# Convergence and the Matthew Effect in the European Union Based on the DESI Index

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Hypothesis**

**1**

**(H1).**

**Hypothesis**

**2**

**(H2).**

**Hypothesis**

**3**

**(H3).**

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. σ- and β-Convergence

- σ
_{t}= the standard deviation of the DESI indices at time t; - $\overline{x}$ = the mean of the DESI index.

- $\frac{1}{T}$ ln$\left(\frac{{X}_{it}}{{X}_{i{t}_{0}}}\right)$ = the average annual growth rate of the DESI index values for the i Member State;
- T = representing the interval in years between observations;
- ${x}_{i{t}_{0}}$ = initial level of development of the DESI index;
- ${\epsilon}_{i}$ = the error term assumed to be identically and independently distributed;
- α and β = the parameters to be estimated.

_{i}is the error of the estimate. We use the least-squares procedure to minimise the squared error to estimate the coefficients α and β. We considered the β-coefficient to be significant only if the empirical significance level (p-value) was less than 5%. However, the concept of conditional convergence suggests that there is an excessive amount of unexplained information hidden in α and ε

_{i}that needs to be extracted on a case-by-case basis. Still, in this case, we only tested absolute convergence.

- β = the value of the regression coefficient;
- T = representing the interval in years between observations.

- θ = the annual rate of convergence.

#### 3.2. Determination of Correlations Coefficients

- s
_{xy}= the sample covariance; - s
_{x}and s_{y}= the sample standard deviations.

#### 3.3. Principal Component Analysis (PCA)

- C
_{j}= jth standardized principal component variable (score); - a
_{ij}= eigenvector matrix elements (loadings); - Z
_{i}= column vectors of standardised scores.

_{1}) contains the most considerable fraction of the variances, represented by the eigenvalue (λ

_{1}). The first principal component is thus constructed in our model:

_{1}) and the variables under investigation. They range from −1 to +1 and should be interpreted similarly as Pearson correlation coefficients. The closer their absolute value is to 1, the closer the correlation. Their practical definition is given by the formula below:

- pc
_{ij}= jth principal component weights (patterns); - a
_{ij}= eigenvector matrix elements (loadings); - λ
_{j}= eigenvalues.

## 4. Results

#### 4.1. σ-Convergence of Digital Economy and Society Index (DESI) Overall Index

_{0}time (2016).

#### 4.2. β-Convergence of DESI Overall Index

^{2}= 0.5), and the regression equation in the figure shows that the annual average rate of development gap shrinkage for the EU was 3.45%, based on a coefficient of the initial development of around −0.0317. The regression coefficient can be considered significant as the p-value (3.69 * 10

^{−5}) is less than α (0.05).

#### 4.3. Correlation Matrix

_{0}–2016 (Figure 5) and repeated with data published in 2021 (Figure 6). The portion of the correlation matrix above the main graph reveals the values of the correlation coefficients and indicates the significance levels (* p < 0.05; ** p < 0.01; *** p < 0.001). The diagonal line indicates the distribution of a given variable. In contrast, the section below the diagonal shows the nature of the relationship between pairs of variables, which is assumed to be linear since the correlation coefficient is only valid at this point.

_{0}. The integration of digital technology at the enterprise level depends on the management approach rather than the degree of network connectivity. In the EU, the SME sector has significant activity: more than 22.8 million registered enterprises, representing 99% of all enterprises in the EU, employing 84 million Europeans and contributing 50% of EU GDP [69]. However, the SME sector is not exploiting the potential of digital technologies, both because of its mindset and because of its limited access to finance.

#### 4.4. Principal Component Analysis (PCA)

#### 4.4.1. PCA Analysis of DESI Main Indicators in 2016

_{1}(Dim1) = desi_1 * 31.28 + desi_2 * 16.3 + desi_3 * 24.66 + desi_4 * 27.76

- The group of those who joined between 1957 and 1980 is the founding six Member States, plus the first round of accessions from 1973: Belgium, Germany, France, Italy, Luxembourg, the Netherlands, Denmark, and Ireland;
- In addition, the countries that joined after the Cold War: Greece, Spain, Portugal, Austria, Finland, and Sweden;
- The group of Member States that joined in the context of the Eastern European enlargements between 2001 and 2013: Cyprus, the Czech Republic, Estonia, Hungary, Lithuania, Latvia, Malta, Poland, Slovenia, Slovakia, Bulgaria, Romania and Croatia.

- The group of Member States with a real GDP per capita between €5000 and €15,000 (marked in red): Bulgaria, Estonia, Croatia, Latvia, Lithuania, Hungary, Poland, Romania, and Slovakia;
- The group of Member States with a real GDP per capita between €15,001–32,000 (marked in green): Cyprus, the Czech Republic, Greece, France, Italy, Malta, Portugal, Slovenia and Spain;
- Member States with a real GDP per capita between €32,001–85,000 (marked in blue): Austria, Belgium, Denmark, Finland, Germany, Ireland, Luxembourg, the Netherlands, Sweden, and Finland.

#### 4.4.2. PCA Analysis of DESI Main Indicators in 2021

_{1}(Dim1) = desi_1 * 29.52 + desi_2 * 18.72 + desi_3 * 27.59 + desi_4 * 24.17

## 5. Discussion

## 6. Limitation

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Pearson correlation coefficients, distribution and correlations of variables, 2016. (*** significance level p < 0.001, ** significance level p < 0.01).

**Figure 6.**Pearson correlation coefficients, distribution and correlations of variables, 2021. (*** significance level p < 0.001, ** significance level p < 0.01, * significance level p < 0.05).

**Figure 7.**The position of countries on the plane of principal components, grouped by the year of joining the EU (2016).

**Figure 8.**The position of countries on the plane of principal components (grouped by the real GDP in euro per capita), 2016.

**Figure 9.**The position of countries on the plane of principal components, grouped by the year of joining the EU (2021).

**Figure 10.**The position of countries on the plane of principal components (grouped by the real GDP euro per capita), 2021.

**Table 1.**Digital Economy and Society Index (DESI) main indicators, sub-dimensions and individual indicators.

Main Indicators of DESI 2021 | Sub-Dimensions and Numbers of Individual Indicators of DESI 2021 |
---|---|

- 1.
- Human capital
| Internet user skills and advanced skills and development (2 sub-dimensions and 7 individual indicators) |

- 2.
- Connectivity
| Fixed and mobile broadband connection coverage and price (4 sub-dimensions and 10 individual indicators) |

- 3.
- Integration of digital technology
| Digital intensity, business digitalisation and e-commerce (3 sub-dimensions and 11 individual indicators) |

- 4.
- Digital public services
| e-Government (1 sub-dimension and 5 individual indicators) |

Cluster 1 (Red) | Cluster 2 (Green) | Cluster 3 (Blue) |
---|---|---|

Information and communication technology (ICT) | DESI | digitalisation |

European Union | digital economy | innovation |

technologies | competitiveness | impact |

digital skills | economy | digital technologies |

education | sustainable development | digital transformation |

digital agenda | Industry 4.0 | economic growth |

digital divide | ||

information |

Observations | R-Square | Constant | β | θ | t Stat | p-Value |
---|---|---|---|---|---|---|

27 | 0.5003 | 0.184 | −0.0317 | 3.45% | −5.0032 | 3.69 × 10^{−5} |

KMO-2016 | KMO-2021 |
---|---|

Kaiser-Meyer-Olkin factor adequacy Call: KMO (r = data) Overall MSA = 0.73 MSA for each item = desi_1 desi_2 desi_3 desi_4 0.68 0.74 0.71 0.83 | Kaiser-Meyer-Olkin factor adequacy Call: KMO (r = data) Overall MSA = 0.8 MSA for each item = desi_1 desi_2 desi_3 desi_4 0.73 0.88 0.78 0.88 |

2016 | 2021 | ||||||
---|---|---|---|---|---|---|---|

Eigenvalue (λ) | Variance | Cumulative Variance | Eigenvalue (λ) | Variance | Cumulative Variance | ||

Dim.1 | 2.836046 | 70.901171 | 70.901171 | Dim.1 | 2.9757839 | 74.394597 | 74.394597 |

Dim.2 | 0.740158 | 17.600395 | 88.50157 | Dim.2 | 0.5673334 | 14.183336 | 88.57793 |

2016 | 2021 | ||||
---|---|---|---|---|---|

Dim1 | Dim2 | Dim1 | Dim2 | ||

desi_1 | 0.9419029 | −0.09252315 | desi_1 | 0.9371991 | −0.06188844 |

desi_2 | 0.6798614 | 0.71318097 | desi_2 | 0.7463895 | 0.65385036 |

desi_3 | 0.8363628 | −0.43011026 | desi_3 | 0.9060466 | −0.16668687 |

desi_4 | 0.8872156 | −0.04281727 | desi_4 | 0.8481886 | −0.32893530 |

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## Share and Cite

**MDPI and ACS Style**

Kovács, T.Z.; Bittner, B.; Huzsvai, L.; Nábrádi, A.
Convergence and the Matthew Effect in the European Union Based on the DESI Index. *Mathematics* **2022**, *10*, 613.
https://doi.org/10.3390/math10040613

**AMA Style**

Kovács TZ, Bittner B, Huzsvai L, Nábrádi A.
Convergence and the Matthew Effect in the European Union Based on the DESI Index. *Mathematics*. 2022; 10(4):613.
https://doi.org/10.3390/math10040613

**Chicago/Turabian Style**

Kovács, Tünde Zita, Beáta Bittner, László Huzsvai, and András Nábrádi.
2022. "Convergence and the Matthew Effect in the European Union Based on the DESI Index" *Mathematics* 10, no. 4: 613.
https://doi.org/10.3390/math10040613