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

#### valentina89

##### New Member
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

##### New Member
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.

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#### valentina89

##### New Member
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

##### New Member
Wow!

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