Test for Error Autocorrelation in VAR Models.

Description

Performs LM tests for error AC in VAR models. The code is based on Paul Catani's original R code used in the paper Wild Bootstrap Tests for Autocorrelation in Vector Autoregressive Models (Ahlgren and Catani, 2016).

Usage

ACtest(fit, h = 4, HCtype = c("LM", "HC0", "HC1", "HC2", "HC3"), univariate = FALSE)
## S3 method for class 'ACtest'
print(x, ...)

Arguments

Details

To run the wild bootstrap version of the test, please use the output from this function with the function wildBoot.

Mathematical details

The tests for error AC are based on the least squares (LS) residuals from the K-dimensional vector autoregressive (VAR) model with p lags (abstracting from deterministic terms):

\mathbf{y}_{t}=\mathbf{\Pi }_{1}\mathbf{y}_{t-1}+\cdots +\mathbf{\Pi }_{p} \mathbf{y}_{t-p}+\mathbf{u}_{t},\quad \text{E}(\mathbf{u}_{t})=\mathbf{0} ,\quad \text{E}(\mathbf{u}_{t}\mathbf{u}_{t}^{\prime })=\mathbf{\Omega},\ \ \ \ t=1,\ldots ,N.

The LS residuals are

\widehat{\mathbf{u}}_{t}=\mathbf{y}_{t}-\widehat{\mathbf{\Pi }}_{1}\mathbf{y} _{t-1}-\cdots -\widehat{\mathbf{\Pi }}_{p}\mathbf{y}_{t-p},

where \widehat{\mathbf{\Pi }}_{1},\ldots ,\widehat{\mathbf{\Pi }}_{p} are the LS estimates of the K\times K parameter matrices \mathbf{\Pi } _{1},\ldots ,\mathbf{\Pi }_{p}.

The LM statistic is computed from the auxilary model

\widehat{\mathbf{u}}_{t}=\mathbf{\Pi }_{1}\mathbf{y} _{t-1}+\cdots +\mathbf{\Pi }_{p}\mathbf{y}_{t-p} + \mathbf{D}_{1}\widehat{\mathbf{u}}_{t-1}+\cdots +% \mathbf{D}_{h}\widehat{\mathbf{u}}_{t-h}+\mathbf{e}_{t}

=(\mathbf{Z}_{t-1}^{\prime }\otimes \mathbf{I}_{K})\boldsymbol{\phi }+(\widehat{% \mathbf{U}}_{t-1}^{\prime }\otimes \mathbf{I}_{K})\boldsymbol{\psi }+\mathbf{e}% _{t}, \

where \mathbf{Z}_{t-1}=(\mathbf{y}_{t-1}^{\prime },\ldots ,\mathbf{y}_{t-p}^{\prime })^{\prime }, \boldsymbol{\phi } =\text{vec}(\mathbf{Pi}_{1},\ldots ,\mathbf{Pi}_{p})^{\prime }, \widehat{\mathbf{U}}% _{t-1}=(\widehat{\mathbf{u}}_{t-1}^{\prime },\ldots ,\widehat{% \mathbf{u}}_{t-h}^{\prime })^{\prime } and \boldsymbol{\psi } =\text{vec}(\mathbf{D}_{1},\ldots ,\mathbf{D}_{h})^{\prime }. The symbol \otimes denotes the Kronecker product and the symbol vec denotes the column vectorisation operator. The first h values of the residuals \widehat{\mathbf{u}}_{t} are set to zero in the auxiliary model, so that the series length is equal to the series length in the original VAR model.

The LM statistic for error AC of order h is given by

Q_{\text{LM}}(h) = N \widehat{\boldsymbol{\psi}}^{\prime} \bigl(\widehat{\boldsymbol{\Sigma}}^{\psi\psi}\bigr)^{-1} \widehat{\boldsymbol{\psi}}

where \widehat{\boldsymbol{\psi }} is the LS estimate of \boldsymbol{\psi } and \widehat{\boldsymbol{\Sigma }}^{\psi \psi } is the block of

\left( N^{-1}\sum_{t=1}^{N}\left[ \begin{array}{c} \mathbf{Z}_{t-1}\otimes \mathbf{I}_{K} \\ \widehat{\mathbf{U}}_{t-1}\otimes \mathbf{I}_{K}% \end{array}% \right] \widehat{\boldsymbol{\Sigma }}_{\mathbf{u}}^{-1}\left[ \begin{array}{cc} \mathbf{Z}_{t-1}^{\prime }\otimes \mathbf{I}_{K} & \widehat{\mathbf{U}}% _{t-1}^{\prime }\otimes \mathbf{I}_{K}% \end{array}% \right] \right) ^{-1}

corresponding to \boldsymbol{\psi }. Here \widehat{\boldsymbol{\Sigma }}_{\mathbf{u}% }=N^{-1}\sum_{t=1}^{N}\widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}% _{t}^{\prime } is the estimator of the error covariance matrix from the VAR model.

The multivariate heteroskedasticity-consistent covariance matrix estimator (HCCME) for the auxilary model is given by (Hafner and Herwartz 2009)

\mathbf{V}_{N}^{-1}\mathbf{W}_{N}\mathbf{V}_{N}^{-1} = (\boldsymbol{\Gamma}_{N}\otimes \mathbf{I}_{K})^{-1} \mathbf{W}_{N} (\boldsymbol{\Gamma}_{N}\otimes \mathbf{I}_{K})^{-1}

where

\mathbf{V}_{N} =\boldsymbol{\Gamma }_{N}\otimes \mathbf{I}_{K},

\boldsymbol{\Gamma }_{N} =\frac{1}{N}\sum_{t=1}^{N}\left( \begin{array}{c} \widehat{\mathbf{U}}_{t-1} \\ \mathbf{Z}_{t-1}% \end{array}% \right) \left( \begin{array}{cc} \widehat{\mathbf{U}}_{t-1}^{\prime } & \mathbf{Z}_{t-1}^{\prime }% \end{array}% \right) ,

\mathbf{W}_{N} =\frac{1}{N}\sum_{t=1}^{N}\left( \begin{array}{c} \widehat{\mathbf{U}}_{t-1} \\ \mathbf{Z}_{t-1}% \end{array}% \right) \left( \begin{array}{cc} \widehat{\mathbf{U}}_{t-1}^{\prime } & \mathbf{Z}_{t-1}^{\prime }% \end{array}% \right) \otimes (\widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}_{t}^{\prime }).

The HCCME-based LM statistics for error AC are obtained from the expression for Q_{\text{LM}}(h) by replacing \widehat{\boldsymbol{\Sigma }}^{\psi \psi } by the block of % \mathbf{V}_{N}^{-1}\mathbf{W}_{N}\mathbf{V}_{N}^{-1}=(\Gamma _{N}\otimes \mathbf{I}_{K})^{-1}\mathbf{W}_{N}(\Gamma _{N}\otimes \mathbf{I}_{K})^{-1} corresponding to \boldsymbol{\psi } and with % \widehat{\mathbf{u}}_{t} defined by HC_{0}, HC_{1}, HC_{2} and HC_{3} , respectively.

HC_{0} uses \widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}_{t}^{\prime }. HC_{1} multiplies the elements of \widehat{\mathbf{u}}_{t}\widehat{\mathbf{u}}_{t}^{\prime } by N/(N-Kp). HC_{2} replaces \widehat{% \mathbf{u}}_{t} by \widehat{\mathbf{u}}_{t}/(1-h_{t})^{1/2}, where h_{t}=\mathbf{Z}_{t}(\mathbf{Z}^{\prime }\mathbf{Z})^{-1}\mathbf{Z}_{t}^{\prime } is the tth diagonal element of \mathbf{Z}(\mathbf{Z}^{\prime }\mathbf{Z})^{-1}\mathbf{Z}^{\prime }, and \mathbf{% Z}=(\mathbf{Z}_{0},\ldots ,\mathbf{Z}_{N-1}). HC_{3} replaces \widehat{\mathbf{u}}_{t} by \widehat{\mathbf{u}}_{t}/(1-h_{t}).

See MacKinnon and White (1985) for details.

The recursive-design wild bootstrap (WB) tests for error AC are computed using Algorithm 1 in Ahlgren and Catani (2016). The Fixed-design WB tests for error AC are computed using Algorithm 2 in Ahlgren and Catani (2016).

Value

a list of class "ACtest".

References

Ahlgren, N. & Catani, P. (2016). Wild bootstrap tests for autocorrelation in vector autoregressive models. Stat Papers, <doi:10.1007/s00362-016-0744-0>.

Hafner, C. M. and Herwartz, H., (2009). Testing for Linear Vector Autoregressive Dynamics under Multivariate Generalized Autoregressive Heteroskedasticity. Stat Neerl, 63, 294–323

MacKinnon, J. G. and White, H. (1985). Some Heteroskedasticity Consistent Covariance Matrix Estimators with Improved Finite Sample Properties. J Econom, 29, 305–325

See Also

VARfit to estimate a VAR(p), and wildBoot to run the Wild Bootstrap versions of the tests.

Examples

fit <- VARfit(y = VodafoneCDS, p = 3, const = TRUE, trend = FALSE)
test <- ACtest(fit = fit, h = 1, HCtype = c("LM", "HC0", "HC1", "HC2", "HC3"))
test

Multiple Time Series Data Set

Description

The data matrix VodafoneCDS is used for the examples. It consists of 804 daily observations, from 1 January 2009 to 31 January 2012, of Vodafone's Credit Default Swap prices (the 'CDS' column) and the credit spread on its bond over the risk-free rate (the 'SWSP' column). For more information, see Ahlgren and Catani (2016), who used the same data set.

References

Ahlgren, N. & Catani, P. (2016). Wild bootstrap tests for autocorrelation in vector autoregressive models. Stat Papers, <doi:10.1007/s00362-016-0744-0>.

VAR(p) (Vector Autoregression) Model Fitting.

Description

Estimates a VAR(p) model from data. The function's purpose is to be used before the ACtest or archBootTest functions, and doesn't have the same full functionality as for example the function vars::VAR in the vars package.

Usage

VARfit(y, p = 1, const = TRUE, trend = FALSE, exogen = NULL, univariate = FALSE)

Arguments

Value

a list of class "VARfit" with the following slots:

See Also

ACtest to test for error autocorrelations, and archBootTest to test for ARCH errors.

Examples

fit <- VARfit(y = VodafoneCDS, p = 3, const = TRUE, trend = FALSE)
fit

Methods for Objects of Class VARfit

Description

residuals.VARfit() returns the residuals, and coef.VARfit() returns the coefficients of a fitted VAR model of class VARfit. print.VARfit() prints the estimated model parameters.

Usage

## S3 method for class 'VARfit'
coef(object, ...)
## S3 method for class 'VARfit'
print(x, ...)
## S3 method for class 'VARfit'
residuals(object, ...)

Arguments

Value

coef.VARfit: A numeric matrix containing the estimated coefficients of the VAR model, including intercepts and lagged autoregressive terms.

print.VARfit: No return value. Called for its side effects (printing model information to the console).

residuals.VARfit: A numeric matrix of residuals from the fitted VAR model. Each column corresponds to one of the time series in the system.

See Also

VARfit

Simulates vector autoregressive (VAR) series

Description

This function simulates VAR(p) series. For quick use, the function can use the estimated model returned by VARfit or vars::VAR as the DGP (Data Generating Process), if passed to the fittedModel argument.

Usage

VARsim(N = 200, K = 2, p = 1, const = TRUE, trend = FALSE, exogen = NULL, 
       coef = NULL, dist = "normal", Ystart = NULL, errors = NULL, fittedModel = NULL)

Arguments

Value

an N x K matrix of the simulated VAR(p).

Examples

# Simulates from a fitted DGP:
fit <- VARfit(y = VodafoneCDS, p = 1, const = TRUE)
y <- VARsim(fittedModel = fit)
VARfit(y = y, p = 1, const = TRUE)

# Simulates from user given parameters. 
#  Includes an exogenous variable:
VARcoef <- matrix(c(1,     2,
                    1,   0.5,
                    0.1,  0.3,
                    0.2,  0.5), 
                  ncol = 2, byrow = TRUE)
exo <- matrix(rnorm(500))
y <- VARsim(N = 500, K = 2, p = 1, const = TRUE, 
            trend = FALSE, exogen = exo,
            coef = VARcoef, dist = "normal")
VARfit(y = y, p = 1, const = TRUE, exogen = exo)

Combined LM test for ARCH errors in VAR models.

Description

Performs the bootstrap combined Lagrange multiplier (LM) test for autoregressive conditional heteroskedastic (ARCH) errors in vector autoregressive (VAR) models of Catani and Ahlgren (2016).

The tests of Eklund and Teräsvirta (2007), as well as the Multivariate LM test for ARCH as described for example in Lütkepohl (2006, sect. 16.5), are also included if the arguments ET respectively MARCH are set to TRUE. The bootstrap procedure for those are the same as in Catani and Ahlgren (2016).

Usage

archBootTest(fit, h = 2, B = 499, CA = TRUE, ET = TRUE, MARCH = TRUE,
             dist = "norm", skT.param = c(0, 1, 0, 5), verbose = TRUE)
## S3 method for class 'archBootTest'
print(x, ...)

Arguments

Details

All tests for ARCH are based on Cholesky-standardised least squares (LS) residuals from the K-dimensional vector autoregressive (VAR) model with p lags (abstracting from deterministic terms):

\mathbf{y}_{t}=\mathbf{\Pi }_{1}\mathbf{y}_{t-1}+\cdots +\mathbf{\Pi }_{p} \mathbf{y}_{t-p}+\mathbf{u}_{t},\quad \text{E}(\mathbf{u}_{t})=\mathbf{0} ,\quad \text{E}(\mathbf{u}_{t}\mathbf{u}_{t}^{\prime })=\mathbf{\Omega},\ \ \ \ t=1,\ldots ,N.

The LS residuals are

\widehat{\mathbf{u}}_{t}=\mathbf{y}_{t}-\widehat{\mathbf{\Pi }}_{1}\mathbf{y} _{t-1}-\cdots -\widehat{\mathbf{\Pi }}_{p}\mathbf{y}_{t-p},

where \widehat{\mathbf{\Pi }}_{1},\ldots ,\widehat{\mathbf{\Pi }}_{p} are the LS estimates of the K\times K parameter matrices \mathbf{\Pi } _{1},\ldots ,\mathbf{\Pi }_{p}. The multivariate LS residuals are \widehat{\mathbf{U}}=(\widehat{\mathbf{u}}_{1},\ldots ,\widehat{\mathbf{u}} _{K}), which is an N\times K matrix. The Cholesky-standardised LS residuals are

\widetilde{\mathbf{w}}_{t}=(\mathbf{S}_{\widehat{\mathbf{U}}}^{-1})^{\prime } \widehat{\mathbf{u}}_{t},

where \mathbf{S}_{\widehat{\mathbf{U}}} is the Cholesky factor of N^{-1} \widehat{\mathbf{U}}^{\prime }\widehat{\mathbf{U}}, i.e. \mathbf{S}_{ \widehat{\mathbf{U}}} is the (unique) upper triangular matrix such that

\widehat{\mathbf{\Omega }}=\mathbf{S}_{\widehat{\mathbf{U}}}^{\prime } \mathbf{S}_{\widehat{\mathbf{U}}},\quad \widehat{\mathbf{\Omega }} ^{-1}=(N^{-1}\widehat{\mathbf{U}}^{\prime }\widehat{\mathbf{U}})^{-1}= \mathbf{S}_{\widehat{\mathbf{U}}}^{-1}(\mathbf{S}_{\widehat{\mathbf{U}} }^{-1})^{\prime }.

The LM test for ARCH of order h (Engle 1982) in equation i, i=1,\ldots ,K, is a test of H_{0}:b_{1}=\cdots =b_{h} against H_{1}:b_{j}\neq 0 for at least one j\in \{1,\ldots ,h\} in the auxiliary regression

\widetilde{w}_{it}^{2}=b_{0}+b_{1}\widetilde{w}_{i,t-1}^{2}+\cdots +b_{h} \widetilde{w}_{i,t-h}^{2}+e_{it}.

The LM statistic has the form

LM_{i}=(N-p)R_{i}^{2},

where R_{i}^{2} is R^{2} from the auxiliary regression for equation i.

The combined LM statistic (Dufour et al. 2010, Catani and Ahlgren 2016) is given by

\widetilde{LM}=1-\min_{1\leq i\leq K}(p(LM_{i})),

where p(LM_{i}) is the p-value of the LM_{i} statistic, derived from the asymptotic \chi ^{2}(h) distibution. The test is only available as a bootstrap test. The bootstrap p-value is simulated using Bootstrap Algorithm 1 of Catani and Ahlgren (2016) if the errors are normal,

w_{i1},\ldots ,w_{iT}\sim \text{N}(0,1),

and Bootstrap Algorithm 2 if the errors are skew-t (by setting the function argument dist = "skT"),

w_{i1},\ldots ,w_{iT}\sim \text{skT}(0,1;\lambda ,v),

where \lambda is the skewness parameter and v is the degrees-of-freedom parameter of the skew-t distribution. These parameters can be set with the skT.param argument.

The multivariate LM test for ARCH of order h is a generalisation of the univariate test, and is based on the auxiliary regression

\text{vech}(\widetilde{\mathbf{u}}_{t}\widetilde{\mathbf{u}}_{t}^{\prime })=\mathbf{b} _{0}+\mathbf{B}_{1} \text{vech}(\widetilde{\mathbf{u}}_{t-1}\widetilde{\mathbf{u}}_{t-1}^{\prime }) +\cdots+\mathbf{B}_{h} \text{vech}(\widetilde{\mathbf{u}}_{t-h}\widetilde{\mathbf{u}}_{t-h}^{\prime }) +\mathbf{e}_{t},

where \text{vech} is the half-vectorisation operator. The null hypothesis is H_{0}:\mathbf{B }_{1}=\cdots =\mathbf{B}_{h}=\mathbf{0} against H_{1}:\mathbf{B}_{j}\neq \mathbf{0\!} for at least one j\in \{1,\ldots ,h\}. The multivariate LM statistic has the form

MLM=\frac{1}{2}(N-p)K(K+1)-(N-p)\text{tr}(\widehat{\mathbf{\Omega}}_{\text{vech}}\widehat{\mathbf{\Omega}}^{-1}),

where \widehat{\mathbf{\Omega }}_{\text{vech}} is the estimator of the error covariance matrix from the auxiliary regression and \widehat{\mathbf{ \Omega }} =N^{-1}\sum_{t=1}^{N}\widetilde{\mathbf{u}}_{t}\widetilde{ \mathbf{u}}_{t}^{\prime } is the estimator of the error covariance matrix from the VAR model (see Lütkepohl 2006, sect. 16.5). The MLM statistic is asymptotically distributed as \chi ^{2}(K^{2}(K+1)^{2}h/4). The test is available as an asymptotic test using the asymptotic \chi ^{2}(K^{2}(K+1)^{2}h/4) distribution to derive the p-value, and as a bootstrap test. The bootstrap p-value is simulated using Bootstrap Algorithms 1 and 2 of Catani and Ahlgren (2016). The asymptotic validity of the bootstrap multivariate LM test has not been established.

The Eklund and Teräsvirta (2007) LM test of constant error covariance matrix assumes the alternative hypothesis is a constant conditional correlation autoregressive conditional heteroskedasticity (CCC-ARCH) process of order h: \mathbf{H}_{t}=\mathbf{D}_{t}\mathbf{PD}_{t}, where \mathbf{ D}_{t}=\text{diag}(h_{1t}^{1/2},\ldots ,h_{Kt}^{1/2}) is a diagonal matrix of conditional standard deviations of the errors \{\mathbf{u}_{t}\} and \mathbf{P}=(\rho _{ij}), i,j=1,\ldots ,K, is a positive definite matrix of conditional correlations. The conditional variance \mathbf{h}_{t}=(h_{1t},\ldots ,h_{Kt})^{\prime } is assumed to follow a CCC-ARCH(h) process:

\mathbf{h}_{t}=\mathbf{a}_{0}+\sum_{j=1}^{h}\mathbf{A}_{j}\boldsymbol{u} _{t-j}^{(2)},

where \mathbf{a}_{0}=(a_{01},\ldots ,a_{0K})^{\prime } is a K-dimensional vector of positive constants, \mathbf{A}_{1},\ldots ,\mathbf{A} _{h} are K\times K diagonal matrices and \boldsymbol{u} _{t}^{(2)}=(u_{1t}^{2},\ldots ,u_{Kt}^{2})^{\prime }.

The null hypothesis is H_{0}:\text{diag}(\mathbf{A}_{1})=\cdots =\text{diag}(\mathbf{A}_{h})=\mathbf{0} against H_{1}: \text{diag}(\mathbf{A}_{j})\neq \mathbf{0\!} for at least one j\in \{1,\ldots ,h\}. The LM statistic has the form

LM_{CCC}=(N-p)\mathbf{s}(\widehat{\boldsymbol{\theta }})^{\prime }\mathbf{I}( \widehat{\boldsymbol{\theta }})^{-1}\mathbf{s}(\widehat{\boldsymbol{\theta }} ),

where \mathbf{s}(\widehat{\boldsymbol{\theta }}) and \mathbf{I}(\widehat{ \boldsymbol{\theta }}) are the score vector and information matrix, respectively, estimated under the null hypothesis (see Eklund and Teräsvirta 2007 for details). The asymptotic distribution of the LM_{CCC} statistic is \chi ^{2}(Kh). The test is available as an asymptotic test using the asymptotic \chi ^{2}(Kh) distribution to derive the p-value, and as a bootstrap test. The bootstrap p-value is simulated using Bootstrap Algorithms 1 and 2 of Catani and Ahlgren (2016). The asymptotic validity of the bootstrap LM_{CCC} test has not been established.

Value

a list of class "ACtest".

References

Catani, P. and Ahlgren, N. (2016). Combined Lagrange multiplier test for ARCH in vector autoregressive models, Economics and Statistics, <doi:10.1016/j.ecosta.2016.10.006>.

Dufour, J.-M., Khalaf, L., and Beaulieu, M.-C. (2010). Multivariate residual-based finite-sample tests for serial dependence and arch effects with applications to asset pricing models, Journal of Applied Econometrics, 25 (2010) 263–285.

Eklund, B. and Teräsvirta, T. (2007). Testing constancy of the error covariance matrix in vector models, Journal of Econometrics, 140, 753-780.

Engle, R.F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50, 987-1007.

Lütkepohl, H. (2006), New Introduction to Multiple Time Series Analysis, Springer, New York.

See Also

VARfit to estimate a VAR(p).

Examples

fit <- VARfit(y = VodafoneCDS, p = 3, const = TRUE, trend = FALSE)
test <- archBootTest(fit = fit, h = 5, B = 199, CA = TRUE, ET = TRUE, MARCH = TRUE, dist = "norm")
test

Bootstrap Determination of Cointegration Rank in VAR Models

Description

This function uses the bootstrap and wild bootstrap to test the cointegration rank of a VAR model. The test is an implementation of Cavaliere, Rahbek & Taylor (2012, 2014), and is used in Ahlgren & Catani (2018).

Usage

cointBootTest(y, r = "sequence", p, model = 1, signif = 0.05, dummies = NULL, B = 999,
boot_type = c("B", "WB"), WB_dist = c("rademacher", "normal", "mammen"), verbose = TRUE)
## S3 method for class 'cointBootTest'
print(x, ...)

Arguments

Details

Consider the K-dimensional heteroskedastic cointegrated VAR model of Cavaliere, Rahbek and Taylor (2014):

\Delta\mathbf{y}_{t}=\boldsymbol{\alpha}\boldsymbol{\beta}^{\prime}\mathbf{y}_{t-1}+\sum_{i=1}^{p-1}\mathbf{\Gamma}_{i}\Delta\mathbf{y}_{t-i}+\boldsymbol{\alpha}\boldsymbol{\rho}^{\prime}D_{t}+\bm{\phi}d_{t}+\bm{\varepsilon}_{t},\ \ \ \ t=1,\ldots ,T,

where \boldsymbol{\alpha} and \boldsymbol{\beta} are \left( K \times r \right) matrices of rank r<K, the number r being the cointegration rank. D_{t} and d_{t} differ according to the model argument in the following manner:

model 1: D_{t}=0 and d_{t}=0 (no deterministic terms) model 2: D_{t}=1 and d_{t}=0 (restricted constant) model 3: D_{t}=1 and d_{t}=1 (restricted linear trend)

The likelihood ratio (LR) statistic for testing cointegration rank r against K is

Q_{r,T}=-T\sum_{i=r+1}^{K}\text{log}( 1 - \widehat{\lambda}_{i},),

where the eigenvalues \mathbf{\widehat{\lambda}}_{1}>\ldots >\mathbf{\widehat{\lambda}}_{K} are the K largest solutions to a certain eigenvalue problem (see Johansen 1996).

Bootstrap and wild bootstrap algorithm of Cavaliere, et al. (2012, 2014):

1. Estimate the model under H(r) using Gaussian PMLE yielding the estimates \boldsymbol{\widehat{\beta}}^{(r)}, \boldsymbol{\widehat{\alpha}}^{(r)}, \boldsymbol{\widehat{\rho}}^{(r)}, \mathbf{\widehat{\Gamma}}_{1}^{(r)},\ldots,\mathbf{\widehat{\Gamma}}_{p-1}^{(r)} and \widehat{\bm{\phi}}^{(r)}, together with the corresponding residuals, \widehat{\bm{\varepsilon}}_{r,t}. 2. Check that the equation | \widehat{\mathbf{A}}^{(r)}(z) |=0, with \widehat{\mathbf{A}}^{(r)}(z):=(1-z)\mathbf{I}_{K}-\boldsymbol{\widehat{\alpha}}^{(r)}\boldsymbol{\widehat{\beta}}^{(r)\prime}z-\sum_{i=1}^{p-1}\mathbf{\widehat{\Gamma}}_{i}^{(r)}(1-z)z^{i}, has K-r roots equal to 1 and all other roots outside the unit circle. If so, procede to step 3. 3. Construct the bootstrap sample recursively from

\Delta\mathbf{y}_{r,t}^{\ast}=\boldsymbol{\widehat{\alpha}}^{(r)}\boldsymbol{\widehat{\beta}}^{(r)\prime}\mathbf{y}_{r,t-1}^{\ast}+\sum_{i=1}^{p-1}\mathbf{\widehat{\Gamma}}_{i}^{(r)}\Delta\mathbf{y}_{r,t-i}^{\ast}+\boldsymbol{\widehat{\alpha}}^{(r)}\boldsymbol{\widehat{\rho}}^{(r)\prime}D_{t}+\widehat{\bm{\phi}}^{(r)}d_{t}+\bm{\varepsilon}_{r,t}^{\ast},\ \ \ \ t=1,\ldots ,T,

initialized at \mathbf{y}_{r,j}^{\ast}=\mathbf{y}_{j}, j=1-p,\ldots,0, and with the T bootstrap errors \bm{\varepsilon}_{r,t}^{\ast} generated using the residuals \widehat{\bm{\varepsilon}}_{r,t}. The bootstrap errors are generated depending on the boot_type argument in the following manner: boot_type = "B": The i.i.d. bootstrap, such that \bm{\varepsilon}_{r,t}^{\ast}:=\mathbf{\widehat{\bm{\varepsilon}}}_{r,\mathcal{U}_{t}}, where \mathcal{U}_{t}, t=1,\ldots ,T is an i.i.d. sequence of discrete uniform distributions on \{1,2,\ldots, T\}. boot_type = "WB": The wild bootstrap, where for each t=1,\ldots ,T, \bm{\varepsilon}_{r,t}^{\ast}:=\widehat{\bm{\varepsilon}}_{r,t}w_{t}, where w_{t}, t=1,\ldots ,T, is an i.i.d. sequence distributed according to the WB_dist argument. 4. Using the bootstrap sample, \{\mathbf{y}_{r,t}^{\ast}\}, and denoting by \mathbf{\widehat{\lambda}}_{1}^{\ast}>\ldots >\mathbf{\widehat{\lambda}}_{K}^{\ast} the ordered solutions to the bootstrap analogue of the eigenvalue problem, compute the bootstrap LR statistic Q_{r,T}^{\ast}:=-T\sum_{i=r+1}^{K}\text{log}( 1 - \widehat{\lambda}_{i}^{\ast}). Define the corresponding p-value as p_{r,T}^{\ast}:=1-G_{r,T}^{\ast}(Q_{r,T}), G_{r,T}^{\ast}(.) denoting the conditional (on the original data) cdf of Q_{r,T}^{\ast}. 5. The bootstrap test of H(r) against H(K) at level \mathbf{\eta} rejects H(r) if p_{r,T}^{\ast}\leq\mathbf{\eta}.

If r = "sequence", the algorithm is repeated for each null hypothesis H(r), r=0,\ldots ,K-1, and the first null hypothesis with a p_{r,T}^{\ast}>\mathbf{\eta} is selected as the cointegration rank. If p_{r,T}^{\ast}\leq\mathbf{\eta}, r=0,\ldots ,K-1, the rank selected is \widehat{r}=K.

Value

a list of class "cointBootTest".

References

Ahlgren, N. & Catani, P. (2018). Practical Problems with Tests of Cointegration Rank with Strong Persistence and Heavy-Tailed Errors. In Corazza, M., Durábn, M., Grané, A., Perna, C., Sibillo, M. (eds) Mathematical and Statistical Methods for Actuarial Sciences and Finance, Cham, Springer.

Cavaliere, G., Rahbek, A., & Taylor, A. M. R. (2012). Bootstrap determination of the co-integration rank in vector autoregressive models, Econometrica, 80, 1721-1740.

Cavaliere, G., Rahbek, A., & Taylor, A. M. R. (2014). Bootstrap determination of the co-integration rank in heteroskedastic VAR models, Econometric Reviews, 33, 606-650.

Johansen, S. (1996). Likelihood-based inference in cointegrated vector autoregressive models, Oxford, Oxford University Press.

Examples

test <- cointBootTest(y = VodafoneCDS, r = "sequence", p = 2, model = 3, signif = 0.05, 
  dummies = NULL, B = 999, boot_type = c("B", "WB"), WB_dist = "rademacher")
test

Wild Bootstrap Tests for Error Autocorrelation

Description

This function runs Ahlgren and Catani's (2016) wild bootstrap (WB) test for error autocorrelation in VAR models. The function uses the test results from the function ACtest for the argument 'test', so that ACtest must be used before running wildBoot.

Usage

wildBoot(test, WBtype = c("recursive", "fixed"), B = 199, 
         WBdist = c("rademacher", "normal", "mammen"), 
         HCtype = c("LM", "HC0", "HC1", "HC2", "HC3"), 
         univariate = FALSE, verbose = TRUE)
## S3 method for class 'wildBoot'
print(x, ...)

Arguments

Value

a list of class "wildBoot".

References

Ahlgren, N. & Catani, P. (2016). Wild bootstrap tests for autocorrelation in vector autoregressive models. Stat Papers, <doi:10.1007/s00362-016-0744-0>.

See Also

ACtest for the test that is required to be run before the wildBoot function.

Examples

fit <- VARfit(y = VodafoneCDS, p = 3, const = TRUE, trend = FALSE)
test <- ACtest(fit = fit, h = 1, HCtype = c("LM", "HC0", "HC1", "HC2", "HC3"))
test
wb <- wildBoot(test = test, WBtype = "recursive", B = 199, 
               WBdist = "rademacher",  HCtype = c("LM", "HC3"))
wb