In the multiple regression model with two explanatory variables

Y _ { i } = \mathcal { P } _ { 0 } + \mathcal { p } _ { 1 } X _ { 1 i } + \mathcal { \rho } _ { 2 } X _ { 2 i } + 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 _ { 2 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 _ { 23 } } { \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 _ { 1 i }

)and the saving rate (

X _ { 2 i }

)(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
(a)What are your expected signs for the regression coefficient? Calculate the coefficients and see if their signs correspond to your intuition.
(b)Find the regression

R ^ { 2 }

, and interpret it. What other factors can you think of that might have an influence on productivity?

In the Multiple Regression Model, the Least Squares Estimator Is

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