A partialCI Guide

Model definition

Based on Engle and Granger (1987), Clegg and Krauss (2016) define the concept of partial cointegration as follows: : “The components of the vector \(X_t\) are said to be partially cointegrated of order \(d\), \(b\), denoted \(X_t \sim PCI\left(d,b\right)\), if (i) all components of \(X_t\) are \(I\left(d\right)\); (ii) there exists a vector \(\alpha\) so that \(Z_t = \alpha \prime X_t\) and \(Z_t\) can be decomposed as a sum \(Z_t = R_t + M_t\), where \(R_t \sim I\left(d\right)\) and \(M_t \sim I\left(d - b\right)\).”

Let \(Y_t\) denote the target time series and \(X_{j,t}\) the \(j^{th}\) factor time series at time \(t\), where \(j = \lbrace 1, 2, \dots, k \rbrace\). The target time series and the \(k\) factor time series are partially cointegrated, if a parameter vector \(\iota = \left\lbrace\beta_1, \beta_2, \dots, \beta_k, \rho, \sigma_M, \sigma_R, M_0, M_R\right\rbrace\) exists such that the subsequent model equations are satisfied:

\[ \begin{aligned} Y_{t} &= \beta_1 X_{1,t} + \beta_2 X_{2,t} + ... + \beta_k X_{k,t} + W_t \\ W_t &= M_t + R_t \\ M_t &= \rho M_{t-1} + \varepsilon_{M,t}\\ R_t &= R_{t-1} + \varepsilon_{R,t}\\ \varepsilon_{M,t} &\sim \mathcal{N}\left(0, \sigma^2_M\right)\\ \varepsilon_{R,t} &\sim \mathcal{N}\left(0, \sigma^2_R\right)\\ \beta_j \in \mathbb{R}; \rho &\in \left(-1, 1\right);\sigma^2_M, \sigma^2_R \in \mathbb{R}_0^+. \\ \end{aligned} \] Thereby, \(W_t\) denotes the partially autoregressive process, \(R_t\) the permanent component, \(M_t\) the transient component and \(\beta = \lbrace \beta_1, \beta_2, \dots, \beta_k \rbrace\) is the partially cointegrating vector. The permanent component is modeled as a random walk and the transient component as an AR(1)-process with \(AR(1)\)-coefficient \(\rho\). The corresponding error terms \(\varepsilon_{M,t}\) and \(\varepsilon_{R,t}\) are assumed to follow mutually independent, normally distributed white noise processes with mean zero and variances \(\sigma^2_M\) and \(\sigma^2_R\). A key advantage of modeling the cointegrating process as a partially autoregressive process is that we are able to calculate the proportion of variance attributable to mean-reversion (PVMR), defined as (Clegg and Krauss (2016)), \[ R^2_{MR} = \frac{VAR\left[\left(1-B\right)M_t\right]}{VAR\left[\left(1-B\right)W_t\right]} = \frac{2\sigma^2_M}{2\sigma^2_M + \left(1+\rho\right)\sigma^2_R} , \hspace{0.2cm} R^2_{MR} \in \left[0,1\right], \] where \(B\) denotes the backshift operator. The statistic \(R^2_{MR}\) is useful to assess how close the cointegration process is to either a pure random walk \(\left(R^2_{MR} = 0\right)\) or a pure AR(1)-process \(\left(R^2_{MR} = 1\right)\).

State space represenation

The applied state space transformation is in line with Clegg and Krauss (2016). Given that the PAR process \(W_t\) is not observable, we convert the PCI model into the following state space model, consisting of an observation and a state equation:
\[ \begin{align} X_t &= H Z_t \\ Z_t &= FZ_{t-1} + W_t. \end{align} \] Thereby, \(Z_t\) denotes the state which is assumed to be influenced linearly by the state in the last period and a noise term \(W_t\). The matrix \(F\) is assumed to be time invariant. The observable part is denoted by \(X_t\). By assumption, there is a linear dependence between \(X_t\) and \(Z_t\), captured in the time invariant matrix \(H\).

Estimation of a partial cointegration model

Parameters are estimated via the maximum likelihood (ML) method. Using a quasi-Newton algorithm, the ML method searches for the parameters \(\rho\), \(\sigma^2_M\), \(\sigma^2_R\) and the parameter vector \(\beta\) which maximizes the likelihood function of the associated Kalman filter.

A likelihood ratio test routine for partial cointegration}

The likelihood ratio test (LRT) implemented in the partialCI package adopts the LRT routine for PAR models proposed by Clegg (2015a). In a PCI scenario the null hypothesis consists of two conditions – namely the hypothesis that the residual series is a pure random walk (\(\mathcal{H}^R_0\)) or a pure AR(1)-process \((\mathcal{H}^M_0)\). The two conditions are separately tested. Only if both, \(\mathcal{H}^R_0\) and \(\mathcal{H}^M_0\) are individually rejected, the null hypothesis of no partial cointegration is rejected.

fit.pci()

The function fit.pci() fits a partial cointegration model to a given 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, include_alpha=FALSE)} 

• Y: Denotes the target time series and X is a matrix containing the k factors used to model Y.
• pci_opt_method: Specifies, whether the joint-penalty method ("jp") or the ("twostep") method is applied to obtain the model with the best fit. If pci_opt_method is specified as "twostep", a two-step procedure similar to the method introduced by Engle and Granger (1987) is performed. Which model is fitted to the residual series, depends on the specification for the argument par_model. In case of "par", a partial autoregressive model is used, in case of "ar1", an AR(1)-process and in case of "rw" a random walk (default: par_model = "par"). On the other hand, if the pci_opt_method is specified as "jp", the joint-penalty method is applied, to estimate \(\beta\), \(\rho\), \(\sigma_M^2\) and \(\sigma_R^2\) jointly via ML. The likelihood score of the associated Kalman filter is extended by a penalty value \(\lambda\sigma_R^2\) (default: lambda = 0), where \(\lambda \in \mathbb{R}_0^+\) (default: pci_opt_method = "jp").
• robust: Determines whether the residuals are assumed to be normally (FALSE) or \(t\)-distributed (TRUE) (default: robust = TRUE). If robust is set to TRUE the degrees of freedom can be specified, using the argument nu (default: nu = 5).
• include_alpha: If TRUE, an intercept \(\alpha\) is added to the PCI relationship (default: include_alpha = FALSE).

test.pci()

The test.pci() function tests the goodness of fit of a PCI model.

test.pci(Y, X, alpha = 0.05, null_hyp = c("rw", "ar1"),  robust = FALSE, pci_opt_method = c("jp", "twostep"))}

• alpha: Determines at which significance level the null hypothesis is rejected (default: alpha = 0.05).
• null_hyp: Specifies whether the null hypothesis is a random walk ("rw"), an AR(1)-process ("ar1") or a union of both hypotheses (c("rw", "ar1")) (default: null_hyp = c("rw", "ar1")).

statehistory.pci()

To estimate the sequence of hidden states the statehistory.pci() function can be applied.

statehistory.pci(A, data = A\$data, basis = A\$basis)}

• A: Denotes a fit.pci() object.
• data: Is a matrix consisting of the target time series and the k factor time series (default: data = A\$data).
• code{basis: Captures the coefficients of the factor time series (default: basis = A\$basis).

hedge.pci()

The function hedge.pci() finds those k factors from a predefined set of factors which yield the best fit to the target time series.

hedge.pci(Y, X, maxfact = 10, lambda = 0, use.multicore = TRUE, minimum.stepsize = 0, verbose = TRUE, exclude.cols = c(), search_type = c("lasso", "full", "limited"), pci_opt_method=c("jp", "twostep"))}

• maxfact: Denotes the maximum number of considered factors (default: * maxfact = 10).
• use.multicore: If TRUE, parallel processing is activated (default: * use.multicore = TRUE).
• verbose: Controls whether detailed information are printed (default: verbose = TRUE).
• exclude.cols: Defines a set of factors which should be excluded from the search routine (default: exclude.cols = c()).
• search_type: Determines the search algorithm applied to find the model that fits best to the target time series. The likelihood ratio score (LRT score) is used to compare the model fits, whereby lower scores are associated with better fits. If the option "lasso" is specified the lasso algorithm as implemented in the R package glmnet (Friedman, Hastie, and Tibshirani 2010) is deployed to search for the portfolio of factors that yields the best linear fit to the target time series. If the option "full" is specified, then at each step, all possible additions to the portfolio are considered and the one which yields the highest likelihood score improvement is chosen. If the option "limited" is specified, then at each step, the correlation of the residuals of the current portfolio is computed with respect to each of the candidate series in the input set \(X\), and the top \(B\) series are chosen for further consideration. Among these top \(B\) candidates, the one which improves the likelihood score by the greatest amount is chosen. The parameter \(B\) can be controled via maxfact (default: search_type = "lasso").