The Solow Growth Model

Mankiw, Romer & Weil (1992), A Contribution to the Empirics of Economic Growth, QJE v.107, n.2, p407-437

The Solow Growth Model (QJE, 1956) predicts that the steady state level of income per capita depends positively on the saving rate, and negatively on the rate of popluation growth.

This is described by the following linear relationship:

\[ln\ \left(\frac{Y}{L} \right)=a+\frac{\alpha}{1-\alpha} ln S_k + \frac{\alpha}{1-\alpha} ln \left(n+g+\delta \right)+ e\]

Where \(ln\ \left(\frac{Y}{L} \right)\) is the log of income per capita; \(lnS_k\) is the log of saving (investment) as a share of GDP; \(ln\left(n+g+\delta \right)\) is the log of the population growth rate (\(n_i\)) plus the steady state growth rate (\(g\)) and the rate of capital depreciation (\(\delta\))

The model tests the validity of the restrictions; \[\beta_2>0\] \[\beta_3<0\] and \[\beta_3 = -\beta_2\]

Regression 1: Income per capita against saving rate

reg.1.nonoil <- lm(ln.y ~ ln.sk + ln.ngd, data = solow.data.nonoil)
summary(reg.1.nonoil)
## 
## Call:
## lm(formula = ln.y ~ ln.sk + ln.ngd, data = solow.data.nonoil)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.79144 -0.39367  0.04124  0.43368  1.58046 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   5.4299     1.5839   3.428 0.000900 ***
## ln.sk         1.4240     0.1431   9.951  < 2e-16 ***
## ln.ngd       -1.9898     0.5634  -3.532 0.000639 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6891 on 95 degrees of freedom
## Multiple R-squared:  0.6009, Adjusted R-squared:  0.5925 
## F-statistic: 71.51 on 2 and 95 DF,  p-value: < 2.2e-16

Test restrictions using heteroskedasticity-Consistent Covariance Matrix Estimation 

linearHypothesis(reg.1.nonoil, "ln.sk + ln.ngd = 0", vcov = vcovHC(reg.1.nonoil, type = "HC0"))
## Linear hypothesis test
## 
## Hypothesis:
## ln.sk  + ln.ngd = 0
## 
## Model 1: restricted model
## Model 2: ln.y ~ ln.sk + ln.ngd
## 
## Note: Coefficient covariance matrix supplied.
## 
##   Res.Df Df      F Pr(>F)
## 1     96                 
## 2     95  1 0.8714 0.3529
linearHypothesis(reg.1.nonoil, c("ln.sk = 0", "ln.ngd = 0"), vcov = vcovHC(reg.1.nonoil,type = "HC0"))
## Linear hypothesis test
## 
## Hypothesis:
## ln.sk = 0
## ln.ngd = 0
## 
## Model 1: restricted model
## Model 2: ln.y ~ ln.sk + ln.ngd
## 
## Note: Coefficient covariance matrix supplied.
## 
##   Res.Df Df      F    Pr(>F)    
## 1     97                        
## 2     95  2 106.19 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1