Prediction intervals based on AR-perc algorithm

Description

Computes a bootstrap percentile-based prediction interval using the AR-perc algorithm for AR(p) model over a forecast horizon h.

Usage

pi_AR_perc(
  series,
  p = 1,
  h = 3,
  B = 1000,
  alpha = 0.05,
  tau = 0.5,
  seed = NULL
)

Arguments

Details

This function implements the AR-perc algorithm described in Novo and Sánchez-Sellero (2025).

Value

A list with elements:

bfor

Numeric matrix of bootstrap forecasts with dimension h x B.

lpi

Numeric vector of lower bounds (length h).

upi

Numeric vector of upper bounds (length h).

len

Numeric vector of interval lengths (length h).

References

Novo, S., & Sánchez-Sellero, C. (2025). Prediction intervals for quantile autoregression. arXiv:2512.22018. https://arxiv.org/abs/2512.22018

Examples

set.seed(123)

# Simulation parameters
burn_in <- 300
n <- 25
m <- burn_in + n
coeff <- 0.6

# Simulate AR(1) process
e <- rnorm(m)
x <- numeric(m)
x[1] <- rnorm(1)

for (t in 2:m) {
  x[t] <- coeff * x[t - 1] + e[t]
}

series <- ts(x[(burn_in + 1):m])

# Compute prediction interval
out <- pi_AR_perc(series)

out$lpi
out$upi
out$len

Prediction intervals based on AR-proot algorithm

Description

Computes a bootstrap predictive-root-based prediction interval using the AR-proot algorithm for an AR(p) model over a forecast horizon h.

Usage

pi_AR_proot(
  series,
  p = 1,
  h = 3,
  B = 1000,
  alpha = 0.05,
  tau = 0.5,
  seed = NULL
)

Arguments

Details

This function implements the AR-proot algorithm described in Novo and Sánchez-Sellero (2025). Predictive residuals are first obtained using a leave-one-out quantile autoregressive fit. To account for the estimation uncertainty of the autoregressive coefficients, bootstrap coefficient estimates are generated through a multiplier (random weights) bootstrap scheme. Conditional on these bootstrap coefficients, future bootstrap predictions and bootstrap observations are constructed, and predictive root replicates are defined as the difference between bootstrap observations and their corresponding bootstrap predictions. Equal-tailed prediction intervals are obtained from the empirical quantiles of the bootstrap predictive roots.

Value

A list with elements:

pfor

Numeric vector of point forecasts (length h).

lpi

Numeric vector of lower bounds (length h).

upi

Numeric vector of upper bounds (length h).

len

Numeric vector of interval lengths (length h).

References

Novo, S., & Sánchez-Sellero, C. (2025). Prediction intervals for quantile autoregression. arXiv:2512.22018. https://arxiv.org/abs/2512.22018

Examples

set.seed(123)

# Simulation parameters
burn_in <- 300
n <- 25
m <- burn_in + n
coeff <- 0.6

# Simulate AR(1) process
e <- rnorm(m)
x <- numeric(m)
x[1] <- rnorm(1)

for (t in 2:m) {
  x[t] <- coeff * x[t - 1] + e[t]
}

series <- ts(x[(burn_in + 1):m])

# Compute prediction interval
out <- pi_AR_proot(series)

out$lpi
out$upi
out$len

Prediction intervals based on the Box-Jenkins method

Description

Implements classical Box-Jenkins Gaussian prediction intervals for an AR(p) model fitted by ordinary least squares.

Usage

pi_BJ(series, p = 1, h = 3, alpha = 0.05)

Arguments

Details

This function implements the classical Box-Jenkins prediction intervals. The method fits an AR(p) model to the observed time series via ordinary least squares (OLS) and derives analytical prediction intervals under the assumption of Gaussian innovations.

Value

A list with elements:

pfor

Numeric vector of point forecasts (length h).

lpi

Numeric vector of lower bounds (length h).

upi

Numeric vector of upper bounds (length h).

len

Numeric vector of interval lengths (length h).

References

Box, G. E. and Jenkins, G. M. (1976). Time Series Analysis: Forecasting and Control. Holden–Day, San Francisco.

Examples

set.seed(123)

# Simulation parameters
burn_in <- 300
n <- 25
m <- burn_in + n
coeff <- 0.6

# Simulate AR(1) process
e <- rnorm(m)
x <- numeric(m)
x[1] <- rnorm(1)

for (t in 2:m) {
  x[t] <- coeff * x[t - 1] + e[t]
}

series <- ts(x[(burn_in + 1):m])

# Compute prediction interval
out <- pi_BJ(series,h=2)

out$lpi
out$upi
out$len

Prediction intervals based on the CB algorithm

Description

Computes bootstrap percentile-based prediction intervals using the conditional bootstrap algorithm of Cao et al. (1997) for an AR(p) model over a forecast horizon h.

Usage

pi_CB(
  series,
  p = 1,
  h = 3,
  B = 1000,
  alpha = 0.05,
  method = c("OLS", "LAD"),
  seed = NULL
)

Arguments

Details

This function implements the conditional bootstrap algorithm described in Cao et al. (1997).

Value

A list with elements:

bfor

Numeric matrix of bootstrap forecasts with dimension h x B.

lpi

Numeric vector of lower bounds (length h).

upi

Numeric vector of upper bounds (length h).

len

Numeric vector of interval lengths (length h).

References

Cao, R., Febrero-Bande, M., González-Manteiga, W., Prada-Sánchez, J., and García-Jurado, I. (1997). Saving computer time in constructing consistent bootstrap prediction intervals for autoregressive processes. Communications in Statistics-Simulation and Computation, 26(3):961–978.

Examples

set.seed(123)

# Simulation parameters
burn_in <- 300
n <- 25
m <- burn_in + n
coeff <- 0.6

# Simulate AR(1) process
e <- rnorm(m)
x <- numeric(m)
x[1] <- rnorm(1)

for (t in 2:m) {
  x[t] <- coeff * x[t - 1] + e[t]
}

series <- ts(x[(burn_in + 1):m])

# Compute prediction interval
out <- pi_CB(series,h=4,alpha=0.01)

out$lpi
out$upi
out$len

Prediction intervals based on PP algorithm

Description

Computes a bootstrap predictive-root-based prediction interval using the forward algorithm of Pan and Politis (2016) for an AR(p) model over a forecast horizon h.

Usage

pi_PP(series, p = 1, h = 3, B = 1000, alpha = 0.05, seed = NULL)

Arguments

Details

This function implements the PP algorithm of Pan and Politis (2016), referred to in their article as Fp (forward bootstrap with predictive residuals).

Value

A list with elements:

pfor

Numeric vector of point forecasts (length h).

lpi

Numeric vector of lower bounds (length h).

upi

Numeric vector of upper bounds (length h).

len

Numeric vector of interval lengths (length h).

References

Pan, L. and Politis, D. N. (2016). Bootstrap prediction intervals for linear, nonlinear and nonpara- metric autoregressions. Journal of Statistical Planning and Inference, 177:1–27.

Examples

set.seed(123)

# Simulation parameters
burn_in <- 300
n <- 25
m <- burn_in + n
coeff <- 0.6

# Simulate AR(1) process
e <- rnorm(m)
x <- numeric(m)
x[1] <- rnorm(1)

for (t in 2:m) {
  x[t] <- coeff * x[t - 1] + e[t]
}

series <- ts(x[(burn_in + 1):m])

# Compute prediction interval
out <- pi_PP(series,alpha=0.10)

out$lpi
out$upi
out$len

Prediction intervals based on the PRR algorithm

Description

Computes bootstrap percentile-based prediction intervals using the forward algorithm of Pascual, Romo and Ruiz (2004) for an AR(p) model over a forecast horizon h.

Usage

pi_PRR(
  series,
  p = 1,
  h = 3,
  B = 1000,
  alpha = 0.05,
  method = c("LAD", "OLS"),
  seed = NULL
)

Arguments

Details

This function implements the forward bootstrap algorithm described in Pascual, Romo and Ruiz (2004).

Value

A list with elements:

bfor

Numeric matrix of bootstrap forecasts with dimension h x B.

lpi

Numeric vector of lower bounds (length h).

upi

Numeric vector of upper bounds (length h).

len

Numeric vector of interval lengths (length h).

References

Pascual, L., Romo, J., and Ruiz, E. (2004). Bootstrap predictive inference for ARIMA processes. Journal of Time Series Analysis, 25(4):449–465.

Examples

set.seed(123)

# Simulation parameters
burn_in <- 300
n <- 25
m <- burn_in + n
coeff <- 0.6

# Simulate AR(1) process
e <- rnorm(m)
x <- numeric(m)
x[1] <- rnorm(1)

for (t in 2:m) {
  x[t] <- coeff * x[t - 1] + e[t]
}

series <- ts(x[(burn_in + 1):m])

# Compute prediction interval
out <- pi_PRR(series,h=4)

out$lpi
out$upi
out$len

Prediction intervals based on QAR-perc algorithm

Description

Computes bootstrap percentile-based prediction intervals using the QAR-perc algorithm for a QAR(p) model over a forecast horizon h.

Usage

pi_QAR_perc(series, p = 1, h = 3, B = 1000, alpha = 0.05, seed = NULL)

Arguments

Details

This function implements the QAR-perc algorithm described in Novo and Sánchez-Sellero (2025).

Value

A list with elements:

bfor

Numeric matrix of bootstrap forecasts with dimension h x B.

lpi

Numeric vector of lower bounds (length h).

upi

Numeric vector of upper bounds (length h).

len

Numeric vector of interval lengths (length h).

References

Novo, S., & Sánchez-Sellero, C. (2025). Prediction intervals for quantile autoregression. arXiv:2512.22018. https://arxiv.org/abs/2512.22018

Examples

set.seed(123)

# Simulation parameters
burn_in <- 300
n <- 25
m <- burn_in + n
coeff <- 0.6

# Simulate AR(1) process
e <- rnorm(m)
x <- numeric(m)
x[1] <- rnorm(1)

for (t in 2:m) {
  x[t] <- coeff * x[t - 1] + e[t]
}

series <- ts(x[(burn_in + 1):m])

# Compute prediction interval
out <- pi_QAR_perc(series)

out$lpi
out$upi
out$len

# Simulate QAR(1) process
n <- 100
e<-runif(m)
x <- numeric(m)
x[c(1,2)] <- rt(2,3)

for (t in 3:m) {x[t]<-0.3*x[t-1]+0.7*e[t]*x[t-2]+qt(e[t],3)}
series2<-ts(x[301:m])

# Compute prediction interval

out2<-pi_QAR_perc(series2,h=4)
out2$lpi
out2$upi
out2$len

Prediction intervals based on QAR-proot algorithm

Description

Computes a bootstrap predictive-root-based prediction interval using the QAR-proot algorithm for a QAR(p) model over a forecast horizon h.

Usage

pi_QAR_proot(
  series,
  p = 1,
  h = 3,
  B = 1000,
  alpha = 0.05,
  tau = 0.5,
  seed = NULL
)

Arguments

Details

This function implements the QAR-proot algorithm described in Novo and Sánchez-Sellero (2025).

Value

A list with elements:

pfor

Numeric vector of point forecasts (length h).

lpi

Numeric vector of lower bounds (length h).

upi

Numeric vector of upper bounds (length h).

len

Numeric vector of interval lengths (length h).

References

Novo, S., & Sánchez-Sellero, C. (2025). Prediction intervals for quantile autoregression. arXiv:2512.22018. https://arxiv.org/abs/2512.22018

Examples

set.seed(123)

# Simulation parameters
burn_in <- 300
n <- 25
m <- burn_in + n
coeff <- 0.6

# Simulate AR(1) process
e <- rnorm(m)
x <- numeric(m)
x[1] <- rnorm(1)

for (t in 2:m) {
  x[t] <- coeff * x[t - 1] + e[t]
}

series <- ts(x[(burn_in + 1):m])

# Compute prediction interval
out <- pi_QAR_proot(series)

out$lpi
out$upi
out$len

#' # Simulate QAR(1) process
n <- 100
e<-runif(m)
x <- numeric(m)
x[c(1,2)] <- rt(2,3)

for (t in 3:m) {x[t]<-0.3*x[t-1]+0.7*e[t]*x[t-2]+qt(e[t],3)}
series2<-ts(x[301:m])

# Compute prediction interval

out2<-pi_QAR_proot(series2,h=1)
out2$lpi
out2$upi
out2$len

Prediction intervals based on the TS algorithm

Description

Computes bootstrap percentile-based prediction intervals using the backward algorithm of Thombs and Schucany (1990) for an AR(p) model over a forecast horizon h.

Usage

pi_TS(series, p = 1, h = 3, B = 1000, alpha = 0.05, seed = NULL)

Arguments

Details

This function implements the backward bootstrap algorithm proposed by Thombs and Schucany (1990).

Value

A list with elements:

bfor

Numeric matrix of bootstrap forecasts with dimension h x B.

lpi

Numeric vector of lower bounds (length h).

upi

Numeric vector of upper bounds (length h).

len

Numeric vector of interval lengths (length h).

References

Thombs, L. A. and Schucany, W. R. (1990). Bootstrap prediction intervals for autoregression. Journal of the American Statistical Association, 85(410):486–492.

Examples

set.seed(123)

# Simulation parameters
burn_in <- 300
n <- 25
m <- burn_in + n
coeff <- 0.6

# Simulate AR(1) process
e <- rnorm(m)
x <- numeric(m)
x[1] <- rnorm(1)

for (t in 2:m) {
  x[t] <- coeff * x[t - 1] + e[t]
}

series <- ts(x[(burn_in + 1):m])

# Compute prediction interval
out <- pi_TS(series,h=4)

out$lpi
out$upi
out$len

Prediction intervals based on X algorithm

Description

Computes a bootstrap percentile-based prediction interval using the algorithm suggested by Xiao (2012) for a QAR(p) model over a forecast horizon h.

Usage

pi_X(series, p = 1, h = 3, B = 1000, alpha = 0.05, seed = NULL)

Arguments

Details

This function implements the X algorithm suggested by Xiao (2012).

Value

A list with elements:

bfor

Numeric matrix of bootstrap forecasts with dimension h x B.

lpi

Numeric vector of lower bounds (length h).

upi

Numeric vector of upper bounds (length h).

len

Numeric vector of interval lengths (length h).

References

Xiao, Z. (2012). Time series quantile regressions. Handbook of Statistics. Time Series Analysis: Methods and Applications, 30:213–257.

Examples

set.seed(123)

# Simulation parameters
burn_in <- 300
n <- 25
m <- burn_in + n
coeff <- 0.6

# Simulate AR(1) process
e <- rnorm(m)
x <- numeric(m)
x[1] <- rnorm(1)

for (t in 2:m) {
  x[t] <- coeff * x[t - 1] + e[t]
}

series <- ts(x[(burn_in + 1):m])

# Compute prediction interval
out <- pi_X(series)

out$lpi
out$upi
out$len

# Simulate QAR(1) process
n <- 100
e<-runif(m)
x <- numeric(m)
x[c(1,2)] <- rt(2,3)

for (t in 3:m) {x[t]<-0.3*x[t-1]+0.7*e[t]*x[t-2]+qt(e[t],3)}
series2<-ts(x[301:m])

# Compute prediction interval

out2<-pi_X(series2,h=4)
out2$lpi
out2$upi
out2$len