In the Multiple Regression Model with Two Explanatory Variables
Y_{i $\bar { Z }$}

question 4

Essay

In the multiple regression model with two explanatory variables
Y_i = ?₀ + ?₁X_1i + ?₂X_2i + u_i
the OLS estimators for the three parameters are as follows (small letters refer to deviations from means as in z_i = Z_i - $\bar { Z }$ ): $$\hat { \beta } _ { 0 } = \bar { Y } - \hat { \beta } _ { 1 } \bar { X } _ { 1 } - \hat { \beta } _ { 2 } \bar { X } _ { 2 }$$ $$\hat { \beta } _ { 1 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$$ $$\hat { \beta } _ { 2 } = \frac { \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } - \sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i } \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } } { \sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 } \sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 } - \left( \sum _ { i = 1 } ^ { n } x _ { 1 i } x _ { 2 i } \right) ^ { 2 } }$$ You have collected data for 104 countries of the world from the Penn World Tables and want to estimate the effect of the population growth rate (X_1i)and the saving rate (X_2i)(average investment share of GDP from 1980 to 1990)on GDP per worker (relative to the U.S.)in 1990. The various sums needed to calculate the OLS estimates are given below: $$\sum _ { i = 1 } ^ { n } Y _ { i }$$ = 33.33; $\sum _ { i = 1 } ^ { n } X _ { 1 i }$ = 2.025; $\sum _ { i = 1 } ^ { n } X _ { 2 i }$ =17.313 $\sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 }$ = 8.3103; $\sum _ { i = 1 } ^ { n } x _ { 1 i } ^ { 2 }$ = .0122; $\sum _ { i = 1 } ^ { n } x _ { 2 i } ^ { 2 }$ = 0.6422 $\sum _ { i = 1 } ^ { n } y _ { i } x _ { 1 i }$ = - 0.2304; $\sum _ { i = 1 } ^ { n } y _ { i } x _ { 2 i }$ = 1.5676; $\sum _ { i = 1 } ^ { n } x _ { 1 \mathrm { i } } x _ { 2 i }$ = -0.0520
The heteroskedasticity-robust standard errors of the two slope coefficients are 1.99 (for population growth)and 0.23 (for the saving rate). Calculate the 95% confidence interval for both coefficients. How many standard deviations are the coefficients away from zero?

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Definitions:

Dependent

Relying on someone or something else for support, care, or sustenance.

Limit Setting

The process of establishing clear boundaries and rules to guide acceptable behavior and protect the well-being of individuals and groups.

Provocative

Causing a strong reaction, often deliberately, by presenting challenging or stimulating ideas, situations, or information.

Hypervigilant

An enhanced state of sensory sensitivity accompanied by an exaggerated intensity of behaviors whose purpose is to detect threats.