Monthly time series used to test VLSTAR models.

Description

This data set contains the series of realized covariances in 4 stock market indices, i.e. SP-500, Nikkei, DAX, and FTSE, Dividend Yield and Earning Price growth rate, inflation growth rates for U.S., U.K., Japan and Germany, from August 1990 to June 2018.

Usage

  data(Realized)

Format

A zoo data frame with 334 monthly observations, ranging from 1990:M8 until 2018:M6.

Author(s)

Andrea Bucci

See Also

rcov to build realized covariances from stock prices or returns.

Sum of squared error

Description

Sum of squared error to be optimized in the NLS method

Usage

SSQ(param, data)

Ten simulated prices series for 19 trading days in January 2010.

Description

Ten hypothetical price series were simulated according to the factor diffusion process discussed in Barndorff-Nielsen et al.

Usage

data("Sample5minutes")

Format

xts object

Author(s)

Andrea Bucci

VLSTAR- Estimation

Description

This function allows the user to estimate the coefficients of a VLSTAR model with m regimes through maximum likelihood or nonlinear least squares. The set of starting values of Gamma and C for the convergence algorithm can be either passed or obtained via searching grid.

Usage

VLSTAR(
  y,
  exo = NULL,
  p = 1,
  m = 2,
  st = NULL,
  constant = TRUE,
  starting = NULL,
  method = c("ML", "NLS"),
  n.iter = 500,
  singlecgamma = FALSE,
  epsilon = 10^(-3),
  ncores = NULL
)

Arguments

Details

The multivariate smooth transition model is an extension of the smooth transition regression model introduced by Bacon and Watts (1971) (see also Anderson and Vahid, 1998). The general model is

y_{t} = \mu_0+\sum_{j=1}^{p}\Phi_{0,j}\,y_{t-j}+A_0 x_t \cdot G_t(s_t;\gamma,c)[\mu_{1}+\sum_{j=1}^{p}\Phi_{1,j}\,y_{t-j}+A_1x_t]+\varepsilon_t

where \mu_{0} and \mu_{1} are the \tilde{n} \times 1 vectors of intercepts, \Phi_{0,j} and \Phi_{1,j} are square \tilde{n}\times\tilde{n} matrices of parameters for lags j=1,2,\dots,p, A_0 and A_1 are \tilde{n}\times k matrices of parameters, x_t is the k \times 1 vector of exogenous variables and \varepsilon_t is the innovation. Finally, G_t(s_t;\gamma,c) is a \tilde{n}\times \tilde{n} diagonal matrix of transition function at time t, such that

G_t(s_t;\gamma,c)=\{G_{1,t}(s_{1,t};\gamma_{1},c_{1}),G_{2,t}(s_{2,t};\gamma_{2},c_{2}), \dots,G_{\tilde{n},t}(s_{\tilde{n},t};\gamma_{\tilde{n}},c_{\tilde{n}})\}.

Each diagonal element G_{i,t}^r is specified as a logistic cumulative density functions, i.e.

G_{i,t}^r(s_{i,t}^r; \gamma_i^r, c_i^r) = \left[1 + \exp\big\{-\gamma_i^r(s_{i,t}^r-c_i^r)\big\}\right]^{-1}

for latex and r=0,1,\dots,m-1, so that the first model is a Vector Logistic Smooth Transition AutoRegressive (VLSTAR) model. The ML estimator of \theta is obtained by solving the optimization problem

\hat{\theta}_{ML} = arg \max_{\theta}log L(\theta)

where log L(\theta) is the log-likelihood function of VLSTAR model, given by

ll(y_t|I_t;\theta)=-\frac{T\tilde{n}}{2}\ln(2\pi)-\frac{T}{2}\ln|\Omega|-\frac{1}{2}\sum_{t=1}^{T}(y_t-\tilde{G}_tB\,z_t)'\Omega^{-1}(y_t-\tilde{G}_tB\,z_t)

The NLS estimators of the VLSTAR model are obtained by solving the optimization problem

\hat{\theta}_{NLS} = arg \min_{\theta}\sum_{t=1}^{T}(y_t - \Psi_t'B'x_t)'(y_t - \Psi_t'B'x_t).

Generally, the optimization algorithm may converge to some local minimum. For this reason, providing valid starting values of \theta is crucial. If there is no clear indication on the initial set of parameters, \theta, this can be done by implementing a grid search. Thus, a discrete grid in the parameter space of \Gamma and C is create to obtain the estimates of B conditionally on each point in the grid. The initial pair of \Gamma and C producing the smallest sum of squared residuals is chosen as initial values, then the model is linear in parameters. The algorithm is the following:

1. Construction of the grid for \Gamma and C, computing \Psi for each poin in the grid

2. Estimation of \hat{B} in each equation, calculating the residual sum of squares, Q_t

3. Finding the pair of \Gamma and C providing the smallest Q_t

4. Once obtained the starting-values, estimation of parameters, B, via nonlinear least squares (NLS)

5. Estimation of \Gamma and C given the parameters found in step 4

6. Repeat step 4 and 5 until convergence.

Value

An object of class VLSTAR, with standard methods.

Author(s)

Andrea Bucci

References

Anderson H.M. and Vahid F. (1998), Testing multiple equation systems for common nonlinear components. Journal of Econometrics. 84: 1-36

Bacon D.W. and Watts D.G. (1971), Estimating the transition between two intersecting straight lines. Biometrika. 58: 525-534

Terasvirta T. and Yang Y. (2014), Specification, Estimation and Evaluation of Vector Smooth Transition Autoregressive Models with Applications. CREATES Research Paper 2014-8

Examples

data(Realized)
y <- Realized[-1,1:10]
y <- y[-nrow(y),]
st <- Realized[-nrow(Realized),1]
st <- st[-length(st)]
stvalues <- startingVLSTAR(y, p = 1, n.combi = 3,
 singlecgamma = FALSE, st = st, ncores = 1)
fit.VLSTAR <- VLSTAR(y, p = 1, singlecgamma = FALSE, starting = stvalues,
 n.iter = 1, st = st, method ='NLS', ncores = 1)
# a few methods for VLSTAR
print(fit.VLSTAR)
summary(fit.VLSTAR)
plot(fit.VLSTAR)
predict(fit.VLSTAR, st.num = 1, n.ahead = 1)
logLik(fit.VLSTAR, type = 'Univariate')
coef(fit.VLSTAR)

Joint linearity test

Description

This function allows the user to test linearity against a Vector Smooth Transition Autoregressive Model with a single transition variable.

Usage

VLSTARjoint(y, exo = NULL, st, st.choice = FALSE, alpha = 0.05)

Arguments

Details

Given a VLSTAR model with a unique transition variable, s_{1t} = s_{2t} = \dots = s_{\widetilde{n}t} = s_t, a generalization of the linearity test presented in Luukkonen, Saikkonen and Terasvirta (1988) may be implemented.

Assuming a 2-state VLSTAR model, such that

y_t = B_{1}z_t + G_tB_{2}z_t + \varepsilon_t.

Where the null H_{0} : \gamma_{j} = 0, j = 1, \dots, \widetilde{n}, is such that G_t \equiv (1/2)/\widetilde{n} and the previous Equation is linear. When the null cannot be rejected, an identification problem of the parameter c_{j} in the transition function emerges, that can be solved through a first-order Taylor expansion around \gamma_{j} = 0.

The approximation of the logistic function with a first-order Taylor expansion is given by

G(s_t; \gamma_{j},c_{j}) = (1/2) + (1/4)\gamma_{j}(s_t-c_{j}) + r_{jt}

= a_{j}s_t + b_{j} + r_{jt}

where a_{j} = \gamma_{j}/4, b_{j} = 1/2 - a_{j}c_{j} and r_{j} is the error of the approximation. If G_t is specified as follows

G_t = diag\big\{a_{1}s_t + b_{1} + r_{1t}, \dots, a_{\widetilde{n}}s_t+b_{\widetilde{n}} + r_{\widetilde{n}t}\big\}

= As_t + B + R_t

where A = diag(a_{1}, \dots, a_{\widetilde{n}}), B = diag(b_{1},\dots, b_{\widetilde{n}}) e R_t = diag(r_{1t}, \dots, r_{\widetilde{n}t}), y_t can be written as

y_t = B_{1}z_t + (As_t + B + R_t)B_{2}z_t+\varepsilon_t

= (B_{1} + BB_{2})z_t+AB_{2}z_ts_t + R_tB_{2}z_t + \varepsilon_t

= \Theta_{0}z_t + \Theta_{1}z_ts_t+\varepsilon_t^{*}

where \Theta_{0} = B_{1} + B_{2}'B, \Theta_{1} = B_{2}'A and \varepsilon_t^{*} = R_tB_{2} + \varepsilon_t. Under the null, \Theta_{0} = B_{1} and \Theta_{1} = 0, while the previous model is linear, with \varepsilon_t^{*} = \varepsilon_t. It follows that the Lagrange multiplier test, under the null, is derived from the score

\frac{\partial \log L(\widetilde{\theta})}{\partial \Theta_{1}} = \sum_{t=1}^{T}z_ts_t(y_t - \widetilde{B}_{1}z_t)'\widetilde{\Omega}^{-1} = S(Y - Z\widetilde{B}_{1})\widetilde{\Omega}^{-1},

where

S = z_{1}'s_{1}\\\vdots\\ z_t's_t

and where \widetilde{B}_{1} and \widetilde{\Omega} are estimated from the model in H_{0}. If P_{Z} = Z(Z'Z)^{-1}Z' is the projection matrix of Z, the LM test is specified as follows

LM = tr\big\{\widetilde{\Omega}^{-1}(Y - Z\widetilde{B}_{1})'S\big[S'(I_t - P_{Z})S\big]^{-1}S'(Y-Z\widetilde{B}_{1})\big\}.

Under the null, the test statistics is distributed as a \chi^{2} with \widetilde{n}(p\cdot\widetilde{n} + k) degrees of freedom.

Value

An object of class VLSTARjoint.

Author(s)

Andrea Bucci

References

Luukkonen R., Saikkonen P. and Terasvirta T. (1988), Testing Linearity Against Smooth Transition Autoregressive Models. Biometrika, 75: 491-499

Terasvirta T. and Yang Y. (2015), Linearity and Misspecification Tests for Vector Smooth Transition Regression Models. CREATES Research Paper 2014-4

Coefficient method for objects of class VLSTAR

Description

Returns the coefficients of a VLSTAR model for objects generated by VLSTAR()

Usage

## S3 method for class 'VLSTAR'
coef(object, ...)

Arguments

Value

Estimated coefficients of the VLSTAR model

Author(s)

Andrea Bucci

References

Terasvirta T. and Yang Y. (2014), Specification, Estimation and Evaluation of Vector Smooth Transition Autoregressive Models with Applications. CREATES Research Paper 2014-8

Examples

mean(1:3)

Log-Likelihood method

Description

Returns the log-Likelihood of a VLSTAR object.

Usage

## S3 method for class 'VLSTAR'
logLik(object, type = c('Univariate', 'Multivariate'), ...)

Arguments

Details

The log-likelihood of a VLSTAR model is defined as:

\log l(y_t|I_t;\theta)=-\frac{T\tilde{n}}{2}\ln(2\pi)-\frac{T}{2}\ln|\Omega|-\frac{1}{2}\sum_{t=1}^{T}(y_t-\tilde{G}_tB\,z_t)'\Omega^{-1}(y_t-\tilde{G}_tB\,z_t)

Value

An object with class attribute logLik.

Author(s)

Andrea Bucci

See Also

VLSTAR

Multivariate log-likelihood

Description

Log-likelihood to be optimized in the ML method and used to check convergence in both methods

Usage

loglike(param, data)

Long-run variance using Bartlett kernel

Description

Function returns the long-run variance of a time series, relying on the Bartlett kernel. The window size of the kernel is the cube root of the sample size.

Usage

lrvarbart(x)

Arguments

Value

Author(s)

Andrea Bucci

References

Hamilton J. D. (1994), Time Series Analysis. Princeton University Press; Tsay R.S. (2005), Analysis of Financial Time Series. John Wiley & SONS

Examples

data(Realized)
lrvarbart(Realized[,1])

Multivariate CUMSUM test

Description

Function returns the test statistics for the presence of co-breaks in a set of multivariate time series.

Usage

multiCUMSUM(data, conf.level = 0.95, max.breaks = 7)

Arguments

Value

Author(s)

Andrea Bucci and Giulio Palomba

References

Aue A., Hormann S., Horvath L.and Reimherr M. (2009), Break detection in the covariance structure of multivariate time series models. The Annals of Statistics. 37: 4046-4087 Bai J., Lumsdaine R. L. and Stock J. H. (1998), Testing For and Dating Common Breaks in Multivariate Time Series. Review of Economic Studies. 65: 395-432 Barassi M., Horvath L. and Yuqian Z. (2018), Change-Point Detection in the Conditional Correlation Structure of Multivariate Volatility Models. Journal of Business \& Economic Statistics

Examples

data(Realized)
testCS <- multiCUMSUM(Realized[,1:10], conf.level = 0.95)
print(testCS)

Plot methods for a VLSTAR object

Description

Plot method for objects with class attribute VLSTAR.

Usage

## S3 method for class 'VLSTAR'
plot(
  x,
  names = NULL,
  main.fit = NULL,
  main.acf = NULL,
  main.pacf = NULL,
  main.logi = NULL,
  ylim.fit = NULL,
  ylim.resid = NULL,
  lty.fit = NULL,
  lty.resid = NULL,
  lty.logi = NULL,
  lwd.fit = NULL,
  lwd.resid = NULL,
  lwd.logi = NULL,
  lag.acf = NULL,
  lag.pacf = NULL,
  col.fit = NULL,
  col.resid = NULL,
  col.logi = NULL,
  ylab.fit = NULL,
  ylab.resid = NULL,
  ylab.acf = NULL,
  ylab.pacf = NULL,
  ylab.logi = NULL,
  xlab.fit = NULL,
  xlab.resid = NULL,
  xlab.logi = NULL,
  mar = par("mar"),
  oma = par("oma"),
  adj.mtext = NA,
  padj.mtext = NA,
  col.mtext = NA,
  ...
)

Arguments

Details

When the plot function is applied to a VLSTAR object, the values of the logistic function, given the estimated values of gamma and c through VLSTAR, are reported.

Value

Plot of VLSTAR fitted values, residuals, ACF, PACF and logistic function

Author(s)

Andrea Bucci

See Also

VLSTAR

Plot methods for a vlstarpred object

Description

Plot method for objects with class attribute vlstarpred.

Usage

## S3 method for class 'vlstarpred'
plot(
  x,
  type = c("single", "multiple"),
  names = NULL,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  lty.obs = 2,
  lty.pred = 1,
  lty.ci = 3,
  lty.vline = 1,
  lwd.obs = 1,
  lwd.pred = 1,
  lwd.ci = 1,
  lwd.vline = 1,
  col.obs = NULL,
  col.pred = NULL,
  col.ci = NULL,
  col.vline = NULL,
  ylim = NULL,
  mar = par("mar"),
  oma = par("oma"),
  ...
)

Arguments

Value

Plot of predictions from VLSTAR with their prediction interval

Author(s)

Andrea Bucci

See Also

predict.VLSTAR

VLSTAR Prediction

Description

One-step or multi-step ahead forecasts, with interval forecast, of a VLSTAR object.

Usage

## S3 method for class 'VLSTAR'
predict(
  object,
  ...,
  n.ahead = 1,
  conf.lev = 0.95,
  st.new = NULL,
  M = 5000,
  B = 1000,
  st.num = NULL,
  newdata = NULL,
  method = c("naive", "Monte Carlo", "bootstrap")
)

Arguments

Value

A list containing:

Author(s)

Andrea Bucci and Eduardo Rossi

References

Granger C.W.J. and Terasvirta T. (1993), Modelling Non-Linear Economic Relationships. Oxford University Press;

Lundbergh S. and Terasvirta T. (2007), Forecasting with Smooth Transition Autoregressive Models. John Wiley and Sons;

Terasvirta T. and Yang Y. (2014), Specification, Estimation and Evaluation of Vector Smooth Transition Autoregressive Models with Applications. CREATES Research Paper 2014-8

See Also

VLSTAR for log-likehood and nonlinear least squares estimation of the VLSTAR model.

Print method for objects of class VLSTAR

Description

‘print’ methods for class ‘VLSTAR’.

Usage

## S3 method for class 'VLSTAR'
print(x, ...)

Arguments

Value

Print of VLSTAR results

Author(s)

Andrea Bucci

References

Terasvirta T. and Yang Y. (2014), Specification, Estimation and Evaluation of Vector Smooth Transition Autoregressive Models with Applications. CREATES Research Paper 2014-8

See Also

VLSTAR

Realized Covariance

Description

Function returns the vectorization of the lowest triangular of the Realized Covariance matrices for different frequencies.

Usage

rcov(
  data,
  freq = c("daily", "monthly", "quarterly", "yearly"),
  make.ret = TRUE,
  cholesky = FALSE
)

Arguments

Value

Author(s)

Andrea Bucci

References

Andersen T.G., Bollerslev T., Diebold F.X. and Labys P. (2003), Modeling and Forecasting Realized Volatility. Econometrica. 71: 579-625

Barndorff-Nielsen O.E. and Shephard N. (2002), Econometric analysis of realised volatility and its use in estimating stochastic volatility models Journal of the Royal Statistical Society. 64(2): 253-280

Examples

data(Sample5minutes)
rc <- rcov(Sample5minutes, freq = 'daily', cholesky = TRUE, make.ret = TRUE)
print(rc)

Starting parameters for a VLSTAR model

Description

This function allows the user to obtain the set of starting values of Gamma and C for the convergence algorithm via searching grid.

Usage

startingVLSTAR(
  y,
  exo = NULL,
  p = 1,
  m = 2,
  st = NULL,
  constant = TRUE,
  n.combi = NULL,
  ncores = 2,
  singlecgamma = FALSE
)

Arguments

Details

The searching grid algorithm allows for the optimal choice of the parameters \gamma and c by minimizing the sum of the Squared residuals for each possible combination.

The parameter c is initialized by using the mean of the dependent(s) variable, while \gamma is sampled between 0 and 100.

Value

An object of class startingVLSTAR.

Author(s)

Andrea Bucci

References

Anderson H.M. and Vahid F. (1998), Testing multiple equation systems for common nonlinear components. Journal of Econometrics. 84: 1-36

Bacon D.W. and Watts D.G. (1971), Estimating the transition between two intersecting straight lines. Biometrika. 58: 525-534

Terasvirta T. and Yang Y. (2014), Specification, Estimation and Evaluation of Vector Smooth Transition Autoregressive Models with Applications. CREATES Research Paper 2014-8

See Also

VLSTAR

Examples

data(Realized)
y <- Realized[-1,1:10]
y <- y[-nrow(y),]
st <- Realized[-nrow(Realized),1]
st <- st[-length(st)]
starting <- startingVLSTAR(y, p = 1, n.combi = 3,
                           singlecgamma = FALSE, st = st,
                           ncores = 1)

Summary method for objects of class VLSTAR

Description

‘summary’ methods for class ‘VLSTAR’.

Usage

## S3 method for class 'VLSTAR'
summary(object, ...)

## S3 method for class 'summary.VLSTAR'
print(x, ...)

Arguments

Value

An object of class summary.VLSTAR containing a list of summary information from VLSTAR estimates. When print is applied to this object, summary information are printed

Functions

• print.summary.VLSTAR: Print of the summary

Author(s)

Andrea Bucci

References

Terasvirta T. and Yang Y. (2014), Specification, Estimation and Evaluation of Vector Smooth Transition Autoregressive Models with Applications. CREATES Research Paper 2014-8

See Also

VLSTAR

Daily closing prices of 3 tech stocks.

Description

This data set contains the series of daily prices of Google, Microsof and Amazon stocks from January 3, 2005 to June 16, 2020, gathered from Yahoo.

Usage

data("techprices")

Format

An xts object with 3890 daily observations, ranging from from January 3, 2005 to June 16, 2020.

Author(s)

Andrea Bucci