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For the linear regression model,

  y_i = beta_0 + summation(x_i,j * beta_j )+ e_j     

from the above equation, How I will find the value of beta(beta_0,beta_1...beta_j)?

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    $\begingroup$ If you augment your $X$ matrix by a column of 1s (to account for $\beta_0$), then you can get the entire vector of $\beta$ with the estimation $\frac{X^ty}{X^tX}$. $\endgroup$
    – Ami Tavory
    Jul 1, 2017 at 5:52
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    $\begingroup$ Yes, that's the standard way of getting the coefficients in OLS. $\endgroup$
    – Ami Tavory
    Jul 1, 2017 at 6:16
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    $\begingroup$ @AmiTavory You shouldn't use a division line to denote matrix multiplication by an inverse. Since matrix multiplication is not commutative, the meaning of your expression is ambiguous. $\endgroup$ Jul 1, 2017 at 6:18
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    $\begingroup$ @SudipDas If you add an l1 penalty, beta will tend to become sparse (under certain conditions), but the solution is not closed form. I suggest you read up on Elements Of Statistical Learning, which has a free online version at the authors' website. $\endgroup$
    – Ami Tavory
    Jul 1, 2017 at 6:43
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    $\begingroup$ @SudipDas No problem. All the best. $\endgroup$
    – Ami Tavory
    Jul 1, 2017 at 7:18

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