# Spatial Dependence of Average Prices for Product Categories and Its Change over Time: Evidence from Daily Data

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data Sources

#### 2.2. Spatial Analysis Method and Study Design

_{it}is the average price, by index of region i and day t, and $\overline{A{P}_{t}}$ is the average price for the day t for all regions,

_{ij}is the elements of spatial weight matrix for regions i and j.

#### 2.3. Seasonality Analysis Methods

## 3. Results

#### 3.1. Spatial Variation

#### 3.2. Spatial Autocorrelation

^{b}estimates for spatial autocorrelation are presented in Table 3. The highest Moran’s I value (0.88) was registered for Potatoes on 12 August 2021. The closest to its expected value Moran’s I (−0.006 for binary contiguity matrix and −0.027 for inverse distance weights matrix) was recorded for Sugar on 31 October 2020.

^{b}(based on binary contiguity matrix) and Moran’s I

^{id}(based on inverse distance weights matrix) are relatively close. For this reason, we present here the main results for the case with binary contiguity matrix only. The corresponding estimates for the inverse distance weights matrix could be found in Appendix A.

^{b}remained in the interval between 0.29 and 0.65.

#### 3.3. Seasonality and Cycles in Spatial Autocorrelation

^{b}values for all categories, with the only exception of QS test for Potatoes (p-value = 0.13). The annual (for the period of 365.5 days) seasonality of Moran’s I

^{b}values for Potatoes was revealed by all tests except the Friedman Rank test. The non-seasonality is not rejected for Potatoes by Kruskal–Wallis test either (p-value = 0.0096), but the p-value is relatively lower than those for other product categories (Table 4).

^{id}(based on inverse distance weights matrix). The only difference is that the annual seasonality of Moran’s I

^{id}values for Potatoes was revealed by Welch’s ANOVA test, QS test, Ollech and Webel’s combined seasonality tests and rejected by the Friedman Rank test and the Kruskal–Wallis test (Table A1).

^{b}for all categories (Figure 3, Figure A2, Figure A3, Figure A4, Figure A5 and Figure A6). The ACF assesses the correlation between observations in a time series for a set of lags. The partial correlation for each lag shows the pairwise correlation between two observations after partial elimination of intermediate correlations.

^{b}for Potatoes changes in a pattern of a sine curve (Figure 3a). The autocorrelation coefficients are noticeably larger for lags at multiples of the seasonal frequency than for other lags, indicating seasonal patterns.

^{b}for Potatoes prices. As for other product categories, no pronounced signs of annual seasonality were revealed from Moran’s I

^{b}dynamics. Some of tests showed annual seasonality for Pasta (Table 4), but these results were not confirmed by ACF/PACF graphs (Figure A2).

## 4. Discussion

#### 4.1. Holidays and Special Events

^{b}= 0.75) was recorded on 8 March 2022, for Candies.

#### 4.2. Weekly Cycles

#### 4.3. Annual Seasonality

^{b}= 0.88 on 10 August 2021, compared to Moran’s I

^{b}= 0.44 on 5 June 2021) as a result, because the regions are geographically structured according to their specialization in agriculture (Figure 6a,b).

#### 4.4. External Shocks and State Regulation of Prices

#### 4.5. Spatial Variety vs. Spatial Autocorrelation

^{b}= 0.61, p-value < 0.001) and low (Moran’s I

^{b}= 0.12, p-value < 0.04) levels of spatial autocorrelation (Figure 7).

^{b}and Moran’s I

^{id}indexes (Figure 2 and Figure A1).

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Daily average regional price of Pasta (

**a**), Potatoes (

**b**), Sugar (

**c**), Candies (

**d**), Poultry meat (

**e**), Butter (

**f**) and global Moran’s I

^{id}index (based on inverse distance weights matrix), from 1 January 2019 to 31 March 2022.

## Appendix B

Pasta | Potatoes | Sugar | Candies | Poultry Meat | Butter | |
---|---|---|---|---|---|---|

Weekly cycle | ||||||

Kruskal–Wallis | 36.4, p-value < 0.000 | 36.7, p-value < 0.000 | 61.8, p-value < 0.000 | 290.3, p-value < 0.000 | 351.5, p-value < 0.000 | 256.3, p-value < 0.000 |

Welch’s ANOVA test | 5.4, p-value < 0.000 | 5.1, p-value < 0.000 | 7.9, p-value < 0.000 | 54.4, p-value < 0.000 | 74.1, p-value < 0.000 | 62.5, p-value < 0.000 |

Friedman Rank test | 31.6, p-value < 0.000 | 38.1, p-value < 0.000 | 55.3, p-value < 0.000 | 233.1, p-value < 0.000 | 307.4, p-value < 0.000 | 198.3, p-value < 0.000 |

QS test | 11.7, p-value = 0.003 | 5.2, p-value = 0.075 | 13.7, p-value = 0.001 | 244.9, p-value < 0.000 | 301.3, p-value < 0.000 | 142.1, p-value < 0.000 |

Ollech and Webel’s combined seasonality test | TRUE | TRUE | TRUE | TRUE | TRUE | TRUE |

Annual seasonality | ||||||

Kruskal–Wallis | 456.1, p-value = 0.0008 | 393.8, p-value = 0.139 | 350.9, p-value = 0.686 | 346.9, p-value = 0.738 | 413.5, p-value = 0.039 | 296.5, p-value = 0.996 |

Welch’s ANOVA test | 5.4, p-value < 0.000 | 5.1, p-value < 0.000 | 7.9, p-value < 0.000 | 54.4. p-value < 0.000 | 74.1, p-value < 0.000 | 62.5, p-value < 0.000 |

Friedman Rank test | 432.5, p-value = 0.0082 | 388.8, p-value = 0.183 | 346.4, p-value = 0.744 | 325.5, p-value = 0.93 | 394.5, p-value = 0.134 | 284.9, p-value = 0.999 |

QS test | 0, p-value = 1 | 11.4, p-value = 0.003 | 1.1, p-value = 0.58 | 0, p-value = 1 | 0, p-value = 1 | 0, p-value = 1 |

Ollech and Webel’s combined seasonality test | FALSE | TRUE | FALSE | FALSE | FALSE | FALSE |

## Appendix C

**Figure A2.**Autocorrelation function (ACF) of the Moran’s I

^{b}for Pasta prices of the original time series (

**a**) and of the first differences (

**c**); Partial autocorrelation function (PACF) of the Moran’s I

^{b}for Pasta prices of the original time series (

**b**) and of the first differences (

**d**). The Moran’s I

^{b}indexes are calculated based on binary contiguity matrices.

**Figure A3.**Autocorrelation function (ACF) of the Moran’s I

^{b}for Sugar prices of the original time series (

**a**) and of the first differences (

**c**); Partial autocorrelation function (PACF) of the Moran’s I

^{b}for Sugar prices of the original time series (

**b**) and of the first differences (

**d**). The Moran’s I

^{b}indexes are calculated based on binary contiguity matrices.

**Figure A4.**Autocorrelation function (ACF) of the Moran’s I

^{b}for Candies prices of the original time series (

**a**) and of the first differences (

**c**); Partial autocorrelation function (PACF) of the Moran’s I

^{b}for Candies prices of the original time series (

**b**) and of the first differences (

**d**). The Moran’s I

^{b}indexes are calculated based on binary contiguity matrices.

**Figure A5.**Autocorrelation function (ACF) of the Moran’s I

^{b}for Poultry meat prices of the original time series (

**a**) and of the first differences (

**c**); Partial autocorrelation function (PACF) of the Moran’s I

^{b}for Poultry meat prices of the original time series (

**b**) and of the first differences (

**d**). The Moran’s I

^{b}indexes are calculated based on binary contiguity matrices.

**Figure A6.**Autocorrelation function (ACF) of the Moran’s I

^{b}for Butter prices of the original time series (

**a**) and of the first differences (

**c**); Partial autocorrelation function (PACF) of the Moran’s I

^{b}for Butter prices of the original time series (

**b**) and of the first differences (

**d**). The Moran’s I

^{b}indexes are calculated based on binary contiguity matrices.

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**Figure 1.**Daily average regional price of Pasta (

**a**), Potatoes (

**b**), Sugar (

**c**), Candies (

**d**), Poultry meat (

**e**), Butter (

**f**) and respective coefficients of variation from 1 January 2019 to 31 March 2022.

**Figure 2.**Daily average regional price of Pasta (

**a**), Potatoes (

**b**), Sugar (

**c**), Candies (

**d**), Poultry meat (

**e**), Butter (

**f**) and global Moran’s I

^{b}index (based on binary contiguity matrix), from 1 January 2019 to 31 March 2022.

**Figure 3.**Autocorrelation function (ACF) of the Moran’s I

^{b}for Potatoes prices for original time series (

**a**) and for the first differences (

**c**); partial autocorrelation function (PACF) of the Moran’s I

^{b}for Potatoes prices for original time series (

**b**) and for the first differences (

**d**).

**Figure 4.**Daily average price and global Moran’s I

^{b}index for Candies (

**a**), Poultry meat (

**b**), and Butter (

**c**), from 15 March 2021 to 31 April 2022.

**Figure 5.**Spatial price of Butter on Sunday: 28 March 2021 (

**a**), 11 April 2021 (

**c**), 25 April 2021 (

**e**), and Monday: 29 March 2021 (

**b**), 12 April 2021 (

**d**), 26 April 2021 (

**f**).

**Figure 6.**Spatial price of Potatoes price, 5 June 2021 (

**a**), 10 August 2021 (

**b**). All regions, for which the data are not available for these dates, are marked as “undefined”.

**Figure 7.**Spatial distribution of Sugar price, 10 June 2021 (

**a**) and 7 September 2021 (

**b**). All regions, for which the data are not available for these dates, are marked as “undefined”.

Pasta | Potatoes | Sugar | Candies | Poultry Meat | Butter | |
---|---|---|---|---|---|---|

Price, ruble for kg | ||||||

Min | 32.04 | 5.68 | 6.71 | 77.88 | 47.96 | 117.45 |

Max | 214.12 | 151.44 | 188.57 | 2034.55 | 423.81 | 1041.46 |

Average | 82.78 | 35.29 | 43.95 | 310.48 | 175.88 | 531.98 |

SD | 20.0 | 16.4 | 13.3 | 80.6 | 37.5 | 97.6 |

Number of observations | ||||||

Total | 97,524 | 96,545 | 95,901 | 97,700 | 97,387 | 97,116 |

Min./day | 79 | 77 | 78 | 81 | 80 | 78 |

Pasta | Potatoes | Sugar | Candies | Poultry Meat | Butter | |
---|---|---|---|---|---|---|

Total spatial variation for all regions | 0.24 | 0.46 | 0.30 | 0.26 | 0.21 | 0.18 |

Minimum value of spatial variation | 0.151 | 0.152 | 0.102 | 0.142 | 140 | 0.115 |

Maximum value of spatial variation | 0.268 | 0.599 | 0.404 | 0.372 | 0.252 | 0.260 |

Pasta | Potatoes | Sugar | Candies | Poultry Meat | Butter | |
---|---|---|---|---|---|---|

Moran’s I^{b} (binary contiguity matrix) | ||||||

Min | 0.29 | 0.39 | −0.01 | 0.11 | 0.29 | 0.17 |

Max | 0.65 | 0.88 | 0.68 | 0.75 | 0.77 | 0.77 |

Mean | 0.53 | 0.68 | 0.38 | 0.44 | 0.63 | 0.47 |

Median | 0.54 | 0.69 | 0.45 | 0.46 | 0.64 | 0.48 |

SD | 0.04 | 0.08 | 0.18 | 0.10 | 0.08 | 0.09 |

Moran’s I^{id} (invers distance matrix) | ||||||

Min | 0.30 | 0.42 | −0.03 | 0.12 | 0.29 | 0.20 |

Max | 0.61 | 0.88 | 0.64 | 0.72 | 0.74 | 0.78 |

Mean | 0.51 | 0.68 | 0.36 | 0.46 | 0.60 | 0.50 |

Median | 0.51 | 0.68 | 0.41 | 0.48 | 0.61 | 0.51 |

SD | 0.03 | 0.07 | 0.17 | 0.09 | 0.07 | 0.09 |

**Table 4.**Tests results for seasonality in spatial autocorrelation (based on binary contiguity matrix).

Pasta | Potatoes | Sugar | Candies | Poultry Meat | Butter | |
---|---|---|---|---|---|---|

Weekly cycle | ||||||

Kruskal–Wallis | 72.7 p-value < 0.000 | 35.5 p-value < 0.000 | 65.1 p-value < 0.000 | 327.4 p-value < 0.000 | 361.1 p-value < 0.000 | 304 p-value < 0.000 |

Welch’s ANOVA test | 12.1 p-value < 0.000 | 4.8 p-value = 0.001 | 8.2 p-value < 0.000 | 63.4 p-value < 0.000 | 77 p-value < 0.000 | 8.7 p-value < 0.000 |

Friedman Rank test | 62.6 p-value < 0.000 | 32.8 p-value < 0.000 | 54.4 p-value < 0.000 | 267.9 p-value < 0.000 | 308.1 p-value < 0.000 | 239.5 p-value < 0.000 |

QS test | 27.7 p-value < 0.000 | 4.1 p-value = 0.13 | 9.0 p-value = 0.011 | 311.6 p-value < 0.000 | 304.3 p-value < 0.000 | 206.4 p-value < 0.000 |

Ollech and Webel’s combined seasonality test | TRUE | TRUE | TRUE | TRUE | TRUE | TRUE |

Annual seasonality | ||||||

Kruskal–Wallis | 447.3 p-value = 0.002 | 400.1 p-value = 0.096 | 339.6 p-value = 0.82 | 342.4 p-value = 0.79 | 395.4 p-value = 0.127 | 273.7 p-value = 0.99 |

Welch’s ANOVA test | 12.1 p-value < 0.000 | 4.8 p-value = 0.001 | 8.2 p-value < 0.000 | 63.4 p-value < 0.000 | 77.0 p-value < 0.000 | 8.7 p-value < 0.000 |

Friedman Rank test | 425.3 p-value = 0.015 | 391.6 p-value = 0.158 | 340 p-value = 0.817 | 322.7 p-value = 0.944 | 373.2 p-value = 0.37 | 26.6 p-value = 1 |

QS test | 0 p-value = 1 | 9.4 p-value = 0.009 | 0.5 p-value = 0.798 | 0 p-value = 1 | 6.2 p-value = 0.045 | 0 p-value = 1 |

Ollech and Webel’s combined seasonality test | FALSE | TRUE | FALSE | FALSE | FALSE | FALSE |

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## Share and Cite

**MDPI and ACS Style**

Timiryanova, V.; Lakman, I.; Prudnikov, V.; Krasnoselskaya, D.
Spatial Dependence of Average Prices for Product Categories and Its Change over Time: Evidence from Daily Data. *Forecasting* **2023**, *5*, 102-126.
https://doi.org/10.3390/forecast5010004

**AMA Style**

Timiryanova V, Lakman I, Prudnikov V, Krasnoselskaya D.
Spatial Dependence of Average Prices for Product Categories and Its Change over Time: Evidence from Daily Data. *Forecasting*. 2023; 5(1):102-126.
https://doi.org/10.3390/forecast5010004

**Chicago/Turabian Style**

Timiryanova, Venera, Irina Lakman, Vadim Prudnikov, and Dina Krasnoselskaya.
2023. "Spatial Dependence of Average Prices for Product Categories and Its Change over Time: Evidence from Daily Data" *Forecasting* 5, no. 1: 102-126.
https://doi.org/10.3390/forecast5010004