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

## Description

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

## Usage

 1 2 3 4 5 6 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)

## 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 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

## 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 (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.

## Value

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

## 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.

## Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 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, :66], 2012, conditions = NULL, print.results = TRUE) # Three years (2010-2012), no conditions (Absolute beta convergence) betaconv.ols (G.counties.gdp\$gdppc2010, 2010, :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, :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

### Example output

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

REAT documentation built on Nov. 21, 2019, 5:08 p.m.