Change regression model ($x^*_i = x_i -10$)

[valentina89]

Hi there.:wave:
I'm solving an exercise on multiple linear regression. Near the end I will be asked for the same data as the previous model, it is the maximum likelihood estimates.

The previous model:
I have the matrix \((X'X)^{-1}\) and the matrix \(X'y\) and the model is:

\(Y_i = B_0 + B_1x_i + B_2x^2_i + e_i\)

\(i= 1,...,10\)

Now I have:

\(Y_i = g_0 + g_1x^*_i + g_2(x^*_i)^2 + e_i\)

\( x^*_i = x_i -10\)

\(i=1,...,10\)
To transform the model can I decrease the Matrix data for 10?

The same thing can I make it also for the matrix \(X'y\)?
I have many doubts.
Publishing the text so it is more understandable.

Consider the regression model linear:

\(Y_i = B_0 + B_1x_i + B_2x^2_i + e_i\)

with \( e_1, ...., e_n \) independent and identically distributed random variables and x_i, i = 1, ..., 0 constant fix.

The only data I have are these:

https://s32.postimg.org/o20r1i5et/Immagine.png

I have to rewrite the whole thing ... I tried so:

\((X) =
\begin{bmatrix}
1 & x_1 & x^2_1\\
. & . & . \\
. & . & . \\
. & . & . \\
1 & x_{10} & x^2_{10}\\
\end{bmatrix}\)

\((X^*) =
\begin{bmatrix}
1 & x_1-10 & (x_{1}-10)^2\\
. & . & . \\
. & . & . \\
. & . & . \\
1 & x_{10}-10 & (x_{10}-10)^2\\
\end{bmatrix}\)

\(X^{*'}X^* =
\begin{bmatrix}
10 & \sum_{i=1}^{10} x_i-10 & \sum_{i=1}^{10}(x_{i}-10)^2\\
\sum_{i=1}^{10} x_i-10 & \sum_{i=1}^{10} (x_{i}-10)^2 & \sum_{i=1}^{10}(x_{i}-10)^3\\
\sum_{i=1}^{10} (x_{i}-10)^2 & \sum_{i=1}^{10}(x_{i}-10)^3 & \sum_{i=1}^{10}(x_{i}-10)^4\\
\end{bmatrix}\)

Now I have to calculate the inverse? I'm following proper solution?


Excuse me , but does not write well in English
Thanks.

 

[Dason]

Are you just trying to find the estimates of the parameters for the model using the transformed data? I would just figure out what the new parameters are in terms of the old parameters and then use the invariance property of MLEs to get the estimates.

 

[valentina89]

The question is:
Consider , for the same data , the model

\(Y_i = g_0 + g_1x^*_i + g_2(x^*_i)^2 + e_i\) with \(x ^* _i = x_i - 10 , i = 1 , . . . , 10\) , independent random variables and identical cally distributed N ( 0 , σ2 ) . It is the maximum likelihood estimates γ0 , γ1 , γ2.

 

[valentina89]

and then use the invariance property of MLEs to get the estimates.

In this case the estimate remains the same?

\( B_0 = g_0 \) ecc..
I do not have a real transformation of the initial parameters , but only a "data processing " . What I did is right ?

 

[Dason]

Consider just a simple linear regression and the transformation you had. Our original model would be
\( E[y] = \beta_0 + \beta_1x_i\)

With \(x_i^* = x_i - 10\) our model is

\( E[y] = \beta_0^* + \beta_1^*x_i^*\)

Replace with our definition of \(x_i^*\) to get

\( E[y] = \beta_0^* + \beta_1^*(x_i - 10)\)

Expand and rearrange and we have

\( E[y] = \beta_0^* - 10\beta_1^* + \beta_1^*x_i\)

Now since we already know \(E[y] = \beta_0 + \beta_1x_i\)

we can now say that

\(\beta_0 = \beta_0^* - 10\beta_1^*\)
and
\(\beta_1 = \beta_1^*\)

So using this (since we're assuming we know the values for \(\beta_0\( and \(\beta_1\) we can solve for the new coefficients. It's the process with your problem but since you have a quadratic the equations take a little bit more work.\)\)

 

[valentina89]

Wow!

Thank you so much.
I try now to do the exercise right!