Finding the least squares estimates of $β1$ and $β2$

Question

I am having trouble beginning the below question.

Suppose that $Y_1, Y_2, Y_3$ are uncorrelated random variables with common variance $σ^2$ and expected values $2β_1 + 2β_2$, $β_1 −2β_2$, and $2β_1 −β_2$, respectively. let $y_1,y_2,$ and $y_3$ be the realisations of $y_1, y_2, y_3$.

b) Find the least squares estimates of $β_1$ and $β_2$.

I am unsure as to begin this as a simple linear regression problem or a multiple linear regression problem.

Do I start with $Y_i = β_1 + β_2X_i + e_i$

Or $Y_i = β_0 + β_1X_i + β_2X_i + e_i$

Answer 1

I think you should list your model as $Y_i = β_1X_{i1} + β_2X_{i2} + e_i$ without a intercept. Your equations will look like:

$\begin{pmatrix} Y_1\\ Y_2\\ Y_3 \end{pmatrix}=\begin{pmatrix} 2 & 2\\ 1 & -2 \\ 2 & -1 \end{pmatrix}\begin{pmatrix} \beta_1\\ \beta_2 \end{pmatrix}+\begin{pmatrix} e_1 \\ e_2\\ e_3 \end{pmatrix}$

You have your design matrix without intercept, otherwise you need a column of 1s then your expected values of $Y_i$ will have the formats $1*\beta_1+a*\beta_2$, $a$ can be different values.

The least square estimation of $\beta_1,\beta_2$ will be$(X'X)^{-1}X'Y$

Suppose you realization of $y_1, y_2, y_3$ is$(4,5,6)$

Then use the following R code you can calculate the least square estimate $\beta_1, \beta_2$

X<-matrix(c(2,2,1,-2,2,-1),nrow=3,ncol=2)
Y<-matrix(c(4,5,6),nrow=3,ncol=1)
solve((t(X)%*%X))%*%t(X)%*%Y

 #       [,1]
 #  [1,]  2.65
 #  [2,] -0.15

Note I suppose your $e$ has a $N(0,\sigma^2)$ distribution.