This function provides the analysis of absolute and conditional regional economic beta convergence for cross-sectional data using a nonlineaer least squares (NLS) technique.
betaconv.nls(gdp1, time1, gdp2, time2, conditions = NULL, conditions.formula = NULL,
conditions.startval = NULL, 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",
print.results = TRUE)A numeric vector containing the GDP per capita (or another economic variable) at time t
A single value of time t (= the initial year)
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
A single value of time t+1 or t_n, respectively
A data frame containing the conditions for conditional beta convergence
A formula for the functional linkage of the conditions in the case of conditional beta convergence
Starting values for the parameters of the conditions in the case of conditional beta convergence
Boolean argument that indicates if a plot of beta convergence has to be created
If beta.plot = TRUE: Point size in the beta convergence plot
If beta.plot = TRUE: Point color in the beta convergence plot
If beta.plot = TRUE: Logical argument that indicates if a regression line has to be added to the plot
If beta.plot = TRUE and beta.plotLine = TRUE: Line color of regression line
If beta.plot = TRUE: Name of the X axis
If beta.plot = TRUE: Name of the Y axis
If beta.plot = TRUE: Plot title
If beta.plot = TRUE: Plot background color
If beta.plot = TRUE: Logical argument that indicates if the plot contains a grid
If beta.plot = TRUE and beta.bgrid = TRUE: Color of the grid
If beta.plot = TRUE and beta.bgrid = TRUE: Size of the grid
If beta.plot = TRUE and beta.bgrid = TRUE: Type of the grid
Logical argument that indicates if the function shows the results or not
A list containing the following objects:
A data frame containing the regression data, including the \(ln\)-transformed economic variables
A list containing the estimates of the absolute beta convergence regression model, including lambda and half-life
If conditions are stated: a list containing the estimates of the conditional beta convergence regression model, including lambda and half-life
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).
This function calculates absolute and/or conditional beta convergence using a nonlinear least squares approach for estimation. It needs at least two vectors (GDP p.c. or another economic variable, \(y\), for \(i\) regions) and the related two points in time (\(t\) and \(t+T\)). If the beta coefficient is negative (using OLS) or positive (using NLS), there is beta convergence.
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.
rca, betaconv.ols, betaconv.speed, sigmaconv, sigmaconv.t, cv, sd2, var2
# NOT RUN {
data (G.counties.gdp)
# Loading GDP data for Germany (counties = Landkreise)
betaconv.nls (G.counties.gdp$gdppc2010, 2010, G.counties.gdp$gdppc2011, 2011,
conditions = NULL, print.results = TRUE)
# Two years, no conditions (Absolute beta convergence)
# }
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