Consider a linear regression model, i.e., Y = β 0 + β 1 x i + ϵ i , where ϵ i satisfies the classical assumptions. The estimation method of the coefficients ( ( β 0 , β )) is the least-squared method. What would be an intuitive explanation of why the sum of residuals is 0?

Question

Consider a linear regression model, i.e.,   Y  =      β    0    +      β    1        x    i    +      ϵ    i  , where       ϵ    i   satisfies the classical assumptions. The estimation method of the coefficients (  (      β    0    ,  β  )) is the least-squared method. What would be an intuitive explanation of why the sum of residuals is   0?

Barbara Klein · Accepted Answer

The residuals should sum to zero. Notice this is the same as the residuals having zero mean. If the residuals did not have zero mean, in effect the average error is not zero in the sample. Thus an easy way to get a better estimate of the desired parameter is to subtract out this average error from our estimate.

metodystap9 · Accepted Answer

A number       x    i  , is equal to the mean of all data       x    ¯   plus its residue       r    i  :      x    i    =      x    ¯    +      r    i  If the sum of all residuals was not R=0 then the mean of data contradicts:  R  =  0      x    ¯    =      ∑          x      i          /    n    =    ∑    (            x      ¯        +          r      i        )      /    n    =            x      ¯        +    n    ∑          r      i        =            x      ¯        +    n    R    →            x      ¯        =            x      ¯        +    R    →  R  =  ∑      r    i    =  0This means the sum of the distance between values and their mean is zero because otherwise, mean is a noncentral parameter and as a central parameter contradicts.The mean&#039;s intrinsic property is having central tendency and this means it is equal to all data being centered.

Consider a linear regression model, i.e., Y=beta_0+beta_1 x_i+epsilon_i, where epsilon_i satisfies the classical assumptions. The estimation method of the coefficients (beta_0,beta) is the least-squared method. What would be an intuitive explanation of why the sum of residuals is 0?

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