Pruned Skewed Kalman filter codes in R, Python, Julia and MATLAB

This is the documentation for the Pruned Skewed Kalman filter(PSKF) implementation in different languages.

This filter was introduced in the paper "Pruned Skewed Kalman Filter and Smoother: With Applications to the Yield Curve" by Gaygysyz Guljanov, Willi Mutschler and Mark Trede (2022). The complete replications codes for this paper is given at Willi Mutschler's github repository.

In this repo, the following functions are present for each language:

• phip(): Evaluates the univariate Normal probability density function
• phid(): Evaluates the univariate Normal cumulative density function
• logcdf_ME(): Evaluates the multivariate Normal cumulative density function
• dim_red4(): Reduces the skewness dimension of csn based on correlations
• kalman_csn(): Evaluates the log-likelihood using PSKF

phip(), phid() and logcdf_ME() functions are slightly modified versions of phip(), phid() and mcdfmvna_ME() functions by @dbauer72, Dietmar Bauer, see the repo.

All the functions above are used by kalman_csn().

Data generating process is the following state-space model:

$$Y_t = F X_t + \epsilon_t, ~~~ \epsilon_t \sim N(\mu_\epsilon, \Sigma_\epsilon), ~~~ \rightarrow \text{measurement equation}$$ $$X_t = G X_{t-1} + R \eta_t, ~~~ \eta_t \sim CSN( \mu_\eta, \Sigma_\eta, \Gamma_\eta, \nu_\eta, \Delta_\eta ), ~~~ \rightarrow \text{state equation}$$

$Y_t$ is y_nbr dimensional observable random variable.
$X_t$ is x_nbr dimensional latent random (state) variable.
$\epsilon_t$ is y_nbr dimensional Normal random variable.
$\eta_t$ is eta_nbr dimensional CSN random variable with skewness dimension of skeweta_dim.
Time $t$ runs up to obs_nbr.

Input arguments of kalman_csn() are roughly as follows:

• Y
  [y_nbr by obs_nbr]
  matrix with data
• mu_tm1_tm1
  [x_nbr by 1]
  initial value of location parameter of CSN distributed states x (does not equal expectation vector unless Gamma_tm1_tm1=0)
• Sigma_tm1_tm1
  [x_nbr by x_nbr]
  initial value of scale parameter of CSN distributed states x (does not equal covariance matrix unless Gamma_tm1_tm1=0)
• Gamma_tm1_tm1
  [skewx_dim by x_nbr]
  initial value of first skewness parameter of CSN distributed states x
• nu_tm1_tm1
  [skewx_dim by x_nbr]
  initial value of second skewness parameter of CSN distributed states x
• Delta_tm1_tm1
  [skewx_dim by skewx_dim]
  initial value of third skewness parameter of CSN distributed states x
• G:
  [x_nbr by x_nbr]
  state transition matrix mapping previous states to current states
• R:
  [x_nbr by eta_nbr]
  state transition matrix mapping current innovations to current states
• F:
  [y_nbr by x_nbr]
  observation equation matrix mapping current states into current observables
• mu_eta:
  [eta_nbr by 1]
  location parameter of CSN distributed innovations eta (does not equal expectation vector unless Gamma_eta=0)
• Sigma_eta:
  [eta_nbr by eta_nbr]
  scale parameter of CSN distributed innovations eta (does not equal covariance matrix unless Gamma_eta=0)
• Gamma_eta:
  [skeweta_dim by eta_nbr]
  first skewness parameter of CSN distributed innovations eta
• nu_eta:
  [skeweta_dim by 1]
  second skewness parameter of CSN distributed innovations eta
• Delta_eta:
  [skeweta_dim by skeweta_dim]
  third skewness parameter of CSN distributed innovations eta
• mu_eps:
  [y_nbr by 1]
  location parameter of normally distributed measurement errors eps (equals expectation vector)
• Sigma_eps:
  [y_nbr by y_nbr]
  scale parameter of normally distributed measurement errors eps (equals covariance matrix)
• logcdfmvna_fct:
  [Function]
  name of function with which the log of multivariate normal cdf is calculated
• cut_tol
  [scalar]
  correlation threshold to cut redundant skewness dimensions as outlined in the paper, if set to 0 no cutting will be done
• eval_lik [boolean]
  true: carries out the log-likelihood computations

Please report any bugs or weird behavior of the functions.