R/sim.R

In exuber: Econometric Analysis of Explosive Time Series

#' Simulation of a single-bubble process
#'
#' The following function generates a time series which switches from a martingale to a mildly explosive
#' process and then back to a martingale.
#'
#' @param n A positive integer specifying the length of the simulated output series.
#' @param te A scalar in (0, tf) specifying the observation in which the bubble originates.
#' @param tf A scalar in (te, n) specifying the observation in which the bubble collapses.
#' @param c A positive scalar determining the autoregressive coefficient in the explosive regime.
#' @param alpha A positive scalar in (0, 1) determining the value of the expansion rate in the autoregressive coefficient.
#' @param sigma A positive scalar indicating the standard deviation of the innovations.
#' @inheritParams radf_mc_cv
#'
#' @details
#' The data generating process is described by the following equation:
#' \deqn{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\}
#' }{X[t] = X[t-1] 1{t < te}+ \delta[T] * X[t-1] 1{te \le t \le tf} +
#' (\sum [k=tf+1]^t \epsilon[k] + X[tf]) 1{t > tf} + \epsilon[t] 1{t \le tf},}
#'
#' where the autoregressive coefficient \eqn{\delta_T}{\delta[T]} is given by:
#'
#' \deqn{\delta_T = 1 + cT^{-a}}{\delta[T] = 1 + c*T^{-a}}
#'
#' with \eqn{c>0}, \eqn{\alpha \in (0,1)}{\alpha in (0,1)},
#' \eqn{\epsilon \sim iid(0, \sigma^2)}{\epsilon - iid(0, \sigma^2)} and
#' \eqn{X_{\tau_f} = X_{\tau_e} + X'}{X[tf] = X[te] + X'} with \eqn{X' = O_p(1)}{X'= 0p(1)},
#' \eqn{\tau_e = [T r_e]}{te = [T re]} dates the origination of the bubble,
#'  and \eqn{\tau_f = [T r_f]}{tf = [T rf]} dates the collapse of the bubble.
#' During the pre- and post- bubble periods, \eqn{[1, \tau_e)}{[1, te)},
#' \eqn{X_t}{Xt} is a pure random walk process. During the bubble expansion period
#'  \eqn{\tau_e, \tau_f]}{[te,tf]} becomes a mildly explosive process with expansion rate
#'  given by the autoregressive coefficient \eqn{\delta_T}{\delta[T]}; and, finally
#'  during the post-bubble period, \eqn{(\tau_f, \tau]}{(tf, t]}  \eqn{X_t}{Xt} reverts to a martingale.
#'
#'
#' For further details see Phillips et al. (2015) p. 1054.
#'
#' @return A numeric vector of length n.
#' @export
#'
#' @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.
#'
#' @seealso \code{\link{sim_psy2}}, \code{\link{sim_blan}}, \code{\link{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()
sim_psy1 <- function(n, te = 0.4 * n, tf = 0.15 * n + te, c = 1,
                     alpha = 0.6, sigma = 6.79, seed = NULL) {
  assert_positive_int(n)
  assert_between(te, 0, n)
  assert_between(tf, te, n)
  assert_positive_int(c)
  assert_between(alpha, 0, 1)
  stopifnot(sigma >= 0)

  set_rng(seed)

  delta <- 1 + c * n ^ (-alpha)
  y <- 100

  for (t in 2:n) {
    if (t < te) {
      y[t] <- y[t - 1] + rnorm(1, sd = sigma)
    } else if (t >= te & t <= tf) {
      y[t] <- delta * y[t - 1] + rnorm(1, sd = sigma)
    } else if (t == tf + 1) {
      y[t] <- y[te] + rnorm(1, sd = sigma)
    } else {
      y[t] <- y[t - 1] + rnorm(1, sd = sigma)
    }
  }

  y %>%
    add_attr(seed = get_rng_state(seed)) %>%
    add_class(class = "sim")
}


#' Simulation of a two-bubble process
#'
#' The following data generating process is similar to  \code{\link{sim_psy1}}, with the difference that
#' there are two episodes of mildly explosive dynamics.
#'
#' @inheritParams sim_psy1
#' @param te1 A scalar in (0, n) specifying the observation in which the first bubble originates.
#' @param tf1 A scalar in  (te1, n) specifying the observation in which the first bubble collapses.
#' @param te2 A scalar in (tf1, n) specifying the observation in which the second bubble originates.
#' @param tf2 A scalar in (te2, n) specifying the observation in which the second bubble collapses.
#'
#' @details
#' The two-bubble data generating process is given by (see also \code{sim_psy1}):
#'
#' \deqn{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\} }{
#' X[t]=X[t-1] 1{t in N[0]}+ \delta[T] * X[t-1] 1{t in B[1] union B[2]} +
#' (\sum[k=t1f+1]^t \epsilon[k] + X'[t1f]) 1{t in N[1]} +
#' }
#'
#' \deqn{ + \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\}}{(\sum[l=t2f+1]^t \epsilon[l] + X'[t2f]) 1{t in N[2]} +
#' \epsilon[t] 1{t in N[0] union B[1] union B[2]},}
#'
#' where the autoregressive coefficient \eqn{\delta_T}{\delta[T]} is:
#'
#' \deqn{\delta_T = 1 + cT^{-a}}{\delta[T] = 1 + c*T^{-a},}
#'
#' with \eqn{c>0}, \eqn{\alpha \in (0,1)}{\alpha in (0,1)},
#' \eqn{\epsilon \sim iid(0, \sigma^2)}{\epsilon - iid(0, \sigma^2)},
#' \eqn{N_0 = [1, \tau_{1e})}{N0 = [1, t1e)},
#' \eqn{B_1 = [\tau_{1e}, \tau_{1f}]}{B1 = [te1, t1f]},
#' \eqn{N_1 = (\tau_{1f}, \tau_{2e})}{N0 = (t1f, t2e)},
#' \eqn{B_2 = [\tau_{2e}, \tau_{2f}]}{N0 = [t2e, t2f]},
#' \eqn{N_2 = (\tau_{2f}, \tau]}{N0 = [t2f, t]},
#' where \eqn{\tau}{t} is the last observation of the sample.
#' The observations \eqn{\tau_{1e} = [T r_{1e}]}{te1 = [T re1]}
#' and \eqn{\tau_{1f} = [T r_{1f}]}{tf = [T r1f]}
#' are the origination and termination dates of the first bubble;
#' \eqn{\tau_{2e} = [T r_{2e}]}{te2 = [T re2]} and \eqn{\tau_{2f} = [T r_{2f}]}{tf = [T r2f]}
#' are the origination and termination dates of the second bubble.
#' After the collapse of the first bubble, \eqn{X_t}{X[t]} resumes a martingale path until time
#' \eqn{\tau_{2e}-1}{t2e - 1}, and a second episode of exuberance begins at \eqn{\tau_{2e}}{t2e}.
#' Exuberance lasts lasts until \eqn{\tau_{2f}}{t2f} at which point the process collapses to a value of
#' \eqn{X_{\tau_{2f}}}{X[t2f]}. The process then continues on a martingale path until the end of the
#' sample period \eqn{\tau}{t}. The duration of the first bubble is assumed to be longer than
#' that of the second bubble, i.e. \eqn{\tau_{1f}-\tau_{1e}>\tau_{2f}-\tau_{2e}}{t1f - t1e > t2f - t2e}.
#'
#' For further details you can refer to Phillips et al., (2015) p. 1055.
#'
#' @return A numeric vector of length \code{n}.
#' @export
#'
#' @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.
#'
#' @seealso \code{\link{sim_psy1}}, \code{\link{sim_blan}}, \code{\link{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()
sim_psy2 <- function(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) {
  assert_positive_int(n)
  assert_between(te1, 0, n)
  assert_between(tf1, te1, n)
  assert_between(te2, tf1, n)
  assert_between(tf2, te2, n)
  assert_between(alpha, 0, 1)
  stopifnot(sigma >= 0)

  set_rng(seed)

  delta <- 1 + c * n ^ (-alpha)
  y <- 100

  for (i in 2:n) {
    if (i < te1) {
      y[i] <- y[i - 1] + rnorm(1, sd = sigma)
    } else if (i >= te1 & i <= tf1) {
      y[i] <- delta * y[i - 1] + rnorm(1, sd = sigma)
    } else if (i == tf1 + 1) {
      y[i] <- y[te1] + rnorm(1, sd = sigma)
    } else if (i > tf1 + 1 & i < te2) {
      y[i] <- y[i - 1] + rnorm(1, sd = sigma)
    } else if (i >= te2 & i <= tf2) {
      y[i] <- delta * y[i - 1] + rnorm(1, sd = sigma)
    } else if (i == tf2 + 1) {
      y[i] <- y[te2] + rnorm(1, sd = sigma)
    } else {
      y[i] <- y[i - 1] + rnorm(1, sd = sigma)
    }
  }

  y %>%
    add_attr(seed = get_rng_state(seed)) %>%
    add_class("sim")
}

#' 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):
#'
#' \emph{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.
#'
#'
#' @inheritParams sim_psy1
#' @param tr A scalar in  (tf, n) specifying the observation in which market recovers
#' @param c A positive scalar determining the drift in the normal market periods.
#' @param c1 A positive scalar determining the autoregressive coefficient in the explosive regime.
#' @param c2 A positive scalar determining the autoregressive coefficient in the collapse regime.
#' @param eta A positive scalar (>0.5) determining the drift in the normal market periods.
#' @param alpha A positive scalar in (0, 1) determining the autoregressive coefficient in the bubble period.
#' @param beta A positive scalar in (0, 1) determining the autoregressive coefficient in the collapse period.
#'
#'
#' @return A numeric vector of length \code{n}.
#'
#' @references Phillips, Peter CB, and Shu-Ping Shi. "Financial bubble implosion
#' and reverse regression." Econometric Theory 34.4 (2018): 705-753.
#'
#' @seealso \code{\link{sim_psy1}}
#' @export
#' @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)
#'
sim_ps1 <- function(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) {

  assert_positive_int(n)
  assert_between(te, 0, n)
  assert_between(tf, te, n)
  assert_between(tr, tf, n)
  assert_positive_int(c)
  assert_positive_int(c1)
  assert_positive_int(c2)
  assert_between(alpha, 0, 1)
  assert_between(beta, 0, 1)
  stopifnot(eta > 0.5, sigma >= 0)

  set_rng(seed)
  drift <- c*n^(-eta)
  delta <- 1 + c1 * n^(-alpha)
  gamma <- 1 - c2 * n^(-beta)
  y <- 100

  for (t in 2:n) {
    if (t < te) {
      y[t] <- drift + y[t - 1] + rnorm(1, sd = sigma)
    } else if (t >= te & t <= tf) {
      y[t] <- delta * y[t - 1] + rnorm(1, sd = sigma)
    } else if (t > tf & t <= tr ) {
      y[t] <- gamma * y[t - 1] + rnorm(1, sd = sigma)
    } else {
      y[t] <- drift + y[t - 1] + rnorm(1, sd = sigma)
    }
  }
  y %>%
    add_attr(seed = get_rng_state(seed)) %>%
    add_class("sim")
}


sim_ps2 <- function(n,
                    te1 = 0.2 * n, tf1 = te1 + 0.2 * n , tr1 = tf1 + 0.1*n,
                    te2 = 0.6 * n, tf2 = te2 + 0.15 * n , tr2 = tf2 + 0.1*n,
                    c = 1, c1 = 1, c2 = 1, eta = 0.6, alpha = 0.6, beta = 0.5,
                    sigma = 6.79, seed = NULL) {

  assert_positive_int(n)
  assert_between(te1, 0, n)
  assert_between(tf1, te1, n)
  assert_between(tr1, tf1, n)
  assert_between(te2, tf1, n)
  assert_between(tf2, te2, n)
  assert_between(tr2, tf2, n)
  assert_between(alpha, 0, 1)
  assert_positive_int(c)
  assert_positive_int(c1)
  assert_positive_int(c2)
  assert_between(alpha, 0, 1)
  assert_between(beta, 0, 1)
  stopifnot(eta > 0.5, sigma >= 0)

  set_rng(seed)
  drift <- c*n^(-eta)
  delta <- 1 + c1 * n^(-alpha)
  gamma <- 1 - c2 * n^(-beta)
  y <- 100

  for (t in 2:n) {
    if (t < te1) {
      y[t] <- drift + y[t - 1] + rnorm(1, sd = sigma) # normal
    } else if (t >= te1 & t <= tf1) {
      y[t] <- delta * y[t - 1] + rnorm(1, sd = sigma) # bubble1
    } else if (t > tf1 & t <= tr1 ) {
      y[t] <- gamma * y[t - 1] + rnorm(1, sd = sigma) # collapse 1
    }  else if (t > tr1 + 1 & t < te2) {
      y[t] <- drift + y[t - 1] + rnorm(1, sd = sigma) # normal 2
    }  else if (t >= te2 + 1 & t <= tf2) {
      y[t] <- delta * y[t - 1] + rnorm(1, sd = sigma) # bubble 2
    }  else if (t > tf2 + 1 & t <= tr2) {
      y[t] <- gamma * y[t - 1] + rnorm(1, sd = sigma) # collapse 2
    } else {
      y[t] <- drift + y[t - 1] + rnorm(1, sd = sigma) # normal 3
    }
  }
  y %>%
    add_attr(seed = get_rng_state(seed)) %>%
    add_class("sim")
}




#' Simulation of a Blanchard (1979) bubble process
#'
#' Simulation of a Blanchard (1979) rational bubble process.
#'
#' @inheritParams sim_psy1
#' @param pi A positive value in (0, 1) which governs the probability of the bubble continuing to grow.
#' @param r A positive scalar that determines the growth rate of the bubble process.
#' @param b0 The initial value of the bubble.
#'
#' @export
#' @return A numeric vector of length \code{n}.
#'
#' @importFrom stats rbinom
#' @details
#' Blanchard's bubble process has two regimes, which occur with probability \eqn{\pi} and \eqn{1-\pi}.
#' In the first regime, the bubble grows exponentially, whereas in the second regime, the bubble
#' collapses to a white noise.
#'
#' With probability \eqn{\pi}:
#' \deqn{B_{t+1} = \frac{1+r}{\pi}B_t+\epsilon_{t+1}}{B[t+1]=(1+r)/\pi*B[t]+\epsilon[t+1],}
#' With probability \eqn{1 - \pi}:
#' \deqn{B_{t+1} = \epsilon_{t+1}}{B[t+1] = \epsilon[t+1],}
#'
#' where \code{r} is a positive constant and \eqn{\epsilon \sim iid(0, \sigma^2)}{\epsilon - iid(0, \sigma^2)}.
#'
#'
#' @references Blanchard, O. J. (1979). Speculative bubbles, crashes and rational expectations.
#' Economics letters, 3(4), 387-389.
#'
#' @seealso \code{\link{sim_psy1}}, \code{\link{sim_psy2}}, \code{\link{sim_evans}}
#'
#' @examples
#' sim_blan(n = 100, seed = 123) %>%
#'   autoplot()
sim_blan <- function(n, pi = 0.7, sigma = 0.03, r = 0.05, b0 = 0.1,
                     seed = NULL) {

  assert_positive_int(n)
  assert_between(pi, 0, 1)
  stopifnot(sigma >= 0)
  stopifnot(r >= 0)

  set_rng(seed)

  b <- b0
  theta <- rbinom(n, 1, pi)
  i <- 1
  while (i < n) {
    if (b[i] > 0) {
      if (theta[i] == 1) {
        b[i + 1] <- (1 + r) / pi * b[i] + rnorm(1, 0, sigma)
      } else {
        b[i + 1] <- rnorm(1, 0, sigma)
      }
      i <- i + 1
    } else {
      i <- i - 1
    }
  }

  b %>%
    add_attr(seed = get_rng_state(seed)) %>%
    add_class("sim")
}

#' Simulation of an Evans (1991) bubble process
#'
#' Simulation of an Evans (1991) rational periodically collapsing bubble process.
#'
#' @inheritParams sim_blan
#' @param delta A positive scalar, with restrictions (see details).
#' @param tau The standard deviation of the innovations.
#' @param alpha A positive scalar, with restrictions (see details).
#' @param b1 A positive scalar, the initial value of the series. Defaults to \code{delta}.
#'
#' @return A numeric vector of length \code{n}.
#'
#' @importFrom stats rbinom
#'
#' @details
#'
#' \code{delta} and \code{alpha} are positive parameters which satisfy \eqn{0 < \delta < (1+r)\alpha}.
#' \code{delta} represents the size of the bubble after collapse.
#' The default value of \code{r} is 0.05.
#' The function checks whether \code{alpha} and \code{delta} satisfy this condition and will return an error if not.
#'
#' The Evans bubble has two regimes. If \eqn{B_t \leq \alpha}{B[t] \le \alpha} the bubble grows at an average rate of \eqn{1 + r}:
#'
#' \deqn{B_{t+1} = (1+r) B_t u_{t+1},}{B[t+1]= (1+r)*B[t]*u[t+1].}
#'
#' When \eqn{B_t > \alpha}{B[t] > \alpha} the bubble expands at the increased rate of \eqn{(1+r)\pi^{-1}}:
#'
#' \deqn{B_{t+1} =  [\delta + (1+r)\pi^{-1} \theta_{t+1}(B_t -  (1+r)^{-1}\delta B_t )]u_{t+1},}{B[t+1] = \delta*(1+r)/\pi* (B[t]-\delta/(1+r))) *u[t+1],}
#'
#' where \eqn{\theta} theta is a binary variable that takes the value 0 with probability \eqn{1-\pi} and 1 with probability \eqn{\pi}.
#' In the second phase, there is a (\eqn{1-\pi})  probability of the bubble process collapsing to \code{delta}.
#' By modifying the values of \code{delta}, \code{alpha} and \code{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.
#'
#' @export
#'
#' @seealso \code{\link{sim_psy1}}, \code{\link{sim_psy2}}, \code{\link{sim_blan}}
#'
#' @references Evans, G. W. (1991). Pitfalls in testing for explosive
#' bubbles in asset prices. The American Economic Review, 81(4), 922-930.
#'
#' @examples
#' sim_evans(100, seed = 123) %>%
#'   autoplot()
sim_evans <- function(n, alpha = 1, delta = 0.5, tau = 0.05, pi = 0.7,
                      r = 0.05, b1 = delta, seed = NULL) {

  # checks here
  assert_positive_int(n)
  stopifnot(alpha > 0)
  if (delta < 0 | delta > (1 + r) * alpha) {
    stop_glue("alpha and delta should satisfy: 0 < delta < (1+r)*alpha")
  }
  assert_between(pi, 0, 1)
  stopifnot(r >= 0)

  set_rng(seed)

  y <- rnorm(n, 0, tau)
  u <- exp(y - tau ^ 2 / 2)
  theta <- rbinom(n, 1, pi)
  b <- b1

  for (i in 1:(n - 1)) {
    if (b[i] <= alpha) {
      b[i + 1] <- (1 + r) * b[i] * u[i + 1]
    } else {
      b[i + 1] <- (delta + pi ^ (-1) * (1 + r) * theta[i + 1] * (b[i] -
        (1 + r) ^ (-1) * delta)) * u[i + 1]
    }
  }

  b %>%
    add_attr(seed = get_rng_state(seed)) %>%
    add_class("sim")
}

#' Simulation of dividends
#'
#' Simulate (log) dividends from a random walk with drift.
#'
#' @inheritParams sim_psy1
#' @param mu A scalar indicating the drift.
#' @param r A positive value indicating the discount factor.
#' @param log Logical. If true dividends follow a lognormal distribution.
#' @param output A character string giving the fundamental price("pf") or
#' dividend series("d"). Default is `pf'.
#'
#' @return A numeric vector of length n.
#' @export
#'
#' @details
#'
#' If log is set to FALSE (default value) dividends follow:
#'
#' \deqn{d_t = \mu + d_{t-1} + \epsilon_t}{d[t] = \mu + d[t-1] + \epsilon[t],}
#'
#' where \eqn{\epsilon \sim \mathcal{N}(0, \sigma^2)}{\epsilon - N(0, \sigma^2)}. The default parameters
#' are \eqn{\mu = 0.0373}, \eqn{\sigma^2 = 0.1574} and \eqn{d[0] = 1.3} (the initial value of the dividend sequence).
#' The above equation can be solved to yield the fundamental price:
#'
#' \deqn{F_t = \mu(1+r)r^{-2} + r^{-1}d_t}{F[t] = \mu * (1 + r)/r^2 + d[t]/r.}
#'
#' If log is set to TRUE then dividends follow a lognormal distribution or log(dividends) follow:
#'
#' \deqn{\ln(d_t) = \mu + \ln(d_{t-1}) + \epsilon_t}{ln(d[t]) = \mu + ln(d[t-1]) + \epsilon[t],}
#'
#' where \eqn{\epsilon \sim \mathcal{N}(0, \sigma^2)}{\epsilon - N(0, \sigma^2)}. Default parameters are
#' \eqn{\mu = 0.013}, \eqn{\sigma^2 = 0.16}. The fundamental price in this case is:
#'
#' \deqn{F_t = \frac{1+g}{r-g}d_t}{F[t] = (1 + g)/(r -g) * d[t],}
#'
#' where \eqn{1+g=\exp(\mu+\sigma^2/2)}{1 + g = exp(\mu + \sigma^2/2)}.
#' All default parameter values are those suggested by West (1988).
#'
#' @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)
sim_div <- function(n, mu, sigma, r = 0.05,
                    log = FALSE, output = c("pf", "d"), seed = NULL) {
  initval <- 1.3
  # Values obtained from West(1988, p53)
  if (missing(mu)) if (log) mu <- 0.013 else mu <- 0.0373
  if (missing(sigma)) if (log) sigma <- sqrt(0.16) else sigma <- sqrt(0.1574)

  assert_positive_int(n)
  stopifnot(sigma >= 0)
  stopifnot(r >= 0)
  stopifnot(is.logical(log))
  return <- match.arg(output)

  set_rng(seed)
  d <- stats::filter(mu + c(initval, rnorm(n - 1, 0, sigma)),
    c(1),
    init = 1.3, method = "recursive"
  ) %>%
    as.numeric() # filter coerces to time-series

  if (log) {
    g <- exp(mu + sigma ^ 2 / 2) - 1
    pf <- (1 + g) * d / (r - g)
  } else {
    pf <- mu * (1 + r) * r ^ (-2) + d / r
  }

  out <- if (return == "pf") pf else d

  out %>%
    add_attr(seed = get_rng_state(seed)) %>%
    add_class("sim")

}


# Methods -----------------------------------------------------------------


#' @export
print.sim <- function(x, ...) {
  attributes(x) <- NULL
  print(x)
}

#' @export
format.sim <- function(x, ...) {
  out <- signif(x, 3)
  out[is.na(x)] <- NA
  out
}

#' @export
#' @keywords internal
autoplot.sim <- function(object, ...) {
  object %>%
    enframe() %>%
    ggplot(aes(name, value)) +
    geom_line() +
    theme_exuber()
}