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Source code for statsmodels.tsa.stattools

"""
Statistical tools for time series analysis
"""

import numpy as np
from scipy import stats, signal
from statsmodels.regression.linear_model import OLS, yule_walker
from statsmodels.tools.tools import add_constant
from tsatools import lagmat, lagmat2ds, add_trend
#from statsmodels.sandbox.tsa import var
from adfvalues import *
#from statsmodels.sandbox.rls import RLS

#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

def _autolag(mod, endog, exog, startlag, maxlag, method, modargs=(),
        fitargs=(), regresults=False):
    """
    Returns the results for the lag length that maximimizes the info criterion.

    Parameters
    ----------
    mod : Model class
        Model estimator class.
    modargs : tuple
        args to pass to model.  See notes.
    fitargs : tuple
        args to pass to fit.  See notes.
    lagstart : int
        The first zero-indexed column to hold a lag.  See Notes.
    maxlag : int
        The highest lag order for lag length selection.
    method : str {"aic","bic","t-stat"}
        aic - Akaike Information Criterion
        bic - Bayes Information Criterion
        t-stat - Based on last lag

    Returns
    -------
    icbest : float
        Best information criteria.
    bestlag : int
        The lag length that maximizes the information criterion.


    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.iteritems())
    elif method == "bic":
        icbest, bestlag = min((v.bic,k) for k,v in results.iteritems())
    elif method == "t-stat":
        lags = sorted(results.keys())[::-1]
#        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
[docs]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 : str {'c','ct','ctt','nc'} Constant and trend order to include in regression * 'c' : constant only * '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' or 'BIC', then the number of lags is chosen to minimize the corresponding information criterium * 't-stat' based choice of maxlag. Starts with maxlag and drops a lag until the t-statistic on the last lag length is significant at the 95 % level. store : bool If True, then a result instance is returned additionally to the adf statistic regresults : bool If True, the full regression results are returned. Returns ------- adf : float Test statistic pvalue : float MacKinnon's approximate p-value based on MacKinnon (1994) 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. regresults : RegressionResults instance The resstore : (optional) instance of ResultStore an instance of 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 accept or reject the null. The autolag option and maxlag for it are described in Greene. Examples -------- see example script References ---------- Greene Hamilton P-Values (regression surface approximation) MacKinnon, J.G. 1994. "Approximate asymptotic distribution functions for unit-root and cointegration tests. `Journal of Business and Economic Statistics` 12, 167-76. Critical values 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): 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] if maxlag is None: #from Greene referencing Schwert 1989 maxlag = int(np.ceil(12. * np.power(nobs/100., 1/4.))) 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 return adfstat, pvalue, critvalues, resstore else: if not autolag: return adfstat, pvalue, usedlag, nobs, critvalues else: return adfstat, pvalue, usedlag, nobs, critvalues, icbest
[docs]def acovf(x, unbiased=False, demean=True, fft=False): ''' Autocovariance for 1D Parameters ---------- x : array time series data unbiased : bool if True, then denominators is n-k, otherwise n fft : bool If True, use FFT convolution. This method should be preferred for long time series. Returns ------- acovf : array autocovariance function ''' n = len(x) if demean: xo = x - x.mean(); else: xo = x if unbiased: # xi = np.ones(n); # d = np.correlate(xi, xi, 'full') xi = np.arange(1,n+1) d = np.hstack((xi,xi[:-1][::-1])) # faster, is correlate more general? else: d = n if fft: nobs = len(xo) Frf = np.fft.fft(xo, n=nobs*2) acov = np.fft.ifft(Frf*np.conjugate(Frf))[:nobs]/d return acov.real else: return (np.correlate(xo, xo, 'full')/d)[n-1:]
[docs]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
[docs]def acf(x, unbiased=False, nlags=40, confint=None, qstat=False, fft=False): ''' 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. confint : float or None, optional If True, the confidence intervals for the given level are returned. For instance if confint=95, 95 % confidence intervals are returned. 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. Returns ------- acf : array autocorrelation function confint : array, optional Confidence intervals for the ACF. Returned if confint 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. 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. ''' nobs = len(x) d = nobs # changes if unbiased if not fft: avf = acovf(x, unbiased=unbiased, demean=True) #acf = np.take(avf/avf[0], range(1,nlags+1)) acf = avf[:nlags+1]/avf[0] else: #JP: move to acovf x0 = x - x.mean() Frf = np.fft.fft(x0, n=nobs*2) # zero-pad for separability if unbiased: d = nobs - np.arange(nobs) acf = np.fft.ifft(Frf * np.conjugate(Frf))[:nobs]/d acf /= acf[0] #acf = np.take(np.real(acf), range(1,nlags+1)) acf = np.real(acf[:nlags+1]) #keep lag 0 if not (confint or qstat): return acf # Based on Bartlett's formula for MA(q) processes #NOTE: not sure if this is correct, or needs to be centered or what. if not confint is None: varacf = np.ones(nlags+1)/nobs #varacf[1:] *= 1 + 2*np.cumsum(acf[1:-1]**2) #TODO: test this, are my changes correct varacf[0] = 0 varacf[1:] *= 1 + 2*np.cumsum(acf[1:]**2) interval = stats.norm.ppf(1-(100-confint)/200.)*np.sqrt(varacf) confint = np.array(zip(acf-interval, acf+interval)) if not qstat: return acf, confint if qstat: qstat, pvalue = q_stat(acf[1:], nobs=nobs) #drop lag 0 if confint is not None: return acf, confint, qstat, pvalue else: return acf, qstat
[docs]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 maxlag : 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 Notes ----- This solves yule_walker for each desired lag and contains currently duplicate calculations. ''' xm = x - x.mean() pacf = [1.] for k in range(1, nlags+1): pacf.append(yule_walker(x, k, method=method)[0][-1]) return np.array(pacf) #NOTE: this is incorrect.
[docs]def pacf_ols(x, nlags=40): '''Calculate partial autocorrelations 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. Returns ------- pacf : 1d array partial autocorrelations, maxlag+1 elements Notes ----- This solves a separate OLS estimation for each desired lag. ''' #TODO: add warnings for Yule-Walker #NOTE: demeaning and not using a constant gave incorrect answers? #JP: demeaning should have a better estimate of the constant #maybe we can compare small sample properties with a MonteCarlo xlags, x0 = lagmat(x, nlags, original='sep') #xlags = sm.add_constant(lagmat(x, nlags), prepend=True) xlags = add_constant(xlags, prepend=True) pacf = [1.] for k in range(1, nlags+1): res = OLS(x0[k:], xlags[k:,:k+1]).fit() #np.take(xlags[k:], range(1,k+1)+[-1], pacf.append(res.params[-1]) return np.array(pacf)
[docs]def pacf(x, nlags=40, method='ywunbiased'): '''Partial autocorrelation estimated Parameters ---------- x : 1d array observations of time series for which pacf is calculated maxlag : int largest lag for which pacf is returned method : 'ywunbiased' (default) or 'ywmle' or 'ols' specifies which method for the calculations to use, - yw or ywunbiased : yule walker with bias correction in denominator for acovf - ywm or ywmle : yule walker without bias correction - ols - regression of time series on lags of it and on constant - ld or ldunbiased : Levinson-Durbin recursion with bias correction - ldb or ldbiased : Levinson-Durbin recursion without bias correction Returns ------- pacf : 1d array partial autocorrelations, nlags elements, including lag zero Notes ----- This solves yule_walker equations or ols for each desired lag and contains currently duplicate calculations. ''' if method == 'ols': return pacf_ols(x, nlags=nlags) elif method in ['yw', 'ywu', 'ywunbiased', 'yw_unbiased']: return pacf_yw(x, nlags=nlags, method='unbiased') elif method in ['ywm', 'ywmle', 'yw_mle']: return pacf_yw(x, nlags=nlags, method='mle') elif method in ['ld', 'ldu', 'ldunbiase', 'ld_unbiased']: acv = acovf(x, unbiased=True) ld_ = levinson_durbin(acv, nlags=nlags, isacov=True) #print 'ld', ld_ return ld_[2] elif method in ['ldb', 'ldbiased', 'ld_biased']: #inconsistent naming with ywmle acv = acovf(x, unbiased=False) ld_ = levinson_durbin(acv, nlags=nlags, isacov=True) return ld_[2] else: raise ValueError('method not available')
[docs]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:]
[docs]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))
[docs]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 and Davis. """ 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 #copied from nitime and scikits\statsmodels\sandbox\tsa\examples\try_ld_nitime.py #TODO: check what to return, for testing and trying out returns everything
[docs]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 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) with a leading 1 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 #rename compared to nitime #from nitime ## if sxx is not None and type(sxx) == np.ndarray: ## sxx_m = sxx[:order+1] ## else: ## sxx_m = ut.autocov(s)[:order+1] if isacov: sxx_m = s else: sxx_m = acovf(s)[: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 xrange(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 xrange(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) pacf_[0] = 1. return sigma_v, arcoefs, pacf_, sig, phi #return everything
[docs]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, (nobs,2) 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 # lazy import resli = {} for mlg in range(1, maxlag+1): result = {} if verbose: print '\nGranger Causality' print 'number of lags (no zero)', mlg mxlg = mlg #+ 1 # Note number of lags starting at zero in lagmat # 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]) dtajoint = add_constant(dta[:,1:]) else: raise ValueError('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-1,mxlg-1)), np.eye(mxlg-1, mxlg-1),\ np.zeros((mxlg-1, 1)))) 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(y1, y2, regression="c"): """ This is a simple cointegration test. Uses unit-root test on residuals to test for cointegrated relationship See Hamilton (1994) 19.2 Parameters ---------- y1 : array_like, 1d first element in cointegrating vector y2 : array_like remaining elements in cointegrating vector c : str {'c'} Included in regression * 'c' : Constant Returns ------- coint_t : float t-statistic of unit-root test on residuals pvalue : float MacKinnon's approximate p-value based on MacKinnon (1994) crit_value : dict Critical values for the test statistic at the 1 %, 5 %, and 10 % levels. 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 are obtained through regression surface approximation from MacKinnon 1994. References ---------- MacKinnon, J.G. 1994. "Approximate asymptotic distribution functions for unit-root and cointegration tests. `Journal of Business and Economic Statistics` 12, 167-76. """ regression = regression.lower() if regression not in ['c','nc','ct','ctt']: raise ValueError("regression option %s not understood") % regression y1 = np.asarray(y1) y2 = np.asarray(y2) if regression == 'c': y2 = add_constant(y2) st1_resid = OLS(y1, y2).fit().resid #stage one residuals lgresid_cons = add_constant(st1_resid[0:-1]) uroot_reg = OLS(st1_resid[1:], lgresid_cons).fit() coint_t = (uroot_reg.params[0]-1)/uroot_reg.bse[0] pvalue = mackinnonp(coint_t, regression="c", N=2, lags=None) crit_value = mackinnoncrit(N=1, regression="c", nobs=len(y1)) return coint_t, pvalue, crit_value __all__ = ['acovf', 'acf', 'pacf', 'pacf_yw', 'pacf_ols', 'ccovf', 'ccf', 'periodogram', 'q_stat', 'coint'] if __name__=="__main__": import statsmodels.api as sm data = sm.datasets.macrodata.load().data x = data['realgdp'] # adf is tested now. adf = adfuller(x,4, autolag=None) adfbic = adfuller(x, autolag="bic") adfaic = adfuller(x, autolag="aic") adftstat = adfuller(x, autolag="t-stat") # acf is tested now acf1,ci1,Q,pvalue = acf(x, nlags=40, confint=95, qstat=True) acf2, ci2,Q2,pvalue2 = acf(x, nlags=40, confint=95, fft=True, qstat=True) acf3,ci3,Q3,pvalue3 = acf(x, nlags=40, confint=95, qstat=True, unbiased=True) acf4, ci4,Q4,pvalue4 = acf(x, nlags=40, confint=95, fft=True, qstat=True, unbiased=True) # pacf is tested now # pacf1 = pacorr(x) # pacfols = pacf_ols(x, nlags=40) # pacfyw = pacf_yw(x, nlags=40, method="mle") y = np.random.normal(size=(100,2)) grangercausalitytests(y,2)