Stable Generalized Regression Models
stablereg fits user specified generalized linear and nonlinear regression models based on the stable distribution to (uncensored, right and/or left censored) data. This allows the location, the dispersion, the skewness and the tails of the fitted stable distribution to vary with explanatory variables.
stablereg( y = NULL, loc = 0, disp = 1, skew = 0, tail = 1.5, oloc = TRUE, odisp = TRUE, oskew = TRUE, otail = TRUE, noopt = FALSE, iloc = NULL, idisp = NULL, iskew = NULL, itail = NULL, loc_h = NULL, disp_h = NULL, skew_h = NULL, tail_h = NULL, weights = 1, exact = FALSE, delta = 1, envir = parent.frame(), integration = "Romberg", eps = 1e-06, up = 10, npoint = 501, hessian = TRUE, llik.output = FALSE, print.level = 0, ndigit = 10, steptol = 1e-05, gradtol = 1e-05, fscale = 1, typsize = abs(p0), stepmax = sqrt(p0 %*% p0), iterlim = 100 )
y: The response vector or a repeated data object. If the repeated data object contains more than one response variable, give that object in envir and give the name of the response variable to be used here.
For censored data, two columns with the second being the censoring indicator (1: uncensored, 0: right censored, -1: left censored.)
loc, loc_h, oloc, iloc: Describe the regression model fitted for the location parameter of the stable distribution, perhaps after transformation by the link function loc_g (set to the identity by default. The inverse link function is denoted by loc_h. Note that these functions cannot contain unknown parameters).
Two specifications are possible:
(1) loc is a linear or nonlinear language expression beginning with ~ or an R function, describing the regression function for the location parameter (after transformation by loc_g, the link function).
iloc is a vector of initial conditions for the parameters in the regression for this parameter.
oloc is a boolean indicating if an optimization of the likelihood has to be carried out on these parameters. If oloc is set to TRUE, a default zero value is considered for the starting values iloc. But if no optimization is desired on the location parameters, i.e. when the likelihood has to be evaluated or optimized at a fixed location, then iloc has to be explicitely specified.
(2) loc is a numeric expression (i.e. a scalar or a vector of the same size as the data vector y, or y[,1] when censoring is considered).
If oloc is set to TRUE, i.e. when an optimization of the likelihood has to be carried out on the location parameter, then the location parameter (after transformation by the link function loc_g) is set to an unknown parameter with initial value equal to iloc[1] or loc[1] when iloc is not specified.
But when oloc is set to FALSE, i.e. when the likelihood has to be evaluated or optimized at a fixed location, then the transformed location is assumed to be equal to loc when it is of the same length as the data vector y (or y[,1] when censoring is considered), and to loc[1] otherwise.
Specification (1) is especially useful in ANOVA-like situations where the location is assumed to change with the levels of some factor variable.
disp, disp_h, odisp, idisp: describe the regression model for the dispersion parameter of the fitted stable distribution, after transformation by the link function disp_g (set to the log function by default). The inverse link function is denoted by disp_h. Again these functions cannot contain unknown parameters. The same rules as above apply when specifying the generalized regression model for the dispersion parameter.
skew, skew_h, oskew, iskew: describe the regression model for the skewness parameter of the fitted stable distribution, after transformation by the link function skew_g (set to log{(1 + [.])/(1 - [.])}
by default). The inverse link function is denoted by skew_h. Again these functions cannot contain unknown parameters. The same rules as above apply when specifying the generalized regression model for the skewness parameter.
tail, tail_h, otail, itail: describe the regression model considered for the tail parameter of the fitted stable distribution, after transformation by the link function tail_g (set to log{([.] - 1)/(2 - [.])} by default. The inverse link function is denoted by tail_h. Again these functions cannot contain unknown parameters). The same rules as above apply when specifying the generalized regression model for the tail parameter.
noopt: When set to TRUE, it forces oloc, odisp, oskew and otail to FALSE, whatever the user choice for these last three arguments. It is especially useful when looking for appropriate initial values for the regression model parameters, before undertaking the optimization of the likelihood.
weights: Weight vector.
exact: If TRUE, fits the exact likelihood function for continuous data by integration over intervals of observation, i.e. interval censoring.
delta: Scalar or vector giving the unit of measurement for each response value, set to unity by default. For example, if a response is measured to two decimals, delta=0.01. If the response is transformed, this must be multiplied by the Jacobian. For example, with a log transformation, delta=1/y. (The delta values for the censored response are ignored.) The transformation cannot contain unknown parameters.
envir: Environment in which model formulae are to be interpreted or a data object of class, repeated, tccov, or tvcov; the name of the response variable should be given in y. If y has class repeated, it is used as the environment.
integration, eps, up, npoint: integration indicates which algorithm must be used to evaluate the stable density when the likelihood is computed with exact set to FALSE. See the man page on stable for extra information.
hessian: Arguments controlling the optimization procedure nlm.
llik.output: is TRUE when the likelihood has to be displayed at each iteration of the optimization.
print.level: Arguments controlling the optimization procedure nlm.
ndigit: Arguments controlling the optimization procedure nlm.
steptol: Arguments controlling the optimization procedure nlm.
gradtol: Arguments controlling the optimization procedure nlm.
fscale: Arguments controlling the optimization procedure nlm.
typsize: Arguments controlling the optimization procedure nlm.
stepmax: Arguments controlling the optimization procedure nlm.
iterlim: Arguments controlling the optimization procedure nlm.
A list of class stable is returned. The printed output includes the -log-likelihood, the corresponding AIC, the maximum likelihood estimates, standard errors, and correlations. It also include all the relevant information calculated, including error codes.
Because of the numerical integrations involved, convergence can be very sensitive to the initial parameter values supplied and to the settings of the arguments controlling nlm. If nlm feeds extreme parameter values in the tails of the distribution to the likelihood function, the integration may hang for a long time.
## Share return over a 50 day period (see reference above) # shares y <- c(296,296,300,302,300,304,303,299,293,294,294,293,295,287,288,297, 305,307,307,304,303,304,304,309,309,309,307,306,304,300,296,301,298, 295,295,293,292,297,294,293,306,303,301,303,308,305,302,301,297,299) # returns ret <- (y[2:50]-y[1:49])/y[1:49] # hist(ret, breaks=seq(-0.035,0.045,0.01)) day <- seq(0,0.48,by=0.01) # time measured in days/100 x <- seq(1,length(ret))-1 # Classic stationary normal model tail=2 print(z1 <- stablereg(y = ret, delta = 1/y[1:49], loc = ~1, disp= ~1, skew = ~1, tail = tail_g(1.9999999), iloc = 0, idisp = -3, iskew = 0, oskew = FALSE, otail = FALSE)) # Normal model (tail=2) with dispersion=disp_h(b0+b1*day) print(z2 <- stablereg(y = ret, delta = 1/y[1:49], loc = ~day, disp = ~1, skew = ~1, tail = tail_g(1.999999), iloc = c(0.003,0), idisp = -4.5, iskew = 0, oskew = FALSE, otail = FALSE)) # Stable model with loc(ation)=loc_h(b0+b1*day) print(z3 <- stablereg(y = ret, delta = 1/y[1:49], loc = ~day, disp = ~1, skew = ~1, tail = ~1, iloc = c(0.001,-0.004), idisp = -4.8, iskew = 0, itail = 0.6)) # Stable model with disp(ersion)=disp_h(b0+b1*day) print(z4 <- stablereg(y = ret, delta = 1/y[1:49], loc = ~1, disp = ~day, skew = ~1, tail = ~1, iloc = 0.003, idisp = c(-4.8,0), iskew = -0.03, itail = 1.6)) # Stable model with skew(ness)=skew_h(b0+b1*day) # Evaluation at fixed parameter values (because noopt is set to TRUE) print(z5 <- stablereg(y = ret, delta = 1/y[1:49], loc = ~1, disp = ~1, skew = ~day, tail = ~1, iloc = 5.557e-04, idisp = -4.957, iskew = c(2.811,-2.158), itail = 1.57, noopt=TRUE)) # Stable model with tail=tail_h(b0+b1*day) print(z6 <- stablereg(y = ret, delta = 1/y[1:49], loc = ret ~ 1, disp = ~1, skew = ~1, tail = ~day, iloc = 0.002, idisp = -4.8, iskew = -2, itail = c(2.4,-4), hessian=FALSE))
Lambert, P. and Lindsey, J.K. (1999) Analysing financial returns using regression models based on non-symmetric stable distributions. Applied Statistics 48, 409-424.
lm, glm, stable
and stable.mode.
Philippe Lambert (Catholic University of Louvain, Belgium, phlambert@stat.ucl.ac.be ) and Jim Lindsey.