In loelschlaeger/fHMM: Fitting Hidden Markov Models to Financial Data

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.dim = c(10, 6),
  out.width = "80%",
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library("fHMM")

This vignette^[This vignette was build using R r paste(R.Version()[c("major", "minor")], collapse = ".") with the {fHMM} r utils::packageVersion("fHMM") package.] discusses model checking in {fHMM}, that means the task of verifying whether the fitted model describes the data well.

Model checking using pseudo-residuals

Since the observations are explained by different distributions (depending on the active state), model checking cannot be done by analyzing standard residuals. Instead, we consider "pseudo-residuals". To transform all observations on a common scale, we proceed as follows: If $X_t$ has the invertible distribution function $F_{X_t}$, then

\begin{align} Z_t=\Phi^{-1}(F_{X_t} (X_t)) \end{align}

is standard normally distributed, where $\Phi$ denotes the cumulative distribution function of the standard normal distribution. The observations, $(X_t)_t$, are modeled well if the so-called pseudo-residuals, $(Z_t)_t$, are approximately standard normally distributed, which can be visually assessed using quantile-quantile plots or further investigated using statistical tests such as the Jarque-Bera test [@zuc16].

For HHMMs, we first decode the coarse-scale state process using the Viterbi algorithm. Subsequently, we assign each coarse-scale observation its distribution function under the fitted model and perform the transformation described above. Using the Viterbi-decoded coarse-scale states, we then treat the fine-scale observations analogously.

Implementation

In {fHMM}, pseudo-residuals can be computed via the compute_residuals() function, provided that the states have been decoded beforehand.

We revisit the DAX example:

data(dax_model_3t)

The following line computes the residuals and saves them into the model object:

dax_model_3t <- compute_residuals(dax_model_3t)

The residuals can be visualized as follows:

plot(dax_model_3t, plot_type = "pr")

For additional normality tests, the residuals can be extracted from the model object via the residuals() method. The following lines exemplary perform a Jarque–Bera test [@jar87]:

tseries::jarque.bera.test(residuals(dax_model_3t))

References

loelschlaeger/fHMM documentation built on June 15, 2025, 9:30 a.m.