2 On Convergence
Several measures of convergence have been recently proposed by Eurofound (Eurofound (2018), Upward convergence in the EU: Concepts, measurements and indicators, Publications Office of the European Union, Luxembourg; by: Massimiliano Mascherini, Martina Bisello, Hans Dubois and Franz Eiffe)
In this section each each measure is considered by one or more examples.
2.1 Beta-convergence
Let’s assume we have a dataset (tibble) of sorted times by countries values. The calculations are performed according to the following linear model: \[ ln(y_{m,i,t+\tau})-ln(y_{m,i,t}) = \beta_0 + \beta_1 ln(y_{m,i,t}) +\epsilon_{m,i,t} \] where \(m\) represent the member state of EU (country), \(i\) refers to an indicator of interest, \(t\) is the reference time and \(\tau \in \{1,2,\ldots\}\) the length of the time window (typically \(1\) or more years).
In the simplest case, just two time values are considered, \(t\) and \(t+\tau\), while in a more general setup all
observed times in set \(\{t,t+1,\ldots,t+\tau-1, t+\tau\}\) are
included into regression.
In this more general case, the current implementation of
beta-convergence function always maintain the same reference time across
different years and it divides the left hand side by the amount of time
elasped as an option, that is the alternative formula: \[
\tau^{-1}(ln(y_{m,i,t+\tau})-ln(y_{m,i,t})) = \beta_0 + \beta_1
ln(y_{m,i,t}) +\epsilon_{m,i,t}
\] is available.
The output of beta_conv() is a list in which transformed data, the point estimate of \(\beta_1\) and a standard two tails test is reported (p-value and adjusted R squared). One tail test \(H_0: \beta_1 \geq 0\) against \(H_1: \beta1< 0\) might be of some interest, but it is not implemented.
Below an example on how to invoke the function:
#library(ggplot2)
#library(dplyr)
#library(tibble)
testTB <- tribble(
~time, ~countryA , ~countryB, ~countryC,
2000, 0.8, 2.7, 3.9,
2001, 1.2, 3.2, 4.2,
2002, 0.9, 2.9, 4.1,
2003, 1.3, 2.9, 4.0,
2004, 1.2, 3.1, 4.1,
2005, 1.2, 3.0, 4.0
)
res <- beta_conv(tavDes = testTB, time_0 = 2002, time_t = 2004,
all_within = TRUE,
timeName = "time")
res
#> $res
#> $res$workTB
#> # A tibble: 6 × 3
#> deltaIndic indic countries
#> <dbl> <dbl> <chr>
#> 1 0.184 -0.105 countryA
#> 2 0 1.06 countryB
#> 3 -0.0123 1.41 countryC
#> 4 0.144 -0.105 countryA
#> 5 0.0333 1.06 countryB
#> 6 0 1.41 countryC
#>
#> $res$model
#>
#> Call:
#> stats::lm(formula = deltaIndic ~ indic, data = workTB)
#>
#> Coefficients:
#> (Intercept) indic
#> 0.1495 -0.1156
#>
#>
#> $res$summary
#> # A tibble: 2 × 5
#> term estimate std.error statistic p.value
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 (Intercept) 0.149 0.0134 11.1 0.000371
#> 2 indic -0.116 0.0131 -8.80 0.000920
#>
#> $res$beta1
#> [1] -0.1156032
#>
#> $res$adj.r.squared
#> [1] 0.9386146
#>
#>
#> $msg
#> NULL
#>
#> $err
#> NULLbut note that this is not the common practice, which considers the first and last time instead.
In order to consider just two times, starting and ending times, the option all_within = FALSE must be specified
res <- beta_conv(tavDes = testTB, time_0 = 2002, time_t = 2004,
all_within = FALSE,
timeName = "time")
res
#> $res
#> $res$workTB
#> # A tibble: 3 × 3
#> deltaIndic indic countries
#> <dbl> <dbl> <chr>
#> 1 0.144 -0.105 countryA
#> 2 0.0333 1.06 countryB
#> 3 0 1.41 countryC
#>
#> $res$model
#>
#> Call:
#> stats::lm(formula = deltaIndic ~ indic, data = workTB)
#>
#> Coefficients:
#> (Intercept) indic
#> 0.13393 -0.09475
#>
#>
#> $res$summary
#> # A tibble: 2 × 5
#> term estimate std.error statistic p.value
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 (Intercept) 0.134 0.000353 380. 0.00168
#> 2 indic -0.0948 0.000345 -275. 0.00232
#>
#> $res$beta1
#> [1] -0.09475194
#>
#> $res$adj.r.squared
#> [1] 0.9999735
#>
#>
#> $msg
#> NULL
#>
#> $err
#> NULLNote that all_within = FALSE is the default.
2.2 Sigma-convergence
The key concept in sigma-convergence is variability with respect to the mean. Let \(Y_{m,i,t}\) be the value of indicator \(i\) for member state \(m\) at time \(t\), and \(\overline{Y}_{A,i,t}\) the average over aggregation \(A\), for example \(A = EU27_2020\), than:
- the average is \(\overline{Y}_{A,i,t} =
n(A)^{-1}\sum_{m \in A} Y_{m,i,t}\), where \(n(A)\) is the number of member states
within aggregation \(A\);
- the standard deviation is \(s_{A,i,t} =
\sqrt(n(A)^{-1} \sum_{m\in A} (Y_{m,i,t} -
\overline{Y}_{A,i,t})^2)\);
- the coefficient of variation is \(CV(A,i,t) = 100\cdot \frac{s_{A,i,t}}{\overline{Y}_{A,i,t}}\).
For each year, the above summaries are calculated to quantify if a reduction in heterogeneity took place.
In this section we assume that all member states contributing to the unweighted mean are contained into the dataset, for example:
testTB <- tribble(
~time, ~countryA , ~countryB, ~countryC,
2000, 0.8, 2.7, 3.9,
2001, 1.2, 3.2, 4.2,
2002, 0.9, 2.9, 4.1,
2003, 1.3, 2.9, 4.0,
2004, 1.2, 3.1, 4.1,
2005, 1.2, 3.0, 4.0
)
sigma_conv(testTB,timeName="time")
#> $res
#> # A tibble: 6 × 5
#> time stdDev CV mean devianceT
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2000 1.28 0.517 2.47 4.89
#> 2 2001 1.25 0.435 2.87 4.67
#> 3 2002 1.32 0.501 2.63 5.23
#> 4 2003 1.11 0.406 2.73 3.69
#> 5 2004 1.20 0.430 2.8 4.34
#> 6 2005 1.16 0.424 2.73 4.03
#>
#> $msg
#> NULL
#>
#> $err
#> NULLIt is possible to select a time window, as follows:
sigma_conv(testTB,timeName="time",time_0 = 2002,time_t = 2004)
#> $res
#> # A tibble: 3 × 5
#> time stdDev CV mean devianceT
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2002 1.32 0.501 2.63 5.23
#> 2 2003 1.11 0.406 2.73 3.69
#> 3 2004 1.20 0.430 2.8 4.34
#>
#> $msg
#> NULL
#>
#> $err
#> NULL
sigma_conv(testTB,time_0 = 2002,time_t = 2004)
#> $res
#> # A tibble: 3 × 5
#> time stdDev CV mean devianceT
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2002 1.32 0.501 2.63 5.23
#> 2 2003 1.11 0.406 2.73 3.69
#> 3 2004 1.20 0.430 2.8 4.34
#>
#> $msg
#> NULL
#>
#> $err
#> NULLMore interesting calculations deal with an Eurofound dataset emp_20_64_MS. Note that all and only countries in EU28 are included, those that contribute to the average:
data(emp_20_64_MS)
mySTB <- sigma_conv(emp_20_64_MS)
mySTB
#> $res
#> # A tibble: 17 × 5
#> time stdDev CV mean devianceT
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2002 6.34 0.0939 67.5 1125.
#> 2 2003 5.95 0.0878 67.8 991.
#> 3 2004 5.70 0.0839 67.9 909.
#> 4 2005 5.54 0.0809 68.4 858.
#> 5 2006 5.57 0.0801 69.6 869.
#> 6 2007 5.47 0.0775 70.6 838.
#> 7 2008 5.36 0.0755 71.0 804.
#> 8 2009 5.03 0.0730 69.0 710.
#> 9 2010 5.24 0.0769 68.1 768.
#> 10 2011 5.59 0.0821 68.1 875.
#> 11 2012 5.98 0.0880 68 1002.
#> 12 2013 6.28 0.0922 68.0 1103.
#> 13 2014 5.98 0.0867 69.0 1000.
#> 14 2015 5.74 0.0820 70.0 922.
#> 15 2016 5.60 0.0789 71.0 879.
#> 16 2017 5.37 0.0741 72.5 808.
#> 17 2018 5.30 0.0717 73.8 786.
#>
#> $msg
#> NULL
#>
#> $err
#> NULLAs a first step, the departure from the mean is characterized
res <- departure_mean(oriTB = emp_20_64_MS, sigmaTB = mySTB$res)
names(res$res)
#> [1] "departures" "squaredContrib" "devianceContrib"
res$res$departures
#> # A tibble: 17 × 33
#> time stdDev CV mean devianceT AT BE BG CY CZ DE DK
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2002 6.34 0.0939 67.5 1125. 0 0 -1 1 0 0 1
#> 2 2003 5.95 0.0878 67.8 991. 0 0 -1 1 0 0 1
#> 3 2004 5.70 0.0839 67.9 909. 0 0 -1 1 0 0 1
#> 4 2005 5.54 0.0809 68.4 858. 0 0 -1 1 0 0 1
#> 5 2006 5.57 0.0801 69.6 869. 0 0 0 1 0 0 1
#> 6 2007 5.47 0.0775 70.6 838. 0 0 0 1 0 0 1
#> 7 2008 5.36 0.0755 71.0 804. 0 0 0 1 0 0 1
#> 8 2009 5.03 0.0730 69.0 710. 0 0 0 1 0 1 1
#> 9 2010 5.24 0.0769 68.1 768. 1 0 0 1 0 1 1
#> 10 2011 5.59 0.0821 68.1 875. 1 0 0 0 0 1 1
#> 11 2012 5.98 0.0880 68 1002. 1 0 0 0 0 1 1
#> 12 2013 6.28 0.0922 68.0 1103. 1 0 0 0 0 1 0
#> 13 2014 5.98 0.0867 69.0 1000. 0 0 0 0 0 1 0
#> 14 2015 5.74 0.0820 70.0 922. 0 0 0 0 0 1 0
#> 15 2016 5.60 0.0789 71.0 879. 0 0 0 0 1 1 0
#> 16 2017 5.37 0.0741 72.5 808. 0 0 0 0 1 1 0
#> 17 2018 5.30 0.0717 73.8 786. 0 0 0 0 1 1 0
#> # ℹ 21 more variables: EE <dbl>, EL <dbl>, ES <dbl>, FI <dbl>, FR <dbl>,
#> # HR <dbl>, HU <dbl>, IE <dbl>, IT <dbl>, LT <dbl>, LU <dbl>, LV <dbl>,
#> # MT <dbl>, NL <dbl>, PL <dbl>, PT <dbl>, RO <dbl>, SE <dbl>, SI <dbl>,
#> # SK <dbl>, UK <dbl>where \(-1,0,1\) indicates values respectively below \(-1\), within the interval \((-1,1)\) and above \(+1\). Details on the contribution of each MS to the variance at a given time \(t\) is evaluate by the square of the difference \((Y_{m,i,t} - \overline{Y}_{EU27,i,t})^2\) between the indicator \(i\) of country \(m\) at time \(t\) and the unweighted average over member states, say EU27:
res$res$squaredContrib
#> # A tibble: 17 × 28
#> AT BE BG CY CZ DE DK EE EL ES FI
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 11.4 7.94 121. 57.5 17.5 1.64e+0 116. 0.612 22.3 18.6 32.3
#> 2 12.5 10.7 82.3 58.2 10.4 3.95e-1 92.7 1.77 15.8 12.1 26.3
#> 3 0.243 4.44 45.0 60.7 4.81 5.10e-5 104. 5.26 13.0 7.33 21.1
#> 4 3.91 3.69 42.5 35.7 5.19 9.58e-1 91.7 12.8 16.2 0.849 21.0
#> 5 4.16 9.37 19.9 38.9 2.69 2.37e+0 96.8 40.2 15.7 0.314 18.8
#> 6 4.84 8.41 4.84 38.4 1.96 5.29e+0 70.6 39.7 23.0 0.810 17.6
#> 7 7.98 8.85 0.0756 30.5 2.03 9.15e+0 59.7 37.5 21.9 6.13 23.3
#> 8 19.3 3.62 0.0414 39.6 3.60 2.70e+1 50.4 0.993 11.6 25.0 20.2
#> 9 33.5 0.264 11.7 47.4 5.22 4.74e+1 46.0 1.73 18.6 28.2 23.9
#> 10 37.7 0.579 26.6 28.5 8.06 7.12e+1 45.4 6.45 71.6 36.7 32.9
#> 11 41.0 0.640 25 4.84 12.2 7.92e+1 39.7 17.6 169 70.6 36
#> 12 43.0 0.710 20.6 0.710 19.9 8.57e+1 39.2 27.6 229. 89.2 27.6
#> 13 27.3 2.81 15.0 1.89 20.5 7.61e+1 32.8 28.4 246. 82.4 17.0
#> 14 18.6 7.74 8.31 4.34 23.2 6.43e+1 29.4 42.5 227. 63.7 8.51
#> 15 14.4 11.0 11.0 5.34 32.4 5.76e+1 24.9 31.2 219. 50.6 5.71
#> 16 8.37 16.1 1.46 2.91 35.9 4.48e+1 16.8 38.4 216. 49.1 2.87
#> 17 5.52 17.2 2.10 0.00250 36.6 3.66e+1 13.3 31.9 206. 46.9 6.00
#> # ℹ 17 more variables: FR <dbl>, HR <dbl>, HU <dbl>, IE <dbl>, IT <dbl>,
#> # LT <dbl>, LU <dbl>, LV <dbl>, MT <dbl>, NL <dbl>, PL <dbl>, PT <dbl>,
#> # RO <dbl>, SE <dbl>, SI <dbl>, SK <dbl>, UK <dbl>It is also possible to decompose the numerator of the variance, called deviance, at each time in order to appreciate the percentage of contribution provided by each member state to the total deviance, \[100 \cdot \frac{(Y_{m,i,t} - \overline{Y}_{EU27,i,t})^2}{ \sum_{m} (Y_{m,i,t} - \overline{Y}_{EU27,i,t})^2 }\] for the indicator \(i\) of country \(m\) at time \(t\).
## sigma_conv(testTB,timeName="time",time_0 = 2002,time_t = 2004)
res$res$devianceContrib
#> # A tibble: 17 × 28
#> AT BE BG CY CZ DE DK EE EL ES FI
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1.02 0.706 10.8 5.11 1.56 1.46e-1 10.3 0.0544 1.98 1.66 2.87
#> 2 1.26 1.08 8.30 5.87 1.05 3.99e-2 9.35 0.178 1.59 1.22 2.65
#> 3 0.0267 0.488 4.95 6.68 0.529 5.61e-6 11.4 0.578 1.43 0.806 2.32
#> 4 0.456 0.430 4.95 4.16 0.605 1.12e-1 10.7 1.49 1.88 0.0989 2.44
#> 5 0.479 1.08 2.29 4.48 0.309 2.73e-1 11.1 4.63 1.81 0.0362 2.17
#> 6 0.578 1.00 0.578 4.59 0.234 6.31e-1 8.42 4.74 2.75 0.0967 2.11
#> 7 0.993 1.10 0.00941 3.80 0.253 1.14e+0 7.43 4.67 2.72 0.762 2.90
#> 8 2.72 0.511 0.00584 5.59 0.507 3.81e+0 7.10 0.140 1.63 3.53 2.85
#> 9 4.36 0.0344 1.52 6.17 0.680 6.17e+0 6.00 0.225 2.42 3.68 3.11
#> 10 4.31 0.0662 3.04 3.26 0.922 8.14e+0 5.19 0.737 8.18 4.20 3.77
#> 11 4.09 0.0639 2.50 0.483 1.22 7.91e+0 3.96 1.76 16.9 7.04 3.59
#> 12 3.90 0.0644 1.87 0.0644 1.80 7.77e+0 3.55 2.51 20.8 8.08 2.51
#> 13 2.73 0.280 1.50 0.189 2.05 7.61e+0 3.28 2.83 24.6 8.23 1.70
#> 14 2.02 0.839 0.901 0.470 2.52 6.97e+0 3.18 4.61 24.7 6.91 0.923
#> 15 1.63 1.25 1.25 0.608 3.68 6.55e+0 2.83 3.56 25.0 5.75 0.650
#> 16 1.04 1.99 0.180 0.360 4.44 5.54e+0 2.07 4.74 26.8 6.07 0.354
#> 17 0.703 2.19 0.268 0.000318 4.66 4.66e+0 1.70 4.06 26.2 5.97 0.764
#> # ℹ 17 more variables: FR <dbl>, HR <dbl>, HU <dbl>, IE <dbl>, IT <dbl>,
#> # LT <dbl>, LU <dbl>, LV <dbl>, MT <dbl>, NL <dbl>, PL <dbl>, PT <dbl>,
#> # RO <dbl>, SE <dbl>, SI <dbl>, SK <dbl>, UK <dbl>thus each row adds to \(100\).
It is possible to produce a graphical output about the main features of country time series, as shown below:
myGG <- graph_departure(res$res$departures,
timeName = "time",
displace = 0.25,
displaceh = 0.45,
dimeFontNum = 4,
myfont_scale = 1.35,
x_angle = 45,
color_rect = c("-1"='red1', "0"='gray80',"1"='lightskyblue1'),
axis_name_y = "Countries",
axis_name_x = "Time",
alpha_color = 0.9
)
myGG
#> $res
#>
#> $msg
#> NULL
#>
#> $err
#> NULLAny selection of countries is feasible:
#myWW1<- warnings()
myGG <- graph_departure(res$res$departures[1:10],
timeName = "time",
displace = 0.25,
displaceh = 0.45,
dimeFontNum = 4,
myfont_scale = 1.35,
x_angle = 45,
color_rect = c("-1"='red1', "0"='gray80',"1"='lightskyblue1'),
axis_name_y = "Countries",
axis_name_x = "Time",
alpha_color = 0.29
)
myGG
#> $res
#>
#> $msg
#> NULL
#>
#> $err
#> NULL2.3 Gamma-convergence
We now introduce gamma convergence by an index based on ranks.
Let \(y_{m,i,t}\) be the value of indicator \(i\) for member state \(m\) at time \(t=0,1,\ldots, T\), and \(\{ \tilde{y}_{m,i,t}: m \in A )\) the ranks for indicator \(i\) over member states in the reference set \(A\), for example \(A = EU27\), at a given time \(t\). The sum of ranks within member state \(m\) is: \[ \tilde{y}^{(s)}_{m,i} = \sum_{t=0}^T \tilde{y}_{m,i,t} \] thus the variance of the sum of ranks over the given interval \[ Var\left[ \{\tilde{y}^{(s)}_{m,i}: m \in A \} \right] \] may be compared to the variance of ranks in the reference time \(t=0\): \[ Var\left[ \{\tilde{y}_{m,i,0}: m \in A \} \right] \]
The Kendall index KI, with respect to aggregation \(A\) of member states for the indicator \(i\) over a given time interval is: \[ KI(A,i,T) = \frac{Var\left[ \{\tilde{y}^{(s)}_{m,i}: m \in A \} \right] }{ (T+1)^2 ~~Var\left[\{\tilde{y}_{m,i,0}: m \in A \}\right] } \]
The measure of gamma-convergence is obtained with the following function:
Note the starting time is zero, the reference, but first a copy of the dataset is performed.
(timeCounTB <- testTB)
#> # A tibble: 6 × 4
#> time countryA countryB countryC
#> <dbl> <dbl> <dbl> <dbl>
#> 1 2000 0.8 2.7 3.9
#> 2 2001 1.2 3.2 4.2
#> 3 2002 0.9 2.9 4.1
#> 4 2003 1.3 2.9 4
#> 5 2004 1.2 3.1 4.1
#> 6 2005 1.2 3 4Now we move to ranks within time using rank():
therefore with the above data:
# debug(gamma_conv)
(gamma_conv(timeCounTB,ref=2000,last=2005,timeName = "time"))
#> $res
#> [1] 0.7346939
#>
#> $msg
#> NULL
#>
#> $err
#> NULL
(gamma_conv(timeCounTB,ref=2000,last=2004,timeName = "time"))
#> $res
#> [1] 0.6944444
#>
#> $msg
#> NULL
#>
#> $err
#> NULL
(gamma_conv(timeCounTB,ref=2000,last=2003,timeName = "time"))
#> $res
#> [1] 0.64
#>
#> $msg
#> NULL
#>
#> $err
#> NULL
(gamma_conv(timeCounTB,ref=2000,last=2002,timeName = "time"))
#> $res
#> [1] 0.5625
#>
#> $msg
#> NULL
#>
#> $err
#> NULL
(gamma_conv(timeCounTB,ref=2000,last=2001,timeName = "time"))
#> $res
#> [1] 0.4444444
#>
#> $msg
#> NULL
#>
#> $err
#> NULLand changing reference year:
(gamma_conv(timeCounTB,ref=2001,last=2005,timeName = "time"))
#> $res
#> [1] 0.7346939
#>
#> $msg
#> NULL
#>
#> $err
#> NULL
(gamma_conv(timeCounTB,ref=2002,last=2004,timeName = "time"))
#> $res
#> [1] 0.6944444
#>
#> $msg
#> NULL
#>
#> $err
#> NULLNow we exchange values and calculate gamma-convergence:
timeCounTB2 <- timeCounTB
timeCounTB2[2,2:4] <- timeCounTB[2,4:2]
timeCounTB2[4,2:4] <- timeCounTB[4,c(4,2,3)]
timeCounTB2
#> # A tibble: 6 × 4
#> time countryA countryB countryC
#> <dbl> <dbl> <dbl> <dbl>
#> 1 2000 0.8 2.7 3.9
#> 2 2001 4.2 3.2 1.2
#> 3 2002 0.9 2.9 4.1
#> 4 2003 4 1.3 2.9
#> 5 2004 1.2 3.1 4.1
#> 6 2005 1.2 3 4
gamma_conv(timeCounTB2,last=2005,ref=2000, timeName = "time",printRanks = T)
#> Ranks:
#> countryA countryB countryC
#> [1,] 1 2 3
#> [2,] 3 2 1
#> [3,] 1 2 3
#> [4,] 3 1 2
#> [5,] 1 2 3
#> [6,] 1 2 3
#> $res
#> [1] 0.1428571
#>
#> $msg
#> NULL
#>
#> $err
#> NULLand after random permutation:
timeCounTB3 <- cbind(timeCounTB[1],t(apply(timeCounTB,1,
function(vet)vet[sample(2:4,3)])))
timeCounTB3
#> time 1 2 3
#> 1 2000 0.8 2.7 3.9
#> 2 2001 1.2 3.2 4.2
#> 3 2002 4.1 2.9 0.9
#> 4 2003 1.3 4.0 2.9
#> 5 2004 4.1 3.1 1.2
#> 6 2005 1.2 4.0 3.0
(gamma_conv(timeCounTB3,last=2005,ref=2000, timeName = "time",printRanks = T))
#> Ranks:
#> 1 2 3
#> [1,] 1 2 3
#> [2,] 1 2 3
#> [3,] 3 2 1
#> [4,] 1 3 2
#> [5,] 3 2 1
#> [6,] 1 3 2
#> $res
#> [1] 0.08163265
#>
#> $msg
#> NULL
#>
#> $err
#> NULL2.4 Delta-convergence
Delta-convergence can be calculated as follows:
2.5 Absolute change
Absolute change as described in the reserved Eurofound Annex is defined as: \[ \Delta y_{m,i,t} = y_{m,i,t} - y_{m,i,t-1} \] for country \(m\), indicator \(i\) at time \(t\).
The R function abso_change calculates the above quantity, for example in the emp_20_64_MS dataset
data(emp_20_64_MS)
mySTB <- abso_change(emp_20_64_MS,
time_0 = 2005,
time_t = 2010,
all_within=TRUE,
timeName = "time")
names(mySTB$res)
#> [1] "abso_change" "sum_abs_change" "average_abs_change"thus the above equation results in:
mySTB$res$abso_change
#> time AT BE BG CY CZ DE DK EE EL ES FI FR HR HU IE
#> 1 2003 0.4 -0.2 2.2 0.3 -0.7 -0.4 -0.9 0.8 1.0 1.1 -0.3 1.1 0.5 1.0 -0.4
#> 2 2004 -2.9 1.3 2.5 0.3 -0.9 -0.5 0.7 1.1 0.5 0.9 -0.4 -0.5 1.3 -0.4 0.6
#> 3 2005 2.0 0.7 0.7 -1.3 0.6 1.5 -0.1 1.8 0.1 2.3 0.5 0.2 0.3 0.2 1.6
#> 4 2006 1.2 0.0 3.2 1.4 0.5 1.7 1.4 3.9 1.2 1.5 0.9 0.0 0.6 0.4 0.8
#> 5 2007 1.2 1.2 3.3 1.0 0.8 1.8 -0.4 1.0 0.2 0.7 0.9 0.5 3.3 -0.3 1.7
#> IT LT LU LV MT NL PL PT RO SE SI SK UK
#> 1 0.9 2.7 -1.2 1.1 -0.4 -0.5 -0.4 -1.1 0.5 -0.3 -1.9 1.8 0.4
#> 2 1.6 -1.1 0.5 0.0 -0.5 -0.4 -0.3 -0.4 -0.1 -0.7 2.9 -1.5 0.2
#> 3 -0.2 1.1 1.3 1.7 0.1 -2.2 1.3 -0.4 -1.1 0.3 0.1 1.0 0.3
#> 4 0.9 0.6 0.1 4.1 0.5 1.0 1.8 0.4 1.2 0.7 0.4 1.5 0.0
#> 5 0.3 1.4 0.5 2.0 0.7 1.8 2.6 -0.1 -0.4 1.3 0.9 1.2 0.0The sum of absolute values \[ \sum_{t=t_0+1}^{} | \Delta y_{m,i,t}| \] is:
round(mySTB$res$sum_abs_change,4)
#> AT BE BG CY CZ DE DK EE EL ES FI FR HR HU IE IT
#> 7.7 3.4 11.9 4.3 3.5 5.9 3.5 8.6 3.0 6.5 3.0 2.3 6.0 2.3 5.1 3.9
#> LT LU LV MT NL PL PT RO SE SI SK UK
#> 6.9 3.6 8.9 2.2 5.9 6.4 2.4 3.3 3.3 6.2 7.0 0.9and such sum can be divided by the number of pair of years so that the result is an average per pair of years:
2.6 Convergence measures on Eurofound lifesatisf indicator
Here we assume that larger the index, better the performance.
Let’s load the Eurofound indicator lifesatisf:
workDF <- extract_indicator_EUF(
indicator_code ="lifesatisf", #Code_in_database
fromTime=2000,
toTime =2018,
gender= c("Total","Females","Males")[1],
countries = convergEU_glb()$EU27_2020$memberStates$codeMS)
workDF
#> $res
#> # A tibble: 4 × 29
#> time sex AT BE BG CY CZ DE DK EE EL ES FI
#> <dbl> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2003 Total 7.85 7.52 4.47 7.22 6.57 7.36 8.47 5.94 6.69 7.49 8.14
#> 2 2007 Total 6.95 7.54 5.01 7.05 6.59 7.16 8.48 6.72 6.58 7.25 8.23
#> 3 2011 Total 7.66 7.38 5.55 7.16 6.43 7.20 8.37 6.28 6.16 7.47 8.08
#> 4 2016 Total 7.92 7.31 5.62 6.54 6.48 7.31 8.19 6.73 5.26 6.95 8.07
#> # ℹ 16 more variables: FR <dbl>, HR <dbl>, HU <dbl>, IE <dbl>, IT <dbl>,
#> # LT <dbl>, LU <dbl>, LV <dbl>, MT <dbl>, NL <dbl>, PL <dbl>, PT <dbl>,
#> # RO <dbl>, SE <dbl>, SI <dbl>, SK <dbl>
#>
#> $msg
#> NULL
#>
#> $err
#> NULL
wDF <- workDF$resthen we ask if it is complete or some missing values are present:
check_data(select(wDF,-sex),timeName="time")
#> $res
#> NULL
#>
#> $msg
#> NULL
#>
#> $err
#> [1] "Error: one or more missing values in the dataframe."thus at least one missing value is present. In the next step, imputation of missing values is performed:
wDFI <- impute_dataset(select(wDF,-sex),
countries= names(select(wDF,-sex,-time)),
timeName = "time",
tailMiss = c("cut", "constant")[2],
headMiss = c("cut", "constant")[1])and some checking is done:
which returns TRUE.
First, we calculate the EU unweighted average of emp:
wwTB <- (wDFI$res %>%
average_clust(timeName="time",cluster="EU27"))$res
wwTB$EU27
#> [1] 6.829626 6.984789 7.014334 6.978321Time series can be plotted:
mini_EU <- min(wwTB$EU27)
maxi_EU <- max(wwTB$EU27)
qplot(time, EU27, data=wwTB,
ylim=c(mini_EU,maxi_EU))+geom_line(colour="navy blue")+
ylab("lifesatisf")
#> Warning: `qplot()` was deprecated in ggplot2 3.4.0.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.
2.6.1 Beta convergence
Now the beta-convergence is calculated for just two years:
betaRes <- beta_conv(wDFI$res,time_0=2007, time_t=2011, all_within=FALSE)
betaRes
#> $res
#> $res$workTB
#> # A tibble: 27 × 3
#> deltaIndic indic countries
#> <dbl> <dbl> <chr>
#> 1 0.0244 1.94 AT
#> 2 -0.00553 2.02 BE
#> 3 0.0256 1.61 BG
#> 4 0.00403 1.95 CY
#> 5 -0.00623 1.89 CZ
#> 6 0.00138 1.97 DE
#> 7 -0.00316 2.14 DK
#> 8 -0.0171 1.91 EE
#> 9 -0.0165 1.88 EL
#> 10 0.00719 1.98 ES
#> # ℹ 17 more rows
#>
#> $res$model
#>
#> Call:
#> stats::lm(formula = deltaIndic ~ indic, data = workTB)
#>
#> Coefficients:
#> (Intercept) indic
#> 0.11155 -0.05679
#>
#>
#> $res$summary
#> # A tibble: 2 × 5
#> term estimate std.error statistic p.value
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 (Intercept) 0.112 0.0314 3.56 0.00153
#> 2 indic -0.0568 0.0162 -3.51 0.00171
#>
#> $res$beta1
#> [1] -0.05678881
#>
#> $res$adj.r.squared
#> [1] 0.3037316
#>
#>
#> $msg
#> NULL
#>
#> $err
#> NULLA plot of transformed data and the straight line may be useful:
mybetaplot<-beta_conv_graph(betaRes,
indiName = 'Mean Life Satisfaction',
time_0 = 2007,
time_t = 2011)
mybetaplot
Note that label are replicated as many times as the number of included
subsequent years.
2.6.2 Sigma convergence
Here we go with calculating the sigma-convergence:
It is also possible to obtain a graphical representation of the standard deviation and the coefficient of variation obtained for the Sigma convergence by invoking the sigma_conv_graph function as follows:
mysigmaplot<-sigma_conv_graph(sigmaconvOut=mysigmares,
time_0 = 2007,
time_t = 2011,
aggregation='EU27_2020')
mysigmaplot
2.6.3 Gamma convergence
Let’s reload Eurofound data:
workDF <- extract_indicator_EUF(
indicator_code ="lifesatisf", #Code_in_database
fromTime=2000,
toTime =2018,
gender= c("Total","Females","Males")[1],
countries = convergEU_glb()$EU27_2020$memberStates$codeMS)
wDFI <- impute_dataset(select(workDF$res,-sex),
countries= names(select(wDF,-sex,-time)),
timeName = "time",
tailMiss = c("cut", "constant")[2],
headMiss = c("cut", "constant")[1])
check_data(wDFI$res,timeName="time")
#> $res
#> [1] TRUE
#>
#> $msg
#> NULL
#>
#> $err
#> NULLNow gamma-convergence is computed:
gamma_conv(wDFI$res,ref=2003,last=2016,timeName = "time")
#> $res
#> [1] 0.5879853
#>
#> $msg
#> NULL
#>
#> $err
#> NULLor equivalently:
Indeed there is the possibility of performing calculation for each pair of subsequent years in the dataset, that is, each year is the reference of the subsequent year:
wDFI$res
#> # A tibble: 4 × 28
#> time AT BE BG CY CZ DE DK EE EL ES FI FR
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2003 7.85 7.52 4.47 7.22 6.57 7.36 8.47 5.94 6.69 7.49 8.14 6.96
#> 2 2007 6.95 7.54 5.01 7.05 6.59 7.16 8.48 6.72 6.58 7.25 8.23 7.32
#> 3 2011 7.66 7.38 5.55 7.16 6.43 7.20 8.37 6.28 6.16 7.47 8.08 7.23
#> 4 2016 7.92 7.31 5.62 6.54 6.48 7.31 8.19 6.73 5.26 6.95 8.07 7.17
#> # ℹ 15 more variables: HR <dbl>, HU <dbl>, IE <dbl>, IT <dbl>, LT <dbl>,
#> # LU <dbl>, LV <dbl>, MT <dbl>, NL <dbl>, PL <dbl>, PT <dbl>, RO <dbl>,
#> # SE <dbl>, SI <dbl>, SK <dbl>2.6.4 Delta convergence
Let \(y_{m,i,t}\) be the value of indicator \(i\) for member state \(m\) at time \(t\), and \(y^{(M)}_{i,t}\) the maximum value over member states in the reference set \(A\), for example \(A = EU27\): \[ y^{(M)}_{i,t} = max(\{ y_{m,i,t}: m \in A\}) \]
The distance of a member state \(m\) from the top performer at time \(i\) is: \[ y^{(M)}_{i,t} - y_{m,i,t} \] thus the overall distance at time \(t\), called delta, is the sum of distances over the reference set \(A\) of MS: \[ \delta_{i,t} = \sum_{m \in A} (y^{(M)}_{i,t} - y_{m,i,t}) \] for the considered indicator \(i\).
The measure of delta-convergence is obtained as follows:
delta_conv(wwTB)
#> $res
#> # A tibble: 4 × 2
#> time delta
#> <dbl> <dbl>
#> 1 2003 46.0
#> 2 2007 41.8
#> 3 2011 38.0
#> 4 2016 34.0
#>
#> $msg
#> NULL
#>
#> $err
#> NULLIt must be noted that the delta_conv function allows to obtain also the declaration of convergence. To this end, the argument extended should be specified as TRUE. For example, for the wwTB indicator the syntax is as follows:
delta_conv(wwTB,"time", extended=TRUE)
#> $res
#> $res$delta_conv
#> # A tibble: 4 × 2
#> time delta
#> <dbl> <dbl>
#> 1 2003 46.0
#> 2 2007 41.8
#> 3 2011 38.0
#> 4 2016 34.0
#>
#> $res$differences
#> AT BE BG CY CZ DE DK EE
#> [1,] 0.6245208 0.9556756 4.003694 1.250645 1.904858 1.113082 0 2.534678
#> [2,] 1.5314507 0.9352727 3.469857 1.430196 1.883980 1.314598 0 1.756762
#> [3,] 0.7128654 0.9938769 2.823287 1.209136 1.939715 1.168355 0 2.095294
#> [4,] 0.2691503 0.8841901 2.568287 1.657201 1.713479 0.885087 0 1.464275
#> EL ES FI FR HR HU IE IT
#> [1,] 1.779027 0.9809632 0.3344421 1.508648 2.035054 2.534398 0.7697177 1.255543
#> [2,] 1.896363 1.2247257 0.2450380 1.154050 2.041460 2.884686 0.8919230 1.889798
#> [3,] 2.211353 0.9066143 0.2946043 1.145797 1.591188 2.598653 0.9794941 1.488251
#> [4,] 2.930435 1.2417102 0.1190214 1.025349 1.860298 1.677889 0.5078416 1.635120
#> LT LU LV MT NL PL PT
#> [1,] 3.028715 0.7834425 2.897604 1.1496062 0.9260612 2.3119373 2.481104
#> [2,] 2.156122 0.5780816 2.438259 0.9191999 0.6090589 1.5864158 2.288289
#> [3,] 1.671255 0.5828137 2.129854 1.1383724 0.6792879 1.3019018 1.604687
#> [4,] 1.709892 0.2899728 1.869513 0.6240749 0.4548769 0.9941764 1.321342
#> RO SE SI SK EU27
#> [1,] 2.348566 0.5827713 1.429040 2.815222 1.642186
#> [2,] 2.003722 0.1419497 1.250872 1.800439 1.493428
#> [3,] 1.638209 0.3387566 1.419971 1.986291 1.357403
#> [4,] 1.692563 0.2608399 1.338965 1.789471 1.214260
#>
#> $res$difference_last_first
#> AT BE BG CY CZ DE
#> -0.35537052 -0.07148552 -1.43540668 0.40655661 -0.19137859 -0.22799492
#> DK EE EL ES FI FR
#> 0.00000000 -1.07040215 1.15140772 0.26074696 -0.21542072 -0.48329926
#> HR HU IE IT LT LU
#> -0.17475605 -0.85650873 -0.26187611 0.37957668 -1.31882334 -0.49346972
#> LV MT NL PL PT RO
#> -1.02809095 -0.52553129 -0.47118425 -1.31776094 -1.15976238 -0.65600348
#> SE SI SK EU27
#> -0.32193136 -0.09007549 -1.02575064 -0.42792575
#>
#> $res$strict_conv_ini_last
#> [1] FALSE
#>
#> $res$label_strict
#> [1] " "
#>
#> $res$converg_ini_last
#> [1] TRUE
#>
#> $res$label_conver
#> [1] "convergence"
#>
#> $res$diffe_delta
#> [1] -11.98192
#>
#>
#> $msg
#> NULL
#>
#> $err
#> NULLIt is also useful to evaluate how much a collection of MS deviates from the EU mean for a given indicator and a period of time. In order to obtain this further information the demea_change function has been implemented in the convergEU package:
res1<-demea_change(wwTB,
timeName="time",
time_0 = 2003,
time_t = 2016,
sele_countries= NA,
doplot=TRUE)
res1
#> $res
#> $res$resDiffe
#> # A tibble: 4 × 29
#> time AT BE BG CY CZ DE DK EE EL ES
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2003 1.02 0.687 -2.36 0.392 -0.263 0.529 1.64 -0.892 -0.137 0.661
#> 2 2007 -0.0380 0.558 -1.98 0.0632 -0.391 0.179 1.49 -0.263 -0.403 0.269
#> 3 2011 0.645 0.364 -1.47 0.148 -0.582 0.189 1.36 -0.738 -0.854 0.451
#> 4 2016 0.945 0.330 -1.35 -0.443 -0.499 0.329 1.21 -0.250 -1.72 -0.0275
#> # ℹ 18 more variables: FI <dbl>, FR <dbl>, HR <dbl>, HU <dbl>, IE <dbl>,
#> # IT <dbl>, LT <dbl>, LU <dbl>, LV <dbl>, MT <dbl>, NL <dbl>, PL <dbl>,
#> # PT <dbl>, RO <dbl>, SE <dbl>, SI <dbl>, SK <dbl>, EU27 <dbl>
#>
#> $res$diffe_abs_diff
#> # A tibble: 3 × 29
#> time AT BE BG CY CZ DE DK EE EL ES
#> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2007 -0.980 -0.128 -0.385 -0.328 0.128 -0.350 -0.149 -0.629 0.266 -0.393
#> 2 2011 0.607 -0.195 -0.511 0.0850 0.192 0.0102 -0.136 0.475 0.451 0.182
#> 3 2016 0.301 -0.0335 -0.112 0.295 -0.0831 0.140 -0.143 -0.488 0.862 -0.423
#> # ℹ 18 more variables: FI <dbl>, FR <dbl>, HR <dbl>, HU <dbl>, IE <dbl>,
#> # IT <dbl>, LT <dbl>, LU <dbl>, LV <dbl>, MT <dbl>, NL <dbl>, PL <dbl>,
#> # PT <dbl>, RO <dbl>, SE <dbl>, SI <dbl>, SK <dbl>, EU27 <dbl>
#>
#> $res$stats
#> # A tibble: 28 × 4
#> MS negaSum posiSum posi
#> <chr> <dbl> <dbl> <int>
#> 1 AT -0.980 0.907 1
#> 2 BE -0.356 0 2
#> 3 BG -1.01 0 3
#> 4 CY -0.328 0.380 4
#> 5 CZ -0.0831 0.320 5
#> 6 DE -0.350 0.150 6
#> 7 DK -0.428 0 7
#> 8 EE -1.12 0.475 8
#> 9 EL 0 1.58 9
#> 10 ES -0.816 0.182 10
#> # ℹ 18 more rows
#>
#> $res$miniX
#> [1] -1.117033
#>
#> $res$maxiX
#> [1] 1.579333
#>
#> $res$res_graph
#>
#>
#> $msg
#> NULL
#>
#> $err
#> NULLTo plot the calculated differences, the user should invoke the plot function as follows:
