from scipy.stats import norm
from numpy import array, polyval, inf, asarray
__all__ = ['mackinnonp','mackinnoncrit']
# These are the cut-off values for the left-tail vs. the rest of the
# tau distribution, for getting the p-values
tau_star_nc = [-1.04, -1.53, -2.68, -3.09, -3.07, -3.77]
tau_min_nc = [-19.04,-19.62,-21.21,-23.25,-21.63,-25.74]
tau_max_nc = [inf,1.51,0.86,0.88,1.05,1.24]
tau_star_c = [-1.61, -2.62, -3.13, -3.47, -3.78, -3.93]
tau_min_c = [-18.83,-18.86,-23.48,-28.07,-25.96,-23.27]
tau_max_c = [2.74,0.92,0.55,0.61,0.79,1]
tau_star_ct = [-2.89, -3.19, -3.50, -3.65, -3.80, -4.36]
tau_min_ct = [-16.18,-21.15,-25.37,-26.63,-26.53,-26.18]
tau_max_ct = [0.7,0.63,0.71,0.93,1.19,1.42]
tau_star_ctt = [-3.21,-3.51,-3.81,-3.83,-4.12,-4.63]
tau_min_ctt = [-17.17,-21.1,-24.33,-24.03,-24.33,-28.22]
tau_max_ctt = [0.54,0.79,1.08,1.43,3.49,1.92]
small_scaling = array([1,1,1e-2])
tau_nc_smallp = [ [0.6344,1.2378,3.2496],
[1.9129,1.3857,3.5322],
[2.7648,1.4502,3.4186],
[3.4336,1.4835,3.19],
[4.0999,1.5533,3.59],
[4.5388,1.5344,2.9807]]
tau_nc_smallp = asarray(tau_nc_smallp)*small_scaling
tau_c_smallp = [ [2.1659,1.4412,3.8269],
[2.92,1.5012,3.9796],
[3.4699,1.4856,3.164],
[3.9673,1.4777,2.6315],
[4.5509,1.5338,2.9545],
[5.1399,1.6036,3.4445]]
tau_c_smallp = asarray(tau_c_smallp)*small_scaling
tau_ct_smallp = [ [3.2512,1.6047,4.9588],
[3.6646,1.5419,3.6448],
[4.0983,1.5173,2.9898],
[4.5844,1.5338,2.8796],
[5.0722,1.5634,2.9472],
[5.53,1.5914,3.0392]]
tau_ct_smallp = asarray(tau_ct_smallp)*small_scaling
tau_ctt_smallp = [ [4.0003,1.658,4.8288],
[4.3534,1.6016,3.7947],
[4.7343,1.5768,3.2396],
[5.214,1.6077,3.3449],
[5.6481,1.6274,3.3455],
[5.9296,1.5929,2.8223]]
tau_ctt_smallp = asarray(tau_ctt_smallp)*small_scaling
large_scaling = array([1,1e-1,1e-1,1e-2])
tau_nc_largep = [ [0.4797,9.3557,-0.6999,3.3066],
[1.5578,8.558,-2.083,-3.3549],
[2.2268,6.8093,-3.2362,-5.4448],
[2.7654,6.4502,-3.0811,-4.4946],
[3.2684,6.8051,-2.6778,-3.4972],
[3.7268,7.167,-2.3648,-2.8288]]
tau_nc_largep = asarray(tau_nc_largep)*large_scaling
tau_c_largep = [ [1.7339,9.3202,-1.2745,-1.0368],
[2.1945,6.4695,-2.9198,-4.2377],
[2.5893,4.5168,-3.6529,-5.0074],
[3.0387,4.5452,-3.3666,-4.1921],
[3.5049,5.2098,-2.9158,-3.3468],
[3.9489,5.8933,-2.5359,-2.721]]
tau_c_largep = asarray(tau_c_largep)*large_scaling
tau_ct_largep = [ [2.5261,6.1654,-3.7956,-6.0285],
[2.85,5.272,-3.6622,-5.1695],
[3.221,5.255,-3.2685,-4.1501],
[3.652,5.9758,-2.7483,-3.2081],
[4.0712,6.6428,-2.3464,-2.546],
[4.4735,7.1757,-2.0681,-2.1196] ]
tau_ct_largep = asarray(tau_ct_largep)*large_scaling
tau_ctt_largep = [ [3.0778,4.9529,-4.1477,-5.9359],
[3.4713,5.967,-3.2507,-4.2286],
[3.8637,6.7852,-2.6286,-3.1381],
[4.2736,7.6199,-2.1534,-2.4026],
[4.6679,8.2618,-1.822,-1.9147],
[5.0009,8.3735,-1.6994,-1.6928]]
tau_ctt_largep = asarray(tau_ctt_largep)*large_scaling
#NOTE: The Z-statistic is used when lags are included to account for
# serial correlation in the error term
z_star_nc = [-2.9,-8.7,-14.8,-20.9,-25.7,-30.5]
z_star_c = [-8.9,-14.3,-19.5,-25.1,-29.6,-34.4]
z_star_ct = [-15.0,-19.6,-25.3,-29.6,-31.8,-38.4]
z_star_ctt = [-20.7,-25.3,-29.9,-34.4,-38.5,-44.2]
# These are Table 5 from MacKinnon (1994)
# small p is defined as p in .005 to .150 ie p = .005 up to z_star
# Z* is the largest value for which it is appropriate to use these
# approximations
# the left tail approximation is
# p = norm.cdf(d_0 + d_1*log(abs(z)) + d_2*log(abs(z))**2 + d_3*log(abs(z))**3
# there is no Z-min, ie., it is well-behaved in the left tail
z_nc_smallp = array([[.0342, -.6376,0,-.03872],
[1.3426,-.7680,0,-.04104],
[3.8607,-2.4159,.51293,-.09835],
[6.1072,-3.7250,.85887,-.13102],
[7.7800,-4.4579,1.00056,-.14014],
[4.0253, -.8815,0,-.04887]])
z_c_smallp = array([[2.2142,-1.7863,.32828,-.07727],
[1.1662,.1814,-.36707,0],
[6.6584,-4.3486,1.04705,-.15011],
[3.3249,-.8456,0,-.04818],
[4.0356,-.9306,0,-.04776],
[13.9959,-8.4314,1.97411,-.22234]])
z_ct_smallp = array([ [4.6476,-2.8932,0.5832,-0.0999],
[7.2453,-4.7021,1.127,-.15665],
[3.4893,-0.8914,0,-.04755],
[1.6604,1.0375,-0.53377,0],
[2.006,1.1197,-0.55315,0],
[11.1626,-5.6858,1.21479,-.15428]])
z_ctt_smallp = array([ [3.6739,-1.1549,0,-0.03947],
[3.9783,-1.0619,0,-0.04394],
[2.0062,0.8907,-0.51708,0],
[4.9218,-1.0663,0,-0.04691],
[5.1433,-0.9877,0,-0.04993],
[23.6812,-14.6485,3.42909,-.33794]])
# These are Table 6 from MacKinnon (1994).
# These are well-behaved in the right tail.
# the approximation function is
# p = norm.cdf(d_0 + d_1 * z + d_2*z**2 + d_3*z**3 + d_4*z**4)
z_large_scaling = array([1,1e-1,1e-2,1e-3,1e-5])
z_nc_largep = array([ [0.4927,6.906,13.2331,12.099,0],
[1.5167,4.6859,4.2401,2.7939,7.9601],
[2.2347,3.9465,2.2406,0.8746,1.4239],
[2.8239,3.6265,1.6738,0.5408,0.7449],
[3.3174,3.3492,1.2792,0.3416,0.3894],
[3.729,3.0611,0.9579,0.2087,0.1943]])
z_nc_largep *= z_large_scaling
z_c_largep = array([ [1.717,5.5243,4.3463,1.6671,0],
[2.2394,4.2377,2.432,0.9241,0.4364],
[2.743,3.626,1.5703,0.4612,0.567],
[3.228,3.3399,1.2319,0.3162,0.3482],
[3.6583,3.0934,0.9681,0.2111,0.1979],
[4.0379,2.8735,0.7694,0.1433,0.1146]])
z_c_largep *= z_large_scaling
z_ct_largep = array([ [2.7117,4.5731,2.2868,0.6362,0.5],
[3.0972,4.0873,1.8982,0.5796,0.7384],
[3.4594,3.6326,1.4284,0.3813,0.4325],
[3.806,3.2634,1.0689,0.2402,0.2304],
[4.1402,2.9867,0.8323,0.16,0.1315],
[4.4497,2.7534,0.6582,0.1089,0.0773]])
z_ct_largep *= z_large_scaling
z_ctt_largep = array([ [3.4671,4.3476,1.9231,0.5381,0.6216],
[3.7827,3.9421,1.5699,0.4093,0.4485],
[4.052,3.4947,1.1772,0.2642,0.2502],
[4.3311,3.1625,0.9126,0.1775,0.1462],
[4.594,2.8739,0.707,0.1181,0.0838],
[4.8479,2.6447,0.5647,0.0827,0.0518]])
z_ctt_largep *= z_large_scaling
#TODO: finish this and then integrate them into adf function
[docs]def mackinnonp(teststat, regression="c", N=1, lags=None):
"""
Returns MacKinnon's approximate p-value for teststat.
Parameters
----------
teststat : float
"T-value" from an Augmented Dickey-Fuller regression.
regression : str {"c", "nc", "ct", "ctt"}
This is the method of regression that was used. Following MacKinnon's
notation, this can be "c" for constant, "nc" for no constant, "ct" for
constant and trend, and "ctt" for constant, trend, and trend-squared.
N : int
The number of series believed to be I(1). For (Augmented) Dickey-
Fuller N = 1.
Returns
-------
p-value : float
The p-value for the ADF statistic estimated using MacKinnon 1994.
References
----------
MacKinnon, J.G. 1994 "Approximate Asymptotic Distribution Functions for
Unit-Root and Cointegration Tests." Journal of Business & Economics
Statistics, 12.2, 167-76.
Notes
-----
For (A)DF
H_0: AR coefficient = 1
H_a: AR coefficient < 1
"""
maxstat = eval("tau_max_"+regression)
minstat = eval("tau_min_"+regression)
starstat = eval("tau_star_"+regression)
if teststat > maxstat[N-1]:
return 1.0
elif teststat < minstat[N-1]:
return 0.0
if teststat <= starstat[N-1]:
tau_coef = eval("tau_" + regression + "_smallp["+str(N-1)+"]")
# teststat = np.log(np.abs(teststat))
#above is only for z stats
else:
tau_coef = eval("tau_" + regression + "_largep["+str(N-1)+"]")
return norm.cdf(polyval(tau_coef[::-1], teststat))
# These are the new estimates from MacKinnon 2010
# the first axis is N -1
# the second axis is 1 %, 5 %, 10 %
# the last axis is the coefficients
tau_nc_2010 = [[ [-2.56574,-2.2358,-3.627,0], # N = 1
[-1.94100,-0.2686,-3.365,31.223],
[-1.61682, 0.2656, -2.714, 25.364]]]
tau_nc_2010 = asarray(tau_nc_2010)
tau_c_2010 = [[ [-3.43035,-6.5393,-16.786,-79.433], # N = 1, 1%
[-2.86154,-2.8903,-4.234,-40.040], # 5 %
[-2.56677,-1.5384,-2.809,0]], # 10 %
[ [-3.89644,-10.9519,-33.527,0], # N = 2
[-3.33613,-6.1101,-6.823,0],
[-3.04445,-4.2412,-2.720,0]],
[ [-4.29374,-14.4354,-33.195,47.433], # N = 3
[-3.74066,-8.5632,-10.852,27.982],
[-3.45218,-6.2143,-3.718,0]],
[ [-4.64332,-18.1031,-37.972,0], # N = 4
[-4.09600,-11.2349,-11.175,0],
[-3.81020,-8.3931,-4.137,0]],
[ [-4.95756,-21.8883,-45.142,0], # N = 5
[-4.41519,-14.0405,-12.575,0],
[-4.13157,-10.7417,-3.784,0]],
[ [-5.24568,-25.6688,-57.737,88.639], # N = 6
[-4.70693,-16.9178,-17.492,60.007],
[-4.42501,-13.1875,-5.104,27.877]],
[ [-5.51233,-29.5760,-69.398,164.295],# N = 7
[-4.97684,-19.9021,-22.045,110.761],
[-4.69648,-15.7315,-5.104,27.877]],
[ [-5.76202,-33.5258,-82.189,256.289], # N = 8
[-5.22924,-23.0023,-24.646,144.479],
[-4.95007,-18.3959,-7.344,94.872]],
[ [-5.99742,-37.6572,-87.365,248.316],# N = 9
[-5.46697,-26.2057,-26.627,176.382],
[-5.18897,-21.1377,-9.484,172.704]],
[ [-6.22103,-41.7154,-102.680,389.33],# N = 10
[-5.69244,-29.4521,-30.994,251.016],
[-5.41533,-24.0006,-7.514,163.049]],
[ [-6.43377,-46.0084,-106.809,352.752],# N = 11
[-5.90714,-32.8336,-30.275,249.994],
[-5.63086,-26.9693,-4.083,151.427]],
[ [-6.63790,-50.2095,-124.156,579.622],# N = 12
[-6.11279,-36.2681,-32.505,314.802],
[-5.83724,-29.9864,-2.686,184.116]]]
tau_c_2010 = asarray(tau_c_2010)
tau_ct_2010 = [[ [-3.95877,-9.0531,-28.428,-134.155], # N = 1
[-3.41049,-4.3904,-9.036,-45.374],
[-3.12705,-2.5856,-3.925,-22.380]],
[ [-4.32762,-15.4387,-35.679,0], # N = 2
[-3.78057,-9.5106,-12.074,0],
[-3.49631,-7.0815,-7.538,21.892]],
[ [-4.66305,-18.7688,-49.793,104.244], # N = 3
[-4.11890,-11.8922,-19.031,77.332],
[-3.83511,-9.0723,-8.504,35.403]],
[ [-4.96940,-22.4694,-52.599,51.314], # N = 4
[-4.42871,-14.5876,-18.228,39.647],
[-4.14633,-11.2500,-9.873,54.109]],
[ [-5.25276,-26.2183,-59.631,50.646], # N = 5
[-4.71537,-17.3569,-22.660,91.359],
[-4.43422,-13.6078,-10.238,76.781]],
[ [-5.51727,-29.9760,-75.222,202.253], # N = 6
[-4.98228,-20.3050,-25.224,132.03],
[-4.70233,-16.1253,-9.836,94.272]],
[ [-5.76537,-33.9165,-84.312,245.394], # N = 7
[-5.23299,-23.3328,-28.955,182.342],
[-4.95405,-18.7352,-10.168,120.575]],
[ [-6.00003,-37.8892,-96.428,335.92], # N = 8
[-5.46971,-26.4771,-31.034,220.165],
[-5.19183,-21.4328,-10.726,157.955]],
[ [-6.22288,-41.9496,-109.881,466.068], # N = 9
[-5.69447,-29.7152,-33.784,273.002],
[-5.41738,-24.2882,-8.584,169.891]],
[ [-6.43551,-46.1151,-120.814,566.823], # N = 10
[-5.90887,-33.0251,-37.208,346.189],
[-5.63255,-27.2042,-6.792,177.666]],
[ [-6.63894,-50.4287,-128.997,642.781], # N = 11
[-6.11404,-36.4610,-36.246,348.554],
[-5.83850,-30.1995,-5.163,210.338]],
[ [-6.83488,-54.7119,-139.800,736.376], # N = 12
[-6.31127,-39.9676,-37.021,406.051],
[-6.03650,-33.2381,-6.606,317.776]]]
tau_ct_2010 = asarray(tau_ct_2010)
tau_ctt_2010 = [[ [-4.37113,-11.5882,-35.819,-334.047], # N = 1
[-3.83239,-5.9057,-12.490,-118.284],
[-3.55326,-3.6596,-5.293,-63.559]],
[ [-4.69276,-20.2284,-64.919,88.884], # N =2
[-4.15387,-13.3114,-28.402,72.741],
[-3.87346,-10.4637,-17.408,66.313]],
[ [-4.99071,-23.5873,-76.924,184.782], # N = 3
[-4.45311,-15.7732,-32.316,122.705],
[-4.17280,-12.4909,-17.912,83.285]],
[ [-5.26780,-27.2836,-78.971,137.871], # N = 4
[-4.73244,-18.4833,-31.875,111.817],
[-4.45268,-14.7199,-17.969,101.92]],
[ [-5.52826,-30.9051,-92.490,248.096], # N = 5
[-4.99491,-21.2360,-37.685,194.208],
[-4.71587,-17.0820,-18.631,136.672]],
[ [-5.77379,-34.7010,-105.937,393.991], # N = 6
[-5.24217,-24.2177,-39.153,232.528],
[-4.96397,-19.6064,-18.858,174.919]],
[ [-6.00609,-38.7383,-108.605,365.208], # N = 7
[-5.47664,-27.3005,-39.498,246.918],
[-5.19921,-22.2617,-17.910,208.494]],
[ [-6.22758,-42.7154,-119.622,421.395], # N = 8
[-5.69983,-30.4365,-44.300,345.48],
[-5.42320,-24.9686,-19.688,274.462]],
[ [-6.43933,-46.7581,-136.691,651.38], # N = 9
[-5.91298,-33.7584,-42.686,346.629],
[-5.63704,-27.8965,-13.880,236.975]],
[ [-6.64235,-50.9783,-145.462,752.228], # N = 10
[-6.11753,-37.056,-48.719,473.905],
[-5.84215,-30.8119,-14.938,316.006]],
[ [-6.83743,-55.2861,-152.651,792.577], # N = 11
[-6.31396,-40.5507,-46.771,487.185],
[-6.03921,-33.8950,-9.122,285.164]],
[ [-7.02582,-59.6037,-166.368,989.879], # N = 12
[-6.50353,-44.0797,-47.242,543.889],
[-6.22941,-36.9673,-10.868,418.414]]]
tau_ctt_2010 = asarray(tau_ctt_2010)
[docs]def mackinnoncrit(N=1, regression ="c", nobs=inf):
"""
Returns the critical values for cointegrating and the ADF test.
In 2010 MacKinnon updated the values of his 1994 paper with critical values
for the augmented Dickey-Fuller tests. These new values are to be
preferred and are used here.
Parameters
----------
N : int
The number of series of I(1) series for which the null of
non-cointegration is being tested. For N > 12, the critical values
are linearly interpolated (not yet implemented). For the ADF test,
N = 1.
reg : str {'c', 'tc', 'ctt', 'nc'}
Following MacKinnon (1996), these stand for the type of regression run.
'c' for constant and no trend, 'tc' for constant with a linear trend,
'ctt' for constant with a linear and quadratic trend, and 'nc' for
no constant. The values for the no constant case are taken from the
1996 paper, as they were not updated for 2010 due to the unrealistic
assumptions that would underlie such a case.
nobs : int or np.inf
This is the sample size. If the sample size is numpy.inf, then the
asymptotic critical values are returned.
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
"""
reg = regression
if reg not in ['c','ct','nc','ctt']:
raise ValueError("regression keyword %s not understood") % reg
if nobs is inf:
return eval("tau_"+reg+"_2010["+str(N-1)+",:,0]")
else:
return polyval(eval("tau_"+reg+"_2010["+str(N-1)+",:,::-1].T"),1./nobs)
if __name__=="__main__":
pass