Do poor countries grow faster than rich countries?

Dataset

The dataset contains growth data of Barro-Lee. The Barro Lee data consists of a panel of 138 countries for the period 1960 to 1985. The dependent variable is national growth rates in GDP per capita for the periods 1965-1975 and 1975-1985. The growth rate in GDP over a period from t1 to t2 is commonly defined as log(GDPt1 /GDPt2 ). The number of variables are p=62. The number of complete observations is n=90.

The full data set and further details can be found at http://www.nber.org/pub/barro.lee, http: //www.barrolee.com, and, http://www.bristol.ac.uk//Depts//Economics//Growth//barlee.htm.

1. Outcome : national growth rates in GDP per capita for the periods 1965-1975.
2. intercept: Constant.
3. gdpsh465 : Real GDP per capita (1980 international prices) in 1965
4. bmp1l : Black market premium. Log (1+BMP)
5. freeop : Measure of "Free trade openness
6. freetar : Measure of tariff restriction
7. h65 : Total gross enrollment ratio for higher education in 1965.
8. hm65 : Male gross enrollment ratio for higher education in 1965.
9. hf65 : Female gross enrollment ratio for higher education in 1965.
10. p65 : Total gross enrollment ratio for primary education in 1965.
11. pm65 : Male gross enrollment ratio for primary education in 1965.
12. pf65 : Female gross enrollment ratio for primary education in 1965.
13. s65 : Total gross enrollment ratio for secondary education in 1965.
14. sm65 : Male gross enrollment ratio for secondary education in 1965.
15. sf65 : Female gross enrollment ratio for secondary education in 1965.
16. fert65 : Total fertility rate (children per woman) in 1965.
17. mort65 : Infant Mortality Rate in 1965.
18. lifee065 : Life expectancy at age 0 in 1965.
19. gpop1 : Growth rate of population.
20. fert1 : Total fertility rate (children per woman).
21. mort1 : Infant Mortality Rate (ages 0-1).
22. invsh41 : Ratio of real domestic investment (private plus public) to real GDP.
23. geetot1 : Ratio of total nominal government expenditure on education to nominal GDP.
24. geerec1 : Ratio of recurring nominal government expenditure on education to nominal GDP.
25. gde1 : Ratio of nominal government expenditure on defense to nominal GDP.
26. govwb1 : Ratio of nominal government "consumption" expenditure to nominal GDP (using current local currency).
27. govsh41 : Ratio of real government "consumption" expenditure to real GDP. (Period average).
28. gvxdxe41 : Ratio of real government "consumption" expenditure net of spending on defense and on education to real GDP.
29. high65 : Percentage of "higher school attained" in the total pop in 1965.
30. highm65 : Percentage of "higher school attained" in the male pop in 1965.
31. highf65 : Percentage of "higher school attained" in the female pop in 1965.
32. highc65 : Percentage of "higher school complete" in the total pop.
33. highcm65 : Percentage of "higher school complete" in the male pop.
34. highcf65 : Percentage of "higher school complete" in the female pop.
35. human65 : Average schooling years in the total population over age 25 in 1965.
36. humanm65 : Average schooling years in the male population over age 25 in 1965.
37. humanf65 : Average schooling years in the female population over age 25 in 1965.
38. hyr65 : Average years of higher schooling in the total population over age 25.
39. hyrm65 : Average years of higher schooling in the male population over age 25.
40. hyrf65 : Average years of higher schooling in the female population over age 25.
41. no65 : Percentage of "no schooling" in the total population.
42. nom65 : Percentage of "no schooling" in the male population.
43. nof65 : Percentage of "no schooling" in the female population.
44. pinstab1 : Measure of political instability.
45. pop65 : Total Population in 1965.
46. worker65 : Ratio of total Workers to population.
47. pop1565 : Population Proportion under 15 in 1965.
48. pop6565 : Population Proportion over 65 in 1965.
49. sec65 : Percentage of "secondary school attained" in the total pop in 1965.
50. secm65 : Percentage of "secondary school attained" in male total pop in 1965.
51. secf65 : Percentage of "secondary school attained" in female total pop in 1965.
52. secc65 : Percentage of "secondary school complete" in the total pop in 1965.
53. seccm65 : Percentage of "secondary school complete" in the total pop in 1965.
54. seccf65 : Percentage of "secondary school complete" in female pop in 1965.
55. syr65 : Average years of secondary schooling in the total population over age 25 in 1965.
56. syrm65 : Average years of secondary schooling in the male population over age 25 in 1965.
57. syrf65 : Average years of secondary schooling in the female population over age 25 in 1965.
58. teapri65 : Pupil/Teacher Ratio in primary school.
59. teasex65 : Pupil/Teacher Ratio in secondary school
60. ex1 : Ratio of export to GDP (in current international prices)
61. im1 : Ratio of import to GDP (in current international prices)
62. xr65 : Exchange rate (domestic currency per U.S. dollar) in 1965.
63. tot1 : Terms of trade shock (growth rate of export prices minus growth rate of import prices).

Importing the necessary libraries and overview of the dataset

#Import required libraries
import numpy as np
import pandas as pd
import warnings
warnings.filterwarnings("ignore")

from google.colab import drive
drive.mount('/content/drive')

Mounted at /content/drive

# Load data
df = pd.read_csv('growth.csv')

# See variables in the dataset
df.head()

	Outcome	intercept	gdpsh465	bmp1l	freeop	freetar	h65	hm65	hf65	p65	...	seccf65	syr65	syrm65	syrf65	teapri65	teasec65	ex1	im1	xr65	tot1
0	-0.024336	1	6.591674	0.2837	0.153491	0.043888	0.007	0.013	0.001	0.29	...	0.04	0.033	0.057	0.010	47.6	17.3	0.0729	0.0667	0.348	-0.014727
1	0.100473	1	6.829794	0.6141	0.313509	0.061827	0.019	0.032	0.007	0.91	...	0.64	0.173	0.274	0.067	57.1	18.0	0.0940	0.1438	0.525	0.005750
2	0.067051	1	8.895082	0.0000	0.204244	0.009186	0.260	0.325	0.201	1.00	...	18.14	2.573	2.478	2.667	26.5	20.7	0.1741	0.1750	1.082	-0.010040
3	0.064089	1	7.565275	0.1997	0.248714	0.036270	0.061	0.070	0.051	1.00	...	2.63	0.438	0.453	0.424	27.8	22.7	0.1265	0.1496	6.625	-0.002195
4	0.027930	1	7.162397	0.1740	0.299252	0.037367	0.017	0.027	0.007	0.82	...	2.11	0.257	0.287	0.229	34.5	17.6	0.1211	0.1308	2.500	0.003283

5 rows × 63 columns

# Dimensions of the dataset
df.shape

(90, 63)

• In this segment, we provide an empirical example of using partialling-out with Lasso to estimate the regression coefficient $β_1$ in the high-dimensional linear regression model. For any inference question, we can write Y as

Y = $β_1D$ + $β^r_2W$ + ε.

• Specifically we are interested in, how the rates at which economies of different countries grow denoted by Y , are related to the initial wealth levels in each country, denoted by D, controlling for country’s institutional, educational, and other similar characteristics, denoted by W.

• The relationship is captured by the regression coefficient $β_1$.

• In this example, this coefficient is called the “speed of convergence/divergence”, as it measures the speed at which poor countries catch up ($β_1$<0) or fall ($β_1$>0) behind wealthy countries, controlling for W.

• Our inference question here is: Do poor countries grow faster than rich countries, controlling for educational and other characteristics? In other words, is the speed of convergence negative: $β_1$<0?

• This is the Convergence Hypothesis predicted by the Solow Growth Model.Robert M. Solow is a world-renowned MIT economist who won the Nobel Prize in Economics.

• The dataset contains 90 countries and about 60 controls. Thus p is approximately 60 and n = 90 and p over n is not small. This means that we operate in the high-dimensional setting.

• Therefore, we expect the least squares method to provide a poor, very noisy estimate of $β_1$.

• In contrast, we expect the method based on partialling-out with Lasso to provide a high quality estimate of $β_1$.

# Extract the names of control and treatment variables from varnames
xnames = df.columns[3:] # names of X variables
dandxnames = df.columns[2:] # names of D and X variables
print(xnames)
print(dandxnames)

Index(['bmp1l', 'freeop', 'freetar', 'h65', 'hm65', 'hf65', 'p65', 'pm65',
       'pf65', 's65', 'sm65', 'sf65', 'fert65', 'mort65', 'lifee065', 'gpop1',
       'fert1', 'mort1', 'invsh41', 'geetot1', 'geerec1', 'gde1', 'govwb1',
       'govsh41', 'gvxdxe41', 'high65', 'highm65', 'highf65', 'highc65',
       'highcm65', 'highcf65', 'human65', 'humanm65', 'humanf65', 'hyr65',
       'hyrm65', 'hyrf65', 'no65', 'nom65', 'nof65', 'pinstab1', 'pop65',
       'worker65', 'pop1565', 'pop6565', 'sec65', 'secm65', 'secf65', 'secc65',
       'seccm65', 'seccf65', 'syr65', 'syrm65', 'syrf65', 'teapri65',
       'teasec65', 'ex1', 'im1', 'xr65', 'tot1'],
      dtype='object')
Index(['gdpsh465', 'bmp1l', 'freeop', 'freetar', 'h65', 'hm65', 'hf65', 'p65',
       'pm65', 'pf65', 's65', 'sm65', 'sf65', 'fert65', 'mort65', 'lifee065',
       'gpop1', 'fert1', 'mort1', 'invsh41', 'geetot1', 'geerec1', 'gde1',
       'govwb1', 'govsh41', 'gvxdxe41', 'high65', 'highm65', 'highf65',
       'highc65', 'highcm65', 'highcf65', 'human65', 'humanm65', 'humanf65',
       'hyr65', 'hyrm65', 'hyrf65', 'no65', 'nom65', 'nof65', 'pinstab1',
       'pop65', 'worker65', 'pop1565', 'pop6565', 'sec65', 'secm65', 'secf65',
       'secc65', 'seccm65', 'seccf65', 'syr65', 'syrm65', 'syrf65', 'teapri65',
       'teasec65', 'ex1', 'im1', 'xr65', 'tot1'],
      dtype='object')

Barro-Lee GrowthData

• The outcome (Y) is the realized annual growth rate of a country's wealth (Gross Domestic Product per capita).

• The target regressor (D) is the initial level of the country's wealth.

• The target parameter $β_1$ is the speed of convergence, which measures the speed at which poor countries catch up with rich countries.

• The controls (W) include measures of education levels, quality of institutions, trade openness, and political stability in the country.

from Scikit-learn.linear_model import LinearRegression #import linear regression from Scikit-learn
from Scikit-learn import metrics
from statsmodels.regression.linear_model import OLS #import Ordinary Least Squares from statsmodels
import statsmodels.api as sm

Y = df['Outcome'] # target variable
D = df['gdpsh465'] # target regressors
D_X = df[dandxnames] # control variables with target regressor
X = df[xnames] # control variables
D_X = sm.add_constant(D_X) # adding constants for intercept to control variables with target regressor
X = sm.add_constant(X) # adding constants for intercept to control variables
model = sm.OLS(Y, D_X) # OLS model object
results = model.fit() # training the model
results.summary() # summary of the trained model

OLS Regression Results

Dep. Variable:	Outcome	R-squared:	0.887
Model:	OLS	Adj. R-squared:	0.641
Method:	Least Squares	F-statistic:	3.607
Date:	Sat, 07 May 2022	Prob (F-statistic):	0.000200
Time:	08:45:19	Log-Likelihood:	238.24
No. Observations:	90	AIC:	-352.5
Df Residuals:	28	BIC:	-197.5
Df Model:	61
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
const	0.2472	0.785	0.315	0.755	-1.360	1.854
gdpsh465	-0.0094	0.030	-0.314	0.756	-0.071	0.052
bmp1l	-0.0689	0.033	-2.117	0.043	-0.135	-0.002
freeop	0.0801	0.208	0.385	0.703	-0.346	0.506
freetar	-0.4890	0.418	-1.169	0.252	-1.346	0.368
h65	-2.3621	0.857	-2.755	0.010	-4.118	-0.606
hm65	0.7071	0.523	1.352	0.187	-0.364	1.779
hf65	1.6934	0.503	3.365	0.002	0.663	2.724
p65	0.2655	0.164	1.616	0.117	-0.071	0.602
pm65	0.1370	0.151	0.906	0.373	-0.173	0.447
pf65	-0.3313	0.165	-2.006	0.055	-0.670	0.007
s65	0.0391	0.186	0.211	0.835	-0.341	0.419
sm65	-0.0307	0.117	-0.263	0.795	-0.270	0.209
sf65	-0.1799	0.118	-1.523	0.139	-0.422	0.062
fert65	0.0069	0.027	0.254	0.801	-0.049	0.062
mort65	-0.2335	0.817	-0.286	0.777	-1.908	1.441
lifee065	-0.0149	0.193	-0.077	0.939	-0.411	0.381
gpop1	0.9702	1.812	0.535	0.597	-2.742	4.682
fert1	0.0088	0.035	0.252	0.803	-0.063	0.081
mort1	0.0666	0.685	0.097	0.923	-1.336	1.469
invsh41	0.0745	0.108	0.687	0.498	-0.148	0.297
geetot1	-0.7151	1.680	-0.426	0.674	-4.157	2.726
geerec1	0.6300	2.447	0.257	0.799	-4.383	5.643
gde1	-0.4436	1.671	-0.265	0.793	-3.867	2.980
govwb1	0.3375	0.438	0.770	0.447	-0.560	1.235
govsh41	0.4632	1.925	0.241	0.812	-3.481	4.407
gvxdxe41	-0.7934	2.059	-0.385	0.703	-5.012	3.425
high65	-0.7525	0.906	-0.831	0.413	-2.608	1.103
highm65	-0.3903	0.681	-0.573	0.571	-1.786	1.005
highf65	-0.4177	0.561	-0.744	0.463	-1.568	0.732
highc65	-2.2158	1.481	-1.496	0.146	-5.249	0.818
highcm65	0.2797	0.658	0.425	0.674	-1.069	1.628
highcf65	0.3921	0.766	0.512	0.613	-1.177	1.961
human65	2.3373	3.307	0.707	0.486	-4.437	9.112
humanm65	-1.2092	1.619	-0.747	0.461	-4.525	2.106
humanf65	-1.1039	1.685	-0.655	0.518	-4.555	2.347
hyr65	54.9139	23.887	2.299	0.029	5.983	103.845
hyrm65	12.9350	23.171	0.558	0.581	-34.529	60.400
hyrf65	9.0926	17.670	0.515	0.611	-27.102	45.287
no65	0.0372	0.132	0.282	0.780	-0.233	0.308
nom65	-0.0212	0.065	-0.326	0.747	-0.154	0.112
nof65	-0.0169	0.067	-0.252	0.803	-0.154	0.120
pinstab1	-0.0500	0.031	-1.616	0.117	-0.113	0.013
pop65	1.032e-07	1.32e-07	0.783	0.440	-1.67e-07	3.73e-07
worker65	0.0341	0.156	0.218	0.829	-0.286	0.354
pop1565	-0.4655	0.471	-0.988	0.332	-1.431	0.500
pop6565	-1.3575	0.635	-2.138	0.041	-2.658	-0.057
sec65	-0.0109	0.308	-0.035	0.972	-0.641	0.619
secm65	0.0033	0.151	0.022	0.983	-0.306	0.313
secf65	-0.0023	0.158	-0.015	0.988	-0.326	0.321
secc65	-0.4915	0.729	-0.674	0.506	-1.985	1.002
seccm65	0.2596	0.356	0.730	0.471	-0.469	0.988
seccf65	0.2207	0.373	0.591	0.559	-0.544	0.985
syr65	-0.7556	7.977	-0.095	0.925	-17.095	15.584
syrm65	0.3109	3.897	0.080	0.937	-7.671	8.293
syrf65	0.7593	4.111	0.185	0.855	-7.661	9.180
teapri65	3.955e-05	0.001	0.051	0.959	-0.002	0.002
teasec65	0.0002	0.001	0.213	0.833	-0.002	0.003
ex1	-0.5804	0.242	-2.400	0.023	-1.076	-0.085
im1	0.5914	0.250	2.363	0.025	0.079	1.104
xr65	-0.0001	5.42e-05	-1.916	0.066	-0.000	7.18e-06
tot1	-0.1279	0.113	-1.136	0.266	-0.359	0.103

Omnibus:	0.439	Durbin-Watson:	1.982
Prob(Omnibus):	0.803	Jarque-Bera (JB):	0.417
Skew:	0.158	Prob(JB):	0.812
Kurtosis:	2.896	Cond. No.	7.52e+08

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 7.52e+08. This might indicate that there are
strong multicollinearity or other numerical problems.

• Model performance is good as the r2 is coming to be 88.7 percentage.
• Low value to adjusted r2 makes sense as we have a lot of features and they may be capturing the noise in the dataset.

# Lasso model with target and control variables
model = sm.OLS(Y, X) # OLS model object
results = model.fit() # training the model
results_Y = model.fit_regularized(alpha = 0.002, L1_wt=1.0,start_params=results.params)
# alpha = learning parameter, L1_wt = 1 - represents lasso


final = sm.regression.linear_model.OLSResults(model,
                                              results_Y.params,
                                              model.normalized_cov_params)
r_Y = final.resid # find the residuals

# Lasso model with target regressor and control variables
model = sm.OLS(D, X)
results = model.fit()
results_D = model.fit_regularized(alpha = 0.002, L1_wt=1.0,start_params=results.params)
final = sm.regression.linear_model.OLSResults(model,
                                              results_D.params,
                                              model.normalized_cov_params)
r_D = final.resid # find the residuals

# Linear model between the residuals of the lasso models created above
Y = r_Y # target variable
X = r_D
X = sm.add_constant(X)
model = sm.OLS(Y, X)
results = model.fit() # train the model
print(results.summary())

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.045
Model:                            OLS   Adj. R-squared:                  0.034
Method:                 Least Squares   F-statistic:                     4.143
Date:                Sat, 07 May 2022   Prob (F-statistic):             0.0448
Time:                        08:45:20   Log-Likelihood:                 128.89
No. Observations:                  90   AIC:                            -253.8
Df Residuals:                      88   BIC:                            -248.8
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          0.0011      0.006      0.177      0.860      -0.011       0.013
x1            -0.0431      0.021     -2.035      0.045      -0.085      -0.001
==============================================================================
Omnibus:                        2.328   Durbin-Watson:                   1.681
Prob(Omnibus):                  0.312   Jarque-Bera (JB):                2.020
Skew:                           0.250   Prob(JB):                        0.364
Kurtosis:                       2.463   Cond. No.                         3.44
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

	Estimate	Standard Error	95% Confidence Interval
Least squares	-0.0094	0.030	[-0.071 0.052]
Partialling-out via lasso	-0.0431	0.021	[-0.085 -0.001]

• As expected, least squares provides a rather noisy estimate of the speed of convergence, and does not allow us to answer the question about the convergence hypothesis.

• In sharp contrast, partialling-out via Lasso provides a more precise estimate.

• The lasso based point estimate is -0.043 and the 95% confidence interval for the (annual) rate of convergence is -0.085 to -0.001

• This empirical evidence does support the convergence hypothesis.

Conclusions

• In this segment, we have examined an empirical example in the high-dimensional setting.

• Least squares yields a very noisy estimate of the target regression coefficient and does not allow us to answer an important empirical question.

• Lasso does yield a precise estimate of the regression coefficient and does allow us to answer that question.

• We have found significant empirical evidence supporting the convergence hypothesis of Solow.

# Convert notebook to html
!jupyter nbconvert --to html "/content/drive/My Drive/Colab Notebooks/Copy of FDS_Project_LearnerNotebook_FullCode.ipynb"