""" Statistical tools for time series analysis """ from __future__ import division from statsmodels.compat.python import (iteritems, range, lrange, string_types, lzip, zip, long) from statsmodels.compat.numpy import lstsq from statsmodels.compat.scipy import _next_regular import numpy as np import pandas as pd from numpy.linalg import LinAlgError from scipy import stats from statsmodels.regression.linear_model import OLS, yule_walker from statsmodels.tools.sm_exceptions import (InterpolationWarning, MissingDataError, CollinearityWarning) from statsmodels.tools.tools import add_constant, Bunch from statsmodels.tsa._bds import bds from statsmodels.tsa.adfvalues import mackinnonp, mackinnoncrit from statsmodels.tsa.arima_model import ARMA from statsmodels.tsa.tsatools import lagmat, lagmat2ds, add_trend __all__ = ['acovf', 'acf', 'pacf', 'pacf_yw', 'pacf_ols', 'ccovf', 'ccf', 'periodogram', 'q_stat', 'coint', 'arma_order_select_ic', 'adfuller', 'kpss', 'bds', 'pacf_burg', 'innovations_algo', 'innovations_filter', 'levinson_durbin_pacf', 'levinson_durbin'] SQRTEPS = np.sqrt(np.finfo(np.double).eps) #NOTE: now in two places to avoid circular import #TODO: I like the bunch pattern for this too. class ResultsStore(object): def __str__(self): return self._str # pylint: disable=E1101 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 iteritems(results)) elif method == "bic": icbest, bestlag = min((v.bic, k) for k, v in iteritems(results)) elif method == "t-stat": #stop = stats.norm.ppf(.95) stop = 1.6448536269514722 for lag in range(startlag + maxlag, startlag - 1, -1): icbest = np.abs(results[lag].tvalues[-1]) if np.abs(icbest) >= stop: bestlag = lag icbest = icbest break else: raise ValueError("Information Criterion %s not understood.") % method 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=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 data series maxlag : int Maximum lag which is included in test, default 12*(nobs/100)^{1/4} regression : {'c','ct','ctt','nc'} Constant and trend order to include in regression * 'c' : constant only (default) * 'ct' : constant and trend * 'ctt' : constant, and linear and quadratic trend * 'nc' : no constant, no trend autolag : {'AIC', 'BIC', 't-stat', None} * 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 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 Test statistic pvalue : float MacKinnon's approximate p-value based on MacKinnon (1994, 2010) usedlag : int Number of lags used nobs : int 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. Examples -------- See example notebook References ---------- .. [*] W. Green. "Econometric Analysis," 5th ed., Pearson, 2003. .. [*] Hamilton, J.D. "Time Series Analysis". Princeton, 1994. .. [*] MacKinnon, J.G. 1994. "Approximate asymptotic distribution functions for unit-root and cointegration tests. `Journal of Business and Economic Statistics` 12, 167-76. .. [*] 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 """ if regresults: store = True trenddict = {None: 'nc', 0: 'c', 1: 'ct', 2: 'ctt'} if regression is None or isinstance(regression, (int, long)): regression = trenddict[regression] regression = regression.lower() if regression not in ['c', 'nc', 'ct', 'ctt']: raise ValueError("regression option %s not understood") % regression x = np.asarray(x) nobs = x.shape[0] ntrend = len(regression) if regression != 'nc' else 0 if maxlag is None: # from Greene referencing Schwert 1989 maxlag = int(np.ceil(12. * np.power(nobs / 100., 1 / 4.))) # -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: resstore = ResultsStore() if autolag: if regression != 'nc': 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 != 'nc': 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 def acovf(x, unbiased=False, demean=True, fft=None, missing='none', nlag=None): """ Autocovariance for 1D Parameters ---------- x : array Time series data. Must be 1d. unbiased : bool If True, then denominators is n-k, otherwise n demean : bool If True, then subtract the mean x from each element of x fft : bool If True, use FFT convolution. This method should be preferred for long time series. missing : str A string in ['none', 'raise', 'conservative', 'drop'] specifying how the NaNs are to be treated. nlag : {int, 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 ------- acovf : array autocovariance function References ----------- .. [*] Parzen, E., 1963. On spectral analysis with missing observations and amplitude modulation. Sankhya: The Indian Journal of Statistics, Series A, pp.383-392. """ if fft is None: import warnings msg = 'fft=True will become the default in a future version of ' \ 'statsmodels. To suppress this warning, explicitly set ' \ 'fft=False.' warnings.warn(msg, FutureWarning) fft = False x = np.squeeze(np.asarray(x)) if x.ndim > 1: raise ValueError("x must be 1d. Got %d dims." % x.ndim) missing = missing.lower() if missing not in ['none', 'raise', 'conservative', 'drop']: raise ValueError("missing option %s not understood" % missing) 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 unbiased: acov /= (n - np.arange(lag_len + 1)) else: acov /= n else: if unbiased: 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 unbiased and deal_with_masked and missing == 'conservative': d = np.correlate(notmask_int, notmask_int, 'full') d[d == 0] = 1 elif unbiased: 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, type="ljungbox"): """ Return's Ljung-Box Q Statistic x : array-like Array of autocorrelation coefficients. Can be obtained from acf. nobs : int Number of observations in the entire sample (ie., not just the length of the autocorrelation function results. Returns ------- q-stat : array Ljung-Box Q-statistic for autocorrelation parameters p-value : array P-value of the Q statistic Notes ----- Written to be used with acf. """ x = np.asarray(x) if type == "ljungbox": ret = (nobs * (nobs + 2) * np.cumsum((1. / (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, unbiased=False, nlags=40, qstat=False, fft=None, alpha=None, missing='none'): """ Autocorrelation function for 1d arrays. Parameters ---------- x : array Time series data unbiased : bool If True, then denominators for autocovariance are n-k, otherwise n nlags: int, optional Number of lags to return autocorrelation for. qstat : bool, optional If True, returns the Ljung-Box q statistic for each autocorrelation coefficient. See q_stat for more information. fft : bool, optional If True, computes the ACF via FFT. alpha : scalar, 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 Bartlett\'s formula. missing : str, optional A string in ['none', 'raise', 'conservative', 'drop'] specifying how the NaNs are to be treated. Returns ------- acf : array autocorrelation function confint : array, optional Confidence intervals for the ACF. Returned if alpha is not None. qstat : array, optional The Ljung-Box Q-Statistic. Returned if q_stat is True. pvalues : array, optional The p-values associated with the Q-statistics. 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 unbiased is true, the denominator for the autocovariance is adjusted but the autocorrelation is not an unbiased estimator. References ---------- .. [*] Parzen, E., 1963. On spectral analysis with missing observations and amplitude modulation. Sankhya: The Indian Journal of Statistics, Series A, pp.383-392. """ if fft is None: import warnings msg = 'fft=True will become the default in a future version of ' \ 'statsmodels. To suppress this warning, explicitly set ' \ 'fft=False.' warnings.warn(msg, FutureWarning) fft = False nobs = len(x) # should this shrink for missing='drop' and NaNs in x? avf = acovf(x, unbiased=unbiased, demean=True, fft=fft, missing=missing) acf = avf[:nlags + 1] / avf[0] if not (qstat or alpha): return acf if alpha is not None: varacf = np.ones(nlags + 1) / nobs varacf[0] = 0 varacf[1] = 1. / nobs varacf[2:] *= 1 + 2 * np.cumsum(acf[1:-1]**2) interval = stats.norm.ppf(1 - alpha / 2.) * np.sqrt(varacf) confint = np.array(lzip(acf - interval, acf + interval)) if not qstat: return acf, confint if qstat: 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, nlags=40, method='unbiased'): '''Partial autocorrelation estimated with non-recursive yule_walker Parameters ---------- x : 1d array observations of time series for which pacf is calculated nlags : int largest lag for which pacf is returned method : 'unbiased' (default) or 'mle' method for the autocovariance calculations in yule walker Returns ------- pacf : 1d array partial autocorrelations, maxlag+1 elements See Also -------- statsmodels.tsa.stattools.pacf statsmodels.tsa.stattools.pacf_burg statsmodels.tsa.stattools.pacf_ols Notes ----- This solves yule_walker for each desired lag and contains currently duplicate calculations. ''' pacf = [1.] 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, nlags=None, demean=True): """ 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 compute the partial autocorrelations. If omitted, uses the smaller of 10(log10(nobs)) or nobs - 1 demean : bool, optional 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 statsmodels.tsa.stattools.pacf_yw statsmodels.tsa.stattools.pacf_ols References ---------- .. [*] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series and forecasting. Springer. """ x = np.squeeze(np.asarray(x)) if x.ndim != 1: raise ValueError('x must be 1-d or squeezable to 1-d.') 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) 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. * (nobs - np.arange(0, p + 1))) pacf[0] = 1 # Insert the 0 lag partial autocorrel return pacf, sigma2 def pacf_ols(x, nlags=40, efficient=True, unbiased=False): """ Calculate partial autocorrelations via OLS Parameters ---------- x : 1d array observations of time series for which pacf is calculated nlags : int Number of lags for which pacf is returned. Lag 0 is not returned. 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. unbiased : bool, optional Adjust each partial autocorrelation by n / (n - lag) Returns ------- pacf : 1d array partial autocorrelations, (maxlag,) array corresponding to lags 0, 1, ..., maxlag 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. See Also -------- statsmodels.tsa.stattools.pacf statsmodels.tsa.stattools.pacf_yw statsmodels.tsa.stattools.pacf_burg 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 """ pacf = np.empty(nlags + 1) pacf[0] = 1.0 x = np.squeeze(np.asarray(x)) if x.ndim != 1: raise ValueError('x must be squeezable to a 1-d array') 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] = 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] = params[-1] if unbiased: n = len(x) pacf *= n / (n - np.arange(nlags + 1)) return pacf def pacf(x, nlags=40, method='ywunbiased', alpha=None): """ Partial autocorrelation estimated Parameters ---------- x : 1d array observations of time series for which pacf is calculated nlags : int largest lag for which the pacf is returned method : str specifies which method for the calculations to use: - 'yw' or 'ywunbiased' : Yule-Walker with bias correction in denominator for acovf. Default. - 'ywm' or 'ywmle' : Yule-Walker without bias correction - '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-unbiased' : regression of time series on lags with a bias adjustment - 'ld' or 'ldunbiased' : Levinson-Durbin recursion with bias correction - 'ldb' or 'ldbiased' : Levinson-Durbin recursion without bias correction 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 : 1d array partial autocorrelations, nlags elements, including lag zero confint : array, optional Confidence intervals for the PACF. Returned if confint is not None. See Also -------- statsmodels.tsa.stattools.acf statsmodels.tsa.stattools.pacf_yw statsmodels.tsa.stattools.pacf_burg statsmodels.tsa.stattools.pacf_ols 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-unbiased. Yule-Walker (unbiased) and Levinson-Durbin (unbiased) performed consistently worse than the other options. """ if method in ('ols', 'ols-inefficient', 'ols-unbiased'): efficient = 'inefficient' not in method unbiased = 'unbiased' in method ret = pacf_ols(x, nlags=nlags, efficient=efficient, unbiased=unbiased) elif method in ('yw', 'ywu', 'ywunbiased', 'yw_unbiased'): ret = pacf_yw(x, nlags=nlags, method='unbiased') elif method in ('ywm', 'ywmle', 'yw_mle'): ret = pacf_yw(x, nlags=nlags, method='mle') elif method in ('ld', 'ldu', 'ldunbiased', 'ld_unbiased'): acv = acovf(x, unbiased=True, fft=False) ld_ = levinson_durbin(acv, nlags=nlags, isacov=True) ret = ld_[2] # inconsistent naming with ywmle elif method in ('ldb', 'ldbiased', 'ld_biased'): acv = acovf(x, unbiased=False, fft=False) ld_ = levinson_durbin(acv, nlags=nlags, isacov=True) ret = ld_[2] else: raise ValueError('method not available') if alpha is not None: varacf = 1. / len(x) # for all lags >=1 interval = stats.norm.ppf(1. - alpha / 2.) * 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 def ccovf(x, y, unbiased=True, demean=True): ''' crosscovariance for 1D Parameters ---------- x, y : arrays time series data unbiased : boolean if True, then denominators is n-k, otherwise n Returns ------- ccovf : array autocovariance function Notes ----- This uses np.correlate which does full convolution. For very long time series it is recommended to use fft convolution instead. ''' n = len(x) if demean: xo = x - x.mean() yo = y - y.mean() else: xo = x yo = y if unbiased: xi = np.ones(n) d = np.correlate(xi, xi, 'full') else: d = n return (np.correlate(xo, yo, 'full') / d)[n - 1:] def ccf(x, y, unbiased=True): '''cross-correlation function for 1d Parameters ---------- x, y : arrays time series data unbiased : boolean if True, then denominators for autocovariance is n-k, otherwise n Returns ------- ccf : array cross-correlation function of x and y Notes ----- This is based np.correlate which does full convolution. For very long time series it is recommended to use fft convolution instead. If unbiased is true, the denominator for the autocovariance is adjusted but the autocorrelation is not an unbiased estimtor. ''' cvf = ccovf(x, y, unbiased=unbiased, demean=True) return cvf / (np.std(x) * np.std(y)) def periodogram(X): """ Returns the periodogram for the natural frequency of X Parameters ---------- X : array-like Array for which the periodogram is desired. Returns ------- pgram : array 1./len(X) * np.abs(np.fft.fft(X))**2 References ---------- .. [*] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series and forecasting. Springer. """ X = np.asarray(X) # if kernel == "bartlett": # w = 1 - np.arange(M+1.)/M #JP removed integer division pergr = 1. / len(X) * np.abs(np.fft.fft(X))**2 pergr[0] = 0. # what are the implications of this? return pergr # 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 : integer largest lag to include in recursion or order of the autoregressive process isacov : boolean flag to indicate whether the first argument, s, contains the autocovariances or the data series. Returns ------- sigma_v : float estimate of the error variance ? arcoefs : ndarray estimate of the autoregressive coefficients for a model including nlags pacf : ndarray partial autocorrelation function sigma : ndarray entire sigma array from intermediate result, last value is sigma_v phi : ndarray 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 = np.asarray(s) 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. 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 acf computed from the partial autocorrelations. Array returned contains the autocorelations corresponding to lags 0, 1, ..., p References ---------- .. [*] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series and forecasting. Springer. """ pacf = np.squeeze(np.asarray(pacf)) if pacf.ndim != 1: raise ValueError('pacf must be 1-d or squeezable to 1-d.') 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 innovations_algo(acov, nobs=None, rtol=None): """ Innovations algorithm to convert autocovariances to MA parameters Parameters ---------- acov : array-like Array containing autocovariances including lag 0 nobs : int, optional Number of periods to run the algorithm. If not provided, nobs is equal to the length of acovf rtol : float, optional Tolerance used to check for convergence. Default value is 0 which will never prematurely end the algorithm. Checks after 10 iterations and stops if sigma2[i] - sigma2[i - 10] < rtol * sigma2[0]. When the stopping condition is met, the remaining values in theta and sigma2 are forward filled using the value of the final iteration. Returns ------- theta : ndarray Innovation coefficients of MA representation. Array is (nobs, q) where q is the largest index of a non-zero autocovariance. theta corresponds to the first q columns of the coefficient matrix in the common description of the innovation algorithm. sigma2 : ndarray The prediction error variance (nobs,). Examples -------- >>> import statsmodels.api as sm >>> data = sm.datasets.macrodata.load_pandas() >>> rgdpg = data.data['realgdp'].pct_change().dropna() >>> acov = sm.tsa.acovf(rgdpg) >>> nobs = activity.shape[0] >>> theta, sigma2 = innovations_algo(acov[:4], nobs=nobs) See Also -------- innovations_filter References ---------- .. [*] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series and forecasting. Springer. """ acov = np.squeeze(np.asarray(acov)) if acov.ndim != 1: raise ValueError('acov must be 1-d or squeezable to 1-d.') rtol = 0.0 if rtol is None else rtol if not isinstance(rtol, float): raise ValueError('rtol must be a non-negative float or None.') if nobs is not None and (nobs != int(nobs) or nobs < 1): raise ValueError('nobs must be a positive integer') n = acov.shape[0] if nobs is None else int(nobs) max_lag = int(np.max(np.argwhere(acov != 0))) v = np.zeros(n + 1) v[0] = acov[0] # Retain only the relevant columns of theta theta = np.zeros((n + 1, max_lag + 1)) for i in range(1, n): for k in range(max(i - max_lag, 0), i): sub = 0 for j in range(max(i - max_lag, 0), k): sub += theta[k, k - j] * theta[i, i - j] * v[j] theta[i, i - k] = 1. / v[k] * (acov[i - k] - sub) v[i] = acov[0] for j in range(max(i - max_lag, 0), i): v[i] -= theta[i, i - j] ** 2 * v[j] # Break if v has converged if i >= 10: if v[i - 10] - v[i] < v[0] * rtol: # Forward fill all remaining values v[i + 1:] = v[i] theta[i + 1:] = theta[i] break theta = theta[:-1, 1:] v = v[:-1] return theta, v def innovations_filter(endog, theta): """ Filter observations using the innovations algorithm Parameters ---------- endog : array-like The time series to filter (nobs,). Should be demeaned if not mean 0. theta : ndarray Innovation coefficients of MA representation. Array must be (nobs, q) where q order of the MA. Returns ------- resid : ndarray Array of filtered innovations Examples -------- >>> import statsmodels.api as sm >>> data = sm.datasets.macrodata.load_pandas() >>> rgdpg = data.data['realgdp'].pct_change().dropna() >>> acov = sm.tsa.acovf(rgdpg) >>> nobs = activity.shape[0] >>> theta, sigma2 = innovations_algo(acov[:4], nobs=nobs) >>> resid = innovations_filter(rgdpg, theta) See Also -------- innovations_algo References ---------- .. [*] Brockwell, P.J. and Davis, R.A., 2016. Introduction to time series and forecasting. Springer. """ orig_endog = endog endog = np.squeeze(np.asarray(endog)) if endog.ndim != 1: raise ValueError('endog must be 1-d or squeezable to 1-d.') nobs = endog.shape[0] n_theta, k = theta.shape if nobs != n_theta: raise ValueError('theta must be (nobs, q) where q is the moder order') is_pandas = isinstance(orig_endog, (pd.DataFrame, pd.Series)) if is_pandas: if len(orig_endog.index) != nobs: msg = 'If endog is a Series or DataFrame, the index must ' \ 'correspond to the number of time series observations.' raise ValueError(msg) u = np.empty(nobs) u[0] = endog[0] for i in range(1, nobs): if i < k: hat = (theta[i, :i] * u[:i][::-1]).sum() else: hat = (theta[i] * u[i - k:i][::-1]).sum() u[i] = endog[i] - hat if is_pandas: u = pd.Series(u, index=orig_endog.index.copy()) return u def grangercausalitytests(x, maxlag, addconst=True, verbose=True): """four tests for granger non causality of 2 timeseries 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, 2d data for test whether the time series in the second column Granger causes the time series in the first column maxlag : integer the Granger causality test results are calculated for all lags up to maxlag verbose : bool print results if true Returns ------- results : dictionary 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 teststatistic, 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 ---------- http://en.wikipedia.org/wiki/Granger_causality Greene: Econometric Analysis """ from scipy import stats x = np.asarray(x) if x.shape[0] <= 3 * maxlag + int(addconst): raise ValueError("Insufficient observations. Maximum allowable " "lag is {0}".format(int((x.shape[0] - int(addconst)) / 3) - 1)) resli = {} for mlg in range(1, maxlag + 1): 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) 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) 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='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 ---------- y1 : array_like, 1d first element in cointegrating vector y2 : array_like remaining elements in cointegrating vector trend : str {'c', 'ct'} trend term included in regression for cointegrating equation * 'c' : constant * 'ct' : constant and linear trend * also available quadratic trend 'ctt', and no constant 'nc' method : string currently only 'aeg' for augmented Engle-Granger test is available. default might change. maxlag : None or int keyword for `adfuller`, largest or given number of lags autolag : string keyword 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 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 ---------- MacKinnon, J.G. 1994 "Approximate Asymptotic Distribution Functions for Unit-Root and Cointegration Tests." Journal of Business & Economics Statistics, 12.2, 167-76. 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 """ trend = trend.lower() if trend not in ['c', 'nc', 'ct', 'ctt']: raise ValueError("trend option %s not understood" % trend) y0 = np.asarray(y0) y1 = np.asarray(y1) if y1.ndim < 2: y1 = y1[:, None] nobs, k_vars = y1.shape k_vars += 1 # add 1 for y0 if trend == 'nc': 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='nc') else: import warnings warnings.warn("y0 and y1 are (almost) perfectly colinear." "Cointegration test is not reliable in this case.", CollinearityWarning) # Edge case where series are too similar res_adf = (-np.inf,) # no constant or trend, see egranger in Stata and MacKinnon if trend == 'nc': 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 don't 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): try: return ARMA(y, order=order, **model_kw).fit(disp=0, trend=trend, 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: # don't 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 = [.1] * sum(order) if trend == 'c': start_params = [.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): """ Returns information criteria for many ARMA models Parameters ---------- y : array-like 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 ------- obj : Results object 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``. 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='nc') >>> res.aic_min_order >>> res.bic_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. """ from pandas import DataFrame ar_range = lrange(0, max_ar + 1) ma_range = lrange(0, max_ma + 1) if isinstance(ic, string_types): 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 = np.asarray(y) for ar in ar_range: for ma in ma_range: if ar == 0 and ma == 0 and trend == 'nc': results[:, ar, ma] = np.nan continue mod = _safe_arma_fit(y_arr, (ar, 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 = [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 iteritems(res): mins = np.where(result.min().min() == result) min_res.update({i + '_min_order': (mins[0][0], mins[1][0])}) 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='c', lags=None, store=False): """ 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 Data series regression : str{'c', 'ct'} Indicates the null hypothesis for the KPSS test * 'c' : The data is stationary around a constant (default) * 'ct' : The data is stationary around a trend lags : {None, str, int}, optional Indicates the number of lags to be used. If None (default), lags is calculated using the legacy method. If 'auto', 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 None, 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. References ---------- Andrews, D.W.K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59: 817-858. Hobijn, B., Frances, B.H., & Ooms, M. (2004). Generalizations of the KPSS-test for stationarity. Statistica Neerlandica, 52: 483-502. 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. Newey, W.K., & West, K.D. (1994). Automatic lag selection in covariance matrix estimation. Review of Economic Studies, 61: 631-653. Schwert, G. W. (1989). Tests for unit roots: A Monte Carlo investigation. Journal of Business and Economic Statistics, 7 (2): 147-159. """ from warnings import warn nobs = len(x) x = np.asarray(x) hypo = regression.lower() # if m is not one, n != m * n if nobs != x.size: raise ValueError("x of shape {0} not understood".format(x.shape)) 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] elif 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] else: raise ValueError("hypothesis '{0}' not understood".format(hypo)) if lags is None: lags = 'legacy' msg = 'The behavior of using lags=None will change in the next ' \ 'release. Currently lags=None is the same as ' \ 'lags=\'legacy\', and so a sample-size lag length is used. ' \ 'After the next release, the default will change to be the ' \ 'same as lags=\'auto\' which uses an automatic lag length ' \ 'selection method. To silence this warning, either use ' \ '\'auto\' or \'legacy\'' warn(msg, FutureWarning) if lags == 'legacy': lags = int(np.ceil(12. * np.power(nobs / 100., 1 / 4.))) lags = min(lags, nobs - 1) elif lags == 'auto': # autolag method of Hobijn et al. (1998) lags = _kpss_autolag(resids, nobs) else: lags = int(lags) if lags >= nobs: raise ValueError("lags ({}) must be < number of observations ({})" .format(lags, 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, lags) kpss_stat = eta / s_hat p_value = np.interp(kpss_stat, crit, pvals) if p_value == pvals[-1]: warn("p-value is smaller than the indicated p-value", InterpolationWarning) elif p_value == pvals[0]: warn("p-value is greater than the indicated p-value", InterpolationWarning) crit_dict = {'10%': crit[0], '5%': crit[1], '2.5%': crit[2], '1%': crit[3]} if store: rstore = ResultsStore() rstore.lags = lags rstore.nobs = nobs stationary_type = "level" if hypo == 'c' else "trend" rstore.H0 = "The series is {0} stationary".format(stationary_type) rstore.HA = "The series is not {0} stationary".format(stationary_type) return kpss_stat, p_value, crit_dict, rstore else: return kpss_stat, p_value, lags, 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. - (i / (lags + 1.))) 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. / 9.)) 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.) s0 += resids_prod s1 += i * resids_prod s_hat = s1 / s0 pwr = 1. / 3. gamma_hat = 1.1447 * np.power(s_hat * s_hat, pwr) autolags = np.amin([nobs, int(gamma_hat * np.power(nobs, pwr))]) return autolags