fit.pci: Fits the partial cointegration model to a collection of time...
In partialCI: Partial Cointegration

Description Usage Arguments Details Value Author(s) References See Also Examples

Fits the partial cointegration model to a collection of time series

fit.pci(Y, X, 
  pci_opt_method = c("jp", "twostep"), 
  par_model = c("par", "ar1", "rw"), 
  lambda = 0, 
  robust = FALSE, nu = 5)

`Y`	The time series that is to be modeled. A plain or `zoo` vector of length `n`.
`X`	A (possibly `zoo`) matrix of dimensions `n` x `k`. If `k=1`, then this may be a plain or `zoo` vector.
`pci_opt_method`	Specifies the method that will be used for finding the best fitting model. One of the following: "jp" The joint-penalty method (see below) "twostep" The two-step method (see below) Default: `jp`
`par_model`	The model used for the residual series. One of the following: "par" The residuals are assumed to follow a partially autoregressive model. "ar1" The residuals are assumed to be autoregressive of order one. "rw" The residuals are assumed to follow a random walk. Default: `par`
`lambda`	The penalty parameter to be used in the joint-penalty (`jp`) estimation method. Default: 0.
`robust`	If `TRUE`, then the residuals are assumed to follow a t-distribution with `nu` degrees of freedom. Default: `FALSE`.
`nu`	The degrees-of-freedom parameter to be used in robust estimation. Default: 5.

The partial cointegration model is given by the equations:

Y[t] = beta[1] * X[t,1] + beta[2] * X[t,2] + ... + beta[k] * X[t,k] + M[t] + R[t]

M[t] = rho * M[t-1] + epsilon_M[t]

R[t] = R[t-1] + epsilon_R[t]

-1 < rho < 1

epsilon_M[t] ~ N(0, sigma_M^2)

epsilon_R[t] ~ N(0, sigma_R^2)

Given the input series Y and X, this function searches for the parameter values beta, rho that give the best fit of this model when using a Kalman filter.

If pci_opt_method is twostep, then a two-step procedure is used. In the first step, a linear regression is performed of X on Y to determine the parameter beta. From this regression, a series of residuals is determined. In the second step, a model is fit to the residual series. If par_model is par, then a partially autoregressive model is fit to the residual series. If par_model is ar1, then an autoregressive model is fit to the residual series. If par_model is rw then a random walk model is fit to the residual series. Note that if pci_opt_method is twostep and par_model is ar1, then this reduces to the Engle-Granger two-step procedure.

If pci_opt_method is jp, then the joint-penalty procedure is used. In this method, the parameterbeta are estimated jointly with the parameter rho using a gradient-search optimization function. In addition, a penalty value of lambda * sigma_R^2 is added to the Kalman filter likelihood score when searching for the optimum solution. By choosing a positive value for lambda, you can drive the solution towards a value that places greater emphasis on the mean-reverting component.

Because the joint-penalty method uses gradient search, the final parameter values found are dependent upon the starting point. There is no guarantee that a global optimum will be found. However, the joint-penalty method chooses several different starting points, so as to increase the chance of finding a global optimum. One of the chosen starting points consists of the parameters found through the two-step procedure. Because of this, the joint-penalty method is guaranteed to find parameter values which give a likelihood score at least as good as those found using the two-step procedure. Sometimes the improvement over the two-step procedure is substantial.

An object of class pci.fit containing the fit that was found. The following components may be of interest

`beta`	The vector of weights
`beta.se`	The standard errors of the components of `beta`
`rho`	The estimated coefficient of mean reversion
`rho.se`	The standard error of `rho`
`negloglik`	The negative of the log likelihood
`pvmr`	The proportion of variance attributable to mean reversion

Matthew Clegg matthewcleggphd@gmail.com

Christopher Krauss christopher.krauss@fau.de

Jonas Rende jonas.rende@fau.de

Clegg, Matthew, 2015. Modeling Time Series with Both Permanent and Transient Components using the Partially Autoregressive Model. Available at SSRN: http://ssrn.com/abstract=2556957

Clegg, Matthew and Krauss, Christopher, 2018. Pairs trading with partial cointegration. Quantitative Finance, 18(1), 121 - 138.

egcm Engle-Granger cointegration model

partialAR Partially autoregressive models

##---- Should be DIRECTLY executable !! ----
##-- ==>  Define data, use random,
##--	or do  help(data=index)  for the standard data sets.


YX <- rpci(n=1000, beta=c(2,3,4), sigma_C=c(1,1,1), rho=0.9,sigma_M=0.1, sigma_R=0.2)
fit.pci(YX[,1], YX[,2:ncol(YX)])