1. INTRODUCTION
The digital transformation of the economy changes its measuring methods and develops new issues for researchers. According to the article of (Abroskin, 2019), a key feature in defining the digital economy is the use of the Internet, digital data, and digital technologies. The motivation for this research was the wellknown scientists work (Kabanov et al., 2020). The authors of the paper made a conclusion that digital transformation is one of the main elements of scientific and technological progress and technology transfer in the context of competitive imports from lowcost territories to ensure sustainable macroeconomic growth. Indeed, economic development based on low cost and natural resources are not able to last long. During 20092017 period in Russia the average growth rate of technological innovations expenditures in 2009 prices was 108.88%. The technological innovation cost is an essential component of the economy’s digital transformation. In the framework of competitive imports from territories with low costs, these costs allow managing the economy’s competitiveness. According to the endogenous growth theory, technological progress is the only source of sustainable longterm economic growth (Alalehto, 2018;Romer, 1986;Schwab and Davis, 2018). However, the technological inequality of the regions hinders balanced economic development.
Economy digitalization is impossible without the knowledge exchange – technological innovations – between regions. Digital technologies development actualizes the technological achievements exchange between territories. Interregional interactions that bring impulses from other territories processes (Abre et al., 2008) are an essential external determinant of the acquisition, exchange, transformation and use of knowledge (in particular, technological innovations) in the modern digital economy. On the one hand, the mobility of production factors and results between regions increases significantly, and on the other hand, the role of knowledge as an endogenous factor of economic growth also grows. Theoretical and empirical studies have revealed spatial dependence in regional innovation (Bottazzi and Peri, 2003;Monjon and Waelbroeck, 2003;Fischer and Varga, 2003;Qiu et al., 2018).
A theoretical and empirical study of the impact of access to innovation and knowledge on economic growth is quite popular. For example, in (PerezTrujillo and Lacalle Calderon, 2020), based on the extended Solow Cygnus growth model and panel data from 138 countries for the period 19902014, the positive impact of Internet access on innovation on economic growth and acceleration is shown process of economic convergence. The convergence of actually innovation growth rates and the assessment of their interregional externalities has not been studied sufficiently. Thus, in the article of (Qiu et al., 2018), based on data from 2000 to 2015, σconvergence of innovations in the provinces of China, measured as the number of approved patents per 10 thousand people, was defined. The assessment of Russian regions spatial interaction in technological innovations remains outside the scope of the main research body. Therefore, it is relevant to assess the spatial correlation of expenditures on technological innovations taking into account regional specifics and identify the growth determinants.
To analyze the growth rates and spatial interactions of regions in technological innovations, the study uses global and local spatial correlation indices, spatial economic models. Crosssections of 83 Russian regions from 2009 to 2017 are used for building models. The maximum likelihood method is used to evaluate model parameters, and the Lagrange multiplier test and the AIC information criterion are used for model verification.
The major goal of the study is to test the assumption of possible technological inequality of regions taking into account the spatial connection.
Based on the literature analysis, two main research questions were formulated:

1. Is there a spatial dependence on the studied indicator – the cost of technological innovations – in the regions?

2. Is there a βconvergence of the technological innovation costs average growth rate in the short term per capita in the Russian regions?
It is expected that the costs of technological innovations and their growth rates in neighboring territories differ, which indicates the technological inequality of Russian regions.
During the study, the following results were obtained: for a number of years, a negative spatial dependence of technological innovation costs per capita in the Russian regions was revealed, most regions are concentrated in the LowLow (LL) and LowHigh (LH) quadrants of the Moran diagram with a low level of technological innovation costs, in recent years there has been a positive spatial lag costs and the growth of the regions number High High (HH) and HighLow (HL)quadrants, local spatial clusters of regions with a focus on the raw materials extraction, where the level and growth rate of costs for technological innovation is higher were revealed, conditional βdivergence of the costs growth rate for technological innovation in the short term was found.
The paper consists of an introduction, two main sections, and a conclusion. The first section describes the statistical indicators used in the Russian regions and defines the spatial economic models used. The second section presents the results of evaluating models and their discussion. The conclusion contains conclusions and recommendations for further research in the field of analytical econometric tools for analyzing the connections and growth rates of technological innovations in the regions.
2. RESEARCH METHODS
A data sample was obtained on the official website of the Federal State Statistics Service of the Russian Federation (www.gks.ru) for 83 regions from 2009 to 2017.
In this study, traditionally, to measure the conditional βconvergence of technological innovation costs per capita, the logarithm of this variable in 2009 prices was used. Table 1 presents descriptive statistics of variables in 2017.
In this study, a boundary neighbor matrix was constructed in regression models, obtained from the database on the administrative borders location – GADM, to calculate spatial connections in the R software environment. When two regions have a common border, the matrix element is equal to one, and zero if the opposite is true.
To estimate the spatial dependence for each year, global Moran indices are defined (Anselin, 1988;Anselin, 1995):
where N is the number of regions, X is the average value of X indicator: growth rate of technological innovation per capita, w_{ij}  elements of the boundary weighing matrix.
Geary spatial correlation indices were also calculated (Anselin, 1988;Semerikova, 2014):
where W denotes the sum over all w_{ij}, other notations correspond to the Moran index notations.
The Geary index from 0 to 1 indicates a positive spatial correlation, from 1 to 2 – a negative one (Semerikova, 2014). A positive spatial correlation coefficient means that a growing region contributes to the growth of its neighbors; a negative value means that a growing region "takes" the resources of its neighbors. The insignificance of the coefficient indicates that there is no correlation between processes in different regions.
To identify spatial clustering of regions, local Moran indices (LISA – Local Index Spatial Autocorrelation) are defined for each year (Anselin, 1995):
If the region is significantly different from its neighbors (outlier), then it has a negative value of the local Moran index. A positive correlation indicates that the region is similar to neighboring territories (cluster). The larger the LISA modulo is, the stronger is the similarity / difference of the region with its neighbors.
To formalize the mechanism of interaction between the determinants of technological innovation costs and their βconvergence, the study uses the model of conditional βconvergence of Barro and SalaiMartin as a base (Barro and SalaiMartin, 1992):
where Y_{i} is the dependent variable, i = 1,…, n – regions number, i_{0} – value of the dependent variable in the base year, ${X}_{ji},j=\text{1},\dots ,k$ – explanatory variables, $\alpha ,\text{}\beta ,\text{}{\gamma}_{\text{j}},j=\text{1},\hspace{0.17em}\dots ,\hspace{0.17em}k$ – coefficient estimates, ${\epsilon}_{i}\sim \hspace{0.17em}\text{iid}\left(0,\hspace{0.17em}{\sigma}_{\sigma}^{2}\right)$i = 1, …, n – model error.
A negative value of the β coefficient indicates the growth rates convergence and predicts a potentially higher increase in the technological innovation cost in regions with a lower initial level of development. A positive value of β coefficient indicates a divergence in growth rates and predicts a potentially higher increase in the cost of technological innovation in the leading regions with a higher initial level of development.
The maximum likelihood study evaluated static models of conditional βconvergence with a spatial component (spdep package) in the R software environment (Anselin, 1988;LeSage and Pace, 2009;Elhorst, 2014) to identify shortterm (annual) spatial effects. In this study, modified βconvergence models were used with the addition of spatial lags on cross sections to account for the mutual influence of regions. This is a model with a spatial autoregression lag (SAR) and a model with spatial interaction in errors (SEM) (Anselin, 1988;Elhorst, 2014;Ivanova, 2018;Demidova and Prokopov, 2019):
where y_{i} is the cost of technological innovation per capita in region i; α is the parameter to be evaluated; β is the convergence rate; W_{ij} is the weighing matrix, normalized by rows in this study, and is also boundary (the diagonal elements of the boundary matrix are equal to zero, and the nondiagonal elements are equal to one, if the corresponding pair of regions has a common border and zero in the opposite case); ρ is the spatial autoregression coefficient for a dependent variable; λ is the spatial autocorrelation coefficient for shock; γ_{j}  spatial coefficients for the independent variables; ε_{i} – random errors.
The dependent variable autoregression coefficient ρ for the spatial lag allows one to identify the influence of the costs growth rate for technological innovations in other regions on the studied region. The statistical insignificance of the spatial autoregression coefficient means that the processes of increasing costs for technological innovations in different regions are not related to each other, a positive value indicates regional cooperation, and a negative value indicates regional competition. The spatial autocorrelation coefficient for shock λ reveals the influence of the spatial structure of errors. The statistical insignificance of λ means that the shocks of neighboring regions that affect the growth rate of technological innovation costs in a given region are not related to each other.
3. ANALYSIS RESULT
Local Moran indices confirmed the regional local spatial clusters with a higher level and growth rate of technological innovations expenditures in Moscow, the Ural Federal district (YamaloNenets Autonomous district, KhantyMansi Autonomous district, Tyumen region), the Siberian Federal district (Omsk region, Tomsk region), and the NorthWestern Federal district (Nenets Autonomous district). Most of these clusters are characterized by the economy orientation to the raw materials extraction, presence of Federal financial support for the technologies development (Figure 1, Figure 2). Spatial Moran diagrams based on the original sample indicated the “emission points” in terms of costs for technological innovations: Moscow, Saint Petersburg, the Republic of Sakha  Yakutia, Tyumen region, YamaloNenets Autonomous district, KhantyMansi Autonomous district, Nenets Autonomous district, Chukotka Autonomous district, Sakhalin region, Chechen Republic, Republic of Ingushetia, Republic of Kalmykia. After their elimination, global statistically significant indexes (Table 2) highlighted the negative spatial correlation (with the exception of 2015 and 2016), when the costs of technological innovation in neighboring territories differ, and strong regions “pull” resources from weak neighbors.
In spatial Moran diagrams (Figure 3) the HH and HL quadrants contain a small number of regions. In the HH quadrant there are regions with high costs for technological innovation surrounded by the same neighbors, that is, “nonextreme” “pulsar stars”, with positive autocorrelation for neighbors. In the HL quadrant, there are several atypical, “supporting framework” “extreme” successful regions with a high concentration of technological innovation costs surrounded by neighbors with low technological innovation costs. These are the “cores” of technological innovations with superiority and negative autocorrelation over the neighbors. The Republic of Tatarstan is one of these regions. However, most Russian regions are concentrated in the LH and LL quadrants. This means that there are atypical peripheral regions with low costs for technological innovation surrounded by successful neighbors from “cores” and “stars” (LH quadrant with negative autocorrelation) and lowcost regions surrounded by the same neighbors (LL quadrant with positive autocorrelation). In terms of negative global spatial correlation, the above shows the influence of “cores” and “stars” on the periphery and the “pull” of innovation costs by strong regions from weak neighbors.
In the short term, all static spatial models on crosssections predict the process of the growth rates conditional βdivergence of expenditures on technological innovations per capita in the regions (Table 3). Estimates of coefficients for exogenous variables in most cases were not statistically significant, except for the found investments correlation in 2016. The expected correlation between the growth rate of expenditures on technological innovations and the number of university students, the use of the Internet in organizations, and the number of patents issued for inventions was not confirmed. A positive statistically significant spatial lag in the growth rate of expenditures on technological innovations in 2015 and 2016 indicates regional cooperation: the growth of expenditures on technological innovations among neighbors causes their growth in this region.
4. CONCLUSION
This paper analyzes the costs of technological innovations in the regions, taking into account spatial relationships. We started from theoretical arguments in favor of linking technological innovations with endogenous knowledge factors of economic growth. The paper uses annual Russian statistical data of Russian regions from 2009 to 2017. The study used spatial correlation indices, a model with a spatial autoregression lag, and a model with spatial interaction in errors. Based on the results of modeling the growth rates βconvergence process of technological innovation costs for crosssections, the following conclusions can be formulated:

1. Over the years, a negative spatial dependence has been revealed for the cost of technological innovation per capita in the Russian regions: the leading regions are “pulling” the resources of their neighbors. However, in 2015 and 2016, a positive spatial lag in the costs growth rate for technological innovations indicates the cooperation of regions in terms of rising costs for technological innovations: leading regions “pull” the lagging neighbors from the periphery.

2. Local spatial clusters of regions with a higher level and growth rate of expenditures on technological innovations were found in Moscow, the Ural Federal district (YamaloNenets Autonomous district, KhantyMansi Autonomous district, Tyumen region), the Siberian Federal district (Omsk region, Tomsk region), and the NorthWestern Federal district (Nenets Autonomous district). Over the period from 2009 to 2017, the number of successful regions of “cores” and “stars” with high costs for technological innovations increases (the HL and HH quadrant of the spatial Moran diagram). The Republic of Tatarstan is among such successful regions.

3. A conditional βdivergence in the growth rate of expenditures on technological innovations in the short term is revealed. This predicts higher shortterm growth in regions with higher initial levels of technological innovation, assuming that the regions have their own sustainable state. At the same time, spatial cooperation of regions has emerged in recent years. This means that the leading regions “pull” the lagging regions after them.

4. Clustering costs for technological innovations, divergence of their growth rates, and regional cooperation reveal that technological innovations are concentrated in strong regions with a high concentration of production factors. This can predict a technological breakthrough thanks to the leading regions. The concentration of most regions in LL and LH quadrants of the Moran diagram predicts a fairly long time lag for technological breakthroughs and economic development of the regions.

5. The results of this research can be used in the policy formation to encourage spending on technological innovations. It is useful to influence the costs of technological innovations in neighboring regions in order to manage the region through the mechanism of regional cooperation.

6. Further research can be aimed at measuring the spatial effects of the growth rate of technological innovation costs in the long term based on spatial econometric models based on panel data. It is interesting to answer the following question: will the βdivergence in the growth rate of technological innovation costs last long? It is also useful to expand the specifications of spatial econometric models and control variables.