fitGSMAR: Estimate Gaussian or Student's t Mixture Autoregressive model

Description

fitGSMAR estimates GMAR, StMAR or G-StMAR model in two phases: in the first phase, the genetic algorithm is employed to find starting values for the gradient based variable metric algorithm (also known as quasi-Newton method). In the second phase, the variable metric algorithm accurately converges to a nearby local maximum or saddle point. Parallel computing is used to perform multiple rounds of estimations in parallel.

Usage

fitGSMAR(data, p, M, model = c("GMAR", "StMAR", "G-StMAR"),
  restricted = FALSE, constraints = NULL, conditional = TRUE,
  parametrization = c("intercept", "mean"), ncalls = round(10 + 9 *
  log(sum(M))), ncores = min(ncalls, parallel::detectCores()),
  maxit = 300, printRes = TRUE, runTests = FALSE, ...)

Arguments

data

a numeric vector class 'ts' object containing the data. NA values are not supported.

a positive integer specifying the order of AR coefficients.

For GMAR and StMAR models:: a positive integer specifying the number of mixture components.
For G-StMAR model:: a size (2x1) vector specifying the number of GMAR-type components M1 in the first element and StMAR-type components M2 in the second. The total number of mixture components is M=M1+M2.

model

is "GMAR", "StMAR" or "G-StMAR" model considered? In G-StMAR model the first M1 components are GMAR-type and the rest M2 components are StMAR-type.

restricted

a logical argument stating whether the AR coefficients $\phi_{m,1},...,\phi_{m,p}$ are restricted to be the same for all regimes.

constraints

specifies linear constraints applied to the autoregressive parameters.

For non-restricted models:: a list of size $(pxq_{m})$ constraint matrices $C_{m}$ of full column rank satisfying $\phi_{m}$$=$$C_{m}\psi_{m}$ for all $m=1,...,M$, where $\phi_{m}$$=(\phi_{m,1},...,\phi_{m,p})$ and $\psi_{m}$$=(\psi_{m,1},...,\psi_{m,q_{m}})$.
For restricted models:: a size $(pxq)$ constraint matrix $C$ of full column rank satisfying $\phi$$=$$C\psi$, where $\phi$$=(\phi_{1},...,\phi_{p})$ and $\psi$$=\psi_{1},...,\psi_{q}$.

Symbol $\phi$ denotes an AR coefficient. Note that regardless of any constraints, the nominal order of AR coefficients is alway p for all regimes. Ignore or set to NULL if applying linear constraints is not desired.

conditional

a logical argument specifying whether the conditional or exact log-likelihood function should be used.

parametrization

is the model parametrized with the "intercepts" $\phi_{m,0}$ or "means" $\mu_m = \phi_{m,0}/(1-\sum\phi_{i,m})$?

ncalls

a positive integer specifying how many rounds of estimation should be conducted. The estimation results may vary from round to round because of multimodality of the log-likelihood function and randomness associated with the genetic algorithm.

ncores

a positive integer specifying the number of cores to be used in the estimation process. Default is that the number of available cores is detected with parallel::detectCores and all of them are used.

maxit

maximum number of iterations in the variable metric algorithm.

printRes

should the estimation results be printed?

runTests

should quantile residuals tests be performed after the estimation?

...

additional settings passed to the function GAfit employing the genetic algorithm.

Value

Returns an object of class 'gsmar' defining the estimated GMAR, StMAR or G-StMAR model. The returned object contains empirical mixing weights, conditional means and variances, quantile residuals, and quantile residual test results if the tests were performed. Note that the first p observations are taken as the initial values so mixing weights, conditional moments and qresiduals start from the p+1:th observation (interpreted as t=1). In addition, the returned object contains the estimates and log-likelihood values from all the estimation rounds. The estimated parameter vector can be obtained at gsmar$params (and the corresponding approximate standard errors at gsmar$std_errors) and it's...

For non-restricted models:

For GMAR model:: Size $(M(p+3)-1x1)$ vector $\theta$$=$($\upsilon_{1}$,...,$\upsilon_{M}$, $\alpha_{1},...,\alpha_{M-1}$), where $\upsilon_{m}$$=(\phi_{m,0},$$\phi_{m}$$, \sigma_{m}^2)$ and $\phi_{m}$=$(\phi_{m,1},...,\phi_{m,p}), m=1,...,M$.
For StMAR model:: Size $(M(p+4)-1x1)$ vector ($\theta, \nu$)$=$($\upsilon_{1}$,...,$\upsilon_{M}$, $\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M}$).
For G-StMAR model:: Size $(M(p+3)+M2-1x1)$ vector ($\theta, \nu$)$=$($\upsilon_{1}$,...,$\upsilon_{M}$, $\alpha_{1},...,\alpha_{M-1}, \nu_{M1+1},...,\nu_{M}$).
With linear constraints:: Replace the vectors $\phi_{m}$ with vectors $\psi_{m}$ and provide a list of constraint matrices C that satisfy $\phi_{m}$$=$$C_{m}\psi_{m}$ for all $m=1,...,M$, where $\psi_{m}$$=(\psi_{m,1},...,\psi_{m,q_{m}})$.

For restricted models:

For GMAR model:: Size $(3M+p-1x1)$ vector $\theta$$=(\phi_{1,0},...,\phi_{M,0},$$\phi$$, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1})$, where $\phi$=$(\phi_{1},...,\phi_{M})$.
For StMAR model:: Size $(4M+p-1x1)$ vector ($\theta, \nu$)$=(\phi_{1,0},...,\phi_{M,0},$$\phi$$, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{1},...,\nu_{M})$.
For G-StMAR model:: Size $(3M+M2+p-1x1)$ vector ($\theta, \nu$)$=(\phi_{1,0},...,\phi_{M,0},$$\phi$$, \sigma_{1}^2,...,\sigma_{M}^2,\alpha_{1},...,\alpha_{M-1}, \nu_{M1+1},...,\nu_{M})$.
With linear constraints:: Replace the vector $\phi$ with vector $\psi$ and provide a constraint matrix $C$ that satisfies $\phi$$=$$C\psi$, where $\psi$$=(\psi_{1},...,\psi_{q})$.

Symbol $\phi$ denotes an AR coefficient, $\sigma^2$ a variance, $\alpha$ a mixing weight and $\nu$ a degrees of freedom parameter. If parametrization=="mean" just replace each intercept term $\phi_{m,0}$ with regimewise mean $\mu_m = \phi_{m,0}/(1-\sum\phi_{i,m})$. In the G-StMAR model the first M1 components are GMAR-type and the rest M2 components are StMAR-type. Note that in the case M=1 the parameter $\alpha$ is dropped, and in the case of StMAR or G-StMAR model the degrees of freedom parameters $\nu_{m}$ have to be larger than $2$.

S3 methods

The following S3 methods are supported for class 'gsmar' objects: print, summary, plot, logLik, residuals.

Suggested packages

For faster evaluation of the quantile residuals of StMAR and G-StMAR models, install the suggested package "gsl". Note that for large StMAR and G-StMAR models with large data the evaluations of the quantile residual tests may take significantly long time without the package "gsl".

Details

Because of complexity and multimodality of the log-likelihood function, it's not guaranteed that the estimation algorithm will end up in the global maximum point. It's often expected that most of the estimation rounds will end up in some local maximum point instead, and therefore a number of estimation rounds is required for reliable results. Because of the nature of the models, the estimation may fail particularly in the cases where the number of mixture components is chosen too large.

If the iteration limit in the variable metric algorithm (maxit) is reached, one can continue the estimation by iterating more with the function iterate_more.

The genetic algorithm is mostly based on the description by Dorsey and Mayer (1995). It uses (slightly modified) individually adaptive crossover and mutation rates described by Patnaik and Srinivas (1994) and employs (50%) fitness inheritance discussed by Smith, Dike and Stegmann (1995). Large (in absolute value) but stationary AR parameter values are generated with the algorithm proposed by Monahan (1984).

The variable metric algorithm (or quasi-Newton method, Nash (1990, algorithm 21)) used in the second phase is implemented with function the optim from the package stats.

Some mixture components of the StMAR model may sometimes get very large estimates for degrees of freedom parameters. Such estimates may, for example, cause computing the quantile residual tests to fail. However, such mixture components are very much similar to the components of the GMAR model. It's hence advisable to further estimate a G-StMAR model by allowing the mixture components with large degrees of freedom parameter estimates to be GMAR type.

References

Dorsey R. E. and Mayer W. J. 1995. Genetic algorithms for estimation problems with multiple optima, nondifferentiability, and other irregular features. Journal of Business & Economic Statistics, 13, 53-66.
Kalliovirta L., Meitz M. and Saikkonen P. 2015. Gaussian Mixture Autoregressive model for univariate time series. Journal of Time Series Analysis, 36, 247-266.
Meitz M., Preve D., Saikkonen P. 2018. A mixture autoregressive model based on Student's t-distribution. arXiv:1805.04010 [econ.EM].
Monahan J.F. 1984. A Note on Enforcing Stationarity in Autoregressive-Moving Average Models. Biometrica 71, 403-404.
Nash J. 1990. Compact Numerical Methods for Computers. Linear algebra and Function Minimization. Adam Hilger.
Patnaik L.M. and Srinivas M. 1994. Adaptive Probabilities of Crossover and Mutation in Genetic Algorithms. Transactions on Systems, Man and Cybernetics 24, 656-667.
Smith R.E., Dike B.A., Stegmann S.A. 1995. Fitness inheritance in genetic algorithms. Proceedings of the 1995 ACM Symposium on Applied Computing, 345-350.
There are currently no published references for the G-StMAR model, but it's a straightforward generalization with theoretical properties similar to the GMAR and StMAR models.

Examples

Run this code

# NOT RUN {
# These are long running examples and use parallel computing

# GMAR model
fit12 <- fitGSMAR(data=logVIX, p=1, M=2, model="GMAR")
fit12
summary(fit12)
plot(fit12)

# Restricted GMAR model
fit12r <- fitGSMAR(logVIX, 1, 2, model="GMAR", restricted=TRUE,
 parametrization="mean", ncalls=10)
fit12r
summary(fit12r)

# Non-mixture version of StMAR model
fit11t <- fitGSMAR(logVIX, 1, 1, model="StMAR", ncores=1, ncalls=1)
fit11t

# StMAR model
fit12t <- fitGSMAR(logVIX, 1, 2, model="StMAR")
fit12t

# G-StMAR model with one GMAR type and one StMAR type regime
fit12gs <- fitGSMAR(logVIX, 1, M=c(1, 1), model="G-StMAR")
fit12gs

# Restricted G-StMAR model
fit12gsr <- fitGSMAR(logVIX, 1, c(1, 1), model="G-StMAR", restricted=TRUE)
fit12gsr

# The following three examples demonstrate how to apply linear constraints
# to the AR parameters.

# GMAR model that is a mixture of AR(1) and such AR(3) that the
# second AR coeffiecient is constrained to zero.
constraints <- list(matrix(c(1, 0, 0, 0, 0, 1), ncol=2), as.matrix(c(1, 0, 0)))
fit32c <- fitGSMAR(logVIX, 3, 2, constraints=constraints)
fit32c

# Such constrained StMAR(3, 1) model that the second order AR coefficient
# is constrained to zero.
constraints <- list(matrix(c(1, 0, 0, 0, 0, 1), ncol=2))
fit31tc <- fitGSMAR(logVIX, 3, 1, model="StMAR", constraints=constraints)
fit31tc

# Such StMAR(3, 2) model that the AR coefficients are restricted to be
# the same for both regimes and that the second AR coefficients are
# constrained to zero.
fit32trc <- fitGSMAR(logVIX, 3, 2, model="StMAR", restricted=TRUE,
                     constraints=matrix(c(1, 0, 0, 0, 0, 1), ncol=2))
fit32trc
# }

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