Problem 1 :

In this problem we will replicate and reexamine the results from Table 3 of "Electoral Institutions, Cleavage Structures, and the Number of Parties", O. Neto and G. Cox, American Journal of Political Science, Vol. 41, No. 1 (1997). Refer to this paper for all variable definitions and for an explanation of the substantive question addressed by Table 3

1. Load the data file netocox.csv into Stata or R. You should find the variables ENPRES, ENETH, and RUNOFF. Use this data to replicate all results from Table 3 of the aforementioned paper.
2. Produce a scatterplot with the ENPRES variable on the \(y\) -axis and the ENETH variable on the \(x\) -axis. Add regression lines according to Model 1 in Table 3 , but use some method (e.g. different colors or line types) to differentiate between the regression lines for the countries with RUNOFF =0 and countries with RUNOFF =1. Include a caption or legend so that we can distinguish between the lines.
   Provide non-causal interpretations for the ENETH and RUNOFF coefficients from Model 1. Does this model make sense in terms of the theoretical discussion of the paper? Explain why or why not.
3. Produce a scatterplot with the ENPRES variable on the \(y\) -axis and the ENETH variable on the \(x\) -axis, but use different colors or symbols to indicate the runoff and the non-runoff countries. Add regression lines according to Model 3 in Table 3 and use some method (e.g. different colors or line types) to differentiate between the regression lines for the countries. Include a caption or legend so that we can distinguish between the lines.
   Provide non-causal interpretations of the slopes of these regression lines and the vertical distances between these lines. Also, discuss whether this model seems to fit the data well.
4. Produce a scatterplot with the ENPRES variable on the \(y\) -axis and the ENETH variable on the \(x\) -axis and use different colors or symbols to indicate the runoff and the non-runoff countries. Add regression lines according to Model 2 in Table 3 and use some method (e.g. different colors or line types) to differentiate between the regression lines for the countries. Include a caption or legend so that we can distinguish between the lines.

Exploring the RUNOFF "effect":

1. Provide a non-causal interpretation of the RUNOFF coefficient from Model 2 . Is this coefficient statistically significant at the \(5 \%\) level? Is this coefficient substantively meaningful? Explain why or why not.
2. Write an expression for the expected marginal "effect" of RUNOFF on ENPRES, conditional on ENETH. (The term "marginal effect" simply means the expected difference in predicted ENPRES given a change in RUNOFF, conditioning on the level of ENETH.)
3. Write an expression for the variance of the expected marginal "effect" function you calculated in Part 2 using the properties of variance. Note: this variance will also be conditional on ENETH.
4. Use you answer from Part 2 to calculate the RUNOFF marginal "effect" when ENETH is equal to 1.5. Calculate the estimated standard error for this marginal effect using your answer from Part 3 when ENETH is equal to 1.5. Calculate the \(95 \%\) confidence interval when ENETH is equal to 1.5.
5. Repeat the procedure from Part 4 with ENETH equal to 2 and discuss the differences in your findings.
6. Provide a non-causal interpretation of the result you produced in Part 5. Explain why it may be problematic to interpret this result causally.
7. Using your answer from Part 2, plot the marginal "effect" of RUNOFF as a function of ENETH, with ENETH on the \(\mathrm{x}\) -axis and the marginal "effect" on the \(\mathrm{y}\) -axis. Add to this plot the curves represented by the \(95 \%\) confidence interval for this marginal "effect": marginal "effect" \(\pm 1.96 \times \sqrt{\operatorname{Var}(\text { marginal effect) }}\), where the marginal "effect" is the linear function calculated in part 2 and the Var(marginal effect) is the variance function calculated in part 3 . What does this plot tell us about the statistical significance of RUNOFF conditional on ENETH?

Exploring the ENETH "effect":

1. Provide a non-causal interpretation of the ENETH coefficient from Model 2 . Is this coefficient statistically significant at the \(10 \%\) level? Is this coefficient substantively meaningful? Explain why or why not.
2. Estimate the expected difference in ENPRES between two countries with runoff elections that have a one unit difference in ENETH. Estimate the standard error for this estimated expected difference. Derive the p-value for a two-sided test of the null hypothesis that this expected difference is zero. Is this result statistically significant at the \(10 \%\) level?
3. Divide the data set into the countries with a runoff election and the countries without a runoff election. Within each of these separate data sets, run the regression of ENPRES on ENETH and report the results in a table.
4. Compare the ENETH coefficients from these two regressions with the estimates from Parts 1 and 2 above.
5. Compare the standard errors and significance of these ENETH coefficients with the standard errors and significance of the estimates in Parts 1 and 2 above.
6. The discrepancy in standard errors that you found in Part 5 of this question is due to a violation of which model assumption? (Hint: look closely at the scatterplot you produced for Model 2.) Present some kind of residual plot (or some other statistical summary of residuals) that clearly demonstrates the violation of this modeling assumption.

Problem 2: For this problem we will continue to use the data from Neto and Cox (1997) used in Problem 1 (netocox.csv).

1. Again replicate Model 1 from Table 3 in the Neto and Cox (1997) paper, by regressing ENPRES on ENETH and RUNOFF. Report the results in a well formatted table.
2. Consider the null hypothesis that \(\beta_{E N E T H}=0\) and the alternative hypothesis that \(\beta_{E N E T H} \neq 0\). Calculate the t-test statistic and rejection region for this test. Can you reject this null hypothesis at the \(\alpha=0.05\) level?
3. Consider the null hypothesis that \(\beta_{R U N O F F}=0\) and the alternative hypothesis that \(\beta_{R U N O F F} \neq 0\). Calculate the t-test statistic and rejection region for this test. Can you reject this null hypothesis at the \(\alpha=0.05\) level?
4. Consider the null hypothesis that \(\beta_{E N E T H}=0\) and \(\beta_{R U N O F F}=0\). Calculate the F-test statistic and rejection region for this test. Can you reject this null hypothesis at the \(\alpha=0.05\) level?
5. Gov 2000/E-2000 Only Write a function to calculate the F-test statistic for the null hypothesis that all coefficients from a regression model are equal to 0 . In addition your function should also output whether we reject or fail to reject the null hypothesis for any significance level (Note: one of the inputs of your function should be the desired significance level).

Problem 3: Again use the netocox.csv data for this question. Regress ENPRES on ENETH and report your results. Consider the null hypothesis that \(\beta_{E N E T H}=0\) and the alternative hypothesis that \(\beta_{E N E T H} \neq 0\). Calculate the F-test statistic and its corresponding \(\mathrm{p}\) -value for this test. Calculate the t-test statistic and its corresponding p-value for this test. Compare the two p-values and explain intuitively what this tells you about the relationship between an F-statistic [used to test significance] for a single coefficient and the t-statistic used to test significance for the same coefficient.

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