Consider the Multiple Regression Model from Chapter 5, Where K $\hat{\beta}$

question 8

Essay

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 (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. 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 + $\hat { \beta }$ ₁X_1i + $\hat { \beta }$ ₂X_2i + $\hat {u}$ _i $\bar { Y }$ = $\hat { \beta }$ ₀ + $\hat { \beta }$ ₁
$\bar { X }$ ₁ + $\hat { \beta }$ ₂
$\bar { X }$ ₂.
Subtracting the second equation from the first, you get
y_i = $\hat { \beta }$ ₁x_1i + $\hat { \beta }$ ₂x_2i + $\hat {u}$ _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?

Understand the concept and implications of anxiety sensitivity.

Recognize effective nonpharmacologic treatments for panic attacks and the rationale behind them.

Identify cognitive explanations and biological aspects of panic disorders.

Comprehend the role and effect of neurotransmitters in panic disorder.

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Definitions:

Paradoxical Sleep

Another term for REM (Rapid Eye Movement) sleep, characterized by the paradox of the brain being highly active while the body is immobilized.

Retina

A light-sensitive layer at the back of the eyeball that contains cells responsive to visual stimuli.

Circadian Rhythm

A 24-hour internal clock cycle that regulates physiological and behavioral processes in living organisms.

Sleep Cycle

A progression through various stages of NREM and REM sleep that occurs several times during a typical night, important for restorative processes.