This paper describes testing the law of one price in 76 Russian regions for 69 goods belonging to a fixed set of goods and services, on the monthly data for the period from 2003 to 2015. To test the law of a one price, the approaches associated with testing the nonstationarity of panel data are applied. The literature devoted to statistical methods for panel unit roots is reviewed. The problem of cross-sectional correlation across time series and its influence on the statistical tests is discussed and various methods are proposed for taking into account this correlation. We review econometric papers in which methods for determining the proportion of stationary and non-stationary time series in a panel are proposed. These methods are robust to cross-sectional correlation. For Russian regional data, the modern tests for panel unit roots, which take into account the cross-sectional correlation between time series and allow to determine the fraction of stationary time series in the panel are applied. First, the data are tested for panel unit roots, and then the proportion of the time series in the panel are estimated. The results show the evidence in favor of the law of one price for most food products, medicines, household chemicals and some of the services provided by public companies. The relative prices of vegetables, cereals and gasoline are stationary in more than 70% of the regions. Violation of the law of one price is typical for garments, footwear, furniture, services provided by private companies. The reason is the heterogeneity of functional and consumer properties of the products in the regional context.