Simulate the designs of the papers — simulate

Data-generating processes used in the Monte Carlo studies of Perron and González-Coya (2026) (simulate_serial(), simulate_combined(), simulate_predictive()) and González-Coya and Perron (2024) (simulate_hetero()). They are handy for examples, teaching and replication.

Usage

simulate_serial(
  n,
  ar = 0.8,
  ma = numeric(0),
  rho_x = 0,
  gamma = 1,
  beta = c(1, 1),
  burn = 0,
  seed = NULL
)

simulate_hetero(
  n,
  skedastic = "x2",
  beta = c(1, 1),
  n_extra = 0L,
  extra_corr = 0,
  seed = NULL
)

simulate_combined(
  n,
  ar = 0.5,
  ma = numeric(0),
  skedastic = "x2",
  beta = c(1, 1),
  driver = c("x", "mixture"),
  phi_prob = 0.75,
  gamma = 0,
  burn = 100L,
  seed = NULL
)

simulate_predictive(
  n,
  theta = 0.5,
  rho_x = 0,
  gamma = 0,
  beta = c(1, 1),
  burn = 100L,
  seed = NULL
)

Arguments

n: Number of observations returned.
ar, ma: Autoregressive and moving-average coefficients of the errors.
rho_x: Autoregressive coefficient of the regressor.
gamma: Feedback from past innovations to the regressor; the regressor is not exogenous when gamma != 0.
beta: Intercept and slope.
burn: Number of initial observations discarded.
seed: Optional seed. The random number generator state of the session is restored afterwards.
skedastic: Skedastic function: one of "homoskedastic", "x", "x2", "log2" (\([\log x]^2\)), "exp" (\(\exp(0.2x + 0.2x^2)\)), "step", or a function of x.
n_extra: Number of extra candidate covariates w1, w2, ... (columns of the result, not regressors).
extra_corr: Weight \(c\) of x in the extra covariates.
driver: Variance driver of simulate_combined(): "x" or "mixture".
phi_prob: Probability that the variance driver equals x when driver = "mixture".
theta: MA(1) coefficient of simulate_predictive().

Value

A data frame with columns y and x (and w or w1, w2, ... where relevant), with attributes beta (true coefficients), u (errors), e (innovations) and info (design parameters and, for the heteroskedastic designs, the true variances v).

Details

simulate_serial(): \(y_t = \beta_0 + \beta_1 x_t + u_t\) with ARMA errors driven by standard normal innovations \(e_t\) and a regressor \(x_t = \rho_x x_{t-1} + v_t + \gamma e_{t-1}\), which is pre-determined and exogenous only when \(\gamma = 0\). The defaults (\(\rho_x = 0\), \(\gamma = 1\), AR(1) coefficient 0.8, no burn-in, \(u_0 = 0\)) give the design of Table 1 of the 2026 paper.
simulate_hetero(): \(y_t = \alpha + \beta x_t + \sqrt{v(x_t)} \varepsilon_t\), \(x_t \sim U[1, 4]\) (eq. 9 of the 2024 paper), optionally with extra candidate covariates \(w_j = c x + (1 - c) q_j\), \(q_j \sim U[1, 4]\).
simulate_combined(): ARMA errors whose innovations are heteroskedastic, \(e_t = \sqrt{v(w_t)} \varepsilon_t\), with \(w_t = x_t\) (driver = "x") or \(w_t = \phi_t x_t + (1 - \phi_t) q_t\), \(\phi_t \sim\) Bernoulli(phi_prob) (driver = "mixture") (Supplement S-4).
simulate_predictive(): the predictive regression \(y_{t+2} = \alpha + \beta x_t + u_{t+2}\) with MA(1) errors (Supplement S-3), returned aligned so that row \(t\) holds \(y_{t+2}\) and \(x_t\).

References

Perron, P. and González-Coya, E. (2026). Feasible GLS for time series regression. Journal of Econometrics, 258, 106346. doi:10.1016/j.jeconom.2026.106346

González-Coya, E. and Perron, P. (2024). Estimation in the presence of heteroskedasticity of unknown form: a Lasso-based approach. Journal of Econometric Methods, 13(1), 29–48. doi:10.1515/jem-2023-0007

Examples

sim <- simulate_serial(200, seed = 1)
head(sim)
#>           y          x
#> 1 2.2460628  1.0624195
#> 2 2.0815178  1.7702318
#> 3 0.8777732 -1.1665364
#> 4 1.4750008 -0.6899547
#> 5 3.9386654  2.8271694
#> 6 1.4232236 -0.1534022
attr(sim, "beta")
#> [1] 1 1

het <- simulate_hetero(100, skedastic = "log2", seed = 2)
summary(attr(het, "info")$v)
#>      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
#> 0.0004453 0.2643021 0.7135063 0.8469443 1.4413330 1.8987080