I have the following questions (all related):
Suppose we have variables $\mathrm{x}^1, \mathrm{x}^2, \mathrm{y}$ and they are normalized to have mean 0 and variance 1 . It is known that the sample Pearson correlation coefficient between $\mathbf{x}_1$ and $\mathbf{x}_2$ are $0.7$.
- Consider a univariate model, that is, $$ y_i=x_{i 1} \beta_1+\epsilon_{i 1} \quad \text { and } \quad y_i=x_{i 2} \beta_2+\epsilon_{i 2} . $$
The sample Pearson correlation coefficients (i) between $\mathbf{x}_1$ and $\mathbf{y}$ and (ii) between $\mathbf{x}_2$ and $\mathbf{y}$ are $0.3$ and $0.75$. Find the $\hat{\beta}_1$ and $\hat{\beta}_2$ obtained by least squares.
Hint: What do $\mathrm{x}_1^{\prime} \mathrm{y}$ and $\mathrm{x}_2^{\prime} \mathrm{y}$ mean?
2) Consider a multiple linear regression model, that is, $$ y_i=x_{i 1} \alpha_1+x_{i 2} \alpha_2+\epsilon_i . $$ Compute $\hat{\alpha}_1$ and $\hat{\alpha}_2$ when they are estimated using the least squares.
3) In general, which condition(s) of $\mathrm{x}^1$ and $\mathrm{x}^2$ are sufficient to have $\hat{\alpha}_1=\beta_1^{(1)}$ and $\hat{\alpha}_2=\beta_1^{(2)}$ ?
For 1), I am using the formula $\hat{\beta}=r\left(\frac{S_y}{S_x}\right)$ but I'm not sure if it applies to a linear regression model without an intercept.
For 2), I am thinking of using the multiple correlation formula but to no avail since I'm not sure how to get the regression coeffificnets into the formula:
$R=\sqrt{\frac{r_{y x_1}^2+r_{y x_2}^2-2 r_{y x_1} \cdot r_{y x_2} \cdot r_{x_1 x_2}}{1-r_{x_1 x_2}^2}}$
For 3), I am thinking it is when $\mathrm{x}^1$ and $\mathrm{x}^2$ are independent.
Any help and guidance is much appreciated, thank you!
