Kalman Filter Beta Data Ordering

Article Quant Q&A · Author: Lisa Ann

Summary

The document presents an example of estimating time-varying betas between several exchange-traded funds with a state-space model and Kalman filtering. It describes preparing adjusted price-change data, fitting a separate regression for each ordered pair of assets, and plotting the resulting beta series. The apparent problem is that some estimated relationships seem inconsistent with familiar views of how the funds co-move.

The accepted explanation points to data ordering rather than a flaw in the Kalman filter. Applying an explicit symbol order to the list returned by the data-preparation step ensures that each series is paired with the intended asset name. The response reports that, after this correction, the time-varying estimates broadly match the corresponding ordinary least squares beta. This is a practical warning that silent column-order changes can invalidate asset comparisons. The document gives no detailed statistical diagnostics or performance evaluation, and the result depends on the specified data and implementation.

Key ideas

A Kalman filter can estimate a regression beta that changes through time.
The example calculates pairwise beta series across a group of exchange-traded funds.
An unordered data extraction can associate price series with the wrong symbols.
Explicitly preserving symbol order resolves the reported counterintuitive beta results.
The document reports comparison with ordinary least squares but gives no broader validation.

Tags

Equities Statistics Machine learning

Full text

# Counterintuitive time varying Beta with Kalman filter


# Counterintuitive time varying Beta with Kalman filter












If you're used to play with `R`, you'll enjoy the following reproducible code:

```
# =================================================== #
# An example of state-space monitor via Kalman filter #
# =================================================== #

op <- par(no.readonly = TRUE)
Sys.setenv(TZ = 'UTC')

# Contents:

# 1. Installing packages
# 2. Loading packages
# 3. Custom functions
# 4. Downloading and preparing data
# 5. Cross-Betas Kalman filtering

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

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

# *********************************
# 2. Loading packages
# *********************************

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

# *********************************
# 3. Custom functions
# *********************************

# This function returns the time varying state-space representation 
# parameters of a linear model which represents y ~ X

Kalman.beta <- cmpfun(function(y, X)
{
  model <- regSSM(y = y, X = cbind(1, X), H = NA, Q = diag(NA, 2))
  object <- fitSSM(inits = rep(0, 3), model = model)$model
      KFAS <- KFS(object = object)
      alpha.beta <- xts(t(KFAS$alphahat), index(y))
  colnames(alpha.beta) <- rep(paste(colnames(y), 'vs' , colnames(X)), 2)
  return(alpha.beta)
})  

# *********************************
# 4. Downloading and preparing data
# *********************************

env <- new.env()
Symbols <- c('SPY', 'QQQ', 'XLF', 'TLT')
getSymbols(Symbols = Symbols, env = env, from = '1950-01-01')
args <- eapply(env = env, FUN = function(x){ClCl(x)})
X <- na.omit(do.call(what = merge, args = args))
colnames(X) <- Symbols
xyplot(X)

# *********************************
# 5. Cross-Betas Kalman filtering
# *********************************

Betas <- NULL
k <- 0

for(i in 1:ncol(X))
{
  for(j in (1:ncol(X))[-i])
  {
    k <- k + 1
    Betas[[k]] <- Kalman.beta(y = X[,i], X = X[,j])[,2]
  }
}

Beta.matrix <- do.call(what = merge, args = Betas)
colnames(Beta.matrix) <- gsub(pattern = '.', fixed = TRUE, 
                              x = colnames(Beta.matrix), replacement = ' ')
xyplot(Beta.matrix, superpose = FALSE, auto.key = FALSE,
       main = '')
```

What does this code do? It basically uses Kalman filter to estimate time varying $\beta_{t}$ of each asset against each other and plot them.

What's the matter with that?

If you use a simple linear regression model to estimate $\beta$ constant over time you will see it often happens, as instance, that $\beta_{t}<1<\beta$ or $\beta_{t}>0>\beta$ for the most of the time series... which is really counterintuitive!

How could SPY be negatively correlated with QQQ while it's quite obvious they are strongly correlated and $\beta \approx 1$? And so on...

How would you explain this?

Is there anything wrong with my code?

## Answer by Lisa Ann (score 2, accepted)

https://quant.stackexchange.com/a/8441

This is definitely not a Kalman filter's issue: if you replace this line of code

```
args <- eapply(env = env, FUN = function(x){ClCl(x)})
```

with this one

```
args <- eapply(env = env, FUN = function(x){ClCl(x)})[Symbols]
```

`eapply()` will keep the order of the original Yahoo query from `quantmod`. You can check and you will see each $\beta_{t}$ matches about the $\beta$ from simple OLS linear regression CAPM-like.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.

Fixing Asset Order in Kalman Filter Beta Estimates

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