partialAR (version 1.0.12)

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", "kfas", "ss"), 
  rho.max = 1, 
  nu = par.nu.default())

Arguments

Y

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++

• "kfas" Kalman filter implementation of the KFAS 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:

data

The input vector Y

robust

The input parameter robust

nu

The input parameter nu

model

The input parameter model

lambda

The input parameter lambda

opt_method

The input parameter opt_method

rho.max

The input parameter rho.max

rho

The estimate of the parameter rho

sigma_M

The estimate of the parameter sigma_M

sigma_R

The estimate of the parameter sigma_R

M0

The estimate of the parameter M[0]

R0

The estimate of the parameter R[0]

par

The vector (rho, sigma_M, sigma_R, M0, R0)

stderr

The vector of standard errors

negloglik

The negative of the log likelihood score for these parameters

pvmr

The 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 "kfas", the KFAS Kalman Filter package KFAS is used. Because the Kalman gain matrix takes some time to converge to its steady state value, the "kfas" 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

# NOT RUN {
    set.seed(1)
    x <- rpar(1000, 0.8, 1, 0.5)  # Generate a random PAR sequence
    fit.par(x)                  # Estimate its parameters
    
# }
# NOT RUN {
plot(fit.par(x)
# }
# NOT RUN {
   # Plot the estimate
    test.par(x)                 # Test the goodness of fit

# }
# NOT RUN {
    # An example involving European stock market data
    data(EuStockMarkets)   # European Stock Markets 1991-1998
    
    # Check for cointegration between German DAX and Swiss SMI
    library(egcm)
    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:
    plot(fit.par(egcm(EuStockMarkets[,c("DAX", "SMI")])$residuals))
    
    # 74% of the variance is attributed to a mean-reverting
    # 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.
# }
# NOT RUN {
    
# }
# NOT RUN {
    # 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% of
    # 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.   
# }