betaconv.ols: Analysis of regional beta convergence using OLS regression
In REAT: Regional Economic Analysis Toolbox

Description Usage Arguments Details Value Author(s) References See Also Examples

This function provides the analysis of absolute and conditional regional economic beta convergence for cross-sectional data using ordinary least squares (OLS) technique.

betaconv.ols(gdp1, time1, gdp2, time2, conditions = 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 = FALSE)

`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
`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
`print.results`	Logical argument that indicates if the function shows the results or not

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 (σ) means a harmonization of regional economic output or income over time, while beta convergence (β) 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 (β < 0), it is possible to calculate the speed of convergence, λ, 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 (σ), 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 ordinary least squares regression (OLS) 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.

A list containing the following objects:

`regdata`	A data frame containing the regression data, including the ln-transformed economic variables
`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

Thomas Wieland

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

data (G.counties.gdp)

betaconv.ols (G.counties.gdp$gdppc2010, 2010, G.counties.gdp$gdppc2011, 2011, 
conditions = NULL, print.results = 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

betaconv.ols (G.counties.gdp$gdppc2010, 2010, G.counties.gdp$gdppc2011, 2011, 
conditions = G.counties.gdp[c(70,71)], print.results = TRUE)
# Two years, with condition (dummy for West/East)
# (Absolute and conditional beta convergence)

betaconverg1 <- betaconv.ols (G.counties.gdp$gdppc2010, 2010, G.counties.gdp$gdppc2011, 2011,
conditions = G.counties.gdp[c(70,71)], print.results = TRUE)
# Store results in object
betaconverg1$cbeta$estimates
# Addressing estimates for the conditional beta model


betaconv.ols (G.counties.gdp$gdppc2010, 2010, G.counties.gdp[65:66], 2012, 
conditions = NULL, print.results = TRUE)
# Three years (2010-2012), no conditions (Absolute beta convergence)

betaconv.ols (G.counties.gdp$gdppc2010, 2010, G.counties.gdp[65:66], 2012, 
conditions = G.counties.gdp[c(70,71)], print.results = TRUE)
# Three years (2010-2012), with conditions (Absolute and conditional beta convergence)

betaconverg2 <- betaconv.ols (G.counties.gdp$gdppc2010, 2010, G.counties.gdp[65:66],
2012, conditions = G.counties.gdp[c(70,71)], print.results = TRUE)
# Store results in object
betaconverg2$cbeta$estimates
# Addressing estimates for the conditional beta model

Absolute Beta Convergence 
Model coefficients (Estimation method: OLS)
             Estimate  Std. Error   t value   Pr (>|t|)
Alpha     0.132453460 0.046001403  2.879335 0.004199233
Beta     -0.008070533 0.004490136 -1.797391 0.073027611
Lambda    0.008103276          NA        NA          NA
Halflife 85.539129957          NA        NA          NA
Model summary 
             Estimate  F value df 1 df 2    Pr (>F)
R-Squared 0.008011833 3.230616    1  400 0.07302761
Absolute Beta Convergence 
Model coefficients (Estimation method: OLS)
             Estimate  Std. Error   t value   Pr (>|t|)
Alpha     0.132453460 0.046001403  2.879335 0.004199233
Beta     -0.008070533 0.004490136 -1.797391 0.073027611
Lambda    0.008103276          NA        NA          NA
Halflife 85.539129957          NA        NA          NA
Model summary 
             Estimate  F value df 1 df 2    Pr (>F)
R-Squared 0.008011833 3.230616    1  400 0.07302761

Conditional Beta Convergence 
Model coefficients (Estimation method: OLS) 
             Estimate  Std. Error   t value    Pr (>|t|)
Alpha     0.159073910 0.047661233  3.337595 0.0009244152
Beta     -0.011348095 0.004755033 -2.386544 0.0174728159
West      0.008582473 0.004226534  2.030617 0.0429564904
Lambda    0.011412976          NA        NA           NA
Halflife 60.733254185          NA        NA           NA
Model summary 
            Estimate  F value df 1 df 2    Pr (>F)
R-Squared 0.01815853 3.689625    2  399 0.02583769
Absolute Beta Convergence 
Model coefficients (Estimation method: OLS)
             Estimate  Std. Error   t value   Pr (>|t|)
Alpha     0.132453460 0.046001403  2.879335 0.004199233
Beta     -0.008070533 0.004490136 -1.797391 0.073027611
Lambda    0.008103276          NA        NA          NA
Halflife 85.539129957          NA        NA          NA
Model summary 
             Estimate  F value df 1 df 2    Pr (>F)
R-Squared 0.008011833 3.230616    1  400 0.07302761

Conditional Beta Convergence 
Model coefficients (Estimation method: OLS) 
             Estimate  Std. Error   t value    Pr (>|t|)
Alpha     0.159073910 0.047661233  3.337595 0.0009244152
Beta     -0.011348095 0.004755033 -2.386544 0.0174728159
West      0.008582473 0.004226534  2.030617 0.0429564904
Lambda    0.011412976          NA        NA           NA
Halflife 60.733254185          NA        NA           NA
Model summary 
            Estimate  F value df 1 df 2    Pr (>F)
R-Squared 0.01815853 3.689625    2  399 0.02583769
             Estimate  Std. Error   t value    Pr (>|t|)
Alpha     0.159073910 0.047661233  3.337595 0.0009244152
Beta     -0.011348095 0.004755033 -2.386544 0.0174728159
West      0.008582473 0.004226534  2.030617 0.0429564904
Lambda    0.011412976          NA        NA           NA
Halflife 60.733254185          NA        NA           NA
Absolute Beta Convergence 
Model coefficients (Estimation method: OLS)
              Estimate  Std. Error   t value    Pr (>|t|)
Alpha      0.140753319 0.028937288  4.864081 1.656289e-06
Beta      -0.010423460 0.002824531 -3.690334 2.549763e-04
Lambda     0.005239083          NA        NA           NA
Halflife 132.303161395          NA        NA           NA
Model summary 
            Estimate  F value df 1 df 2      Pr (>F)
R-Squared 0.03292541 13.61856    1  400 0.0002549763
Absolute Beta Convergence 
Model coefficients (Estimation method: OLS)
              Estimate  Std. Error   t value    Pr (>|t|)
Alpha      0.140753319 0.028937288  4.864081 1.656289e-06
Beta      -0.010423460 0.002824531 -3.690334 2.549763e-04
Lambda     0.005239083          NA        NA           NA
Halflife 132.303161395          NA        NA           NA
Model summary 
            Estimate  F value df 1 df 2      Pr (>F)
R-Squared 0.03292541 13.61856    1  400 0.0002549763

Conditional Beta Convergence 
Model coefficients (Estimation method: OLS) 
              Estimate  Std. Error   t value    Pr (>|t|)
Alpha      0.151849900 0.030068175  5.050187 6.727653e-07
Beta      -0.011789693 0.002999821 -3.930132 1.000613e-04
West       0.003577554 0.002666405  1.341714 1.804517e-01
Lambda     0.005929871          NA        NA           NA
Halflife 116.890761789          NA        NA           NA
Model summary 
            Estimate  F value df 1 df 2      Pr (>F)
R-Squared 0.03726904 7.723002    2  399 0.0005119593
Absolute Beta Convergence 
Model coefficients (Estimation method: OLS)
              Estimate  Std. Error   t value    Pr (>|t|)
Alpha      0.140753319 0.028937288  4.864081 1.656289e-06
Beta      -0.010423460 0.002824531 -3.690334 2.549763e-04
Lambda     0.005239083          NA        NA           NA
Halflife 132.303161395          NA        NA           NA
Model summary 
            Estimate  F value df 1 df 2      Pr (>F)
R-Squared 0.03292541 13.61856    1  400 0.0002549763

Conditional Beta Convergence 
Model coefficients (Estimation method: OLS) 
              Estimate  Std. Error   t value    Pr (>|t|)
Alpha      0.151849900 0.030068175  5.050187 6.727653e-07
Beta      -0.011789693 0.002999821 -3.930132 1.000613e-04
West       0.003577554 0.002666405  1.341714 1.804517e-01
Lambda     0.005929871          NA        NA           NA
Halflife 116.890761789          NA        NA           NA
Model summary 
            Estimate  F value df 1 df 2      Pr (>F)
R-Squared 0.03726904 7.723002    2  399 0.0005119593
              Estimate  Std. Error   t value    Pr (>|t|)
Alpha      0.151849900 0.030068175  5.050187 6.727653e-07
Beta      -0.011789693 0.002999821 -3.930132 1.000613e-04
West       0.003577554 0.002666405  1.341714 1.804517e-01
Lambda     0.005929871          NA        NA           NA
Halflife 116.890761789          NA        NA           NA