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REAT (version 3.0.2)

rca: Analysis of regional beta and sigma convergence

Description

This function provides the analysis of absolute and conditional regional economic beta convergence and sigma convergence for cross-sectional data. Beta convergence can be estimated using an OLS or NLS technique. Sigma convergence can be analyzed using ANOVA or trend regression.

Usage

rca(gdp1, time1, gdp2, time2, 
conditions = NULL, conditions.formula = NULL, conditions.startval = NULL, 
beta.estimate = "ols", beta.plot = FALSE, beta.plotPSize = 1, beta.plotPCol = "black", 
beta.plotLine = FALSE, beta.plotLineCol = "red", beta.plotX = "Ln (initial)", 
beta.plotY = "Ln (growth)", beta.plotTitle = "Beta convergence", beta.bgCol = "gray95", 
beta.bgrid = TRUE, beta.bgridCol = "white", beta.bgridSize = 2, beta.bgridType = "solid", 
sigma.type = "anova", sigma.measure = "sd", sigma.log = TRUE, sigma.weighting = NULL, 
sigma.issample = FALSE, sigma.plot = FALSE, sigma.plotLSize = 1, 
sigma.plotLineCol = "black", sigma.plotRLine = FALSE, sigma.plotRLineCol = "blue", 
sigma.Ymin = 0, sigma.plotX = "Time", sigma.plotY = "Variation", 
sigma.plotTitle = "Sigma convergence", sigma.bgCol = "gray95", sigma.bgrid = TRUE, 
sigma.bgridCol = "white", sigma.bgridSize = 2, sigma.bgridType = "solid")

Arguments

gdp1

A numeric vector containing the GDP per capita (or another economic variable) at time t

time1

A single value of time t (= the initial year)

gdp2

A numeric vector containing the GDP per capita (or another economic variable) at time t+1 or a data frame containing the GDPs per capita (or another economic variable) at time t+1, t+2, t+3, ..., t+n

time2

A single value of time t+1 or t_n, respectively

conditions

A data frame containing the conditions for conditional beta convergence

conditions.formula

If beta.estimate = "nls": A formula for the functional linkage of the conditions in the case of conditional beta convergence

conditions.startval

If beta.estimate = "nls": Starting values for the parameters of the conditions in the case of conditional beta convergence

beta.estimate

Beta estimate via ordinary least squares (OLS) or nonlinear least squares (NLS). Default: beta.estimate = "ols"

beta.plot

Boolean argument that indicates if a plot of beta convergence has to be created

beta.plotPSize

If beta.plot = TRUE: Point size in the beta convergence plot

beta.plotPCol

If beta.plot = TRUE: Point color in the beta convergence plot

beta.plotLine

If beta.plot = TRUE: Logical argument that indicates if a regression line has to be added to the plot

beta.plotLineCol

If beta.plot = TRUE and beta.plotLine = TRUE: Line color of regression line

beta.plotX

If beta.plot = TRUE: Name of the X axis

beta.plotY

If beta.plot = TRUE: Name of the Y axis

beta.plotTitle

If beta.plot = TRUE: Plot title

beta.bgCol

If beta.plot = TRUE: Plot background color

beta.bgrid

If beta.plot = TRUE: Logical argument that indicates if the plot contains a grid

beta.bgridCol

If beta.plot = TRUE and beta.bgrid = TRUE: Color of the grid

beta.bgridSize

If beta.plot = TRUE and beta.bgrid = TRUE: Size of the grid

beta.bgridType

If beta.plot = TRUE and beta.bgrid = TRUE: Type of the grid

sigma.type

Estimating sigma convergence via ANOVA (two years) or trend regression (more than two years). Default: sigma.type = "anova"

sigma.measure

argument that indicates how the sigma convergence should be measured. The default is output = "sd", which means that the standard deviation is used. If output = "var" or output = "cv", the variance or the coefficient of variation is used, respectively.

sigma.log

Logical argument. Per default (sigma.log = TRUE), also in the sigma convergence analysis, the economic variables are transformed by natural logarithm. If the original values should be used, state sigma.log = FALSE

sigma.weighting

If the measure of statistical dispersion in the sigma convergence analysis (coefficient of variation or standard deviation) should be weighted, a weighting vector has to be stated

sigma.issample

Logical argument that indicates if the dataset is a sample or the population (default: is.sample = FALSE, so the denominator of variance is \(n\))

sigma.plot

Logical argument that indicates if a plot of sigma convergence has to be created

sigma.plotLSize

If sigma.plot = TRUE: Line size of the sigma convergence plot

sigma.plotLineCol

If sigma.plot = TRUE: Line color of the sigma convergence plot

sigma.plotRLine

If sigma.plot = TRUE: Logical argument that indicates if a regression line has to be added to the plot

sigma.plotRLineCol

If sigma.plot = TRUE and sigma.plotRLine = TRUE: Color of the regression line

sigma.Ymin

If sigma.plot = TRUE: start value of the Y axis in the plot

sigma.plotX

If sigma.plot = TRUE: Name of the X axis

sigma.plotY

If sigma.plot = TRUE: Name of the Y axis

sigma.plotTitle

If sigma.plot = TRUE: Title of the plot

sigma.bgCol

If sigma.plot = TRUE: Plot background color

sigma.bgrid

If sigma.plot = TRUE: Logical argument that indicates if the plot contains a grid

sigma.bgridCol

If sigma.plot = TRUE and sigma.bgrid = TRUE: Color of the grid

sigma.bgridSize

If sigma.plot = TRUE and sigma.bgrid = TRUE: Size of the grid

sigma.bgridType

If sigma.plot = TRUE and sigma.bgrid = TRUE: Type of the grid

Value

A list containing the following objects:

betaconv

A list containing the following objects:

regdata

A data frame containing the regression data, including the \(ln\)-transformed economic variables

tinterval

The time interval

abeta

A list containing the estimates of the absolute beta convergence regression model, including lambda and half-life

cbeta

If conditions are stated: a list containing the estimates of the conditional beta convergence regression model, including lambda and half-life

sigmaconv

A list containing the following objects:

sigmaconv

A matrix containing either the standard deviations, their quotient and the results of the significance test (F-statistic) or the results of trend regression

Details

From the regional economic perspective (in particular the neoclassical growth theory), regional disparities are expected to decline. This convergence can have different meanings: Sigma convergence (\(\sigma\)) means a harmonization of regional economic output or income over time, while beta convergence (\(\beta\)) means a decline of dispersion because poor regions have a stronger economic growth than rich regions (Capello/Nijkamp 2009). Regardless of the theoretical assumptions of a harmonization in reality, the related analytical framework allows to analyze both types of convergence for cross-sectional data (GDP p.c. or another economic variable, \(y\), for \(i\) regions and two points in time, \(t\) and \(t+T\)), or one starting point (\(t\)) and the average growth within the following \(n\) years (\(t+1, t+2, ..., t+n\)), respectively. Beta convergence can be calculated either in a linearized OLS regression model or in a nonlinear regression model. When no other variables are integrated in this model, it is called absolute beta convergence. Implementing other region-related variables (conditions) into the model leads to conditional beta convergence. If there is beta convergence (\(\beta < 0\)), it is possible to calculate the speed of convergence, \(\lambda\), and the so-called Half-Life \(H\), while the latter is the time taken to reduce the disparities by one half (Allington/McCombie 2007, Goecke/Huether 2016). There is sigma convergence, when the dispersion of the variable (\(\sigma\)), e.g. calculated as standard deviation or coefficient of variation, reduces from \(t\) to \(t+T\). This can be measured using ANOVA for two years or trend regression with respect to several years (Furceri 2005, Goecke/Huether 2016).

The rca function is a wrapper for the functions betaconv.ols, betaconv.nls, sigmaconv and sigmaconv.t. This function calculates (absolute and/or conditional) beta convergence and sigma convergence. Regional disparities are measured by the standard deviation (or variance, coefficient of variation) for all GDPs per capita (or another economic variable) for the given years. Beta convergence is estimated either using ordinary least squares (OLS) or nonlinear least squares (NLS). If the beta coefficient is negative (using OLS) or positive (using NLS), there is beta convergence. Sigma convergence is analyzed either using an analysis of variance (ANOVA) for these deviation measures (year 1 divided by year 2, F-statistic) or a trend regression (F-statistic, t-statistic). In the former case, if \(\sigma_t1/\sigma_t2 > 0\), there is sigma convergence. In the latter case, if the slope of the trend regression is negative, there is sigma convergence.

References

Allington, N. F. B./McCombie, J. S. L. (2007): “Economic growth and beta-convergence in the East European Transition Economies”. In: Arestis, P./Baddely, M./McCombie, J. S. L. (eds.): Economic Growth. New Directions in Theory and Policy. Cheltenham: Elgar. p. 200-222.

Capello, R./Nijkamp, P. (2009): “Introduction: regional growth and development theories in the twenty-first century - recent theoretical advances and future challenges”. In: Capello, R./Nijkamp, P. (eds.): Handbook of Regional Growth and Development Theories. Cheltenham: Elgar. p. 1-16.

Dapena, A. D./Vazquez, E. F./Morollon, F. R. (2016): “The role of spatial scale in regional convergence: the effect of MAUP in the estimation of beta-convergence equations”. In: The Annals of Regional Science, 56, 2, p. 473-489.

Furceri, D. (2005): “Beta and sigma-convergence: A mathematical relation of causality”. In: Economics Letters, 89, 2, p. 212-215.

Goecke, H./Huether, M. (2016): “Regional Convergence in Europe”. In: Intereconomics, 51, 3, p. 165-171.

Young, A. T./Higgins, M. J./Levy, D. (2008): “Sigma Convergence versus Beta Convergence: Evidence from U.S. County-Level Data”. In: Journal of Money, Credit and Banking, 40, 5, p. 1083-1093.

See Also

betaconv.ols, betaconv.nls, betaconv.speed, sigmaconv, sigmaconv.t, cv, sd2, var2

Examples

Run this code
# NOT RUN {
data (G.counties.gdp)
# Loading GDP data for Germany (counties = Landkreise)

rca (G.counties.gdp$gdppc2010, 2010, G.counties.gdp$gdppc2011, 2011, 
conditions = NULL, beta.plot = TRUE)
# Two years, no conditions (Absolute beta convergence)


regionaldummies <- to.dummy(G.counties.gdp$regional)
# Creating dummy variables for West/East
G.counties.gdp$West <- regionaldummies[,2]
G.counties.gdp$East <- regionaldummies[,1]
# Adding dummy variables to data

rca (G.counties.gdp$gdppc2010, 2010, G.counties.gdp$gdppc2011, 2011, 
conditions = G.counties.gdp[c(70,71)])
# Two years, with conditions
# (Absolute and conditional beta convergence)

converg1 <- rca (G.counties.gdp$gdppc2010, 2010, G.counties.gdp$gdppc2011, 2011, 
conditions = G.counties.gdp[c(70,71)])
# Store results in object
converg1$betaconv$abeta
# Addressing estimates for the conditional beta model


rca (G.counties.gdp$gdppc2010, 2010, G.counties.gdp[65:68], 2014, conditions = NULL, 
sigma.type = "trend", beta.plot = TRUE, sigma.plot = TRUE)
# Five years, no conditions (Absolute beta convergence)
# with plots for both beta and sigma convergence
# }

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