# Question

An analyst at the United Nations is developing a model that describes GDP (gross domestic product per capita, a measure of the overall production in an economy per citizen) among developed countries. For this analysis, she uses national data for 30 countries from the 2005 report of the Organization for Economic Cooperation and Development (OECD). Her current equation is

Estimated per capita GDP = β0 + β1 Trade Balance

+ β2 Waste per capita

Trade balance is measured as a percentage of GDP. Exporting countries tend to have large positive trade balances. Importers have negative balances. The other explanatory variable is the annual number of kilograms of municipal waste per person.

(a) Is there any natural reason to expect these explanatory variables to be correlated? Suppose the analyst had formulated her model using national totals as

GDP = β0 + β1 Net Export ($)

+ β2 Total Waste (kg) + e

Would this model have more or less collinearity? (You should not need to explicitly form these variables to answer this question.)

(b) One nation is particularly leveraged in the marginal relationship between per capita GDP and trade balance. Which is it?

(c) Does collinearity exert a strong influence on the standard errors of the estimates in the analyst’s multiple regression?

(d) Because multiple regression estimates the partial effect of an explanatory variable rather than its marginal effect, we cannot judge the effect of outliers on the partial slope from their position in the scatterplot of y on x. We can, however, see their effect by constructing a plot that shows the partial slope. To do this, we have to remove the effect of one of the explanatory variables from the other variables. Here’s how to make a so-called partial regression leverage plot for these data.First, regress per capita GDP on per capita Waste and save the residuals. Second, regress Trade Balance on per capita Waste and save these residuals. These regressions remove the effects of waste from the other two variables. Now, make a scatterplot of the residuals from the regression of per capita GDP on per capita Waste on the residuals from the regression of Trade Balance on per capita Waste. Fit the simple regression for this scatterplot, and com-pare the slope in this ft to the partial slope for Trade Balance in the multiple regression. Are they different?

(e) Which nation, if any, is leveraged in the partial regression leverage plot constructed in part (d)? What would happen to the estimate for this partial slope if the outlier were excluded?

Estimated per capita GDP = β0 + β1 Trade Balance

+ β2 Waste per capita

Trade balance is measured as a percentage of GDP. Exporting countries tend to have large positive trade balances. Importers have negative balances. The other explanatory variable is the annual number of kilograms of municipal waste per person.

(a) Is there any natural reason to expect these explanatory variables to be correlated? Suppose the analyst had formulated her model using national totals as

GDP = β0 + β1 Net Export ($)

+ β2 Total Waste (kg) + e

Would this model have more or less collinearity? (You should not need to explicitly form these variables to answer this question.)

(b) One nation is particularly leveraged in the marginal relationship between per capita GDP and trade balance. Which is it?

(c) Does collinearity exert a strong influence on the standard errors of the estimates in the analyst’s multiple regression?

(d) Because multiple regression estimates the partial effect of an explanatory variable rather than its marginal effect, we cannot judge the effect of outliers on the partial slope from their position in the scatterplot of y on x. We can, however, see their effect by constructing a plot that shows the partial slope. To do this, we have to remove the effect of one of the explanatory variables from the other variables. Here’s how to make a so-called partial regression leverage plot for these data.First, regress per capita GDP on per capita Waste and save the residuals. Second, regress Trade Balance on per capita Waste and save these residuals. These regressions remove the effects of waste from the other two variables. Now, make a scatterplot of the residuals from the regression of per capita GDP on per capita Waste on the residuals from the regression of Trade Balance on per capita Waste. Fit the simple regression for this scatterplot, and com-pare the slope in this ft to the partial slope for Trade Balance in the multiple regression. Are they different?

(e) Which nation, if any, is leveraged in the partial regression leverage plot constructed in part (d)? What would happen to the estimate for this partial slope if the outlier were excluded?

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