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    Question WLS Confidence interval




    Guys can anyone assist?

    I've googled and read lots on the matter of WLS (mainly lecture notes from asorted university courses google finds). However I've not come across any that go in depth into WLS and specifically address CI and PI production.

    Im looking for how to calculate a confidence interval for the mean value at a point x_0 from a regression carried out using weighted least squares. Preferably from a reliable source I can quote to people at work if Im quizzed on the matter!
    The situation I am examining is combining data from a number of sources. To help me understand the situation I am starting with the scenario of having one good source of data (low e_y) and another not so good data source (highere_y) (but which it is desirable to include either due to a small amount of good data being available or because it reduces the amount of extrapolation from in a multivariable space).
    I think Ive got my head around WLS parameter estimation (Im visualiing it to myself as being conducted in a space where the each error term has been standardised by dividing by σ_i).
    However I now wish to make a prediction in real space for a vector of X.
    Under OLS \hat{\beta }=\left(X^T X \right)^{-1}X^T Y
    And the CI around the prediction for point x_0 follows a t-distrubution multiplied by s\sqrt{(x^T_o (X^T X) x_o)}
    and the PI s\sqrt{(x^T_o (X^T X) x_o +1)}
    For WLS \hat{\beta }=\left(X^T W^{-1} X \right)^{-1}X^TW^{-1} Y where W is a matrix with diagonal elements which are the (expected/estimated/guessed) variance of the residuals
    as its based on rescaling so that Y'_i=\frac{\alpha}{\sigma_i}+\frac {\beta x_i}{\sigma_i}+ \frac {\epsilon_i}{\sigma_i} this transformation makes the error term NIID(0,1). (Well it does if we are assuming it was normal independant and had variance of \sigma_i ^{2}

    But What happens to the CI and PI calculations?! \left(X^T W^{-1} X \right)^{-1} seems like a logical replacement for \left(X^T X \right)^{-1} (which. if the data is mean cantered. replaces a variance/covariance matrix with a correlation matrix), but x_0 is not standardised so x^T_o (X^T W^{-1} X) x_o would not be correct. I could divide x_0 by some value, maybe some sort of average?
    (But then what happens if the Hetroscedasticity is a function of X for some sort of systematic issue or W contains covariance as well as variance terms?)
    But I cant be the first person want to do this? So could anyone give me some pointers?!

    I'm happy that once the CI has been estimated adding an appropriate amount of variance for the independant "error term" should handle the PI.

    (Sorry for the long post!)

    Many Thanks
    JP
    Last edited by dj_johnphillips; 06-23-2015 at 10:42 AM.

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    Re: WLS Confidence interval

    Additional note, I did check my logic by trying W= nI which confimed my CI was out by a factor of n as I expected. I'm stumped on how to handle it when W is more complex!

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    Re: WLS Confidence interval

    Quote Originally Posted by dj_johnphillips View Post
    Additional note, I did check my logic by trying W= nI which confimed my CI was out by a factor of n as I expected.
    What exactly do you mean by this?
    I don't have emotions and sometimes that makes me very sad.

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    Re: WLS Confidence interval


    Thanks for the reply.

    Quote Originally Posted by Dason View Post
    What exactly do you mean by this?
    Sorry was having notation trouble and running out of time at work!

    To check my logic I tried using different weighting matrices which were diagonal with values n to examine what impact that had on the CI calculations. As expected as it increased from 1 the CI calculations went wonky along with the estimate of e_y but the values of the coefficents and their standard errors remained the same. Once the value on the diagonals reached the OLS estimated e_y the WLS gave a value of e_y of 1 because I'd rescaled the points to effectively standardise the regression.
    Last edited by dj_johnphillips; 06-24-2015 at 02:14 AM.

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