fit.par: Fit a partially autoregressive model

Description

Fit a partially autoregressive model

Usage

fit.par(Y, 
  robust = FALSE, 
  model = c("par", "ar1", "rw"), 
  lambda = 0, 
  opt_method = c("css", "fkf", "ss"), 
  rho.max = 1, 
  nu = par.nu.default())

Arguments

A numeric vector or zoo vector representing the time series whose parameters are to be estimated

robust

If TRUE, then the error terms in the fit are assumed to follow a Student's t-distribution with degrees of freedom parameter given by nu. Otherwise, the error terms are assumed to be normally distributed. Default: FALSE

model

Specifies the model that is to be fit.  Possible values are
"par" The partially autoregressive model is fit.
"ar1" An autoregressive model of order one is fit.
"rw"  A random walk is fit.
Default: par

lambda

A penalty term $lambda sigma_R^2$ is added to the likelihood function.
Default: $lambda = 0.$

opt_method

Specifies the Kalman filter that will be used for optimization:
"ss" Steady-state Kalman filter
"css" Steady-state Kalman filter coded in C++
"fkf" Kalman filter implementation of the FKF package
Default: css

rho.max

Specifies an upper limit on the value of rho that will
be returned.

nu

If robust is TRUE, this specifies the value of the
degrees-of-freedom parameter used by the t-distribution.  
Default: 5

`Value`

An S3 object of class fit.par is returned.  The object
contains the following values:
dataThe input vector Y
robustThe input parameter robust
nuThe input parameter nu
modelThe input parameter model
lambdaThe input parameter lambda
opt_methodThe input parameter opt_method
rho.maxThe input parameter rho.max
rhoThe estimate of the parameter rho
sigma_MThe estimate of the parameter sigma_M
sigma_RThe estimate of the parameter sigma_R
M0The estimate of the parameter M[0]
R0The estimate of the parameter R[0]
parThe vector (rho, sigma_M, sigma_R, M0, R0)
stderrThe vector of standard errors
negloglikThe negative of the log likelihood score for these parameters
pvmrThe proportion of variance attributable to mean reversion (see pvmr.par)

`Disclaimer`

DISCLAIMER: The software in this package is for general information purposes
only. It is hoped that it will be useful, but it is provided WITHOUT ANY
WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A
PARTICULAR PURPOSE. It is not intended to form the basis of any investment
decision. USE AT YOUR OWN RISK!

`Details`

This routine determines the maximum likelihood fit of a time series
to the partially autoregressive model, which is given by the 
specification:
$$X_t = M_t + R_t$$
$$M_t = \rho M_{t-1} + \epsilon_{M,t}$$
$$R_t = R_{t-1} + \epsilon_{R,t}$$
$$-1 < \rho < 1$$
$$\epsilon_{M,t} \sim N(0,\sigma_M^2)$$
$$\epsilon_{R,t} \sim N(0,\sigma_R^2)$$
The partially autoregressive model is a candidate for working with time series 
having both permanent and transient
components.
If robust is TRUE, then a form of robust estimation is used.
The error term is assumed to follow a Student's t-distribution with
nu degrees of freedom.
The model parameter is used to alter the model that is fit.
If model is "par", then the partially autoregressive model
is fit.  If model is "ar1", then an AR(1) model is fit.
This is performed by fitting the partially autoregressive model with
the restriction that $sigma_R = 0$.  If model is "rw",
then a random walk model is fit.  This is performed by fitting the
partially autoregressive model with the restriction that $sigma_M = 0$.
The parameter lambda specifies the weighting of a penalty term that is added
to the likelihood function.
When $lambda > 0$, this drives the optimizer towards a solution 
that places a greater weight on the transient (mean-reverting) component, and 
when $lambda < 0$, this drives the optimizer towards a solution that
places a greater weight on the permanent (random walk) component.
The fit is performed using maximum likelihood estimation for a 
Kalman filter representation of the model.  When opt_method is
"ss" or "css", a steady-state Kalman filter is used.
These two methods should give the same result, although "css" is
to be preferred because the implementation is much faster.  When opt_method
is "fkf", the Fast Kalman Filter package fkf is used.
Because the Kalman gain matrix takes some time to converge to its steady state
value, the "fkf" implementation will yield values that are close to
but not the same as those of "ss" and "css".
This routine prints the model that is found.  
The following is an example of the output obtained in one 
particular run:
Fitted model:
    X[t] = M[t] + R[t]
    M[t] = 0.9427 M[t-1] + eps_M,t,  eps_M,t ~ N(0, 0.8843^2)
          (0.0302)                                 (0.0685)
    R[t] = R[t-1] + eps_R,t,         eps_R,t ~ N(0, 0.2907^2)
                                                   (0.1710)
    M_0 = 0.0000, R_0 = -5.2574
             (NA)       (0.9625)
  Proportion of variance attributable to mean reversion (pvmr) = 0.9050
  Negative log likelihood = 339.51
In this ouptut, the coefficient of mean reversion rho is
found to be 0.9427 with a standard error of 0.0302.  This corresponds
to a half-life of mean reversion of log(0.5)/log(0.9427) = 11.7 days.
The parameter sigma_M is found to be 0.8843 with a standard error 0.0685.
The parameter sigma_R is found to be 0.2907 with a 
standard error of 0.1710.  The parameters M[0] and R[0]
are 0.0 and -5.2574, respectively.
An important measure of the quality of fit of the partially autoregressive
model is the proportion of variance attributable to mean reversion.
This is a number between zero and one.  When it is zero, the best fit
is a pure random walk, and when it is one, the best fit is a pure
mean-reverting series.  In this case, it is found to be 0.9050, indicating
that the mean-reverting component dominates.
The negative log likelihood of this particular fit is 339.51.
A plot method is available for plotting the fit, and the
test.par method is available for testing the null hypotheses
that an adequate fit can be obtained with a pure random walk or pure
autoregressive series.

`References`

Summers, Lawrence H.
  Does the stock market rationally reflect fundamental values?
  Journal of Finance, 41(3), 591-601.
  
  Poterba, James M. and Lawrence H. Summers.
  Mean reversion in stock market prices:  Evidence and implications.
  Journal of Financial Economics, 22(1), 27-59.
  
  Clegg, Matthew.
  Modeling Time Series with Both Permanent and Transient Components
  using the Partially Autoregressive Model.
  Available at SSRN: http://ssrn.com/abstract=2556957

`See Also`

arima ARIMA modeling of time series
egcm Engle-Granger cointegration model

`Examples`

Run this codeset.seed(1)
    x <- rpar(1000, 0.8, 1, 0.5)  # Generate a random PAR sequence
    fit.par(x)                  # Estimate its parameters
    plot(fit.par(x)   # Plot the estimate
    test.par(x)                 # Test the goodness of fit

# An example involving European stock market data
    data(EuStockMarkets)   # European Stock Markets 1991-1998
    
    # Check for cointegration between German DAX and Swiss SMI
    egcm(log(EuStockMarkets[,c("DAX", "SMI")]))
    
    # The series are not found to be cointegrated.
    # Perhaps they are partially cointegrated?  Check the residuals
    # of the cointegration fit for partial autoregression:
    fit.par(egcm(EuStockMarkets[,c("DAX", "SMI")])$residuals)
    
    # A plot of the model looks promising:
    \dontrun{plot(fit.par(egcm(EuStockMarkets[,c("DAX", "SMI")])$residuals))}
    
    # 74    # AR(1) process.  However, it is important to check whether this is
    # a better explanation than a simple random walk:
    test.par(egcm(EuStockMarkets[,c("DAX", "SMI")])$residuals)
    
    # The p-value is found to be 0.36, so the random walk hypothesis
    # cannot be rejected.
    
# Another example involving a potential pairs trade between
    # Coca-Cola and Pepsi.
    
    # Fetch the price series for Coca-Cola (KO) and Pepsi (PEP) in 2014
    library(TTR)
    KO <- getYahooData("KO", 20140101, 20141231)$Close
    PEP <- getYahooData("PEP", 20140101, 20141231)$Close
    
    # Check whether they were cointegrated
    library(egcm)
    egcm(KO,PEP)
    
    # It turns out that they are not cointegrated.  Perhaps a better
    # fit can be obtained with the partially autoregressive model:
    fit.par(egcm(KO,PEP)$residuals)
    
    # The mean-reverting component of the above fit explains 90    # the variance of the daily returns.  Thus, it appears that the
    # two series are close to being cointegrated.  A plot further
    # confirms this:
    plot(fit.par(egcm(KO,PEP)$residuals))
    
    # Still, it is important to check whether or not the residual
    # series is simply a random walk:
    test.par(egcm(KO,PEP)$residuals)
    
    # In this case, the p-value associated with the hypothesis that
    # the series is partially autoregressive is 0.12.  Thus, the
    # evidence of partial autoregression is marginal. The random walk
    # may be a better explanation.
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