Problem Solving 01

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Replication of a paper published by Dr. Jessica Goldberg - "Experimental Evidence about Labor Supply in Rural Malawi"

Original paper, published by American Economic Journal in 2016 can be found here

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In this session, we work on a paper that analysis labor supply in Malawi in an experimental setting.

Abstract: I use a field experiment to estimate the wage elasticity of employment in the day labor market in rural Malawi. Once a week for 12 consecutive weeks, I make job offers for a workfare-type program to 529 adults. The daily wage varies from the tenth to the ninetieth percentile of the wage distribution, and individuals are entitled to work a maximum of one day per week. In this context (the low agricultural season), 74 percent of individuals worked at the lowest wage, and consequently the estimated labor supply elasticity is low (0.15), regardless of observable characteristics.

The paper uses data collected by the author and her colleagues in Malawi. They randomly offered jobs with different wages to individuals in 10 villages for 12 consecutive weeks and recorded acceptance rates. They used data to estimate the elasticity of labor supply to wage. The variable of interest in this paper is acceptance rate in each village in each week and the explanatory variable is wage.

The other explanatory variables may be variables that distinguish each village from the others.

The main finding of the paper is that in the off-season, the majority of workers accept the job offer at the lowest wage, and the elasticity of labor is low, meaning that the main force of accepting job is probably factors beyond wage.

loading all the neccessary tools needed for the project:

%%capture
! pip install seaborn
import seaborn as sns
import matplotlib.pyplot as plt
%matplotlib inline
import numpy as np
import pandas as pd
from matplotlib.ticker import StrMethodFormatter

Part 1

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loading the dataset

path = "PS03.dta"
df = pd.read_stata(path)
df.head(5)

	week	village	vil_labor	vil_hh	badshock_prev	goodshock_prev	badshock_prev_lnwage	goodshock_prev_lnwage	wage	lnwage	...	wagedum3	wagedum4	wagedum5	wagedum6	wagedum7	wagedum8	wagedum9	wagedum10	wagedum11	wagedum12
0	1	CHIMOWA	1.000000	1.0000	NaN	NaN	NaN	NaN	100.0	4.605170	...	0	0	0	0	0	1	0	0	0	0
1	2	CHIMOWA	0.857143	0.7500	0.285714	0.326531	1.169813	1.336929	60.0	4.094345	...	0	1	0	0	0	0	0	0	0	0
2	3	CHIMOWA	0.000000	0.0000	0.448980	0.306122	2.149486	1.465559	120.0	4.787492	...	0	0	0	0	0	0	0	1	0	0
3	4	CHIMOWA	0.897959	0.8125	0.918367	0.183673	3.123549	0.624710	30.0	3.401197	...	0	0	0	0	0	0	0	0	0	0
4	5	CHIMOWA	1.000000	1.0000	0.204082	0.183673	0.959282	0.863354	110.0	4.700480	...	0	0	0	0	0	0	1	0	0	0

5 rows × 29 columns

1.a) First, we have to know our variable of interest, acceptance rate, and the main explanatory variable, wage.

df['vil_labor'].describe()

count    120.000000
mean       0.845476
std        0.171205
min        0.000000
25%        0.809310
50%        0.907692
75%        0.958333
max        1.000000
Name: vil_labor, dtype: float64

By observing dependent variable 'vil_labor' we see that the average acceptance rate among the villages was 84%

df['wage'].describe()

count    120.000000
mean      85.000000
std       34.665264
min       30.000000
25%       57.500000
50%       85.000000
75%      112.500000
max      140.000000
Name: wage, dtype: float64

By observing independent variable 'wage' we see that the average offered wage among the villages was 85 Malawi Kwacha

y = df['vil_labor']
plt.hist(y, color='green', zorder=2, rwidth=0.95)
plt.grid(axis='y', alpha=0.75)
plt.grid(axis='x', alpha=0.75)
plt.ylabel('Frequency',fontsize=12)
plt.xlabel('Acceptance Rate',fontsize=12)
plt.title('Histogram for the A-Rate ',fontsize=15)

Text(0.5, 1.0, 'Histogram for the A-Rate ')

1.b) I'll Find the mean and standard deviation of acceptance rate for the 20 observations with the lowest wage, 20 observations with the highest wage.

df_wage_labor = df[['wage', 'vil_labor']]
top20 = df_wage_labor.nlargest(20, ['wage', 'vil_labor'])
top20.describe()

	wage	vil_labor
count	20.000000	20.000000
mean	135.000000	0.936237
std	5.129892	0.048356
min	130.000000	0.836735
25%	130.000000	0.902071
50%	135.000000	0.950820
75%	140.000000	0.978941
max	140.000000	1.000000

bottom20 = df_wage_labor.nsmallest(20, ['wage', 'vil_labor'])
bottom20.describe()

	wage	vil_labor
count	20.000000	20.000000
mean	35.000000	0.749579
std	5.129892	0.166202
min	30.000000	0.302326
25%	30.000000	0.652975
50%	35.000000	0.794005
75%	40.000000	0.856280
max	40.000000	0.959184

It is a good idea to compare the acceptance rate for the bottom and upper 20 villages. Based on this observations, even that the mean wage for the top 20 is 286% more than for the bottom (avg. for the top 20 is 135 Kwacha; avg. for the bottom is 35 Kwacha), we see that there is a small gap of 18.66% in acceptance rate between those two groups (93.62% acceptance rate for the top and 74.96% for the bottom)

There is not enough evidence to assume that wages influence if residents of the village accept the job or not

1.c.) Find the correlation coefficient between wage and acceptance rate.

x1_corr = df['wage'].corr(df['vil_labor'])

if x1_corr > 0 and x1_corr < 0.5:
    print('We\'ve a WEAK POSITIVE correlation with coeficient of:', x1_corr)
elif x1_corr > 0.5 and x1_corr <= 1:
    print('We\'ve a STRONG POSITIVE correlation with coeficient of:', x1_corr)
elif x1_corr < 0 and x1_corr > -0.5:
    print('We\'ve a WEAK NEGATIVE correlation with coeficient of:', x1_corr)
else:
    print('We\'ve a STRONG NEGATIVE correlation with coeficient of:', x1_corr)

We've a WEAK POSITIVE correlation with coeficient of: 0.34560122231666063

1.d) Let’s draw a scatter-graph to visualize the relationship between wage and job acceptance rate

Let's see if this confirms our assumptions that despite the wage differences, people are still willing to accept the job

sns.regplot(x="wage", y="vil_labor", data=df, color='purple')
plt.ylabel('Acceptance Rate', fontsize=12)
plt.xlabel('Wage', fontsize=12)
plt.title('Scatter-Graph of A-Rate and Wage')
plt.grid(axis='y', alpha=0.75)
plt.grid(axis='x', alpha=0.75)

Yes! graph confirms that there is a small influence of the wage offered on if villager accepted the job or not.

Part 2:

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2.a) We run a regression of wage on acceptance rate.

from sklearn.model_selection import train_test_split 
from sklearn.linear_model import LinearRegression
from sklearn import metrics

import statsmodels.formula.api as smf

#The coefficients for wage seems to be small. Generated a new variable that is 1000 times smaller than wage (wage/1000) and use it instead of wage
x = df[['wage']]
y = df['vil_labor']

#running regression acceptance rate on wage
reg = smf.ols('y ~ x', data=df).fit()
reg.summary()

OLS Regression Results

Dep. Variable:	y	R-squared:	0.119
Model:	OLS	Adj. R-squared:	0.112
Method:	Least Squares	F-statistic:	16.01
Date:	Mon, 16 Dec 2019	Prob (F-statistic):	0.000111
Time:	23:38:21	Log-Likelihood:	49.648
No. Observations:	120	AIC:	-95.30
Df Residuals:	118	BIC:	-89.72
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
Intercept	0.7004	0.039	17.894	0.000	0.623	0.778
x	0.0017	0.000	4.001	0.000	0.001	0.003

Omnibus:	83.233	Durbin-Watson:	1.811
Prob(Omnibus):	0.000	Jarque-Bera (JB):	475.521
Skew:	-2.436	Prob(JB):	5.52e-104
Kurtosis:	11.448	Cond. No.	244.

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

Our explanatory 'wage' showed to have a strong explanatory power as it's p-value is approx. 0 (which means that we are 99% confident it is statistically significant). Also, explanatory variable shows that offering of an additional 1 Kwacha increases acceptance rate by 0.0017

However, if we look that the R-squared, which is the measure of goodness of fit of our model, we see that our model is the explanation of the outcome only by approx. 12%

#now once again we regress acceptance rate on wages
reg = LinearRegression()

x = df[['wage']]
y = df['vil_labor']

#fitting the model
reg = reg.fit(x, y)

#creating yhat or the prediction line
yhat = reg.predict(x)

2.b) We visualize our results:

plt.plot(x, yhat, color='green', label='Regression Line')
plt.scatter(x, y, c='red', label='Actual Data')
plt.ylabel('Acceptance Rate', fontsize=12)
plt.xlabel('Wage', fontsize=12)
plt.grid(axis='y', alpha=0.75)
plt.grid(axis='x', alpha=0.75)
plt.legend()
plt.show()

2.c) Coefficient for the wage is tiny. I will generate new variable 1000x smaller than wage and use it instead

wage1000 = df['wage'] / 1000

reg = smf.ols('y ~ wage1000', data=df).fit()
reg.summary()

OLS Regression Results

Dep. Variable:	y	R-squared:	0.119
Model:	OLS	Adj. R-squared:	0.112
Method:	Least Squares	F-statistic:	16.01
Date:	Mon, 16 Dec 2019	Prob (F-statistic):	0.000111
Time:	23:38:21	Log-Likelihood:	49.648
No. Observations:	120	AIC:	-95.30
Df Residuals:	118	BIC:	-89.72
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
Intercept	0.7004	0.039	17.894	0.000	0.623	0.778
wage1000	1.7069	0.427	4.001	0.000	0.862	2.552

Omnibus:	83.233	Durbin-Watson:	1.811
Prob(Omnibus):	0.000	Jarque-Bera (JB):	475.521
Skew:	-2.436	Prob(JB):	5.52e-104
Kurtosis:	11.448	Cond. No.	29.2

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

Now we see that any additional 10 Kwacha increases acceptance rate by 1.7%

2.d) to make sure that our model is a good model we check if sum of the residuals is equal 0.

print('sum of the residuals is:\t', np.sum(reg.resid))
print('average of the residuals is:\t', np.mean(reg.resid))

sum of the residuals is:	 -4.5075054799781356e-14
average of the residuals is:	 -3.811765717879704e-16

in both cases we see that they are very very close to 0!

2.e) Now instead of 'wage' we will run regression using 'lnwage' which is interpreted as a percentage change in wage

lnwage = df['lnwage']

reg = smf.ols('y ~ lnwage', data=df).fit()
reg.summary()

OLS Regression Results

Dep. Variable:	y	R-squared:	0.116
Model:	OLS	Adj. R-squared:	0.109
Method:	Least Squares	F-statistic:	15.49
Date:	Mon, 16 Dec 2019	Prob (F-statistic):	0.000140
Time:	23:38:21	Log-Likelihood:	49.418
No. Observations:	120	AIC:	-94.84
Df Residuals:	118	BIC:	-89.26
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
Intercept	0.3060	0.138	2.220	0.028	0.033	0.579
lnwage	0.1242	0.032	3.936	0.000	0.062	0.187

Omnibus:	82.986	Durbin-Watson:	1.832
Prob(Omnibus):	0.000	Jarque-Bera (JB):	464.288
Skew:	-2.439	Prob(JB):	1.52e-101
Kurtosis:	11.311	Cond. No.	42.9

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

Here we see that 1% change in the wage changes acceptance rate by 12.42%.

2.f) Now we just manually compute the regression which was used in the study to calculate the elasticity of the wages

regression used in the model is labor = α + β*ln(wage) + e

x = 1
labor = 0.3060 + 0.1242*x
print('Elasticity of the job acceptance is based on wage:',labor)

Elasticity of the job acceptance is based on wage: 0.4302

Even though I used exactly the same model with the same variables from the same dataset my elasticity came to be a little higher than found in the original paper (0.35). This can explain by a small difference in the data. In original data, it appears they got rid of the village which had a 0% acceptance rate (probably there was something happening at that time, so nobody accepted the job), while this data is present in my analysis. However, we still can canclude that findings are somewhat inelastic

Part 3:

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3.a) Now in trying to reproduce the other columns of Table 4 of the paper, we would like to calculate two sets of variables called “village effects” and “week effects”. We work only on village effects. Week effects are similar.

lnwage = df['lnwage']
village = df['village']

reg = smf.ols('y ~ lnwage + C(village)', data=df).fit()
reg.summary()

OLS Regression Results

Dep. Variable:	y	R-squared:	0.202
Model:	OLS	Adj. R-squared:	0.128
Method:	Least Squares	F-statistic:	2.752
Date:	Mon, 16 Dec 2019	Prob (F-statistic):	0.00461
Time:	23:38:21	Log-Likelihood:	55.524
No. Observations:	120	AIC:	-89.05
Df Residuals:	109	BIC:	-58.38
Df Model:	10
Covariance Type:	nonrobust

	coef	std err	t	P>|t|	[0.025	0.975]
Intercept	0.3228	0.143	2.255	0.026	0.039	0.607
C(village)[T.HASHAMU]	-0.1058	0.065	-1.622	0.108	-0.235	0.023
C(village)[T.KACHULE]	-0.0631	0.065	-0.966	0.336	-0.192	0.066
C(village)[T.KADZINJA/KHUNGWA]	0.0347	0.065	0.531	0.596	-0.095	0.164
C(village)[T.KAFOTOKOZA]	0.0257	0.065	0.394	0.694	-0.104	0.155
C(village)[T.KAMWENDO]	0.0128	0.065	0.195	0.845	-0.117	0.142
C(village)[T.KUNFUNDA]	-0.0855	0.065	-1.310	0.193	-0.215	0.044
C(village)[T.MANASE]	-0.0409	0.065	-0.626	0.532	-0.170	0.088
C(village)[T.MSANGU/KALUTE]	0.0272	0.065	0.418	0.677	-0.102	0.157
C(village)[T.NJONJA]	0.0272	0.065	0.417	0.678	-0.102	0.157
lnwage	0.1242	0.031	3.980	0.000	0.062	0.186

Omnibus:	94.244	Durbin-Watson:	2.025
Prob(Omnibus):	0.000	Jarque-Bera (JB):	715.205
Skew:	-2.719	Prob(JB):	4.96e-156
Kurtosis:	13.652	Cond. No.	51.1

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

After controlling for individual village, we see that effect of a wage on the acceptance rate didn't change, and none of the village variables is statistically significant and the effect on acceptance rate is marginally or non-existence.