Ran Wang
Linear Regression
\[ y_t=c+\alpha x_t \]
change
\[ y_1=c+\alpha x_1, y_0=c+\alpha x_0 \]
\[ y_1-y_0=\alpha (x_1-x_0) \]
\[ \Delta y~(increase~in~y)=\alpha \Delta x~(increase~in~x) \]
explaination
\( x \) changes by one unit in value, \( y \) will change by \( \alpha \) units in value.
Log-Linear Regression
\[ log(y_t)=c+\alpha x_t \]
change
\[ log(y_1)=c+\alpha x_1, log(y_0)=c+\alpha x_0 \]
\[ log(y_1)-log(y_0)=\alpha (x_1-x_0) \]
\[ \Delta y/y=\alpha \Delta x \]
explaination
\( x \) changes by one unit in value, \( y \) will change by \( \alpha \) units in percentage.
Linear-Log Regression
\[ y_t=c+\alpha log(x_t) \]
change
\[ y_1=c+\alpha log(x_1), y_0=c+\alpha log(x_0) \]
\[ y_1-y_0=\alpha (log(x_1)-log(x_0)) \]
\[ \Delta y=\alpha \Delta x/x \]
explaination
\( x \) changes by one unit in percentage, \( y \) will change by \( \alpha \) units in value.
Log-log Regression
\[ log(y_t)=c+\alpha log(x_t) \]
change
\[ log(y_1)=c+\alpha log(x_1), log(y_0)=c+\alpha log(x_0) \]
\[ log(y_1)-log(y_0)=\alpha (log(x_1)-log(x_0)) \]
\[ \Delta y/y=\alpha \Delta x/x \]
explaination
\( x \) changes by one unit in percentage, \( y \) will change by \( \alpha \) units in percentage.
Polynomial Regression
\[ y_t=c+\alpha x_t+\beta x_t^2 \]
change
\[ y_1=c+\alpha x_1+\beta x_1^2, y_0=c+\alpha x_0+\beta x_0^2 \]
\[ y_1-y_0=\alpha (x_1-x_0)+\beta(x_1^2-x_0^2) \]
\[ \Delta y=\alpha (x_1-x_0)+\beta(x_1+x_0)(x_1-x_0) \]
\[ \Delta y=(\alpha +\beta(x_1+x_0))(x_1-x_0) \]
\[ \Delta y=(\alpha +\beta(x_1+x_0))\Delta x \]
Polynomial Regression
\[ \Delta y=(\alpha +\beta(x_1+x_0))\Delta x \]
explaination
\( x \) changes by one unit in value, \( y \) will change by \( (\alpha +\beta(x_1+x_0)) \) units in value (The coefficient is a function of \( x_1,x_0 \)).
Polynomial Regression with Product Term
\[ y_t=c+\alpha x_t+\beta z_t+\gamma z_t\times x_t \]
change
Fix \( z \)
\[ y_1=c+\alpha x_1+\beta z_0+\gamma z_0\times x_1 \]
\[ y_0=c+\alpha x_0+\beta z_0+\gamma z_0\times x_0 \]
\[ y_1-y_0=\alpha(x_1-x_0)+\gamma z_0(x_1-x_0) \]
\[ \Delta y=(\alpha+\gamma z_0)\Delta x \]
Polynomial Regression with Product Term
\[ \Delta y=(\alpha+\gamma z_0)\Delta x \]
explaination
\( x \) changes by one unit in value, \( y \) will change by \( (\alpha +\gamma z_0) \) units in value (The coefficient is a function of \( z_0 \)).
Stata Code:
twoway (scatter y x)
Stata Code (with fitted line):
twoway (scatter y x) (lfit y x)
Does economic growth improve with more education and political stability?
Data: Growth.dta
Data Analysis:
Your dataset contains information on average growth rates for 65 countries between 1960 and 1995.
First, construct a scatterplot of \( Growth \) on \( YearsSchool \).
Then use the outreg2 command to create a table containing:
(1) \( Growth \) on \( TradeShare \) and \( YearsSchool \)
(2) \( Growth \) on \( TradeShare \) and \( log(YearsSchool) \)
(3) \( Growth \) on \( TradeShare \), \( log(YearsSchool) \), \( RevCoups \), \( Assasinations \) and \( log(RGDP60) \)
(4) \( Growth \) on \( TradeShare \), \( log(YearsSchool) \), \( RevCoups \), \( Assasinations \), \( log(RGDP60) \) and \( TradeShare\times log(YearsSchool) \)
(5) \( Growth \) on \( TradeShare \), \( TradeShare^2 \), \( TradeShare^3 \), \( log(YearsSchool) \), \( RevCoups \), \( Assasinations \) and \( log(RGDP60) \)
Write-Up
(a) Interpret the regression coefficients in regression (3) above.
(b) Using your plot, is the relationship between \( Growth \) and \( YearsSchool \) nonlinear?
( c) In 1960, a country contemplated an education policy that would increase average years of schooling from 4 to 6 years. What is the predicted increase in \( Growth \) according to regressions (1) and (2)? Explain why your predictions are different.
Write-Up
(d) Does the effect of \( TradeShare \) on \( Growth \) depends on the level of education (\( YearsSchool \))?
(e) Using regression (5), is the relationship between \( TradeShare \) and \( Growth nonlinear \)?
(f) In 1960, a country contemplated a trade policy that would increase the average value of \( TradeShare \) from 0.5 to 1. Use regressions (3) and (5) to predict the increase in Growth.