0
$\begingroup$

A table is given the data: table

Based on this table, we computed

$\sum_{i=1}^{n} X_{i1}^2 = 471$, $\sum_{i=1}^{n} X_{i2}^2 = 163.84$, $\sum_{i=1}^{n} X_{i1}X_{i2} = 235$

$\sum_{i=1}^{n} X_{i1}Y_i = 4915.3$, $\sum_{i=1}^{n} X_{i2}Y_i = 3103.66$

We consider the following model involving both independent variables and an intercept: $$Y_i = \beta_0 + \beta_1X_{i1} + \beta_2X_{i2} + \epsilon_i$$

where $\beta_j, j = 0,1,2$ are $3$ parameters and $\epsilon_i$ are pairwise indepedent random errors with mean $0$ and common variance $\sigma^2$. In the matrix notation, the model is

$$Y = X \beta + \epsilon$$

$$ X= \begin{bmatrix} 1 & 7 & 2.6\\ 1 & 1 & 2.9\\ 1 & 11 & 5.6\\ 1 & 11 & 3.1\\ 1 & 7 & 5.2\\ 1 & 11 & 5.5\\ 1 & 3 & 7.1 \end{bmatrix} $$

$$ Y= \begin{bmatrix} 78.5 \\ 74.3 \\ 104.3 \\ 87.6 \\ 95.9 \\ 109.2 \\ 102.7 \end{bmatrix} $$

The least square estimator of $\hat\beta$ of $\beta$ is

$\hat\beta = (X'X)^{-1}X'Y = \begin{bmatrix} 51.7 \\ 1.5 \\ 6.6 \end{bmatrix} $

question:

(a) write the estimated regression surface and interpret each regression coefficient in the context of the data

How do I do that? I have calculated most of $(X'X)$, $(X'X)^{-1}$, $X'Y$. But not sure how to answer this question.

$\endgroup$
0
$\begingroup$

The regression surface is given by

$$\hat{y}=51.7+1.5x_1+6.6x_2.$$

If you plot this bivariate function (independent variables: $x_1$ and $x_2$; dependent variable: $\hat{y}$) you will get something similar to this surface

enter image description here

The coefficients can be interpreted in the following way.

If $x_1$ and $x_2$ are zero then the output is given by the intercept/bias $51.7$ units, which is the coefficient $\beta_0$. If you only increase $x_1$ by one unit, then the output will increase by $1.5$ units, this is the coefficient $\beta_1$. And if you only increase $x_2$ by one unit, then the output will increase by $6.6$ units, this number is the coefficient $\beta_2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.