# StatisticsIn this problem we w

StatisticsIn this problem we will return to the dataset from Electoral Institutions, Cleavage Structures, and theNumber of Parties?, O. Neto and G. Cox,American Journal of Political Science, Vol. 41, No. 1 (1997)that we used in Problem Set 7. Refer to this paper for all variable de nitions and for an explanation of thesubstantive question addressed by Table 3 (The paper is available on the course website).A) Load the data lenetocox.csvinto R. You should nd the variables ENPRES, ENETH, and RUNOFF.Use this data to replicate the results from Model 2, Table 3 in the paper. Present your results (pointestimates and standard errors) in a neatly formatted table with an informative caption.B) 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 2 in Table 3, but use some method (e.g. di erent colors or linetypes) to di erentiate between the regression lines for the countries with RUNOFF = 0 and countries withRUNOFF = 1. Include a caption or legend so that we can distinguish between the lines.C)i) Provide a non-causal interpretation of the RUNOFF coe cient from Model 2. Is this coe cient sta-tistically signi cant at the 5% level? Is this coe cient substantivelymeaningful? Explain why or whynot.1ii) Write an expression for the expected marginal e ect? of RUNOFF on ENPRES, conditional onENETH. (The term marginal e ect? simply means the expected di erence in predicted ENPRES givena change in RUNOFF, conditioning on the level of ENETH.) Write an expression for the variance of theexpected marginal e ect.?iii) Use your answer from Part ii to calculate the RUNOFF marginal e ect? when ENETH is equal toits sample mean. Calculate the estimated standard error for this marginal e ect and form a 95% con denceinterval for it.D)i) Using your regression from A, calculate the expected average change in ENPRES for a unit change inENETH when RUNOFF = 0. Calculate the expected average change in ENPRES for a unit change inENETH when RUNOFF = 1. Compute the standard errors of these estimates and report the results in aneatly formatted table.ii) On Problem Set 7, you estimated the regression of ENPRES on ENETH for two subsets of the data(when RUNOFF = 1 and RUNOFF = 0). Table 1 shows the results of those two regressions. Compare thepoint estimates and standard errors in i) with the estimates and standard errors reported in table 1.Runo = 0 Runo = 1Intercept 4.303 1.812(.6498) (1.262)Eneth -.979 1.026(.407) (0.707)Note: Standard errors in parenthesesTable 1: Bivariate regression of e ective number of parties on ethnic heterogeneity for divided data.iii) The discrepancy in standard errors that you nd in ii) 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 someother statistical summary of residuals) that clearly demonstrates the violation of this modeling assumption.Problem 2 (Gov 2000/1000/Stat E-190 Only)Again, consider the results you obtained in Problem 1 replicating Model 2, Table 3 of Neto and Cox (1997).A) Consider the null hypothesis that ENETH = 0 and the alternative hypothesis that ENETH 6= 0.Calculate the t-test statistic and rejection region for this test. Can you reject this null hypothesis at the= 0:05 level?2B) Calculate the F-test statistic for the hypothesis test you conducted in part A and the corresponding p-value for this test. Compare the two p-values and explainintuitively what this tells you about the relationshipbetween an F-statistic [used to test signi cance] for a single coe cient and the t-statistic used to testsigni cance for the same coe cient.C) Consider the null hypothesis that ENETH= 0,RUNOFF= 0, ENETH RUNOFF= 0 (that is, thatall coe cients besides the intercept are 0). Calculate the F-test statistic and rejection region for this test.Can you reject this null hypothesis at the = 0:05 level?Problem 4 (Gov 2000e Only)Wooldridge 3.4, 4.6, 6.4!

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