clipped between -0.49 and 0.49.

Source code in src\arfima\core.py 
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def _gph_estimator(time_series:NDArray[np.float64] | np.ndarray, bandwidth_power:float = 0.5):
    """
    Approximates d(fractional differencing parameter) using the Geweke and Porter Hudak(GPH) semiparametric
    estimator.

    Parameters
    ----------
    time_series: NDArray[np.float64] | np.ndarray
        1D time series array
    bandwidth_power: float (default = 0.5)
        determines the number of periodogram frequencies included in the regression. Ideally between 0.5 and 0.7

    Returns
    -------
    float
        returns the estimated fractional difference parameter d.
        d is clipped between -0.49 and 0.49.
    """
    n = len(time_series)

    freqs, pgram = signal.periodogram(time_series)

    m = int(np.floor(n ** bandwidth_power))
    if m < 2: 
        return 0.1

    lam = freqs[1 : m+1]
    i_lam = pgram[1 : m+1]

    x = 2 * np.log(2 * np.sin(lam / 2))
    y = np.log(i_lam)

    slope, _ = np.polyfit(x, y, 1)
    d_estimate = -slope

    return float(np.clip(d_estimate, -0.49, 0.49))

_score_function(params, time_series, p)

Calculates negative log-likelihood for a given set of ARFIMA parameters.

Parameters

params : list[float] The optimization parameters containing \(d\), followed by AR and MA coefficients. time_series : NDArray[np.float64] The 1D input time series data. p : int The autoregressive (AR) lag order. Returns

float The negative log-likelihood value, or scalar (\(1^{10}\)) for invalid variances.

Source code in src\arfima\core.py 
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def _score_function(params:list[float], time_series:NDArray[np.float64], p:int):
    """
    Calculates negative log-likelihood for a given set of ARFIMA parameters.

    Parameters
    ----------
    params : list[float]
        The optimization parameters containing $d$, followed by AR and MA coefficients.
    time_series : NDArray[np.float64]
        The 1D input time series data.
    p : int
        The autoregressive (AR) lag order.
    Returns
    -------
    float
        The negative log-likelihood value, or scalar ($1^{10}$) for invalid variances.
    """
    n = len(time_series)

    d_guess = float(params[0])
    ar_coeffs = params[1 : p + 1]
    ma_coeffs = params[p + 1 :]

    y_differenced = _frac_diff(time_series, d_guess)

    residuals = _fast_residuals(y_differenced, ar_coeffs, ma_coeffs)

    variance = np.sum(residuals ** 2) / n


    if variance <= 1e-10 or np.isnan(variance):
        return 1e10  

    neg_ll = (n / 2) * np.log(2 * np.pi) + (n / 2) * np.log(variance) + (n / 2)

    if np.isnan(neg_ll) or np.isinf(neg_ll):
        return 1e10

    return neg_ll