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I have two questions about multi linear regression model.

First question. Suppose 2 independent samples

Sample1 : $y_1$, ... $y_{n_1}$ and $x_1$, ..., $x_{n_1}$

Sample2 : $y_{n_1 +1}$, ... $y_{n_1 +n_2}$ and $x_{n_1 +1}$, ..., $x_{n_1 +n_2}$

and each samples fit in the models as follows :

Sample 1 : $y_i$ = $\beta_0$ + $\beta_1$$x_i$ + $\epsilon$ for i = 1,2,...,$n_1$

Sample 2 : $y_i$ = $\gamma_0$ + $\gamma_1$$x_i$ + $\epsilon$ for i = $n_1 +1$,...,$n_1+n_2$

I want to unite these two models into a single model. How can i do this? I have considered a multi linear regression model with two regressor x and x' where x is from sample1, x' is from sample2. but I can't deal with the constant part.

Second question.

In my text book, the definition of $SS_R$, the regression sum of square is ($\hat{y}-\bar{y})^t$($\hat{y}-\bar{y}$) using matrix notation and its degree of freedom is k which is the number of regressors in the model. but I have also seen that in the text book, it is written that the regression sum of square $SS_R (\beta)$ = $\hat{\beta}^tX^t$y which is equal to $\hat{y}^t$$\hat{y}$ and its degree of freedom is k+1. Why $SS_R$ has two different formula?

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1 Answer 1

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Hoping that I properly understood the first question : for each data point create a binary variable $z_i$ which is $1$ if the data point come from the first sample and $2$ if the data point come from the second sample. Since the model is multilinear, you now have to fit $$y=(a+bz)+(c+dz)x=\alpha+\beta z+\gamma x+\delta x z$$ so three regressors ($x,z,xz$).

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