Consider the multiple regression model from Chapter 5, where k = 2 and the assumptions of the multiple regression model hold.
(a)Show what the X matrix and the β vector would look like in this case.
(b)Having collected data for 104 countries of the world from the Penn World Tables, you want to estimate the effect of the population growth rate (X₁_i)and the saving rate (X₂_i)(average investment share of GDP from 1980 to 1990)on GDP per worker (relative to the U.S.)in 1990. What are your expected signs for the regression coefficient? What is the order of the (X'X)here?
(c)You are asked to find the OLS estimator for the intercept and slope in this model using the
formula

\hat{\beta}

= (X'X)^-¹ X'^Y. Since you are more comfortable in inverting a 2×2 matrix (the inverse of a 2×2 matrix is,

\left( \begin{array} { l l l l l } a&b\\c&d\end{array} \right)^{-1}

\frac { 1 } { a d - b c }

\left( \begin{array} { c c } d & - b \\- c & a\end{array} \right)

)
you decide to write the multiple regression model in deviations from mean form. Show what the X matrix, the (X'X)matrix, and the X'Y matrix would look like now.
(Hint: use small letters to indicate deviations from mean, i.e., z_i = Z_i -

\bar { Z }

and note that
Y_i =

\hat { \beta }

₀ +

\hat { \beta }

₁X₁_i +

\hat { \beta }

₂X₂_i +

\hat {u}

\bar { Y }

\hat { \beta }

₀ +

\hat { \beta }

₁

\bar { X }

₁ +

\hat { \beta }

₂

\bar { X }

₂.
Subtracting the second equation from the first, you get
y_i =

\hat { \beta }

₁x₁_i +

\hat { \beta }

₂x₂_i +

\hat {u}

_i)
(d)Show that the slope for the population growth rate is given by

\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_{2 i}^{2}-(\left.\sum_{i=1}^{n} x_{1 i} x_{2 i}\right)^{2}}

(e)The various sums needed to calculate the OLS estimates are given below:

\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
Find the numerical values for the effect of population growth and the saving rate on per capita income and interpret these.
(f)Indicate how you would find the intercept in the above case. Is this coefficient of interest in the interpretation of the determinants of per capita income? If not, then why estimate it?

The Linear Multiple Regression Model Can Be Represented in Matrix

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