**I want to fit a SARIMA model** and have daily sales data with a weekly seasonal pattern (frequency = 7) with this auto correlation function (ACF):

Clearly, there are **seasonal effects** as the spikes of the ACF indicate. As explained in *(Shumway, Stoffer (2011): Time Series Analysis and its applications. Example 3.46, Figure 3.23.)* the ACF of my time series also indicates seasonal non-stationarity and therefore a unit root in the seasonal component.

After **differencing** at lag $k=7$ ($y_t = y_t – y_{t-7}$) the ACF looks like this:

Since I wanted to automate this process I wanted to **test for a unit root** with the R package `uroot`

using the Canova and Hansen test. This test assumes no unit root in $H_0$

However, when I execute:

`z <- ts(data[,1], frequency=7) res <- ch.test(z, type = "dummy", sid = 7) res `

I get the result:

` Canova and Hansen test for seasonal stability data: z statistic pvalue [1,] 0.1584 0.4785 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Test type: seasonal dummies NW covariance matrix lag order: 12 First order lag: no Other regressors: no P-values: interpolation in original tables `

Am I right assuming that the test gives me a $text{p-value}=0.4785$ and therefore I cannot reject $H_0$ on a 95% confidence level and therefore have to assume there is no unit root in the seasonal component? And if yes, how does this fit the ACF indicating non-stationarity in the seasonal component? Is there a difference between unit root in the seasonal component and non-stationary seasonal component?

Am I missing/confusing something here?

Exemplary plot of the first 30 days:

All data:

`[ 0, 5530, 4327, 4486, 4997, 0, 7176, 5580, 5471, 4892, 4881, 4952, 0, 4717, 3900, 4008, 4044, 4127, 5182, 0, 5394, 5720, 5578, 5195, 5586, 5598, 0, 4055, 3725, 4601, 4709, 5633, 5970, 0, 7032, 6049, 6140, 5499, 5681, 5370, 0, 4409, 4015, 4252, 4241, 4809, 6154, 0, 6407, 5386, 5660, 5261, 5000, 5237, 0, 4038, 3794, 4558, 4676, 4611, 5350, 0, 7675, 6300, 5973, 5637, 5853, 5578, 0, 4949, 3853, 4341, 5108, 4925, 5003, 0, 7072, 6563, 5598, 5179, 5506, 5603, 0, 6729, 6686, 6660, 7285, 0, 7132, 0, 0, 5484, 4625, 4293, 4390, 5075, 0, 6046, 5514, 4903, 4366, 5263, 4773, 0, 3941, 3357, 3649, 2952, 4303, 4350, 0, 5672, 4492, 4309, 3939, 4841, 5726, 0, 5821, 5925, 0, 6486, 6027, 5912, 0, 4568, 4624, 5230, 0, 5409, 5064, 0, 6106, 5083, 4790, 4448, 4856, 4413, 0, 0, 4789, 3559, 4030, 4232, 4408, 0, 5519, 4995, 5784, 0, 7893, 5693, 0, 5422, 5220, 5012, 4881, 4315, 4262, 0, 4291, 3784, 3563, 3661, 4150, 4766, 0, 5337, 4633, 4180, 5011, 5173, 4341, 0, 3549, 4536, 3827, 3593, 4407, 5804, 0, 5614, 5868, 5875, 5701, 4986, 4090, 0, 4475, 4047, 3963, 4053, 4134, 4015, 0, 6377, 5643, 4946, 4904, 4275, 4421, 0, 4149, 3937, 3849, 3502, 4221, 3352, 0, 6290, 5773, 5572, 4994, 4494, 4461, 0, 4086, 3582, 4143, 3680, 3257, 3768, 0, 5326, 5299, 4724, 4575, 4552, 4078, 0, 3954, 3492, 3096, 3703, 3493, 4752, 0, 5482, 5156, 4583, 4804, 5469, 5317, 0, 4195, 3928, 3343, 3883, 3836, 4663, 0, 5893, 5403, 5103, 5079, 5187, 4663, 0, 4249, 3685, 3946, 3717, 3516, 3909, 0, 4770, 4274, 4212, 4481, 4669, 4175, 0, 4088, 4178, 4001, 0, 5447, 5352, 0, 6004, 5452, 4957, 5286, 5896, 4954, 0, 3773, 3355, 3383, 3223, 3566, 3667, 0, 5055, 4068, 4612, 4044, 4573, 4347, 0, 3557, 3737, 4119, 4035, 5471, 5233, 0, 5861, 5348, 5572, 5009, 5242, 4681, 0, 4014, 4017, 3310, 3795, 4016, 5071, 0, 5927, 5143, 4936, 4623, 5058, 5297, 0, 4600, 4201, 4561, 4614, 5623, 6373, 0, 6943, 6293, 5688, 5691, 6112, 5799, 0, 4462, 4228, 4301, 4074, 4703, 6119, 0, 8277, 7356, 7821, 6788, 8414, 8043, 0, 9528, 3204, 0, 0, 6110, 5659, 0, 7193, 2362, 0, 4969, 4190, 5173, 0, 6194, 5539, 4931, 4396, 4486, 4454, 0, 3994, 3621, 3776, 3632, 3803, 5128, 0, 6148, 5151, 4562, 4597, 5098, 4546, 0, 3601, 3581, 3789, 4549, 4906, 5292, 0, 5839, 5540, 5031, 4823, 5596, 4726, 0, 3591, 3461, 3662, 3638, 4647, 5258, 0, 5393, 4602, 4935, 4563, 4727, 4820, 0, 3826, 3499, 4108, 3826, 5031, 5307, 0, 6198, 5397, 4916, 4585, 5340, 4400, 0, 3573, 3588, 3805, 3224, 3883, 5042, 0, 5563, 5200, 4036, 4023, 4510, 5241, 0, 3850, 3136, 3225, 3847, 3987, 4583, 0, 6008, 5407, 4834, 4556, 4738, 4599, 0, 3751, 3990, 3919, 3948, 3733, 4435, 0, 6803, 6196, 6381, 6160, 0, 5871, 0, 0, 4133, 3482, 3376, 3710, 3989, 0, 5488, 5923, 5870, 0, 6790, 5498, 0, 5325, 5055, 5075, 5222, 5237, 5996, 0, 3957, 3535, 3710, 3589, 3863, 3785, 0, 5637, 4909, 4298, 3801, 4968, 4574, 0, 3912, 3971, 4616, 0, 5407, 4985, 0, 6071, 5614, 5229, 5278, 4957, 4923, 0, 0, 4607, 4207, 3702, 3901, 4757, 0, 4788, 5770, 5256, 0, 6362, 3982, 0, 3755, 3351, 3727, 2749, 4019, 5162, 0, 5738, 5138, 5161, 4756, 4025, 5176, 0, 4223, 4081, 3547, 3741, 4020, 4409, 0, 5083, 5298, 5424, 4685, 4609, 4084, 0, 4507, 3933, 3806, 3557, 3858, 4266, 0, 5372, 5252, 5487, 5106, 5038, 4731, 0, 5655, 5433, 5337, 4154, 4451, 4174, 0, 3886, 3185, 4069, 3906, 3746, 4425, 0, 5623, 5190, 4280, 4327, 3971, 3582, 0, 3414, 3396, 3148, 3920, 3869, 4094, 0, 5464, 5008, 4978, 4346, 4706, 3959, 0, 3676, 3377, 3275, 3392, 3906, 4270, 0, 4611, 4381, 4383, 3740, 4128, 3911, 0, 3982, 3407, 3405, 2462, 3518, 3914, 0, 5280, 4919, 4712, 5400, 0, 5355, 0, 5402, 5439, 4944, 4654, 4396, 4743, 0, 4033, 3681, 3775, 3430, 3720, 4169, 0, 4652, 4678, 4868, 4466, 4196, 4596, 0, 4260, 3364, 3761, 4042, 4161, 6532, 0, 5857, 5253, 4838, 4785, 5220, 4720, 0, 5474, 4479, 4677, 3869, 5334, 4967, 0, 3582, 3890, 3894, 4963, 4594, 5849, 0, 6527, 5815, 5328, 6144, 7195, 7066, 0, 7380, 6467, 6454, 7016, 6207, 6185, 0, 4884, 4345, 4915, 5060, 5269, 7263, 0, 8069, 7739, 7523, 7785, 7558, 8367, 0, 9331, 7959, 3659, 0, 0, 6057, 0, 6463, 6466, 2605, 0, 5509, 5023, 0, 6239, 4574, 4796, 3716, 3998, 4599, 0, 5346, 4924, 4541, 4295, 4161, 5255, 0, 3721, 3680, 3299, 3492, 3586, 4840, 0, 4781, 4806, 4310, 5171, 5577, 5363, 0, 6038, 4901, 4672, 4394, 5022, 4663, 0, 3965, 3136, 3735, 3900, 4726, 5015, 0, 4303, 4833, 4180, 4460, 4651, 4475, 0, 3598, 4054, 3875, 4042, 4708, 5289, 0, 5942, 5451, 5568, 4419, 5397, 4592, 0, 3701, 3805, 4170, 3141, 3725, 5225, 0, 5695, 4806, 3858, 4748, 4057, 3909, 0, 3565, 3547, 3531, 3932, 4005, 5208, 0, 6714, 6206, 6816, 6574, 0, 6709, 0, 0, 4163, 4194, 3467, 3549, 4173, 0, 5377, 4648, 4110, 4116, 4718, 4594, 0, 3722, 3037, 3319, 3076, 3198, 4318, 0, 5575, 5199, 5775, 6228, 0, 5850, 0, 5591, 4564, 4960, 4529, 4683, 4945, 0, 3551, 3547, 3998, 0, 4178, 4431, 0, 5280, 5235, 4735, 3755, 4459, 4276, 0, 0, 4211, 4083, 4111, 4656, 5592, 0, 5774, 5450, 5809, 0, 5384, 4183, 0, 4071, 4102, 3591, 3627, 3695, 4256, 0, 5518, 4852, 4000, 4645, 4202, 4097, 0, 3846, 3762, 3346, 3533, 3317, 4019, 0, 5197, 5735, 5223, 5558, 4665, 4797, 0, 4359, 3650, 3797, 3897, 3808, 3530, 0, 5054, 5042, 4767, 4427, 4852, 4406, 0, 4395, 3558, 3464, 3769, 3706, 4364, 0, 6102, 5011, 4782, 5020, 5263] `

EDIT:

I edited the question a couple of times in order to feedback the comments below.

EDIT:

Differencing like suggested above seems the wrong way to go (see comments). Furthermore the statement there is a unit root could not be backed further (see comments). I will look into the OCSB test before writing an answer just to be sure.

**Contents**hide

#### Best Answer

Just because the ACF shows seasonal spikes with relatively slow decay it can not automatically be concluded that there is a unit-root in the seasonal componend.

Actually (Ghysels, Osborn (2001): The Econometic Analysis of Seasonal Time Series, p.29) states:

"[…] for the sample sizes often observed in practice, it may be difficult to discriminate between deterministic seasonality and a seasonal unit root process."

Furthermore on page 42 they write:

"It is now well known that a series generated by a unit root process can wander widely and smoothly over time without any inherent tendency to return to its underlying mean value […]. In the seasonal context, there are S unit root processes, none of which has an inherent tendency to return to a deterministic pattern. As a result, the values for the seasons can wander widely and smoothly in relation to each other […]"

Therefore it is useful to visualize the time series of the different seasons $S$ and see if they look non-stationary. In our case we have daily data and model each day as a season. So we get seasons $S=7$. Plotting gives:

None of the single seasonal time series looks particularly non-stationary.

Therefore the Canova and Hansen test statistics is perfectly fine. The same result is obtained when using the `forecast::nsdiffs()`

function or the `OCSBtest()`

function from the `R forecast package`

source code on github.

### Similar Posts:

- Solved – Interpretation of Canova and Hansen test for seasonal unit-root with R uroot package
- Solved – Interpretation of Canova and Hansen test for seasonal unit-root with R uroot package
- Solved – Treating non-stationarity of time series in seasonal adjusted data with R
- Solved – How does R’s auto.arima() function determine the order of differencing when estimating a regression with seasonal ARIMA errors
- Solved – How does R’s auto.arima() function determine the order of differencing when estimating a regression with seasonal ARIMA errors