# What is the intuition for testing seasonal difference with OCSB test and its correct application?

I have daily time series data of a shop's revenue. Now I would like to test for seasonal differencing with the OCSB test originally intrduced in (Osborn et al. (1988): Seasonality and the Order of Integration for Consumption).

This test is explained in more detail in (Ghysels, Osborn (2001): The Econometic Analysis of Seasonal Time Series, p. 66ff.). There they discuss the test regression

$$\Delta_1 \Delta_S y_t = \beta_1 \Delta_S y_{t-1} + \beta_2\Delta_1 y_{t-S} + \epsilon_t, \quad \quad t=1,...,T.$$

Where $S$ is the seasonality, $y_t$ is the time series values and $\Delta$ the difference operator.

By using $\Delta_S = \Delta_1(1+L+...+L^{S-1})$ and $S(L)=\sum_{i=0}^{S-1}L^i$ they get:

$$\Delta_1 \Delta_S y_t = \beta_1 (1+L+...+L^{S-1}) \Delta_1 y_{t-1} + \beta_2\Delta_1 y_{t-S} + \epsilon_t=$$ $$\Delta_1 \Delta_S y_t = \beta_1 S(L) \Delta_1 y_{t-1} + \beta_2\Delta_1 y_{t-S} + \epsilon_t.$$

Note that now all variables impose a first difference $\Delta_1$. When only interested in testing seasonal differencing (Osborn, Rodriguez (2002): Asymptotic Distributions Of Seasonal Unit Root Tests: A Unifying Approach, p.224) tell us that we can simplify to:

$$\Delta_S y_t = \beta_1 S(L) y_{t-1} + \beta_2 y_{t-S} + u_t=$$ $$\Delta_S y_t = \beta_1 (y_{t-1}+y_{t-2}+...+y_{t-S}) + \beta_2 y_{t-S} + u_t=$$ $$\Delta_S y_t = \beta_1 (y_{t-1}+y_{t-2}+...+y_{t-S+1}) + (\beta_1+\beta_2) y_{t-S} + u_t.$$

The test has null hypothesis $H_0: \beta_1 = \beta_2= 0$ and we have to test the regression model's F-Statistics $F(\hat{\beta}_1\cap\hat\beta_2)$ and t-Statistics $t(\hat\beta_1), t(\hat\beta_2)$.

The critical test values are given in (Osborn et al. (1988): Seasonality and the Order of Integration for Consumption, Appendix 1).

So far the theory. I would like to implement this in python using the following code:

import statsmodels as sm

def OCSBtest(timeseries, period):

# timeseries is a pandas dataframe, period is an integer

# calculate Δ_s y_t:
diff_series = timeseries.diff(period)
# calculate S(l)y_t-1:
y1 = timeseries.shift(1).rolling(window=(period-1)).sum()
# calculate y_(t-s):
y2 = timeseries.shift(period)
# Ensure all arrays have same length:
y1 = y1[period:]
y2 = y2[period:]
diff_series = diff_series[(period):]
# construct regression matrix:
x_reg = np.c_[y1, y2]

# build regression model and fit:
try:
except:
# In case of numerical problems treat residuals as AR(1) model:
res = sm.tsa.statespace.SARIMAX(diff_series, x_reg, order=(1,0,0), seasonal_order=(0,0,0,period), trend='c').fit()

# Check t-statistics of the results
if (res.tvalues[1] < -1.94) or (res.tvalues[2] < -1.93):
D = 0
else:
D = 1

return D


To be honest I am not very confident I understood the test properly. However I tried to follow all of their steps.

My quesitons are:

1. Does anyone have any intution for the original approach of the OCSB test? Why do we build this particular regression model and what is an interpretation for $\beta_1\neq0$ or $\beta_2 \neq 0$?
2. Is there anyone who has experience with the implementation of this test and can give me feedback on my piece of code?

My seasonality is $S=7$ and my data is:

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5578, 5195, 5586, 5598,    0, 4055, 3725, 4601, 4709, 5633, 5970,
0, 7032, 6049, 6140, 5499, 5681, 5370,    0, 4409, 4015, 4252,
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4038, 3794, 4558, 4676, 4611, 5350,    0, 7675, 6300, 5973, 5637,
5853, 5578,    0, 4949, 3853, 4341, 5108, 4925, 5003,    0, 7072,
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6071, 5614, 5229, 5278, 4957, 4923,    0,    0, 4607, 4207, 3702,
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5623, 5190, 4280, 4327, 3971, 3582,    0, 3414, 3396, 3148, 3920,
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0, 4033, 3681, 3775, 3430, 3720, 4169,    0, 4652, 4678, 4868,
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3846, 3762, 3346, 3533, 3317, 4019,    0, 5197, 5735, 5223, 5558,
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5042, 4767, 4427, 4852, 4406,    0, 4395, 3558, 3464, 3769, 3706,
4364,    0, 6102, 5011, 4782, 5020, 5263]