[R-SIG-Finance] PerformanceAnalytics - Style Analysis

[Eric Zivot]

You are right. The correct way to enforce that the regression coefficients
sum to unity is to impose this in the regression instead of just normalizing
the coefficients to sum to unity. It is not difficult to do this because it
is a linear restriction and the restriction can be substituted into the
linear regression to create another linear regression. I was supposed to add
this fix to the style.fit() function last year. I'll do it now and send the
fix off to Brian so he can put it in the PerformanceAnalytics package.
Briefly, consider the simple linear regression

Y = x1*b1 + x2*b2 + e

The restriction to impose is b1 + b2 = 1. Equivalently, b1 = 1 - b2.
Substituting into the regression gives

Y = x1*(1 - b2) + x2*b2 + e = x1 + b2*(x2 - x1) + e => Y - x1 = b2*(x2 - x1)
+ e

Therefore, the linear regression to run to impose the restriction b1+b2 = 1
is

(Y - x1) = b2*(x2 - x1) + e

Once you have b2.hat, then just solve for b1 using the constraint: b1.hat =
1 - b2.hat. (Note that it doesn't matter if you define the restriction in
terms of b1 = 1 - b2 or b2 = 1 - b1.)  Of course, to get the right R2 you
have to base it on Y as the dependent variable and not Y - x1. That is,
compute the R2 from the fitted values

Y.hat = b1.hat*x1 + b2.hat*x2 

Where b1.hat and b2.hat are the restricted least squares coefficients. 

Eric Zivot                  			               
Professor and Gary Waterman Distinguished Scholar       
Department of Economics                                 
Adjunct Professor of Finance                            
Adjunct Professor of Statistics
Box 353330                  email:  ezivot at u.washington.edu 
University of Washington    phone:  206-543-6715            
Seattle, WA 98195-3330
www:  http://faculty.washington.edu/ezivot                  




-----Original Message-----
From: r-sig-finance-bounces at stat.math.ethz.ch
[mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of Thomas Etheber
Sent: Tuesday, March 23, 2010 4:03 AM
To: r-sig-finance at stat.math.ethz.ch
Subject: [R-SIG-Finance] PerformanceAnalytics - Style Analysis

Dear List,

I had a look at the code of the PerformanceAnalytics package and I use 
this package on a regular basis.

The style.fit function for Style Analysis provides thrree supported 
methods, which can be chosen via a parameter. I am talking of the 
normalized method here. This method requires the regression coefficients 
to sum to 1.
Indeed the code in Version 1.0.0 says something like this:

[... Default implementation ...]
column.weights = as.data.frame(coef(column.lm))
[...]
 if (method == "normalized") {
                column.weights = column.weights/sum(column.weights)
 }

I suppose here we are just scaling the coefficients to sum to 1, in my 
view the regression should a priori deal with this restriction (some 
sort of constrained regression or the like).
At least if you look at Tabel 2 of Sharpe (1992) the coefficients do not 
support the actual implementation and I think somebody might want to 
have a look at this code fragment.
Unfortunately I do not know how to implement this kind of restricted 
regression in R, but I believe there are already ways of doing this. 
Perhaps somebody from the list can guide us to the right direction.

Anyway great thx to the authors of this packkage!
Thomas

PS: In Stata the right command seems to be cnsreg.

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