# Good criteria to sort state-space $\beta_{t}$ according to Kalman filter output

Let's assume the usual state-space linear model without constant term for simplicity:

$$y_{t}=\beta_{t} X_{t}+\epsilon_{t}$$

If we apply Gaussian Kalman filter to estimate $$\beta_{t}$$ we get $$P_{t}$$, which is the covariance matrices of predicted states, and $$v_{t}$$, which is the prediction error.

The following simple R code allows you to download pair of tickers (QQQ and XLK for instance) from Yahoo Finance and estimate $$P_{t}$$ and $$v_{t}$$ while plotting them:

# ======================================== #
# Kalman filter errors and states variance #
# ======================================== #

Sys.setenv(TZ = 'UTC')

# Contents:

# 1. Installing packages
# 4. Kalman filtering of linear regression Beta

# *********************************
# 1. Installing packages
# *********************************

#install.packages('KFAS')
#install.packages('latticeExtra')
#install.packages('quantmod')

# *********************************
# *********************************

require(compiler)
require(latticeExtra)
require(KFAS)
require(quantmod)

# *********************************
# *********************************

Symbols <- c('QQQ', 'XLK')
getSymbols(Symbols, from = '1950-01-01')
data <- na.omit(merge(Cl(QQQ), Cl(XLK)))
colnames(data) <- Symbols
xyplot(data)

# *********************************
# 4. Kalman filtering of linear regression Beta
# *********************************

y <- na.omit(merge(ClCl(QQQ), ClCl(XLK)))[,1]
X <- na.omit(merge(ClCl(QQQ), ClCl(XLK)))[,2]
model <- regSSM(y = y, X = X, H = NA, Q = NA)
object <- fitSSM(inits = rep(0, 2), model = model)$$model KFAS <- KFS(object = object) P <- xts(as.vector(KFAS$$P)[-1], index(y))
v <- xts(t(KFAS$v), index(y)) Z <- cbind(P, v) colnames(Z) <- c('Covariance of predicted state', 'Prediction error') xyplot(tail(Z, 1000))  Now let's iterate this procedure over several pairs of securities to estimate their $$\beta_{t}$$, $$P_{T}$$ and $$v_{t}$$ and you want to sort these pairs by the stability and accuracy of $$\beta_{t}$$, that is, low variance and low prediction error. I would like to know suitable criteria to make this ranking system having available $$P_{t}$$ and $$v_{t}$$, i.e. how to penalize a linear relationship because of too high variance and prediction errors? For instance: Replacing QQQ and XLK in my code with VXX and TLT, you will see greater $$P_{t}$$ and $$v_{t}$$, which are linearly related between VXX and TLT and are more volatile and have weaker predictive power than the one between QQQ and XLK. This is similar to a ranking system and I would like to know how to produce some numeric criteria. • I am not exactly clear what your question is, do you mind rephrasing it a little or expanding on it? Thanks – Matthias Wolf Aug 14 '13 at 2:03 • @MattWolf, what's exactly that you find not clear in my question? I'll be happy to add further details. As instance: replace QQQ and XLK in my code with VXX and TLT, you will see greater$P_{t}$and$v_{t}$, that is, the linear relationship between VXX and TLT is more volatile and has weaker predictive power than the one between QQQ and XLK. This is like a ranking system, and I would like to know how to produce a numeric criterion. – Lisa Ann Aug 14 '13 at 6:08 • Ok, I get that but that is not related in any way to Kalman filters (other than that the prior algorithm involved a Kalman filter), correct? I was just confused because you spend 90% of your question on data acquisition and Kalman filter but your real question seems to be one of how to rank results. But maybe that is just my wrong impression – Matthias Wolf Aug 14 '13 at 6:16 • The attached code purpose is to start from a common output before answering my question. This common output is made up by$P_{t}$and$v_{t}\$ arrays. In fact, the most important part of my question lies at the end. – Lisa Ann Aug 14 '13 at 6:33
• well, what have you tried so far in R re your real question, result ranking? (Re earlier messages, do not get me wrong please, I do like your question and +1, just wanted to clarify...) – Matthias Wolf Aug 14 '13 at 6:43