增长模型：R 语言复现
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课程开始时，鉴于电子商务专业的同学编程能力很强，教师承诺过会给一个关于增长模型的代码复现例子。
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Here we go!
MRW 1992
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数据和模型均来自论文， “A Contribution to the Empirics of Economic Growth”
- Greg Mankiw, David Romer and David Weil (QJE, 1992)
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索洛模型：储蓄率和人口增长率决定了稳态时的人均收入
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问：现实中，储蓄率和人口增长率低的国家，人均 GDP 会更高么？
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MRW: 对，但不完全对
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储蓄率和人口增长率只能解释不同国家质检约 50% 的收入差距
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如果加入人力资本，可以解释 80% 的收入差距
模型设定
\[Y(t) = K(t)^\alpha [A(t) L(t)]^{1-\alpha} \; \text{where} \; 0 < \alpha < 1
\]
\[L(t) = L(0) e^{nt}
\]
\[A(t) = A(0) e^{gt}
\]
\[\dot{k}(t) = sk(t)^\alpha - (n+g+\delta)k(t)
\]
\[k^* = [\frac{s}{n+g+\delta}]^{\frac{1}{1-a}}
\]
- 其中 \(\alpha = 2/3\)
Exploratory data analysis
library(AER)
library(dplyr)
library(skimr)
library(moderndive)
data("GrowthDJ")
skim(GrowthDJ)
── Data Summary ────────────────────────
Values
Name GrowthDJ
Number of rows 121
Number of columns 10
_______________________
Column type frequency:
factor 3
numeric 7
________________________
Group variables None
── Variable type: factor ──────────────────────────────────────────────────────────────────────────
skim_variable n_missing complete_rate ordered n_unique top_counts
1 oil 0 1 FALSE 2 no: 98, yes: 23
2 inter 0 1 FALSE 2 yes: 75, no: 46
3 oecd 0 1 FALSE 2 no: 99, yes: 22
── Variable type: numeric ─────────────────────────────────────────────────────────────────────────
skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
1 gdp60 5 0.959 3682. 7493. 383 973. 1962 4274. 77881 ▇▁▁▁▁
2 gdp85 13 0.893 5683. 5689. 412 1209. 3484. 7719. 25635 ▇▂▂▁▁
3 gdpgrowth 4 0.967 4.09 1.89 -0.9 2.8 3.9 5.3 9.2 ▁▅▇▃▁
4 popgrowth 14 0.884 2.28 0.999 0.3 1.7 2.4 2.9 6.8 ▃▇▃▁▁
5 invest 0 1 18.2 7.85 4.1 12 17.7 24.1 36.9 ▅▇▇▅▂
6 school 3 0.975 5.53 3.53 0.4 2.4 4.95 8.17 12.1 ▇▆▅▅▅
7 literacy60 18 0.851 48.2 35.4 1 15 39 83.5 100 ▇▃▂▃▆
Estimation
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参数校准：\(g + \delta = 0.5\)
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下面以 MRW 中数据中的样本 1 为例进行估计
nonoil <- subset(GrowthDJ, oil == "no")
inter <- subset(GrowthDJ, inter == "yes")
oecd <- subset(GrowthDJ, oecd == "yes")
mrw1 <- lm(log(gdp85) ~ log(invest/100) + log((popgrowth/100) + 0.05), data = nonoil)
mrw2 <- lm(log(gdp85) ~ log(invest/100) + log((popgrowth/100) + 0.05), data = inter)
mrw3 <- lm(log(gdp85) ~ log(invest/100) + log((popgrowth/100) + 0.05), data = oecd)
summary(mrw1)
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 ***
log(invest/100) 1.4240 0.1431 9.951 < 2e-16 ***
log((popgrowth/100) + 0.05) -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
Restricted regression
- Restriction: \(\ln (s)\) 与 \(\ln(g+n+\delta)\) 的和为0
mrw4 <- lm(log(gdp85) ~ I(log(invest/100) - log((popgrowth/100) + 0.05)), data = nonoil)
mrw5 <- lm(log(gdp85) ~ I(log(invest/100) - log((popgrowth/100) + 0.05)), data = inter)
mrw6 <- lm(log(gdp85) ~ I(log(invest/100) - log((popgrowth/100) + 0.05)), data = oecd)
summary(mrw4)
Residuals:
Min 1Q Median 3Q Max
-1.87388 -0.43133 0.03757 0.51698 1.49645
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 6.8724 0.1206 56.99 <2e-16 ***
I(log(invest/100) - log((popgrowth/100) + 0.05)) 1.4880 0.1247 11.93 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.6885 on 96 degrees of freedom
Multiple R-squared: 0.5974, Adjusted R-squared: 0.5932
F-statistic: 142.4 on 1 and 96 DF, p-value: < 2.2e-16
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运算结果表明，\((N+g+\delta)\) 的系数显著为负，而储蓄 \(s\) 的系数显著为正
- 这和索洛模型的预测一致
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但是，restricted regression 中计算得到的 \(\alpha= 0.6\), 和实际不符
考虑加入人力资本的有效劳动人口
- 衡量人力资本：使用不同时间段的“劳动人口中完成中学教育的比例”作为代理变量
- 从 \(R^2\) 可以看出，加入人力资本后的索洛模型，可以解释不同国家之间 80% 的收入差距
- 同时，\(\alpha\) 的取值也接近 1/3. 这和实际数据一致
- 因此，加入人力资本后的索洛模型，能更好地解释不通国家之间的收入差距
mrw7 <- lm(log(gdp85) ~ log(invest/100) + log((popgrowth/100) + 0.05) + log(school/100), data = nonoil)
mrw8 <- lm(log(gdp85) ~ log(invest/100) + log((popgrowth/100) + 0.05) + log(school/100), data = inter)
mrw9 <- lm(log(gdp85) ~ log(invest/100) + log((popgrowth/100) + 0.05) + log(school/100), data = oecd)
mrw10 <- lm(log(gdp85) ~ I(log(invest/100) - log((popgrowth/100) + 0.05)) + I(log(school/100) - log((popgrowth/100) + 0.05)), data = nonoil)
mrw11 <- lm(log(gdp85) ~ I(log(invest/100) - log((popgrowth/100) + 0.05)) + I(log(school/100) - log((popgrowth/100) + 0.05)), data = inter)
mrw12 <- lm(log(gdp85) ~ I(log(invest/100) - log((popgrowth/100) + 0.05)) + I(log(school/100) - log((popgrowth/100) + 0.05)), data = oecd)
收敛和条件收敛
- 索洛模型的另一个主要预测，是有类似条件的国家，稳态时的人均 GDP 会收敛 (conditional convergence)
- 此外，它还定量说明了收敛的速度
- 我们用 1960--1985 年之间不同国家的相关数据，对这个预测进行检验
converge1 <- lm(I(log(gdp85)-log(gdp60)) ~ log(gdp60), data = nonoil)
summary(converge1)
Residuals:
Min 1Q Median 3Q Max
-1.09784 -0.27467 -0.02826 0.25975 1.17747
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.26658 0.37960 -0.702 0.4842
log(gdp60) 0.09431 0.04962 1.901 0.0603 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.4405 on 96 degrees of freedom
Multiple R-squared: 0.03627, Adjusted R-squared: 0.02623
F-statistic: 3.613 on 1 and 96 DF, p-value: 0.06033
converge4 <- lm(I(log(gdp85)-log(gdp60)) ~ log(gdp60) + log(invest/100) + log(popgrowth/100 + 0.05), data = nonoil)
summary(converge4)
Residuals:
Min 1Q Median 3Q Max
-1.07648 -0.15215 0.01185 0.19595 0.96056
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.91938 0.83367 2.302 0.02352 *
log(gdp60) -0.14090 0.05202 -2.709 0.00803 **
log(invest/100) 0.64724 0.08670 7.465 4.16e-11 ***
log(popgrowth/100 + 0.05) -0.30235 0.30438 -0.993 0.32311
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.3507 on 94 degrees of freedom
Multiple R-squared: 0.4019, Adjusted R-squared: 0.3828
F-statistic: 21.05 on 3 and 94 DF, p-value: 1.622e-10
converge7 <- lm(I(log(gdp85)-log(gdp60)) ~ log(gdp60) + log(invest/100) + log(popgrowth/100 + 0.05) + log(school/100), data = nonoil)
summary(converge7)
Residuals:
Min 1Q Median 3Q Max
-0.91041 -0.17599 0.01789 0.18439 0.93846
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.02152 0.82748 3.651 0.000431 ***
log(gdp60) -0.28837 0.06158 -4.683 9.62e-06 ***
log(invest/100) 0.52374 0.08687 6.029 3.30e-08 ***
log(popgrowth/100 + 0.05) -0.50566 0.28861 -1.752 0.083061 .
log(school/100) 0.23112 0.05946 3.887 0.000190 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.327 on 93 degrees of freedom
Multiple R-squared: 0.4855, Adjusted R-squared: 0.4633
F-statistic: 21.94 on 4 and 93 DF, p-value: 8.987e-13
回归结果：
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不存在绝对收敛
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控制储蓄率和人口增长后，存在条件收敛的趋势
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进一步控制人力资本变量后，条件收敛的趋势非常明显
收敛图示
library(ggplot2)
library(gridExtra)
controlx1 <- lm(log(gdp60) ~ log(invest/100) + log(popgrowth/100 + 0.05), data = nonoil)
controlx1res <- residuals(controlx1)
controly1 <- lm(gdpgrowth ~ log(invest/100) + log(popgrowth/100 + 0.05), data = nonoil)
controly1res <- residuals(controly1)
controlx2 <- lm(log(gdp60) ~ log(invest/100) + log(popgrowth/100 + 0.05) + log(school/100), data = nonoil)
controlx2res <- residuals(controlx2)
controly2 <- lm(gdpgrowth ~ log(invest/100) + log(popgrowth/100 + 0.05) + log(school/100), data = nonoil)
controly2res <- residuals(controly2)
p1 <- ggplot(data = nonoil, mapping = aes(x=log(gdp60), y=gdpgrowth)) + geom_point() + geom_smooth(method = "lm") + labs(title = "A. Unconditional", x = "", y="")
p2 <- ggplot(data = nonoil, mapping = aes(x=(mean(log(gdp60))+controlx1res), y=(mean(gdpgrowth)+controly1res))) + geom_point() + geom_smooth(method = "lm") + labs(title = "B. Conditional on Savings and Population Growth", x = "", y="")
p3 <- ggplot(data = nonoil, mapping = aes(x=(mean(log(gdp60))+controlx2res), y=(mean(gdpgrowth)+controly2res))) + geom_point() + geom_smooth(method = "lm") + labs(title = "C. Conditional on Savings, Population Growth and Human Capital", x = "", y="")
grid.arrange(p1, p2, p3, ncol=1, top = "Unconditional versus Conditional Convergence", left = "% Growth Rate of GDP 1960-1985", bottom = "Log GDP per capita in 1960")
posted on 2021-06-18 20:19 Albert_Lei 阅读(394) 评论(0) 收藏 举报
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