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Question is the following:

For analyzing the observed data, the following regression model \begin{eqnarray} y_i=\beta_0+\beta_1x_{i1}+\beta_2x_{i2}+\beta_3x_{i3}+\epsilon_i \end{eqnarray} is chosen by a researcher. When it is known that $\beta_2=4$, how can a researcher obtain the least squares estimates for all the other unknown parameters?

Well, in my viewpoint, since the parameter $\beta_2$ is known, thus we only need to get the other unknown parameter, that is, obtain the least squares estimator focus on $\beta_0,\beta_1,\beta_3$ only but not $\beta_2$, thus the LSE well be the same as usual, that is, \begin{eqnarray} \hat{\beta}_{LSE}=\hat{\beta_{0,1,3}}=(X^{T}X)^{-1}X^{T}y \end{eqnarray} Is this right?

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  • $\begingroup$ Hi: That's correct but note that the $X$ matrix and the $y$ vector change because the known piece has to be subtracted from both sides of the equation. Maybe this was obvious in which case, my apologies. $\endgroup$
    – mlofton
    Commented Nov 9, 2019 at 12:45
  • $\begingroup$ Subtract $\beta_2 x_{i2}$ from $y_i$ and regress this on the remaining covariates. The least squares estimator will be $(\tilde X’\tilde X)^{-1} \tilde X’ \tilde y$ where $\tilde X$ is the covariate matrix without $x_{i2}$ and $\tilde y$ is the outcome vector with $\beta_2 x_{i2}$ subtracted. $\endgroup$
    – Student
    Commented Nov 9, 2019 at 12:46
  • $\begingroup$ Appreciate, I actually forget to alter the matrix representation, this is my fault. $\endgroup$
    – 連振宇
    Commented Nov 11, 2019 at 1:06
  • $\begingroup$ This question is also answered at stats.stackexchange.com/a/434554/919 and stats.stackexchange.com/a/136602/919. $\endgroup$
    – whuber ♦
    Commented Nov 13, 2019 at 22:28

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You are right that under the constraint that $\beta_2 = 4$, you do not need to estimate $\beta_2$. But you do need to apply the constraint.

  • Step 1: subtract $\beta_2 x_{i2}$ from $y_i$. Denote the transformed outcome by $\tilde y \equiv y_i - 4 x_{i2}$.
  • Step 2: remove the column correspond to $x_{i2}$ from the covariate matrix $X$. Denote the resulting covariate matrix by $\tilde X$.
  • Step 3: calculate the constrained least squares estimates as $\hat\beta_{\rm LSE} = (\tilde X' \tilde X)^{-1} \tilde X' \tilde y$.
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    $\begingroup$ It is equivalent, and more practical, to use an offset term in the model formula (in R, should exist also in other systems.) y ~ x1 + offset(I(4*x2)) + x3, data=your_data_frame, ...) $\endgroup$
    – kjetil b halvorsen ♦
    Commented Nov 13, 2019 at 22:05

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