Description Usage Arguments Details Value Author(s) References See Also Examples
Fits the partial cointegration model to a collection of time series
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Y |
The time series that is to be modeled. A plain or |
X |
A (possibly |
pci_opt_method |
Specifies the method that will be used for finding the best fitting model. One of the following:
Default: |
par_model |
The model used for the residual series. One of the following:
Default: |
lambda |
The penalty parameter to be used in the joint-penalty ( |
robust |
If |
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 |
rho |
The estimated coefficient of mean reversion |
rho.se |
The standard error of |
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
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