# Spurious Seasonality Detection: A Non-Parametric Test Proposal

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Ordinal Pattern Analysis

- (1)
- Forbidden pattern: a pattern that does not appear within the sample.
- (2)
- Rare pattern: a pattern that seldom appears.
- (3)
- Preferred pattern: a pattern that emerges more often than expected by the uniform distribution.

## 3. Day-of-the-Week Effect: A Redefinition of the Problem

**Definition**

**1.**

**Hypothesis**

**1.**

**Definition**

**2.**

**Hypothesis**

**2.**

**Hypothesis**

**3.**

**Hypothesis**

**4.**

**Hypothesis**

**5.**

## 4. Simulation of Fractional Brownian Motion

`wfbm`function in order to simulate fractional Brownian motion for $\mathcal{H}=\{0.1,\dots ,0.9\}$, where $\mathcal{H}$ is the Hurst exponent. Then, we take first differences in the time series in order to obtain the corresponding fractional Gaussian noise (fGn). The Hurst exponent H characterizes the scaling behavior of the range of cumulative departures of a time series from its mean. The study of long-range dependence can be traced back to a seminal paper by Hurst (1951), whose original methodology was applied to detect long memory in hydrologic time series. This method was also explored by Mandelbrot and Wallis (1968) and later introduced in the study of economic time series by Mandelbrot (1972). If the series of first differences is a white noise, then its $\mathcal{H}=0.5$. Alternatively, Hurst exponents greater than 0.5 reflect persistent processes and less than 0.5 define antipersistent processes.

## 5. Empirical Application

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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1 | |

2 | Let us assume that the time series is characterized by a continuous distribution. |

**Table 1.**Ordinal patterns. Each number $\{0,1,2,3,4\}$ of a pattern represents a day of the week, beginning on Monday. The position of the numbers in a pattern represents the increasing order of returns within a week.

${\mathit{P}}_{\mathit{xx}}$ | Pattern | ${\mathit{P}}_{\mathit{xx}}$ | Pattern | ${\mathit{P}}_{\mathit{xx}}$ | Pattern | ${\mathit{P}}_{\mathit{xx}}$ | Pattern | ${\mathit{P}}_{\mathit{xx}}$ | Pattern |
---|---|---|---|---|---|---|---|---|---|

1 | 01234 | 25 | 10234 | 49 | 20134 | 73 | 30124 | 97 | 40123 |

2 | 01243 | 26 | 10243 | 50 | 20143 | 74 | 30142 | 98 | 40132 |

3 | 01324 | 27 | 10324 | 51 | 20314 | 75 | 30214 | 99 | 40213 |

4 | 01342 | 28 | 10342 | 52 | 20341 | 76 | 30241 | 100 | 40231 |

5 | 01423 | 29 | 10423 | 53 | 20413 | 77 | 30412 | 101 | 40312 |

6 | 01432 | 30 | 10432 | 54 | 20431 | 78 | 30421 | 102 | 40321 |

7 | 02134 | 31 | 12034 | 55 | 21034 | 79 | 31024 | 103 | 41023 |

8 | 02143 | 32 | 12043 | 56 | 21043 | 80 | 31042 | 104 | 41032 |

9 | 02314 | 33 | 12304 | 57 | 21304 | 81 | 31204 | 105 | 41203 |

10 | 02341 | 34 | 12340 | 58 | 21340 | 82 | 31240 | 106 | 41230 |

11 | 02413 | 35 | 12403 | 59 | 21403 | 83 | 31402 | 107 | 41302 |

12 | 02431 | 36 | 12430 | 60 | 21430 | 84 | 31420 | 108 | 41320 |

13 | 03124 | 37 | 13024 | 61 | 23014 | 85 | 32014 | 109 | 42013 |

14 | 03142 | 38 | 13042 | 62 | 23041 | 86 | 32041 | 110 | 42031 |

15 | 03214 | 39 | 13204 | 63 | 23104 | 87 | 32104 | 111 | 42103 |

16 | 03241 | 40 | 13240 | 64 | 23140 | 88 | 32140 | 112 | 42130 |

17 | 03412 | 41 | 13402 | 65 | 23401 | 89 | 32401 | 113 | 42301 |

18 | 03421 | 42 | 13420 | 66 | 23410 | 90 | 32410 | 114 | 42310 |

19 | 04123 | 43 | 14023 | 67 | 24013 | 91 | 34012 | 115 | 43012 |

20 | 04132 | 44 | 14032 | 68 | 24031 | 92 | 34021 | 116 | 43021 |

21 | 04213 | 45 | 14203 | 69 | 24103 | 93 | 34102 | 117 | 43102 |

22 | 04231 | 46 | 14230 | 70 | 24130 | 94 | 34120 | 118 | 43120 |

23 | 04312 | 47 | 14302 | 71 | 24301 | 95 | 34201 | 119 | 43201 |

24 | 04321 | 48 | 14320 | 72 | 24310 | 96 | 34210 | 120 | 43210 |

$\mathcal{H}$ | $\mathcal{H}$ | ||||
---|---|---|---|---|---|

0.1 | ${\chi}^{2}$ | 224.82856 *** | 0.6 | ${\chi}^{2}$ | 20.42699 |

#rejections | 1000 | #rejections | 348 | ||

0.2 | ${\chi}^{2}$ | 140.50932 * | 0.7 | ${\chi}^{2}$ | 86.57005 |

#rejections | 1000 | #rejections | 996 | ||

0.3 | ${\chi}^{2}$ | 67.89548 | 0.8 | ${\chi}^{2}$ | 204.51209 *** |

#rejections | 970 | #rejections | 1000 | ||

0.4 | ${\chi}^{2}$ | 18.10740 | 0.9 | ${\chi}^{2}$ | 386.14031 *** |

#rejections | 317 | #rejections | 1000 | ||

0.5 | ${\chi}^{2}$ | 0.13315 | |||

#rejections | 50 |

$\mathcal{H}$ | M | T | X | T | F | |
---|---|---|---|---|---|---|

0.1 | ${\chi}^{2}$ | 5.47646 | 1.94578 | 5.40877 | 2.06613 | 5.80607 |

#rejections | 461 | 176 | 437 | 176 | 464 | |

0.2 | ${\chi}^{2}$ | 3.83407 | 1.34267 | 3.30037 | 1.33138 | 3.83153 |

#rejections | 308 | 124 | 275 | 139 | 318 | |

0.3 | ${\chi}^{2}$ | 1.95798 | 0.62585 | 1.79598 | 0.59399 | 2.03637 |

#rejections | 186 | 87 | 146 | 86 | 162 | |

0.4 | ${\chi}^{2}$ | 0.62419 | 0.17731 | 0.47961 | 0.22011 | 0.60417 |

#rejections | 82 | 54 | 78 | 57 | 71 | |

0.5 | ${\chi}^{2}$ | 0.00666 | 0.00304 | 0.00078 | 0.00500 | 0.00159 |

#rejections | 51 | 53 | 54 | 57 | 59 | |

0.6 | ${\chi}^{2}$ | 0.76829 | 0.25922 | 0.67148 | 0.25509 | 0.87231 |

#rejections | 95 | 68 | 79 | 66 | 93 | |

0.7 | ${\chi}^{2}$ | 3.94460 | 1.16245 | 3.40845 | 1.22150 | 3.72159 |

#rejections | 318 | 123 | 293 | 114 | 320 | |

0.8 | ${\chi}^{2}$ | 9.90281 ** | 3.44099 | 8.96147 * | 3.61403 | 9.96604 ** |

#rejections | 706 | 270 | 657 | 295 | 719 | |

0.9 | ${\chi}^{2}$ | 20.42595 *** | 7.95607 * | 20.65903 *** | 7.86588 * | 20.71557 *** |

#rejections | 960 | 572 | 966 | 596 | 966 |

$\mathcal{H}$ | Worst Return | Best Return | ||||
---|---|---|---|---|---|---|

0.1 | ${\chi}^{2}$ | 5.66761 | 2.40628 | 4.47306 | 2.35024 | 5.80600 |

#rejections | 461 | 193 | 357 | 188 | 471 | |

0.2 | ${\chi}^{2}$ | 3.69734 | 1.58367 | 2.83837 | 1.46318 | 4.05746 |

#rejections | 299 | 133 | 220 | 144 | 335 | |

0.3 | ${\chi}^{2}$ | 2.08654 | 0.68270 | 1.44706 | 0.80686 | 1.98700 |

#rejections | 187 | 95 | 117 | 82 | 181 | |

0.4 | ${\chi}^{2}$ | 0.48517 | 0.21129 | 0.42396 | 0.22292 | 0.76205 |

#rejections | 73 | 60 | 68 | 51 | 89 | |

0.5 | ${\chi}^{2}$ | 0.00390 | 0.00301 | 0.00335 | 0.00009 | 0.00673 |

#rejections | 58 | 57 | 59 | 49 | 37 | |

0.6 | ${\chi}^{2}$ | 0.78880 | 0.34282 | 0.58457 | 0.24124 | 0.86898 |

#rejections | 95 | 66 | 85 | 56 | 91 | |

0.7 | ${\chi}^{2}$ | 3.72085 | 1.27291 | 3.09096 | 1.35108 | 4.02279 |

p-value | 0.44510 | 0.86595 | 0.54272 | 0.85265 | 0.40293 | |

#rejections | 294 | 115 | 264 | 127 | 315 | |

0.8 | ${\chi}^{2}$ | 10.08089 ** | 3.67766 | 8.59256 * | 3.66076 | 9.87346 ** |

#rejections | 700 | 306 | 631 | 296 | 700 | |

0.9 | ${\chi}^{2}$ | 20.63329 *** | 8.33357 * | 20.41387 *** | 7.67672 | 20.56505 *** |

#rejections | 965 | 617 | 963 | 570 | 962 |

Hurst | ${\mathit{p}}_{\mathit{e}}$ | ${\mathit{p}}_{\mathbf{0}}$ | ${\mathit{q}}_{\mathbf{0}}$ | z | # of Rejections | # ${\mathit{p}}_{\mathit{o}}\mathbf{>}{\mathit{p}}_{\mathit{e}}$ |
---|---|---|---|---|---|---|

0.10 | 0.20 | 0.18850 | 0.81150 | 0.97729 | 306 | 161 |

0.20 | 0.20 | 0.19034 | 0.80966 | 0.81787 | 232 | 222 |

0.30 | 0.20 | 0.19397 | 0.80603 | 0.50693 | 172 | 275 |

0.40 | 0.20 | 0.19742 | 0.80258 | 0.21572 | 124 | 359 |

0.50 | 0.20 | 0.20243 | 0.79757 | −0.20089 | 117 | 507 |

0.60 | 0.20 | 0.20842 | 0.79158 | −0.68858 | 107 | 670 |

0.70 | 0.20 | 0.21406 | 0.78594 | −1.13915 | 225 | 839 |

0.80 | 0.20 | 0.22125 | 0.77875 | −1.70120 ** | 380 | 924 |

0.90 | 0.20 | 0.22867 | 0.77133 | −2.26805 ** | 627 | 983 |

Hurst | ${\mathit{p}}_{\mathit{e}}$ | ${\mathit{p}}_{\mathbf{0}}$ | ${\mathit{q}}_{\mathbf{0}}$ | z | # of Rejections | # ${\mathit{p}}_{\mathit{o}}\mathbf{>}{\mathit{p}}_{\mathit{e}}$ |
---|---|---|---|---|---|---|

0.1 | 0.05 | 0.04013 | 0.95987 | 1.67154 ** | 562 | 45 |

0.2 | 0.05 | 0.04263 | 0.95737 | 1.21287 | 427 | 91 |

0.3 | 0.05 | 0.04531 | 0.95469 | 0.74944 | 302 | 155 |

0.4 | 0.05 | 0.04824 | 0.95176 | 0.27363 | 174 | 290 |

0.5 | 0.05 | 0.05197 | 0.94803 | −0.29438 | 78 | 507 |

0.6 | 0.05 | 0.05588 | 0.94412 | −0.85028 | 119 | 698 |

0.7 | 0.05 | 0.06074 | 0.93926 | −1.49392 * | 309 | 873 |

0.8 | 0.05 | 0.06681 | 0.93319 | −2.23719 ** | 589 | 983 |

0.9 | 0.05 | 0.07376 | 0.92624 | −3.01987 *** | 856 | 998 |

**Table 7.**Absolute frequency of each pattern. Whole period: 3 January 1966–8 December 2017. Each number $\{0,1,2,3,4\}$ of a pattern represents a day of the week, beginning on Monday. The position of the numbers in a pattern represents the increasing order of returns within a week.

Pattern | Abs. Freq. | Pattern | Abs. Freq. | Pattern | Abs. Freq. | Pattern | Abs. Freq. |
---|---|---|---|---|---|---|---|

42013 | 7 | 14320 | 16 | 42031 | 20 | 21403 | 24 |

23041 | 10 | 20314 | 16 | 42301 | 20 | 32140 | 24 |

24013 | 10 | 23140 | 16 | 43012 | 20 | 42310 | 24 |

02413 | 11 | 31204 | 16 | 43210 | 20 | 02431 | 25 |

13240 | 11 | 04213 | 17 | 10324 | 21 | 04123 | 25 |

13402 | 11 | 20413 | 17 | 12043 | 21 | 01423 | 26 |

23104 | 11 | 40312 | 17 | 13042 | 21 | 03412 | 26 |

30124 | 11 | 41230 | 17 | 14032 | 21 | 20134 | 26 |

40123 | 11 | 20431 | 18 | 34201 | 21 | 34012 | 26 |

40132 | 11 | 23410 | 18 | 02134 | 22 | 34210 | 26 |

41203 | 11 | 40231 | 18 | 03124 | 22 | 01342 | 27 |

43102 | 11 | 02143 | 19 | 10432 | 22 | 12403 | 27 |

20143 | 12 | 02341 | 19 | 21340 | 22 | 31420 | 27 |

24310 | 12 | 03142 | 19 | 21430 | 22 | 34021 | 27 |

42103 | 12 | 10342 | 19 | 24301 | 22 | 43201 | 27 |

20341 | 13 | 12340 | 19 | 32014 | 22 | 02314 | 28 |

23014 | 13 | 23401 | 19 | 34102 | 22 | 10423 | 28 |

13024 | 14 | 24031 | 19 | 41302 | 22 | 21304 | 28 |

14203 | 14 | 30421 | 19 | 01243 | 23 | 01234 | 29 |

24103 | 14 | 32041 | 19 | 03241 | 23 | 34120 | 29 |

24130 | 14 | 41032 | 19 | 10234 | 23 | 10243 | 30 |

31024 | 14 | 42130 | 19 | 12034 | 23 | 14023 | 30 |

41320 | 14 | 43120 | 19 | 14302 | 23 | 01432 | 31 |

30142 | 15 | 01324 | 20 | 21034 | 23 | 04132 | 31 |

30214 | 15 | 12430 | 20 | 32104 | 23 | 04231 | 31 |

30412 | 15 | 31042 | 20 | 03214 | 24 | 30241 | 31 |

40213 | 15 | 31402 | 20 | 04321 | 24 | 31240 | 31 |

41023 | 15 | 32401 | 20 | 13204 | 24 | 43021 | 31 |

12304 | 16 | 32410 | 20 | 14230 | 24 | 03421 | 34 |

13420 | 16 | 40321 | 20 | 21043 | 24 | 04312 | 34 |

**Table 8.**Absolute frequency of each day in each position. Whole period: 3 January 1966–8 December 2017. Columns 2 to 6 reflect the frequency a given day in terms of the worst return of its week, the next to worst return, etc. until the best return of its week. Q is the ${\chi}^{2}$ statistic defined in Equation (6). Hurst = 0.5471.

Worst Return | Best Return | Q | ||||
---|---|---|---|---|---|---|

Monday | 637 | 503 | 537 | 501 | 532 | 22.78967 *** |

Tuesday | 557 | 572 | 475 | 509 | 597 | 17.94834 *** |

Wednesday | 479 | 540 | 564 | 564 | 563 | 9.92989 ** |

Thursday | 570 | 505 | 564 | 581 | 490 | 12.66052 ** |

Friday | 467 | 590 | 570 | 555 | 528 | 16.74908 *** |

Total | 2710 | 2710 | 2710 | 2710 | 2710 | |

Q | 36.21402 *** | 11.25092 ** | 11.56089 ** | 9.12177 * | 11.92989 ** |

**Table 9.**Absolute frequency of each day in each position with shuffled data for the whole period. Columns 2 to 6 reflect the frequency a given day in terms of the worst return of its week, the next to worst return, etc. until the best return of its week. Q is the ${\chi}^{2}$ statistic defined in Equation (6).

Worst Return | Best Return | Q | ||||
---|---|---|---|---|---|---|

Mo | 506 | 469 | 501 | 484 | 480 | 1.91393 |

Tu | 488 | 508 | 472 | 498 | 474 | 1.95082 |

We | 513 | 475 | 491 | 471 | 490 | 2.24590 |

Th | 479 | 491 | 509 | 482 | 479 | 1.32787 |

Fr | 454 | 497 | 467 | 505 | 517 | 5.75410 |

Total | 2440 | 2440 | 2440 | 2440 | 2440 | |

Q | 4.47951 | 2.09016 | 2.69672 | 1.49590 | 2.43033 |

**Table 10.**Absolute frequency of each day in each position. Subperiod 1: 3 January 1966–9 September 1977. Columns 2 to 6 reflect the frequency a given day in terms of the worst return of its week, the next to worst return, etc. until the best return of its week. Q is the ${\chi}^{2}$ statistic defined in Equation (6). Hurst = 0.5633.

Worst Return | Best Return | Q | ||||
---|---|---|---|---|---|---|

Mo | 182 | 121 | 105 | 97 | 105 | 39.37705 *** |

Tu | 108 | 145 | 107 | 124 | 126 | 7.95082 * |

We | 108 | 99 | 120 | 142 | 141 | 12.21311 ** |

Th | 116 | 119 | 138 | 132 | 105 | 5.65574 |

Fr | 96 | 126 | 140 | 115 | 133 | 9.72131 ** |

Total | 610 | 610 | 610 | 610 | 610 | |

Q | 38.55738 *** | 8.88525 * | 9.00000 * | 9.65574 ** | 8.81967 * |

**Table 11.**Absolute frequency of each day in each position. Subperiod 2: 12 September 1977–19 May 1989. Columns 2 to 6 reflect the frequency a given day is the worst return of its week, the next to worst return, etc. until the best return of its week. Q is the ${\chi}^{2}$ statistic defined in Equation (6). Hurst = 0.5168.

Worst Return | Best Return | Q | ||||
---|---|---|---|---|---|---|

Mo | 165 | 111 | 110 | 107 | 117 | 19.37705 *** |

Tu | 138 | 117 | 111 | 104 | 140 | 8.60656 * |

We | 100 | 125 | 129 | 131 | 125 | 5.18033 |

Th | 112 | 119 | 129 | 148 | 102 | 10.11475 ** |

Fr | 95 | 138 | 131 | 120 | 126 | 8.90164 * |

Total | 610 | 610 | 610 | 610 | 610 | |

Q | 28.01639 *** | 3.44262 | 3.63934 | 10.73770 ** | 6.34426 |

**Table 12.**Absolute frequency of each day in each position. Subperiod 3: 22 May 1989–26 January 2001. Columns 2 to 6 reflect the frequency a given day in terms of the worst return of the week, the next to worst return, etc. until the best return of its week. Q is the ${\chi}^{2}$ statistic defined in Equation (6). Hurst = 0.4453.

Worst Return | Best Return | Q | ||||
---|---|---|---|---|---|---|

Mo | 109 | 105 | 133 | 126 | 137 | 6.72131 |

Tu | 136 | 124 | 103 | 119 | 128 | 4.96721 |

We | 89 | 141 | 147 | 125 | 108 | 18.68852 *** |

Th | 154 | 117 | 108 | 123 | 108 | 11.81967 ** |

Fr | 122 | 123 | 119 | 117 | 129 | 0.68852 |

Total | 610 | 610 | 610 | 610 | 610 | |

Q | 20.31148 *** | 5.57377 | 10.75410 ** | 0.49180 | 5.75410 |

**Table 13.**Absolute frequency of each day in each position. Subperiod 4: 29 January 2001–5 October 2012. Columns 2 to 6 reflect the frequency a given day in terms of the worst return of the week, the next to worst return, etc. until the best return of its week. Q is the ${\chi}^{2}$ statistic defined in Equation (6). Hurst = 0.4983

Worst Return | Best Return | Q | ||||
---|---|---|---|---|---|---|

Mo | 134 | 106 | 121 | 128 | 121 | 3.59016 |

Tu | 112 | 139 | 117 | 106 | 136 | 7.09836 |

We | 126 | 115 | 125 | 115 | 129 | 1.40984 |

Th | 131 | 105 | 121 | 125 | 128 | 3.40984 |

Fr | 107 | 145 | 126 | 136 | 96 | 13.45902 *** |

Total | 610 | 610 | 610 | 610 | 610 | |

Q | 4.63934 | 11.57377 ** | 0.42623 | 4.47541 | 7.85246 * |

**Table 14.**Absolute frequency of each day in each position. Subperiod 5: 8 October 2012–8 December 2017. Columns 2 to 6 reflect the frequency a given day in terms of the worst return of the week, the next to worst return, etc. until the best return of its week. Q is the ${\chi}^{2}$ statistic defined in Equation (6). Hurst = 0.4914.

Worst Return | Best Return | Q | ||||
---|---|---|---|---|---|---|

Mo | 47 | 60 | 68 | 43 | 52 | 7.51852 |

Tu. | 63 | 47 | 37 | 56 | 67 | 10.96296 ** |

We | 56 | 60 | 43 | 51 | 60 | 3.81481 |

Th | 57 | 45 | 68 | 53 | 47 | 6.22222 |

Fr | 47 | 58 | 54 | 67 | 44 | 6.18519 |

Total | 270 | 270 | 270 | 270 | 270 | |

Q | 3.55556 | 4.03704 | 14.85185 *** | 5.62963 | 6.62963 |

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**MDPI and ACS Style**

Bariviera, A.F.; Plastino, A.; Judge, G. Spurious Seasonality Detection: A Non-Parametric Test Proposal. *Econometrics* **2018**, *6*, 3.
https://doi.org/10.3390/econometrics6010003

**AMA Style**

Bariviera AF, Plastino A, Judge G. Spurious Seasonality Detection: A Non-Parametric Test Proposal. *Econometrics*. 2018; 6(1):3.
https://doi.org/10.3390/econometrics6010003

**Chicago/Turabian Style**

Bariviera, Aurelio F., Angelo Plastino, and George Judge. 2018. "Spurious Seasonality Detection: A Non-Parametric Test Proposal" *Econometrics* 6, no. 1: 3.
https://doi.org/10.3390/econometrics6010003