## 1 Introduction

## 2 Theoretical background

## 3 Data

Year | LAU 1 | NUTS 3 | ||||||
---|---|---|---|---|---|---|---|---|

Min | Mean | Max | CV | Min | Mean | Max | CV | |

2003 | 76.9 | 98.0 | 132.5 | 9.5 | 87.7 | 99.4 | 117.9 | 7.5 |

2006 | 85.2 | 97.4 | 121.1 | 6.3 | 91.3 | 99.5 | 121.1 | 6.0 |

2009 | 82.6 | 97.7 | 120.2 | 6.3 | 90.9 | 99.4 | 120.2 | 5.8 |

2012 | 85.2 | 97.6 | 119.2 | 6.1 | 91.2 | 99.4 | 119.2 | 5.8 |

2015 | 84.3 | 97.2 | 121.2 | 6.6 | 89.0 | 99.1 | 121.2 | 6.4 |

Year | LAU 1 | NUTS 3 | ||||||
---|---|---|---|---|---|---|---|---|

Min | Mean | Max | Cv | Min | Mean | Max | Cv | |

2009 | 45.8 | 82.9 | 234.2 | 29.8 | 57.2 | 88.0 | 171.3 | 25.0 |

2012 | 49.3 | 89.2 | 228.7 | 25.9 | 64.5 | 91.9 | 150.6 | 20.3 |

2015 | 50.4 | 91.6 | 236.8 | 25.5 | 66.8 | 92.6 | 146.1 | 18.2 |

2018 | 44.2 | 92.8 | 229.7 | 23.7 | 69.9 | 92.9 | 143.9 | 16.8 |

## 4 Parallel convergence within the distribution dynamics framework

^{1}:

_{x}(x) is the marginal distribution of the variable in the initial period, while f(y, x) is the joint distribution of y and x. To estimate the conditional density function, one needs to replace the numerator and denominator of the above expression with nonparametric estimators. The marginal distribution of the variable in the initial period is estimated using a two-step adaptive kernel density estimation for one-dimensional distributions:

_{x}is the optimal estimation bandwidth for the initial distribution of the variable, and K(.) is the kernel function. In the first stage of the adaptive method weights w

_{i}assume the value of 1 for all observations. The joint distribution of the variable in the initial and final period, i.e. the numerator of the Eq. (3), is estimated using the formula:

_{y}is the optimal bandwidth for the distribution of the variable in the final period. Also the joint density function is initially estimated without differentiating weights for individual observations. Then, the initial estimation of the joint density is used to calculate the weights differentiating the bandwidth locally according to the expression:

^{2}goodness of fit test might be applied—see, e.g. Cochran (1952) and Anderson and Goodman (1957). The use of this test to verify the null hypothesis that the estimated (empirical) transition matrix is equal to an exogenously defined matrix is discussed e.g. by Bickenbach and Bode (2001). The test statistic has the form:

_{i}is the number of observations in the i-th row of the empirical transition matrix, p

_{ij}means probabilities from the estimated (empirical) transition matrix, p

_{ij}

^{0}—probability from the exogenous transition matrix, F

_{i}= {j: p

_{ij}

^{0}> 0}—the set of column indices of the exogenous transition matrix, in which elements of the row i are greater than zero, and f

_{i}—the size of the set F

_{i}. The test is therefore carried out only for those transition probabilities that are non-zero in the matrix exogenously specified. In the case of testing the equality of two empirical transition matrices, I propose that the test should be applied twice, each time taking one of the compared matrices as exogenous.

^{2}test of compatibility of ergodic distributions (proportions of observations) in two subgroups can be used. Due to the fact that the sample (number of territorial units) might be small in this case, I propose usage of the tests discussed by Wilcox et al. (2013). The authors describe two new methods for comparing discrete distributions for a small sample. The first of the proposed new approaches (referred to as ‘method M’) is the extension of the Storer and Kim (1990) method for comparing two independent binomial ratios. The second method (called ‘method B’) is a procedure of multiple comparisons for all corresponding pairs of probabilities from both measurements. It uses the Storer and Kim (1990) method in combination with the Hochberg (1988) approach, which allows for controlling the probability of one or more errors of type I.

## 5 Convergence measured within transition matrices

Group 1 < = 90% | Group 2 (90%, 95.2%] | Group 3 (95.2%, 100.1%] | Group 4 (100.1%, 105.6%] | Group 5 > 105.6% | |
---|---|---|---|---|---|

group 1 | 23.7 | 57.9 | 17.1 | 1.3 | 0.0 |

group 2 | 6.6 | 43.4 | 36.8 | 13.2 | 0.0 |

group 3 | 4.0 | 22.7 | 37.3 | 32.0 | 4.0 |

group 4 | 0.0 | 10.5 | 44.7 | 35.5 | 9.2 |

group 5 | 1.3 | 6.6 | 32.9 | 23.7 | 35.5 |

ergodic | 4.8 | 25.2 | 38.0 | 26.4 | 6.1 |

Group 1 < = 92.1% | Group 2 (92.1%, 97.1%] | Group 3 (97.1%, 101.9%] | Group 4 (101.9%, 105.7%] | Group 5 > 105.7% | |
---|---|---|---|---|---|

group 1 | 26.7 | 66.7 | 6.7 | 0.0 | 0.0 |

group 2 | 0.0 | 64.3 | 35.7 | 0.0 | 0.0 |

group 3 | 0.0 | 21.4 | 50.0 | 28.6 | 0.0 |

group 4 | 0.0 | 7.1 | 42.9 | 35.7 | 14.3 |

group 5 | 0.0 | 6.7 | 46.7 | 13.3 | 33.3 |

ergodic | 0.0 | 31.3 | 43.9 | 20.4 | 4.4 |

Group 1 < = 72.9% | Group 2 (72.9%, 82.2%] | Group 3 (82.2%, 91.9%] | Group 4 (91.9%, 102.8%] | Group 5 > 102.8% | |
---|---|---|---|---|---|

group 1 | 61.8 | 32.9 | 5.3 | 0.0 | 0.0 |

group 2 | 7.9 | 47.4 | 42.1 | 2.6 | 0.0 |

group 3 | 0.0 | 13.3 | 49.3 | 33.3 | 4.0 |

group 4 | 0.0 | 1.3 | 14.5 | 54.0 | 30.3 |

group 5 | 1.3 | 0.0 | 1.3 | 10.5 | 86.8 |

ergodic | 3.1 | 5.8 | 13.0 | 22.5 | 55.6 |

Group 1 < = 78.7% | Group 2 (78.7%, 84.7%] | Group 3 (84.7%, 93.3%] | Group 4 (93.3%, 103.9%] | Group 5 > 103.9% | |
---|---|---|---|---|---|

group 1 | 66.7 | 26.7 | 6.7 | 0.0 | 0.0 |

group 2 | 0.0 | 64.29 | 28.6 | 7.1 | 0.0 |

group 3 | 0.0 | 7.1 | 71.4 | 21.4 | 0.0 |

group 4 | 0.0 | 0.0 | 7.1 | 78.6 | 14.3 |

group 5 | 0.0 | 0.0 | 0.0 | 26.7 | 73.3 |

ergodic | 0.0 | 3.3 | 16.4 | 52.3 | 28.0 |

## 6 Convergence within kernel density estimates

## 7 Testing parallel convergence between income and exam results

Level or variable | Variant | Direct transition, 10 years lag | 3-yearly transitions, 6 years lag | ||
---|---|---|---|---|---|

Statistic | p value | Statistic | p value | ||

NUTS 3 | Matrix income = matrix exams | 45.2 | 0.00 | 77.1 | 0.00 |

NUTS 3 | Matrix exams = matrix income | 51.7 | 0.00 | 152.6 | 0.00 |

LAU 1 | Matrix income = matrix exams | 233.9 | 0.00 | 302.1 | 0.00 |

LAU 1 | Matrix exams = matrix income | 815.0 | 0.00 | 1388.5 | 0.00 |

Exams | Matrix NUTS 3 = matrix LAU 1 | 51.3 | 0.00 | 93.4 | 0.00 |

Exams | Matrix LAU 1 = matrix NUTS 3 | 6.5 | 0.98 | 22.4 | 0.44 |

Income | Matrix NUTS 3 = matrix LAU 1 | 78.2 | 0.00 | 110.1 | 0.00 |

Income | Matrix LAU 1 = matrix NUTS 3 | 11.3 | 0.66 | 20.5 | 0.16 |

Level or variable | Direct transition, 10 years lag | 3-yearly transitions, 6 years lag | ||
---|---|---|---|---|

Statistic | p value | Statistic | p value | |

LAU 1 (exams vs income) | 0.35 | 0.00 | 0.40 | 0.00 |

NUTS 3 (exams vs income) | 0.31 | 0.00 | 0.40 | 0.00 |

Exams (LAU 1 vs NUTS 3) | 0.01 | 0.37 | 0.00 | 0.35 |

Income (LAU 1 vs NUTS 3) | 0.17 | 0.00 | 0.03 | 0.00 |

Transition | Level or variable | Ergodic probability for group | ||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||

Direct transition, 10 years lag | LAU 1 (exams vs income) | 0.498 | 0.000 | 0.000 | 0.212 | 0.000 |

NUTS 3 (exams vs income) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

Exams (LAU 1 vs NUTS 3) | 0.074 | 0.244 | 0.421 | 0.325 | 0.540 | |

Income (LAU 1 vs NUTS 3) | 0.127 | 0.301 | 0.396 | 0.000 | 0.000 | |

3-Yearly transitions, 6 years lag | LAU 1 (exams vs income) | 0.000 | 0.000 | 0.000 | 0.430 | 0.000 |

NUTS 3 (exams vs income) | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | |

Exams (LAU 1 vs NUTS 3) | 0.546 | 0.217 | 0.838 | 0.106 | 0.596 | |

Income (LAU 1 vs NUTS 3) | 0.214 | 0.142 | 0.118 | 0.000 | 0.015 |

Distribution | Level | Direct transition, 10 years lag | 3-Yearly transitions, 6 years lag | ||||||
---|---|---|---|---|---|---|---|---|---|

KS test | AD test | KS test | AD test | ||||||

Statistic | pvalue | statistic | p value | Statistic | p value | Statistic | p value | ||

initial | LAU 1 | 0.100 | 0.044 | 13.976 | 0.009 | 0.058 | 0.043 | 5.283 | 0.002 |

initial | NUTS 3 | 0.125 | 0.631 | 1.179 | 0.275 | 0.042 | 0.992 | 0.312 | 0.931 |

ergodic | LAU 1 | 0.386 | 0.000 | 62.526 | 0.000 | 0.538 | 0.000 | 413.620 | 0.000 |

ergodic | NUTS 3 | 0.285 | 0.000 | 4.941 | 0.003 | 0.495 | 0.000 | 57.616 | 0.000 |