PACF function in statsmodels.tsa.stattools gives numbers greater than 1 when using ywunbiased?

Question

I have a dataframe which is of length 177 and I want to calculate and plot the partial auto-correlation function (PACF).

I have the data imported etc and I do:

from statsmodels.tsa.stattools import pacf
ys = pacf(data[key][array].diff(1).dropna(), alpha=0.05, nlags=176, method="ywunbiased")
xs = range(lags+1)
plt.figure()
plt.scatter(xs,ys[0])
plt.grid()
plt.vlines(xs, 0, ys[0])
plt.plot(ys[1])

The method used results in numbers greater than 1 for very long lags (90ish) which is incorrect and I get a RuntimeWarning: invalid value encountered in sqrtreturn rho, np.sqrt(sigmasq) but since I can't see their source code I don't know what this means.

To be honest, when I search for PACF, all the examples only carry out PACF up to 40 lags or 60 or so and they never have any significant PACF after lag=2 and so I couldn't compare to other examples either.

But when I use:

method="ols"
# or
method="ywmle"

the numbers are corrected. So it must be the algo they use to solve it.

I tried importing inspect and getsource method but its useless it just shows that it uses another package and I can't find that.

If you also know where the problem arises from, I would really appreciate the help.

For your reference, the values for data[key][array] are:

[1131.130005, 1144.939941, 1126.209961, 1107.300049, 1120.680054, 1140.839966, 1101.719971, 1104.23999, 1114.579956, 1130.199951, 1173.819946, 1211.920044, 1181.27002, 1203.599976, 1180.589966, 1156.849976, 1191.5, 1191.329956, 1234.180054, 1220.329956, 1228.810059, 1207.01001, 1249.47998, 1248.290039, 1280.079956, 1280.660034, 1294.869995, 1310.609985, 1270.089966, 1270.199951, 1276.660034, 1303.819946, 1335.849976, 1377.939941, 1400.630005, 1418.300049, 1438.23999, 1406.819946, 1420.859985, 1482.369995, 1530.619995, 1503.349976, 1455.27002, 1473.98999, 1526.75, 1549.380005, 1481.140015, 1468.359985, 1378.550049, 1330.630005, 1322.699951, 1385.589966, 1400.380005, 1280.0, 1267.380005, 1282.829956, 1166.359985, 968.75, 896.23999, 903.25, 825.880005, 735.090027, 797.869995, 872.8099980000001, 919.1400150000001, 919.320007, 987.4799800000001, 1020.6199949999999, 1057.079956, 1036.189941, 1095.630005, 1115.099976, 1073.869995, 1104.48999, 1169.430054, 1186.689941, 1089.410034, 1030.709961, 1101.599976, 1049.329956, 1141.199951, 1183.26001, 1180.550049, 1257.640015, 1286.119995, 1327.219971, 1325.829956, 1363.609985, 1345.199951, 1320.640015, 1292.280029, 1218.890015, 1131.420044, 1253.300049, 1246.959961, 1257.599976, 1312.410034, 1365.680054, 1408.469971, 1397.910034, 1310.329956, 1362.160034, 1379.319946, 1406.579956, 1440.670044, 1412.160034, 1416.180054, 1426.189941, 1498.109985, 1514.680054, 1569.189941, 1597.569946, 1630.73999, 1606.280029, 1685.72998, 1632.969971, 1681.550049, 1756.540039, 1805.810059, 1848.359985, 1782.589966, 1859.449951, 1872.339966, 1883.949951, 1923.569946, 1960.22998, 1930.6700440000002, 2003.369995, 1972.290039, 2018.050049, 2067.560059, 2058.899902, 1994.9899899999998, 2104.5, 2067.889893, 2085.51001, 2107.389893, 2063.110107, 2103.840088, 1972.180054, 1920.030029, 2079.360107, 2080.409912, 2043.939941, 1940.2399899999998, 1932.22998, 2059.73999, 2065.300049, 2096.949951, 2098.860107, 2173.600098, 2170.949951, 2168.27002, 2126.149902, 2198.810059, 2238.830078, 2278.8701170000004, 2363.639893, 2362.719971, 2384.199951, 2411.800049, 2423.409912, 2470.300049, 2471.649902, 2519.360107, 2575.26001, 2584.840088, 2673.610107, 2823.810059, 2713.830078, 2640.8701170000004, 2648.050049, 2705.27002, 2718.3701170000004, 2816.290039, 2901.52002, 2913.97998]

Semidome

Could you put somewhere 1d array of input data? – Candice

@SeverinPappadeux Sorry for the delayed response. I added the data. Thanks for looking into this :) – Semidome

Answer 1

Your time series is pretty clearly not stationary, so that Yule-Walker assumptions are violated.

More generally, PACF is usually appropriate with stationary time series. You might difference your data first, before considering the partial autocorrelations.

Esemplastic

I differenced the series once and to make sure I carried out an ADF test and it showed no unit root but I still got the same problem. I differenced it twice just to make sure and still the same results. Also, I don't see any seasonality but I still took a moving average as well to cover that base. Also tried OLS as well but that too gives values over 1 :'(((((( Any ideas? – Semidome

You're right. I suppose that the problem may be that there are relatively few datapoints being used to compute these values for lags that long, and so the estimates are numerically not very stable. It's possible that the other methods are slightly less prone to numerical problems, or maybe they just don't run into trouble here. In any case, I would not put too much weight on any partial autocorrelation value for very long lags. – Esemplastic

Just to add a reference, Enders (2014) suggests that with sample size T, the PACF should only be computed up to lags T / 4. Since you have 176 datapoints, this rule of thumb would suggest not considering the PACF for lags greater than 44. – Esemplastic