Part One:

Generating a dataset that contains Average Investment Rate, Annual Population Growth rate, and income per capita.

# Loading Penn World Table Data.

pwt <- read.csv("pwt71.csv")

# Creating a new dataset titled pwt.ss (subset).

pwt.ss <- pwt %>% filter(between (year, 1985, 2010)) %>%
  group_by(isocode) %>% 
  summarise(n = log(last(POP)/first(POP))/n(),
            inv = mean(ki)/100,
            y = last(y)) %>% filter(!is.na(inv))


# n = Continuous Compounding Growth Rate.

# inv = Average Investment as a Percentage of GDP scaled to same order of magnitude.

# y = income per capita (in purchasing power terms) relative to the United States in 2010.
        
# Scatterplot of Investment on Population Growth rate.    
pwt.ss %>% ggplot(aes(x = inv, y = n)) +
  geom_point()

# Scatterplot of Investment on Income per Capita.
pwt.ss %>% ggplot(aes(x = inv, y = y)) +
  geom_point() 

# Scatterplot of Population Growth Rate on Income per Capita.
pwt.ss %>% ggplot(aes(x = n, y = y)) +
  geom_point()

Part Two:

Creating Linear and Non-Linear Models

Basic Linear Solow Model:

pwt.ss <- pwt.ss %>% 
  mutate(logY = log(y),
            logS = log(inv),
            g = 0.03,
            delta = 0.02,
            logngd = log(n+g+delta)) 

# The logic here is to take the dataset generated in part one and then add new columns logY, logS, g, delta, alpha and logngd (this being (n + g+ delta))

# The values for delta and g are taken from assumptions used in 'Contributions to the Empirics of Economic Growth (1992)' by Mankiw, Romer and Weil.



# This is the generation of the linear model.

linmod <- lm(logY ~ logS + logngd, data = pwt.ss)

summary(linmod)
## 
## Call:
## lm(formula = logY ~ logS + logngd, data = pwt.ss)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.58394 -0.74338 -0.00635  0.64977  3.06547 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -8.9330     1.5663  -5.703 5.73e-08 ***
## logS          1.0019     0.1928   5.198 6.24e-07 ***
## logngd       -4.8203     0.5439  -8.863 1.67e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.002 on 156 degrees of freedom
## Multiple R-squared:  0.4467, Adjusted R-squared:  0.4396 
## F-statistic: 62.98 on 2 and 156 DF,  p-value: < 2.2e-16

The non-linear Solow model with previously used variables:

nonlinmod <- nls(logY ~ (alpha/(1-alpha))*logS - (alpha/(1-alpha))*logngd, data = pwt.ss, start = list(alpha = 0.4), control = nls.control(warnOnly = T), subset = !is.na(logS))

summary(nonlinmod)
## 
## Formula: logY ~ (alpha/(1 - alpha)) * logS - (alpha/(1 - alpha)) * logngd
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## alpha 0.683072   0.007119   95.96   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.152 on 158 degrees of freedom
## 
## Number of iterations to convergence: 4 
## Achieved convergence tolerance: 6.132e-10
# This is the same model with an intercept term of A0, where A0 is starting at 1

nonlinmodA0 <- nls(logY ~ A0 + (alpha/(1-alpha))*logS - (alpha/(1-alpha))*logngd, data = pwt.ss, start = list(alpha = 0.4, A0 = 1), control = nls.control(warnOnly = T), subset = !is.na(logS))

summary(nonlinmodA0)
## 
## Formula: logY ~ A0 + (alpha/(1 - alpha)) * logS - (alpha/(1 - alpha)) * 
##     logngd
## 
## Parameters:
##       Estimate Std. Error t value Pr(>|t|)    
## alpha  0.61202    0.02847   21.50  < 2e-16 ***
## A0     0.79943    0.24373    3.28  0.00128 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.118 on 157 degrees of freedom
## 
## Number of iterations to convergence: 4 
## Achieved convergence tolerance: 1.671e-07

Moment Comparison

# Adding a predicted value for y in the table
pwt.ss <- pwt.ss %>% 
  mutate(ypredicted = predict(nonlinmodA0))

# plotting logy against the predicted value of y
pwt.ss %>% ggplot(aes(x = logY, y = ypredicted)) +
  geom_point() + geom_smooth(method = "lm")

The plotting of \(y\) vs. predicted value of y show that there is a strong correlation between the model’s predictions and it’s actual results.

Moment checking

#finding the Residuals
pwt.ss <- pwt.ss %>% 
  mutate(residual = logY - ypredicted)

#Historgram of residuals
pwt.ss %>% ggplot(aes(x = residual)) +
  geom_histogram()

# plotting the residuals
pwt.ss %>% ggplot(aes(x = ypredicted, y = residual)) +
  geom_point() +  geom_hline(yintercept = 0)

The scatterplot seems random enough to indicate that the residuals and the fitted values are uncorrelated.

#Finding the standard deviation
sigma <- sd(pwt.ss$residual, na.rm = TRUE)

# Standard Deviation is 1.14681921927611

# Generating new version of countries
pwt.ss <- pwt.ss %>%
  mutate(Newcountry = exp(rnorm(n = n(), mean = ypredicted, sd = sigma)))

# Histogram of GDP per capita relative to the US
pwt.ss %>% ggplot(aes(x = y)) +
  geom_histogram()

# Histogram of new countries
    
pwt.ss %>% ggplot(aes(x = Newcountry)) +
  geom_histogram()

Conclusion

The “new countries” and the actual GDP having a very similar look, with most of the data on the right hand side in both. The y vs predicted y shows a symetrical diagnal distribution around the line and the residual plot seems sufficiently random. This information means with some certainty we can say the model is likely a good fit.