The cointReg package

Description

Parameter Estimation and Inference in a Cointegrating Regression

Details

See the vignette:
vignette("cointReg")

See the DESCRIPTION:
help(package = cointReg)

See the README:
https://github.com/aschersleben/cointReg/blob/master/README.md

Open the package documentation page:
package?cointReg

Further information and bug reporting:
https://github.com/aschersleben/cointReg

Functions

• cointReg(method = c("FM", "D", "IM"), ...)
  General function to estimate parameters of the given model. Three methods are possible; they can be choosen directly by using one of the following functions:

  • cointRegFM: Fully Modified OLS

  • cointRegD: Dynamic OLS

  • cointRegIM: Integrated Modified OLS

• print
  Print clear results.

• plot
  Plot the residuals of a cointReg model.

• Helper functions:

  • Checking inputs and arguments:
    checkObject, checkVars

  • Calculation of bandwidth and long run variance:
    getBandwidth, getBandwidthAnd, getBandwidthNW
getLongRunVar, getLongRunWeights

  • Additional D-OLS functions:
    getLeadLag, makeLeadLagMatrix, getModD, checkDoptions

Check list D.options.

Description

Checking the list D.options, that is an argument of cointRegD.

Usage

checkDoptions(n.lag = NULL, n.lead = NULL, kmax = c("k4", "k12"),
  info.crit = c("AIC", "BIC"))

Arguments

Value

[list]. List with the checked and (if necessary) converted arguments.

If one of n.lag and n.lead is NULL, only kmax and info.crit will be not NULL.

See Also

Other check: checkObject, checkVars

Examples

checkDoptions(n.lag = 3, n.lead = 4)
checkDoptions(info.crit = "BIC")
checkDoptions()

# It's not sufficient to include only one of "n.lag" and "n.lead":
checkDoptions(n.lag = 2)

Variable check for single objects.

Description

Checking the variable and convert it for internal use, if necessary. (Also used by the cointmonitoR package.)

Usage

checkObject(obj, obj.name, ..., out = "return", .env)

Arguments

Details

Possible values of obj.name to check:

"y", "x.stat":

Of type numeric, matrix or data.frame. Only the first row/column will be used.
Converted to object: column matrix

"y.fm", "x.coint", "deter":

Of type numeric, matrix or data.frame.
Converted to object: column matrix

"m":

Of type numeric(1), has to be greater than 0.

"model":

One of c("FM", "D", "IM").

"signif.level":

Of type numeric(1), has to be in the interval [0.01, 0.1].

"trend", "return.stats", "return.input", "demeaning", "t.test":

Converted to object: logical(1).

"kernel":

One of c("ba", "bo", "da", "pa", "qs", "th", "tr").

"bandwidth":

One of c("and", "nw").

"selector":

One or both c(1, 2).

Value

The checked and converted argument is assigned to the given environment (.env) and/or returned (depending on the argument out).

See Also

Other check: checkDoptions, checkVars

Examples

x = matrix(1:20, nrow = 2)
x2 = checkObject(x, "x.coint")
x2

env = environment()
y = 1:10
checkObject(y, out = "assign", .env = env)
y

# example for the use of the ... argument:
det = rbind(1, 1:10)
x3 = sin(10:20)
det2 = checkObject(deter = det)
det2
(checkObject(deter = det, x.stat = x3))

Multiple variable checks for certain values.

Description

Checking the arguments and convert them for internal use, if necessary.

Usage

checkVars(..., out = "assign", .env)

Arguments

Details

See checkObject for details.

Value

The checked and converted arguments are assigned to the given environment (.env) or invisibly returned as a list.

See Also

Other check: checkDoptions, checkObject

Examples

env = environment()
x.data = data.frame(a = 1:10, b = 20:11)
y.data = 1:10
checkVars(x.coint = x.data, y = y.data, .env = env)
x.coint
y

test = checkVars(x.coint = x.data, y = y.data, out = "return")
str(test)

# If the variables already have the "right" name,
# there's no need to do something like
# checkVars(kernel = kernel, bandwidth = bandwidth) -
# just call checkVars without specifing the arguments:
kernel = "q"
bandwidth = "a"
(checkVars(kernel, bandwidth, out = "return"))

Estimation and Inference for cointegrating regressions

Description

Computes either the Phillips and Hansen (1990) Fully Modified OLS estimator, or the Saikkonen (1990) Dynamic OLS estimator, or the Vogelsang and Wagner (2014) Integrated Modified OLS estimator.

Usage

cointReg(method = c("FM", "D", "IM"), x, y, ...)

Arguments

Value

[cointReg] object.

References

• Phillips, P.C.B. and B. Hansen (1990): "Statistical Inference in Instrumental Variables Regression with I(1) Processes," Review of Economic Studies, 57, 99–125, DOI:10.2307/2297545.

• Phillips, P.C.B. and M. Loretan (1991): "Estimating Long Run Economic Equilibria," Review of Economic Studies, 58, 407–436, DOI:10.2307/2298004.

• Saikkonen, P. (1991): "Asymptotically Efficient Estimation of Cointegrating Regressions," Econometric Theory, 7, 1–21, DOI:10.1017/S0266466600004217.

• Stock, J.H. and M.W. Watson (1993): "A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems," Econometrica, 61, 783–820, DOI:10.2307/2951763.

• Vogelsang, T.J. and M. Wagner (2014): "Integrated Modified OLS Estimation and Fixed-b Inference for Cointegrating Regressions," Journal of Econometrics, 148, 741–760, DOI:10.1016/j.jeconom.2013.10.015.

See Also

Other cointReg: cointRegD, cointRegFM, cointRegIM, plot.cointReg, print.cointReg

Examples

set.seed(1909)
x1 = cumsum(rnorm(100, mean = 0.05, sd = 0.1))
x2 = cumsum(rnorm(100, sd = 0.1)) + 1
x3 = cumsum(rnorm(100, sd = 0.2)) + 2
x = cbind(x1, x2, x3)
y = x1 + x2 + x3 + rnorm(100, sd = 0.2) + 1
deter = cbind(level = 1, trend = 1:100)
cointReg("FM", x = x, y = y, deter = deter, kernel = "ba",
         bandwidth = "and")

# Compare the results of all three models:
res = sapply(c("FM", "D", "IM"), cointReg, x = x, y = y, deter = deter)
do.call(cbind, lapply(res, "[[", "theta"))

Dynamic OLS

Description

Computes the Saikkonen (1990) Dynamic OLS estimator.

Usage

cointRegD(x, y, deter, kernel = c("ba", "pa", "qs", "tr"),
  bandwidth = c("and", "nw"), n.lead = NULL, n.lag = NULL,
  kmax = c("k4", "k12"), info.crit = c("AIC", "BIC"), demeaning = FALSE,
  check = TRUE, ...)

Arguments

Details

The equation for which the FM-OLS estimator is calculated:

y = \delta \cdot D + \beta \cdot x + u

with D as the deterministics matrix. Then \theta = (\delta', \beta')' is the full parameter vector.

Information about the D-OLS specific arguments:

n.lag, n.lead

A positive number to set the number of lags and leads. If at least one of them is equal to NULL (default), the function getLeadLag will be used to calculate them automatically (see Choi and Kurozumi (2012) for details). In that case, the following two arguments are needed.

kmax

Maximal value for lags and leads, when they are calculated automatically. If "k4", then the maximum is equal to floor(4 * (x.T/100)^(1/4)), else it's floor(12 * (x.T/100)^(1/4)) with x.T is equal to the data's length. One of "k4" or "k12". Default is "k4".

info.crit

Information criterion to use for the automatical calculation of lags and leads. One of "AIC" or "BIC". Default is "AIC".

Value

[cointReg]. List with components:

beta [numeric]

coefficients of the regressors

delta [numeric]

coefficients of the deterministics

theta [numeric]

combined coefficients of beta and delta

sd.theta [numeric]

standard errors for theta

t.theta [numeric]

t-values for theta

p.theta [numeric]

p-values for theta

theta.all [numeric]

combined coefficients of beta, delta and the auxiliary leads-and-lags regressors

residuals [numeric]

D-OLS residuals (length depends on leads and lags)

omega.u.v [numeric]

conditional long-run variance based on OLS residuals

varmat [matrix]

variance-covariance matrix

Omega [list]

the whole long-run variance matrix and parts of it

bandwidth [list]

number and name of the calculated bandwidth

kernel [character]

abbr. name of kernel type

lead.lag [list]

leads-and-lags parameters

References

• Phillips, P.C.B. and M. Loretan (1991): "Estimating Long Run Economic Equilibria," Review of Economic Studies, 58, 407–436, DOI:10.2307/2298004.

• Saikkonen, P. (1991): "Asymptotically Efficient Estimation of Cointegrating Regressions," Econometric Theory, 7, 1–21, DOI:10.1017/S0266466600004217.

• Stock, J.H. and M.W. Watson (1993): "A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems," Econometrica, 61, 783–820, DOI:10.2307/2951763.

See Also

Other cointReg: cointRegFM, cointRegIM, cointReg, plot.cointReg, print.cointReg

Other D-OLS: getLeadLag, getModD, makeLeadLagMatrix

Examples

set.seed(1909)
x1 <- cumsum(rnorm(100, mean = 0.05, sd = 0.1))
x2 <- cumsum(rnorm(100, sd = 0.1)) + 1
x3 <- cumsum(rnorm(100, sd = 0.2)) + 2
x <- cbind(x1, x2, x3)
y <- x1 + x2 + x3 + rnorm(100, sd = 0.2) + 1
deter <- cbind(level = 1, trend = 1:100)
test <- cointRegD(x, y, deter, n.lead = 2, n.lag = 2,
                    kernel = "ba", bandwidth = "and")
print(test)
test2 <- cointRegD(x, y, deter, kmax = "k4", info.crit = "BIC",
                     kernel = "ba", bandwidth = "and")
print(test2)

Fully Modified OLS

Description

Computes the Phillips and Hansen (1990) Fully Modified OLS estimator.

Usage

cointRegFM(x, y, deter, kernel = c("ba", "pa", "qs", "tr"),
  bandwidth = c("and", "nw"), demeaning = FALSE, check = TRUE, ...)

Arguments

Details

The equation for which the FM-OLS estimator is calculated:

y = \delta \cdot D + \beta \cdot x + u

with D as the deterministics matrix. Then \theta = (\delta', \beta')' is the full parameter vector.

The calculation of t-values and the variance-covariance matrix is only possible, if y is one-dimensional.

Value

[cointReg]. List with components:

delta [numeric | matrix]

coefficients as vector / matrix

beta [numeric | matrix]

coefficients as vector / matrix

theta [numeric | matrix]

combined coefficients of beta and delta as vector / matrix

sd.theta [numeric]

standard errors for theta

t.theta [numeric]

t-values for theta

p.theta [numeric]

p-values for theta

residuals [numeric]

FM-OLS residuals (first value is always missing)

omega.u.v [numeric]

conditional long-run variance based on OLS residuals.

varmat [matrix]

variance-covariance matrix

Omega [list]

the whole long-run variance matrix and parts of it

beta.OLS [numeric | matrix]

OLS coefficients as vector / matrix

delta.OLS [numeric | matrix]

OLS coefficients as vector / matrix

u.OLS [numeric]

OLS residuals

bandwidth [list]

number and name of bandwidth

kernel [character]

abbr. name of kernel type

References

• Phillips, P.C.B. and B. Hansen (1990): "Statistical Inference in Instrumental Variables Regression with I(1) Processes," Review of Economic Studies, 57, 99–125, DOI:10.2307/2297545.

See Also

Other cointReg: cointRegD, cointRegIM, cointReg, plot.cointReg, print.cointReg

Examples

set.seed(1909)
x1 = cumsum(rnorm(100, mean = 0.05, sd = 0.1))
x2 = cumsum(rnorm(100, sd = 0.1)) + 1
x3 = cumsum(rnorm(100, sd = 0.2)) + 2
x = cbind(x1, x2, x3)
y = x1 + x2 + x3 + rnorm(100, sd = 0.2) + 1
deter = cbind(level = 1, trend = 1:100)
test = cointRegFM(x, y, deter, kernel = "ba", bandwidth = "and")
print(test)

Integrated Modified OLS

Description

Computes the Vogelsang and Wagner (2014) Integrated Modified OLS estimator.

Usage

cointRegIM(x, y, deter, selector = 1, t.test = TRUE, kernel = c("ba",
  "pa", "qs", "tr"), bandwidth = c("and", "nw"), check = TRUE, ...)

Arguments

Details

The equation for which the IM-OLS estimator is calculated (type 1):

S_y = \delta \cdot S_{D} + \beta \cdot S_{x} + \gamma \cdot x + u

where S_y, S_x and S_D are the cumulated sums of y, x and D (with D as the deterministics matrix). Then \theta = (\delta', \beta', \gamma')' is the full parameter vector.

The equation for which the IM-OLS estimator is calculated (type 2):

S_y = \delta \cdot S_D + \beta \cdot S_x + \gamma \cdot x + \lambda \cdot Z + u

where S_y, S_x and S_D are the cumulated sums of y, x and D (with D as the deterministics matrix) and Z as defined in equation (19) in Vogelsang and Wagner (2015). Then \theta = (\delta', \beta', \gamma', \lambda')' is the full parameter vector.

Value

[cointReg]. List with components:

delta [numeric]

coefficients of the deterministics (cumulative sum S_{deter})

beta [numeric]

coefficients of the regressors (cumulative sum S_{x})

gamma [numeric]

coefficients of the regressors (original regressors x)

theta [numeric]

combined coefficients of beta, delta

sd.theta [numeric]

standard errors for the theta coefficients

t.theta [numeric]

t-values for the theta coefficients

p.theta [numeric]

p-values for the theta coefficients

theta.all [numeric]

combined coefficients of beta, delta, gamma

residuals [numeric]

IM-OLS residuals. Attention: These are the first differences of S_u – the original residuals are stored in u.plus.

u.plus [numeric]

IM-OLS residuals, not differenced. See residuals above.

omega.u.v [numeric]

conditional long-run variance based on OLS residuals, via cointRegFM (in case of argument t.test is TRUE) or NULL

varmat [matrix]

variance-covariance matrix

Omega [matrix]

NULL (no long-run variance matrix for this regression type)

bandwidth [list]

number and name of bandwidth if t.test = TRUE

kernel [character]

abbr. name of kernel type if t.test = TRUE

delta2 [numeric]

coefficients of the deterministics (cumulative sum S_{deter}) for regression type 2

beta2 [numeric]

coefficients of the regressors (cumulative sum S_{x}) for regression type 2

gamma2 [numeric]

coefficients of the regressors (original regressors x) for regression type 2

lambda2 [numeric]

coefficients of the Z regressors for regression type 2

theta2 [numeric]

combined coefficients of beta2, delta2, gamma2 and lambda2 for regression type 2

u.plus2 [numeric]

IM-OLS residuals for regression type 2

References

• Vogelsang, T.J. and M. Wagner (2014): "Integrated Modified OLS Estimation and Fixed-b Inference for Cointegrating Regressions," Journal of Econometrics, 148, 741–760, DOI:10.1016/j.jeconom.2013.10.015.

See Also

Other cointReg: cointRegD, cointRegFM, cointReg, plot.cointReg, print.cointReg

Examples

set.seed(1909)
x1 = cumsum(rnorm(100, mean = 0.05, sd = 0.1))
x2 = cumsum(rnorm(100, sd = 0.1)) + 1
x3 = cumsum(rnorm(100, sd = 0.2)) + 2
x = cbind(x1, x2, x3)
y = x1 + x2 + x3 + rnorm(100, sd = 0.2) + 1
deter = cbind(level = 1, trend = 1:100)
test = cointRegIM(x, y, deter, selector = c(1, 2), t.test = TRUE,
                    kernel = "ba", bandwidth = "and")
print(test)

Automatic Bandwidth Selection

Description

Automatic bandwidth selection of Andrews (1991) and of Newey and West (1994).

Usage

getBandwidth(u, bandwidth = c("and", "nw"), kernel, ..., check = TRUE)

getBandwidthAnd(u, kernel = c("ba", "pa", "qs", "th", "tr"), check = TRUE)

getBandwidthNW(u, kernel = c("ba", "pa", "qs"), inter = FALSE,
  u.weights = NULL, check = TRUE)

Arguments

Details

For Andrews (1991), the AR(1) individual version is implemented.

The kernel that is used for calculating the long-run variance can be one of the following:

• "ba": Bartlett kernel

• "pa": Parzen kernel

• "qs": Quadratic Spectral kernel

• "th": Tukey-Hanning kernel (only if bandwidth = "and")

• "tr": Truncated kernel (only if bandwidth = "and")

Value

[numeric(1)]. Bandwidth

Functions

• getBandwidthAnd: Automatic bandwidth selection of Andrews (1991).

• getBandwidthNW: Automatic bandwidth selection of Newey and West (1994).

References

• Andrews, D.W.K. (1991): "Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation," Econometrica, 59, 817–854, DOI:10.2307/2938229.

• Newey, W.K. and K.D. West (1994): "Automatic Lag Selection in Covariance Matrix Estimation", Review of Economic Studies, 61, 631–653, DOI:10.2307/2297912.

See Also

getLongRunVar

Examples

set.seed(1909)
x <- rnorm(100)
getBandwidth(x, kernel = "ba")
getBandwidth(x, bandwidth = "nw", kernel = "ba")

x2 <- arima.sim(model = list(ar = c(0.7, 0.2)), innov = x, n = 100)
getBandwidth(x2, kernel = "qs")
getBandwidth(x2, bandwidth = "nw", kernel = "qs")

Leads and Lags

Description

Generates "optimal" numbers of leads and lags for the Dynamic OLS estimator.

Usage

getLeadLag(x, y, deter, max.lag, max.lead, ic = c("AIC", "BIC"),
  symmet = FALSE, check = FALSE)

Arguments

Value

[numeric(2)]. "Optimal" numbers of leads and lags.

See Also

Other D-OLS: cointRegD, getModD, makeLeadLagMatrix

Examples

set.seed(1909)
y <- matrix(cumsum(rnorm(100)), ncol = 1)
x <- matrix(rep(y, 4) + rnorm(400, mean = 3, sd = 2), ncol = 4)
deter <- cbind(1, 1:100)
cointReg:::getLeadLag(x = x, y = y, deter = deter, max.lag = 5,
                      max.lead = 5, ic = "AIC", symmet = FALSE)

Long-Run Variance

Description

This function computes the long-run variance Omega, the one sided long-run variance Delta (starting with lag 0) and the variance Sigma from an input matrix of residuals.

Usage

getLongRunVar(u, bandwidth = c("and", "nw"), kernel = c("ba", "bo", "da",
  "pa", "qs", "tr"), demeaning = FALSE, check = TRUE, ...)

Arguments

Details

The bandwidth can be one of the following:

• "ba": Bartlett kernel

• "bo": Bohmann kernel

• "da": Daniell kernel

• "pa": Parzen kernel

• "qs": Quadratic Spectral kernel

• "tr": Truncated kernel

Value

[list] with components:

Omega [matrix]

Long-run variance matrix

Delta [matrix]

One-sided long-run variance matrix

Sigma [matrix]

Variance matrix

See Also

getBandwidth

Examples

set.seed(1909)
x <- rnorm(100)
band <- getBandwidthAnd(x, kernel = "ba")
getLongRunVar(x, kernel = "ba", bandwidth = band)
# shorter:
getLongRunVar(x, kernel = "ba", bandwidth = "and")

x2 <- arima.sim(model = list(ar = c(0.7, 0.2)), innov = x, n = 100)
x2 <- cbind(a = x2, b = x2 + rnorm(100))
getLongRunVar(x2, kernel = "ba", bandwidth = "nw")

Weights for Long-Run Variance

Description

Compute the weights corresponding to some kernel funtions.

Usage

getLongRunWeights(n, bandwidth, kernel)

Arguments

Value

[list] with components:

w [numeric]

Vector of weights

upper [numeric(1)]

Index to largest non-zero entry in w

See Also

getLongRunVar

Examples

lrw.ba = cointReg:::getLongRunWeights(100, kernel = "ba", bandwidth = 25)
plot(lrw.ba$w)

Get D OLS model.

Description

Generates an lm model for the Dynamic OLS estimator.

Usage

getModD(x, y, deter, n.lag, n.lead, check = FALSE)

Arguments

Value

[lm]. An lm object, containing an additional list element (aux) with D-OLS specific objects:

Z [matrix]

jointed matrix of deterministics and x

x.delta [matrix]

differences of x

dx.all [matrix]

leads-and-lags matrix

all.trunc [matrix]

truncated version of jointed matrix of Z and dx.all

y.trunc [matrix]

truncated version of y

See Also

Other D-OLS: cointRegD, getLeadLag, makeLeadLagMatrix

Examples

set.seed(1909)
y <- matrix(cumsum(rnorm(100)), ncol = 1)
x <- matrix(rep(y, 4) + rnorm(400, mean = 3, sd = 2), ncol = 4)
deter <- cbind(1, 1:100)
cointReg:::getModD(x = x, y = y, deter = deter, n.lag = 2, n.lead = 3)

Leads-and-Lags Matrix

Description

Generates leads-and-lags matrix for the Dynamic OLS estimator.

Usage

makeLeadLagMatrix(x, n.lag, n.lead)

Arguments

Value

[matrix]. Leads-and-lags matrix.

See Also

Other D-OLS: cointRegD, getLeadLag, getModD

Examples

x <- matrix(1:20, 2, byrow = TRUE)
cointReg:::makeLeadLagMatrix(x = x, n.lag = 2, n.lead = 3)

Plot Method for Cointegration Models (Modified OLS).

Description

Plotting objects of class "cointReg". Currently, only the residuals will be plotted.

Usage

## S3 method for class 'cointReg'
plot(x, type, main, xlab, ylab, axes = TRUE, ...)

Arguments

See Also

Other cointReg: cointRegD, cointRegFM, cointRegIM, cointReg, print.cointReg

Examples

set.seed(42)
x = data.frame(x1 = cumsum(rnorm(200)), x2 = cumsum(rnorm(200)))
eps1 = rnorm(200, sd = 2)
y = x$x1 - x$x2 + 10 + eps1
deter = cbind(level = rep(1, 200))
test = cointRegFM(x = x, y = y, deter = deter)
plot(test)

Print Method for Cointegration Models (Modified OLS).

Description

Printing objects of class "cointReg".

Usage

## S3 method for class 'cointReg'
print(x, ..., digits = getOption("digits"),
  all.coeffs = FALSE)

Arguments

Value

The invisible x object.

See Also

Other cointReg: cointRegD, cointRegFM, cointRegIM, cointReg, plot.cointReg

Examples

set.seed(42)
x = data.frame(x1 = cumsum(rnorm(200)), x2 = cumsum(rnorm(200)))
eps1 = rnorm(200, sd = 2)
y = x$x1 - x$x2 + 10 + eps1
deter = cbind(level = rep(1, 200))

test.fm = cointRegFM(x = x, y = y, deter = deter)
print(test.fm)

test.d = cointRegD(x = x, y = y, deter = deter)
print(test.d)

test.im2 = cointRegIM(x = x, y = y, deter = deter)
print(test.im2)