suppressMessages({
  library(stargazer)
  library(hdm)
  library(AER)
  })

Model

\[\begin{equation} (\Delta\log GDP)_i=\alpha \cdot GDP^0_i+U_i. \end{equation}\]

\((\Delta\log GDP)_i\) is the change in the log of GDP per capita of country \(i\) between periods \(t_0\) and \(t_1\), \(t_0<t_1\).
\(GDP^0_i\) is the GDP per capita of country \(i\) in period \(t_0\)
The Catching Up Hypothesis: \[\begin{equation}\alpha <0.\end{equation}\] The rate of growth slows down, less developed countries catch up with more developed.

Data

data("GrowthData")

Initial GDP per capita: gdpsh465.
GDP per capita growth rate: Outcome.

Testing the hypothesis

simple<-lm(Outcome~gdpsh465,data=GrowthData)
suppressWarnings(
  stargazer(simple,
            header=FALSE, 
            title="Testing the simple catching up hypothesis",
            omit.stat = "all",
            #type="text"
            type="html",notes.append = FALSE,notes = c("<sup>&sstarf;</sup>p<0.1; <sup>&sstarf;&sstarf;</sup>p<0.05; <sup>&sstarf;&sstarf;&sstarf;</sup>p<0.01")
  )
)

**Testing the simple catching up hypothesis**

	Dependent variable:

	Outcome

gdpsh465	0.001
	(0.006)

Constant	0.035
	(0.047)



Note:	^⋆p<0.1; ^⋆⋆p<0.05; ^⋆⋆⋆p<0.01

No support for the catching up hypothesis.

Conditional model:

Institutions and technology matter.
The catching up hypothesis works with similar countries only.
Need to control for the characteristics:\[\begin{equation} (\Delta\log GDP)_i=\alpha \cdot GDP^0_i+X_i'\beta+U_i. \end{equation}\]
\(X_i\) is the vector of controls describing the economic conditions of country \(i\) in period \(t_0\).

There are a lot of potential controls in the data:

dim(GrowthData)

## [1] 90 63

names(GrowthData)

##  [1] "Outcome"   "intercept" "gdpsh465"  "bmp1l"     "freeop"    "freetar"  
##  [7] "h65"       "hm65"      "hf65"      "p65"       "pm65"      "pf65"     
## [13] "s65"       "sm65"      "sf65"      "fert65"    "mort65"    "lifee065" 
## [19] "gpop1"     "fert1"     "mort1"     "invsh41"   "geetot1"   "geerec1"  
## [25] "gde1"      "govwb1"    "govsh41"   "gvxdxe41"  "high65"    "highm65"  
## [31] "highf65"   "highc65"   "highcm65"  "highcf65"  "human65"   "humanm65" 
## [37] "humanf65"  "hyr65"     "hyrm65"    "hyrf65"    "no65"      "nom65"    
## [43] "nof65"     "pinstab1"  "pop65"     "worker65"  "pop1565"   "pop6565"  
## [49] "sec65"     "secm65"    "secf65"    "secc65"    "seccm65"   "seccf65"  
## [55] "syr65"     "syrm65"    "syrf65"    "teapri65"  "teasec65"  "ex1"      
## [61] "im1"       "xr65"      "tot1"

90 observations (countries).
Over 60 potential controls.
This is a “big-data”/“high-dimensional” example.

Estimation of the conditional model

y=as.vector(GrowthData$Outcome)
D=as.vector(GrowthData$gdpsh465)
Controls=as.matrix(GrowthData)[,-c(1,2,3)]

y = GDP per capita growth rate.
D = initial GDP per capita.
-c(1,2,3) instructs to exclude the first 3 variables in GrowthData:
- Outcome
- intercept
- gdpsh465

OLS regression with all controls:

conditional=lm(y~D+Controls)
suppressWarnings(
  stargazer(conditional,
            header=FALSE, 
            title="Testing the conditional catching up hypothesis",
            omit.stat = "all",
            omit="Controls",
            #type="text"
            type="html",notes.append = FALSE,notes = c("<sup>&sstarf;</sup>p<0.1; <sup>&sstarf;&sstarf;</sup>p<0.05; <sup>&sstarf;&sstarf;&sstarf;</sup>p<0.01")
  )
)

**Testing the conditional catching up hypothesis**

	Dependent variable:

	y

D	-0.009
	(0.030)

Constant	0.247
	(0.785)



Note:	^⋆p<0.1; ^⋆⋆p<0.05; ^⋆⋆⋆p<0.01

No support for the conditional catching up hypothesis.

The estimate is negative but the std.err is too large - too many controls, still no support.
The std.err. on the initial GDP per capita increased from 0.006 to 0.030.

Post-Lasso with Double Lasso

?rlassoEffect

Usage:

x= specifies the matrix of controls.
y= specifies the outcome variable.
d= specifies the treatment variable (the main regressor of interest).

Effect<-rlassoEffect(x=Controls,y=y,d=D,method="double selection")
summary(Effect)

## [1] "Estimates and significance testing of the effect of target variables"
##    Estimate. Std. Error t value Pr(>|t|)   
## d1  -0.05001    0.01579  -3.167  0.00154 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

names(Effect)

##  [1] "alpha"            "se"               "t"                "pval"            
##  [5] "no.selected"      "coefficients"     "coefficient"      "coefficients.reg"
##  [9] "selection.index"  "residuals"        "call"             "samplesize"

A negative significant estimate!

Effect$selection.index

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

Included controls:

sum(Effect$selection.index==TRUE)

## [1] 7

Effect$selection.index[Effect$selection.index==TRUE]

##    bmp1l  freetar     hm65     sf65 lifee065 humanf65  pop6565 
##     TRUE     TRUE     TRUE     TRUE     TRUE     TRUE     TRUE

Double Lasso selected 7 controls:

bmp1l: Log of the black market premium.
freetar: Measure of tariff restrictions.
hm65: Male gross enrollment ratio for higher education in 1965.
sf65: Female gross enrollment ratio for secondary education in 1965.
lifee065: Life expectancy at 0 in 1965.
humanf65: Average schooling years in the female population over age 25 in 1965.
pop6565: Population Proportion over 65 in 1965.

?rlasso()

Step 1 of the algorithm:

lasso.Y<-rlasso(Outcome~Controls,data=GrowthData)
names(lasso.Y)

##  [1] "coefficients" "beta"         "intercept"    "index"        "lambda"      
##  [6] "lambda0"      "loadings"     "residuals"    "sigma"        "iter"        
## [11] "call"         "options"      "model"        "tss"          "rss"         
## [16] "dev"

coef(lasso.Y)

##      (Intercept)    Controlsbmp1l   Controlsfreeop  Controlsfreetar 
##       0.05810092      -0.07556548       0.00000000       0.00000000 
##      Controlsh65     Controlshm65     Controlshf65      Controlsp65 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##     Controlspm65     Controlspf65      Controlss65     Controlssm65 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##     Controlssf65   Controlsfert65   Controlsmort65 Controlslifee065 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##    Controlsgpop1    Controlsfert1    Controlsmort1  Controlsinvsh41 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##  Controlsgeetot1  Controlsgeerec1     Controlsgde1   Controlsgovwb1 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##  Controlsgovsh41 Controlsgvxdxe41   Controlshigh65  Controlshighm65 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##  Controlshighf65  Controlshighc65 Controlshighcm65 Controlshighcf65 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##  Controlshuman65 Controlshumanm65 Controlshumanf65    Controlshyr65 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##   Controlshyrm65   Controlshyrf65     Controlsno65    Controlsnom65 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##    Controlsnof65 Controlspinstab1    Controlspop65 Controlsworker65 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##  Controlspop1565  Controlspop6565    Controlssec65   Controlssecm65 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##   Controlssecf65   Controlssecc65  Controlsseccm65  Controlsseccf65 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##    Controlssyr65   Controlssyrm65   Controlssyrf65 Controlsteapri65 
##       0.00000000       0.00000000       0.00000000       0.00000000 
## Controlsteasec65      Controlsex1      Controlsim1     Controlsxr65 
##       0.00000000       0.00000000       0.00000000       0.00000000 
##     Controlstot1 
##       0.00000000

Step 2

lasso.D<-rlasso(D~Controls)
coef(lasso.D)

## (Intercept)       bmp1l      freeop     freetar         h65        hm65 
## -4.48400246  0.00000000  0.00000000 -6.22415497  0.00000000  1.64599221 
##        hf65         p65        pm65        pf65         s65        sm65 
##  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 
##        sf65      fert65      mort65    lifee065       gpop1       fert1 
## -0.16139575  0.00000000  0.00000000  2.91176774  0.00000000  0.00000000 
##       mort1     invsh41     geetot1     geerec1        gde1      govwb1 
##  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 
##     govsh41    gvxdxe41      high65     highm65     highf65     highc65 
##  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 
##    highcm65    highcf65     human65    humanm65    humanf65       hyr65 
##  0.00000000  0.00000000  0.00000000  0.00000000  0.03873902  0.00000000 
##      hyrm65      hyrf65        no65       nom65       nof65    pinstab1 
##  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 
##       pop65    worker65     pop1565     pop6565       sec65      secm65 
##  0.00000000  0.00000000  0.00000000  1.85116114  0.00000000  0.00000000 
##      secf65      secc65     seccm65     seccf65       syr65      syrm65 
##  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 
##      syrf65    teapri65    teasec65         ex1         im1        xr65 
##  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000  0.00000000 
##        tot1 
##  0.00000000

Using the partialling out approach:

Effect_PO<-rlassoEffect(x=Controls,y=y,d=D,method="partialling out")
summary(Effect_PO)

## [1] "Estimates and significance testing of the effect of target variables"
##      Estimate. Std. Error t value Pr(>|t|)    
## [1,]  -0.04981    0.01394  -3.574 0.000351 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

A very similar estimate to the Double Lasso approach.

sum(Effect_PO$selection.index==TRUE)

## [1] 7

Effect_PO$selection.index[Effect_PO$selection.index==TRUE]

##    bmp1l  freetar     hm65     sf65 lifee065 humanf65  pop6565 
##     TRUE     TRUE     TRUE     TRUE     TRUE     TRUE     TRUE

The same selected controls.

lasso.Y<-rlasso(Outcome~Controls,data=GrowthData)
Ytilde<-lasso.Y$residuals

lasso.D<-rlasso(D~Controls,data=GrowthData)
Dtilde<-lasso.D$residuals

Post<-lm(Ytilde~ -1+ Dtilde)
coeftest(Post,vcov. = vcovHC(Post,type="HC0"))

## 
## t test of coefficients:
## 
##         Estimate Std. Error t value Pr(>|t|)   
## Dtilde -0.049811   0.015219 -3.2729 0.001516 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Conclusion

This data set contains many potential controls that are highly correlated among each other:

Many similar education variables.
Many similar demographic variables.
Many similar political variables.
Etc.

These controls are related not only among each other, but also to the main regressor (treatment): The “initial” GDP per capita. As a result, including all potential controls produces insignificant estimates due to the presence of many controls (over 60 controls with only 90 observations).

It is plausible to assume that the model is sparse: only certain demographic, education, and etc. variables matter. This is an appropriate problem for Lasso as we need to select few out of many controls.

Lasso selects the important controls. The double Lasso step also selects the controls that are related to the main regressor to avoid potential omitted variables bias.

Post Lasso produces significant estimates on the main regressor. The result implies that the conditional catching up hypothesis holds: growth rates converge for countries with similar economic and demographic characteristics.

Lab 03: Post-Lasso and double Lasso with growth data