exuber: Econometric Analysis of Explosive Time Series

Description

Testing for and dating periods of explosive dynamics (exuberance) in time series using the univariate and panel recursive unit root tests proposed by Phillips et al. (2015) doi:10.1111/iere.12132 and Pavlidis et al. (2016) doi:10.1007/s11146-015-9531-2.The recursive least-squares algorithm utilizes the matrix inversion lemma to avoid matrix inversion which results in significant speed improvements. Simulation of a variety of periodically-collapsing bubble processes. Details can be found in Vasilopoulos et al. (2022) doi:10.18637/jss.v103.i10.

Package options

exuber.show_progress

• Should lengthy operations such as radf_mc_cv() show a progress bar? Default: TRUE

exuber.parallel

• Should lengthy operations use parallel computation? Default: TRUE

exuber.ncores

• How many cores to use for parallel computation. Default: system cores - 1

exuber.global_seed

• When chosen automatically feeds into all functions with random-number generation. Default: NA

Author(s)

Maintainer: Kostas Vasilopoulos k.vasilopoulo@gmail.com

Authors:

• Efthymios Pavlidis e.pavlidis@lancaster.ac.uk

• Enrique Martínez-García emg.economics@gmail.com

• Simon Spavound simon.spavound@googlemail.com

See Also

Useful links:

• https://github.com/kvasilopoulos/exuber

• Report bugs at https://github.com/kvasilopoulos/exuber/issues

Pipe operator

Description

Pipe operator

Usage

lhs %>% rhs

Plotting a ds_radf object

Description

Takes a ds_radf object and returns a ggplot2 object, with a geom_segment() layer.

Usage

## S3 method for class 'ds_radf'
autoplot(object, trunc = TRUE, ...)

Arguments

Value

A ggplot2::ggplot()

Examples

sim_data_wdate %>%
  radf() %>%
  datestamp() %>%
  autoplot()

# Change the colour manually
sim_data_wdate %>%
  radf() %>%
  datestamp() %>%
  autoplot() +
  ggplot2::scale_colour_manual(values = rep("black", 4))

Plotting a radf_distr object

Description

Takes a radf_distr object and returns a ggplot2 object.

Usage

## S3 method for class 'radf_distr'
autoplot(object, ...)

Arguments

Value

A ggplot2::ggplot()

Plotting radf models

Description

autoplot.radf_obj takes radf_obj and radf_cv and returns a faceted ggplot object. shade is used as an input to shape_opt. shade modifies the geom_rect layer that demarcates the exuberance periods.

Usage

## S3 method for class 'radf_obj'
autoplot(
  object,
  cv = NULL,
  sig_lvl = 95,
  option = c("gsadf", "sadf"),
  min_duration = 0L,
  select_series = NULL,
  nonrejected = FALSE,
  shade_opt = shade(),
  trunc = TRUE,
  include_negative = "DEPRECATED",
  ...
)

## S3 method for class 'radf_obj'
autoplot2(
  object,
  cv = NULL,
  sig_lvl = 95,
  option = c("gsadf", "sadf"),
  min_duration = 0L,
  select_series = NULL,
  nonrejected = FALSE,
  trunc = TRUE,
  shade_opt = shade(),
  ...
)

shade(
  fill = "grey55",
  fill_negative = fill,
  fill_ongoing = NULL,
  opacity = 0.3,
  ...
)

Arguments

Value

A ggplot2::ggplot()

Examples

rsim_data <- radf(sim_data_wdate)

autoplot(rsim_data)

# Modify facet_wrap options through ellipsis
autoplot(rsim_data, scales = "free_y", dir  = "v")

# Modify the shading options
autoplot(rsim_data, shade_opt = shade(fill = "pink", opacity = 0.5))

# Allow for nonrejected series to be plotted
autoplot(rsim_data, nonrejected = TRUE)

# Remove the shading completely (2 ways)
autoplot(rsim_data, shade_opt = NULL)
autoplot(rsim_data, shade_opt = shade(opacity = 0))

# Plot only the series with the shading options
autoplot2(rsim_data)
autoplot2(rsim_data, trunc = FALSE) # keep the minw period

# We will need ggplot2 from here on out
library(ggplot2)

# Change (overwrite) color, size or linetype
autoplot(rsim_data) +
  scale_color_manual(values = c("black", "black")) +
  scale_size_manual(values = c(0.9, 1)) +
  scale_linetype_manual(values = c("solid", "solid"))

# Change names through labeller (first way)
custom_labels <- c("psy1" = "new_name_for_psy1", "psy2" = "new_name_for_psy2")
autoplot(rsim_data, labeller = labeller(.default = label_value, id = as_labeller(custom_labels)))

# Change names through labeller (second way)
custom_labels2 <- series_names(rsim_data)
names(custom_labels2) <- custom_labels2
custom_labels2[c(3,5)] <- c("Evans", "Blanchard")
autoplot(rsim_data, labeller = labeller(id = custom_labels2))

# Or change names before plotting
series_names(rsim_data) <- LETTERS[1:5]
autoplot(rsim_data)

# Change Theme options
autoplot(rsim_data) +
  theme(legend.position = "right")
 

Create a complete ggplot appropriate to a particular data type

Description

autoplot2() uses ggplot2 to draw a particular plot for an object of a particular class in a single command. This defines the S3 generic that other classes and packages can extend.

Usage

autoplot2(object, ...)

Arguments

See Also

autoplot()

Calculate p-values

Description

Calculate p-values from distr object

Usage

calc_pvalue(x, distr = NULL)

Arguments

Examples

## Not run: 
radf_psy1 <- radf(sim_psy1(100))

calc_pvalue(radf_psy1)

# Using the Wild-Bootstrapped
wb_psy1 <- radf_wb_distr(sim_psy1(100))

calc_pvalue(radf_psy1, wb_psy1)

sb_psy1 <- radf_sb_distr(sim_data)

calc_pvalue(radf(sim_data), sb_psy1)


## End(Not run)

Date-stamping periods of mildly explosive behavior

Description

Computes the origination, termination and duration of episodes during which the time series display explosive dynamics.

Usage

datestamp(object, cv = NULL, min_duration = 0L, ...)

## S3 method for class 'radf_obj'
datestamp(
  object,
  cv = NULL,
  min_duration = 0L,
  sig_lvl = 95,
  option = c("gsadf", "sadf"),
  nonrejected = FALSE,
  ...
)

Arguments

Details

Datestamp also stores a vector whose elements take the value of 1 when there is a period of explosive behaviour and 0 otherwise. This output can serve as a dummy variable for the occurrence of exuberance.

Value

Return a table with the following columns:

• Start:

• Peak:

• End:

• Duration:

• Signal:

• Ongoing:

Returns a list containing the estimated origination and termination dates of episodes of explosive behaviour and the corresponding duration.

References

Phillips, P. C. B., Shi, S., & Yu, J. (2015). Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500. International Economic Review, 56(4), 1043-1078.

Examples

rsim_data <- radf(sim_data)

ds_data <- datestamp(rsim_data)
ds_data

# Choose minimum window
datestamp(rsim_data, min_duration = psy_ds(nrow(sim_data)))

autoplot(ds_data)

Diagnostics on hypothesis testing

Description

Provides information on whether the null hypothesis of a unit root is rejected against the alternative of explosive behaviour for each series in a dataset.

Usage

diagnostics(object, cv = NULL, ...)

## S3 method for class 'radf_obj'
diagnostics(object, cv = NULL, option = c("gsadf", "sadf"), ...)

Arguments

Details

Diagnostics also stores a vector whose elements take the value of 1 when there is a period of explosive behaviour and 0 otherwise.

Value

Returns a list with the series that reject (positive) and the series that do not reject (negative) the null hypothesis, and at what significance level.

Examples

rsim_data <- radf(sim_data)
diagnostics(rsim_data)

diagnostics(rsim_data, option = "sadf")

Defunct functions in package exuber.

Description

The functions listed below are defunct. Help pages for defunct functions are available at help("exuber-defunct").

Usage

ggarrange(...)

## S3 method for class 'radf_obj'
fortify(model, data, ...)

## S3 method for class 'ds_radf'
fortify(model, data, ...)

report(x, y, panel = FALSE, ...)

sim_dgp1(
  n,
  te = 0.4 * n,
  tf = 0.15 * n + te,
  c = 1,
  alpha = 0.6,
  sigma = 6.79,
  seed = NULL
)

sim_dgp2(
  n,
  te1 = 0.2 * n,
  tf1 = 0.2 * n + te1,
  te2 = 0.6 * n,
  tf2 = 0.1 * n + te2,
  c = 1,
  alpha = 0.6,
  sigma = 6.79,
  seed = NULL
)

Deprecated functions in package exuber.

Description

The functions listed below are deprecated and will be defunct in the near future. When possible, alternative functions with similar functionality are also mentioned. Help pages for deprecated functions are available at help("exuber-deprecated").

Usage

col_names(x)

mc_cv(n, minw = NULL, nrep = 1000L, seed = NULL)

wb_cv(data, minw = NULL, nboot = 1000L, seed = NULL)

sb_cv(data, minw = NULL, nboot = 1000L, seed = NULL)

Retrieve/Replace the index

Description

Retrieve or replace the index of an object.

Usage

index(x, ...)

index(x) <- value

Arguments

Details

If the user does not specify an index for the estimation a pseudo-index is generated which is a sequential numeric series. After the estimation, the user can use index to retrieve or `index<-` to replace the index. The index can be either numeric or Date.

Install exuberdata Package

Description

This function wraps the install.packages function and offers a faster and more convenient way to install exuberdata.

Usage

install_exuberdata()

Examples

if("exuberdata" %in% loadedNamespaces()) {
 exuberdata::radf_crit2
}

Helper function to find tb from the Phillips and Shi (2020)

Description

This function helps to find the number of observations in the window over which size is to be controlled.

Usage

ps_tb(n, freq = c("monthly", "quarterly", "annual", "weekly"), size = 2)

Arguments

References

Phillips, P. C., & Shi, S. (2020). Real time monitoring of asset markets: Bubbles and crises. In Handbook of Statistics (Vol. 42, pp. 61-80). Elsevier.

Shi, S., Hurn, S., Phillips, P.C.B., 2018. Causal change detection in possibly integrated systems: Revisiting the money-income relationship.

Helper functions in accordance to PSY(2015)

Description

psy_minw and psy_ds use the rules-of- thumb proposed by Phillips et al. (2015) to compute the minimum window size and the minimum duration of an episode of exuberance, respectively.

Usage

psy_minw(n)

psy_ds(n, rule = 1, delta = 1)

Arguments

Details

For the minimum duration period, psy_ds allows the user to choose from two rules:

rule_1 = \delta \log(n) \quad\& \quad rule_2 = \delta \log(n)/n

delta depends on the frequency of the data and the minimal duration condition.

References

Phillips, P. C. B., Shi, S., & Yu, J. (2015). Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500. International Economic Review, 56(4), 1043-1078.

Examples

psy_minw(100)
psy_ds(100)

Recursive Augmented Dickey-Fuller Test

Description

radf returns the recursive univariate and panel Augmented Dickey-Fuller test statistics.

Usage

radf(data, minw = NULL, lag = 0L)

Arguments

Details

The radf() function is vectorized, i.e., it can handle multiple series at once, to improve efficiency. This property also enables the computation of panel statistics internally as a by-product of the univariate estimations with minimal additional cost incurred.

Value

A list that contains the unit root test statistics (sequence):

And attributes:

References

Phillips, P. C. B., Wu, Y., & Yu, J. (2011). Explosive Behavior in The 1990s Nasdaq: When Did Exuberance Escalate Asset Values? International Economic Review, 52(1), 201-226.

Phillips, P. C. B., Shi, S., & Yu, J. (2015). Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500. International Economic Review, 56(4), 1043-1078.

Pavlidis, E., Yusupova, A., Paya, I., Peel, D., Martínez-García, E., Mack, A., & Grossman, V. (2016). Episodes of exuberance in housing markets: in search of the smoking gun. The Journal of Real Estate Finance and Economics, 53(4), 419-449.

Examples

# We will use simulated data that are stored as data
sim_data

rsim <- radf(sim_data)

str(rsim)

# We would also use data that contain a Date column
sim_data_wdate

rsim_wdate <- radf(sim_data_wdate)

tidy(rsim_wdate)

augment(rsim_wdate)

tidy(rsim_wdate, panel = TRUE)

head(index(rsim_wdate))

# For lag = 1 and minimum window = 20
rsim_20 <- radf(sim_data, minw = 20, lag = 1)

Stored Monte Carlo Critical Values

Description

A dataset containing Monte Carlo critical values for up to 600 observations generated using the default minimum window. The critical values have been simulated and stored as data to save computation time for the user. The stored critical values can be obtained with the radf_mc_cv() function, using nrep = 2000 and the seed = 123.

Usage

radf_crit

Format

A list with lower level lists that contain

adf_cv:

Augmented Dickey-Fuller

badf_cv:

Backward Augmented Dickey-Fuller

sadf_cv:

Supremum Augmented Dickey-Fuller

bsadf_cv:

Backward Supremum Augmented Dickey-Fuller

gsadf_cv:

Generalized Supremum Augmented Dickey Fuller

Source

Simulated from exuber package function radf_mc_cv().

Examples

## Not run: 
all.equal(radf_crit[[50]], radf_mc_cv(50, nrep = 2000, seed = 123))

## End(Not run)

Monte Carlo Critical Values

Description

radf_mc_cv computes Monte Carlo critical values for the recursive unit root tests. radf_mc_distr computes the distribution.

Usage

radf_mc_cv(n, minw = NULL, nrep = 1000L, seed = NULL)

radf_mc_distr(n, minw = NULL, nrep = 1000L, seed = NULL)

Arguments

Value

For radf_mc_cv a list that contains the critical values for ADF, BADF, BSADF and GSADF test statistics. For radf_mc_distr a list that contains the ADF, SADF and GSADF distributions.

See Also

radf_wb_cv for wild bootstrap critical values and radf_sb_cv for sieve bootstrap critical values

Examples

# Default minimum window
mc <- radf_mc_cv(n = 100)

tidy(mc)

# Change the minimum window and the number of simulations
mc2 <- radf_mc_cv(n = 100, nrep = 600, minw = 20)

tidy(mc2)

mdist <- radf_mc_distr(n = 100, nrep = 1000)

autoplot(mdist)

Panel Sieve Bootstrap Critical Values

Description

radf_sb_cv computes critical values for the panel recursive unit root test using the sieve bootstrap procedure outlined in Pavlidis et al. (2016). radf_sb_distr computes the distribution.

Usage

radf_sb_cv(data, minw = NULL, lag = 0L, nboot = 500L, seed = NULL)

radf_sb_distr(data, minw = NULL, lag = 0L, nboot = 500L, seed = NULL)

Arguments

Value

For radf_sb_cv A list A list that contains the critical values for the panel BSADF and panel GSADF test statistics. For radf_wb_dist a numeric vector that contains the distribution of the panel GSADF statistic.

References

Pavlidis, E., Yusupova, A., Paya, I., Peel, D., Martínez-García, E., Mack, A., & Grossman, V. (2016). Episodes of exuberance in housing markets: In search of the smoking gun. The Journal of Real Estate Finance and Economics, 53(4), 419-449.

See Also

radf_mc_cv for Monte Carlo critical values and radf_wb_cv for wild Bootstrap critical values

Examples

rsim_data <- radf(sim_data, lag = 1)

# Critical vales should have the same lag length with \code{radf()}
sb <- radf_sb_cv(sim_data, lag = 1)

tidy(sb)

summary(rsim_data, cv = sb)

autoplot(rsim_data, cv = sb)

# Simulate distribution
sdist <- radf_sb_distr(sim_data, lag = 1, nboot = 1000)

autoplot(sdist)

Wild Bootstrap Critical Values

Description

radf_wb_cv performs the Harvey et al. (2016) wild bootstrap re-sampling scheme, which is asymptotically robust to non-stationary volatility, to generate critical values for the recursive unit root tests. radf_wb_distr computes the distribution.

Usage

radf_wb_cv(data, minw = NULL, nboot = 500L, dist_rad = FALSE, seed = NULL)

radf_wb_distr(data, minw = NULL, nboot = 500L, dist_rad = FALSE, seed = NULL)

Arguments

Details

This approach involves applying a wild bootstrap re-sampling scheme to construct the bootstrap analogue of the Phillips et al. (2015) test which is asymptotically robust to non-stationary volatility.

Value

For radf_wb_cv a list that contains the critical values for the ADF, BADF, BSADF and GSADF tests. For radf_wb_distr a list that contains the ADF, SADF and GSADF distributions.

References

Harvey, D. I., Leybourne, S. J., Sollis, R., & Taylor, A. M. R. (2016). Tests for explosive financial bubbles in the presence of non-stationary volatility. Journal of Empirical Finance, 38(Part B), 548-574.

Phillips, P. C. B., Shi, S., & Yu, J. (2015). Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500. International Economic Review, 56(4), 1043-1078.

See Also

radf_mc_cv for Monte Carlo critical values and radf_sb_cv for sieve bootstrap critical values.

Examples

# Default minimum window
wb <- radf_wb_cv(sim_data)

tidy(wb)

# Change the minimum window and the number of bootstraps
wb2 <- radf_wb_cv(sim_data, nboot = 600, minw = 20)

tidy(wb2)

# Simulate distribution
wdist <- radf_wb_distr(sim_data)

autoplot(wdist)

Wild Bootstrap Critical Values

Description

radf_wb_cv performs the Phillips & Shi (2020) wild bootstrap re-sampling scheme, which is asymptotically robust to non-stationary volatility, to generate critical values for the recursive unit root tests. radf_wb_distr2 computes the distribution.

Usage

radf_wb_cv2(
  data,
  minw = NULL,
  nboot = 500L,
  adflag = 0,
  type = c("fixed", "aic", "bic"),
  tb = NULL,
  seed = NULL
)

radf_wb_distr2(
  data,
  minw = NULL,
  nboot = 500L,
  adflag = 0,
  type = c("fixed", "aic", "bic"),
  tb = NULL,
  seed = NULL
)

Arguments

Value

For radf_wb_cv2 a list that contains the critical values for the ADF, BADF, BSADF and GSADF tests. For radf_wb_distr a list that contains the ADF, SADF and GSADF distributions.

References

Phillips, P. C., & Shi, S. (2020). Real time monitoring of asset markets: Bubbles and crises. In Handbook of Statistics (Vol. 42, pp. 61-80). Elsevier.

Phillips, P. C. B., Shi, S., & Yu, J. (2015). Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500. International Economic Review, 56(4), 1043-1078.

See Also

radf_mc_cv for Monte Carlo critical values and radf_sb_cv for sieve bootstrap critical values.

Examples

# Default minimum window
wb <- radf_wb_cv2(sim_data)

tidy(wb)

# Change the minimum window and the number of bootstraps
wb2 <- radf_wb_cv2(sim_data, nboot = 600, minw = 20)

tidy(wb2)

# Simulate distribution
wdist <- radf_wb_distr(sim_data)

autoplot(wdist)

Objects exported from other packages

Description

These objects are imported from other packages. Follow the links below to see their documentation.

generics

augment, tidy

ggplot2

autoplot, fortify

See Also

fortify.radf_obj() fortify.ds_radf()

autoplot.radf_obj() autoplot.ds_radf()

tidy.radf_obj() tidy.radf_cv() tidy.radf_distr()

augment.radf_obj() augment.radf_cv()

Exuber scale and theme functions

Description

scale_exuber_manual allows specifying the color, size and linetype in autoplot.radf_obj mappings. theme_exuber is a complete theme which control all non-data display.

Usage

scale_exuber_manual(
  color_values = c("red", "blue"),
  linetype_values = c(2, 1),
  size_values = c(0.8, 0.7)
)

theme_exuber(
  base_size = 11,
  base_family = "",
  base_line_size = base_size/22,
  base_rect_size = base_size/22
)

Arguments

Retrieve/Replace series names

Description

Retrieve or replace the series names of an object.

Usage

series_names(x, ...)

series_names(x) <- value

## S3 replacement method for class 'radf_obj'
series_names(x) <- value

## S3 replacement method for class 'wb_cv'
series_names(x) <- value

## S3 replacement method for class 'sb_cv'
series_names(x) <- value

Arguments

Examples

# Simulate bubble processes
dta <- data.frame(psy1 = sim_psy1(n = 100), psy2 = sim_psy2(n = 100))

rfd <- radf(dta)

series_names(rfd) <- c("OneBubble", "TwoBubbles")

Simulation of a Blanchard (1979) bubble process

Description

Simulation of a Blanchard (1979) rational bubble process.

Usage

sim_blan(n, pi = 0.7, sigma = 0.03, r = 0.05, b0 = 0.1, seed = NULL)

Arguments

Details

Blanchard's bubble process has two regimes, which occur with probability \pi and 1-\pi. In the first regime, the bubble grows exponentially, whereas in the second regime, the bubble collapses to a white noise.

With probability \pi:

B_{t+1} = \frac{1+r}{\pi}B_t+\epsilon_{t+1}

With probability 1 - \pi:

B_{t+1} = \epsilon_{t+1}

where r is a positive constant and \epsilon \sim iid(0, \sigma^2).

Value

A numeric vector of length n.

References

Blanchard, O. J. (1979). Speculative bubbles, crashes and rational expectations. Economics letters, 3(4), 387-389.

See Also

sim_psy1, sim_psy2, sim_evans

Examples

sim_blan(n = 100, seed = 123) %>%
  autoplot()

Simulated dataset

Description

An artificial dataset containing series simulated from data generating processes widely used in the literature on speculative bubbles.

Usage

sim_data

sim_data_wdate

Format

An object of class tbl_df (inherits from tbl, data.frame) with 100 rows and 5 columns.

An object of class tbl_df (inherits from tbl, data.frame) with 100 rows and 6 columns.

See Also

sim_psy1 sim_psy1 sim_evans sim_div sim_blan

Examples

## Not run: 
# The dataset can be easily replicated with the code below
library(tibble)
set.seed(1122)
sim_data <- tibble(
  sim_psy1 = sim_psy1(100),
  sim_psy2 = sim_psy2(100),
  sim_evans = sim_evans(100),
  sim_div = sim_div(100),
  sim_blan = sim_blan(100)
)
sim_data_wdate <- tibble(
  psy1 = sim_psy1(100),
  psy2 = sim_psy2(100),
  evans = sim_evans(100),
  div = sim_div(100),
  blan = sim_blan(100),
  date = seq(as.Date("2000-01-01"), by = "month", length.out = 100)
)

## End(Not run)

Simulation of dividends

Description

Simulate (log) dividends from a random walk with drift.

Usage

sim_div(
  n,
  mu,
  sigma,
  r = 0.05,
  log = FALSE,
  output = c("pf", "d"),
  seed = NULL
)

Arguments

Details

If log is set to FALSE (default value) dividends follow:

d_t = \mu + d_{t-1} + \epsilon_t

where \epsilon \sim \mathcal{N}(0, \sigma^2). The default parameters are \mu = 0.0373, \sigma^2 = 0.1574 and d[0] = 1.3 (the initial value of the dividend sequence). The above equation can be solved to yield the fundamental price:

F_t = \mu(1+r)r^{-2} + r^{-1}d_t

If log is set to TRUE then dividends follow a lognormal distribution or log(dividends) follow:

\ln(d_t) = \mu + \ln(d_{t-1}) + \epsilon_t

where \epsilon \sim \mathcal{N}(0, \sigma^2). Default parameters are \mu = 0.013, \sigma^2 = 0.16. The fundamental price in this case is:

F_t = \frac{1+g}{r-g}d_t

where 1+g=\exp(\mu+\sigma^2/2). All default parameter values are those suggested by West (1988).

Value

A numeric vector of length n.

References

West, K. D. (1988). Dividend innovations and stock price volatility. Econometrica: Journal of the Econometric Society, p. 37-61.

Examples

# Price is the sum of the bubble and fundamental components
# 20 is the scaling factor
pf <- sim_div(100, r = 0.05, output = "pf", seed = 123)
pb <- sim_evans(100, r = 0.05, seed = 123)
p <- pf + 20 * pb

autoplot(p)

Simulation of an Evans (1991) bubble process

Description

Simulation of an Evans (1991) rational periodically collapsing bubble process.

Usage

sim_evans(
  n,
  alpha = 1,
  delta = 0.5,
  tau = 0.05,
  pi = 0.7,
  r = 0.05,
  b1 = delta,
  seed = NULL
)

Arguments

Details

delta and alpha are positive parameters which satisfy 0 < \delta < (1+r)\alpha. delta represents the size of the bubble after collapse. The default value of r is 0.05. The function checks whether alpha and delta satisfy this condition and will return an error if not.

The Evans bubble has two regimes. If B_t \leq \alpha the bubble grows at an average rate of 1 + r:

B_{t+1} = (1+r) B_t u_{t+1},

When B_t > \alpha the bubble expands at the increased rate of (1+r)\pi^{-1}:

B_{t+1} = [\delta + (1+r)\pi^{-1} \theta_{t+1}(B_t - (1+r)^{-1}\delta B_t )]u_{t+1},

where \theta theta is a binary variable that takes the value 0 with probability 1-\pi and 1 with probability \pi. In the second phase, there is a (1-\pi) probability of the bubble process collapsing to delta. By modifying the values of delta, alpha and pi the user can change the frequency at which bubbles appear, the mean duration of a bubble before collapse and the scale of the bubble.

Value

A numeric vector of length n.

References

Evans, G. W. (1991). Pitfalls in testing for explosive bubbles in asset prices. The American Economic Review, 81(4), 922-930.

See Also

sim_psy1, sim_psy2, sim_blan

Examples

sim_evans(100, seed = 123) %>%
  autoplot()

Simulation of a single-bubble process with multiple forms of collapse regime

Description

The new generating process considered here differs from the sim_psy1 model in three respects - Phillips and Shi (2018):

First, it includes an asymptotically negligible drift in the martingale path during normal periods. Second, the collapse process is modeled directly as a transient mildly integrated process that covers an explicit period of market collapse. Third, a market recovery date is introduced to capture the return to normal market behavior.

• ⁠sudden:⁠ with beta = 0.1 and tr = tf + 0.01*n

• ⁠disturbing:⁠ with beta = 0.5 and tr = tf + 0.1*n

• ⁠smooth:⁠ with beta = 0.9 and tr = tf + 0.2*n

In order to provide the duration of the collapse period tr as ⁠tr = tf + 0.2n⁠, you have to provide tf as well.

Usage

sim_ps1(
  n,
  te = 0.4 * n,
  tf = te + 0.2 * n,
  tr = tf + 0.1 * n,
  c = 1,
  c1 = 1,
  c2 = 1,
  eta = 0.6,
  alpha = 0.6,
  beta = 0.5,
  sigma = 6.79,
  seed = NULL
)

Arguments

Value

A numeric vector of length n.

References

Phillips, Peter CB, and Shu-Ping Shi. "Financial bubble implosion and reverse regression." Econometric Theory 34.4 (2018): 705-753.

See Also

sim_psy1

Examples

# Disturbing collapse (default)
disturbing <- sim_ps1(100)
autoplot(disturbing)

# Sudden collapse
sudden <- sim_ps1(100, te = 40, tf= 60, tr = 61, beta = 0.1)
autoplot(sudden)

Simulation of a single-bubble process

Description

The following function generates a time series which switches from a martingale to a mildly explosive process and then back to a martingale.

Usage

sim_psy1(
  n,
  te = 0.4 * n,
  tf = 0.15 * n + te,
  c = 1,
  alpha = 0.6,
  sigma = 6.79,
  seed = NULL
)

Arguments

Details

The data generating process is described by the following equation:

X_t = X_{t-1}1\{t < \tau_e\}+ \delta_T X_{t-1}1\{\tau_e \leq t\leq \tau_f\} + \left(\sum_{k=\tau_f+1}^t \epsilon_k + X_{\tau_f}\right) 1\{t > \tau_f\} + \epsilon_t 1\{t \leq \tau_f\}

where the autoregressive coefficient \delta_T is given by:

\delta_T = 1 + cT^{-a}

with c>0, \alpha \in (0,1), \epsilon \sim iid(0, \sigma^2) and X_{\tau_f} = X_{\tau_e} + X' with X' = O_p(1), \tau_e = [T r_e] dates the origination of the bubble, and \tau_f = [T r_f] dates the collapse of the bubble. During the pre- and post- bubble periods, [1, \tau_e), X_t is a pure random walk process. During the bubble expansion period \tau_e, \tau_f] becomes a mildly explosive process with expansion rate given by the autoregressive coefficient \delta_T; and, finally during the post-bubble period, (\tau_f, \tau] X_t reverts to a martingale.

For further details see Phillips et al. (2015) p. 1054.

Value

A numeric vector of length n.

References

Phillips, P. C. B., Shi, S., & Yu, J. (2015). Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500. International Economic Review, 5 6(4), 1043-1078.

See Also

sim_psy2, sim_blan, sim_evans

Examples

# 100 periods with bubble origination date 40 and termination date 55
sim_psy1(n = 100, seed = 123) %>%
  autoplot()

# 200 periods with bubble origination date 80 and termination date 110
sim_psy1(n = 200, seed = 123) %>%
  autoplot()

# 200 periods with bubble origination date 100 and termination date 150
sim_psy1(n = 200, te = 100, tf = 150, seed = 123) %>%
  autoplot()

Simulation of a two-bubble process

Description

The following data generating process is similar to sim_psy1, with the difference that there are two episodes of mildly explosive dynamics.

Usage

sim_psy2(
  n,
  te1 = 0.2 * n,
  tf1 = 0.2 * n + te1,
  te2 = 0.6 * n,
  tf2 = 0.1 * n + te2,
  c = 1,
  alpha = 0.6,
  sigma = 6.79,
  seed = NULL
)

Arguments

Details

The two-bubble data generating process is given by (see also sim_psy1):

X_t = X_{t-1}1\{t \in N_0\}+ \delta_T X_{t-1}1\{t \in B_1 \cup B_2\} + \left(\sum_{k=\tau_{1f}+1}^t \epsilon_k + X_{\tau_{1f}}\right) 1\{t \in N_1\}

+ \left(\sum_{l=\tau_{2f}+1}^t \epsilon_l + X_{\tau_{2f}}\right) 1\{t \in N_2\} + \epsilon_t 1\{t \in N_0 \cup B_1 \cup B_2\}

where the autoregressive coefficient \delta_T is:

\delta_T = 1 + cT^{-a}

with c>0, \alpha \in (0,1), \epsilon \sim iid(0, \sigma^2), N_0 = [1, \tau_{1e}), B_1 = [\tau_{1e}, \tau_{1f}], N_1 = (\tau_{1f}, \tau_{2e}), B_2 = [\tau_{2e}, \tau_{2f}], N_2 = (\tau_{2f}, \tau], where \tau is the last observation of the sample. The observations \tau_{1e} = [T r_{1e}] and \tau_{1f} = [T r_{1f}] are the origination and termination dates of the first bubble; \tau_{2e} = [T r_{2e}] and \tau_{2f} = [T r_{2f}] are the origination and termination dates of the second bubble. After the collapse of the first bubble, X_t resumes a martingale path until time \tau_{2e}-1, and a second episode of exuberance begins at \tau_{2e}. Exuberance lasts lasts until \tau_{2f} at which point the process collapses to a value of X_{\tau_{2f}}. The process then continues on a martingale path until the end of the sample period \tau. The duration of the first bubble is assumed to be longer than that of the second bubble, i.e. \tau_{1f}-\tau_{1e}>\tau_{2f}-\tau_{2e}.

For further details you can refer to Phillips et al., (2015) p. 1055.

Value

A numeric vector of length n.

References

Phillips, P. C. B., Shi, S., & Yu, J. (2015). Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500. International Economic Review, 5 6(4), 1043-1078.

See Also

sim_psy1, sim_blan, sim_evans

Examples

# 100 periods with bubble origination dates 20/60 and termination dates 40/70
sim_psy2(n = 100, seed = 123) %>%
 autoplot()

# 200 periods with bubble origination dates 40/120 and termination dates 80/140
sim_psy2(n = 200, seed = 123) %>%
  autoplot()

Summarizing radf models

Description

summary method for radf models that consist of radf_obj and radf_cv.

Usage

## S3 method for class 'radf_obj'
summary(object, cv = NULL, ...)

Arguments

Value

Returns a list of summary statistics, which include the estimated ADF, SADF, and GSADF test statistics and the corresponding critical values

Examples

# Simulate bubble processes, compute the test statistics and critical values
rsim_data <- radf(sim_data)

# Summary, diagnostics and datestamp (default)
summary(rsim_data)

#Summary, diagnostics and datestamp (wild bootstrap critical values)

wb <- radf_wb_cv(sim_data)

summary(rsim_data, cv = wb)

Tidy a ds_radf object

Description

Summarizes information about ds_radf object.

Usage

## S3 method for class 'ds_radf'
tidy(x, ...)

Arguments

Tidy a radf_cv object

Description

Summarizes information about radf_cv object.

Usage

## S3 method for class 'radf_cv'
tidy(x, format = c("wide", "long"), ...)

## S3 method for class 'radf_cv'
augment(x, format = c("wide", "long"), trunc = TRUE, ...)

Arguments

Value

A tibble::tibble()

• id: The series names.

• sig: The significance level.

• name: The name of the series (when format is "long").

• crit: The critical value (when format is "long").

Examples

mc <- radf_mc_cv(100)

# Get the critical values
tidy(mc)

# Get the critical value sequences
augment(mc)

Tidy a radf_distr object

Description

Summarizes information about radf_distr object.

Usage

## S3 method for class 'radf_distr'
tidy(x, ...)

Arguments

Value

A tibble::tibble()

Examples

## Not run: 
mc <- mc_cv(n = 100)

tidy(mc)

## End(Not run)

Tidy a radf_obj object

Description

Summarizes information about radf_obj object.

Usage

## S3 method for class 'radf_obj'
tidy(x, format = c("wide", "long"), panel = FALSE, ...)

## S3 method for class 'radf_obj'
augment(x, format = c("wide", "long"), panel = FALSE, trunc = TRUE, ...)

Arguments

Value

A tibble::tibble()

Examples

dta <- data.frame(psy1 = sim_psy1(n = 100), psy2 = sim_psy2(n = 100))

rfd <- radf(dta)

# Get the test statistic
tidy(rfd)

# Get the test statisticsequences
augment(rfd)

# Get the panel test statistic
tidy(rfd, panel = TRUE)

Tidy into a joint model

Description

Tidy or augment and then join objects.

Usage

tidy_join(x, y, ...)

augment_join(x, y, ...)

Arguments

Tidy into a joint model

Description

Tidy or augment and then join objects of class radf_obj and radf_cv. The object of reference is the radf_cv. For example, if panel critical values are provided the function will return the panel test statistic.

Usage

## S3 method for class 'radf_obj'
tidy_join(x, y = NULL, ...)

## S3 method for class 'radf_obj'
augment_join(x, y = NULL, trunc = TRUE, ...)

Arguments

Details

tidy_join also calls augment_join when cv is of class sb_cv.