Determinants of Economic Growth

Introduction

• The cross-country regression investigates the contribution of accumulation of physical and human capital and population growth (X_i) to economic growth.

• Besides these, there are many other determinants of economic growth that can be included in a growth regression (Z_i):

log (\frac {Y} {L})_{i,1} - log (\frac {Y} {L})_{i,0} = \alpha + \beta_0 log (\frac {Y} {L})_{i,0} + \beta_1 X_{i} + \beta_2 log Z_i + u_i

• What else might affect economic growth and the main variables in the cross-country growth regression?

Introduction

• Problem: Many theories, so potentially more variables than countries, i.e., k \gg n.

• Therefore, we cannot include all potentially relevant variables in one regression.

• Reduction of model dimension is often needed. Various techniques are available: One of the most prominent is the method of principal components (PCA).

• Furthermore, some of the variables can be endogenous => an IV estimator is needed.

Economic Growth and Institutions

• Institutions: Any constraints that shape economic behaviour in economic and non-economic actions. Hence, institutions are sets of rules governing behaviour that people respect.

• Formal institutions: Laws. Determined through the political process. Most important: property rights ensuring that one can benefit from the products of one’s own production. Property rights enable savings and smooth functioning of market exchange.

• Informal institutions: Habits, customs and norms that are not written, but spontaneously followed. Culture of a given society.

• Importance of institutions: institutions decrease uncertainty as they facilitate anticipation of the behaviour of others. They decrease transaction costs in the economy.

• Different institutions can either encourage innovation, entrepreneurship and growth or redistribution, rent-seeking, criminal activities and thus harm the growth prospects.

• Our task: Estimate the effects of institutions on economic growth.

Measuring institutional quality

How to measure institutional quality?

No explicit and observable indicator of overall institutional quality exists, but various indirect indicators are available. Many of them focus on outcomes rather than on evaluating different components of institutional rules.

This seminar:

Data from 2005. Using data from the beginning of the period makes these variables pre-determined and thus less prone to endogeneity. However, because of the persistence of institutional and cultural variables the endogeneity issue might remain relevant and so we will reestimate the model using TSLS later.

Quick look into the Doing Business data

• Entry costs are percentages of income per capita; contract enforcement is measured in days.
• Doing Business dataset has been depreciated due to data-integrity concerns. Its successor, B-READY, began in 2024. Since PWT11 contain data till 2023, the B-READY dataset is not a useful replacement for our purposes at this point.

Six variables from the World Governance Indicators

Macro variables: levels and growth (PWT11)

Let y denote GDP per person, not GDP per worker.

Methodology

Many variables representing institutions. What we can expect when all appear in one regression?

• Many variables insignificant.

• Degrees of freedom are too low and parsimony is lost.

• Multicollinearity.

Instead of sequential exclusion of variables using t-tests and F-tests, we restrict the dimension using principal components, which aim to retain as much information from all variables as possible.

Methodology: Principal Component Analysis

• Principal component analysis = method for extracting the most important factors in a set of variables.

• Linear transformation of variables into a set of artificial ones.

• It converts data to a new coordinate system so that the highest variance is projected onto the first axis (component), the second highest on the second and so on.

• Each axis represents a new, artificial variable that explains some part of the variance of original data.

• These variables are called the principal components of the system.

• We can use only the most important principal components explaining the highest proportion of variance and decrease the dimension of the model while utilising most of the information from all variables (instead of deleting some).

PCA: the geometric idea

Methodology: Principal Components

Note on eigenvalues and eigenvectors.

• Eigenvalues of a symmetric matrix A are real numbers \lambda that fulfill:

det (\lambda I - A) = 0

• Each eigenvalue \lambda has its corresponding eigenvector \vec u such that

A \vec u = \lambda \vec u

• That is, a vector multiplied by the matrix yields a vector that differs from the original by a scalar multiple.

• For a 2×2 symmetric matrix, the characteristic equation det (\lambda I - A) = 0 has two real solutions, counting multiplicity. There is an orthonormal basis of eigenvectors.

• Formally, PCA can be considered an orthogonal analysis of a symmetric positive semidefinite matrix where eigenvectors correspond to principal components and eigenvalues to the variance explained.

Empirical analysis

library(readxl)
instgrowth <- read_excel("4InstitutionsAndGrowth_students.xlsx", sheet = "Data")
head(instgrowth, 10)

# A tibble: 10 × 33
   iso3  country     gdp_pc_2005 gdp_pc_2023 growth_pct_2005_2023 investment_share_200…¹
   <chr> <chr>             <dbl>       <dbl>                <dbl>                  <dbl>
 1 ALB   Albania           7415.      15924.               3.88                    0.258
 2 DZA   Algeria          13391.      14498.               1.16                    0.298
 3 AGO   Angola            4694.       6148.              -0.411                   0.234
 4 AIA   Anguilla         33666.      22696.              -0.0460                  0.524
 5 ATG   Antigua an…      15678.      27073.               0.957                   0.388
 6 ARG   Argentina        15029.      27221.               0.849                   0.153
 7 ARM   Armenia           6563.      20843.               4.49                    0.151
 8 ABW   Aruba            51564.      44684.              -0.428                   0.329
 9 AUS   Australia        49162.      64022.               1.07                    0.261
10 AUT   Austria          45422.      62777.               0.683                   0.301
# ℹ abbreviated name: ¹​investment_share_2005_2023
# ℹ 27 more variables: hc_mean_2005_2023 <dbl>, pop_growth_2005_2023 <dbl>,
#   db_year <dbl>, wgi_year <dbl>, starting_business_costs_pctincome <dbl>,
#   starting_capital_min_pctincome <dbl>, no_proc_register_property <dbl>,
#   no_proc_start_business <dbl>, resolve_insolvency_costs_pct <dbl>,
#   no_tax_payments <dbl>, contact_enforcement_days <dbl>,
#   contact_enforcement_costs_pct <dbl>, corruption_control <dbl>, …

Empirical analysis

• Omit observations with missing values.

instgrowthdata <- na.omit(instgrowth)
nrow(instgrowthdata)

[1] 114

• Create a set of variables from the Doing Business database.

dbvars <- c("starting_business_costs_pctincome", "starting_capital_min_pctincome",
            "no_proc_register_property", "no_proc_start_business",
            "resolve_insolvency_costs_pct", "no_tax_payments",
            "contact_enforcement_days", "contact_enforcement_costs_pct")
dbdata <- instgrowthdata[dbvars]

PCA: Eigenvalues and Eigenvectors

db_pca <- prcomp(dbdata, center = TRUE, scale. = TRUE)
summary(db_pca)

Importance of components:
                         PC1   PC2   PC3   PC4    PC5    PC6    PC7    PC8
Standard deviation     1.585 1.192 1.040 0.929 0.8405 0.7289 0.6988 0.6289
Proportion of Variance 0.314 0.178 0.135 0.108 0.0883 0.0664 0.0610 0.0494
Cumulative Proportion  0.314 0.491 0.627 0.735 0.8231 0.8895 0.9506 1.0000

• The components PC1 … PC8 are ordered from the most important to the least important.
• Proportion of variance: explanatory power of each PC separately (the first explains 31.4% of the original variance).
• Cumulative proportion: the first four PCs together explain 73.5% of the variation.
• These figures help decide how many principal components to retain.

Rule of thumb: retain components with eigenvalues above one, or enough components to explain the desired proportion of variance.

PCA: Eigenvalues and Eigenvectors

The second part of the output shows the weights assigned to variables within each principal component.

db_pca$rotation # eigenvectors: weights defining the components

                                      PC1       PC2      PC3     PC4      PC5      PC6
starting_business_costs_pctincome 0.38852  0.469157  0.15464  0.2371  0.12453  0.15180
starting_capital_min_pctincome    0.07748  0.003057  0.88806 -0.2702  0.11806  0.23607
no_proc_register_property         0.29699 -0.474335 -0.22144  0.1474  0.60067  0.42826
no_proc_start_business            0.40661 -0.362852  0.14728 -0.1528  0.24986 -0.66999
resolve_insolvency_costs_pct      0.44805  0.239809 -0.16991 -0.2755 -0.13835 -0.29774
no_tax_payments                   0.38728 -0.174748 -0.18721 -0.5409 -0.43300  0.44028
contact_enforcement_days          0.22737 -0.435292  0.22241  0.5842 -0.58099 -0.01447
contact_enforcement_costs_pct     0.43253  0.381712 -0.05767  0.3404  0.05609  0.07894
                                      PC7      PC8
starting_business_costs_pctincome  0.3070 -0.64500
starting_capital_min_pctincome    -0.2055  0.14303
no_proc_register_property         -0.2584 -0.07121
no_proc_start_business             0.3816  0.03231
resolve_insolvency_costs_pct      -0.7149 -0.13471
no_tax_payments                    0.3301  0.03953
contact_enforcement_days          -0.1450 -0.09612
contact_enforcement_costs_pct      0.1009  0.72695

PCA: Eigenvalues and Eigenvectors

• PC1 has positive weights across the measured business burdens, including entry procedures, tax payments, and costs => “Business Obstacles”.

• PC2 contrasts entry and contract costs with procedures and contract duration => “Costs versus procedures”.

• … PC3? What name?

Note that the prcomp function uses covariance matrix, however, with centered means and scaled variances the result is the same as with the correlation matrix.

The sign of a principal component is arbitrary; interpret its weights before naming it.

PCA: Eigenvalues and Eigenvectors

Alternatively, one could use the princomp function. The component directions agree up to sign; score normalizations can differ.

db_pca2 <- princomp(dbdata, cor = TRUE, scores = TRUE)
summary(db_pca2)

Importance of components:
                       Comp.1 Comp.2 Comp.3 Comp.4  Comp.5  Comp.6  Comp.7  Comp.8
Standard deviation     1.5846 1.1920 1.0404 0.9295 0.84053 0.72889 0.69884 0.62886
Proportion of Variance 0.3139 0.1776 0.1353 0.1080 0.08831 0.06641 0.06105 0.04943
Cumulative Proportion  0.3139 0.4915 0.6268 0.7348 0.82311 0.88952 0.95057 1.00000

PCA: Eigenvalues and Eigenvectors

Note that loadings printed with db_pca2$loadings leave blanks for numbers that are close to zero, but not necessarily zero. These are not missing values.

db_pca2$loadings

Loadings:
                                  Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7
starting_business_costs_pctincome  0.389  0.469  0.155  0.237  0.125  0.152  0.307
starting_capital_min_pctincome                   0.888 -0.270  0.118  0.236 -0.205
no_proc_register_property          0.297 -0.474 -0.221  0.147  0.601  0.428 -0.258
no_proc_start_business             0.407 -0.363  0.147 -0.153  0.250 -0.670  0.382
resolve_insolvency_costs_pct       0.448  0.240 -0.170 -0.276 -0.138 -0.298 -0.715
no_tax_payments                    0.387 -0.175 -0.187 -0.541 -0.433  0.440  0.330
contact_enforcement_days           0.227 -0.435  0.222  0.584 -0.581        -0.145
contact_enforcement_costs_pct      0.433  0.382         0.340                0.101
                                  Comp.8
starting_business_costs_pctincome  0.645
starting_capital_min_pctincome    -0.143
no_proc_register_property               
no_proc_start_business                  
resolve_insolvency_costs_pct       0.135
no_tax_payments                         
contact_enforcement_days                
contact_enforcement_costs_pct     -0.727

               Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 Comp.7 Comp.8
SS loadings     1.000  1.000  1.000  1.000  1.000  1.000  1.000  1.000
Proportion Var  0.125  0.125  0.125  0.125  0.125  0.125  0.125  0.125
Cumulative Var  0.125  0.250  0.375  0.500  0.625  0.750  0.875  1.000

PCA: Eigenvalues and Eigenvectors

Here, we can see that principal components are uncorrelated:

pca_corr <- cor(db_pca$x)
print(pca_corr)

           PC1        PC2        PC3        PC4        PC5        PC6        PC7
PC1  1.000e+00  1.775e-18  3.041e-16 -2.620e-16  8.200e-17 -5.525e-17  6.329e-18
PC2  1.775e-18  1.000e+00  1.786e-16  1.476e-16  1.445e-16  5.422e-16 -2.501e-16
PC3  3.041e-16  1.786e-16  1.000e+00 -4.159e-16  5.376e-16  2.337e-16 -3.298e-16
PC4 -2.620e-16  1.476e-16 -4.159e-16  1.000e+00  3.732e-16 -3.355e-17  9.561e-18
PC5  8.200e-17  1.445e-16  5.376e-16  3.732e-16  1.000e+00 -6.869e-16  1.245e-16
PC6 -5.525e-17  5.422e-16  2.337e-16 -3.355e-17 -6.869e-16  1.000e+00  2.966e-16
PC7  6.329e-18 -2.501e-16 -3.298e-16  9.561e-18  1.245e-16  2.966e-16  1.000e+00
PC8  3.677e-17  1.453e-16 -4.883e-17  2.477e-16  2.113e-15 -6.792e-17 -1.042e-16
           PC8
PC1  3.677e-17
PC2  1.453e-16
PC3 -4.883e-17
PC4  2.477e-16
PC5  2.113e-15
PC6 -6.792e-17
PC7 -1.042e-16
PC8  1.000e+00

Summary of the results of the PCA

The results imply that 3 PCs have eigenvalues above one. The first three explain 62.7% of the original variance.

Now we can run a regression with two institutional variables—PC1 as Business Obstacles and PC2 as Costs versus Procedures—replacing the eight original variables. These two PCs retain 49.1% of the variance.

To do so, we need to create a data frame of principal components.

db_pcs <- as.data.frame(db_pca$x)

Principal component analysis: WGI

Now redo PCA for the governance indicators:

givars <- c("corruption_control", "government_effectiveness", "political_stability",
            "regulatory_quality", "rule_of_law", "voice_accountability")
gidata <- instgrowthdata[givars]
gi_pca <- prcomp(gidata, center = TRUE, scale. = TRUE)
summary(gi_pca)

Importance of components:
                         PC1    PC2    PC3    PC4     PC5   PC6
Standard deviation     2.323 0.5723 0.3838 0.2575 0.18601 0.173
Proportion of Variance 0.899 0.0546 0.0245 0.0111 0.00577 0.005
Cumulative Proportion  0.899 0.9536 0.9782 0.9892 0.99500 1.000

gi_pca$rotation # weights defining the governance components

                            PC1      PC2      PC3      PC4      PC5      PC6
corruption_control       0.4179  0.15965  0.32692  0.55813 -0.60765  0.11065
government_effectiveness 0.4194  0.24075  0.25074 -0.12457  0.49355  0.66644
political_stability      0.3656 -0.92168  0.02147 -0.07283 -0.02756  0.10158
regulatory_quality       0.4172  0.21841 -0.03440 -0.76346 -0.40643 -0.17025
rule_of_law              0.4226  0.07004  0.27310  0.14979  0.46430 -0.70987
voice_accountability     0.4039  0.12021 -0.86835  0.24971  0.07519  0.02009

gi_pcs <- as.data.frame(gi_pca$x)
names(gi_pcs) <- paste0("PC", 11:16)

Now, we merge the principal components with the original dataset.

rd <- cbind(instgrowthdata, db_pcs, gi_pcs)

Do institutions add explanatory power? Solow-like regression

\log\left(\frac{Y}{L}\right)_i = \alpha + \beta_1 X_i + \beta_2 Z_i + u_i

Take logs of the appropriate variables.

rd$l_gdp <- log(rd$gdp_pc_2023)
rd$l_sk <- log(rd$investment_share_2005_2023)
rd$l_sh <- log(rd$hc_mean_2005_2023)
rd$l_ngd <- log(rd$pop_growth_2005_2023 + 0.05)
rd$l_gdp0 <- log(rd$gdp_pc_2005)

model1 <- lm(l_gdp ~ l_sk + l_sh + l_ngd, data = rd)

model2 <- lm(l_gdp ~ l_sk + l_sh + l_ngd + PC1 + PC2 + PC11, data = rd)

model3 <- lm(growth_pct_2005_2023 ~ l_gdp0 + l_sk + l_sh + l_ngd +
               PC1 + PC2 + PC11, data = rd)

Methodology: IV Estimator

• We estimated income and growth equations with institutional variables among the regressors Z_i:

log (\frac {Y} {L})_{i,1} - log (\frac {Y} {L})_{i,0} = \alpha + \beta_0 log (\frac {Y} {L})_{i,0} + \beta_1 X_{i} + \beta_2 log Z_i + u_i

• Potential endogeneity: Rich countries might afford better institutions; better institutions might foster growth even further.

• Solution: a two-stage estimator (TSLS), where Z represents institutions and S contains variables correlated with institutions but assumed uncorrelated with the outcome-equation error.

Z_i = \gamma + \theta S_i + \nu_i

log(Y_i/L_i) = \alpha + \beta \hat Z_i + \varepsilon_i

• The idea of the TSLS estimator is that the first step provides fitted values \hat Z that are driven solely by exogenous variables. This part enters the second step estimation.

Instruments

Instruments are supposed to be correlated with institutions but exogenous to the outcome-equation error. Being determined before the outcome period helps establish timing but does not by itself ensure validity.

• Openness

• Dummy indicating whether a country is landlocked.

• Distance from equator

• Percentages of the population using English or European language.

• Proxy for trade

• Number of revolutions (average for decade in the second half of the 20th century)

• Freedom House index of political liberties

• Average casualties in wars

Initial income is not included in the baseline instrument set: earlier income can directly affect later income and need not be a valid excluded instrument.

Data: Dollar and Kraay: Institutions, trade, and growth : revisiting the evidence, Journal of Monetary Economics (2002). Additional variables in the dataset (International Country Risk Guide score and Mortality of population in European settlements) were excluded due to many missing observations.

IV regressions

library(AER)
model4 <- ivreg(l_gdp ~ PC11 | LNOPEN + LANDLOCK + DISTEQ + ENGFRAC +
  EURFRAC + LNFRINSTEX + REVOLAV + FREEDOMAV + WARDEADAV, data = rd)

model5 <- ivreg(l_gdp ~ l_sk + l_sh + l_ngd + PC11 |
  LNOPEN + LANDLOCK + DISTEQ + ENGFRAC + EURFRAC + LNFRINSTEX + REVOLAV +
  FREEDOMAV + WARDEADAV + l_sk + l_sh + l_ngd, data = rd)

# Original specification: initial income is an additional excluded instrument.
model6 <- ivreg(l_gdp ~ l_sk + l_sh + l_ngd + PC11 |
  LNOPEN + LANDLOCK + DISTEQ + ENGFRAC + EURFRAC + LNFRINSTEX + REVOLAV +
  FREEDOMAV + WARDEADAV + l_sk + l_sh + l_ngd + l_gdp0, data = rd)

# All institutional principal components included => but the results break down due to collinearity and weak instruments for PC1 and PC2.
model7 <- ivreg(l_gdp ~ l_sk + l_sh + l_ngd + PC11 + PC1 + PC2 |
                  LNOPEN + LANDLOCK + DISTEQ + ENGFRAC + EURFRAC + LNFRINSTEX + REVOLAV +
                  FREEDOMAV + WARDEADAV + l_sk + l_sh + l_ngd + l_gdp0, data = rd)

IV regressions

summary(model5, diagnostics = TRUE)

Call:
ivreg(formula = l_gdp ~ l_sk + l_sh + l_ngd + PC11 | LNOPEN + 
    LANDLOCK + DISTEQ + ENGFRAC + EURFRAC + LNFRINSTEX + REVOLAV + 
    FREEDOMAV + WARDEADAV + l_sk + l_sh + l_ngd, data = rd)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.9177 -0.2378  0.0348  0.2679  1.0390 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   6.6489     0.9985    6.66  1.2e-09 ***
l_sk          0.4430     0.2106    2.10    0.038 *  
l_sh          1.8048     0.3377    5.34  5.0e-07 ***
l_ngd        -0.6838     0.3913   -1.75    0.083 .  
PC11          0.2234     0.0462    4.84  4.3e-06 ***

Diagnostic tests:
                 df1 df2 statistic p-value    
Weak instruments   9 101     15.23 2.2e-15 ***
Wu-Hausman         1 108      0.86    0.36    
Sargan             8  NA      7.20    0.52    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.591 on 109 degrees of freedom
Multiple R-Squared: 0.779,  Adjusted R-squared: 0.77 
Wald test:   94 on 4 and 109 DF,  p-value: <2e-16 

IV regressions

Detailed summary reveals low F statistics in Weak instrument test (reliable benchmark is at least 10) and Wu-Hausman rejects validity of instruments. Along with high correlation between PC1,2 with PC11 from WGI those features make this model unreliable.

summary(model7, diagnostics = TRUE)

Call:
ivreg(formula = l_gdp ~ l_sk + l_sh + l_ngd + PC11 + PC1 + PC2 | 
    LNOPEN + LANDLOCK + DISTEQ + ENGFRAC + EURFRAC + LNFRINSTEX + 
        REVOLAV + FREEDOMAV + WARDEADAV + l_sk + l_sh + l_ngd + 
        l_gdp0, data = rd)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.3303 -0.2937 -0.0549  0.2572  4.0826 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)    7.990      1.454    5.49  2.7e-07 ***
l_sk           0.102      0.301    0.34    0.736    
l_sh           0.769      0.632    1.22    0.226    
l_ngd         -0.358      0.535   -0.67    0.504    
PC11           0.118      0.124    0.95    0.342    
PC1           -0.463      0.293   -1.58    0.117    
PC2           -0.358      0.189   -1.89    0.061 .  

Diagnostic tests:
                        df1 df2 statistic p-value    
Weak instruments (PC11)  10 100     16.80 < 2e-16 ***
Weak instruments (PC1)   10 100      3.91 0.00017 ***
Weak instruments (PC2)   10 100      2.11 0.03054 *  
Wu-Hausman                3 104      3.16 0.02787 *  
Sargan                    7  NA      4.76 0.68958    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.762 on 107 degrees of freedom
Multiple R-Squared: 0.639,  Adjusted R-squared: 0.619 
Wald test: 39.5 on 6 and 107 DF,  p-value: <2e-16 

Alternatives to Principal Component Analysis

• Model averaging methods: many models with different variables are estimated, and final coefficients are calculated as weighted averages of coefficients from these models (the weights depend on model fit and, in Bayesian methods, prior assumptions).

• Xavier Sala-i-Martin: I Just Ran Two Million Regressions (AER, 1997) is one of the most famous papers in this field. Although the author does not identify set of few relevant variables, equally important is which variables are not likely associated with growth: government spending, inflation, financial sophistication or scale effects.

• Fernandez, Ley, Steel (2001): Model uncertainty in cross‐country growth regressions. They used the same dataset as Sala-i-Martin (1997) but considered 2^{41} models within the BMA framework. The following set of variables was selected: GDP in 1960, life expectancy, openness, rule of law, civil liberties, several variables representing schooling and investment + religion.

• The prominent role of institutions was confirmed by Magnus, Powell, and Prüfer (2010). Using alternative model averaging techniques (BMA and WALS) they show that robust growth theories should include not only neoclassical growth variables but also institutions. However, the choice of variables representing institutional quality matters for the results as well.

Conclusions: Do Institutions Matter for Economic Growth?

• Institutional quality has been found to have predictive power when estimating future growth prospects.

• Jonathan Temple and Paul A. Johnson (QJE, 1998, Social Capability and Economic Growth) challenge the frequent conclusion in the literature that the differences in economic growth in the 20th century were unpredictable (recall our regression on convergence).

• They found that the Adelman-Morris social development index from 1967 - focused on non-economic factors with social and political indicators – has strong predictive power for the future growth. In particular, it was able to predict success in East Asia and growth disasters in Africa.

Insights from recent Nobel Prize winners