使用RStudio调试(debug)基础学习(二)和fGarch包中的garchFit函数估计GARCH模型的原理和源码
一、garchFit函数的参数---------------------------------------------
algorithm
a string parameter that determines the algorithm used for maximum likelihood estimation.
设定最大似然估计所用的算法
cond.dist
a character string naming the desired conditional distribution. Valid values are "dnorm", "dged", "dstd", "dsnorm", "dsged", "dsstd" and "QMLE". The default value is the normal distribution. See Details for more information.
条件扰动(新息)的分布
control
control parameters, the same as used for the functions from nlminb, and 'bfgs' and 'Nelder-Mead' from optim.
data
an optional timeSeries or data frame object containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which armaFit is called. If data is an univariate series, then the series is converted into a numeric vector and the name of the response in the formula will be neglected(被忽略).
包含变量的timeSeries或者数据框类型，如果打他中没有这些变量，则将去环境(formula)中找,armaFit的环境通常也会被调用。如果是单变量序列，则序列被转为数值向量，formula那些就被忽略。
delta
a numeric value, the exponent delta of the variance recursion. By default, this value will be fixed, otherwise the exponent will be estimated together with the other model parameters if include.delta=FALSE.
变量定义的delta指数？
description
a character string which allows for a brief description.
简短描述
formula
formula object describing the mean and variance equation of the ARMA-GARCH/APARCH model. A pure GARCH(1,1) model is selected when e.g. formula=~garch(1,1). To specify for example an ARMA(2,1)-APARCH(1,1) use formula = ~arma(2,1)+apaarch(1,1).
描述ARMA-GARCH/APARCH模型的均值方程和波动率方程的对象
~garch(1,1) ——表示GARCH(1,1) 模型
~arma(2,1)+apaarch(1,1) ——表示ARMA(2,1)-APARCH(1,1)模型
gamma
APARCH leverage parameter entering into the formula for calculating the expectation value.
杠杆参数
hessian
a string denoting how the Hessian matrix should be evaluated, either hessian ="rcd", or "ropt", the default, "rcd" is a central difference approximation implemented in R and "ropt" use the internal R function optimhess.
海赛矩阵被计算的方式，默认为rcd(中心差分近似？)
include.delta
a logical flag which determines if the parameter for the recursion equation delta will be estimated or not. If include.delta=FALSE then the shape parameter will be kept fixed during the process of parameter optimization.
逻辑标志，表示迭代法处delta参数是否被估计
include.mean
this flag determines if the parameter for the mean will be estimated or not. If include.mean=TRUE this will be the case, otherwise the parameter will be kept fixed durcing(应该是during) the process of parameter optimization.
include.mean=TRUE表示均值是否被估计，否则均值参数在参数最优化过程中将是固定的
include.shape
a logical flag which determines if the parameter for the shape of the conditional distribution will be estimated or not. If include.shape=FALSE then the shape parameter will be kept fixed during the process of parameter optimization.
一个参数，决定条件分布的自由度shape是否被估计
include.shape=FALSE则在参数最优化过程中，t分布的自由度shape是固定的
include.skew
a logical flag which determines if the parameter for the skewness of the conditional distribution will be estimated or not. If include.skew=FALSE then the skewness parameter will be kept fixed during the process of parameter optimization.
skew=FALSE则shew(偏度)在参数最优化过程中是固定的
init.rec
a character string indicating the method how to initialize the mean and varaince recursion relation.
一个字符串，指明如何初始化均值和方差迭代关系。
取值呢？
"mci", "uev"
leverage
a logical flag for APARCH models. Should the model be leveraged? By default leverage=TRUE.
是否带杠杆
shape
a numeric value, the shape parameter of the conditional distribution.
条件分布的自由度值
skew
a numeric value, the skewness parameter of the conditional distribution.
条件分布的偏度
title
a character string which allows for a project title.
标题
trace
a logical flag. Should the optimization process of fitting the model parameters be printed? By default trace=TRUE.
参数最优化过程是否被打印出来。
...
additional arguments to be passed.
二、查看数据----------------------------------------------
双击da(相当于View(da))，可以看到数据是这样的
经过对数转化后的收益率，是我们建模用的原始数据。
intc=log(da$intc+1)intc[1] 0.0099998346 -0.1500127529 0.0670640792[4] 0.0829486352 -0.1103484905 0.1251628488[7] 0.4855078158 0.1112255825 0.2109235908..........
library(fGarch)da=read.table("m-intcsp7309.txt",header=T)intc=log(da$intc+1)m4=garchFit(~1+garch(1,1),data=intc)summary(m4)Title:GARCH ModellingCall:garchFit(formula = ~1 + garch(1, 1), data = intc)Mean and Variance Equation:data ~ 1 + garch(1, 1)<environment: 0x000000001d475658>[data = intc]Conditional Distribution:normCoefficient(s):mu omega alpha1 beta10.01126568 0.00091902 0.08643831 0.85258554Std. Errors:based on HessianError Analysis:Estimate Std. Error t value Pr(>|t|)mu 0.0112657 0.0053931 2.089 0.03672 *omega 0.0009190 0.0003888 2.364 0.01808 *alpha1 0.0864383 0.0265439 3.256 0.00113 **beta1 0.8525855 0.0394322 21.622 < 2e-16 ***---Signif. codes:0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1Log Likelihood:312.3307 normalized: 0.7034475Description:Fri Jan 27 21:49:06 2017 by user: xuanStandardised Residuals Tests:Statistic p-ValueJarque-Bera Test R Chi^2 174.904 0Shapiro-Wilk Test R W 0.9709615 1.030282e-07Ljung-Box Test R Q(10) 8.016844 0.6271916Ljung-Box Test R Q(15) 15.5006 0.4159946Ljung-Box Test R Q(20) 16.41549 0.6905368Ljung-Box Test R^2 Q(10) 0.8746345 0.9999072Ljung-Box Test R^2 Q(15) 11.35935 0.7267295Ljung-Box Test R^2 Q(20) 12.55994 0.8954573LM Arch Test R TR^2 10.51401 0.5709617Information Criterion Statistics:AIC BIC SIC HQIC-1.388877 -1.351978 -1.389037 -1.374326
三、主要的函数结构----------------------------------------------
garchFit函数
match.xxx函数....
各种判断可以忽略
.garchArgsParser
.garchFit函数
ans最终的返回结果
.garchFit函数
.garchInitSeries函数
.garchInitParameters函数
.garchOptimizeLLH函数
.garchRnlminb函数
nlminb函数
.garchLLH函数
.Fortran("garchllh")
四、调试----------------------------------------------
(4.0)garchFit函数
function (formula = ~garch(1, 1), data = dem2gbp, init.rec = c("mci","uev"), delta = 2, skew = 1, shape = 4, cond.dist = c("norm","snorm", "ged", "sged", "std", "sstd", "snig", "QMLE"),include.mean = TRUE, include.delta = NULL, include.skew = NULL,include.shape = NULL, leverage = NULL, trace = TRUE, algorithm = c("nlminb","lbfgsb", "nlminb+nm", "lbfgsb+nm"), hessian = c("ropt","rcd"), control = list(), title = NULL, description = NULL,...){DEBUG = FALSEinit.rec = match.arg(init.rec)cond.dist = match.arg(cond.dist)hessian = match.arg(hessian)algorithm = match.arg(algorithm)CALL = match.call()Name = capture.output(substitute(data))if (is.character(data)) {eval(parse(text = paste("data(", data, ")")))data = eval(parse(text = data))}data <- as.data.frame(data)if (isUnivariate(data)) {colnames(data) <- "data"}else {uniqueNames = unique(sort(colnames(data)))if (is.null(colnames(data))) {stop("Column names of data are missing.")}if (length(colnames(data)) != length(uniqueNames)) {stop("Column names of data are not unique.")}}if (length(formula) == 3 && isUnivariate(data))formula[2] <- NULLif (length(formula) == 2) {if (isUnivariate(data)) {formula = as.formula(paste("data", paste(formula,collapse = " ")))}else {stop("Multivariate data inputs require lhs for the formula.")}}robust.cvar <- (cond.dist == "QMLE")args = .garchArgsParser(formula = formula, data = data,trace = FALSE)if (DEBUG)print(list(formula.mean = args$formula.mean, formula.var = args$formula.var,series = args$series, init.rec = init.rec, delta = delta,skew = skew, shape = shape, cond.dist = cond.dist,include.mean = include.mean, include.delta = include.delta,include.skew = include.skew, include.shape = include.shape,leverage = leverage, trace = trace, algorithm = algorithm,hessian = hessian, robust.cvar = robust.cvar, control = control,title = title, description = description))ans = .garchFit(formula.mean = args$formula.mean, formula.var = args$formula.var,series = args$series, init.rec, delta, skew, shape,cond.dist, include.mean, include.delta, include.skew,include.shape, leverage, trace, algorithm, hessian,robust.cvar, control, title, description, ...)ans@call = CALLattr(formula, "data") <- paste("data = ", Name, sep = "")ans@formula = formulaans}
以下四句都是对传入的参数是否math(合法、规范)做检验的
init.rec = match.arg(init.rec)cond.dist = match.arg(cond.dist)hessian = match.arg(hessian)algorithm = match.arg(algorithm)CALL = match.call()
我们可以按下
进入函数查看
进入函数查看截取match.arg函数中的几句作说明：
formal.args <- formals(sys.function(sys.parent())).......................................if (identical(arg, choices))return(arg[1L])
如果传入和默认的一样，说明没有传入，就返回第一个作为默认值
可以点击
跳出函数
跳出函数(4.0.1).garchArgsParser函数
args = .garchArgsParser(formula = formula, data = data, trace = FALSE)
function (formula, data, trace = FALSE){allVars = unique(sort(all.vars(formula)))allVarsTest = mean(allVars %in% colnames(data))if (allVarsTest != 1) {print(allVars)print(colnames(data))stop("Formula and data units do not match.")}formula.lhs = as.character(formula)[2]mf = match.call(expand.dots = FALSE)if (trace) {cat("\nMatched Function Call:\n ")print(mf)}m = match(c("formula", "data"), names(mf), 0)mf = mf[c(1, m)]mf[[1]] = as.name(".garchModelSeries")mf$fake = FALSEmf$lhs = TRUEif (trace) {cat("\nModelSeries Call:\n ")print(mf)}x = eval(mf, parent.frame())if (trace)print(x)x = as.vector(x[, 1])names(x) = rownames(data)if (trace)print(x)allLabels = attr(terms(formula), "term.labels")if (trace) {cat("\nAll Term Labels:\n ")print(allLabels)}if (length(allLabels) == 2) {formula.mean = as.formula(paste("~", allLabels[1]))formula.var = as.formula(paste("~", allLabels[2]))}else if (length(allLabels) == 1) {formula.mean = as.formula("~ arma(0, 0)")formula.var = as.formula(paste("~", allLabels[1]))}if (trace) {cat("\nMean Formula:\n ")print(formula.mean)cat("\nVariance Formula:\n ")print(formula.var)}ans <- list(formula.mean = formula.mean, formula.var = formula.var,formula.lhs = formula.lhs, series = x)ans}
提下这个函数里面的一些信息:
data是数据
%in%运算符就是math的作用 ?'%in%'
x就是原始数据
(需要特别说明的是,x是从data里取出来后as.vector的，被转化为向量后，会四舍五入显示了，所以如果直接在environment pane中查看有点不橡原始数据，比如intc的第一个值是0.0099998346，而这里x的值显示是0.01)
函数运行到最后一行的时候，environment pane中的部分信息。
(方程的基本形式已经被解析出来了)
(4.0.2).garchFit函数
ans = .garchFit(formula.mean = args$formula.mean, formula.var = args$formula.var,series = args$series, init.rec, delta, skew, shape,cond.dist, include.mean, include.delta, include.skew,include.shape, leverage, trace, algorithm, hessian,robust.cvar, control, title, description, ...)
我们为什么分析了.garchArgsParser这个解析函数，就是因为.garchFit函数用到的许多参数都是从args中取出的。
另外提到一下，带.开头的函数和变量都是内部的，不会直接显示出来的。
如果是变量名.xxx，这样的形式，那么需要我们自己在控制台输入.xxx查看，不能在envir pane中查看。
如果要查看内部函数，则
单击可以切换到对应的源代码和变量
可以选择环境，一般情况是在global environment.garchFit函数的源码
function (formula.mean = ~arma(0, 0), formula.var = ~garch(1,1), series, init.rec = c("mci", "uev"), delta = 2, skew = 1,shape = 4, cond.dist = c("norm", "snorm", "ged", "sged","std", "sstd", "QMLE"), include.mean = TRUE, include.delta = NULL,include.skew = NULL, include.shape = NULL, leverage = NULL,trace = TRUE, algorithm = c("sqp", "nlminb", "lbfgsb", "nlminb+nm","lbfgsb+nm"), hessian = c("ropt", "rcd"), robust.cvar,control = list(), title = NULL, description = NULL, ...){DEBUG <- FALSEif (DEBUG)print("Formula Specification ...")fcheck = rev(all.names(formula.mean))[1]if (fcheck == "ma") {stop("Use full formula: arma(0,q) for ma(q)")}else if (fcheck == "ar") {stop("Use full formula expression: arma(p,0) for ar(p)")}if (DEBUG)print("Recursion Initialization ...")if (init.rec[1] != "mci" & algorithm[1] != "sqp") {stop("Algorithm only supported for mci Recursion")}.StartFit <- Sys.time()if (DEBUG)print("Generate Control List ...")con <- .garchOptimizerControl(algorithm, cond.dist)con[(namc <- names(control))] <- controlif (DEBUG)print("Initialize Time Series ...")data <- seriesscale <- if (con$xscale)sd(series)else 1series <- series/scale.series <- .garchInitSeries(formula.mean = formula.mean,formula.var = formula.var, cond.dist = cond.dist[1],series = series, scale = scale, init.rec = init.rec[1],h.start = NULL, llh.start = NULL, trace = trace).setfGarchEnv(.series = .series)if (DEBUG)print("Initialize Model Parameters ...").params <- .garchInitParameters(formula.mean = formula.mean,formula.var = formula.var, delta = delta, skew = skew,shape = shape, cond.dist = cond.dist[1], include.mean = include.mean,include.delta = include.delta, include.skew = include.skew,include.shape = include.shape, leverage = leverage,algorithm = algorithm[1], control = con, trace = trace).setfGarchEnv(.params = .params)if (DEBUG)print("Select Conditional Distribution ...").setfGarchEnv(.garchDist = .garchSetCondDist(cond.dist[1]))if (DEBUG)print("Estimate Model Parameters ...").setfGarchEnv(.llh = 1e+99).llh <- .getfGarchEnv(".llh")fit = .garchOptimizeLLH(hessian, robust.cvar, trace)if (DEBUG)print("Add to fit ...").series <- .getfGarchEnv(".series").params <- .getfGarchEnv(".params")names(.series$h) <- NULLfit$series = .seriesfit$params = .paramsif (DEBUG)print("Retrieve Residuals and Fitted Values ...")residuals = .series$zfitted.values = .series$x - residualsh.t = .series$hif (.params$includes["delta"])deltainv = 1/fit$par["delta"]else deltainv = 1/fit$params$deltasigma.t = (.series$h)^deltainvif (DEBUG)print("Standard Errors and t-Values ...")fit$cvar <- if (robust.cvar)(solve(fit$hessian) %*% (t(fit$gradient) %*% fit$gradient) %*%solve(fit$hessian))else -solve(fit$hessian)fit$se.coef = sqrt(diag(fit$cvar))fit$tval = fit$coef/fit$se.coeffit$matcoef = cbind(fit$coef, fit$se.coef, fit$tval, 2 *(1 - pnorm(abs(fit$tval))))dimnames(fit$matcoef) = list(names(fit$tval), c(" Estimate"," Std. Error", " t value", "Pr(>|t|)"))if (DEBUG)print("Add Title and Description ...")if (is.null(title))title = "GARCH Modelling"if (is.null(description))description = description()Time = Sys.time() - .StartFitif (trace) {cat("\nTime to Estimate Parameters:\n ")print(Time)}new("fGARCH", call = as.call(match.call()), formula = as.formula(paste("~",formula.mean, "+", formula.var)), method = "Max Log-Likelihood Estimation",data = data, fit = fit, residuals = residuals, fitted = fitted.values,h.t = h.t, sigma.t = as.vector(sigma.t), title = as.character(title),description = as.character(description))}
在正式分析函数的源码之前
先看一下模型变量m4的结构：
> str(m4)
Formal class 'fGARCH' [package "fGarch"] with 11 slots
..@ call : language garchFit(formula = ~1 + garch(1, 1), data = intc, trace = F)
..@ formula :Class 'formula' language data ~ 1 + garch(1, 1)
.. .. ..- attr(*, ".Environment")=<environment: 0x0000000003e60360>
.. .. ..- attr(*, "data")= chr "data = intc"
..@ method : chr "Max Log-Likelihood Estimation"
..@ data : Named num [1:444] 0.01 -0.15 0.0671 0.0829 -0.1103 ...
.. ..- attr(*, "names")= chr [1:444] "1" "2" "3" "4" ...
..@ fit :List of 17
.. ..$ par : Named num [1:4] 0.011266 0.000919 0.086438 0.852586 #参数
.. .. ..- attr(*, "names")= chr [1:4] "mu" "omega" "alpha1" "beta1"
.. ..$ objective : num 604
.. ..$ convergence: int 1
.. ..$ iterations : int 19
.. ..$ evaluations: Named int [1:2] 36 105
.. .. ..- attr(*, "names")= chr [1:2] "function" "gradient"
.. ..$ message : chr "singular convergence (7)"
.. ..$ value : Named num -312 (为什么显示是负的，而输出是正的？，因为nlminb是取最小值的)
.. .. ..- attr(*, "names")= chr "LogLikelihood"
.. ..$ coef : Named num [1:4] 0.011266 0.000919 0.086438 0.852586
.. .. ..- attr(*, "names")= chr [1:4] "mu" "omega" "alpha1" "beta1"
.. ..$ llh : Named num -312 #似然函数的最大值取负数
.. .. ..- attr(*, "names")= chr "LogLikelihood"
.. ..$ hessian : num [1:4, 1:4] -35115 108140 -253 919 108140 ...
.. .. ..- attr(*, "dimnames")=List of 2
.. .. .. ..$ : chr [1:4] "mu" "omega" "alpha1" "beta1"
.. .. .. ..$ : chr [1:4] "mu" "omega" "alpha1" "beta1"
.. .. ..- attr(*, "time")=Class 'difftime' atomic [1:1] 0.021
.. .. .. .. ..- attr(*, "units")= chr "secs"
.. ..$ ics : Named num [1:4] -1.39 -1.35 -1.39 -1.37
.. .. ..- attr(*, "names")= chr [1:4] "AIC" "BIC" "SIC" "HQIC"
.. ..$ series :List of 10
.. .. ..$ model : chr [1:2] "arma" "garch"
.. .. ..$ order : Named num [1:4] 0 0 1 1
.. .. .. ..- attr(*, "names")= chr [1:4] "u" "v" "p" "q"
.. .. ..$ max.order: num 1
.. .. ..$ z : num [1:444] -0.00127 -0.16128 0.0558 0.07168 -0.12161 ... 残差
.. .. ..$ h : num [1:444] 0.016 0.0146 0.0156 0.0145 0.0137 ...
.. .. ..$ x : Named num [1:444] 0.01 -0.15 0.0671 0.0829 -0.1103 ...
.. .. .. ..- attr(*, "names")= chr [1:444] "1" "2" "3" "4" ...
.. .. ..$ scale : num 0.127
.. .. ..$ init.rec : chr "mci"
.. .. ..$ h.start : num 2
.. .. ..$ llh.start: num 1
.. ..$ params :List of 14
.. .. ..$ params : Named num [1:8] 0.011266 0.000919 0.086438 0.1 0.852586 ...
.. .. .. ..- attr(*, "names")= chr [1:8] "mu" "omega" "alpha1" "gamma1" ...
.. .. ..$ U : Named num [1:8] -1.13 1.00e-06 1.00e-08 -1.00 1.00e-08 ...
.. .. .. ..- attr(*, "names")= chr [1:8] "mu" "omega" "alpha1" "gamma1" ...
.. .. ..$ V : Named num [1:8] 1.13 100 1 1 1 ...
.. .. .. ..- attr(*, "names")= chr [1:8] "mu" "omega" "alpha1" "gamma1" ...
.. .. ..$ includes : Named logi [1:8] TRUE TRUE TRUE FALSE TRUE FALSE ...
.. .. .. ..- attr(*, "names")= chr [1:8] "mu" "omega" "alpha1" "gamma1" ...
.. .. ..$ index : Named int [1:4] 1 2 3 5
.. .. .. ..- attr(*, "names")= chr [1:4] "mu" "omega" "alpha1" "beta1"
.. .. ..$ mu : Named num 0.113
.. .. .. ..- attr(*, "names")= chr "mu"
.. .. ..$ delta : num 2 #默认是2
.. .. ..$ skew : num 1
.. .. ..$ shape : num 4
.. .. ..$ cond.dist : chr "norm"
.. .. ..$ leverage : logi FALSE
.. .. ..$ persistence: Named num 0.9
.. .. .. ..- attr(*, "names")= chr "persistence"
.. .. ..$ control :List of 19
.. .. .. ..$ fscale : logi TRUE
.. .. .. ..$ xscale : logi TRUE
.. .. .. ..$ algorithm: chr "nlminb"
.. .. .. ..$ llh : chr "internal"
.. .. .. ..$ tol1 : num 1
.. .. .. ..$ tol2 : num 1
.. .. .. ..$ MIT : num 2000
.. .. .. ..$ MFV : num 5000
.. .. .. ..$ MET : num 5
.. .. .. ..$ MEC : num 2
.. .. .. ..$ MER : num 1
.. .. .. ..$ MES : num 4
.. .. .. ..$ XMAX : num 1000
.. .. .. ..$ TOLX : num 1e-10
.. .. .. ..$ TOLC : num 1e-06
.. .. .. ..$ TOLG : num 1e-06
.. .. .. ..$ TOLD : num 1e-06
.. .. .. ..$ TOLS : num 1e-04
.. .. .. ..$ RPF : num 0.01
.. .. ..$ llh : num 604
.. ..$ cvar : num [1:4, 1:4] 2.91e-05 1.06e-07 -1.51e-05 6.60e-06 1.06e-07 ...
.. .. ..- attr(*, "dimnames")=List of 2
.. .. .. ..$ : chr [1:4] "mu" "omega" "alpha1" "beta1"
.. .. .. ..$ : chr [1:4] "mu" "omega" "alpha1" "beta1"
.. ..$ se.coef : Named num [1:4] 0.005393 0.000389 0.026544 0.039432
.. .. ..- attr(*, "names")= chr [1:4] "mu" "omega" "alpha1" "beta1"
.. ..$ tval : Named num [1:4] 2.09 2.36 3.26 21.62
.. .. ..- attr(*, "names")= chr [1:4] "mu" "omega" "alpha1" "beta1"
.. ..$ matcoef : num [1:4, 1:4] 0.011266 0.000919 0.086438 0.852586 0.005393 ...
.. .. ..- attr(*, "dimnames")=List of 2
.. .. .. ..$ : chr [1:4] "mu" "omega" "alpha1" "beta1"
.. .. .. ..$ : chr [1:4] " Estimate" " Std. Error" " t value" "Pr(>|t|)"
..@ residuals : num [1:444] -0.00127 -0.16128 0.0558 0.07168 -0.12161 ...
..@ fitted : Named num [1:444] 0.0113 0.0113 0.0113 0.0113 0.0113 ...
#是.series$x - residuals 其实这个x就是r(i)，
#那么fitted就是mu了(只不过这里的值是被四舍五入处理了不明显)
代码 fitted.values = .series$x - residuals
fitted = fitted.values
.. ..- attr(*, "names")= chr [1:444] "1" "2" "3" "4" ...
..@ h.t : num [1:444] 0.016 0.0146 0.0156 0.0145 0.0137 ... #条件方差
..@ sigma.t : num [1:444] 0.127 0.121 0.125 0.12 0.117 ... #是条件标准差，等于volatility(m4)
代码 deltainv = 1/fit$par["delta"]=0.
sigma.t = (.series$h)^deltainv
..@ title : chr "GARCH Modelling"
..@ description: chr "Sun Jan 22 10:09:46 2017 by user: xuan"
?'@'
Extract or replace the contents of a slot in a object with a formal (S4) class structure.
从S4类里取出slot可以适应@运算符
hessian
a string denoting how the Hessian matrix should be evaluated, either hessian ="rcd", or "ropt", the default, "rcd" is a central difference approximation implemented in R and "ropt" use the internal R function optimhess.
黑塞矩阵（Hessian Matrix），又译作海森矩阵、海瑟矩阵、海塞矩阵等，是一个多元函数的二阶偏导数构成的方阵，描述了函数的局部曲率
Details
"QMLE" stands for Quasi-Maximum Likelihood Estimation, which assumes normal distribution and uses robust standard errors for inference.
Bollerslev and Wooldridge (1992) proved that if the mean and the volatility equations are correctly specified, the QML estimates are consistent and asymptotically normally distributed. However, the estimates are not efficient and “the efficiency loss can be marked under asymmetric ... distributions” (Bollerslev and Wooldridge (1992), p. 166).
The robust variance-covariance matrix of the estimates equals the (Eicker-White) sandwich estimator, i.e.
V = H^(-1) G' G H^(-1),
where V denotes the variance-covariance matrix, H stands for the Hessian and G represents the matrix of contributions to the gradient(倾斜度), the elements of which are defined as
G_{t,i} = derivative of l_{t} w.r.t. zeta_{i},
where l_{t} is the log likelihood of the t-th observation and zeta_{i} is the i-th estimated parameter. See sections 10.3 and 10.4 in Davidson and MacKinnon (2004) for a more detailed description of the robust variance-covariance matrix.
Value
garchFit
returns a S4 object of class "fGARCH" with the following slots:
@call
the call of the garch function.
@formula
a list with two formula entries, one for the mean and the other one for the variance equation.
@method
a string denoting the optimization method, by default the returneds string is "Max Log-Likelihood Estimation".
@data
a list with one entry named x, containing the data of the time series to be estimated, the same as given by the input argument series.
@fit
a list with the results from the parameter estimation. The entries of the list depend on the selected algorithm, see below.
@residuals
a numeric vector with the (raw, unstandardized) residual values.
@fitted
a numeric vector with the fitted values.
@h.t
a numeric vector with the conditional variances (h.t = sigma.t^delta).
@sigma.t
a numeric vector with the conditional standard deviation.
@title
a title string.
@description
a string with a brief description.
The entries of the @fit slot show the results from the optimization.
其中1
scale <- if (con$xscale)sd(series)series <- series/scale
是对数据进行标准化
（注意，标准化之后，数据和原始数据不同啦，在后面运算时，不要忘了~）
我们可以在调试界面的控制台直接输入命令，来查看变量在当前环境和一定语句代运行完后的值
这个所谓的标准化，并没有减去均值的，要注意！
其中2
.series <- .garchInitSeries(formula.mean = formula.mean,formula.var = formula.var, cond.dist = cond.dist[1],series = series, scale = scale, init.rec = init.rec[1],h.start = NULL, llh.start = NULL, trace = trace).setfGarchEnv(.series = .series)
注意这种.setfGarchEnv的做法，就是对环境变量中的变量赋值
还有.getfGarchEnv，可以提高自己的编程能力。
.garchInitSeries函数
function (formula.mean, formula.var, cond.dist, series, scale,init.rec, h.start, llh.start, trace){mm = length(formula.mean)if (mm != 2)stop("Mean Formula misspecified")end = regexpr("\\(", as.character(formula.mean[mm])) - 1model.mean = substr(as.character(formula.mean[mm]), 1, end)if (!any(c("ar", "ma", "arma") == model.mean))stop("formula.mean must be one of: ar, ma, arma")mv = length(formula.var)if (mv != 2)stop("Variance Formula misspecified")end = regexpr("\\(", as.character(formula.var[mv])) - 1model.var = substr(as.character(formula.var[mv]), 1, end)if (!any(c("garch", "aparch") == model.var))stop("formula.var must be one of: garch, aparch")model.order = as.numeric(strsplit(strsplit(strsplit(as.character(formula.mean),"\\(")[[2]][2], "\\)")[[1]], ",")[[1]])u = model.order[1]v = 0if (length(model.order) == 2)v = model.order[2]maxuv = max(u, v)if (u < 0 | v < 0)stop("*** ARMA orders must be positive.")model.order = as.numeric(strsplit(strsplit(strsplit(as.character(formula.var),"\\(")[[2]][2], "\\)")[[1]], ",")[[1]])p = model.order[1]q = 0if (length(model.order) == 2)q = model.order[2]if (p + q == 0)stop("Misspecified GARCH Model: Both Orders are zero!")maxpq = max(p, q)if (p < 0 | q < 0)stop("*** GARCH orders must be positive.")max.order = max(maxuv, maxpq)if (is.null(h.start))h.start = max.order + 1if (is.null(llh.start))llh.start = 1if (init.rec != "mci" & model.var != "garch") {stop("Algorithm only supported for mci Recursion")}ans = list(model = c(model.mean, model.var), order = c(u = u,v = v, p = p, q = q), max.order = max.order,z = rep(0, times = length(series)), #这里的0，长度为444h = rep(var(series), times = length(series)), #h是方差是1(因为此时的series是被标准化过的)，长度为444x = series, scale = scale, init.rec = init.rec, h.start = h.start,llh.start = llh.start)ans}
其结果
这个函数的作用，主要是形成z、h之类的存储容器
其中3
.params <- .garchInitParameters(formula.mean = formula.mean,formula.var = formula.var, delta = delta, skew = skew,shape = shape, cond.dist = cond.dist[1], include.mean = include.mean,include.delta = include.delta, include.skew = include.skew,include.shape = include.shape, leverage = leverage,algorithm = algorithm[1], control = con, trace = trace).setfGarchEnv(.params = .params)
.garchInitParameters函数
function (formula.mean, formula.var, delta, skew, shape, cond.dist,include.mean, include.delta, include.skew, include.shape,leverage, algorithm, control, trace){.DEBUG = FALSE.series <- .getfGarchEnv(".series")model.order = as.numeric(strsplit(strsplit(strsplit(as.character(formula.mean),"\\(")[[2]][2], "\\)")[[1]], ",")[[1]])u = model.order[1]v = 0if (length(model.order) == 2)v = model.order[2]model.order = as.numeric(strsplit(strsplit(strsplit(as.character(formula.var),"\\(")[[2]][2], "\\)")[[1]], ",")[[1]])p = model.order[1]q = 0if (length(model.order) == 2)q = model.order[2]model.var = .series$model[2]if (is.null(include.delta)) {if (model.var == "garch") {include.delta = FALSE}else {include.delta = TRUE}}if (is.null(leverage)) {if (model.var == "garch") {leverage = FALSE}else {leverage = TRUE}}if (cond.dist == "t")cond.dist = "std"skewed.dists = c("snorm", "sged", "sstd", "snig")if (is.null(include.skew)) {if (any(skewed.dists == cond.dist)) {include.skew = TRUE}else {include.skew = FALSE}}shaped.dists = c("ged", "sged", "std", "sstd", "snig")if (is.null(include.shape)) {if (any(shaped.dists == cond.dist)) {include.shape = TRUE}else {include.shape = FALSE}}Names = c("mu",if (u > 0) paste("ar", 1:u, sep = ""),if (v > 0) paste("ma", 1:v, sep = ""), "omega",if (p > 0) paste("alpha", 1:p, sep = ""),if (p > 0) paste("gamma", 1:p, sep = ""),if (q > 0) paste("beta", 1:q, sep = ""),"delta", "skew", "shape")fit.mean = arima(.series$x, order = c(u, 0, v), include.mean = include.mean)$coefalpha.start = 0.1beta.start = 0.8params = c(if (include.mean) fit.mean[length(fit.mean)] else 0, #最后一项就是intercept，所以就是mu啊！if (u > 0) fit.mean[1:u],if (v > 0) fit.mean[(u + 1):(length(fit.mean) - as.integer(include.mean))], var(.series$x, na.rm = TRUE) * (1 - alpha.start - beta.start),if (p > 0) rep(alpha.start/p, times = p),if (p > 0) rep(0.1, times = p),if (q > 0) rep(beta.start/q, times = q),delta, skew, shape)names(params) = NamesTINY = 1e-08USKEW = 1/10USHAPE = 1if (cond.dist == "snig")USKEW = -0.99U = c(-10 * abs(mean(.series$x)), if (u > 0) rep(-1 + TINY,times = u), if (v > 0) rep(-1 + TINY, times = v), 1e-06 *var(.series$x), if (p > 0) rep(0 + TINY, times = p),if (p > 0) rep(-1 + TINY, times = p), if (q > 0) rep(0 +TINY, times = q), 0, USKEW, USHAPE)names(U) = NamesVSKEW = 10VSHAPE = 10if (cond.dist == "snig")VSKEW = 0.99V = c(10 * abs(mean(.series$x)), if (u > 0) rep(1 - TINY,times = u), if (v > 0) rep(1 - TINY, times = v), 100 *var(.series$x), if (p > 0) rep(1 - TINY, times = p),if (p > 0) rep(1 - TINY, times = p), if (q > 0) rep(1 -TINY, times = q), 2, VSKEW, VSHAPE)names(V) = Namesincludes = c(include.mean, if (u > 0) rep(TRUE, times = u),if (v > 0) rep(TRUE, times = v), TRUE, if (p > 0) rep(TRUE,times = p), if (p > 0) rep(leverage, times = p),if (q > 0) rep(TRUE, times = q), include.delta, include.skew,include.shape)names(includes) = Namesindex = (1:length(params))[includes == TRUE]names(index) = names(params)[includes == TRUE]alpha <- beta <- NULLif (p > 0)alpha = params[substr(Names, 1, 5) == "alpha"]if (p > 0 & leverage)gamma = params[substr(Names, 1, 5) == "gamma"]if (p > 0 & !leverage)gamma = rep(0, times = p)if (q > 0)beta = params[substr(Names, 1, 4) == "beta"]if (.series$model[2] == "garch") {persistence = sum(alpha) + sum(beta)}else if (.series$model[2] == "aparch") {persistence = sum(beta)for (i in 1:p)persistence = persistence + alpha[i] * garchKappa(cond.dist, gamma[i], params["delta"], params["skew"], params["shape"])}names(persistence) = "persistence"list(params = params, U = U, V = V, includes = includes,index = index, mu = params[1], delta = delta, skew = skew,shape = shape, cond.dist = cond.dist, leverage = leverage,persistence = persistence, control = control)}
这个函数的作用，主要是形成参数的初始值吧
fit.mean = arima(.series$x, order = c(u, 0, v), include.mean = include.mean)$coefparams = c(if (include.mean) fit.mean[length(fit.mean)] else 0
include.mean默认是T，fit.mean就相当于是对原始数据标准化后的序列进行arima建模，只不过ar和ma的阶数都是0，只剩下截距项，作为mu的初始值，其中length(fit.mean)就是模型的截距项intercept。
此时的mu是0.1128933
此时的mu是0.1128933
事实上，如果是到底为止，那么就参数的值其实还是没有被求出，为什么后面在.garchFit函数中却直接开始赋值了呢？这是因为后面的.garchOptimizeLLH函数借助.get/.setfGarchEnv()函数，取得了.series和.prameter
其中4 (4.0.2.4).garchOptimizeLLH函数
.garchOptimizeLLH函数(删去了一些用不到的代码)
- function (hessian = hessian, robust.cvar, trace)
{.series <- .getfGarchEnv(".series").params <- .getfGarchEnv(".params")INDEX = .params$indexalgorithm = .params$control$algorithm[1] #algorithm是nlminbTOL1 = .params$control$tol1TOL2 = .params$control$tol2if (algorithm == "nlminb" | algorithm == "nlminb+nm") {fit <- .garchRnlminb(.params, .series, .garchLLH, trace).params$llh = fit$llh.params$params[INDEX] = fit$par.setfGarchEnv(.params = .params)}.params$llh = fit$llh.params$params[INDEX] = fit$par.setfGarchEnv(.params = .params)if (hessian == "ropt") {fit$hessian <- -.garchRoptimhess(par = fit$par, .params = .params,.series = .series)titleHessian = "R-optimhess"}...............if (.params$control$xscale) {.series$x <- .series$x * .series$scale #将标准化后的值还原为原始的数值了if (.params$include["mu"])fit$coef["mu"] <- fit$par["mu"] <- .params$params["mu"] <- .params$params["mu"] *.series$scaleif (.params$include["omega"])fit$coef["omega"] <- fit$par["omega"] <- .params$params["omega"] <- .params$params["omega"] *.series$scale^(.params$params["delta"]).setfGarchEnv(.params = .params).setfGarchEnv(.series = .series)}if (.params$control$xscale) {if (.params$include["mu"]) {fit$hessian[, "mu"] <- fit$hessian[, "mu"]/.series$scalefit$hessian["mu", ] <- fit$hessian["mu", ]/.series$scale}if (.params$include["omega"]) {fit$hessian[, "omega"] <- fit$hessian[, "omega"]/.series$scale^(.params$params["delta"])fit$hessian["omega", ] <- fit$hessian["omega", ]/.series$scale^(.params$params["delta"])}}.llh <- fit$llh <- fit$value <- .garchLLH(fit$par, trace = FALSE,fGarchEnv = TRUE).series <- .getfGarchEnv(".series")if (robust.cvar)fit$gradient <- -.garchRCDAGradient(par = fit$par, .params = .params,.series = .series)N = length(.series$x)NPAR = length(fit$par)fit$ics = c(AIC = c((2 * fit$value)/N + 2 * NPAR/N), BIC = (2 *fit$value)/N + NPAR * log(N)/N, SIC = (2 * fit$value)/N +log((N + 2 * NPAR)/N), HQIC = (2 * fit$value)/N + (2 *NPAR * log(log(N)))/N)names(fit$ics) <- c("AIC", "BIC", "SIC", "HQIC")fit}
其中4 (4.0.2.4.1).garchRnlminb函数
.garchRnlminb函数
function (.params, .series, .garchLLH, trace){if (trace)cat("\nR coded nlminb Solver: \n\n")INDEX = .params$indexparscale = rep(1, length = length(INDEX))names(parscale) = names(.params$params[INDEX])parscale["omega"] = var(.series$x)^(.params$delta/2)parscale["mu"] = abs(mean(.series$x)) #这个并非mu的初始值TOL1 = .params$control$tol1fit <- nlminb(start = .params$params[INDEX], objective = .garchLLH,lower = .params$U[INDEX], upper = .params$V[INDEX],scale = 1/parscale, control = list(eval.max = 2000,iter.max = 1500, rel.tol = 1e-14 * TOL1, x.tol = 1e-14 *TOL1, trace = as.integer(trace)), fGarchEnv = FALSE)fit$value <- fit.llh <- fit$objectivenames(fit$par) = names(.params$params[INDEX]).garchLLH(fit$par, trace = FALSE, fGarchEnv = TRUE)fit$coef <- fit$parfit$llh <- fit$objectivefit}
关于nlminb函数，请看笔记：
.params$params[INDEX]是初始值
objective = .garchLLH是被优化的函数
数据呢？怎么没见传入数据？
.garchRnlminb函数函数运行完，则会估计出参数：
fit$par
mu omega alpha1 beta1
0.08876900 0.05705989 0.08643831 0.85258554
这个值呢，是别标准化后的参数值
fit$coef["mu"] <- fit$par["mu"] <- .params$params["mu"] <- .params$params["mu"] *.series$scale
通过上述语句转化为原始序列对应的参数即可！
Browse[4]> fit$coef
mu omega alpha1 beta1
0.0112656813 0.0009190163 0.0864383140 0.8525855370
这就是模型最终输出的参数！
运行到.llh <- fit$llh <- fit$value <- .garchLLH(fit$par, trace = FALSE,
fGarchEnv = TRUE)
这里跳进去
得到其中用到的参数.garchLLH函数
.garchLLHfunction (params, trace = TRUE, fGarchEnv = FALSE){DEBUG = FALSEif (DEBUG)print("Entering Function .garchLLH").series <- .getfGarchEnv(".series").params <- .getfGarchEnv(".params").garchDist <- .getfGarchEnv(".garchDist") #概率密度函数.llh <- .getfGarchEnv(".llh") #对数似然函数if (DEBUG)print(.params$control$llh)if (.params$control$llh == "internal") {if (DEBUG)print("internal")return(.aparchLLH.internal(params, trace = trace, fGarchEnv = fGarchEnv)) #结束}else if (.params$control$llh == "filter") {if (DEBUG)print("filter")return(.aparchLLH.filter(params, trace = trace, fGarchEnv = fGarchEnv))}else if (.params$control$llh == "testing") {if (DEBUG)print("testing")return(.aparchLLH.testing(params, trace = trace, fGarchEnv = fGarchEnv))}else {stop("LLH is neither internal, testing, nor filter!")}}<environment: namespace:fGarch>
.aparchLLH.internal函数
其中我进去看了下
function (params, trace = TRUE, fGarchEnv = TRUE){DEBUG = FALSEif (DEBUG)print("Entering Function .garchLLH.internal").series <- .getfGarchEnv(".series").params <- .getfGarchEnv(".params").garchDist <- .getfGarchEnv(".garchDist").llh <- .getfGarchEnv(".llh")if (DEBUG)print(.params$control$llh)if (.params$control$llh == "internal") {INDEX <- .params$indexMDIST <- c(norm = 10, QMLE = 10, snorm = 11, std = 20,sstd = 21, ged = 30, sged = 31)[.params$cond.dist]if (.params$control$fscale)NORM <- length(.series$x)else NORM = 1REC <- 1if (.series$init.rec == "uev")REC <- 2MYPAR <- c(REC = REC, LEV = as.integer(.params$leverage),MEAN = as.integer(.params$includes["mu"]), DELTA = as.integer(.params$includes["delta"]),SKEW = as.integer(.params$includes["skew"]), SHAPE = as.integer(.params$includes["shape"]),ORDER = .series$order, NORM = as.integer(NORM))MAX <- max(.series$order)NF <- length(INDEX)N <- length(.series$x)DPARM <- c(.params$delta, .params$skew, .params$shape)fit <- .Fortran("garchllh", N = as.integer(N), Y = as.double(.series$x),Z = as.double(.series$z), H = as.double(.series$h),NF = as.integer(NF), X = as.double(params), DPARM = as.double(DPARM),MDIST = as.integer(MDIST), MYPAR = as.integer(MYPAR),F = as.double(0), PACKAGE = "fGarch")llh <- fit[[10]] #此时llh就是-312if (is.na(llh))llh = .llh + 0.1 * (abs(.llh))if (!is.finite(llh))llh = .llh + 0.1 * (abs(.llh)).setfGarchEnv(.llh = llh)if (fGarchEnv) {.series$h <- fit[[4]].series$z <- fit[[3]] #直到这里，z才从444个0被替换为残差.setfGarchEnv(.series = .series)}}else {stop("LLH is not internal!")}c(LogLikelihood = llh)}
.Fortran是调用Fortran代码的，真是我擦！
Browse[5]> .garchDistfunction (z, hh, skew, shape){dnorm(x = z/hh, mean = 0, sd = 1)/hh}<environment: namespace:fGarch>
写在2017年1月28日00:54:55
农历新年凌晨，新的一年要更加努力~
苍茫大海，旅鼠它来！
