CATS.oxh

CATS : Modelbase : Database : Sample class reference

CATS function members (public by default)
	ACTest_LB(const mResid, iOrder=0)
	Computes the Ljung-Box autocorrelation statistic.
virtual	ACTest_LM(const mResid, const iOrder=1)
	Performs the LM test for autocorrelation.
static	ARCHStats(const mResid, const iOrder=1)
	Univariate ARCH statistics.
static	ARCHTest(const mResid, const iOrder=1)
	Multivariate ARCH test, Luetkepohl, 2004.
static	AlphaBetaStart(const mAlpha, const mBeta, const amG, const amH, const mQtZ0, const mQtZ1, const mQtZ0sq, const cT, const cInitSteps=2, const aAnalysis=<>)
	Get starting values for alpha,beta switching I(1) algorithm using GLS.
static	AlphaBetaStartAnalysis(const mAlpha, const mBeta, const amG, const amH)
	Get starting values for alpha,beta switching I(1) algorithm using GLS.
static	AlphaBetaStartSearch(const mAlpha0, const mBeta0, const fnAlphaBetaStart, const fnEstimate, const cStart, const mxStartIt, const iInfo, const bTrace=FALSE)

static	AlphaBetaSwitching(const mAlpha, const mBeta, const amG, const amH, const mQtZ0, const mQtZ1, const mQtZ0sq, const cT, const dCrit, const iMaxIt, const iInfo, const iLineSearch=2)
	Alpha, beta switching algorithm for estimating restricted Alpha and Beta vectors in the I(1)-model.
static	AlphaKnown(const mA, const mBeta1, const mS00, const mS01, const mS11)
	Estimate under the restriction H0: Alpha = (G,G_orthogonal*Psi) where G is a known matrix
static	ApplyColumnScale(const mA, const mB, const mC, const vNormRow)
	Applies a normalization to columns of a matrix, AsBs = (A . v)(B ./ v) = AB.
static	ApplyColumnScaleVec( vP, const cAlpha, amG, const vOldScale, const bApply=TRUE)
	Apply/remove the scale transformation to theta or phi in vP
static	AutoIdentify1( amH, const iR, const abIsIdentified)
	Automatic identification of the cointegration space.
static	AutoIdentify2(const amH, const iR, const abIsIdentified)
	Automatic identification of the cointegration space.
static	BartlettBeta(const iS, const iType, const mAlpha, const mBeta, const iRank, fnGetGamma, const mS00, const cT, const iDetTrend, const cSeas, const iMethod=0)
	Calculates the Bartlett correction factor for tests on the cointegrating relations.
static	BartlettRank(const mAlpha, const mBeta, const iRank, fnGetGamma, const mS00, const cT, const iDetTrend, const bWarn=TRUE, const iMethod=0)
	Calculates the rank test Bartlett small sample correction of Johansen (2002) given the rank of Pi.
static	Bartlett_c2(const mP, const mVpsi, const iMethod=0)
	Four methods to compute c_2: direct, summation, very large matrices, approximate.
static	BetaStdErrors( amH, const viNorm, const mSM1, const cT)
	Calculates the asymptotic standard errors of beta for the I(1) model if the restrictions identify beta.
static	BetaStdErrors_DiagH( amH, const viNorm)
	Calculates the DiagCat(amH) with normalization columns removed.
static	BetaSwitching(const mG, const amH, const vVpg, const mBeta, const mQtZ0, const mQtZ1, const mQtZ0sq, const cT, const dCrit=1e-6, const iMaxIt=1000, const iInfo=0, const iLineSearch=2)
	Switching algorithm for estimating restricted Beta vectors in the I(1)-model.
static	BlockInvHess(const mHessian, const aSizes)
	Creates an array of matrices with minus the inverse of diagonal blocks of a matrix.
	BootstrapRankTest( cB=399, const iType=0)
	Bootstraps and reports the bootstrap distribution of the rank test for the current specification.
	BootstrapRestrictions( cB=199, const iType=0)
	Bootstraps and reports the bootstrap distribution of the rank test for the current specification.
	CATS()
	Constructor, requires Ox 8.0 or newer.
	CATSminingSearch(const amRes, const bFixDet=TRUE, const iHomogenize=2, const dP1=0.05, const dP2=0.25)
	Finds restrictions on a cointegrating vector.
	CATSminingSelect(const amRes, const dP2=0.05, const dP4=0.05, const mxDepth=20)
	Selects from a set of possible cointegrating vectors.
static	CheckConvergence(const dLogdet, const dLogdetPrev, const mPi, const mPiPrev, const dCrit, const iMaxIt, const iInfo, const iItno, mIterInfo=<>, const dStep=1, const bForceTerminate=FALSE)
	Checks convergence of switching algorithms based on relative
	CheckEstimated()
	Generates a run-time error if the unrestricted reduced rank model has not been estimated.
static	CheckRestrictions(const sH, amH, const cRows, const iR, vVpg=<>, const bNeedRmatrices=TRUE)
	Check restrictions matrices (if any); if possible supply missing items.
static	ClassifyDummies(const mR, const mDR)

static	ColCat(const amA)
	Concatenates an array of matrices along the columns.
	CompanionRoots()
	Prints the roots of the companion matrix based on the current model.
virtual	Covar()

static	DecLQ(const mA)
	LQ decomposition
static	DecQL(const mA)
	QL decomposition
static	DecQRt(const mA)
	QR decomposition
static	DiagCat(const amA)
	Concatenates an array of matrices along the diagonal.
virtual	DoAutoOutput()
	Adds to the standard output (does nothing here).
	DoBartlettBetaFactor(const amH)
	Check if the restrictions on beta are such that the Bartlett correction can be applied.
	DoCompute(const iDo)
	Specify default computations and tabulations.
virtual	DoEstimation( vPar)
	Estimates the model (called by `Modelbase::Estimate`).
static	DoGetCompanion(const mAlpha, const mBeta, const mGamma_i, const amRoots)
	Returns the companion matrix based on the specified coefficients, optionally with roots.
	DoI1RankTable(const dAlpha=0.05, const bDoBartlett=TRUE)
	Computes the table of the I(1)-analysis.
	DoI1Residuals(const mPi)
	Computes the residuals of the I(1) model.
virtual	DoMoments()
	Creates the QR transformations that are needed for estimation.
static	DoRestrictAlphaBeta(const mAlpha0, const mBeta0, const amG, const amH, const bWithStandardErrors, const vNormBeta, const mQtZ0, const mQtZ1, const mQtZ0sq, const cT, const dCrit, const iMaxIt, const iInfo, const dBFGSCrit, const iMaxItBFGS, const iSwitchOptions=0, const cStartSearch=20)

	DoSetI1Rank(const iR)
	Sets the cointegrating rank of Pi to r, 0 <= r <= n, after estimating the VAR.
	DoUpdateEstimates_I1(const mAlpha, const mBeta, dLogDet=.NaN, const viNorm=<>)
	Updates the model estimates with (restricted) alpha, beta, Omega.
	DoUpdateRestrictedSettings(const amG, const amH, const vVpg, const iAlphaFlag, const iBetaFlag, const cResCount, const sResType=0)
	Store the settings for restricted estimation.
	DropDiffLags(const vDropLags, const vDropLagsExo=<>)
	Drop lagged differences in the ECM representation of the VAR.
static	ECMToVAR(const mPi, const mGammas)
	Translates the ECM coefficients to the VAR coefficients.
static	Eigen(const ma, const amval, const amvec=0)
	As Ox eigen function, except that it always includes the imaginary part.
virtual	EstimateAlphaBeta( amG, amH, const vNormBetaOvr=<>)
	Estimates the model using alpha-beta switching but doesn't make it the current model.
static	FixRank(const mHj, const mRj, const mH, const iTargetRank)
	Used for the automatic Beta-space identification scheme.
static	GammaApprox_I1(const dTstat, const iType, const dQuantile, const iDetTrend, const iP, const iR, const iDimX1, const iS=0, const cSD=0, const viSDDate=<>)
	Procedure for calculating the approximate p-values and quantiles of
static	GammaApprox_I2(const dTstat, const iType, const dQuantile, const iDetTrend, const iP, const iR, const iDimX1, const iS=0, const cSD=0, const viSDDate=<>)
	Procedure for calculating the approximate p-values and quantiles of
	GenerateAsY(const mEps, const mAlpha, const mBeta, const mGams, const mPsi)
	Generates a random sample formulated as the current I(1) model, but with specified coefficients.
static	GenerateECM(const cT, const iDet, const mAlpha, const mBeta, const mGams, const mPsi, const mOmegaSqrt=<>, const mYinit=<>, const mYmean=<>)

virtual	GenerateY( mEps=<>)
	Generates a random sample from the current model.
	GetAlpha()
	Gets current alpha.
	GetAlphaSe()
	Gets standard errors of current alpha.
	GetAlphaVAR()
	Gets alpha for the full rank VAR.
static	GetBar(const mA)
	returns A_bar.
	GetBeta()
	Gets current beta.
	GetBetaSe()
	Gets standard errors of current beta.
	GetBetaVAR()
	Gets beta for the full rank VAR.
static	GetColumnScale(const mB, const cRows=0)
	Finds a row elements for each column to normalize a matrix.
virtual	GetCompanion(const amRoots=0)
	Returns the companion matrix based on the current model, optionally with roots.
virtual	GetDimensions(const sType="")
	Get model dimensions.
	GetEval()
	Gets the eigenvalues.
	GetGammaFrom(const mAlpha, const mBeta)
	Gets the current Gamma matrix (coefficient of lagged second difference) in the same format as the I(2) model.
	GetGamsInArray(const mAlpha, const mBeta)
	Calculates the Gamma_i matrices (coefficient of lagged differences) based on the alpha and beta.
virtual	GetGroupLabels()

static	GetInvertibleSubMatrix(const mB, const viNorm=<>)
	Finds a unique row elements for each column to rotate a matrix.
	GetLags()
	Returns the lag length of the dependent variables.
	GetLambdaXiSe()
	Computes standard errors of transformed eigenvalues and their sum.
	GetLogLikUnr()
	Gets the log-likelihood of the unrestricted model (except for the rank-restrictions).
	GetLogLikVAR()
	Gets the log-likelihood of the unrestricted VAR.
	GetMARep()
	Calculates and returns the matrices of the moving average representation.
	GetMaxMethodInfo()
	Gets the maximization method info.
	GetMaxMethodName()
	Gets the maximization method name.
	GetMethodLabel()
	Gets method label.
virtual	GetMethodLabels()

	GetModelLabel()
	Gets model label.
	GetOmega()
	Gets current residual variance matrix Omega.
virtual	GetPackageName()
	Gets the package name.
virtual	GetPackageVersion()
	Gets the package version.
virtual	GetParNames()
	Gets the names of the parameters.
	GetPi()
	Gets current Pi.
	GetPiSe()
	Gets standard errors of current Pi.
virtual	GetRegsLongRun(const fCorrected=TRUE)
	Gets the long-run regressors, optionally corrected for the short-run regressors.
virtual	GetResVar()
	Gets current residual variance matrix Omega.
	GetResiduals()
	Gets the residuals.
virtual	GetS00()
	Gets S_{00}.
	GetSeasonals()

	GetTraceTest()
	Gets the trace-test statistics.
	GetTrend()

virtual	GetUNames()
	Gets the names of the U variables.
virtual	GetXNames()
	Gets the names of the X variables.
virtual	GetYNames()

virtual	GetZ0()
	Gets Z_0, the differenced Y.
virtual	GetZ0Names()
	Gets the names of Z_0, the differenced Y.
virtual	GetZ1()
	Gets Z1, the long-run regressors.
virtual	GetZ2()
	Gets Z2, the short-run regressors.
virtual	GetZ3()
	Gets Z3 which is empty in the I(1) model.
virtual	GetcDfLoss()
	Degrees of freedom lost (no of estimated parameters).
virtual	GraphicAnalysis(const iPlot=0, const bActualFitted=FALSE, const bActualFittedCross=FALSE, const iResiduals=2, const iCointegration=2, const iAcf=1, const iAcfLag=-1, const iDensity=0, const iRootsPlot=0, const iLevelsPlot=0, const iEqn=-1)
	Graphical analysis.
virtual	I1Rank(const iR)
	Sets the cointegrating rank r and forces re-estimation of the CVAR.
	I1RankTable(const bDoBartlett=TRUE)
	Computes and prints the default I(1) rank test table.
	Init()
	Reset the transitory properties.
virtual	InitData()
	Checks the model formulation and creates the data matrices.
virtual	InitPar()
	Initialize the model parameters
static	InsertFixedPhiInPar(const vP, const cAlpha, const vDroppedPhi)
	Reinserts the normalizations that are not in the parameter vector
static	InvertSymQR(const mA, const pdLogLik=0)
	Returns the inverse of symmetric p.d. matrix.
virtual	IsI2()
	Checks if this is an I(2) model
virtual	IsIdentified()
	Get identification status.
static	IsIdentifying(const amH, const bFull=FALSE)
	Tests if the r matrices H_i generically identify an r-dimensional space.
static	IsIdentifyingJac(const mAlpha, const mBeta, const amG, const amH, const vDroppedPhi, const cPar)
	Tests if the G and H matrices identify the cointegrating space using the Jacobian criterion.
	IsOutsideCircle()
	Checks if the current model is inside the circle using the current radius.
virtual	IsUnivariate()

static	JacLinResBeta(const amH, const vDroppedPhi)
	Jacobian of linear restrictions on beta (optionally identified by fixing an element of each phi).
	Lags(const lags, const lagsExo=-1, const lagsDet=-1)
	Set the lag length of the dependent variables in the VAR.
static	LineSearch(const vP, const vPprev, const vP0, const dF, const dFprev, const fnLogLik, const vStepGridUp, const vStepGridDown, cStages)
	Line search algorithm for switching.
	LogDetFromLogLik(const dLogLik)
	Translates the log-likelihood to log\|Omega\|.
	LogLikFromLogDet(const dLogDet)
	Translates log\|Omega\| to the log-likelihood.
static	LogdetI1Model(const mAlpha, const mBeta, const cAlpha, const cBeta, const adFunc, const avScore, const amHessian, const mQtZ0, const mQtZ1, const mQtZ0sq, const cT, const amG, const amH, const vDroppedPhi, const mJacA, const mJacB, const bAsymp=FALSE)
	Evaluates the scaled log-likelihood of the restricted I(1) model.
static	LogdetSymQR(const mA)
	Returns the log determinant of a symmetric p.d. matrix.
virtual	MARep()
	Prints the matrices of the moving average representation.
static	MapParToAlphaBeta( vP, const cAlpha, const amG, const amH, const vDroppedPhi=<>)
	Maps the unrestricted (free) parameters back to alpha and beta using the specified restrictions.
static	MapScoreToAlphaBeta(const mDloglikDpi, const mAlpha, const mBeta, const cAlpha, const cBeta, const amG, const amH, const vDroppedPhi)
	Maps the score of the scaled log-likelihood of the restricted I(1) model
static	MapToPar(const mB, const amH)
	Maps the free parameters in A to a column vector using the specified restrictions.
static	MapVarToAlphaBetaPiSE(const mAlpha, const mBeta, const mMaa, const mMbb, const mJacA, const mJacB)
	Maps the unrestricted (unscaled) variance matrices of the free parameters to
	MaxControl(const iMaxIt=10000, const iInfo=1000)
	Sets maximum iterations and output level for switching.
	MaxControlEps(const dSwitchCrit=1e-9, const dBFGSCrit=1e-5)
	Sets the convergence criterion for switching
	MaxOptions(const iHybridInit=-1, const iOptions=-1, const cStartSearch=20)
	Sets options for switching.
static	NormalizePhiInPar( vP, const cBefore, const vDim, const dCutOff=0, const vNormIdxP=<>)
	Normalize the unrestricted (free) parameters in beta, adjusting alpha accordingly.
static	NormalizeThetaPhiInPar( vP, const cAlpha, const amG, const amH, const bApply=FALSE, const vNormBeta=<>)
	Normalize the unrestricted (free) parameters in beta, adjusting alpha accordingly.
static	Nullspace(const mA)
	Returns the orthogonal complement (null space) of a matrix.
virtual	Output()
	Prints the estimated the model (called by `Modelbase::Estimate`).
	OutputBootstrapHeader(const sTitle, const cB, const iType, const time=0)
	Prints header of bootstraps report.
	OutputHeader(const sTitle, const asR, const asX)
	Prints the estimated the model (called by `CATS::Output`).
	OutputLogLik(const sExtra="")
	Prints the log-likelihood and dimensions.
virtual	OutputOutliers( dSE)

	OutputRestrictionsTest()
	Prints the log-likelihood and dimensions.
static	PartialMoments(const mY, const mX, const cT=-1, const fWarn=0, const amPi=0)
	Returns (R'R)/T where R are the residuals of regressing Y on X.
	PrintAlphaBeta(const mAlpha, const mBeta, const mAlphaSe, const mBetaSe, const asY, const mPiSe=<>, const bPiPrint=0)
	Writes the beta, alpha and Pi matrices and t-values (or standard errors).
	PrintDeterministics()
	Prints the Deterministic specification.
	PrintI1Rank(const sExtra="")
	Prints the estimates for the current rank.
	PrintI1RankTable(const mTable, const dAlpha, const sHeader, const sFooter="", asTraceAlt=0)
	Prints the table of the I(1)-analysis.
static	PrintMat( mA, const mSE, const asRow, const asCol, const asOptions="")
	Prints a matrix with standard errors.
static	ProjectOntoH(const mBeta, const amH, const bIsIdentified=TRUE, const vVpg=<>)
	Unrestricted vectors are projected onto H matrices.
static	ProjectOntoH_reg(const mBeta, const amH)

static	RROlsQ( mQtY, mQtX, const mQtYsq, const cT, const cR, const mQtZ=<>, const aGamma=0)
	Reduced rank regression of Y on X corrected for Z when Q'Y,Q'Z,Q'Z are already available.
static	RROlsQ2( mQtY, mQtX, const mQtYsq, const cT, const cR, const mQtZ=<>, const aGamma=0)

static	RanBootstrap(const mResiduals, const iType=0)

static	RankOrder(const mA)
	Obtains the rank order of a matrix.
virtual	RecursiveAnalysis(const iPlot=0, const cTMin=0, const bEval=FALSE, const bXi=TRUE, const bTrace=TRUE, const bLogLik=TRUE, const bAlpha=FALSE, const bBeta=FALSE, const bFixShortRun=FALSE, bApproxXiSe=TRUE)
	Recursive estimation.
	RenewY(const mY)
	Replaces the Y estimation data in the underlying database with the specified random sample.
	ResidualAnalysis()
	Prints multivariate and unvariate analysis of the estimated residuals.
virtual	Restrict(const amRes, const vNormBetaOvr=<>)
	Estimates the model and prints results for the current rank imposing restrictions on alpha and beta using alpha-beta switching.
virtual	RestrictAlphaBeta( amG, amH, const vNormBetaOvr=<>)
	Estimates the model and prints results for the current rank imposing restrictions on alpha and beta using alpha-beta switching.
	RestrictBeta( amG, amH, vVpg=<>)
	Estimates the model for the current rank imposing restrictions on alpha and beta using beta switching.
static	RowCat(const amA)
	Concatenates an array of matrices along the rows.
	Seasonals(const iUse=1)

	SetAlphaBetaI1Start(const mAlpha, const mBeta)
	Sets current I(1) alpha,beta initial values for the starting value procedures.
static	SetMaths(const iMaths, const sLabel)
	Sets the labels of mathematics symbols.
	SetMaxMethodInfo(const sName, const aInfo=<>)
	Sets the name of the maximization method.
	SetNormalization(const bSet)
	Sets default normalization mode.
	SetPrintMatOptions(const asOptions, const iAdd=0)
	Sets the options for `CATS::PrintMat`.
	SetRadius(const dRadius=1.00001)
	Sets the radius of 'unit' circle (default is 1.00001).
static	SetTrackProgress(const bTrack)
	Sets progress tracking of iterative maximizers.
	SetUseDefaultSeed(const bSet)
	Sets usage of default seed for simulations and bootstraps on (TRUE) or off (FALSE, default setting).
	ShortRun()
	Prints the short-run coefficients.
	ShortRunEval(const mAlpha=0, const mBeta=0, const mOmega=0, const bDoStdDev=FALSE)
	Calculates the shortrun coefficients of the current model.
virtual	ShrinkSample(const cDrop, const cTmin=1, const bFixShortRun=FALSE)
	Shrinks the estimation sample, but doesn't re-estimate yet.
	SimulateRankDet(const cTsim, const bPrint)
	Simulates and reports the asymptotic distribution of the I(1) rank test in the
	SimulateRankTest(const cTsim=400, const cM=1000)
	Simulates and reports the asymptotic distribution of the I(1) rank test in the
static	SolveRRR(const mS00, const mS01, const mS11, const amEigVal, const amEigVec, const cR=-1)
	Solves the I(1) pencil \|\lambda S_{11} - S_{10}S_{00}^{-1}S_{01}\|.
static	SplitRestrictions(const asRes, const asSections, const aaH1, ...)
	Split restrictions array by sections.
static	StatComp(const mAlpha, mBeta, const amGamma, const iLags, const mS00, const amP, const amQ)
	Creates the companion matrix P and the matrix Q for the stationary process Y(t).
	TestAlphaCommon( mG)
	Tests restricted alpha, where alpha = (G*theta_1,theta_2) and G is a known matrix.
	TestAlphaKnown(const mG)
	Tests restricted alpha, where alpha = (G,G_orthogonal psi) and G is a known matrix.
static	TestDiff(const x, const y)
	Computes a scaled norm of the difference of two vectors.
virtual	TestEach(const bAlpha=TRUE, const bBeta=TRUE)
	Tests variable exclusion, variable stationarity, long-run weak exogeneity, unit vector in alpha.
virtual	TestEachAllRanks(const bAlpha=TRUE, const bBeta=TRUE)
	Tests variable exclusion, variable stationarity, long-run weak exogeneity, unit vector in alpha.
	TestEachBetaRestriction( fnGetG, fnGetH, const vVpg, const asZ, const cRes, const sType)
	Tests similar restrictions over a range of elements using beta switching.
virtual	TestForecastGraphics(const bExp)

virtual	TestGraphicAnalysis()
	Plots residuals and ACF.
static	TestRanksAndFix(const amH, const amR, const vIndex, const amHnew, const amRnew)
	Used for the automatic Beta-space identification scheme.
virtual	TestSummary()

	Trend( mode)
	Specifies the trending behaviour of the model.

CATS data members (protected by default)
	m_asA

	m_iAFlag

	m_asAlpha

	m_mAlpha

	m_mAlpha_I1

	m_mAlpha_VAR

	m_mAlphaSE

	m_asB

	m_dBartletBetaCorr

	m_asBeta

	m_mBeta

	m_mBeta_I1

	m_mBeta_VAR

	m_mBetaSE

	m_dBFGSCrit

	m_iBFlag

	m_iDetTrend

	m_asDetTrend

	m_iDoCompute

	m_vDYLagSel

	m_vEigVal

	m_amG

	m_amH

	m_iIFlag

	m_cLags

	m_cLagsDet

	m_cLagsExo

	m_dLogLik_I1

	m_dLogLik_Unr

	m_dLogLik_VAR

static	sm_asMaths

	m_aMaxMethodInfo

	m_sMaxMethodName

	m_asMethods

	m_iModel

	m_iModelClass

	m_bNormalizeOnAlpha

	m_viNormBeta

static	sm_dNormCrit

	m_mOmega

	m_mOmega_I1

	m_mPiSE

	m_asPrintMatOptions

	m_mQtZ0_Z2

	m_mQtZ0sq_Z2

	m_mQtZ1_Z2

	m_iR

	m_iR_preset

	m_dRadius

	m_cResCount

	m_mResidual

	m_sResType

	m_mS00

	m_dS00logdet

	m_mS01

	m_mS11

	m_iSeasType

	m_mSigmaBB

	m_cStartSearch

	m_dSwitchCrit

	m_mxSwitchIter

	m_mxSwitchIter_BFGSstep

	m_iSwitchOptions

	m_iSwitchPrint

	m_vTestOrders

	m_sTexOutput

static	sm_bTrackProgress

static	sm_asTrend

	m_cU

	m_mU

	m_asU

	m_bUseDefaultSeed

	m_vVpg

	m_asX

	m_cX1

	m_cX2

	m_vXIndex

	m_mXlag

	m_vXType

	m_asY

	m_mYlag

	m_cZ0

	m_mZ0

	m_asZ0

	m_mZ0_Z2

	m_asZ1

	m_mZ1

	m_cZ1

	m_mZ1_Z2

	m_cZ2

	m_asZ2

	m_mZ2

	m_cZ2RegCount

CATS enum members (protected by default)
	{ALPHA_NONE, ALPHA_EACH, ALPHA_SAME}
	Type of restriction on beta.
	{BETA_NONE, BETA_EACH, BETA_SAME}
	Type of restriction on beta.
	{XTYPE_EXOG, XTYPE_DETERMINISTIC, XTYPE_D_IMPULSE, XTYPE_D_STEP, XTYPE_D_TREND, XTYPE_D_OTHER}
	Type of X variable.
	{NO_BREAK, LEVEL_BREAK, TREND_BREAK}
	Break options
	{I1, I2}
	model class
	{MAX_ITERATING, MAX_INITIALIZING}
	maximizattion extension
public:	{Y_VAR, X_VAR, U_VAR, R_VAR, SEL_VAR, A_VAR}
	variable types for model specification
public:	{M_NONE, M_OLS, M_RRR}
	estimators
public:	{NONE, CIMEAN, DRIFT, CIDRIFT}
	deterministic specifications
public:	{SW_UNUSED, SW_NO_INIT, SW_NO_RANDOMINIT, SW_SWITCH_ONLY, SW_NO_BFGS, SW_NO_LINESEARCH, SW_LINESEARCH_1, SW_LINESEARCH_1NEG, SW_LINESEARCH_ALT, SW_LINESEARCH_OPT}
	Switching options, not normally used; can be added together
public:	{RANKTEST_TRACE, RANKTEST_LMAX, RANKTEST_BREAK}
	Type of test for cointegration rank, used in GammaApprox
public:	{DO_BARTLETT, DO_BOOTSTRAP, DO_SKIPTABLE}
	Optional actions to do automatically, set by `CATS::DoCompute`
public:	{BS_WILD, BS_IID, BS_SIGN, BS_PARAMETRIC, BS_JAD, BS_JAD2}
	Bootstrap sampling schemes
public:	{MATHS_alpha, MATHS_beta, MATHS_tau, MATHS_phi, MATHS_psi, MATHS_Gamma, MATHS_Phi, MATHS_Pi, MATHS_Psi, MATHS_Omega, MATHS_zeta, MATHS_delta, MATHS_dee, MATHS_ee}
	Index for matrix label

CATSI2 : CATS : Modelbase : Database : Sample class reference

CATSI2 function members (public by default)
virtual	ACTest_LM(const mResid, const iOrder=1)
	Performs the LM test for autocorrelation in an I(2) model.
	BootstrapRankTest( cB=399, const iType=0)
	Bootstraps and reports the bootstrap distribution of the rank test for the current specification.
	CATSI2()
	Constructor.
static	DeltaSwitching(const mAlpha, const mBeta, const amH, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT, const dCrit=1e-6, const iMaxIt=1000, const iInfo=0, const iLineSearch=2)
	Delta-switching algorithm for the I(2)-model.
	DoDeltaSwitching(const mAlpha, const mBeta, const amH=<>, const viNorm=<>)
	Estimation using delta-switching algorithm for ML-estimation of the I(2)-model.
	DoDeltaSwitchingLogLik(const mAlpha, const mBeta, const iS1)
	As `CATSI2::DoDeltaSwitching` but without setting estimates, only returning the log-likelihood, alpha and beta.
virtual	DoEstimation( vPar)
	Estimates the model (called by `Modelbase::Estimate`).
	DoI2RankTable(const dAlpha=0.05, const iDoMLE=1)
	Computes the table of the I(1)-analysis.
	DoI2RankTable_2step()
	Computes the table of the I(1)-analysis using two-step estimation.
	DoI2Residuals(const mPi, const mGamma)
	Computes the residuals of the I(2) model z0_t = Pi z1_t - Gamma z2_t + rest.
virtual	DoMoments()
	Partials out z3 and creates the transformed data matrices QZi that are needed for estimation.
	DoRestrictAlphaBeta( amA, amH, const vNormBetaOvr, bSetZeroV13, const dCrit, const iMaxIt, const iInfo, const dBFGSCrit, const iMaxItBFGS, const iSwitchOptions=0, const cStartSearch=20)
	Estimates the model and prints results for the current rank imposing restrictions on alpha and beta.
	DoSetI2Rank(const iS, const iTriangular=-1)
	Sets the I(2) to s==s1, 0 <= s <= p-r, after estimating the VAR.
	DoTauSwitching(const mAlpha, const mBeta, const mH=<>)
	Estimation using tau-switching algorithm for ML-estimation of the I(2)-model.
	DoTauSwitchingLogLik(const mAlpha, const mBeta, const iS1)
	As `CATSI2::DoTauSwitching` but without setting estimates, only returning the log-likelihood, alpha and beta.
	DoTriSwitching(const mAlpha=<>, const mBeta=<>)
	Estimation using triangular-switching algorithm for ML-estimation of the I(2)-model without restrictions on A,B.
	DoTriSwitchingLogLik(const mAlpha, const mBeta, const iS1)
	As `CATSI2::DoTriSwitching` but without setting estimates, only returning the log-likelihood, alpha and beta.
	DoUpdateEstimates_I2(const mAlpha, const mTau, const mRho, const mPsi, const mKappa, const mWterm, const mOmega, const dLogDet, viNorm=<>)
	Updates the I(2) model after estimation.
	DoUpdateEstimates_I2_AB(const mA, const mB, const mV, const iR, const iS1, const dLogDet, const vNorm=<>)
	Updates the I(2) model after estimation.
virtual	EstimateAlphaBeta( amG, amH, const vNormBetaOvr=<>, bSetZeroV13=FALSE)
	Estimates the model using triangular switching but doesn't make it the current model.
	EstimateTau( mH)
	Test the same linear restrictions on all Tau vectors.
	EvalBetaSe()
	Computes standard errors of current beta, psi, dee.
	GenerateAsY_I2(const mEps, const mAlpha, const mBeta, const mGamma, const mPsis, const mPhi)
	Generates a random sample formulated as the current I(2) model, but with specified coefficients.
static	GenerateECM_I2(const cT, const iDet, const mAlpha, const mBeta, const mGamma, const mPsis=<>, const mPhi=<>, const mOmegaSqrt=<>, const mYinit=<>, const mYmean=<>)

virtual	GenerateY( mEps=<>)
	Generates a random sample from the current model.
	GetAWithSe()
	Gets A with standard errors.
	GetAlphaOrt1WithSe()
	Calculates the alpha_ort1 with asymptotic standard errors.
	GetAlphaOrt2WithSe(const viNorm=<>)
	Calculates the alpha_ort2 with asymptotic standard errors.
	GetBWithSe()
	Gets B with standard errors.
	GetBetaSe()
	Gets standard errors of current beta.
virtual	GetCompanion(const amRoots=0)
	Returns the companion matrix based on the current model, optionally with roots.
	GetDeeEe()
	Gets the current dee and ee matrices.
static	GetDeeEeFrom(const mAlpha, const mTau, const mPsi, const mKappa, const mWterm)

	GetDeeEeSe()
	Gets standard errors of current dee and ee.
	GetDeltaZeta()
	Gets the current delta and zeta matrices.
static	GetDeltaZetaFrom(const mAlpha, const mTau, const mPsi, const mKappa, const mWterm)

	GetDeltaZetaSe()
	Gets standard errors of current delta and zeta.
virtual	GetDimensions(const sType="")
	Get model dimensions.
	GetGamma()
	Gets the current Gamma matrix (coefficient of lagged second difference).
static	GetGammaFrom_I2(const mAlpha, const mTau, const mPsi, const mKappa, const mWterm)
	Gets the Gamma matrix from tau, psi, kappa and w.
	GetPsiKappa()
	Gets the current psi and kappa matrices.
static	GetPsiKappaFrom(const mAlpha, const mTau, const mGamma, const mOmega)

	GetPsiKappaSe()
	Gets standard errors of current psi and kappa.
virtual	GetRegsLongRun(const fCorrected=TRUE)
	Gets the long-run regressors, optionally corrected for the short-run regressors.
virtual	GetRegsMediumRun(const fCorrected=TRUE)
	Gets the medium-run regressors, optionally corrected for the short-run regressors.
	GetTheta()
	Gets the current Theta matrix
static	GetTriFrom(const mAlpha0, const mBeta0, const mGamma0, const iS1, const bFixGamma_c=TRUE)

	GetVWithSe()
	Gets V with standard errors.
	GetWWithSe()
	Gets W with standard errors.
virtual	GetZ0()
	Gets Z_0, the double-differenced Y.
virtual	GetZ0Names()
	Gets the names of Z_0, the double-differenced Y.
virtual	GetZ1()
	Gets Z1, the medium-run regressors.
virtual	GetZ2()
	Gets Z2, the long-run regressors.
virtual	GetZ3()
	Gets Z3, the long-run regressors.
virtual	GetcDfLoss()
	Degrees of freedom lost (no of estimated parameters).
virtual	GraphicAnalysis(const iPlot=0, const bActualFitted=FALSE, const bActualFittedCross=FALSE, const iResiduals=2, const iCointegration=2, const iAcf=1, const iAcfLag=-1, const iDensity=0, const iRootsPlot=0, const iLevelsPlot=0, const iEqn=-1)
	Graphical analysis.
virtual	I1Rank(const iR)
	Sets the I(1) cointegrating rank r and switches to an I(1) model.
	I2Rank(const iS)
	Sets the I(2) rank s1 prior to estimating the VAR and switches to an I(2) model.
	I2RankMLE(const bPrint)

	I2RankTable(const iDoMLE=1)
	Computes and prints the default I(2) rank test table.
	I2RankTablePvals(const mTests, const dAlpha)
	Gets the p-values and fractiles for the test table of the I(2)-analysis.
	I2SwitchStart(const fnSwitching, const mAlpha, const mBeta, const iS1, const mAlpha2=<>, const mBeta2=<>)
	Starting values for tau-switching algorithm.
	Init()
	Reset the transitory properties.
virtual	InitData()
	Checks the model formulation and creates the data matrices.
virtual	IsI2()
	Checks if this is an I2 model.
static	JacI2Model_AB(const vP, const vSelV, const vInitV, const cZ0, const cZ1, const iR, const iS, const amG, const amH, const vDroppedPhi=<>, const bNumerical=FALSE)
	Used in `CATSI2::RestrictAlphaBeta`.
static	LogdetI2Model_AB(const vP, const mA, const mB, const mV, const mW, const vSelV, const iR, const iS, const adFunc, const avScore, const amHessian, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT, const amG, const amH, const vDroppedPhi, const mJac, const bAsympSE)

virtual	MARep()
	Prints the matrices of the moving average representation.
static	MapGammaToPar1(const mAlpha, const mBeta, const mGamma, const mXi, const mEta, const iS1, const mJ, const bUseKappa=TRUE)

static	MapMatToPar_AB(const mA, const mB, const mV, const vSelV, const iR, const iS, const amG, const amH)
	Used in `CATSI2::RestrictAlphaBeta`.
static	MapParToMat_AB( vP, const vSelV, const vInitV, const cZ0, const cZ1, const iR, const iS, const amG, const amH, const vDroppedPhi=<>)

static	MapScore_AB(const mDloglikDpi, const mA, const mB, const amG, const amH, const vDroppedPhi=<>)

virtual	Output()
	Prints the estimated the model (called by `Modelbase::Estimate`).
	PrintI2Rank(const sExtra="")
	Prints the estimates for the current rank.
	PrintI2RankTable(const mTests, const mPvals, const mFrac95, const sType="", const bWarn=TRUE)
	Prints the table of the I(2)-analysis.
virtual	Restrict(const amRes, const vNormBetaOvr=<>)
	Estimates the model and prints results for the current rank imposing restrictions on alpha and beta.
virtual	RestrictAlphaBeta( amA, amH, const vNormBetaOvr=<>, bSetZeroV13=FALSE)
	Estimates the model and prints results for the current rank imposing restrictions on alpha and beta.
	RestrictBeta( mG, amH)
	Estimates the I(2) model with linear restrictions on each beta vector using delta switching.
	RestrictTau( mH)
	Impose the same linear restrictions on all Tau vectors.
static	RotateRho(const iR, const iS, mTau, mKappa, mRho)
	Rotates rho (I \| 0), offsetting tau and kappa.
	ShortRun()
	Prints the short-run coefficients of the I(2) model.
	ShortRunEval_I2(const bDoStdDev=FALSE, const mAlpha=0, const mBeta=0, const mGamma=0, const mTau=0, const mPsi=0, const mOmega=0)
	Calculates the shortrun coefficients of the current I(2) model.
virtual	ShrinkSample(const cDrop, const cTmin=1, const bFixShortRun=FALSE)
	Shrinks the estimation sample, but doesn't re-estimate yet.
	SimulateRankTest(const cTsim=400, const cM=1000)
	Simulates and reports the asymptotic distribution of the I(1) rank test in the
static	SolveConditional(const mAlpha, const mTau, const mAlphaOrt, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)
	Solves the conditional equation from the tau representation.
static	SolveDeltaForAlpha(const mAlpha, const mTau, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)
	Solve the delta representation given tau and tau_ort to get the remaining parameters
static	SolveDeltaForGammaStart(const mAlpha, const mBeta, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)
	Get starting values for I(2) algorithm from alpha,beta with restricted beta from the delta representation.
static	SolveDeltaForGammaStart2(const iR, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)

static	SolveDeltaForTau(const mAlpha, const iR, const iS1, const mDelta, const mZeta, const mOmega, const mH1, const mH2, const mQtZ0, const mQtZ1, const mQtZ2)
	Solve the delta representation given tau_ort, zeta, omega to get tau
static	SolveMarginal(const mAlpha, const mBeta, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)
	Solves the marginal equation from the tau representation.
static	SolveTauForAlpha(const iR, const mTau, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)
	Solve the tau representation to get alpha.
static	SolveTauForTau(const mAlpha, const mAlphaOrt, const iR, const iS1, const mRho, const mW, const mKappa, const mOmega, const mH, const mQtZ0, const mQtZ1, const mQtZ2)
	Solve the delta representation given tau_ort, zeta, omega to get tau
static	TauSwitching(const mAlpha, const mBeta, const mH, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT, const dCrit=1e-6, const iMaxIt=1000, const iInfo=0, const iLineSearch=2)
	Tau-switching algorithm for the I(2)-model.
virtual	TestEach(const bAlpha=TRUE, const bBeta=TRUE)
	Tests variable exclusion, variable stationarity, long-run weak exogeneity, unit vector in alpha.
	TestEachBetaRestriction( fnGetG, fnGetH, const vVpg, const asZ, const cRes, const sType)
	Tests similar restrictions over a range of elements using alpha,beta switching.
static	TriGetVFromAB(const mA, const mB, const iR, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const bSetZeroV13=FALSE)
	Get V from A,B and the data
static	TriNormalize(const mA, const mB, const mV, const iR, const iS1, const bA1IsFree=FALSE, const bA2IsFree=FALSE, const bB1IsFree=FALSE, const bB2IsFree=FALSE, const vNormBOverride=<>)

static	TriOrthogonalize12(const mA, const mB, const mV, const iR, const iS1, const bA1IsFree, const bA2IsFree, const bB1IsFree, const bB2IsFree)

static	TriRegularizeV(const mA, const mB, const mV, const iR, const iS1, const bA1B1IsFree, const bB2IsFree)

static	TriRemoveW(const mA, const mV, const mW, const iR)

	TriStart(const mAlpha, const mBeta, const cAlpha, const cBeta, const iR, const iS1, const amG=<>, const amH=<>, const mAlphaI2=<>, const mBetaI2=<>)
	Get starting values for triangular switching algorithm trying several options.
static	TriSwitching(const mA, const mB, const mV, const bFixW11, const bSetZeroV13, const amG, const amH, const iR, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT, const dCrit, const iMaxIt, const iInfo, const bLineSearch=TRUE, const dAddLogdet=0)
	Triangular switching algorithm for the I(2) model with linear restrictions on A and B.

CATSI2 data members (protected by default)
	m_mA

	m_asAlpha1

	m_asAlpha2

	m_mAlpha_I2

	m_mASE

	m_mB

	m_asBeta1

	m_asBeta2

	m_mBeta_I2

	m_mBSE

	m_asDee

	m_mDeeSE

	m_mEeSE

	m_mGamma

	m_mGamma_I2

	m_mKappa

	m_dLogLik_I2

	m_mOmega_I2

	m_mPsi

	m_mPsiSE

	m_mQtZ0I2_Z3

	m_mQtZ0sqI2_Z3

	m_mQtZ1I2_Z3

	m_mQtZ2I2_Z3

	m_mRho_ort

	m_iS1

	m_iS1_preset

	m_mSigmaZZ

	m_mTau

	m_asTau

	m_mV

	m_mV00

	m_mV01

	m_mV02

	m_mV11

	m_mV12

	m_mV22

	m_mWterm

	m_mZ0I2

	m_asZ0I2

	m_mZ0I2_Z3

	m_mZ1I2

	m_asZ1I2

	m_mZ1I2_Z3

	m_mZ2I2

	m_asZ2I2

	m_mZ2I2_Z3

	m_asZ3I2

	m_mZ3I2

	m_asZeta

CATS::ACTest_LB(const mResid, iOrder=0)

Parameters

mResid	matrix of residuals [T][p]
iOrder	lag length for test (default 0 corresponds to T/4)

Returns

the test statistic

virtual CATS::ACTest_LM(const mResid, const iOrder=1)

Parameters

mResid	matrix of residuals [T][p]
iOrder	integer, lag length for test (default is 1)

Returns

the test statistic

static CATS::ARCHStats(const mResid, const iOrder=1)

Parameters

mResid	matrix of residuals [T][p]
iOrder	lag length for ARCH statistics (default is 1)

Returns

(p x 2) matrix with single equation ARCH statistics and p-values

static CATS::ARCHTest(const mResid, const iOrder=1)

Parameters

mResid	matrix of residuals [T][p]
iOrder	lag length for ARCH test (default is 1)

Returns

(2 x 1) matrix: test statistic and degrees of freedom

static CATS::AlphaBetaStart(const mAlpha, const mBeta, const amG, const amH, const mQtZ0, const mQtZ1, const mQtZ0sq, const cT, const cInitSteps=2, const aAnalysis=<>)

Parameters

mAlpha[p][r]	initial values for alpha
mBeta[p1][r]	initial values for beta
mOmega[p][p]	initial values for omega
amG	array[r] of G_i G matrices
amH	array[r] of H_i matrices
mQtZ0	matrix[n][cz0] Q'Z0 corrected for z2
mQtZ1	matrix[n][cz1] Q'Z1 corrected for z2
mQtZ0sq	matrix[cz0][cz0] rest of Z0'Z0 corrected for z2
cT	int, sample size
cInitSteps	int, number of initialization steps, >= 1 (default is 2)

Returns

array with five elements: alpha, beta, dlogdet, success value (MAX_CONV), vp = theta|phi.

static CATS::AlphaBetaStartAnalysis(const mAlpha, const mBeta, const amG, const amH)

Parameters

mAlpha[p][r]	initial values for alpha
mBeta[p1][r]	initial values for beta
amG	array[r] of G_i G matrices
amH	array[r] of H_i matrices

Returns

array with four elements: {#restrictions on alpha, #restrictions on beta, TRUE if all G are I, TRUE if all H are I, TRUE if common alpha restrictions, TRUE if beta identified}.

static CATS::AlphaBetaStartSearch(const mAlpha0, const mBeta0, const fnAlphaBetaStart, const fnEstimate, const cStart, const mxStartIt, const iInfo, const bTrace=FALSE)

static CATS::AlphaBetaSwitching(const mAlpha, const mBeta, const amG, const amH, const mQtZ0, const mQtZ1, const mQtZ0sq, const cT, const dCrit, const iMaxIt, const iInfo, const iLineSearch=2)

Parameters

mAlpha	matrix [cz0][r] initial alpha
mBeta	matrix [cz1][r] initial beta
cAlpha	int, no of free parameters in alpha.
amG	array[r] of G_i G matrices
amH	array[r] of H_i matrices
mQtZ0,...,mQtZ0sq	transformed data matrices
cT	int, sample size
dCrit	double, convergence tolerance
iMaxIt	int, maximum number of iterations
iInfo	int, >0: print output for every iInfo-th iteration
iLineSearch	int, 0: no search, 1: 1-stage search, 2: 2-stage search (default)

Returns

array with five elements: alpha, beta, dlogdet, success value (MAX_CONV, MAX_WEAK_CONV, MAX_MAXIT, MAX_NOCONV, MAX_FUNC_FAIL), vp= theta|phi.

static CATS::AlphaKnown(const mA, const mBeta1, const mS00, const mS01, const mS11)

Parameters

mG	A matrix
mBeta1	current beta, determines rank and use if alpha=G (fully known)

Returns

array with four elements: alpha, beta, omega, adgf

Comments

Equation numbering refers to Johansen [1996].

static CATS::ApplyColumnScale(const mA, const mB, const mC, const vNormRow)

Parameters

mA	A[p][n] matrix to scale by multiplication
mB	B[m][n] matrix, m >= n, to scale by division
mC	C[m][n] matrix providing scale factors (usually equal to mB)
vNormRow[q]	vector with elements in B for normalization, row indices for columns 0,..,q-1.

Returns

array with three matrices {As, Bs, v}, As has inverse normalization applied, Bs [m][n] normalized matrix, v is normalization vector/matrix.

static CATS::ApplyColumnScaleVec( vP, const cAlpha, amG, const vOldScale, const bApply=TRUE)

Parameters

vP	parameter vector
cAlpha	offset in vP
amG	array[r] of H_i matrices, one for each restriction
vOldScale	old scale to remove
bApply	TRUE: scale (default), FALSE: remove old scale

Returns

{vP, amG, newscale}

static CATS::AutoIdentify1( amH, const iR, const abIsIdentified)

Parameters

amH	array of H matrices
iR	integer, rank for identification
abIsIdentified	address, TRUE on output if amH is identifying, FALSE otherwise
bCheck	boolean indicating if check should be performed

Returns

amH array of H matrices, possibly modified to obtain identification.

static CATS::AutoIdentify2(const amH, const iR, const abIsIdentified)

Parameters

amH	array of H matrices
iR	integer, rank for identification
abIsIdentified	address, TRUE on output if amH is identifying, FALSE otherwise
bCheck	boolean indicating if check should be performed

Returns

amH array of H matrices, possibly modified to obtain identification.

static CATS::BartlettBeta(const iS, const iType, const mAlpha, const mBeta, const iRank, fnGetGamma, const mS00, const cT, const iDetTrend, const cSeas, const iMethod=0)

Parameters

1: (Beta,Rho) known [Corollary 5] 2: Beta = H*Phi, Rho unrestricted [Corollary 6] 3: (Beta,Rho) partly known [Corollary 7] fnGetGammas(const mAlphar, const mBetar)

iS	integer, the number of columns in H.
iType	specifies the type of restriction on Beta being tested:
mAlpha	alpha (adjusted internally to specified iRank)
mBeta	beta (adjusted internally to specified iRank)
iRank	integer of desired rank
fnGetGammas	function that returns array[k-1] with gamma matrices ({} if lag length k <= 1),
mS00	S_{00} matrix
cT	sample size
iDetTrend	specification of constant/mean (1,2,3,4 for NONE,CIMEAN,DRIFT,CIDRIFT)
cSeas	no of seasonal terms

Returns

the correction factor (double)

static CATS::BartlettRank(const mAlpha, const mBeta, const iRank, fnGetGamma, const mS00, const cT, const iDetTrend, const bWarn=TRUE, const iMethod=0)

Parameters

fnGetGammas(const mAlphar, const mBetar) 1: use iteration, 2: calculations with (n^2 x n^2) matrices, 3: neglects a term. The default is 0; other values are just to measure the relative speed (2 will be very slow).

mAlpha	alpha (adjusted internally to specified iRank)
mBeta	beta (adjusted internally to specified iRank)
iRank	integer of desired rank
fnGetGammas	function that returns array[k-1] with gamma matrices ({} if lag length k <= 1),
mS00	S_{00} matrix
cT	sample size
iDetTrend	specification of constant/mean (1,2,3,4 for NONE,CIMEAN,DRIFT,CIDRIFT)
bWarn,	TRUE: print warning when Sigma is not positive definite (the default)
iMethod,	0: use unsymmetric eigen value decomposition,

Returns

The Bartlett correction factor

static CATS::Bartlett_c2(const mP, const mVpsi, const iMethod=0)

Parameters

1: use iteration, 2: calculations with (n^2 x n^2) matrices, 3: neglects a term. The default is 0; other values are just to measure the relative speed (2 will be very slow).

mP	P
mVpsi	V_psi
iMethod,	0: use unsymmetric eigen value decomposition,

Returns

The Bartlett correction factor

static CATS::BetaStdErrors( amH, const viNorm, const mSM1, const cT)

Parameters

viNorm must have same number of rows as beta and the j'th entry should be the index of the element of vector j on which beta is normalized.

amH	array of H_i matrices
viNorm	matrix (vector of integers)
mSM1	matrix, unrestricted variance matrix
cT	sample size (if scaling still needed)

Returns

matrix with standard errors of beta (or <> if computation failed)

static CATS::BetaStdErrors_DiagH( amH, const viNorm)

Parameters

viNorm must have same number of rows as beta and the j'th entry should be the index of the element of vector j on which beta is normalized.

amH	array of H_i matrices
viNorm	matrix (vector of integers)

Returns

large mapping matrix

static CATS::BetaSwitching(const mG, const amH, const vVpg, const mBeta, const mQtZ0, const mQtZ1, const mQtZ0sq, const cT, const dCrit=1e-6, const iMaxIt=1000, const iInfo=0, const iLineSearch=2)

Parameters

of vectors in group i. The sum is the rank of the restricted beta.

mG	matrix, <> or common restriction on alpha
amH	array[q] of H matrices, one for each restriction
vVpg	matrix[q] rank for each restriction. Entry i of vVpg gives the number
mBeta[p1][r]	initial beta
mS00	S00 matrix
mS01	S01 matrix
mS11	S11 matrix
dCrit	double, convergence tolerance (default 1e-6)
iMaxIt	int, maximum number of iterations (default 1000)
iInfo	int, >0: print output for every iInfo-th iteration (default 0)
iLineSearch	int, 0: no search, 1: 1-stage search, 2: 2-stage search (default)

Returns

array with three elements: alpha, beta and success value ((MAX_CONV, MAX_WEAK_CONV, MAX_MAXIT, MAX_NOCONV, MAX_FUNC_FAIL))

Comments

The reported logdet is only the actual log-determinant if there are no restrictions on alpha.

static CATS::BlockInvHess(const mHessian, const aSizes)

Parameters

mHessian	Hessian matrices.
aSizes	array[n] of block sizes, which should add up to the dimension of mHessian (or less).

Returns

array[n] withinverse of minus the the diagonal blocks

CATS::BootstrapRankTest( cB=399, const iType=0)

Parameters

cB	Number of bootstrap replications (B, optional argument, default is 399)
iType	Type of bootstrap sampling: BS_WILD (Wild N[0,1], the default), BS_IID (IID resampling), BS_SIGN (+1,-1)

Returns

array[2] {table with tests and p-values, table of quantiles}

CATS::BootstrapRestrictions( cB=199, const iType=0)

Parameters

cB	Number of bootstrap replications (B, optional argument, default is 399)
iType	Type of bootstrap sampling: BS_WILD (Wild N[0,1]), BS_IID (IID resampling), BS_SIGN (+1,-1)

Returns

array[2] {table with tests and p-values, table of quantiles}

CATS::CATS()

Constructor, requires Ox 8.0 or newer.

CATS::CATSminingSearch(const amRes, const bFixDet=TRUE, const iHomogenize=2, const dP1=0.05, const dP2=0.25)

Parameters

amRes	array with restrictions, sections are [alpha],[beta],[fix] (without sections, [beta] is assumed)
bFixDet	TRUE: fix deterministic variables (default is TRUE)
iHomogenize	1: homogenize (a=1,-1) selection, 2: include a+b=1,-1 (default is 2), 0: off
dP1	restrictions with pvalue < dP1 are rejected (default is 0.05)
dP2	homogenization restrictions with pvalue < dP2 are rejected (default is 0.25), unless it improves on the previous p-value

Returns

array with four elements: {array with restrictions (ordered), matrix with model estimation results for each restriction, amg, amhfix}.

CATS::CATSminingSelect(const amRes, const dP2=0.05, const dP4=0.05, const mxDepth=20)

Parameters

param dP3 interim selections with pvalue < dP3 are rejected (default is 0.05) param dP4 final selections with pvalue < dP4 are rejected (default is 0.05)

amRes	array with restrictions, sections are [alpha],[beta],[fix],[user] (without sections, [beta] is assumed)
mxDepth	int: maximum number extended in each round (default is 20)

Returns

array with three elements: vector of tests for each.

static CATS::CheckConvergence(const dLogdet, const dLogdetPrev, const mPi, const mPiPrev, const dCrit, const iMaxIt, const iInfo, const iItno, mIterInfo=<>, const dStep=1, const bForceTerminate=FALSE)

Parameters

dLogdet	current -log-determinant
dLogdetPrev	previous -log-determinant
mPi	current Pi matrix
mPiPrev	current previous matrix
dCrit	double, convergence criterion (<0 if step is down)
iMaxIt	int, maximum number of iterations (default 1000)
iInfo	int, >0: print output for every iInfo-th iteration (default 0)
mIterInfo	<> or current info, updated if necessary
dStep	int, !=1: current step size (default is 1)

Returns

{convergence criterion based on -logdet, convergence criterion based on pi, TRUE if converged, mIterInfo}

CATS::CheckEstimated()

Generates a run-time error if the unrestricted reduced rank model has not been estimated.

static CATS::CheckRestrictions(const sH, amH, const cRows, const iR, vVpg=<>, const bNeedRmatrices=TRUE)

Parameters

sH	string: label for matrix
amH	array[q] of H matrices, q <= r, or array[q] of strings with restriction expressions
cRows	int, no of rows in each H matrix.
vVpg	vector[q] rank for each restriction, default is <>.
bNeedRmatrices	TRUE: need one entry for each vector, so vVpg must be all ones, default is TRUE

Returns

array with siz items: 1. amH[q] array with restrictions 2. the number of restrictions imposed (-1 for an error), 3. no of free parameters, 4. vnorm[iR] for each vector, either -1 or index to be used for normalization (row vector) 5. vVpg[q], multiplicity of restrictions, and 6. the number of restrictions imposed on each vector

static CATS::ClassifyDummies(const mR, const mDR)

static CATS::ColCat(const amA)

Parameters

amA	array[n] of matrices.

Returns

~ on all the matrices in amA

CATS::CompanionRoots()

Prints the roots of the companion matrix based on the current model.

virtual CATS::Covar()

static CATS::DecLQ(const mA)

Parameters

mA	A[m][n] matrix.

Returns

{L, Q, TRUE/FALSE}

static CATS::DecQL(const mA)

Parameters

mA	A[m][n] matrix.

Returns

Q, L

static CATS::DecQRt(const mA)

Parameters

mA	A[m][n] matrix.

Returns

Q, R'

static CATS::DiagCat(const amA)

Parameters

amA	array[n] of matrices.

Returns

diagcat on all the matrices in amA

virtual CATS::DoAutoOutput()

Adds to the standard output (does nothing here).

CATS::DoBartlettBetaFactor(const amH)

Parameters

amH	array of H matrices

Returns

0 (no Bartlett correction available) or the correction factor.

CATS::DoCompute(const iDo)

Parameters

iDo	int, DO_ flags to specify computations to include

virtual CATS::DoEstimation( vPar)

Parameters

Estimate

vPar	vector of initial parameters (ignored, all zero here)

Returns

vec(alpha)|vec(beta)|vec(Omega)

static CATS::DoGetCompanion(const mAlpha, const mBeta, const mGamma_i, const amRoots)

Parameters

out: if array, amRoots will hold an [n][4] matrix : real part, imaginary part, absolute value and argument.

mAlpha[p][r]	alpha
mBeta[p1][r]	beta
mGamma_i[p][p	* (k-1)] coefficients of lagged differences of y_t
amRoots	in: array or 0 (default if no arguments)

Returns

companion matrix

CATS::DoI1RankTable(const dAlpha=0.05, const bDoBartlett=TRUE)

Parameters

dAlpha	Significance level to be used in table (default is 0.05 for 95% quantiles)
bDoBartlett	boolean, TRUE to compute Bartlett correction (default is TRUE)

Returns

an array[2]: [m_cY][8] matrix with {"p-r", "r", "Eig.Value", "Trace", "Trace*", "Crit5%", "p-value", "p-value*"} and a message.

CATS::DoI1Residuals(const mPi)

Parameters

mPi	matrix Pi[m_cY][cz1]

Returns

residual matrix [m_cT][m_cY]

Comments

Returns, in matrix form: Z0|Z2 - Z1|Z2 Pi_hat' where Pi_hat' = beta alpha'.

virtual CATS::DoMoments()

Comments: Creates m_mQtZ0_Z2,m_mQtZ1_Z2, m_mQtZ0sq_Z2 as well as m_mS00, m_mS01, m_mS11.

static CATS::DoRestrictAlphaBeta(const mAlpha0, const mBeta0, const amG, const amH, const bWithStandardErrors, const vNormBeta, const mQtZ0, const mQtZ1, const mQtZ0sq, const cT, const dCrit, const iMaxIt, const iInfo, const dBFGSCrit, const iMaxItBFGS, const iSwitchOptions=0, const cStartSearch=20)

CATS::DoSetI1Rank(const iR)

Parameters

iR	integer, r

Comments

Use I1Rank to specify the rank prior to estimation. Called by DoEstimation to complete the estimation for the specified rank. DoSetI1Rank can be called after estimation to change the rank.

CATS::DoUpdateEstimates_I1(const mAlpha, const mBeta, dLogDet=.NaN, const viNorm=<>)

Parameters

mAlpha	alpha matrix
mBeta	beta matrix
mOmega	Omega matrix or <>: compute from residuals
dLogDet	double, log\|Omega\| or .NaN: compute from Omega
viNorm	normalization vector or <>: normalize automatically

CATS::DoUpdateRestrictedSettings(const amG, const amH, const vVpg, const iAlphaFlag, const iBetaFlag, const cResCount, const sResType=0)

Parameters

amG	array[r] or array[1] of G_i matrices
amH	array[q] of H_i matrices
vVpg	vector[q] rank for each restriction.
iAlphaFlag	int, type of restrictions imposed on alpha, ALPHA_NONE,ALPHA_EACH,...
iBetaFlag	int, type of restrictions imposed on beta, BETA_NONE,BETA_EACH,...
cResCount	int, no of restrictions imposed
sResType	0 to used standard label (default), or string for specified label

CATS::DropDiffLags(const vDropLags, const vDropLagsExo=<>)

Parameters

vDropLags:	vector with lags to drop.
vDropLagsExo	(optional): vector with DX lags to drop (currently unsupported).

static CATS::ECMToVAR(const mPi, const mGammas)

Parameters

mPi[p][p1]	matrix alpha*beta' mPi[][p:] part is ignored for this
mGammas[p][p	* (k - 1)] matrix with coefficients of lagged differences

Returns

matrix[p][p * k] with parameters for the model in levels: A_1 ~ ... ~ A_k.

static CATS::Eigen(const ma, const amval, const amvec=0)

Parameters

out: if array, amval will hold an [2][m] matrix with eigenvalues of A, first row is real part, second row imaginary part. out: if array, amvec will hold an [2m][m] matrix with eigenvectors of A (real values are in first m rows).

ma	in: m x m matrix A
amval	in: array
amvec	in: array or 0

Returns

0 (success) or 1 (max iterations reached)

virtual CATS::EstimateAlphaBeta( amG, amH, const vNormBetaOvr=<>)

Parameters

amG	array[r] of G_i matrices
amH	array[r] of H_i matrices
vNormBetaOvr	<> (default) or vector[r] with normalization indices (-1 for free to choose) used to override normalization of the restrictions

Returns

array {alpha, beta, retval, loglik, isidentified, no of restrictions, se_alpha, se_beta}

note

does not change any member variables, so can be called in parallel

static CATS::FixRank(const mHj, const mRj, const mH, const iTargetRank)

Parameters

Because the rank can only increased by 1 (a vector is added), the iTargetRank can only be one higher than the current rank.

mHj	matrix, current (jth) H matrix
mRj	matrix, orthogonal complement of current H matrix (zero sizec if mHj is square)
mH	matrix, concatenation of all other H matrices
iTargetRank	integer target rank.

Returns

array[3]: {1, modified mRj, modified mHj} if successful; if failed: {1, original mRj, original mHj}.

static CATS::GammaApprox_I1(const dTstat, const iType, const dQuantile, const iDetTrend, const iP, const iR, const iDimX1, const iS=0, const cSD=0, const viSDDate=<>)

Parameters

dTstat	double trace test statistic
iType	RANKTEST_TRACE, RANKTEST_LMAX or RANKTEST_BREAK
dQuantile	double, 0.95 for alpha = 0.05
iDetTrend	NONE, CIMEAN, DRIFT or CIDRIFT
iP	integer, dimension of system (m_cY)
iR	integer, rank
iDimX1	integer, no of weakly exogenous variables
iS	integer, no of I(1) trends s_1 in the I(2) model (only used for GammaApprox_I2)
cSD	integer, no of breaks (1,2, only for iType==RANKTEST_BREAK)
viSDDate	vector, size of breaks (used for RANKTEST_BREAK, expressed as fraction of sample size)

Returns

array with two elements: {p-value, critical-value}

Comments

References: Doornik: Approximations to the Asymptotic Distribution of Cointegration Tests, Journal of Econometric Surveys, 1998, Vol 12, No 5. Johansen, Mosconi & Nielsen: Cointegration Analysis in the Presence of Structural Breaks in the Deterministic Trend, Econometrics Journal 2000, 3. Harbo, Johansen, Nielsen, Rahbek: Asymptotic Inference on the Cointegrating Rank in Partial Systems, Journal of Business and Economic Statistics 16.

static CATS::GammaApprox_I2(const dTstat, const iType, const dQuantile, const iDetTrend, const iP, const iR, const iDimX1, const iS=0, const cSD=0, const viSDDate=<>)

Procedure for calculating the approximate p-values and quantiles of the asymptotic distribution of the I(2) test for the cointegrating rank. See GammaApprox_I1

CATS::GenerateAsY(const mEps, const mAlpha, const mBeta, const mGams, const mPsi)

Parameters

mEps	matrix[T][p] error term for the random sample (must match the estimated sample size, which can be obtained by Modelbase::GetcT())
mAlpha[p][r]	matrix or <>
mBeta[p1][r]	matrix or <>
mGams[p][n	* (k - 1)] matrix with coefficients on lagged differences
mPhi[p][d]	matrix with coefficients on remaining short-run variables

Returns

matrix[T][p] random sample

static CATS::GenerateECM(const cT, const iDet, const mAlpha, const mBeta, const mGams, const mPsi, const mOmegaSqrt=<>, const mYinit=<>, const mYmean=<>)

virtual CATS::GenerateY( mEps=<>)

Parameters

normality is assumed, as if specifying rann(GetcT(), GetcY()) * choleski(GetResVar())')

mEps	<> or matrix[T][p] with error term for the random sample; if <> (the default)

Returns

matrix[T][p] random sample

Comments

To make mEps match the estimated variance use:

CATS::GetAlpha()

Returns: current alpha matrix

CATS::GetAlphaSe()

Returns: matrix of standard errors of current alpha

CATS::GetAlphaVAR()

Returns: alpha[p][p] matrix for the full rank VAR

static CATS::GetBar(const mA)

Parameters

mA	matrix[p][r], p >= r

Returns

A_bar

CATS::GetBeta()

Returns: current beta matrix

CATS::GetBetaSe()

Returns: matrix of standard errors of current beta (<> if not identified)

CATS::GetBetaVAR()

Returns: beta[p][p] matrix for the full rank VAR, beta'S_{11}beta = I_p

static CATS::GetColumnScale(const mB, const cRows=0)

Parameters

mB	[m][n] matrix, m >= n
bDifferent	each row index should be different (FALSE by default)
cRows	int, >0: only consider rows 0,...,cRows-1 of mB (<= 0: all rows, this is the default)

Returns

[n] column vector with row indices for sizec 0,..,n-1.

Comments

Does not take small pivots into account.

virtual CATS::GetCompanion(const amRoots=0)

Parameters

out: if array, amRoots will hold an [n][4] matrix : real part, imaginary part, absolute value and argument.

amRoots

in: array or 0 (default if no arguments)

Returns

companion matrix

virtual CATS::GetDimensions(const sType="")

Parameters

sType

string: type of dimension ("p","p1","r","s","s2","R") or default "" for array with all

Returns

array with {p, p_1, r, 0, p-r, R (no of restrictions imposed)} or single dimension

Comments

Also see GetLags

CATS::GetEval()

Returns: column vector with n trace tests

CATS::GetGammaFrom(const mAlpha, const mBeta)

Parameters

mAlpha[p][r]	matrix or <>
mBeta[p][r]	matrix or <>

Returns

matrix Gamma[p][p1] = (Gamma_y : Gamma_0)

Comments

Not to be confused with the first return value of ShortRunEval. For the I(1) model: Gamma_y = I - \sum Gamma_i. Gamma_0 = (-\sum Psi_i : -mu_0)

CATS::GetGamsInArray(const mAlpha, const mBeta)

Parameters

mAlpha[p][r]	matrix or <>
mBeta[p][r]	matrix or <>

Returns

{} if lag length k <= 1, else array[k-1] with Gamma_1,.. Gamma_{k-1}

virtual CATS::GetGroupLabels()

static CATS::GetInvertibleSubMatrix(const mB, const viNorm=<>)

Parameters

mB	[m][n] matrix, m >= n
viNorm	[n] specified override of normalization (default is <>: do not override)

Returns

array with three elements: { the submatrix of mB, adjusted mB, [n] column vector with unique row indices for columns 0,..,n-1. }

Comments

Uses pivoting info from the QR decomposition of B' to get the best (n x n) submatrix can be singular if rank(B)<n.

CATS::GetLags()

Returns: current an array with three values: lag length of VAR, lag length of exogenous variables, lag length of restricted deterministics. See Lags

CATS::GetLambdaXiSe()

Returns: matrix[p+1][1] sdev(xi) | sdev(sum(xi))

CATS::GetLogLikUnr()

Returns: double, log-likelihood of the unrestricted model (except for the rank-restrictions)

CATS::GetLogLikVAR()

Returns: double, log-likelihood of the unrestricted VAR

CATS::GetMARep()

Returns: array[9]: {C(1), SE[C(1)], alpha_ort, SE[alpha_ort], beta~, SE[beta~], tau, <>, correlations}

CATS::GetMaxMethodInfo()

Returns: array with information

CATS::GetMaxMethodName()

Returns: string with package name

CATS::GetMethodLabel()

Returns: label of estimation method.

virtual CATS::GetMethodLabels()

CATS::GetModelLabel()

Returns: string with CATS(#) where # is model number.

CATS::GetOmega()

Returns: current Omega matrix

virtual CATS::GetPackageName()

Returns: string with package name

virtual CATS::GetPackageVersion()

Returns: string with package version

virtual CATS::GetParNames()

Returns: array of strings with parameter names

CATS::GetPi()

Returns: current Pi matrix

CATS::GetPiSe()

Returns: matrix of standard errors of current Pi

virtual CATS::GetRegsLongRun(const fCorrected=TRUE)

Parameters

fCorrected,

TRUE: corrected for short-run (default), else uncorrected.

Returns

(T x cZ1) matrix

virtual CATS::GetResVar()

Returns: current Omega matrix

CATS::GetResiduals()

Returns: (T x p) matrix of residuals

virtual CATS::GetS00()

Returns: (cZ0 x cZ0) matrix with S_{00}

CATS::GetSeasonals()

CATS::GetTraceTest()

Returns: column vector with n trace tests

CATS::GetTrend()

Returns: m_iDetTrend, which defines the current trend behaviour, see Trend

virtual CATS::GetUNames()

Returns: array of strings with U variable (exogenous I(0)) names

virtual CATS::GetXNames()

Returns: array of strings with X variable (exogenous I(1)) names

virtual CATS::GetYNames()

virtual CATS::GetZ0()

Returns: (T x p) matrix with Z_0 = DY

virtual CATS::GetZ0Names()

Returns: array[2] with names

virtual CATS::GetZ1()

Returns: (T x cZ1) matrix with Z_1 = Y_1 ~ X

virtual CATS::GetZ2()

Returns: (T x cZ2) matrix with Z_2 = [DY_1 .. DY_{k-1}] ~ [DX1 .. DX1_{m-1}] ~ U

virtual CATS::GetZ3()

Returns: <>

virtual CATS::GetcDfLoss()

Returns: number of estimated parameters

virtual CATS::GraphicAnalysis(const iPlot=0, const bActualFitted=FALSE, const bActualFittedCross=FALSE, const iResiduals=2, const iCointegration=2, const iAcf=1, const iAcfLag=-1, const iDensity=0, const iRootsPlot=0, const iLevelsPlot=0, const iEqn=-1)

Parameters

iPlot	int: index of first plot
bActualFitted	TRUE: include plots of actual and fitted
bActualFittedCross	TRUE: include crossplots of actual and fitted
iResiduals	>0: include plots of 1: residuals, 2: scaled residuals, 3: standardized to IN[0,1] residuals (as in normality test)
iCointegration	>0: cointegrating vectors, 1: one plot, 2: individual plots
iAcf	>0: include ACF plots of 1: residuals, 2: squared residuals
iAcfLag:	lag length for ACF plots (<=0 then using default)
iDensity	>0: include histogram and QQ plots of residuals
iRootsPlot	>0: include roots of companion matrix
iLevelsPlot	>0: include levels plot Y (iLevelsPlot >1 with DY; iLevelsPlot >2 with DDY)
iEqn	<0: plot for all equations, else plot specified equation

Returns

int: number of plots created.

virtual CATS::I1Rank(const iR)

Comments

Use DoSetI1Rank to change the rank after estimation.

Parameters

iR	integer, 0 <= r <= p

Comments

This changes the model class to I1 (relevant for CATSI2).

CATS::I1RankTable(const bDoBartlett=TRUE)

Parameters

bDoBartlett,

boolean, TRUE: include Bartlett correction (default)

Returns

the table.

CATS::Init()

Reset the transitory properties.

virtual CATS::InitData()

Checks the model formulation and creates the data matrices.

virtual CATS::InitPar()

Initialize the model parameters

static CATS::InsertFixedPhiInPar(const vP, const cAlpha, const vDroppedPhi)

Parameters

(assumed sorted in ascending order)

vP,	column vector with unrestricted (free) parameters.
cOffset	int, no of free parameters before (usually alpha).
vDroppedPhi	vector[r] indices of the elements of phi that are fixed at unity

Returns

modified vP (unchanged if vDroppedPhi == <>)

static CATS::InvertSymQR(const mA, const pdLogLik=0)

Parameters

mA	A[n][n] matrix, symmetric p.d. matrix.
pdLogDet	in: address, out: log determinant.

Comments

Like invertsym, but using the QR. Unlike invertsym, this avoids a message on singularity, and will return a value when (close to) singular (which can be a disadvantage).

virtual CATS::IsI2()

Returns: TRUE for I(2): s_1 != p - r, FALSE otherwise

virtual CATS::IsIdentified()

Returns: TRUE if identified

static CATS::IsIdentifying(const amH, const bFull=FALSE)

Parameters

amH	array[r] of matrices of dimension (n x s_i)
bFull	bool, TRUE: check all matrices, FALSE: stop at first failure (default)

Returns

(r x 1) column vector with 1 in position i if H_i space full rank

static CATS::IsIdentifyingJac(const mAlpha, const mBeta, const amG, const amH, const vDroppedPhi, const cPar)

Parameters

mAlpha[p][r]	alpha
mBeta[p1][r]	beta
amG	array[r] of matrices of dimension
amH	array[r] of matrices of dimension
cPar	no of parameters estimated

Returns

{TRUE if identified, rank of Jacobian}

CATS::IsOutsideCircle()

Returns: TRUE if inside (or the radius is >= 2)
Comments: Use SetRadius to set or get the radius.

virtual CATS::IsUnivariate()

static CATS::JacLinResBeta(const amH, const vDroppedPhi)

Parameters

amH	array[r] of H_i matrices, one for each restriction

Returns

Jacobian matrix

CATS::Lags(const lags, const lagsExo=-1, const lagsDet=-1)

Parameters

If this is zero, then no lagged differences are added unrestrictedly. If this is zero, then no lagged differences are added unrestrictedly.

lags:	integer with Y lag length >= 1 (default is 2).
lagsExo	(optional): integer with lag length >= -1 of weakly exogenous variables (default is -1: same as Y).
lagsDet	(optional): integer with lag length >= -1 of restricted deterministic variables (default is -1: same as lagsDet).

Comments

Sets m_cLags, m_cLagsExo, and m_cLagsDet. If lagged Y are added through Select, then InitData will automatically set m_cLags from that instead. If no lags are set, the default is 2.

static CATS::LineSearch(const vP, const vPprev, const vP0, const dF, const dFprev, const fnLogLik, const vStepGridUp, const vStepGridDown, cStages)

Parameters

vP	matrix[m][q] with parameters
vPprev	matrix[m][q] with previous parameters
vP0	matrix[m][q] with previous candidate parameters (may be same as vPprev)
dF	current function value,
dFprev	previous function value,
fnLogLik	function to compute function value (MaxBFGS format)
vStepGridUp	vector with grid points when dF >= dFprev (progress)
vStepGridDown	vector with grid points when dF < dFprev (went down)
cStages	1 or 2, number of linesearch stages (can also be -1, only used for research)

Returns

array with three elements: parameters, function value, step size

CATS::LogDetFromLogLik(const dLogLik)

Parameters

dLogLik

double log-likelihood

Returns

log-likelihoodlog|Omega|.

CATS::LogLikFromLogDet(const dLogDet)

Parameters

dLogDet

double log|Omega|, Omega = (V'V)/T

Returns

log-likelihood.

static CATS::LogdetI1Model(const mAlpha, const mBeta, const cAlpha, const cBeta, const adFunc, const avScore, const amHessian, const mQtZ0, const mQtZ1, const mQtZ0sq, const cT, const amG, const amH, const vDroppedPhi, const mJacA, const mJacB, const bAsymp=FALSE)

Evaluates the scaled log-likelihood of the restricted I(1) model. Only used in DoRestrictAlphaBeta

static CATS::LogdetSymQR(const mA)

Parameters

mA	A[n][n] matrix, symmetric p.d. matrix.

Returns

double: log determinant.

Comments

This uses the QR decomposition, so avoiding a message on singularity, and will return a value when (close to) singular (which can be a disadvantage).

virtual CATS::MARep()

Prints the matrices of the moving average representation.

static CATS::MapParToAlphaBeta( vP, const cAlpha, const amG, const amH, const vDroppedPhi=<>)

Parameters

vP,	column vector with unrestricted (free) parameters.
cAlpha	int, no of free parameters in alpha.
amG	array[r] of G_i G matrices
amH	array[r] of H_i matrices, one for each restriction
vDroppedPhi	vector[r] indices of the elements of phi that are fixed at unity (default is <> for none)

Returns

{alpha, beta}

static CATS::MapScoreToAlphaBeta(const mDloglikDpi, const mAlpha, const mBeta, const cAlpha, const cBeta, const amG, const amH, const vDroppedPhi)

Maps the score of the scaled log-likelihood of the restricted I(1) model w.r.t. Pi to that w.r.t. theta and phi. Used in DoRestrictAlphaBeta

static CATS::MapToPar(const mB, const amH)

Parameters

mB[p][r]	parameter matrix
amH	array[r] of H_i matrices, one for each restriction

Returns

column vector with free parameters

Comments

If each H_i equals I_p, the return value is vec(mB)

static CATS::MapVarToAlphaBetaPiSE(const mAlpha, const mBeta, const mMaa, const mMbb, const mJacA, const mJacB)

Returns: {se_alpha, se_beta, se_pi}

CATS::MaxControl(const iMaxIt=10000, const iInfo=1000)

Parameters

iMaxIt	integer max iterations (default is 10000)
iInfo	integer print every iInfo iterations (default is 1000)

Comments

For all parameters: use -1 to leave current value unchanged.

CATS::MaxControlEps(const dSwitchCrit=1e-9, const dBFGSCrit=1e-5)

Parameters

dSwitchCrit	double, convergence epsilon for switching (default is 1e-8)
dBFGSCrit	double, convergence epsilon for BFGS (default is 1e-5)

CATS::MaxOptions(const iHybridInit=-1, const iOptions=-1, const cStartSearch=20)

Parameters

iMaxItBFGSstep	no of steps for intermediate BFGS in Hybrid method (default -1 to leave current value unchanged; use 0 to reset default)
iOptions	options for switching (default is -1 to leave current value unchanged; use 0 to reset default)
cStartSearch	no of random starts for hybrid alpha-beta switching (default is 20, -1 to leave current value unchanged)

Comments

For all parameters: use -1 to leave current value unchanged.

static CATS::NormalizePhiInPar( vP, const cBefore, const vDim, const dCutOff=0, const vNormIdxP=<>)

Parameters

vP,	column vector with parameters.
cBefore	int, no of parameters before phi starts in vP.
vDim[p]	dimensions of column-parameters in phi
dCutOff	\|scale\| > cutoff is accepted scale (default is 0)
vNormIdxPhi	<> (default) or vector[p] with normalization indices (indices in each column of phi)

Returns

{vP (normalized), vNormIdxP (indices in vP of normalization)}

static CATS::NormalizeThetaPhiInPar( vP, const cAlpha, const amG, const amH, const bApply=FALSE, const vNormBeta=<>)

Parameters

vP,	column vector with unrestricted parameters (including normalizations).
cAlpha	int, no of free parameters in alpha.
amG	array[r] of G_i matrices, only used for dimensions, cannot be empty
amH	array[r] of H_i matrices, only used for dimensions, cannot be empty
bApply	TRUE: remove normalized elements from phi (default is FALSE to leave them)
vNormBeta	<> or vector[r] with valid normalization indices

Returns

{vP (renormalized), vFixPhi (indices in phi), vDropPhi (indices in phi of dropped elements, <> if none), vScale (scale factors used)}

static CATS::Nullspace(const mA)

Parameters

mA	A[m][n] matrix, m >= n with full column rank m.

Comments

Returns the null space of A. If n=0 this is I_m, if n=m this is an empty matrix, else it is the first m-n columns of nullspace(ma). There is currently no error message if A has deficient column rank.

This is different from nullspace:<br> 1. nullspace returns an (m x s) matrix where s is the column rank of A, s >= n;<br> 2. nullspace does not require m >= n.

virtual CATS::Output()

Comments: Switch printing of by calling SetPrint(FALSE).

CATS::OutputBootstrapHeader(const sTitle, const cB, const iType, const time=0)

Prints header of bootstraps report.

CATS::OutputHeader(const sTitle, const asR, const asX)

Comments: Switch printing of by calling SetPrint(FALSE).

CATS::OutputLogLik(const sExtra="")

Prints the log-likelihood and dimensions.

virtual CATS::OutputOutliers( dSE)

CATS::OutputRestrictionsTest()

Prints the log-likelihood and dimensions.

static CATS::PartialMoments(const mY, const mX, const cT=-1, const fWarn=0, const amPi=0)

Parameters

mY	[n][p] matrix
mX	[n][k] matrix (may be <>, may be singular)
cT	if -1 (the default) uses T=sizer(mY) else T=cT
fWarn	TRUE: print warning for singularity or bad scaling (default FALSE)
amPi	in: 0 or address of variable out: coefficient matrix Pi' (if array on input)

Returns

[p][p] matrix (R'R)/T

CATS::PrintAlphaBeta(const mAlpha, const mBeta, const mAlphaSe, const mBetaSe, const asY, const mPiSe=<>, const bPiPrint=0)

Parameters

mAlpha	alpha matrix, may be <>
mBeta	beta matrix, may be <>
mAlphaSe	standard errors of alpha, may be <>
mBetaSe	standard errors of beta, may be <>
cT	integer scale
mPiSe	standard errors of Pi, may be <>
bPrintPi	bool, TRUE: print Pi

CATS::PrintDeterministics()

Prints the Deterministic specification.

CATS::PrintI1Rank(const sExtra="")

Prints the estimates for the current rank.

CATS::PrintI1RankTable(const mTable, const dAlpha, const sHeader, const sFooter="", asTraceAlt=0)

Parameters

mTable	Table produced by DoI1RankTable
dAlpha	Significance level used in table
bDoBartlett	boolean, TRUE to if Bartlett correction used in table

static CATS::PrintMat( mA, const mSE, const asRow, const asCol, const asOptions="")

Parameters

mA	in: matrix to print
mSE	in: matrix with standard errors (conforms to A or <>)
asRow	in: array with row labels
asCol	in: array with column labels
asOptions	in: array of strings with options: "t-values", "separate", "%lwr" (default is {})
bSeparate	in: TRUE: separate matrices, else se/tvals below each row (default FALSE)

static CATS::ProjectOntoH(const mBeta, const amH, const bIsIdentified=TRUE, const vVpg=<>)

Parameters

checked for each vector. of vectors in group i, so adding up to r. Default is <>, in which case the restrictions are analyzed for duplicates.

mBeta[p1][r]	unrestricted beta for rank r
amH	array[q] of H matrices, one for each restriction
bIsIdentified	TRUE: restrictions already known to be identifying, FALSE: rank of rotation
vVpg	matrix[q] rank for each restriction. Entry i of vVpg gives the number

Returns

{rotated beta, rotation matrix, is_full_rank_rotation, beta satisfying the restrictions, parameter vector}.

Comments

See Johansen (1995, JEcts), top of p.120.

static CATS::ProjectOntoH_reg(const mBeta, const amH)

static CATS::RROlsQ( mQtY, mQtX, const mQtYsq, const cT, const cR, const mQtZ=<>, const aGamma=0)

Parameters

mQtY	[n][p] matrix
mQtX	[n][k] matrix
mQtYsq	[p][p] matrix, rest of Q'Y'YQ
cT	in: sample size
cR	in: rank r
mQtZ	[n][q] matrix (may be <>, may be singular), default is <>
aGamma	in: array, out: gamma (if array and mQtZ != <>)

Returns

{alpha[p][r], beta[k][r], log|Omega|, evals

static CATS::RROlsQ2( mQtY, mQtX, const mQtYsq, const cT, const cR, const mQtZ=<>, const aGamma=0)

static CATS::RanBootstrap(const mResiduals, const iType=0)

static CATS::RankOrder(const mA)

Parameters

m	[m][n] matrix.

Returns

returns an [m][n] matrix with the rank order of each element in mX

Comments

The smallest element element has value 0, the largest is mn - 1. Does not handle ties.

virtual CATS::RecursiveAnalysis(const iPlot=0, const cTMin=0, const bEval=FALSE, const bXi=TRUE, const bTrace=TRUE, const bLogLik=TRUE, const bAlpha=FALSE, const bBeta=FALSE, const bFixShortRun=FALSE, bApproxXiSe=TRUE)

Parameters

iPlot	int: index of first plot (-1: no plots, default is 0)
cTMin	int: min no of observations (<=0: decided here, default is 0)
bEval	boolean: TRUE: recursive eigenvalues, FALSE (default): skip
bXi	boolean: TRUE (default): recursive transformed eigenvalues, FALSE: skip
bTrace	boolean: TRUE (default): recursive trace tests, FALSE: skip
bLogLik	boolean: TRUE (default): recursive likelihood constancy, FALSE: skip
bAlpha	boolean: TRUE: recursive alpha (only if identified), FALSE (default): skip
bBeta	boolean: TRUE: recursive beta (only if identified), FALSE (default): skip
bFixShortRun	boolean: TRUE: keep short run fixed (R1 form), FALSE (default): re-estimate short run (X form)
bApproxXiSe	boolean: TRUE (default): approximate from last value, FALSE: re-estimate (only for X-form)

CATS::RenewY(const mY)

Parameters

mY	matrix[T][p] T must match the estimated sample size, which can be obtained by Modelbase::GetcT()

Returns

matrix[T][p] random sample

CATS::ResidualAnalysis()

Prints multivariate and unvariate analysis of the estimated residuals.

virtual CATS::Restrict(const amRes, const vNormBetaOvr=<>)

Parameters

may also contain [method=beta]

amRes	array with restrictions, sections are [alpha],[beta] (without sections, [beta] is assumed)
vNormBetaOvr	<> (default) or vector[r] with normalization indices (-1 for free to choose) used to override normalization of the restrictions

Returns

success value (0 failure, 1 success)

virtual CATS::RestrictAlphaBeta( amG, amH, const vNormBetaOvr=<>)

Parameters

(same as {G,G,..,G}; <> for unrestricted), or array[r] of strings with restriction expressions

amG	array[r] of G_i G matrices ({} for unrestricted alpha), or matrix G with common restriction
amH	array[r] of H matrices, one for each restriction ({} for no restrictions), or array[r] of strings with restriction expressions
vNormBetaOvr	<> (default) or vector[r] with normalization indices (-1 for free to choose) used to override normalization of the restrictions

Returns

success value (0 failure, 1 success)

CATS::RestrictBeta( amG, amH, vVpg=<>)

Parameters

Or a single matrix or string, which is treated as a common restriction on all columns of beta. of vectors in group i. If the argument is omitted, a vector of ones is used, unless amH is a matrix or string. The sum is the rank of the restricted beta.

amG	G matrix (<> for unrestricted alpha) or string with common restrictions on alpha
amH	array[q] of H matrices, one for each restriction ({} for no restrictions), or array[q] of strings with restriction expressions.
vVpg	vector[q] rank for each restriction. Entry i of vVpg gives the number

Returns

success value (0 failure, 1 success)

static CATS::RowCat(const amA)

Parameters

amA	array[n] of matrices.

Returns

| on all the matrices in amA

CATS::Seasonals(const iUse=1)

CATS::SetAlphaBetaI1Start(const mAlpha, const mBeta)

No return value

static CATS::SetMaths(const iMaths, const sLabel)

Parameters

iMaths:	one of MATHS_... constants
sLabel:	new label

CATS::SetMaxMethodInfo(const sName, const aInfo=<>)

Parameters

sName:

string with new name

CATS::SetNormalization(const bSet)

Parameters

bSet:

TRUE (most significant alpha) or FALSE (largest beta)

CATS::SetPrintMatOptions(const asOptions, const iAdd=0)

Parameters

PrintMat

asOptions	in: array of strings with options: "t-values", "separate", "%lwr"
iAdd	in: 0: replace existing options, 1: add to options (default), -1: remove from options

Returns

array with current options

CATS::SetRadius(const dRadius=1.00001)

Parameters

dRadius

in: double, sets new radius if >0

Returns

current radius

Comments

Used by IsOutsideCircle

static CATS::SetTrackProgress(const bTrack)

Parameters

bTrack:

TRUE or FALSE

CATS::SetUseDefaultSeed(const bSet)

Parameters

bSet:

TRUE or FALSE

CATS::ShortRun()

Prints the short-run coefficients.

CATS::ShortRunEval(const mAlpha=0, const mBeta=0, const mOmega=0, const bDoStdDev=FALSE)

Parameters

mAlpha[p][r]	matrix (or int 0 for current alpha, the default)
mBeta[p][r]	matrix (or int 0 for current beta, the default)
mOmega[p][p]	matrix (or int 0 for current omega, the default)
bDoStdDev	TRUE: compute standard errors (default is FALSE)

Returns

array of four matrices: { Gamma_i[p][p * (k-1)] for lagged differences of y_t, Phi[p][] remainder, SE[Gamma_i], SE[Phi] } where m is the maximum lag length (gaps will be set to zero), and the SE matrices will be empty if bDoStdDev is FALSE (the default).

virtual CATS::ShrinkSample(const cDrop, const cTmin=1, const bFixShortRun=FALSE)

Parameters

cDrop	int: number of observations to shrink the sample by
cTmin	int: don't shrink past this sample size
bFixShortRun	boolean: TRUE: keep short run fixed (R1 form), FALSE (default): re-estimate short run (X form)

Returns

TRUE if shrinkage was successful.

CATS::SimulateRankDet(const cTsim, const bPrint)

Parameters

cTsim	Length of random walks (T, optional argument, default is 400)
cM	Number of replications (M, optional argument, default is 1000)

Returns

array[3] {table with tests and p-values, table of Gamma quantiles, table of simulated quantiles }

CATS::SimulateRankTest(const cTsim=400, const cM=1000)

Parameters

cTsim	Length of random walks (T, optional argument, default is 400)
cM	Number of replications (M, optional argument, default is 1000)

Returns

array[3] {table with tests and p-values, table of Gamma quantiles, table of simulated quantiles }

static CATS::SolveRRR(const mS00, const mS01, const mS11, const amEigVal, const amEigVec, const cR=-1)

Parameters

out: (r x 1) column vector with eigenvalues (<> if r==0 or failed) out: (r x r) matrix with eigenvectors in sizec, r=min(p,p1) (<> if r==0 or failed)

mS00	in: symmetric positive definite matrix S_{00}[p][p]
mS01	in: matrix S_{01}[p][p1]
mS11	in: symmetric matrix S_{11}[p1][p1]
cR	in: optional rank R: r=(R,min(p,p1)); ignored if <0 (the default)
amEigVal	in: address
amEigVec	in: address

Returns

log|S_{00}| (.NaN if eigensymgen failed, .Inf if mS00 singular)

static CATS::SplitRestrictions(const asRes, const asSections, const aaH1, ...)

Parameters

asRes	array to split by section
asSections	array[q] strings with section labels
aaH1	first section
..	second section (or {}), etc

static CATS::StatComp(const mAlpha, mBeta, const amGamma, const iLags, const mS00, const amP, const amQ)

Parameters

out: P matrix out: Q matrix

mAlpha	matrix
mBeta	matrix
mGamma	array[lags - 1] of Gamma_i[p][p] matrices
cLags	integer, autoregressive lag length
mS00	unrestricted variance matrix
amP	in: address
amQ	in: address

Comments

Creates the companion matrix P and the matrix Q for the stationary process Y(t) = P*Y(t-1) + Q*epsilon(t) where Y(t) = (X(t)'Beta,DX(t)',....,DX(t-k+2)')'.

CATS::TestAlphaCommon( mG)

Parameters

amG	G matrix of dimension (n x S), 0 < S < r, or string.

Returns

array[6] {test, p-value, alpha, beta, omega, df}

CATS::TestAlphaKnown(const mG)

Parameters

mG	G matrix of dimension (n x S), 0 < S <= r.

Returns

array[6] {test, p-value, alpha, beta, omega, df}

Comments

Known alpha restrictions cannot be combined with restrictions on beta.

static CATS::TestDiff(const x, const y)

Returns: Computes min_i{ |(y[i] - x[i]) / x[i]|, |(y[i] - x[i])|} for each element in the vectors x,y and returns the maximum of that.

virtual CATS::TestEach(const bAlpha=TRUE, const bBeta=TRUE)

Parameters

bAlpha	TRUE: Weak exogeneity and unit vectors (default)
bBeta	TRUE: Variable exclusions and stationarity (default)

Returns

array[4]: matrix of tests results (first column test, 2nd pvalues) for each case: long-run, stationarity, weak exoegenity, unit alpha

virtual CATS::TestEachAllRanks(const bAlpha=TRUE, const bBeta=TRUE)

Parameters

bAlpha	TRUE: Weak exogeneity and unit vectors (default)
bBeta	TRUE: Variable exclusions and stationarity (default)

Returns

array with three elements: vector of tests for each.

CATS::TestEachBetaRestriction( fnGetG, fnGetH, const vVpg, const asZ, const cRes, const sType)

Parameters

fnGetG	function returning common restriction on alpha
fnGetH	function returning restrictions on beta
vVpg	vector with multiplicity of restrictions on beta
cRes	no of restrictions imposed
sType	type of restrictions (for output)

Returns

matrix of tests results (first column test, 2nd pvalues).

virtual CATS::TestForecastGraphics(const bExp)

virtual CATS::TestGraphicAnalysis()

Plots residuals and ACF.

static CATS::TestRanksAndFix(const amH, const amR, const vIndex, const amHnew, const amRnew)

Parameters

amH	array of H matrices
amH	array of R matrices, the orthogonal complements the H matrices
vIndex	vector with indices of H to consider from amH
amHnew	address on input, if successful: new H matrix on output
amRnew	address on input, if successful: new R matrix on output

Returns

-1 if failed, otherwise index in amH, amR of new H,R matrices

virtual CATS::TestSummary()

CATS::Trend( mode)

Parameters

Equivalently, use the strings "H_z","H_c","H_lc","H_l", or the predefined constants CATS::NONE, CATS::CIMEAN, CATS::DRIFT, CATS::CIDRIFT.

mode	string: one of "NONE","CIMEAN","DRIFT","CIDRIFT".

Comments

Sets m_iDetTrend, which is used in InitData to add the Constant and Trend to the model. InitData takes the Constant and Trend that are specified through Select into account (and updates m_iDetTrend if necessary).

m_iDetTrend is set to CATS::NONE by default.

The I(1) hypotheses are:

X_VARU_VARU_VAR

Hypothesis	Constant	Trend
H_z
H_c
H_lc
H_l		X_VAR

The string argument is case sensitive.

virtual CATSI2::ACTest_LM(const mResid, const iOrder=1)

Parameters

mResid	matrix of residuals [T][p]
iOrder	integer, lag length for test (default is 1)

Returns

the test statistic

CATSI2::BootstrapRankTest( cB=399, const iType=0)

Parameters

cB	Number of bootstrap replications (B, optional argument, default is 399)
iType	Type of bootstrap sampling: BS_WILD (Wild N[0,1], the default), BS_IID (IID resampling), BS_SIGN (+1,-1)

Returns

array[2] {table with tests and p-values, table of quantiles}

CATSI2::CATSI2()

Constructor.

static CATSI2::DeltaSwitching(const mAlpha, const mBeta, const amH, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT, const dCrit=1e-6, const iMaxIt=1000, const iInfo=0, const iLineSearch=2)

Parameters

mAlpha	matrix [cz0][r] initial alpha
mBeta	matrix [cz1][r] initial beta
iS	integer 0 <= S < cz0-r
amH	array[r] with H matrices or {}
mQtZ0,...,mQtZ0sq	transformed data matrices
cT	int, sample size
dCrit	double, convergence tolerance (default 1e-6)
iMaxIt	int, maximum number of iterations (default 1000)
iInfo	int, >0: print output for every iInfo-th iteration (default 0)

Returns

array with 10 elements: { alpha, alpha_ort, tau, rho, psi, kappa, w, omega, log-determinant, retval (MAX_CONV, MAX_WEAK_CONV, MAX_NOCONV, MAX_FUNC_FAIL) }

CATSI2::DoDeltaSwitching(const mAlpha, const mBeta, const amH=<>, const viNorm=<>)

Parameters

mAlpha	matrix [cz0][r]
mBeta	matrix [cz1][r]
iS	integer 0 <= s <= cz0-r (s==s1)
amH	array[r] of H matrices
viNorm	normalization vector, or <>: apply automatic normalization

Returns

the log-likelihood

CATSI2::DoDeltaSwitchingLogLik(const mAlpha, const mBeta, const iS1)

As DoDeltaSwitching but without setting estimates, only returning the log-likelihood, alpha and beta. Does not set any member variables of the object.

virtual CATSI2::DoEstimation( vPar)

Parameters

Estimate

vPar	vector of initial parameters (ignored, all zero here)

Returns

vec(alpha)|vec(beta)|vec(Omega)

CATSI2::DoI2RankTable(const dAlpha=0.05, const iDoMLE=1)

Parameters

(default is 0.05 for 95% quantiles)

dAlpha	Significance level to be used in table when p-values are not available
iDoMLE	0: two-step, 1: ML using delta switching (default), 2: ML using triangular switching, 3: ML using tau switching

Returns

array with four matrices: tests, pvalues, quantiles, iteration_counts

CATSI2::DoI2RankTable_2step()

Returns: matrix with the table

CATSI2::DoI2Residuals(const mPi, const mGamma)

Parameters

mPi	matrix Pi[m_cY][cz1]
mGamma	matrix Gamma[m_cY][cz2]

Returns

residual matrix [m_cT][m_cY]

Comments

Runs the regression, in matrix form: Z0 - Z1 Pi_hat' + Z1 Gamma_hat' = Z2 Psi' + epsilon.

virtual CATSI2::DoMoments()

Comments: Creates m_mZ0I2_Z3, m_mZ1I2_Z3, m_mZ2I2_Z3, m_mQtZ0sqI2_Z3, m_mQtZ0I2_Z3, m_mQtZ2I2_Z3, m_mQtZ1I2_Z3.

CATSI2::DoRestrictAlphaBeta( amA, amH, const vNormBetaOvr, bSetZeroV13, const dCrit, const iMaxIt, const iInfo, const dBFGSCrit, const iMaxItBFGS, const iSwitchOptions=0, const cStartSearch=20)

Parameters

(same as {G,G,..,G}; <> for unrestricted), or array[r] of strings with restriction expressions

amG	array[r(+s1+s2)] of G_i G matrices ({} for unrestricted alpha), or matrix G with common restriction
amH	array[r(+s1+s2)] of H matrices, one for each restriction ({} for no restrictions), or array[r] of strings with restriction expressions
bSetZeroV13	TRUE: set V13 (and hence d) to zero, FALSE (default): leave unrestricted
vNormBetaOvr	<> (default) or vector[r] with normalization indices (-1 for free to choose) used to override normalization of the restrictions

Returns

success value (0 failure, 1 success)

CATSI2::DoSetI2Rank(const iS, const iTriangular=-1)

Parameters

iS	integer, I(2) rank s (== s1)
iTriangular	<=1: use delta switching (the default), 2: use Triangular switching, 3: Tau switching

Comments

Use I2Rank and I1Rank to specify the rank prior to estimation. Called by DoEstimation to complete the estimation for the specified rank. DoSetI2Rank can be called after estimation to change the rank.

CATSI2::DoTauSwitching(const mAlpha, const mBeta, const mH=<>)

Parameters

mAlpha	matrix [cz0][r]
mBeta	matrix [cz1][r]
mH	matrix, restrictions matrix H with common restrictions or empty matrix for unrestricted

Returns

the log-likelihood

CATSI2::DoTauSwitchingLogLik(const mAlpha, const mBeta, const iS1)

As DoTauSwitching but without setting estimates, only returning the log-likelihood, alpha and beta. Does not set any member variables of the object.

CATSI2::DoTriSwitching(const mAlpha=<>, const mBeta=<>)

Parameters

mAlpha	matrix [cz0][r]
mBeta	matrix [cz1][r]

Returns

the log-likelihood

CATSI2::DoTriSwitchingLogLik(const mAlpha, const mBeta, const iS1)

Parameters

DoTriSwitching

mAlpha	matrix [cz0][r]
mBeta	matrix [cz1][r]
iS1	I(2) rank

Returns

the log-likelihood, alpha, beta, gamma

CATSI2::DoUpdateEstimates_I2(const mAlpha, const mTau, const mRho, const mPsi, const mKappa, const mWterm, const mOmega, const dLogDet, viNorm=<>)

Parameters

mAlpha,	.., mWterm matrices from estimation
dLogDet	double, new log\|Omega\|
viNorm	normalization vector, or <>: apply automatic normalization

Returns

log-likelihood

CATSI2::DoUpdateEstimates_I2_AB(const mA, const mB, const mV, const iR, const iS1, const dLogDet, const vNorm=<>)

Parameters

mA,mB,mV	matrices from estimation
iR	rank r
iS1	rank s
dLogDet	double, new log\|Omega\|
viNorm	normalization vector or <>: normalize automatically

Returns

log-likelihood

virtual CATSI2::EstimateAlphaBeta( amG, amH, const vNormBetaOvr=<>, bSetZeroV13=FALSE)

Parameters

amG	array[r] of G_i matrices
amH	array[r] of H_i matrices
vNormBetaOvr	<> (default) or vector[r] with normalization indices (-1 for free to choose) used to override normalization of the restrictions
bSetZeroV13	TRUE: set V13 (and hence d) to zero, FALSE (default): leave unrestricted

Returns

array {alpha, beta, retval, loglik, isidentified, no of restrictions, se_alpha, se_beta}

note

does not change any member variables, so can be called in parallel

CATSI2::EstimateTau( mH)

Parameters

mH	matrix of restrictions H (or R if bIsR is TRUE)

Comments

Doesn't change the current model

CATSI2::EvalBetaSe()

Returns: matrix of standard errors of current beta (<> if not identified)

CATSI2::GenerateAsY_I2(const mEps, const mAlpha, const mBeta, const mGamma, const mPsis, const mPhi)

Parameters

if the I(2) model has s_1=p-r: Gamma_{I(2)} = Gamma_{I(1)} : -mu)

mEps	matrix[T][p] error term for the random sample (must match the estimated sample size, which can be obtained by Modelbase::GetcT())
mAlpha[p][r]	matrix or <>
mBeta[p1][r]	matrix or <>
mGamma[p][p1]	matrix, note sign: DDy_t=alpha*beta'y_{t-1} - Gamma Dy_{t-1} + ... (note the optional extra column,
mPsis[p][p	* (k - 2)] matrix with coefficients on lagged double differences or <>
mPhi[p][d]	matrix with coefficients on remaining short-run variables

Returns

matrix[T][p] random sample

static CATSI2::GenerateECM_I2(const cT, const iDet, const mAlpha, const mBeta, const mGamma, const mPsis=<>, const mPhi=<>, const mOmegaSqrt=<>, const mYinit=<>, const mYmean=<>)

virtual CATSI2::GenerateY( mEps=<>)

Parameters

mEps	matrix[T][p] error term for the random sample (must match the estimated sample size, which can be obtained by Modelbase::GetcT())

Returns

matrix[T][p] random sample

CATSI2::GetAWithSe()

Returns: {A, SE[A]}

CATSI2::GetAlphaOrt1WithSe()

Returns: array with 2 matrices: { alpha_ort1, SE[alpha_ort1] }

CATSI2::GetAlphaOrt2WithSe(const viNorm=<>)

Parameters

viNorm

[n] specified override of normalization (optional argument, default is <>: do not override)

Returns

array with 2 matrices: { alpha_ort2, SE[alpha_ort2] }

CATSI2::GetBWithSe()

Returns: {B, SE[B]}

CATSI2::GetBetaSe()

Returns: matrix of standard errors of current beta (<> if not identified)

virtual CATSI2::GetCompanion(const amRoots=0)

Parameters

out: if array, amRoots will hold an [n][4] matrix : real part, imaginary part, absolute value and argument.

amRoots

in: array or 0 (default if no arguments)

Returns

companion matrix

CATSI2::GetDeeEe()

Returns: array with two matrices: d[p1][r], e[p1][p],

static CATSI2::GetDeeEeFrom(const mAlpha, const mTau, const mPsi, const mKappa, const mWterm)

CATSI2::GetDeeEeSe()

Returns: array[2] {matrix of standard errors of current dee, matrix of standard errors of current ee, matrix of standard errors of current ee (not taking restrictions on A1 into account)}

CATSI2::GetDeltaZeta()

Returns: array with two matrices: delta[r][s2*], zeta[p][r+s],
Comments: Note that zeta is not unique when beta is identified: it depends on the chosen normalization of B_1. Same for delta depending on B_2.

static CATSI2::GetDeltaZetaFrom(const mAlpha, const mTau, const mPsi, const mKappa, const mWterm)

CATSI2::GetDeltaZetaSe()

Returns: array[2] {matrix of standard errors of current delta, matrix of standard errors of current zeta}

virtual CATSI2::GetDimensions(const sType="")

Parameters

sType

string: type of dimension ("p","p1","r","s","s2","R") or default "" for array with all

Returns

array with {p, p_1, r, s, s2, R (no of restrictions imposed)} or single dimension

Comments

Also see GetLags

CATSI2::GetGamma()

Returns: matrix Gamma[p][p1]

static CATSI2::GetGammaFrom_I2(const mAlpha, const mTau, const mPsi, const mKappa, const mWterm)

Parameters

mAlpha	matrix
mTau	matrix
mPsi	matrix
mKappa	matrix
mWterm	matrix

Returns

Gamma[p0][p1]

CATSI2::GetPsiKappa()

Returns: array with two matrices: psi[p1][r], kappa[r+s][p-r]

static CATSI2::GetPsiKappaFrom(const mAlpha, const mTau, const mGamma, const mOmega)

CATSI2::GetPsiKappaSe()

Returns: array[2] {matrix of standard errors of current psi, matrix of standard errors of current kappa}

virtual CATSI2::GetRegsLongRun(const fCorrected=TRUE)

Parameters

fCorrected,

TRUE: corrected for short-run (default), else uncorrected.

Returns

(T x cZ1) matrix

virtual CATSI2::GetRegsMediumRun(const fCorrected=TRUE)

Parameters

fCorrected,

TRUE: corrected for short-run (default), else uncorrected.

Returns

(T x cZ1) matrix

CATSI2::GetTheta()

Returns: matrix Theta[p][p]

static CATSI2::GetTriFrom(const mAlpha0, const mBeta0, const mGamma0, const iS1, const bFixGamma_c=TRUE)

CATSI2::GetVWithSe()

Returns: {V, SE[V]}

CATSI2::GetWWithSe()

Returns: {W, <>}

virtual CATSI2::GetZ0()

Returns: (T x p) matrix with Z_0 = DDY

virtual CATSI2::GetZ0Names()

Returns: array[2] with names

virtual CATSI2::GetZ1()

Returns: (T x cZ1) matrix with Z_1 = DY_1 ~ [1] ~ DX_1

virtual CATSI2::GetZ2()

Returns: (T x cZ2) matrix with Z_2 = Y_1 ~ [t] ~ X_1

virtual CATSI2::GetZ3()

Returns: (T x cZ3) matrix with Z_3 = [DDY_1 .. DDY_{k-1}] ~ [DDX1 .. DDX1_{m-1}] ~ U

virtual CATSI2::GetcDfLoss()

Returns: number of estimated parameters

virtual CATSI2::GraphicAnalysis(const iPlot=0, const bActualFitted=FALSE, const bActualFittedCross=FALSE, const iResiduals=2, const iCointegration=2, const iAcf=1, const iAcfLag=-1, const iDensity=0, const iRootsPlot=0, const iLevelsPlot=0, const iEqn=-1)

Parameters

iPlot	int: index of first plot
bActualFitted	TRUE: include plots of actual and fitted
bActualFittedCross	TRUE: include crossplots of actual and fitted
iResiduals	>0: include plots of 1: residuals, 2: scaled residuals, 3: standardized to IN[0,1] residuals (as in normality test)
iCointegration	>0: cointegrating vectors, 1: one plot, 2: individual plots
iAcf	>0: include ACF plots of 1: residuals, 2: squared residuals
iAcfLag:	lag length for ACF plots (<=0 then using default)
iDensity	>0: include histogram and QQ plots of residuals
iRootsPlot	>0: include roots of companion matrix

Returns

int: number of plots created.

virtual CATSI2::I1Rank(const iR)

Parameters

iR	integer, 0 <= r <= p

Comments

This changes the model class to I1.

CATSI2::I2Rank(const iS)

Parameters

iR	integer, 0 <= r + s1 <= p

Comments

This changes the model class to I2.

CATSI2::I2RankMLE(const bPrint)

CATSI2::I2RankTable(const iDoMLE=1)

Parameters

iDoMLE

0: two-step, 1: ML using delta switching (default), 2: ML using triangular switching

Returns

array with three matrices: tests, pvalues, quantiles

CATSI2::I2RankTablePvals(const mTests, const dAlpha)

Parameters

mTests	matrix[cZ0][cZ0+1] with test statistics
dAlpha

Returns

matrix with p-values and matrix with 95% fractiles

CATSI2::I2SwitchStart(const fnSwitching, const mAlpha, const mBeta, const iS1, const mAlpha2=<>, const mBeta2=<>)

Parameters

fnSwitching	function: TauSwitching or DeltaSwitching derived
mAlpha	matrix [cz0][r]
mBeta	matrix [cz1][r]

Returns

{alpha, beta, gamma, log-determinant}

CATSI2::Init()

Reset the transitory properties.

virtual CATSI2::InitData()

Checks the model formulation and creates the data matrices.

virtual CATSI2::IsI2()

Returns: TRUE for I(2), FALSE otherwise

static CATSI2::JacI2Model_AB(const vP, const vSelV, const vInitV, const cZ0, const cZ1, const iR, const iS, const amG, const amH, const vDroppedPhi=<>, const bNumerical=FALSE)

Used in RestrictAlphaBeta

static CATSI2::LogdetI2Model_AB(const vP, const mA, const mB, const mV, const mW, const vSelV, const iR, const iS, const adFunc, const avScore, const amHessian, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT, const amG, const amH, const vDroppedPhi, const mJac, const bAsympSE)

virtual CATSI2::MARep()

Prints the matrices of the moving average representation.

static CATSI2::MapGammaToPar1(const mAlpha, const mBeta, const mGamma, const mXi, const mEta, const iS1, const mJ, const bUseKappa=TRUE)

static CATSI2::MapMatToPar_AB(const mA, const mB, const mV, const vSelV, const iR, const iS, const amG, const amH)

Used in RestrictAlphaBeta

static CATSI2::MapParToMat_AB( vP, const vSelV, const vInitV, const cZ0, const cZ1, const iR, const iS, const amG, const amH, const vDroppedPhi=<>)

static CATSI2::MapScore_AB(const mDloglikDpi, const mA, const mB, const amG, const amH, const vDroppedPhi=<>)

virtual CATSI2::Output()

Comments: Switch printing off by calling SetPrint(FALSE).

CATSI2::PrintI2Rank(const sExtra="")

Prints the estimates for the current rank.

CATSI2::PrintI2RankTable(const mTests, const mPvals, const mFrac95, const sType="", const bWarn=TRUE)

Parameters

mTests	matrix with test statistics
mPvals	matrix with p-values
mFrac95	matrix with 95% fractiles, used when p-values are missing

virtual CATSI2::Restrict(const amRes, const vNormBetaOvr=<>)

Parameters

may also contain [method=beta], [method=delta], [method=tau]

amRes	array with restrictions, sections are [alpha],[beta] (without sections, [beta] is assumed)
vNormBetaOvr	<> (default) or vector[r] with normalization indices (-1 for free to choose) used to override normalization of the restrictions

Returns

success value (0 failure, 1 success)

virtual CATSI2::RestrictAlphaBeta( amA, amH, const vNormBetaOvr=<>, bSetZeroV13=FALSE)

Parameters

(same as {G,G,..,G}; <> for unrestricted), or array[r] of strings with restriction expressions

amG	array[r(+s1+s2)] of G_i G matrices ({} for unrestricted alpha), or matrix G with common restriction
amH	array[r(+s1+s2)] of H matrices, one for each restriction ({} for no restrictions), or array[r] of strings with restriction expressions
bSetZeroV13	TRUE: set V13 (and hence d) to zero, FALSE (default): leave unrestricted
vNormBetaOvr	<> (default) or vector[r] with normalization indices (-1 for free to choose) used to override normalization of the restrictions

Returns

success value (0 failure, 1 success)

CATSI2::RestrictBeta( mG, amH)

Parameters

(same as {G,G,..,G}; <> for unrestricted), or array[r] of strings with restriction expressions

amG	matrix G with common restriction (I1 version only), otherwise ignored
amH	array[r] of H matrices, one for each restriction ({} for no restrictions), or array[r] of strings with restriction expressions

CATSI2::RestrictTau( mH)

Parameters

mH	matrix of restrictions H

static CATSI2::RotateRho(const iR, const iS, mTau, mKappa, mRho)

Returns: array { tau, kappa, rho }

CATSI2::ShortRun()

Prints the short-run coefficients of the I(2) model.

CATSI2::ShortRunEval_I2(const bDoStdDev=FALSE, const mAlpha=0, const mBeta=0, const mGamma=0, const mTau=0, const mPsi=0, const mOmega=0)

Parameters

bDoStdDev	TRUE: compute standard errors (default is FALSE)
mAlpha[p][r]	matrix (or int 0 for current alpha, the default)
mBeta[p][r]	matrix (or int 0 for current beta, the default)
mGamma	matrix[p][p1] (or int 0 for current Gamma, the default)
mTau	matrix (or int 0 for current Tau, the default)
mPsi	matrix (or int 0 for current Psi, the default)
mOmega	matrix[p][p] (or int 0 for current Omega, the default)

Returns

array of four matrices: { Psi_i[p][p * (k-2)] for lagged differences of y_t, Phi[p][] remainder, SE[Gamma_i], SE[Phi] } where k is the maximum lag length, and the SE matrices will be empty if bDoStdDev is FALSE (the default).

virtual CATSI2::ShrinkSample(const cDrop, const cTmin=1, const bFixShortRun=FALSE)

Parameters

cDrop	int: number of observations to shrink the sample by
cTmin	int: don't shrink past this sample size
bFixShortRun	boolean: TRUE: keep short run fixed (R1 form), FALSE (default): re-estimate short run (X form)

Returns

TRUE if shrinkage was successful.

CATSI2::SimulateRankTest(const cTsim=400, const cM=1000)

Parameters

cTsim	Length of random walks (T, optional argument, default is 400)
cM	Number of replications (M, optional argument, default is 1000)

Returns

array[3] {table with tests and p-values, table of Gamma quantiles, table of simulated quantiles }

static CATSI2::SolveConditional(const mAlpha, const mTau, const mAlphaOrt, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)

Parameters

mAlpha[p][r]	initial values for alpha
mBeta[p1][r]	initial values for beta
mAlphaOrt[p][p-r]	mAlpha_ort
iS1	I(2) rank
mQtZ0	... mQtZ0sq transformed data matrices
cT	int, sample size

Returns

array with six elements: {gamma, omega, tau (rotated to beta~B1), kappa, psi, w}

static CATSI2::SolveDeltaForAlpha(const mAlpha, const mTau, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)

Parameters

mAlpha[p][r]	alpha
mTau[p1][r+s]	tau
mQtZ0	... mQtZ0sq transformed data matrices
cT	int, sample size

Returns

array with five elements: { alpha, gamma, omega, delta, zeta }.

static CATSI2::SolveDeltaForGammaStart(const mAlpha, const mBeta, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)

Parameters

mAlpha[p][r]	initial values for alpha
mBeta[p1][r]	initial values for beta
iS1	I(2) rank
mQtZ0	matrix[cz0][cz0] Q'Z0 corrected for z3
mQtZ1	matrix[cz1][cz1] Q'Z1 corrected for z3
mQtZ2	matrix[cz1][cz1] Q'Z2 corrected for z3
mQtZ0sq	matrix[cz0][cz0] rest of Z0'Z0 corrected for z3
cT	int, sample size

Returns

array with four elements: alpha, beta, gamma, omega

static CATSI2::SolveDeltaForGammaStart2(const iR, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)

static CATSI2::SolveDeltaForTau(const mAlpha, const iR, const iS1, const mDelta, const mZeta, const mOmega, const mH1, const mH2, const mQtZ0, const mQtZ1, const mQtZ2)

Parameters

mAlpha[p][r]	alpha
mZeta[p][r+s]	zeta
mOmega[p][p]	Omega
mH1	restrictions on vec(beta) or <>
mH2	restrictions on vec(beta_1) or <> (either have both H1 and H2 empty or neither)
mQtZ0	... mQtZ0sq transformed data matrices
cT	int, sample size

Returns

{tau[p1][r+s], vectorized parameters in tau, delta}.

static CATSI2::SolveMarginal(const mAlpha, const mBeta, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)

Parameters

mAlpha[p][r]	initial values for alpha
mBeta[p1][r]	initial values for beta
iS1	I(2) rank
mQtZ0	... mQtZ0sq transformed data matrices
cT	int, sample size

Returns

array with six elements: {xi, eta, tau, kappa, alpha_ort, beta_ort, eval}

static CATSI2::SolveTauForAlpha(const iR, const mTau, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT)

Parameters

iR	rank r
mAlpha[p][r]	current alpha (will choose rotation to minimize distance of new alpha to old one)
mTau[p1][r+s]	initial values for tau
mQtZ0	... mQtZ0sq adjusted data matrices
cT	int, sample size

Returns

array with two elements: { alpha, alpha_ort }.

static CATSI2::SolveTauForTau(const mAlpha, const mAlphaOrt, const iR, const iS1, const mRho, const mW, const mKappa, const mOmega, const mH, const mQtZ0, const mQtZ1, const mQtZ2)

Parameters

mAlpha[p][r]	alpha
mAlphaOrt[p][p-r]	alpha orthogonal
mRho[r+s][rs]	rho
mPsi[p1][r]	psi
mW[p][p-r]	w
mKappa[r+s][p-r]	kappa
mOmega[p][p]	Omega
mQtZ0	... mQtZ0sq adjusted data matrices
cT	int, sample size

Returns

{tau[p1][r+s], vectorized parameters in tau}.

static CATSI2::TauSwitching(const mAlpha, const mBeta, const mH, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT, const dCrit=1e-6, const iMaxIt=1000, const iInfo=0, const iLineSearch=2)

Parameters

mAlpha	matrix [cz0][r]
mBeta	matrix [cz1][r]
iS	integer 0 <= S < cz0-r
mH	matrix[r+s][q] with H matrix for common restriction on tau
mQtZ0,...,mQtZ0sq	transformed data matrices
cT	int, sample size
dCrit	double, convergence tolerance (default 1e-6)
iMaxIt	int, maximum number of iterations (default 1000)
iInfo	int, >0: print output for every iInfo-th iteration (default 0)

Returns

array with 10 elements: { alpha, alpha_ort, tau, rho, psi, kappa, w, omega, log-determinant, retval (MAX_CONV, MAX_WEAK_CONV, MAX_NOCONV, MAX_FUNC_FAIL) }

virtual CATSI2::TestEach(const bAlpha=TRUE, const bBeta=TRUE)

Parameters

bAlpha	TRUE: Weak exogeneity and unit vectors (default)
bBeta	TRUE: Variable exclusions and stationarity (default)

Returns

array with seven elements: matrix of tests for each (tests in first column, pvalues in second).

CATSI2::TestEachBetaRestriction( fnGetG, fnGetH, const vVpg, const asZ, const cRes, const sType)

Parameters

fnGetG	function returning common restriction on alpha (one for each entry in asZ)
fnGetH	function returning restrictions on beta (one for each entry in asZ)
vVpg	vector with multiplicity of restrictions on beta
asZ	array of test descriptions
cRes	not used
sType	type of restrictions (for output)

Returns

matrix of tests results (first column test, 2nd pvalues).

static CATSI2::TriGetVFromAB(const mA, const mB, const iR, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const bSetZeroV13=FALSE)

Parameters

mA[p][p]
mB[p1][p1]
mV[p][p1]
iR	rank of Pi
iS1	rank of alpha_ort'Gamma*beta_y,ort
mQtZ0,mQtZ1,mQtZ2	: data matrices
bSetZeroV13:	set V13 to zero (default is FALSE)

Returns

static CATSI2::TriNormalize(const mA, const mB, const mV, const iR, const iS1, const bA1IsFree=FALSE, const bA2IsFree=FALSE, const bB1IsFree=FALSE, const bB2IsFree=FALSE, const vNormBOverride=<>)

static CATSI2::TriOrthogonalize12(const mA, const mB, const mV, const iR, const iS1, const bA1IsFree, const bA2IsFree, const bB1IsFree, const bB2IsFree)

static CATSI2::TriRegularizeV(const mA, const mB, const mV, const iR, const iS1, const bA1B1IsFree, const bB2IsFree)

static CATSI2::TriRemoveW(const mA, const mV, const mW, const iR)

CATSI2::TriStart(const mAlpha, const mBeta, const cAlpha, const cBeta, const iR, const iS1, const amG=<>, const amH=<>, const mAlphaI2=<>, const mBetaI2=<>)

Parameters

mAlpha[p][r]	initial values for alpha
mBeta[p1][r]	initial values for beta
cAlpha	int, no of free parameters in alpha (unrestricted: p*r)
cBeta	int, no of free parameters in beta (unrestricted: p1*r)
iR	I(1) rank
iS1	I(2) rank
amG	array of G_i G matrices ({} for unrestricted A)
amH	array of H matrices, one for each restriction ({} for no restrictions)
mAlphaI2[p][r]	initial unrestricted I(2) values for alpha (or <>)
mBetaI2[p1][r]	initial unrestricted I(2) values for beta (or <>)

Returns

array with seven elements: {A,B,V,W,Omega,-logdet,max code}.

static CATSI2::TriSwitching(const mA, const mB, const mV, const bFixW11, const bSetZeroV13, const amG, const amH, const iR, const iS1, const mQtZ0, const mQtZ1, const mQtZ2, const mQtZ0sq, const cT, const dCrit, const iMaxIt, const iInfo, const bLineSearch=TRUE, const dAddLogdet=0)

Parameters

mA	matrix initial A[p][p]
mB	matrix initial B[p1][p1]
mV	matrix initial V[p][p1]
bFixW:	TRUE: W_{11}=I_r alwyas, FALSE: W_{11} is free if possible (but will be rotated back to I_r at the end)
amG	array[r] of G_i G matrices ({} for unrestricted A)
amH	array[r] of H matrices, one for each restriction ({} for no restrictions)
mQtZ0	matrix[cz0][cz0] Q'Z0 corrected for z3
mQtZ1	matrix[cz1][cz1] Q'Z1 corrected for z3
mQtZ2	matrix[cz1][cz1] Q'Z2 corrected for z3
mQtZ0sq	matrix[cz0][cz0] rest of Z0'Z0 corrected for z3
cT	int, sample size
dCrit	double, convergence tolerance
iMaxIt	int, maximum number of iterations
iInfo	int, >0: print output for every iInfo-th iteration
bLineSearch	int, FALSE: no search, TRUE: 2-stage search (default)

Returns

{A,B,V,W,Omega,-logdet,max code}