# A partialCI Guide

## Introduction

The partialCI package fits a partial cointegration model to describe a time series. Partial cointegration (PCI) is a weakening of cointegration, allowing for the residual series to contain a mean-reverting and a random walk component. Analytically, this residual series is described by a partially autoregressive process (PAR – see Summers (1986), Poterba and Summers (1988), and Clegg (2015a)), consisting of a stationary AR-process and a random walk. Whereas classic cointegration in the sense of Engle and Granger (1987) requires all shocks to be transient, PCI is more flexible and allows for permanent shocks as well – a realistic assumption across many (macro)economic applications. Even though neither the residual series, nor its mean-reverting and permanent component are directly observable, estimation is still possible in state space – see Brockwell and Davis (2010) and Durbin and Koopman (2012).

## The partial cointegration framework

### Model definition

Based on Engle and Granger (1987), Clegg and Krauss (2018) 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 (2018)), $R^2_{MR} = \frac{VAR\left[\left(1-B\right)M_t\right]}{VAR\left[\left(1-B\right)W_t\right]} = \frac{2\sigma^2_M}{2\sigma^2_M + \left(1+\rho\right)\sigma^2_R} , \hspace{0.2cm} R^2_{MR} \in \left[0,1\right],$ 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 (2018). 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}

A time series is classiefied as PCI if we can individually reject both of the subsequent null hypothesis: (i) pure random walk $$\left(\mathcal{H}_0^{R}\right)$$ and (ii) pure stationary AR(1)-process $$\left(\mathcal{H}_0^{M}\right)$$ – reflecting the cointegration case. The alternative is in both cases unrestricted PCI $$\left(\mathcal{H}_1^{MR}\right)$$. Since the proposed null models are all nested in the PCI model (i) and (ii) can be tested with a likelihood ratio test scheme (Neyman and Pearson (1933)). Given that under the null hypothesis we are solely testing parameters at the boundaries of the corresponding parameter space the regularity condition underlying the theorem of Wilks (Wilks (1938)) that the true parameter has to be an inner point of the parameter space does not hold. At boundaries standard asymptotics do not hold. There is a stream of the literature adressing the problem of likelihood ratio tests under such non-standard conditions (Shapiro (1988)). The asymptotic distribution of the likelihood test statistic is a mixture of $$\chi^2$$ distributions (Shapiro (1988)): $\begin{equation}\label{eq: chibar} \sum\limits_{m=0}^k w_m \chi^2_{(m)}, \end{equation}$

where the weights $$w_i$$, with $$i \in \{0, ... , k\}$$ sum up to one. Let $$m$$ denote the degrees of freedom assosciated with the $$\chi^2$$ distribution of interest and $$k$$ is equal to the parameter difference of the full and the null model. Note that $$\chi^2_{(0)} \equiv 0$$, indicating a point mass at zero. Under Wilks theorem the weight for the $$\chi^2_{(k)}$$ distribution is set to one. Therefore, the resulting p-values assuming Wilks theorem holds are more conservative compared to the p-values associated with a mixed distribution. For large sample sizes we calculate p-values under Wilks theorem which can be seen as a conservative approximation of the p-values calculated from the mixed $$\chi^2$$ distribution (Stoel et al. (2006)). To appropriately handle small or medium sample sizes p-values are parameterically bootstrapped as proposed by MacKinnon (2009).

To address the multiple testing problem we provide a Bonferroni (Dunn (1961)) and a Holm correction (Holm (1979)).

## Using the PCI package

The main functions of the partialCI package are fit.pci(), test.pci(), statehistory.pci(), and hedge.pci().

### 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"), robust = FALSE, nu = 5} 
• 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.
• 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).

### test.pci()

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

test.pci(Y, X, pci\_opt\_method = c("jp", "twostep"), irobust = FALSE, inu = 5,  null\_hyp = c("par", "rw",  "ar1"), imethod = c("wilk", "boot"), inrep = 999, istart.seed = 1, alpha = 0.05, use.multicore = FALSE)
• Y: Within the test.pci() function Y can also be a fit.pci object.
• null_hyp: Specifies whether the null hypothesis is either a random walk "rw" or an AR(1)-process "ar1", or if both should be tested "par" (default: null_hyp = "par").
• imethod: Determine the method used to calculate p-values associated with the likelihood ratio test. If set to "wilk" p-values are determined under the assumption that theorem of Wilks holds (Wilks (1938)). If set to "boot" a parametric bootstrap method is carried out following the instructions of MacKinnon (2009). The arguments inrep, istart.seed anduse.multicore only have an effect if imethod = "boot" (default: imethod = "wilk").
• inrep: The number of bootstrap iterations (default: inrep = 999).
• istart.seed: Sets the starting seed for the parametric bootstrap (default: istart.seed = 1).
• use.multicore: If TRUE, parallel processing will be used to improve bootstrap performance (default: use.multicore = FALSE).
• alpha: The global significance level used to determine the local siginificance level following (i) the conservative Bonferroni correction and (ii) the more liberal Holm correction to account for multiple testing. Only has an effect if null_hyp = "par" (default: alpha = 0.05).

### 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). • 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, 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").

## Example

As an introductory example, we explore the relationship between Royal Dutch Shell plc A (RDS-A) and Royal Dutch Shell plc B (RDS-B), using daily (closing) price data from 1 January 2006 to 1 January 2018. RDS-A (Royal Dutch Shell plc - A (2018)) and RDS-B (Royal Dutch Shell plc - B (2018)) data are downloaded from Yahoo Finance. To download the price data we use the getYahooData() function, implemented in the R package TTR (Ulrich (2016)).

    library(partialCI)
library(TTR)

RDSA<-getYahooData("RDS-A", 20060101, 20180101)$Close RDSB<-getYahooData("RDS-B", 20060101, 20180101)$Close

A classic cointegration analysis yields that the two time series are not cointegrated.

    library(egcm)

egcm_finance <- egcm(RDSA,RDSB,include.const = T)

In particular, we apply the two-step approach of Engle and Granger (1987) implemented in the R package egcm (Clegg (2015c)).

The following residual plot (code: plot(egcm\_finance\\$residuals,type = "l")) suggests that the residual series is not purely mean-reverting, but rather shows a stochastical trend as well as a mean-reverting behavior. Residual plot classic cointegration: RDS-A and RDS-B (1.01.2006 - 1.01.2018, daily)

Hence, it is not suprising that RDS-A and RDS-B are not cointegrated. Using the PCI framework, we are able to fit a PCI model to RDS-A and RDS-B.

PCI_RDSA_RDSB<-fit.pci(RDSA, RDSB, pci_opt_method = c("jp"), par_model = c("par"), robust = FALSE, nu = 5)

The R output is given as

Fitted values for PCI model
Y[t] = X[t] %*% beta + M[t] + R[t]
M[t] = rho * M[t-1] + eps_M [t], eps_M[t] ~ N(0, sigma_M^2)
R[t] = R[t-1] + eps_R [t], eps_R[t] ~ N(0, sigma_R^2)

Estimate Std. Err
beta_Close   0.9232   0.0037
rho          0.3549   0.0935
sigma_M      0.0937   0.0070
sigma_R      0.1085   0.0061

-LL = -1580.80, R^2[MR] = 0.524,

where beta_Close denotes the partially cointegrating coefficient. The PVMR of 0.52 suggests that the spread time series also exhibits a clear mean-reverting behavior.

In the subsequent step, we utilize the test.pci() function to check whether RDS-A and RDS-B are partially cointegrated. Given the large sample size we will utilize imethod = "wilk".

The R code

test.pci(RDSA, RDSB, alpha = 0.05, null_hyp = "par", imethod = "wilk")

  LR test of [RW or CI(1)] vs Almost PCI(1) (joint penalty,wilk)

data:  RDSA / RDSB

Hypothesis    Statistic  p-value    alpha alpha_bonf alpha_holm
Random Walk      -61.55    0.000    0.050    0.025    0.050
AR(1)            -59.69    0.000    0.050    0.025    0.025.

Recall that a time series is classified as partially cointegrated, if and only if the random walk as well as the AR(1)-hypotheses are rejected. The $$p$$-value of 0.000 for both null model indicates that RDS-A and RDS-B are partially cointegrated in the considered period of time.

Next, we demonstrate the use of the statehistory.pci() function which allows to estimate and extract the hidden states. The R code,

statehistory.pci(PCI_RDSA_RDSB)},


results in the R output:

              Y       Yhat      Z          M         R       eps_M      eps_R
2006-01-03 32.2123 31.6496 0.562708  0.00000000 0.562708  0.0000000  0.0000000
2006-01-04 32.5445 31.9223 0.622142  0.01758541 0.604556  0.0175854  0.0418485
2006-01-05 32.1519 31.6260 0.525826 -0.01890081 0.544726 -0.0251415 -0.0598301
2006-01-06 32.7659 32.1575 0.608460  0.01413479 0.594325  0.0208423  0.0495991
...
2017-12-22 63.0633 59.6017 3.461665 -0.04153269 3.503198 -0.0802318 -0.1909302
2017-12-26 63.8210 60.1070 3.714061  0.05201243 3.662049  0.0667515  0.1588508
2017-12-27 63.5237 60.0804 3.443331 -0.05171763 3.495049 -0.0701758 -0.1669996
2017-12-28 63.7827 60.2666 3.516118 -0.00668903 3.522807  0.0116645  0.0277584
2017-12-29 63.9841 60.5414 3.442713 -0.02536998 3.468083 -0.0229961 -0.0547247.

The latter table covers the estimates of the hidden states $$M$$ and $$R$$ as well as the corresponding error terms eps_M and eps_R. Z is equal to the sum of $$M$$ and $$R$$. The estimate of the target time series is denoted by Yhat.

The subsequent figure illustrates a plot of the extracted mean-reverting component of the spread associated with the RDS-A and RDS-B price time series (plot(statehistory.pci(PCI_RDSA_RDSB)[,4],type = "l",ylab = "", xlab = "")). Mean-reverting component RDS-A and RDS-B (1.01.2006 - 1.01.2018, daily)

The horizontal blue lines are equal to two times the historical standard deviation in absolute terms of the mean-reverting component. A pairs trading strategy could exploit the mean-reverting behavior of $$M_t$$. Note that this example is in-sample; for a true out-of-sample application see Clegg and Krauss (2018).

We continue with using hedge.pci() to find the set of sector ETFs forming the best hedging portfolio for the SPY index (S&P500 index).

Thereby, the R code,

    sectorETFS <- c("XLB", "XLE", "XLF", "XLI", "XLK", "XLP", "XLU", "XLV", "XLY")
prices <- multigetYahooPrices(c("SPY", sectorETFS), start=20060101)

hedge.pci(prices[,"SPY"], prices),

results in the subsequent output:

-LL     LR[rw]  p[rw]   p[mr]  rho  R^2[MR] Fac.| Factor coefficients
2521.2 -19.281  0.000  0.000  0.593  0.407  XLI | 3.033
1891.8 -41.011  0.000  0.000  0.350  0.435  XLY | 1.848 1.182
1558.8 -47.619  0.000  0.000  0.350  0.461  XLV | 1.627 0.915 0.679
1302.6 -52.992  0.000  0.000  0.350  0.483  XLU | 1.490 0.895 0.596 0.583
724.5 -64.165  0.000  0.000  0.382  0.552  XLK | 1.209 0.544 0.479 0.491 1.265.  `

The table summarizes information about the best hedging portfolio, where each row corresponds to an increasing number of factors. Row 1: The best single-factor hedging portfolio comprises XLI (industrials) as only factor. Row 2: The best two-factor hedging portfolio consists of XLI and XLY (consumer discretionary). As such, XLY leads to the best improvement of the LRT score among all remaining factors. Row 3 includes XLV (health care) for the three-factor portfolio and row 4 XLK (technology) for the best four-factor portfolio. The last row corresponds to the overall best fit out of the nine potential sector ETFs, based on the LRT score. Note that for all rows, the union of random walk and AR(1)-null hypothesis is rejected at the 5 percent significant level, so we find a PCI model at each step.