0
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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 Nov 9 '19 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 Nov 9 '19 at 12:46
  • $\begingroup$ Appreciate, I actually forget to alter the matrix representation, this is my fault. $\endgroup$ – 連振宇 Nov 11 '19 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 Nov 13 '19 at 22:28
2
$\begingroup$

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 $\hat\beta_{\rm LSE} = (\tilde X' \tilde X)^{-1} \tilde X' \tilde y$.
$\endgroup$
  • 1
    $\begingroup$ 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 Nov 13 '19 at 22:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.