Sse sumi1ngdpi beta0 beta1polityi2 g now use linear

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SSE = $\sum_{i=1}^n(GDP_i-(\beta_0 + \beta_1Polity_i))^2$ ####(g) Now use linear regression to regress `gdpenl` on `polity2l` using the `lm` function in `R` - make sure you save the model as an object. Show the result using the `summary()` command. Interpret the meaning of the coefficient estimates (both the intercept and the coefficients on `polity2l`). Bonus: Consider the p-values reported on the table in your interpretation, if you want to read ahead and figure out what these mean. ```{r} model1 <- lm(gdpenl~ polity2l, data = fl2) summary(model1) ``` When the polity rating is 0 (the intercept), the estimated GDP is 2.46. For the `polity2l` coefficients, the estimated increase in GDP when there is a one-unit increase in polity is -0.011 (indicating a negative relationship/slope). P-Value: A probability value indicates how probable an outcome is under the curve. Outcomes with a P-value of 0.05 or lower are considered statistically significant and reject the null hypothesis. In this case the P-value at the intercept is 1.27e^-07, meaning that the data at the intercept is statistically significant. However, the P-value for polity2l is 0.86, meaning that it fails to reject the null hypothesis and that the overall data collected is not statistically significant . ####(h) Sometimes we need to transform a variable to make it more suitable to analysis by regression. For example, with income-related variables like `gdpenl`, we usually need to take their log first before using regression. Create a new variable that is equal to the log of `gpdenl`.
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```{r} log.gdpnl <- log(fl2$gdpenl) ``` ####(i) Now remake a scatter plot like before but with `polity2l` on the horizontal access and the log of `gdpenl` on the vertical axis. Add the regression line. ```{r} plot(log.gdpnl ~ polity2l, data = fl2, xlab = "Polity Rating", ylab = "Log of GDP per Capita", main = "Relationship between Levels of Democracy and Log of GDP per Capita") abline(lm(log.gdpnl ~ polity2l, data = fl2), col = "blue") ``` It turns out that when you regress a logged dependent variable on an (unlogged) independent variable, we can roughly interpret the coefficient $\beta$ as meaning "a one-unit shift in the independent variable corresponds to a 100$\beta$ *percent* increase in the dependent variable." For example, a $\beta$ of 0.01 from such a regression would imply that a one- unit change in the independent variable is associated with a $1\%$ higher value of the dependent variable. (This is just an approximation, but for coefficient estimates near zero, it is okay.) ####(j) Using this knowledge, re-run your regression but now regress the (log of) `gdpenl` on `polity2l`. Use `summary()` to show the results. Interpret the new coefficient on `polity2l`. How has it changed compared to your earlier regression? Would you say your results are robust? Bonus: Interpret the new p- value. ```{r} model2 <-lm(log.gdpnl ~ polity2l, data = fl2) summary(model2) ``` The new coefficient of the log of GDP per Capita is 0.049, indicating a weak positive relationship between (the log of) GDP per capita and the polity rating.
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