curvir: Specify Reserve Demand Curves

Description

Automatic specification and estimation of reserve demand curves for central bank operations. The package can help to choose the best demand curve and identify additional explanatory variables. Various plot and predict options are included. For more details, see Chen et al. (2023) https://www.imf.org/en/Publications/WP/Issues/2023/09/01/Modeling-the-Reserve-Demand-to-Facilitate-Central-Bank-Operations-538754.

Author(s)

Maintainer: Nikolaos Kourentzes nikolaos@kourentzes.com

Authors:

• Zhuohui Chen zchen4@imf.org

• Romain Veyrune

Reserve demand curve

Description

Fits the reserve demand curve between excess reserves and normalised rates

Usage

curve(x, y, type = "logistic", dummy = NULL, q = NULL, ...)

Arguments

Details

For a description of the parametric curves, see the provided reference. Below we list their functions:

• logisitc (Logistic)

  r_i = \alpha + \kappa / (1 - \beta e^{g(\bm{C}_i)}) + \varepsilon_i

• redLogistic (Reduced logistic)

  r_i = \alpha + 1 / (1 - \beta e^{g(\bm{C}_i)}) + \varepsilon_i

• fixLogistic (Fixed logistic)

  r_i = \alpha + 1 / (1 - e^{g(\bm{C}_i)}) + \varepsilon_i

• doubleExp (Double exponential)

  r_i = \alpha + \beta e^{\rho e^{g(\bm{C}_i)}} + \varepsilon_i

• exponential (Exponential)

  r_i = \alpha + \beta e^{g(\bm{C}_i)} + \varepsilon_i

• fixExponential (Fixed exponential)

  r_i = \beta e^{g(\bm{C}_i)} + \varepsilon_i

• arctan (Arctangent)

  r_i = \alpha + \beta \arctan ( g(\bm{C}_i)) + \varepsilon_i

• linear (Linear)

  r_i = g(\bm{C}_i) + \varepsilon_i

And g(\bm{C}) = c + \bm{C} w_g, where \alpha, \beta, \kappa, \rho are curve parameters, c is a constant togglable by constant, \bm{C} are the regressors including the excess reserves. w_g their coefficients, and finally \varepsilon_i is the error term of the curve.

Value

Returns a model of class curvir. This includes

• type the type of the curve.

• constant a logical indicating the use of a constant.

• w a list including: mean the curve parameters for the mean of the curve, upper and lower the parameters for the curve at the upper and lower intervals.

• data a list including the y, x, and dummy used for the fitting of the curve.

• mse the MSE from the fitting of the curve (the mean only).

• q the interval used in the fitting of the curve.

Note

An additional column for the constant is automatically generated, unless requested otherwise.

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

See Also

predict.curvir, plot.curvir, and curveopt.

Examples

  # Use ECB example data
  rate <- ecb$rate
  x <- ecb$x[,1,drop=FALSE]
  curve(x,rate)

  # An arctangent curve
  curve(x,rate,type="arctan")
 

Optimise curve parameters

Description

Finds optimal curve parameters.

Usage

curveopt(
  x,
  y,
  type = "logistic",
  constant = c(TRUE, FALSE),
  reps = 3,
  sign = NULL,
  q = NULL,
  winit = NULL,
  yhat = NULL,
  wsel = c("select", "combine"),
  dummy = NULL,
  sameSign = c(TRUE, FALSE)
)

Arguments

Value

Returns a list of

• w The optimal parameters

• mse The Mean Squared Error of the fitted curve.

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com

References

• Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

• Kourentzes, N., Barrow, D., & Petropoulos, F. (2019). Another look at forecast selection and combination: Evidence from forecast pooling. International Journal of Production Economics, 209, 226-235.

See Also

curve, and curvepred.

Examples

  # Use ECB example data
  rate <- ecb$rate
  x <- ecb$x[,1,drop=FALSE]
  curveopt(x,rate)

Reserve demand curve predicted values

Description

Provides the predicted values for the reserve demand curve of choice. For general use prefer the predict() function, which handles the constant internally.

Usage

curvepred(x, w, type = "logistic", dummy = NULL)

Arguments

Details

For a description of the parametric curves, see the provided reference. Below we list their functions:

• logisitc (Logistic)

  r_i = \alpha + \kappa / (1 - \beta e^{g(\bm{C}_i)}) + \varepsilon_i

• redLogistic (Reduced logistic)

  r_i = \alpha + 1 / (1 - \beta e^{g(\bm{C}_i)}) + \varepsilon_i

• fixLogistic (Fixed logistic)

  r_i = \alpha + 1 / (1 - e^{g(\bm{C}_i)}) + \varepsilon_i

• doubleExp (Double exponential)

  r_i = \alpha + \beta e^{\rho e^{g(\bm{C}_i)}} + \varepsilon_i

• exponential (Exponential)

  r_i = \alpha + \beta e^{g(\bm{C}_i)} + \varepsilon_i

• fixExponential (Fixed exponential)

  r_i = \beta e^{g(\bm{C}_i)} + \varepsilon_i

• arctan (Arctangent)

  r_i = \alpha + \beta \arctan ( g(\bm{C}_i)) + \varepsilon_i

• linear (Linear)

  r_i = g(\bm{C}_i) + \varepsilon_i

And g(\bm{C}) = c + \bm{C} w_g, where \alpha, \beta, \kappa, \rho are curve parameters, c is a constant togglable by constant, \bm{C} are the regressors including the excess reserves. w_g their coefficients, and finally \varepsilon_i is the error term of the curve.

Value

Returns a vector of the predicted values.

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

See Also

curve, and curveopt.

curvir:Reserve demand curve modelling toolbox

Description

The curvir package provides tools for building reserve demand curves for central bank operations.

Parametric curve modelling

• curve - fit a parametric curve

• curveopt - optimise curve parameters (called through curve)

• curvepred - predict curve values (called through predict)

• invcurve - produce predictions from the inverse curve

Non-parametric curve modelling

• npcurve - fit a non-parametric curve

Curve specification

• cvfit - Joint automatic specification of curve type and regressors

• varselect - Automatic specification of regressors

• cvnpcurve - Provide cross-validated errors for non-paramteric curves

• cvfitplot - Summary plot for cvfit and cvnpcurve

General

• predict Predict parametric and non-parametric curves

• plot Plot parametric and non-parametric curves

Data

• ecb ECB dataset from Chen et al. (2023)

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com, Zhuohui Chen, zchen4@imf.org, Romain R. Veyrune.

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

Automatically select curve type and explanatory variables

Description

Using cross-validation automatically select explanatory variables jointly with curve type. When running cvfit there is no need to use varselect separately.

Usage

cvfit(
  x,
  y,
  folds = 10,
  constant = c(TRUE, FALSE),
  sign = NULL,
  reps = 3,
  parallel = c(FALSE, TRUE),
  usepbapply = c(FALSE, TRUE),
  alltype = c("logistic", "redLogistic", "doubleExp", "exponential", "arctan", "linear"),
  search = c("backward", "forward"),
  wsel = c("select", "combine"),
  dummy = NULL
)

Arguments

Value

Returns a list with the recommended variable selection choice:

• type the type of selected curve.

• keep a logical vector with which variables to keep.

• varRes the result from varselect for each evaluated curve type.

• cvIndex a matrix detailing how the sample was split for the cross-validation. First column is the fold number and second column is the index of the observation.

Use cvfitplot to visualise the output.

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

See Also

cvfitplot, curve, and varselect.

Examples

  # Use ECB example data
  rate <- ecb$rate
  x <- ecb$x[,1:3,drop=FALSE]
  cvKeep <- cvfit(x,rate,folds=5,alltype=c("logistic","arctan"),parallel=TRUE)
  # Print result
  print(cvKeep)
  # Fit curve with the selected variables
  curve(x[,cvKeep$keep,drop=FALSE],rate,type=cvKeep$type)

Plot output of cvfit to facilitate comparison

Description

Plot summarised error of different curves specification from cvfit. Assuming normal errors, plot the mean cross-validated error and the 95

Usage

cvfitplot(cvKeep, xlock = c("mean", "median"), cvRF = NULL, cvSP = NULL)

Arguments

Value

No returned value. Produces a summary plot of cross-validated errors.

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

See Also

cvfit.

Examples

  # Use ECB example data
  rate <- ecb$rate
  x <- ecb$x[,1:3,drop=FALSE]
  cvKeep <- cvfit(x,rate,folds=5,alltype=c("logistic","arctan"),parallel=TRUE)
  cvfitplot(cvKeep)
  # Add results from non-parameteric curves
  cvRF <- cvnpcurve(x,rate,cvKeep$cvIndex)
  cvSP <- cvnpcurve(x,rate,cvKeep$cvIndex,type="spline")
  cvfitplot(cvKeep,cvRF=cvRF,cvSP=cvSP)

Cross-validated errors for non-parametric curve

Description

Obtain cross-validated errors for a non-parametric curve with given sample splits.

Usage

cvnpcurve(x, y, cvIndex, type = "rforest", dummy = NULL)

Arguments

Value

Returns summary cross-validated errors, comparable with the output from cvfit.

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

See Also

cvfit, and cvfitplot.

Examples

  # Use ECB example data
  rate <- ecb$rate
  x <- ecb$x[,1:3,drop=FALSE]
  cvKeep <- cvfit(x,rate,folds=5,alltype=c("logistic","arctan"),parallel=TRUE)
  # Get non-parametric curve cross-validated errors
  cvRF <- cvnpcurve(x,rate,cvKeep$cvIndex)
  cvSP <- cvnpcurve(x,rate,cvKeep$cvIndex,type="spline")

ECB excess reserve data

Description

The short-term interest rate used for the European Central Bank (ECB) is the volume-weighted Euro Overnight Index Average (EONIA) rate. Various explanatory variables are provided, as listed in the Chen et al. (2023). Data are collected from 1999 to 2019, resulting in 239 maintenance periods where all data is available.

Usage

ecb

Format

ecb a list containing

• rate the normalised EONIA rate.

• x a matrix including the ECB excess reserves and various regressors.

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

Examples

plot(ecb$x[,1],ecb$rate,xlab="Excess Reserves",ylab="Rate")

Calculate the inverse curve prediction

Description

Calculate the predicted reserves given some rate, i.e., calculate the prediction of the inverse curve.

Usage

invcurve(
  object,
  ynew = NULL,
  xnew = NULL,
  dummynew = NULL,
  warn = c(TRUE, FALSE)
)

Arguments

Value

Returns a vector of values of the predicted reserves

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

See Also

curve, and predict.

Examples

  # Use ECB example data
  rate <- ecb$rate
  x <- ecb$x[,1,drop=FALSE]
  fit <- curve(x,rate,type="logistic")
  invcurve(fit)

  # Use a different input rate
  invcurve(fit,ynew=0.1)

Reserve demand non-parametric curve

Description

Fits a non-parametric reserve demand curve between excess reserves and normalised rates

Usage

npcurve(x, y, type = c("rforest", "spline"), dummy = NULL, q = NULL, ...)

Arguments

Value

Returns a model of class npcurvir. This includes

• type the type of the curve.

• fit the non-parametric model for the mean.

• fitQ The non-parametric model for the quantiles.

• data a list including the y, x, and dummy used for the fitting of the curve.

• q the interval used in the fitting of the curve.

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

See Also

predict.npcurvir, and plot.npcurvir.

Examples

# Use ECB example data
rate <- ecb$rate
x <- ecb$x[,1,drop=FALSE]
npcurve(x,rate)

Plot method for curvir reserve demand models

Description

Plot a reserve demand curve estimated using curve.

Plot a non-parametric reserve demand curve estimated using npcurve.

Usage

## S3 method for class 'curvir'
plot(
  x,
  ploty = c(FALSE, TRUE),
  plotq = c(TRUE, FALSE),
  usemean = c(FALSE, TRUE),
  prcmp = c(FALSE, TRUE),
  useline = c(FALSE, TRUE),
  main = NULL,
  pch = 20,
  ...
)

## S3 method for class 'npcurvir'
plot(
  x,
  ploty = c(FALSE, TRUE),
  plotq = c(TRUE, FALSE),
  usemean = c(FALSE, TRUE),
  prcmp = c(FALSE, TRUE),
  useline = c(FALSE, TRUE),
  main = NULL,
  pch = 20,
  ...
)

Arguments

Value

No returned value. Produces a plot of the estimated curve.

Methods (by class)

• plot(curvir): Plot parametric curves

• plot(npcurvir): Plot non-parametric curves

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

See Also

curve.

npcurve.

Examples

  # Use ECB example data
  rate <- ecb$rate
  x <- ecb$x[,1,drop=FALSE]
  fit <- curve(x,rate)
  plot(fit)


# Use ECB example data
rate <- ecb$rate
x <- ecb$x[,1,drop=FALSE]
fit <- npcurve(x,rate)
plot(fit)

Predict method for curvir reserve demand models

Description

Predicted values based on curvir model object

Predicted values based on npcurvir model object

Usage

## S3 method for class 'curvir'
predict(object, newdata = NULL, newdummy = NULL, ...)

## S3 method for class 'npcurvir'
predict(object, newdata = NULL, newdummy = NULL, ...)

Arguments

Value

Returns a matrix of predicted values. If the model has estimates for intervals then it will provide upper and lower intervals.

Returns a matrix of predicted values. If the model has estimates for intervals then it will provide upper and lower intervals.

Methods (by class)

• predict(curvir): Predicted values for parametric curves

• predict(npcurvir): Predicted values for non-parametric curves

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

See Also

curve.

npcurve.

Examples

  # Use ECB example data
  rate <- ecb$rate
  x <- ecb$x[,1,drop=FALSE]
  fit <- curve(x,rate)
  predict(fit)

  # An example with new data
  predict(fit,newdata=tail(x))
 

# Use ECB example data
rate <- ecb$rate
x <- ecb$x[,1,drop=FALSE]
fit <- npcurve(x,rate)
predict(fit)

Automatically select explanatory variables for a curve type

Description

Using cross-validation automatically select explanatory variables for a reserve demand curve type.

Usage

varselect(
  x,
  y,
  type = "logistic",
  folds = 10,
  constant = c(TRUE, FALSE),
  sign = NULL,
  reps = 3,
  search = c("backward", "forward"),
  wsel = c("select", "combine"),
  dummy = NULL
)

Arguments

Value

Returns a list with the recommended variable selection choice:

• keep a logical vector with which variables to keep.

• errors statistics of the cross-validated MSE error.

Author(s)

Nikolaos Kourentzes, nikolaos@kourentzes.com

References

Chen, Z., Kourentzes, N., & Veyrune, R. (2023). Modeling the Reserve Demand to Facilitate Central Bank Operations. IMF Working Papers, 2023(179).

See Also

curve, and cvfit.

Examples

  # Use ECB example data
  rate <- ecb$rate
  x <- ecb$x[,1:3,drop=FALSE]
  varKeep <- varselect(x,rate,folds=5)
  # Print result
  print(varKeep)
  # Fit curve with the selected variables
  curve(x[,varKeep$keep,drop=FALSE],rate)