rishi wrote: > Given two standard normal variables - e1 and e2 - both IID. > Given also x and y, with correlation p between them. > > Let x = e1 > > Then y = p*e1 + ((1-p)^.5)*e2 (I think) > > Here is my question: How do I extend this to the three variable case? > > In this case, there are three variables, x, y, and z with correlations > Pxy, Pxz, and Pyz. You can have as many gaussian e disturbances as you > want, but I think you'd only need e1, e2, and e3. > > I must be stupid, but I can't generalize from the two to the three > variable case. Help!
First: Although your formula for y does cause x and y to have correlation p, it's wrong if you want y to be standard-normal, as you should be able to prove easily. In order to correct your expression in that case, write an expression for the variance of y = p*e1 + d*e2 and then solve for the value of d you'd need in order to make the variance of y equal unity. Second: The generalization that you're looking for can be worked out as follows. Suppose that you've generated jointly-normal r.v. x and y by the method you described and that you know the correlation (Pxy) and variances of x and y. Moreover, suppose that a third r.v. z is generated from z = a*x + b*y +c*e3, where e3 is standard-normal and uncorrelated with both x and y, which will be the case if e1, e2 and e3 are standard-normal and IID, for example. You should be able to write expressions for the variance of z, the correlation Pxz and the correlation Pyz in terms of a, b, c, Pxy, and the variances of x and y. Solving these expressions for a, b and c in terms of the other parameters indicates what values of a, b and c you'll need in order to get the set of variances and correlations you want. The way in which this approach can be extended from three to four r.v., from four to five, etc., should now be obvious. Charles Metz . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
