In federico-m-stefanini/convergEU: Monitoring Convergence of EU Countries

Beta convergence

Beta convergence is a catching-up process in which poorer performing countries grow faster than the best perfoming ones. The red line shows the relationship between the growth of an indicator over a certain period of time and its initial value. Beta convergence exists if that relation is statistically significant and negative. Its coefficient gives an indication of the speed of the process.

The calculation of beta convergence is performed according to the following linear model: $$ \tau^{-1}~ ln(y_{m,i,t+\tau})-ln(y_{m,i,t}) = \tau^{-1}~log\left{\frac{y_{m,i,t+\tau}}{y_{m,i,t}}\right} = \beta_0 + \beta_1 ~~ln(y_{m,i,t}) +\epsilon_{m,i,t} $$ where $m$ represent the member state of EU (country), $i$ refers to the indicator of interest, $t$ is the reference time and $\tau \in {1,2,\ldots}$ the size of the time window.

betaRes <- beta_conv(tavDes = wkDF2, time_0 = ptime_0, time_t = ptime_t, 
                 all_within = FALSE, timeName = timeName,useTau=TRUE)
# Check out with convergeEU team at Eurofound 
if(betaRes$res$beta1 > 0 && params$indiType == 'highBest'){
   diagnosis <- 'divergence'
}else if(betaRes$res$beta1 <= 0 && params$indiType == 'highBest'){
  diagnosis <- 'convergence'
}else if(betaRes$res$beta1 > 0 && params$indiType == 'lowBest'){
   diagnosis <-'divergence'
}else if(betaRes$res$beta1 <= 0 && params$indiType == 'lowBest'){
   diagnosis <- 'convergence'
}else{
  diagnosis <- 'unassigned'
} 

lastRowAverages <- nrow(outMed)

out_packed_list$betaSummary <-  list(
  indicatorType= params$indiType,
  betaCoeff = betaRes$res$beta1,
  diagnosis= diagnosis,
  startingYear = ptime_0,
  endingYear = ptime_t,
  startingAverage = as.numeric(outMed[1,aggregation]),
  endingAverage = as.numeric(outMed[lastRowAverages,aggregation]),
  DeltaAverage =  as.numeric(outMed[lastRowAverages,aggregation])-
                   as.numeric(outMed[1,aggregation])

)

beta_conv_gr1 <- qplot(betaRes$res$workTB$indic,
      betaRes$res$workTB$deltaIndic,
      xlab= paste("log ",indiName ),
      ylab= paste("Log rate of growth:",ptime_0,"-",ptime_t)) +
  geom_abline(intercept = as.numeric(betaRes$res$summary[1,2]),
              slope = as.numeric(betaRes$res$summary[2,2]),
              colour = "red") +
  geom_text(aes(label=betaRes$res$workTB$countries),
            hjust=0, vjust=0,colour="blue")
# export
out_packed_list$beta_conv_gr1 <-  beta_conv_gr1

# plot
beta_conv_gr1

sigCores <- sigma_conv(wkDF2, 
                       timeName = timeName, 
                       time_0 = ptime_0, 
                       time_t=ptime_t)
#sigCores
lastRowAveragesSig <- nrow(sigCores$res)

sd_enne <<- function(vec_obs){
   enne <-  length(vec_obs)
   esse_n <- sd(vec_obs)*sqrt((enne-1)/enne)
   esse_n
}
dichia_con_stddev <- upDo_CoDi(wkDF2,
          timeName = timeName,
          indiType = params$indiType,
          time_0 = ptime_0,
          time_t = ptime_t,
          heter_fun = "sd_enne"
          )


# here stddev
if(dichia_con_stddev$res$declaration_strict != "none"){
  label_dichia_con_stddev <- paste(
       dichia_con_stddev$res$declaration_strict,
       dichia_con_stddev$res$declaration_type
       )
}else{
  label_dichia_con_stddev <- paste(
               dichia_con_stddev$res$declaration_weak,
               dichia_con_stddev$res$declaration_type
               )
}

Beta summary list:

Indicator type: r params$indiType
Beta Coefficient: r betaRes$res$beta1
Change in average:
Year r ptime_0, average: r as.numeric(outMed[1,aggregation])
Year r ptime_t, average: r as.numeric(outMed[lastRowAverages,aggregation])
Difference $\Delta$ between years: r as.numeric(outMed[lastRowAverages,aggregation])-as.numeric(outMed[1,aggregation])
Results for year r ptime_t with reference year r ptime_0:
- For beta convergence: r diagnosis