""" Statistical tools for time series analysis """ from __future__ import annotations from statsmodels.compat.numpy import lstsq from statsmodels.compat.pandas import deprecate_kwarg from statsmodels.compat.python import lzip from statsmodels.compat.scipy import _next_regular from typing import Literal, Union import warnings import numpy as np from numpy.linalg import LinAlgError import pandas as pd from scipy import stats from scipy.interpolate import interp1d from scipy.signal import correlate from statsmodels.regression.linear_model import OLS, yule_walker from statsmodels.tools.sm_exceptions import ( CollinearityWarning, InfeasibleTestError, InterpolationWarning, MissingDataError, ValueWarning, ) from statsmodels.tools.tools import Bunch, add_constant from statsmodels.tools.validation import ( array_like, bool_like, dict_like, float_like, int_like, string_like, ) from statsmodels.tsa._bds import bds from statsmodels.tsa._innovations import innovations_algo, innovations_filter from statsmodels.tsa.adfvalues import mackinnoncrit, mackinnonp from statsmodels.tsa.tsatools import add_trend, lagmat, lagmat2ds ArrayLike1D = Union[np.ndarray, pd.Series, list[float]] __all__ = [ "acovf", "acf", "pacf", "pacf_yw", "pacf_ols", "ccovf", "ccf", "q_stat", "coint", "arma_order_select_ic", "adfuller", "kpss", "bds", "pacf_burg", "innovations_algo", "innovations_filter", "levinson_durbin_pacf", "levinson_durbin", "zivot_andrews", "range_unit_root_test", ] SQRTEPS = np.sqrt(np.finfo(np.double).eps) def _autolag( mod, endog, exog, startlag, maxlag, method, modargs=(), fitargs=(), regresults=False, ): """ Returns the results for the lag length that maximizes the info criterion. Parameters ---------- mod : Model class Model estimator class endog : array_like nobs array containing endogenous variable exog : array_like nobs by (startlag + maxlag) array containing lags and possibly other variables startlag : int The first zero-indexed column to hold a lag. See Notes. maxlag : int The highest lag order for lag length selection. method : {"aic", "bic", "t-stat"} aic - Akaike Information Criterion bic - Bayes Information Criterion t-stat - Based on last lag modargs : tuple, optional args to pass to model. See notes. fitargs : tuple, optional args to pass to fit. See notes. regresults : bool, optional Flag indicating to return optional return results Returns ------- icbest : float Best information criteria. bestlag : int The lag length that maximizes the information criterion. results : dict, optional Dictionary containing all estimation results Notes ----- Does estimation like mod(endog, exog[:,:i], *modargs).fit(*fitargs) where i goes from lagstart to lagstart+maxlag+1. Therefore, lags are assumed to be in contiguous columns from low to high lag length with the highest lag in the last column. """ # TODO: can tcol be replaced by maxlag + 2? # TODO: This could be changed to laggedRHS and exog keyword arguments if # this will be more general. results = {} method = method.lower() for lag in range(startlag, startlag + maxlag + 1): mod_instance = mod(endog, exog[:, :lag], *modargs) results[lag] = mod_instance.fit() if method == "aic": icbest, bestlag = min((v.aic, k) for k, v in results.items()) elif method == "bic": icbest, bestlag = min((v.bic, k) for k, v in results.items()) elif method == "t-stat": # stop = stats.norm.ppf(.95) stop = 1.6448536269514722 # Default values to ensure that always set bestlag = startlag + maxlag icbest = 0.0 for lag in range(startlag + maxlag, startlag - 1, -1): icbest = np.abs(results[lag].tvalues[-1]) bestlag = lag if np.abs(icbest) >= stop: # Break for first lag with a significant t-stat break else: raise ValueError(f"Information Criterion {method} not understood.") if not regresults: return icbest, bestlag else: return icbest, bestlag, results # this needs to be converted to a class like HetGoldfeldQuandt, # 3 different returns are a mess # See: # Ng and Perron(2001), Lag length selection and the construction of unit root # tests with good size and power, Econometrica, Vol 69 (6) pp 1519-1554 # TODO: include drift keyword, only valid with regression == "c" # just changes the distribution of the test statistic to a t distribution # TODO: autolag is untested def adfuller( x, maxlag: int | None = None, regression="c", autolag="AIC", store=False, regresults=False, ): """ Augmented Dickey-Fuller unit root test. The Augmented Dickey-Fuller test can be used to test for a unit root in a univariate process in the presence of serial correlation. Parameters ---------- x : array_like, 1d The data series to test. maxlag : {None, int} Maximum lag which is included in test, default value of 12*(nobs/100)^{1/4} is used when ``None``. regression : {"c","ct","ctt","n"} Constant and trend order to include in regression. * "c" : constant only (default). * "ct" : constant and trend. * "ctt" : constant, and linear and quadratic trend. * "n" : no constant, no trend. autolag : {"AIC", "BIC", "t-stat", None} Method to use when automatically determining the lag length among the values 0, 1, ..., maxlag. * If "AIC" (default) or "BIC", then the number of lags is chosen to minimize the corresponding information criterion. * "t-stat" based choice of maxlag. Starts with maxlag and drops a lag until the t-statistic on the last lag length is significant using a 5%-sized test. * If None, then the number of included lags is set to maxlag. store : bool If True, then a result instance is returned additionally to the adf statistic. Default is False. regresults : bool, optional If True, the full regression results are returned. Default is False. Returns ------- adf : float The test statistic. pvalue : float MacKinnon's approximate p-value based on MacKinnon (1994, 2010). usedlag : int The number of lags used. nobs : int The number of observations used for the ADF regression and calculation of the critical values. critical values : dict Critical values for the test statistic at the 1 %, 5 %, and 10 % levels. Based on MacKinnon (2010). icbest : float The maximized information criterion if autolag is not None. resstore : ResultStore, optional A dummy class with results attached as attributes. Notes ----- The null hypothesis of the Augmented Dickey-Fuller is that there is a unit root, with the alternative that there is no unit root. If the pvalue is above a critical size, then we cannot reject that there is a unit root. The p-values are obtained through regression surface approximation from MacKinnon 1994, but using the updated 2010 tables. If the p-value is close to significant, then the critical values should be used to judge whether to reject the null. The autolag option and maxlag for it are described in Greene. See the notebook `Stationarity and detrending (ADF/KPSS) <../examples/notebooks/generated/stationarity_detrending_adf_kpss.html>`__ for an overview. References ---------- .. [1] W. Green. "Econometric Analysis," 5th ed., Pearson, 2003. .. [2] Hamilton, J.D. "Time Series Analysis". Princeton, 1994. .. [3] MacKinnon, J.G. 1994. "Approximate asymptotic distribution functions for unit-root and cointegration tests. `Journal of Business and Economic Statistics` 12, 167-76. .. [4] MacKinnon, J.G. 2010. "Critical Values for Cointegration Tests." Queen"s University, Dept of Economics, Working Papers. Available at http://ideas.repec.org/p/qed/wpaper/1227.html """ x = array_like(x, "x") maxlag = int_like(maxlag, "maxlag", optional=True) regression = string_like( regression, "regression", options=("c", "ct", "ctt", "n") ) autolag = string_like( autolag, "autolag", optional=True, options=("aic", "bic", "t-stat") ) store = bool_like(store, "store") regresults = bool_like(regresults, "regresults") if x.max() == x.min(): raise ValueError("Invalid input, x is constant") if regresults: store = True trenddict = {None: "n", 0: "c", 1: "ct", 2: "ctt"} if regression is None or isinstance(regression, int): regression = trenddict[regression] regression = regression.lower() nobs = x.shape[0] ntrend = len(regression) if regression != "n" else 0 if maxlag is None: # from Greene referencing Schwert 1989 maxlag = int(np.ceil(12.0 * np.power(nobs / 100.0, 1 / 4.0))) # -1 for the diff maxlag = min(nobs // 2 - ntrend - 1, maxlag) if maxlag < 0: raise ValueError( "sample size is too short to use selected " "regression component" ) elif maxlag > nobs // 2 - ntrend - 1: raise ValueError( "maxlag must be less than (nobs/2 - 1 - ntrend) " "where n trend is the number of included " "deterministic regressors" ) xdiff = np.diff(x) xdall = lagmat(xdiff[:, None], maxlag, trim="both", original="in") nobs = xdall.shape[0] xdall[:, 0] = x[-nobs - 1 : -1] # replace 0 xdiff with level of x xdshort = xdiff[-nobs:] if store: from statsmodels.stats.diagnostic import ResultsStore resstore = ResultsStore() if autolag: if regression != "n": fullRHS = add_trend(xdall, regression, prepend=True) else: fullRHS = xdall startlag = fullRHS.shape[1] - xdall.shape[1] + 1 # 1 for level # search for lag length with smallest information criteria # Note: use the same number of observations to have comparable IC # aic and bic: smaller is better if not regresults: icbest, bestlag = _autolag( OLS, xdshort, fullRHS, startlag, maxlag, autolag ) else: icbest, bestlag, alres = _autolag( OLS, xdshort, fullRHS, startlag, maxlag, autolag, regresults=regresults, ) resstore.autolag_results = alres bestlag -= startlag # convert to lag not column index # rerun ols with best autolag xdall = lagmat(xdiff[:, None], bestlag, trim="both", original="in") nobs = xdall.shape[0] xdall[:, 0] = x[-nobs - 1 : -1] # replace 0 xdiff with level of x xdshort = xdiff[-nobs:] usedlag = bestlag else: usedlag = maxlag icbest = None if regression != "n": resols = OLS( xdshort, add_trend(xdall[:, : usedlag + 1], regression) ).fit() else: resols = OLS(xdshort, xdall[:, : usedlag + 1]).fit() adfstat = resols.tvalues[0] # adfstat = (resols.params[0]-1.0)/resols.bse[0] # the "asymptotically correct" z statistic is obtained as # nobs/(1-np.sum(resols.params[1:-(trendorder+1)])) (resols.params[0] - 1) # I think this is the statistic that is used for series that are integrated # for orders higher than I(1), ie., not ADF but cointegration tests. # Get approx p-value and critical values pvalue = mackinnonp(adfstat, regression=regression, N=1) critvalues = mackinnoncrit(N=1, regression=regression, nobs=nobs) critvalues = { "1%": critvalues[0], "5%": critvalues[1], "10%": critvalues[2], } if store: resstore.resols = resols resstore.maxlag = maxlag resstore.usedlag = usedlag resstore.adfstat = adfstat resstore.critvalues = critvalues resstore.nobs = nobs resstore.H0 = ( "The coefficient on the lagged level equals 1 - " "unit root" ) resstore.HA = "The coefficient on the lagged level < 1 - stationary" resstore.icbest = icbest resstore._str = "Augmented Dickey-Fuller Test Results" return adfstat, pvalue, critvalues, resstore else: if not autolag: return adfstat, pvalue, usedlag, nobs, critvalues else: return adfstat, pvalue, usedlag, nobs, critvalues, icbest @deprecate_kwarg("unbiased", "adjusted") def acovf(x, adjusted=False, demean=True, fft=True, missing="none", nlag=None): """ Estimate autocovariances. Parameters ---------- x : array_like Time series data. Must be 1d. adjusted : bool, default False If True, then denominators is n-k, otherwise n. demean : bool, default True If True, then subtract the mean x from each element of x. fft : bool, default True If True, use FFT convolution. This method should be preferred for long time series. missing : str, default "none" A string in ["none", "raise", "conservative", "drop"] specifying how the NaNs are to be treated. "none" performs no checks. "raise" raises an exception if NaN values are found. "drop" removes the missing observations and then estimates the autocovariances treating the non-missing as contiguous. "conservative" computes the autocovariance using nan-ops so that nans are removed when computing the mean and cross-products that are used to estimate the autocovariance. When using "conservative", n is set to the number of non-missing observations. nlag : {int, None}, default None Limit the number of autocovariances returned. Size of returned array is nlag + 1. Setting nlag when fft is False uses a simple, direct estimator of the autocovariances that only computes the first nlag + 1 values. This can be much faster when the time series is long and only a small number of autocovariances are needed. Returns ------- ndarray The estimated autocovariances. References ---------- .. [1] Parzen, E., 1963. On spectral analysis with missing observations and amplitude modulation. Sankhya: The Indian Journal of Statistics, Series A, pp.383-392. """ adjusted = bool_like(adjusted, "adjusted") demean = bool_like(demean, "demean") fft = bool_like(fft, "fft", optional=False) missing = string_like( missing, "missing", options=("none", "raise", "conservative", "drop") ) nlag = int_like(nlag, "nlag", optional=True) x = array_like(x, "x", ndim=1) missing = missing.lower() if missing == "none": deal_with_masked = False else: deal_with_masked = has_missing(x) if deal_with_masked: if missing == "raise": raise MissingDataError("NaNs were encountered in the data") notmask_bool = ~np.isnan(x) # bool if missing == "conservative": # Must copy for thread safety x = x.copy() x[~notmask_bool] = 0 else: # "drop" x = x[notmask_bool] # copies non-missing notmask_int = notmask_bool.astype(int) # int if demean and deal_with_masked: # whether "drop" or "conservative": xo = x - x.sum() / notmask_int.sum() if missing == "conservative": xo[~notmask_bool] = 0 elif demean: xo = x - x.mean() else: xo = x n = len(x) lag_len = nlag if nlag is None: lag_len = n - 1 elif nlag > n - 1: raise ValueError("nlag must be smaller than nobs - 1") if not fft and nlag is not None: acov = np.empty(lag_len + 1) acov[0] = xo.dot(xo) for i in range(lag_len): acov[i + 1] = xo[i + 1 :].dot(xo[: -(i + 1)]) if not deal_with_masked or missing == "drop": if adjusted: acov /= n - np.arange(lag_len + 1) else: acov /= n else: if adjusted: divisor = np.empty(lag_len + 1, dtype=np.int64) divisor[0] = notmask_int.sum() for i in range(lag_len): divisor[i + 1] = notmask_int[i + 1 :].dot( notmask_int[: -(i + 1)] ) divisor[divisor == 0] = 1 acov /= divisor else: # biased, missing data but npt "drop" acov /= notmask_int.sum() return acov if adjusted and deal_with_masked and missing == "conservative": d = np.correlate(notmask_int, notmask_int, "full") d[d == 0] = 1 elif adjusted: xi = np.arange(1, n + 1) d = np.hstack((xi, xi[:-1][::-1])) elif deal_with_masked: # biased and NaNs given and ("drop" or "conservative") d = notmask_int.sum() * np.ones(2 * n - 1) else: # biased and no NaNs or missing=="none" d = n * np.ones(2 * n - 1) if fft: nobs = len(xo) n = _next_regular(2 * nobs + 1) Frf = np.fft.fft(xo, n=n) acov = np.fft.ifft(Frf * np.conjugate(Frf))[:nobs] / d[nobs - 1 :] acov = acov.real else: acov = np.correlate(xo, xo, "full")[n - 1 :] / d[n - 1 :] if nlag is not None: # Copy to allow gc of full array rather than view return acov[: lag_len + 1].copy() return acov def q_stat(x, nobs): """ Compute Ljung-Box Q Statistic. Parameters ---------- x : array_like Array of autocorrelation coefficients. Can be obtained from acf. nobs : int, optional Number of observations in the entire sample (ie., not just the length of the autocorrelation function results. Returns ------- q-stat : ndarray Ljung-Box Q-statistic for autocorrelation parameters. p-value : ndarray P-value of the Q statistic. See Also -------- statsmodels.stats.diagnostic.acorr_ljungbox Ljung-Box Q-test for autocorrelation in time series based on a time series rather than the estimated autocorrelation function. Notes ----- Designed to be used with acf. """ x = array_like(x, "x") nobs = int_like(nobs, "nobs") ret = ( nobs * (nobs + 2) * np.cumsum((1.0 / (nobs - np.arange(1, len(x) + 1))) * x ** 2) ) chi2 = stats.chi2.sf(ret, np.arange(1, len(x) + 1)) return ret, chi2 # NOTE: Changed unbiased to False # see for example # http://www.itl.nist.gov/div898/handbook/eda/section3/autocopl.htm def acf( x, adjusted=False, nlags=None, qstat=False, fft=True, alpha=None, bartlett_confint=True, missing="none", ): """ Calculate the autocorrelation function. Parameters ---------- x : array_like The time series data. adjusted : bool, default False If True, then denominators for autocovariance are n-k, otherwise n. nlags : int, optional Number of lags to return autocorrelation for. If not provided, uses min(10 * np.log10(nobs), nobs - 1). The returned value includes lag 0 (ie., 1) so size of the acf vector is (nlags + 1,). qstat : bool, default False If True, returns the Ljung-Box q statistic for each autocorrelation coefficient. See q_stat for more information. fft : bool, default True If True, computes the ACF via FFT. alpha : scalar, default None If a number is given, the confidence intervals for the given level are returned. For instance if alpha=.05, 95 % confidence intervals are returned where the standard deviation is computed according to Bartlett"s formula. bartlett_confint : bool, default True Confidence intervals for ACF values are generally placed at 2 standard errors around r_k. The formula used for standard error depends upon the situation. If the autocorrelations are being used to test for randomness of residuals as part of the ARIMA routine, the standard errors are determined assuming the residuals are white noise. The approximate formula for any lag is that standard error of each r_k = 1/sqrt(N). See section 9.4 of [2] for more details on the 1/sqrt(N) result. For more elementary discussion, see section 5.3.2 in [3]. For the ACF of raw data, the standard error at a lag k is found as if the right model was an MA(k-1). This allows the possible interpretation that if all autocorrelations past a certain lag are within the limits, the model might be an MA of order defined by the last significant autocorrelation. In this case, a moving average model is assumed for the data and the standard errors for the confidence intervals should be generated using Bartlett's formula. For more details on Bartlett formula result, see section 7.2 in [2]. missing : str, default "none" A string in ["none", "raise", "conservative", "drop"] specifying how the NaNs are to be treated. "none" performs no checks. "raise" raises an exception if NaN values are found. "drop" removes the missing observations and then estimates the autocovariances treating the non-missing as contiguous. "conservative" computes the autocovariance using nan-ops so that nans are removed when computing the mean and cross-products that are used to estimate the autocovariance. When using "conservative", n is set to the number of non-missing observations. Returns ------- acf : ndarray The autocorrelation function for lags 0, 1, ..., nlags. Shape (nlags+1,). confint : ndarray, optional Confidence intervals for the ACF at lags 0, 1, ..., nlags. Shape (nlags + 1, 2). Returned if alpha is not None. The confidence intervals are centered on the estimated ACF values. This behavior differs from plot_acf which centers the confidence intervals on 0. qstat : ndarray, optional The Ljung-Box Q-Statistic for lags 1, 2, ..., nlags (excludes lag zero). Returned if q_stat is True. pvalues : ndarray, optional The p-values associated with the Q-statistics for lags 1, 2, ..., nlags (excludes lag zero). Returned if q_stat is True. Notes ----- The acf at lag 0 (ie., 1) is returned. For very long time series it is recommended to use fft convolution instead. When fft is False uses a simple, direct estimator of the autocovariances that only computes the first nlag + 1 values. This can be much faster when the time series is long and only a small number of autocovariances are needed. If adjusted is true, the denominator for the autocovariance is adjusted for the loss of data. References ---------- .. [1] Parzen, E., 1963. On spectral analysis with missing observations and amplitude modulation. Sankhya: The Indian Journal of Statistics, Series A, pp.383-392. .. [2] Brockwell and Davis, 1987. Time Series Theory and Methods .. [3] Brockwell and Davis, 2010. Introduction to Time Series and Forecasting, 2nd edition. See Also -------- statsmodels.tsa.stattools.acf Estimate the autocorrelation function. statsmodels.graphics.tsaplots.plot_acf Plot autocorrelations and confidence intervals. """ adjusted = bool_like(adjusted, "adjusted") nlags = int_like(nlags, "nlags", optional=True) qstat = bool_like(qstat, "qstat") fft = bool_like(fft, "fft", optional=False) alpha = float_like(alpha, "alpha", optional=True) missing = string_like( missing, "missing", options=("none", "raise", "conservative", "drop") ) x = array_like(x, "x") # TODO: should this shrink for missing="drop" and NaNs in x? nobs = x.shape[0] if nlags is None: nlags = min(int(10 * np.log10(nobs)), nobs - 1) avf = acovf(x, adjusted=adjusted, demean=True, fft=fft, missing=missing) acf = avf[: nlags + 1] / avf[0] if not (qstat or alpha): return acf _alpha = alpha if alpha is not None else 0.05 if bartlett_confint: varacf = np.ones_like(acf) / nobs varacf[0] = 0 varacf[1] = 1.0 / nobs varacf[2:] *= 1 + 2 * np.cumsum(acf[1:-1] ** 2) else: varacf = 1.0 / len(x) interval = stats.norm.ppf(1 - _alpha / 2.0) * np.sqrt(varacf) confint = np.array(lzip(acf - interval, acf + interval)) if not qstat: return acf, confint qstat, pvalue = q_stat(acf[1:], nobs=nobs) # drop lag 0 if alpha is not None: return acf, confint, qstat, pvalue else: return acf, qstat, pvalue def pacf_yw( x: ArrayLike1D, nlags: int | None = None, method: Literal["adjusted", "mle"] = "adjusted", ) -> np.ndarray: """ Partial autocorrelation estimated with non-recursive yule_walker. Parameters ---------- x : array_like The observations of time series for which pacf is calculated. nlags : int, optional Number of lags to return autocorrelation for. If not provided, uses min(10 * np.log10(nobs), nobs - 1). method : {"adjusted", "mle"}, default "adjusted" The method for the autocovariance calculations in yule walker. Returns ------- ndarray The partial autocorrelations, maxlag+1 elements. See Also -------- statsmodels.tsa.stattools.pacf Partial autocorrelation estimation. statsmodels.tsa.stattools.pacf_ols Partial autocorrelation estimation using OLS. statsmodels.tsa.stattools.pacf_burg Partial autocorrelation estimation using Burg"s method. Notes ----- This solves yule_walker for each desired lag and contains currently duplicate calculations. """ x = array_like(x, "x") nlags = int_like(nlags, "nlags", optional=True) nobs = x.shape[0] if nlags is None: nlags = max(min(int(10 * np.log10(nobs)), nobs - 1), 1) method = string_like(method, "method", options=("adjusted", "mle")) pacf = [1.0] with warnings.catch_warnings(): warnings.simplefilter("once", ValueWarning) for k in range(1, nlags + 1): pacf.append(yule_walker(x, k, method=method)[0][-1]) return np.array(pacf) def pacf_burg( x: ArrayLike1D, nlags: int | None = None, demean: bool = True ) -> tuple[np.ndarray, np.ndarray]: """ Calculate Burg"s partial autocorrelation estimator. Parameters ---------- x : array_like Observations of time series for which pacf is calculated. nlags : int, optional Number of lags to return autocorrelation for. If not provided, uses min(10 * np.log10(nobs), nobs - 1). demean : bool, optional Flag indicating to demean that data. Set to False if x has been previously demeaned. Returns ------- pacf : ndarray Partial autocorrelations for lags 0, 1, ..., nlag. sigma2 : ndarray Residual variance estimates where the value in position m is the residual variance in an AR model that includes m lags. See Also -------- statsmodels.tsa.stattools.pacf Partial autocorrelation estimation. statsmodels.tsa.stattools.pacf_yw Partial autocorrelation estimation using Yule-Walker. statsmodels.tsa.stattools.pacf_ols Partial autocorrelation estimation using OLS. References ---------- .. [1] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series and forecasting. Springer. """ x = array_like(x, "x") if demean: x = x - x.mean() nobs = x.shape[0] p = nlags if nlags is not None else min(int(10 * np.log10(nobs)), nobs - 1) p = max(p, 1) if p > nobs - 1: raise ValueError("nlags must be smaller than nobs - 1") d = np.zeros(p + 1) d[0] = 2 * x.dot(x) pacf = np.zeros(p + 1) u = x[::-1].copy() v = x[::-1].copy() d[1] = u[:-1].dot(u[:-1]) + v[1:].dot(v[1:]) pacf[1] = 2 / d[1] * v[1:].dot(u[:-1]) last_u = np.empty_like(u) last_v = np.empty_like(v) for i in range(1, p): last_u[:] = u last_v[:] = v u[1:] = last_u[:-1] - pacf[i] * last_v[1:] v[1:] = last_v[1:] - pacf[i] * last_u[:-1] d[i + 1] = (1 - pacf[i] ** 2) * d[i] - v[i] ** 2 - u[-1] ** 2 pacf[i + 1] = 2 / d[i + 1] * v[i + 1 :].dot(u[i:-1]) sigma2 = (1 - pacf**2) * d / (2.0 * (nobs - np.arange(0, p + 1))) pacf[0] = 1 # Insert the 0 lag partial autocorrel return pacf, sigma2 @deprecate_kwarg("unbiased", "adjusted") def pacf_ols( x: ArrayLike1D, nlags: int | None = None, efficient: bool = True, adjusted: bool = False, ) -> np.ndarray: """ Calculate partial autocorrelations via OLS. Parameters ---------- x : array_like Observations of time series for which pacf is calculated. nlags : int, optional Number of lags to return autocorrelation for. If not provided, uses min(10 * np.log10(nobs), nobs - 1). efficient : bool, optional If true, uses the maximum number of available observations to compute each partial autocorrelation. If not, uses the same number of observations to compute all pacf values. adjusted : bool, optional Adjust each partial autocorrelation by n / (n - lag). Returns ------- ndarray The partial autocorrelations, (maxlag,) array corresponding to lags 0, 1, ..., maxlag. See Also -------- statsmodels.tsa.stattools.pacf Partial autocorrelation estimation. statsmodels.tsa.stattools.pacf_yw Partial autocorrelation estimation using Yule-Walker. statsmodels.tsa.stattools.pacf_burg Partial autocorrelation estimation using Burg"s method. Notes ----- This solves a separate OLS estimation for each desired lag using method in [1]_. Setting efficient to True has two effects. First, it uses `nobs - lag` observations of estimate each pacf. Second, it re-estimates the mean in each regression. If efficient is False, then the data are first demeaned, and then `nobs - maxlag` observations are used to estimate each partial autocorrelation. The inefficient estimator appears to have better finite sample properties. This option should only be used in time series that are covariance stationary. OLS estimation of the pacf does not guarantee that all pacf values are between -1 and 1. References ---------- .. [1] Box, G. E., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time series analysis: forecasting and control. John Wiley & Sons, p. 66 """ x = array_like(x, "x") nlags = int_like(nlags, "nlags", optional=True) efficient = bool_like(efficient, "efficient") adjusted = bool_like(adjusted, "adjusted") nobs = x.shape[0] if nlags is None: nlags = max(min(int(10 * np.log10(nobs)), nobs // 2), 1) if nlags > nobs//2: raise ValueError(f"nlags must be smaller than nobs // 2 ({nobs//2})") pacf = np.empty(nlags + 1) pacf[0] = 1.0 if efficient: xlags, x0 = lagmat(x, nlags, original="sep") xlags = add_constant(xlags) for k in range(1, nlags + 1): params = lstsq(xlags[k:, : k + 1], x0[k:], rcond=None)[0] pacf[k] = np.squeeze(params[-1]) else: x = x - np.mean(x) # Create a single set of lags for multivariate OLS xlags, x0 = lagmat(x, nlags, original="sep", trim="both") for k in range(1, nlags + 1): params = lstsq(xlags[:, :k], x0, rcond=None)[0] # Last coefficient corresponds to PACF value (see [1]) pacf[k] = np.squeeze(params[-1]) if adjusted: pacf *= nobs / (nobs - np.arange(nlags + 1)) return pacf def pacf( x: ArrayLike1D, nlags: int | None = None, method: Literal[ "yw", "ywadjusted", "ols", "ols-inefficient", "ols-adjusted", "ywm", "ywmle", "ld", "ldadjusted", "ldb", "ldbiased", "burg", ] = "ywadjusted", alpha: float | None = None, ) -> np.ndarray | tuple[np.ndarray, np.ndarray]: """ Partial autocorrelation estimate. Parameters ---------- x : array_like Observations of time series for which pacf is calculated. nlags : int, optional Number of lags to return autocorrelation for. If not provided, uses min(10 * np.log10(nobs), nobs // 2 - 1). The returned value includes lag 0 (ie., 1) so size of the pacf vector is (nlags + 1,). method : str, default "ywunbiased" Specifies which method for the calculations to use. - "yw" or "ywadjusted" : Yule-Walker with sample-size adjustment in denominator for acovf. Default. - "ywm" or "ywmle" : Yule-Walker without adjustment. - "ols" : regression of time series on lags of it and on constant. - "ols-inefficient" : regression of time series on lags using a single common sample to estimate all pacf coefficients. - "ols-adjusted" : regression of time series on lags with a bias adjustment. - "ld" or "ldadjusted" : Levinson-Durbin recursion with bias correction. - "ldb" or "ldbiased" : Levinson-Durbin recursion without bias correction. - "burg" : Burg"s partial autocorrelation estimator. alpha : float, optional If a number is given, the confidence intervals for the given level are returned. For instance if alpha=.05, 95 % confidence intervals are returned where the standard deviation is computed according to 1/sqrt(len(x)). Returns ------- pacf : ndarray The partial autocorrelations for lags 0, 1, ..., nlags. Shape (nlags+1,). confint : ndarray, optional Confidence intervals for the PACF at lags 0, 1, ..., nlags. Shape (nlags + 1, 2). Returned if alpha is not None. See Also -------- statsmodels.tsa.stattools.acf Estimate the autocorrelation function. statsmodels.tsa.stattools.pacf Partial autocorrelation estimation. statsmodels.tsa.stattools.pacf_yw Partial autocorrelation estimation using Yule-Walker. statsmodels.tsa.stattools.pacf_ols Partial autocorrelation estimation using OLS. statsmodels.tsa.stattools.pacf_burg Partial autocorrelation estimation using Burg's method. statsmodels.graphics.tsaplots.plot_pacf Plot partial autocorrelations and confidence intervals. Notes ----- Based on simulation evidence across a range of low-order ARMA models, the best methods based on root MSE are Yule-Walker (MLW), Levinson-Durbin (MLE) and Burg, respectively. The estimators with the lowest bias included included these three in addition to OLS and OLS-adjusted. Yule-Walker (adjusted) and Levinson-Durbin (adjusted) performed consistently worse than the other options. """ nlags = int_like(nlags, "nlags", optional=True) methods = ( "ols", "ols-inefficient", "ols-adjusted", "yw", "ywa", "ld", "ywadjusted", "yw_adjusted", "ywm", "ywmle", "yw_mle", "lda", "ldadjusted", "ld_adjusted", "ldb", "ldbiased", "ld_biased", "burg", ) x = array_like(x, "x", maxdim=2) method = string_like(method, "method", options=methods) alpha = float_like(alpha, "alpha", optional=True) nobs = x.shape[0] if nlags is None: nlags = min(int(10 * np.log10(nobs)), nobs // 2 - 1) nlags = max(nlags, 1) if nlags > x.shape[0] // 2: raise ValueError( "Can only compute partial correlations for lags up to 50% of the " f"sample size. The requested nlags {nlags} must be < " f"{x.shape[0] // 2}." ) if method in ("ols", "ols-inefficient", "ols-adjusted"): efficient = "inefficient" not in method adjusted = "adjusted" in method ret = pacf_ols(x, nlags=nlags, efficient=efficient, adjusted=adjusted) elif method in ("yw", "ywa", "ywadjusted", "yw_adjusted"): ret = pacf_yw(x, nlags=nlags, method="adjusted") elif method in ("ywm", "ywmle", "yw_mle"): ret = pacf_yw(x, nlags=nlags, method="mle") elif method in ("ld", "lda", "ldadjusted", "ld_adjusted"): acv = acovf(x, adjusted=True, fft=False) ld_ = levinson_durbin(acv, nlags=nlags, isacov=True) ret = ld_[2] elif method == "burg": ret, _ = pacf_burg(x, nlags=nlags, demean=True) # inconsistent naming with ywmle else: # method in ("ldb", "ldbiased", "ld_biased") acv = acovf(x, adjusted=False, fft=False) ld_ = levinson_durbin(acv, nlags=nlags, isacov=True) ret = ld_[2] if alpha is not None: varacf = 1.0 / len(x) # for all lags >=1 interval = stats.norm.ppf(1.0 - alpha / 2.0) * np.sqrt(varacf) confint = np.array(lzip(ret - interval, ret + interval)) confint[0] = ret[0] # fix confidence interval for lag 0 to varpacf=0 return ret, confint else: return ret @deprecate_kwarg("unbiased", "adjusted") def ccovf(x, y, adjusted=True, demean=True, fft=True): """ Calculate the cross-covariance between two series. Parameters ---------- x, y : array_like The time series data to use in the calculation. adjusted : bool, optional If True, then denominators for cross-covariance are n-k, otherwise n. demean : bool, optional Flag indicating whether to demean x and y. fft : bool, default True If True, use FFT convolution. This method should be preferred for long time series. Returns ------- ndarray The estimated cross-covariance function: the element at index k is the covariance between {x[k], x[k+1], ..., x[n]} and {y[0], y[1], ..., y[m-k]}, where n and m are the lengths of x and y, respectively. """ x = array_like(x, "x") y = array_like(y, "y") adjusted = bool_like(adjusted, "adjusted") demean = bool_like(demean, "demean") fft = bool_like(fft, "fft", optional=False) n = len(x) if demean: xo = x - x.mean() yo = y - y.mean() else: xo = x yo = y if adjusted: d = np.arange(n, 0, -1) else: d = n method = "fft" if fft else "direct" return correlate(xo, yo, "full", method=method)[n - 1:] / d @deprecate_kwarg("unbiased", "adjusted") def ccf(x, y, adjusted=True, fft=True, *, nlags=None, alpha=None): """ The cross-correlation function. Parameters ---------- x, y : array_like The time series data to use in the calculation. adjusted : bool If True, then denominators for cross-correlation are n-k, otherwise n. fft : bool, default True If True, use FFT convolution. This method should be preferred for long time series. nlags : int, optional Number of lags to return cross-correlations for. If not provided, the number of lags equals len(x). alpha : float, optional If a number is given, the confidence intervals for the given level are returned. For instance if alpha=.05, 95 % confidence intervals are returned where the standard deviation is computed according to 1/sqrt(len(x)). Returns ------- ndarray The cross-correlation function of x and y: the element at index k is the correlation between {x[k], x[k+1], ..., x[n]} and {y[0], y[1], ..., y[m-k]}, where n and m are the lengths of x and y, respectively. confint : ndarray, optional Confidence intervals for the CCF at lags 0, 1, ..., nlags-1 using the level given by alpha and the standard deviation calculated as 1/sqrt(len(x)) [1]. Shape (nlags, 2). Returned if alpha is not None. Notes ----- If adjusted is True, the denominator for the cross-correlation is adjusted. References ---------- .. [1] Brockwell and Davis, 2016. Introduction to Time Series and Forecasting, 3rd edition, p. 242. """ x = array_like(x, "x") y = array_like(y, "y") adjusted = bool_like(adjusted, "adjusted") fft = bool_like(fft, "fft", optional=False) cvf = ccovf(x, y, adjusted=adjusted, demean=True, fft=fft) ret = cvf / (np.std(x) * np.std(y)) ret = ret[:nlags] if alpha is not None: interval = stats.norm.ppf(1.0 - alpha / 2.0) / np.sqrt(len(x)) confint = ret.reshape(-1, 1) + interval * np.array([-1, 1]) return ret, confint else: return ret # moved from sandbox.tsa.examples.try_ld_nitime, via nitime # TODO: check what to return, for testing and trying out returns everything def levinson_durbin(s, nlags=10, isacov=False): """ Levinson-Durbin recursion for autoregressive processes. Parameters ---------- s : array_like If isacov is False, then this is the time series. If iasacov is true then this is interpreted as autocovariance starting with lag 0. nlags : int, optional The largest lag to include in recursion or order of the autoregressive process. isacov : bool, optional Flag indicating whether the first argument, s, contains the autocovariances or the data series. Returns ------- sigma_v : float The estimate of the error variance. arcoefs : ndarray The estimate of the autoregressive coefficients for a model including nlags. pacf : ndarray The partial autocorrelation function. sigma : ndarray The entire sigma array from intermediate result, last value is sigma_v. phi : ndarray The entire phi array from intermediate result, last column contains autoregressive coefficients for AR(nlags). Notes ----- This function returns currently all results, but maybe we drop sigma and phi from the returns. If this function is called with the time series (isacov=False), then the sample autocovariance function is calculated with the default options (biased, no fft). """ s = array_like(s, "s") nlags = int_like(nlags, "nlags") isacov = bool_like(isacov, "isacov") order = nlags if isacov: sxx_m = s else: sxx_m = acovf(s, fft=False)[: order + 1] # not tested phi = np.zeros((order + 1, order + 1), "d") sig = np.zeros(order + 1) # initial points for the recursion phi[1, 1] = sxx_m[1] / sxx_m[0] sig[1] = sxx_m[0] - phi[1, 1] * sxx_m[1] for k in range(2, order + 1): phi[k, k] = ( sxx_m[k] - np.dot(phi[1:k, k - 1], sxx_m[1:k][::-1]) ) / sig[k - 1] for j in range(1, k): phi[j, k] = phi[j, k - 1] - phi[k, k] * phi[k - j, k - 1] sig[k] = sig[k - 1] * (1 - phi[k, k] ** 2) sigma_v = sig[-1] arcoefs = phi[1:, -1] pacf_ = np.diag(phi).copy() pacf_[0] = 1.0 return sigma_v, arcoefs, pacf_, sig, phi # return everything def levinson_durbin_pacf(pacf, nlags=None): """ Levinson-Durbin algorithm that returns the acf and ar coefficients. Parameters ---------- pacf : array_like Partial autocorrelation array for lags 0, 1, ... p. nlags : int, optional Number of lags in the AR model. If omitted, returns coefficients from an AR(p) and the first p autocorrelations. Returns ------- arcoefs : ndarray AR coefficients computed from the partial autocorrelations. acf : ndarray The acf computed from the partial autocorrelations. Array returned contains the autocorrelations corresponding to lags 0, 1, ..., p. References ---------- .. [1] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series and forecasting. Springer. """ pacf = array_like(pacf, "pacf") nlags = int_like(nlags, "nlags", optional=True) pacf = np.squeeze(np.asarray(pacf)) if pacf[0] != 1: raise ValueError( "The first entry of the pacf corresponds to lags 0 " "and so must be 1." ) pacf = pacf[1:] n = pacf.shape[0] if nlags is not None: if nlags > n: raise ValueError( "Must provide at least as many values from the " "pacf as the number of lags." ) pacf = pacf[:nlags] n = pacf.shape[0] acf = np.zeros(n + 1) acf[1] = pacf[0] nu = np.cumprod(1 - pacf ** 2) arcoefs = pacf.copy() for i in range(1, n): prev = arcoefs[: -(n - i)].copy() arcoefs[: -(n - i)] = prev - arcoefs[i] * prev[::-1] acf[i + 1] = arcoefs[i] * nu[i - 1] + prev.dot(acf[1 : -(n - i)][::-1]) acf[0] = 1 return arcoefs, acf def breakvar_heteroskedasticity_test( resid, subset_length=1 / 3, alternative="two-sided", use_f=True ): r""" Test for heteroskedasticity of residuals Tests whether the sum-of-squares in the first subset of the sample is significantly different than the sum-of-squares in the last subset of the sample. Analogous to a Goldfeld-Quandt test. The null hypothesis is of no heteroskedasticity. Parameters ---------- resid : array_like Residuals of a time series model. The shape is 1d (nobs,) or 2d (nobs, nvars). subset_length : {int, float} Length of the subsets to test (h in Notes below). If a float in 0 < subset_length < 1, it is interpreted as fraction. Default is 1/3. alternative : str, 'increasing', 'decreasing' or 'two-sided' This specifies the alternative for the p-value calculation. Default is two-sided. use_f : bool, optional Whether or not to compare against the asymptotic distribution (chi-squared) or the approximate small-sample distribution (F). Default is True (i.e. default is to compare against an F distribution). Returns ------- test_statistic : {float, ndarray} Test statistic(s) H(h). p_value : {float, ndarray} p-value(s) of test statistic(s). Notes ----- The null hypothesis is of no heteroskedasticity. That means different things depending on which alternative is selected: - Increasing: Null hypothesis is that the variance is not increasing throughout the sample; that the sum-of-squares in the later subsample is *not* greater than the sum-of-squares in the earlier subsample. - Decreasing: Null hypothesis is that the variance is not decreasing throughout the sample; that the sum-of-squares in the earlier subsample is *not* greater than the sum-of-squares in the later subsample. - Two-sided: Null hypothesis is that the variance is not changing throughout the sample. Both that the sum-of-squares in the earlier subsample is not greater than the sum-of-squares in the later subsample *and* that the sum-of-squares in the later subsample is not greater than the sum-of-squares in the earlier subsample. For :math:`h = [T/3]`, the test statistic is: .. math:: H(h) = \sum_{t=T-h+1}^T \tilde v_t^2 \Bigg / \sum_{t=1}^{h} \tilde v_t^2 This statistic can be tested against an :math:`F(h,h)` distribution. Alternatively, :math:`h H(h)` is asymptotically distributed according to :math:`\chi_h^2`; this second test can be applied by passing `use_f=False` as an argument. See section 5.4 of [1]_ for the above formula and discussion, as well as additional details. References ---------- .. [1] Harvey, Andrew C. 1990. *Forecasting, Structural Time Series* *Models and the Kalman Filter.* Cambridge University Press. """ squared_resid = np.asarray(resid, dtype=float) ** 2 if squared_resid.ndim == 1: squared_resid = squared_resid.reshape(-1, 1) nobs = len(resid) if 0 < subset_length < 1: h = int(np.round(nobs * subset_length)) elif type(subset_length) is int and subset_length >= 1: h = subset_length numer_resid = squared_resid[-h:] numer_dof = (~np.isnan(numer_resid)).sum(axis=0) numer_squared_sum = np.nansum(numer_resid, axis=0) for i, dof in enumerate(numer_dof): if dof < 2: warnings.warn( "Early subset of data for variable %d" " has too few non-missing observations to" " calculate test statistic." % i, stacklevel=2, ) numer_squared_sum[i] = np.nan denom_resid = squared_resid[:h] denom_dof = (~np.isnan(denom_resid)).sum(axis=0) denom_squared_sum = np.nansum(denom_resid, axis=0) for i, dof in enumerate(denom_dof): if dof < 2: warnings.warn( "Later subset of data for variable %d" " has too few non-missing observations to" " calculate test statistic." % i, stacklevel=2, ) denom_squared_sum[i] = np.nan test_statistic = numer_squared_sum / denom_squared_sum # Setup functions to calculate the p-values if use_f: from scipy.stats import f pval_lower = lambda test_statistics: f.cdf( # noqa:E731 test_statistics, numer_dof, denom_dof ) pval_upper = lambda test_statistics: f.sf( # noqa:E731 test_statistics, numer_dof, denom_dof ) else: from scipy.stats import chi2 pval_lower = lambda test_statistics: chi2.cdf( # noqa:E731 numer_dof * test_statistics, denom_dof ) pval_upper = lambda test_statistics: chi2.sf( # noqa:E731 numer_dof * test_statistics, denom_dof ) # Calculate the one- or two-sided p-values alternative = alternative.lower() if alternative in ["i", "inc", "increasing"]: p_value = pval_upper(test_statistic) elif alternative in ["d", "dec", "decreasing"]: test_statistic = 1.0 / test_statistic p_value = pval_upper(test_statistic) elif alternative in ["2", "2-sided", "two-sided"]: p_value = 2 * np.minimum( pval_lower(test_statistic), pval_upper(test_statistic) ) else: raise ValueError("Invalid alternative.") if len(test_statistic) == 1: return test_statistic[0], p_value[0] return test_statistic, p_value def grangercausalitytests(x, maxlag, addconst=True, verbose=None): """ Four tests for granger non causality of 2 time series. All four tests give similar results. `params_ftest` and `ssr_ftest` are equivalent based on F test which is identical to lmtest:grangertest in R. Parameters ---------- x : array_like The data for testing whether the time series in the second column Granger causes the time series in the first column. Missing values are not supported. maxlag : {int, Iterable[int]} If an integer, computes the test for all lags up to maxlag. If an iterable, computes the tests only for the lags in maxlag. addconst : bool Include a constant in the model. verbose : bool Print results. Deprecated .. deprecated: 0.14 verbose is deprecated and will be removed after 0.15 is released Returns ------- dict All test results, dictionary keys are the number of lags. For each lag the values are a tuple, with the first element a dictionary with test statistic, pvalues, degrees of freedom, the second element are the OLS estimation results for the restricted model, the unrestricted model and the restriction (contrast) matrix for the parameter f_test. Notes ----- TODO: convert to class and attach results properly The Null hypothesis for grangercausalitytests is that the time series in the second column, x2, does NOT Granger cause the time series in the first column, x1. Grange causality means that past values of x2 have a statistically significant effect on the current value of x1, taking past values of x1 into account as regressors. We reject the null hypothesis that x2 does not Granger cause x1 if the pvalues are below a desired size of the test. The null hypothesis for all four test is that the coefficients corresponding to past values of the second time series are zero. `params_ftest`, `ssr_ftest` are based on F distribution `ssr_chi2test`, `lrtest` are based on chi-square distribution References ---------- .. [1] https://en.wikipedia.org/wiki/Granger_causality .. [2] Greene: Econometric Analysis Examples -------- >>> import statsmodels.api as sm >>> from statsmodels.tsa.stattools import grangercausalitytests >>> import numpy as np >>> data = sm.datasets.macrodata.load_pandas() >>> data = data.data[["realgdp", "realcons"]].pct_change().dropna() All lags up to 4 >>> gc_res = grangercausalitytests(data, 4) Only lag 4 >>> gc_res = grangercausalitytests(data, [4]) """ x = array_like(x, "x", ndim=2) if not np.isfinite(x).all(): raise ValueError("x contains NaN or inf values.") addconst = bool_like(addconst, "addconst") if verbose is not None: verbose = bool_like(verbose, "verbose") warnings.warn( "verbose is deprecated since functions should not print results", FutureWarning, ) else: verbose = True # old default try: maxlag = int_like(maxlag, "maxlag") if maxlag <= 0: raise ValueError("maxlag must be a positive integer") lags = np.arange(1, maxlag + 1) except TypeError: lags = np.array([int(lag) for lag in maxlag]) maxlag = lags.max() if lags.min() <= 0 or lags.size == 0: raise ValueError( "maxlag must be a non-empty list containing only " "positive integers" ) if x.shape[0] <= 3 * maxlag + int(addconst): raise ValueError( "Insufficient observations. Maximum allowable " "lag is {}".format(int((x.shape[0] - int(addconst)) / 3) - 1) ) resli = {} for mlg in lags: result = {} if verbose: print("\nGranger Causality") print("number of lags (no zero)", mlg) mxlg = mlg # create lagmat of both time series dta = lagmat2ds(x, mxlg, trim="both", dropex=1) # add constant if addconst: dtaown = add_constant(dta[:, 1 : (mxlg + 1)], prepend=False) dtajoint = add_constant(dta[:, 1:], prepend=False) if ( dtajoint.shape[1] == (dta.shape[1] - 1) or (dtajoint.max(0) == dtajoint.min(0)).sum() != 1 ): raise InfeasibleTestError( "The x values include a column with constant values and so" " the test statistic cannot be computed." ) else: raise NotImplementedError("Not Implemented") # dtaown = dta[:, 1:mxlg] # dtajoint = dta[:, 1:] # Run ols on both models without and with lags of second variable res2down = OLS(dta[:, 0], dtaown).fit() res2djoint = OLS(dta[:, 0], dtajoint).fit() # print results # for ssr based tests see: # http://support.sas.com/rnd/app/examples/ets/granger/index.htm # the other tests are made-up # Granger Causality test using ssr (F statistic) if res2djoint.model.k_constant: tss = res2djoint.centered_tss else: tss = res2djoint.uncentered_tss if ( tss == 0 or res2djoint.ssr == 0 or np.isnan(res2djoint.rsquared) or (res2djoint.ssr / tss) < np.finfo(float).eps or res2djoint.params.shape[0] != dtajoint.shape[1] ): raise InfeasibleTestError( "The Granger causality test statistic cannot be computed " "because the VAR has a perfect fit of the data." ) fgc1 = ( (res2down.ssr - res2djoint.ssr) / res2djoint.ssr / mxlg * res2djoint.df_resid ) if verbose: print( "ssr based F test: F=%-8.4f, p=%-8.4f, df_denom=%d," " df_num=%d" % ( fgc1, stats.f.sf(fgc1, mxlg, res2djoint.df_resid), res2djoint.df_resid, mxlg, ) ) result["ssr_ftest"] = ( fgc1, stats.f.sf(fgc1, mxlg, res2djoint.df_resid), res2djoint.df_resid, mxlg, ) # Granger Causality test using ssr (ch2 statistic) fgc2 = res2down.nobs * (res2down.ssr - res2djoint.ssr) / res2djoint.ssr if verbose: print( "ssr based chi2 test: chi2=%-8.4f, p=%-8.4f, " "df=%d" % (fgc2, stats.chi2.sf(fgc2, mxlg), mxlg) ) result["ssr_chi2test"] = (fgc2, stats.chi2.sf(fgc2, mxlg), mxlg) # likelihood ratio test pvalue: lr = -2 * (res2down.llf - res2djoint.llf) if verbose: print( "likelihood ratio test: chi2=%-8.4f, p=%-8.4f, df=%d" % (lr, stats.chi2.sf(lr, mxlg), mxlg) ) result["lrtest"] = (lr, stats.chi2.sf(lr, mxlg), mxlg) # F test that all lag coefficients of exog are zero rconstr = np.column_stack( (np.zeros((mxlg, mxlg)), np.eye(mxlg, mxlg), np.zeros((mxlg, 1))) ) ftres = res2djoint.f_test(rconstr) if verbose: print( "parameter F test: F=%-8.4f, p=%-8.4f, df_denom=%d," " df_num=%d" % (ftres.fvalue, ftres.pvalue, ftres.df_denom, ftres.df_num) ) result["params_ftest"] = ( np.squeeze(ftres.fvalue)[()], np.squeeze(ftres.pvalue)[()], ftres.df_denom, ftres.df_num, ) resli[mxlg] = (result, [res2down, res2djoint, rconstr]) return resli def coint( y0, y1, trend="c", method="aeg", maxlag=None, autolag: str | None = "aic", return_results=None, ): """ Test for no-cointegration of a univariate equation. The null hypothesis is no cointegration. Variables in y0 and y1 are assumed to be integrated of order 1, I(1). This uses the augmented Engle-Granger two-step cointegration test. Constant or trend is included in 1st stage regression, i.e. in cointegrating equation. **Warning:** The autolag default has changed compared to statsmodels 0.8. In 0.8 autolag was always None, no the keyword is used and defaults to "aic". Use `autolag=None` to avoid the lag search. Parameters ---------- y0 : array_like The first element in cointegrated system. Must be 1-d. y1 : array_like The remaining elements in cointegrated system. trend : str {"c", "ct"} The trend term included in regression for cointegrating equation. * "c" : constant. * "ct" : constant and linear trend. * also available quadratic trend "ctt", and no constant "n". method : {"aeg"} Only "aeg" (augmented Engle-Granger) is available. maxlag : None or int Argument for `adfuller`, largest or given number of lags. autolag : str Argument for `adfuller`, lag selection criterion. * If None, then maxlag lags are used without lag search. * If "AIC" (default) or "BIC", then the number of lags is chosen to minimize the corresponding information criterion. * "t-stat" based choice of maxlag. Starts with maxlag and drops a lag until the t-statistic on the last lag length is significant using a 5%-sized test. return_results : bool For future compatibility, currently only tuple available. If True, then a results instance is returned. Otherwise, a tuple with the test outcome is returned. Set `return_results=False` to avoid future changes in return. Returns ------- coint_t : float The t-statistic of unit-root test on residuals. pvalue : float MacKinnon"s approximate, asymptotic p-value based on MacKinnon (1994). crit_value : dict Critical values for the test statistic at the 1 %, 5 %, and 10 % levels based on regression curve. This depends on the number of observations. Notes ----- The Null hypothesis is that there is no cointegration, the alternative hypothesis is that there is cointegrating relationship. If the pvalue is small, below a critical size, then we can reject the hypothesis that there is no cointegrating relationship. P-values and critical values are obtained through regression surface approximation from MacKinnon 1994 and 2010. If the two series are almost perfectly collinear, then computing the test is numerically unstable. However, the two series will be cointegrated under the maintained assumption that they are integrated. In this case the t-statistic will be set to -inf and the pvalue to zero. TODO: We could handle gaps in data by dropping rows with nans in the Auxiliary regressions. Not implemented yet, currently assumes no nans and no gaps in time series. References ---------- .. [1] MacKinnon, J.G. 1994 "Approximate Asymptotic Distribution Functions for Unit-Root and Cointegration Tests." Journal of Business & Economics Statistics, 12.2, 167-76. .. [2] MacKinnon, J.G. 2010. "Critical Values for Cointegration Tests." Queen"s University, Dept of Economics Working Papers 1227. http://ideas.repec.org/p/qed/wpaper/1227.html """ y0 = array_like(y0, "y0") y1 = array_like(y1, "y1", ndim=2) trend = string_like(trend, "trend", options=("c", "n", "ct", "ctt")) string_like(method, "method", options=("aeg",)) maxlag = int_like(maxlag, "maxlag", optional=True) autolag = string_like( autolag, "autolag", optional=True, options=("aic", "bic", "t-stat") ) return_results = bool_like(return_results, "return_results", optional=True) nobs, k_vars = y1.shape k_vars += 1 # add 1 for y0 if trend == "n": xx = y1 else: xx = add_trend(y1, trend=trend, prepend=False) res_co = OLS(y0, xx).fit() if res_co.rsquared < 1 - 100 * SQRTEPS: res_adf = adfuller( res_co.resid, maxlag=maxlag, autolag=autolag, regression="n" ) else: warnings.warn( "y0 and y1 are (almost) perfectly colinear." "Cointegration test is not reliable in this case.", CollinearityWarning, stacklevel=2, ) # Edge case where series are too similar res_adf = (-np.inf,) # no constant or trend, see egranger in Stata and MacKinnon if trend == "n": crit = [np.nan] * 3 # 2010 critical values not available else: crit = mackinnoncrit(N=k_vars, regression=trend, nobs=nobs - 1) # nobs - 1, the -1 is to match egranger in Stata, I do not know why. # TODO: check nobs or df = nobs - k pval_asy = mackinnonp(res_adf[0], regression=trend, N=k_vars) return res_adf[0], pval_asy, crit def _safe_arma_fit(y, order, model_kw, trend, fit_kw, start_params=None): from statsmodels.tsa.arima.model import ARIMA try: return ARIMA(y, order=order, **model_kw, trend=trend).fit( start_params=start_params, **fit_kw ) except LinAlgError: # SVD convergence failure on badly misspecified models return except ValueError as error: if start_params is not None: # do not recurse again # user supplied start_params only get one chance return # try a little harder, should be handled in fit really elif "initial" not in error.args[0] or "initial" in str(error): start_params = [0.1] * sum(order) if trend == "c": start_params = [0.1] + start_params return _safe_arma_fit( y, order, model_kw, trend, fit_kw, start_params ) else: return except: # no idea what happened return def arma_order_select_ic( y, max_ar=4, max_ma=2, ic="bic", trend="c", model_kw=None, fit_kw=None ): """ Compute information criteria for many ARMA models. Parameters ---------- y : array_like Array of time-series data. max_ar : int Maximum number of AR lags to use. Default 4. max_ma : int Maximum number of MA lags to use. Default 2. ic : str, list Information criteria to report. Either a single string or a list of different criteria is possible. trend : str The trend to use when fitting the ARMA models. model_kw : dict Keyword arguments to be passed to the ``ARMA`` model. fit_kw : dict Keyword arguments to be passed to ``ARMA.fit``. Returns ------- Bunch Dict-like object with attribute access. Each ic is an attribute with a DataFrame for the results. The AR order used is the row index. The ma order used is the column index. The minimum orders are available as ``ic_min_order``. Notes ----- This method can be used to tentatively identify the order of an ARMA process, provided that the time series is stationary and invertible. This function computes the full exact MLE estimate of each model and can be, therefore a little slow. An implementation using approximate estimates will be provided in the future. In the meantime, consider passing {method : "css"} to fit_kw. Examples -------- >>> from statsmodels.tsa.arima_process import arma_generate_sample >>> import statsmodels.api as sm >>> import numpy as np >>> arparams = np.array([.75, -.25]) >>> maparams = np.array([.65, .35]) >>> arparams = np.r_[1, -arparams] >>> maparam = np.r_[1, maparams] >>> nobs = 250 >>> np.random.seed(2014) >>> y = arma_generate_sample(arparams, maparams, nobs) >>> res = sm.tsa.arma_order_select_ic(y, ic=["aic", "bic"], trend="n") >>> res.aic_min_order >>> res.bic_min_order """ max_ar = int_like(max_ar, "max_ar") max_ma = int_like(max_ma, "max_ma") trend = string_like(trend, "trend", options=("n", "c")) model_kw = dict_like(model_kw, "model_kw", optional=True) fit_kw = dict_like(fit_kw, "fit_kw", optional=True) ar_range = [i for i in range(max_ar + 1)] ma_range = [i for i in range(max_ma + 1)] if isinstance(ic, str): ic = [ic] elif not isinstance(ic, (list, tuple)): raise ValueError("Need a list or a tuple for ic if not a string.") results = np.zeros((len(ic), max_ar + 1, max_ma + 1)) model_kw = {} if model_kw is None else model_kw fit_kw = {} if fit_kw is None else fit_kw y_arr = array_like(y, "y", contiguous=True) for ar in ar_range: for ma in ma_range: mod = _safe_arma_fit(y_arr, (ar, 0, ma), model_kw, trend, fit_kw) if mod is None: results[:, ar, ma] = np.nan continue for i, criteria in enumerate(ic): results[i, ar, ma] = getattr(mod, criteria) dfs = [ pd.DataFrame(res, columns=ma_range, index=ar_range) for res in results ] res = dict(zip(ic, dfs)) # add the minimums to the results dict min_res = {} for i, result in res.items(): delta = np.ascontiguousarray(np.abs(result.min().min() - result)) ncols = delta.shape[1] loc = np.argmin(delta) min_res.update({i + "_min_order": (loc // ncols, loc % ncols)}) res.update(min_res) return Bunch(**res) def has_missing(data): """ Returns True if "data" contains missing entries, otherwise False """ return np.isnan(np.sum(data)) def kpss( x, regression: Literal["c", "ct"] = "c", nlags: Literal["auto", "legacy"] | int = "auto", store: bool = False, ) -> tuple[float, float, int, dict[str, float]]: """ Kwiatkowski-Phillips-Schmidt-Shin test for stationarity. Computes the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test for the null hypothesis that x is level or trend stationary. Parameters ---------- x : array_like, 1d The data series to test. regression : str{"c", "ct"} The null hypothesis for the KPSS test. * "c" : The data is stationary around a constant (default). * "ct" : The data is stationary around a trend. nlags : {str, int}, optional Indicates the number of lags to be used. If "auto" (default), lags is calculated using the data-dependent method of Hobijn et al. (1998). See also Andrews (1991), Newey & West (1994), and Schwert (1989). If set to "legacy", uses int(12 * (n / 100)**(1 / 4)) , as outlined in Schwert (1989). store : bool If True, then a result instance is returned additionally to the KPSS statistic (default is False). Returns ------- kpss_stat : float The KPSS test statistic. p_value : float The p-value of the test. The p-value is interpolated from Table 1 in Kwiatkowski et al. (1992), and a boundary point is returned if the test statistic is outside the table of critical values, that is, if the p-value is outside the interval (0.01, 0.1). lags : int The truncation lag parameter. crit : dict The critical values at 10%, 5%, 2.5% and 1%. Based on Kwiatkowski et al. (1992). resstore : (optional) instance of ResultStore An instance of a dummy class with results attached as attributes. Notes ----- To estimate sigma^2 the Newey-West estimator is used. If lags is "legacy", the truncation lag parameter is set to int(12 * (n / 100) ** (1 / 4)), as outlined in Schwert (1989). The p-values are interpolated from Table 1 of Kwiatkowski et al. (1992). If the computed statistic is outside the table of critical values, then a warning message is generated. Missing values are not handled. See the notebook `Stationarity and detrending (ADF/KPSS) <../examples/notebooks/generated/stationarity_detrending_adf_kpss.html>`__ for an overview. References ---------- .. [1] Andrews, D.W.K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59: 817-858. .. [2] Hobijn, B., Frances, B.H., & Ooms, M. (2004). Generalizations of the KPSS-test for stationarity. Statistica Neerlandica, 52: 483-502. .. [3] 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: 159-178. .. [4] Newey, W.K., & West, K.D. (1994). Automatic lag selection in covariance matrix estimation. Review of Economic Studies, 61: 631-653. .. [5] Schwert, G. W. (1989). Tests for unit roots: A Monte Carlo investigation. Journal of Business and Economic Statistics, 7 (2): 147-159. """ x = array_like(x, "x") regression = string_like(regression, "regression", options=("c", "ct")) store = bool_like(store, "store") nobs = x.shape[0] hypo = regression # if m is not one, n != m * n if nobs != x.size: raise ValueError(f"x of shape {x.shape} not understood") if hypo == "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. resids = OLS(x, add_constant(np.arange(1, nobs + 1))).fit().resid crit = [0.119, 0.146, 0.176, 0.216] 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). resids = x - x.mean() crit = [0.347, 0.463, 0.574, 0.739] if nlags == "legacy": nlags = int(np.ceil(12.0 * np.power(nobs / 100.0, 1 / 4.0))) nlags = min(nlags, nobs - 1) elif nlags == "auto" or nlags is None: if nlags is None: # TODO: Remove before 0.14 is released warnings.warn( "None is not a valid value for nlags. It must be an integer, " "'auto' or 'legacy'. None will raise starting in 0.14", FutureWarning, stacklevel=2, ) # autolag method of Hobijn et al. (1998) nlags = _kpss_autolag(resids, nobs) nlags = min(nlags, nobs - 1) elif isinstance(nlags, str): raise ValueError("nvals must be 'auto' or 'legacy' when not an int") else: nlags = int_like(nlags, "nlags", optional=False) if nlags >= nobs: raise ValueError( f"lags ({nlags}) must be < number of observations ({nobs})" ) pvals = [0.10, 0.05, 0.025, 0.01] eta = np.sum(resids.cumsum() ** 2) / (nobs ** 2) # eq. 11, p. 165 s_hat = _sigma_est_kpss(resids, nobs, nlags) kpss_stat = eta / s_hat p_value = np.interp(kpss_stat, crit, pvals) warn_msg = """\ The test statistic is outside of the range of p-values available in the look-up table. The actual p-value is {direction} than the p-value returned. """ if p_value == pvals[-1]: warnings.warn( warn_msg.format(direction="smaller"), InterpolationWarning, stacklevel=2, ) elif p_value == pvals[0]: warnings.warn( warn_msg.format(direction="greater"), InterpolationWarning, stacklevel=2, ) crit_dict = {"10%": crit[0], "5%": crit[1], "2.5%": crit[2], "1%": crit[3]} if store: from statsmodels.stats.diagnostic import ResultsStore rstore = ResultsStore() rstore.lags = nlags rstore.nobs = nobs stationary_type = "level" if hypo == "c" else "trend" rstore.H0 = f"The series is {stationary_type} stationary" rstore.HA = f"The series is not {stationary_type} stationary" return kpss_stat, p_value, crit_dict, rstore else: return kpss_stat, p_value, nlags, crit_dict def _sigma_est_kpss(resids, nobs, lags): """ Computes equation 10, p. 164 of Kwiatkowski et al. (1992). This is the consistent estimator for the variance. """ s_hat = np.sum(resids ** 2) for i in range(1, lags + 1): resids_prod = np.dot(resids[i:], resids[: nobs - i]) s_hat += 2 * resids_prod * (1.0 - (i / (lags + 1.0))) return s_hat / nobs def _kpss_autolag(resids, nobs): """ Computes the number of lags for covariance matrix estimation in KPSS test using method of Hobijn et al (1998). See also Andrews (1991), Newey & West (1994), and Schwert (1989). Assumes Bartlett / Newey-West kernel. """ covlags = int(np.power(nobs, 2.0 / 9.0)) s0 = np.sum(resids ** 2) / nobs s1 = 0 for i in range(1, covlags + 1): resids_prod = np.dot(resids[i:], resids[: nobs - i]) resids_prod /= nobs / 2.0 s0 += resids_prod s1 += i * resids_prod s_hat = s1 / s0 pwr = 1.0 / 3.0 gamma_hat = 1.1447 * np.power(s_hat * s_hat, pwr) autolags = int(gamma_hat * np.power(nobs, pwr)) return autolags def range_unit_root_test(x, store=False): """ Range unit-root test for stationarity. Computes the Range Unit-Root (RUR) test for the null hypothesis that x is stationary. Parameters ---------- x : array_like, 1d The data series to test. store : bool If True, then a result instance is returned additionally to the RUR statistic (default is False). Returns ------- rur_stat : float The RUR test statistic. p_value : float The p-value of the test. The p-value is interpolated from Table 1 in Aparicio et al. (2006), and a boundary point is returned if the test statistic is outside the table of critical values, that is, if the p-value is outside the interval (0.01, 0.1). crit : dict The critical values at 10%, 5%, 2.5% and 1%. Based on Aparicio et al. (2006). resstore : (optional) instance of ResultStore An instance of a dummy class with results attached as attributes. Notes ----- The p-values are interpolated from Table 1 of Aparicio et al. (2006). If the computed statistic is outside the table of critical values, then a warning message is generated. Missing values are not handled. References ---------- .. [1] Aparicio, F., Escribano A., Sipols, A.E. (2006). Range Unit-Root (RUR) tests: robust against nonlinearities, error distributions, structural breaks and outliers. Journal of Time Series Analysis, 27 (4): 545-576. """ x = array_like(x, "x") store = bool_like(store, "store") nobs = x.shape[0] # if m is not one, n != m * n if nobs != x.size: raise ValueError(f"x of shape {x.shape} not understood") # Table from [1] has been replicated using 200,000 samples # Critical values for new n_obs values have been identified pvals = [0.01, 0.025, 0.05, 0.10, 0.90, 0.95] n = np.array( [25, 50, 100, 150, 200, 250, 500, 1000, 2000, 3000, 4000, 5000] ) crit = np.array( [ [0.6626, 0.8126, 0.9192, 1.0712, 2.4863, 2.7312], [0.7977, 0.9274, 1.0478, 1.1964, 2.6821, 2.9613], [0.9070, 1.0243, 1.1412, 1.2888, 2.8317, 3.1393], [0.9543, 1.0768, 1.1869, 1.3294, 2.8915, 3.2049], [0.9833, 1.0984, 1.2101, 1.3494, 2.9308, 3.2482], [0.9982, 1.1137, 1.2242, 1.3632, 2.9571, 3.2842], [1.0494, 1.1643, 1.2712, 1.4076, 3.0207, 3.3584], [1.0846, 1.1959, 1.2988, 1.4344, 3.0653, 3.4073], [1.1121, 1.2200, 1.3230, 1.4556, 3.0948, 3.4439], [1.1204, 1.2295, 1.3303, 1.4656, 3.1054, 3.4632], [1.1309, 1.2347, 1.3378, 1.4693, 3.1165, 3.4717], [1.1377, 1.2402, 1.3408, 1.4729, 3.1252, 3.4807], ] ) # Interpolation for nobs inter_crit = np.zeros((1, crit.shape[1])) for i in range(crit.shape[1]): f = interp1d(n, crit[:, i]) inter_crit[0, i] = f(nobs) # Calculate RUR stat xs = pd.Series(x) exp_max = xs.expanding(1).max().shift(1) exp_min = xs.expanding(1).min().shift(1) count = (xs > exp_max).sum() + (xs < exp_min).sum() rur_stat = count / np.sqrt(len(x)) k = len(pvals) - 1 for i in range(len(pvals) - 1, -1, -1): if rur_stat < inter_crit[0, i]: k = i else: break p_value = pvals[k] warn_msg = """\ The test statistic is outside of the range of p-values available in the look-up table. The actual p-value is {direction} than the p-value returned. """ direction = "" if p_value == pvals[-1]: direction = "smaller" elif p_value == pvals[0]: direction = "larger" if direction: warnings.warn( warn_msg.format(direction=direction), InterpolationWarning, stacklevel=2, ) crit_dict = { "10%": inter_crit[0, 3], "5%": inter_crit[0, 2], "2.5%": inter_crit[0, 1], "1%": inter_crit[0, 0], } if store: from statsmodels.stats.diagnostic import ResultsStore rstore = ResultsStore() rstore.nobs = nobs rstore.H0 = "The series is not stationary" rstore.HA = "The series is stationary" return rur_stat, p_value, crit_dict, rstore else: return rur_stat, p_value, crit_dict class ZivotAndrewsUnitRoot: """ Class wrapper for Zivot-Andrews structural-break unit-root test """ def __init__(self): """ Critical values for the three different models specified for the Zivot-Andrews unit-root test. Notes ----- The p-values are generated through Monte Carlo simulation using 100,000 replications and 2000 data points. """ self._za_critical_values = {} # constant-only model self._c = ( (0.001, -6.78442), (0.100, -5.83192), (0.200, -5.68139), (0.300, -5.58461), (0.400, -5.51308), (0.500, -5.45043), (0.600, -5.39924), (0.700, -5.36023), (0.800, -5.33219), (0.900, -5.30294), (1.000, -5.27644), (2.500, -5.03340), (5.000, -4.81067), (7.500, -4.67636), (10.000, -4.56618), (12.500, -4.48130), (15.000, -4.40507), (17.500, -4.33947), (20.000, -4.28155), (22.500, -4.22683), (25.000, -4.17830), (27.500, -4.13101), (30.000, -4.08586), (32.500, -4.04455), (35.000, -4.00380), (37.500, -3.96144), (40.000, -3.92078), (42.500, -3.88178), (45.000, -3.84503), (47.500, -3.80549), (50.000, -3.77031), (52.500, -3.73209), (55.000, -3.69600), (57.500, -3.65985), (60.000, -3.62126), (65.000, -3.54580), (70.000, -3.46848), (75.000, -3.38533), (80.000, -3.29112), (85.000, -3.17832), (90.000, -3.04165), (92.500, -2.95146), (95.000, -2.83179), (96.000, -2.76465), (97.000, -2.68624), (98.000, -2.57884), (99.000, -2.40044), (99.900, -1.88932), ) self._za_critical_values["c"] = np.asarray(self._c) # trend-only model self._t = ( (0.001, -83.9094), (0.100, -13.8837), (0.200, -9.13205), (0.300, -6.32564), (0.400, -5.60803), (0.500, -5.38794), (0.600, -5.26585), (0.700, -5.18734), (0.800, -5.12756), (0.900, -5.07984), (1.000, -5.03421), (2.500, -4.65634), (5.000, -4.40580), (7.500, -4.25214), (10.000, -4.13678), (12.500, -4.03765), (15.000, -3.95185), (17.500, -3.87945), (20.000, -3.81295), (22.500, -3.75273), (25.000, -3.69836), (27.500, -3.64785), (30.000, -3.59819), (32.500, -3.55146), (35.000, -3.50522), (37.500, -3.45987), (40.000, -3.41672), (42.500, -3.37465), (45.000, -3.33394), (47.500, -3.29393), (50.000, -3.25316), (52.500, -3.21244), (55.000, -3.17124), (57.500, -3.13211), (60.000, -3.09204), (65.000, -3.01135), (70.000, -2.92897), (75.000, -2.83614), (80.000, -2.73893), (85.000, -2.62840), (90.000, -2.49611), (92.500, -2.41337), (95.000, -2.30820), (96.000, -2.25797), (97.000, -2.19648), (98.000, -2.11320), (99.000, -1.99138), (99.900, -1.67466), ) self._za_critical_values["t"] = np.asarray(self._t) # constant + trend model self._ct = ( (0.001, -38.17800), (0.100, -6.43107), (0.200, -6.07279), (0.300, -5.95496), (0.400, -5.86254), (0.500, -5.77081), (0.600, -5.72541), (0.700, -5.68406), (0.800, -5.65163), (0.900, -5.60419), (1.000, -5.57556), (2.500, -5.29704), (5.000, -5.07332), (7.500, -4.93003), (10.000, -4.82668), (12.500, -4.73711), (15.000, -4.66020), (17.500, -4.58970), (20.000, -4.52855), (22.500, -4.47100), (25.000, -4.42011), (27.500, -4.37387), (30.000, -4.32705), (32.500, -4.28126), (35.000, -4.23793), (37.500, -4.19822), (40.000, -4.15800), (42.500, -4.11946), (45.000, -4.08064), (47.500, -4.04286), (50.000, -4.00489), (52.500, -3.96837), (55.000, -3.93200), (57.500, -3.89496), (60.000, -3.85577), (65.000, -3.77795), (70.000, -3.69794), (75.000, -3.61852), (80.000, -3.52485), (85.000, -3.41665), (90.000, -3.28527), (92.500, -3.19724), (95.000, -3.08769), (96.000, -3.03088), (97.000, -2.96091), (98.000, -2.85581), (99.000, -2.71015), (99.900, -2.28767), ) self._za_critical_values["ct"] = np.asarray(self._ct) def _za_crit(self, stat, model="c"): """ Linear interpolation for Zivot-Andrews p-values and critical values Parameters ---------- stat : float The ZA test statistic model : {"c","t","ct"} The model used when computing the ZA statistic. "c" is default. Returns ------- pvalue : float The interpolated p-value cvdict : dict Critical values for the test statistic at the 1%, 5%, and 10% levels Notes ----- The p-values are linear interpolated from the quantiles of the simulated ZA test statistic distribution """ table = self._za_critical_values[model] pcnts = table[:, 0] stats = table[:, 1] # ZA cv table contains quantiles multiplied by 100 pvalue = np.interp(stat, stats, pcnts) / 100.0 cv = [1.0, 5.0, 10.0] crit_value = np.interp(cv, pcnts, stats) cvdict = { "1%": crit_value[0], "5%": crit_value[1], "10%": crit_value[2], } return pvalue, cvdict def _quick_ols(self, endog, exog): """ Minimal implementation of LS estimator for internal use """ xpxi = np.linalg.inv(exog.T.dot(exog)) xpy = exog.T.dot(endog) nobs, k_exog = exog.shape b = xpxi.dot(xpy) e = endog - exog.dot(b) sigma2 = e.T.dot(e) / (nobs - k_exog) return b / np.sqrt(np.diag(sigma2 * xpxi)) def _format_regression_data(self, series, nobs, const, trend, cols, lags): """ Create the endog/exog data for the auxiliary regressions from the original (standardized) series under test. """ # first-diff y and standardize for numerical stability endog = np.diff(series, axis=0) endog /= np.sqrt(endog.T.dot(endog)) series = series / np.sqrt(series.T.dot(series)) # reserve exog space exog = np.zeros((endog[lags:].shape[0], cols + lags)) exog[:, 0] = const # lagged y and dy exog[:, cols - 1] = series[lags : (nobs - 1)] exog[:, cols:] = lagmat(endog, lags, trim="none")[ lags : exog.shape[0] + lags ] return endog, exog def _update_regression_exog( self, exog, regression, period, nobs, const, trend, cols, lags ): """ Update the exog array for the next regression. """ cutoff = period - (lags + 1) if regression != "t": exog[:cutoff, 1] = 0 exog[cutoff:, 1] = const exog[:, 2] = trend[(lags + 2) : (nobs + 1)] if regression == "ct": exog[:cutoff, 3] = 0 exog[cutoff:, 3] = trend[1 : (nobs - period + 1)] else: exog[:, 1] = trend[(lags + 2) : (nobs + 1)] exog[: (cutoff - 1), 2] = 0 exog[(cutoff - 1) :, 2] = trend[0 : (nobs - period + 1)] return exog def run(self, x, trim=0.15, maxlag=None, regression="c", autolag="AIC"): """ Zivot-Andrews structural-break unit-root test. The Zivot-Andrews test tests for a unit root in a univariate process in the presence of serial correlation and a single structural break. Parameters ---------- x : array_like The data series to test. trim : float The percentage of series at begin/end to exclude from break-period calculation in range [0, 0.333] (default=0.15). maxlag : int The maximum lag which is included in test, default is 12*(nobs/100)^{1/4} (Schwert, 1989). regression : {"c","t","ct"} Constant and trend order to include in regression. * "c" : constant only (default). * "t" : trend only. * "ct" : constant and trend. autolag : {"AIC", "BIC", "t-stat", None} The method to select the lag length when using automatic selection. * if None, then maxlag lags are used, * if "AIC" (default) or "BIC", then the number of lags is chosen to minimize the corresponding information criterion, * "t-stat" based choice of maxlag. Starts with maxlag and drops a lag until the t-statistic on the last lag length is significant using a 5%-sized test. Returns ------- zastat : float The test statistic. pvalue : float The pvalue based on MC-derived critical values. cvdict : dict The critical values for the test statistic at the 1%, 5%, and 10% levels. baselag : int The number of lags used for period regressions. bpidx : int The index of x corresponding to endogenously calculated break period with values in the range [0..nobs-1]. Notes ----- H0 = unit root with a single structural break Algorithm follows Baum (2004/2015) approximation to original Zivot-Andrews method. Rather than performing an autolag regression at each candidate break period (as per the original paper), a single autolag regression is run up-front on the base model (constant + trend with no dummies) to determine the best lag length. This lag length is then used for all subsequent break-period regressions. This results in significant run time reduction but also slightly more pessimistic test statistics than the original Zivot-Andrews method, although no attempt has been made to characterize the size/power trade-off. References ---------- .. [1] Baum, C.F. (2004). ZANDREWS: Stata module to calculate Zivot-Andrews unit root test in presence of structural break," Statistical Software Components S437301, Boston College Department of Economics, revised 2015. .. [2] Schwert, G.W. (1989). Tests for unit roots: A Monte Carlo investigation. Journal of Business & Economic Statistics, 7: 147-159. .. [3] Zivot, E., and Andrews, D.W.K. (1992). Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of Business & Economic Studies, 10: 251-270. """ x = array_like(x, "x", dtype=np.double, ndim=1) trim = float_like(trim, "trim") maxlag = int_like(maxlag, "maxlag", optional=True) regression = string_like( regression, "regression", options=("c", "t", "ct") ) autolag = string_like( autolag, "autolag", options=("aic", "bic", "t-stat"), optional=True ) if trim < 0 or trim > (1.0 / 3.0): raise ValueError("trim value must be a float in range [0, 1/3)") nobs = x.shape[0] if autolag: adf_res = adfuller( x, maxlag=maxlag, regression="ct", autolag=autolag ) baselags = adf_res[2] elif maxlag: baselags = maxlag else: baselags = int(12.0 * np.power(nobs / 100.0, 1 / 4.0)) trimcnt = int(nobs * trim) start_period = trimcnt end_period = nobs - trimcnt if regression == "ct": basecols = 5 else: basecols = 4 # normalize constant and trend terms for stability c_const = 1 / np.sqrt(nobs) t_const = np.arange(1.0, nobs + 2) t_const *= np.sqrt(3) / nobs ** (3 / 2) # format the auxiliary regression data endog, exog = self._format_regression_data( x, nobs, c_const, t_const, basecols, baselags ) # iterate through the time periods stats = np.full(end_period + 1, np.inf) for bp in range(start_period + 1, end_period + 1): # update intercept dummy / trend / trend dummy exog = self._update_regression_exog( exog, regression, bp, nobs, c_const, t_const, basecols, baselags, ) # check exog rank on first iteration if bp == start_period + 1: o = OLS(endog[baselags:], exog, hasconst=1).fit() if o.df_model < exog.shape[1] - 1: raise ValueError( "ZA: auxiliary exog matrix is not full rank.\n" " cols (exc intercept) = {} rank = {}".format( exog.shape[1] - 1, o.df_model ) ) stats[bp] = o.tvalues[basecols - 1] else: stats[bp] = self._quick_ols(endog[baselags:], exog)[ basecols - 1 ] # return best seen zastat = np.min(stats) bpidx = np.argmin(stats) - 1 crit = self._za_crit(zastat, regression) pval = crit[0] cvdict = crit[1] return zastat, pval, cvdict, baselags, bpidx def __call__( self, x, trim=0.15, maxlag=None, regression="c", autolag="AIC" ): return self.run( x, trim=trim, maxlag=maxlag, regression=regression, autolag=autolag ) zivot_andrews = ZivotAndrewsUnitRoot() zivot_andrews.__doc__ = zivot_andrews.run.__doc__