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Estimate the HysTAR model using conditional least squares estimation

Source: R/hystar_fit.R
hystar_fit.Rd

Estimate the parameters of the hysteretic threshold autoregressive (HysTAR) model.

Usage

hystar_fit(
  data,
  r = c(0.1, 0.9),
  d = 0L,
  p0 = 1L,
  p1 = 1L,
  p_select = c("bic", "aic", "aicc", "aiccp"),
  thin = FALSE,
  tar = FALSE,
  show_progress = FALSE
)

Arguments

data

a vector, matrix or data.frame containing the outcome variable \(y\) in the first column and the threshold variable \(z\) in the second. Other columns are ignored. A vector, is taken to be both the outcome and control variable, so, in that case, a self-exciting HysTAR is fitted.

r

A vector or a matrix with search values for \(\hat{r}_0, \hat{r}_1\). Defaults to c(.1, .9).

  • A vector r must contain two values \(a\) and \(b\) in \([0, 1]\). The search space for the thresholds will be observed values of z between its \(100a\%\) and \(100b\%\) percentiles.

  • A matrix r allows for a custom search. It must have two columns, such that each row represents a pair \(r_0 \le r_1\) to test. You can use a matrix with one row if you don't want to estimate the thresholds. Note that the values in these matrix should be on the scale of z.

d

A numeric vector with one or more values for the search space of the delay parameter. Defaults to 1. Typically, d is not very large, so a reasonable search space might be 0, 1, 2, ..., 5.

p0

A numeric vector with one or more values for the search space of the autoregressive order of Regime 0. Defaults to 1.

p1

Same as p0, but for regime 1. Note that it does not need to be equal to p0.

p_select

The information criterion that should be minimized to select the orders \(p_0\) and \(p_1\). Choices:

  • "bic" (default, Bayesian Information Criterion)

  • "aic" (Akaike Information Criterion)

  • "aicc" (Corrected Akaike Information Criterion)

  • "aiccp" (Change-point Akaike Information Criterion)

thin

TRUE (default) or FALSE. Only relevant when r is a vector.

  • If TRUE (default), the search space for the thresholds are the \(100a\%, 100(a+0.01)\%, \dots, 100b\%\) percentiles of z. This drastically reduces computation costs while keeping a reasonably large search space for the thresholds. Note that this is a purely practical choice with no theoretical justification.

  • If FALSE, all observed unique values of z between the \(100a\%\) and \(100b\%\) percentiles of z will be considered.

tar

TRUE or FALSE (default). Choose TRUE if you want to fit a traditional 2-regime threshold autoregressive (TAR) model. In this model, there is only one threshold (or equivalently, a HysTAR model with \(r_0 = r_1\)).

show_progress

TRUE or FALSE (default). Do you want to be updated on the progress of the estimation algorithm? This can be desirable when the number of time points, or the search space of d, p0 or p1, are large.

Value

An object of S3 class hystar_fit, which is a list containing the following items:

  • $data. A data.frame containing

    • y, the outcome variable

    • z, the threshold variable

    • H, a logical vector that indicates at which time points the hysteresis effect is happening. Note that this vector starts with NA(s), since not all values can be predicted in the HysTAR model. See Details.

    • R, the regime indicator vector. (Also starts with NA(s).)

  • $residuals. Also accessible with the residuals() S3 method.

  • $coefficients, a vector with the estimated coefficients. With the coef() S3 method, the coefficients are represented in a matrix. Use the confint() method to get the confidence intervals of the estimates.

  • $delay, a scalar with the estimate for the delay parameter.

  • $thresholds, a vector with the estimates of the thresholds.

  • $orders, a vector with the estimates of the orders.

  • $resvar, a vector with the estimates of the residual variances.

  • $rss, the minimized residual sum of squares.

  • $ic, a vector with the aic, the corrected aic and the bic.

  • $n, a vector with the total effective observations and the effective obeservations in regime 0 and regime 1.

  • $eff, a vector with the time indicators of the effective observations.

  • $equiv, a matrix containing equivalent estimates for the delay and thresholds, i.e., estimates that imply exactly the same regime indicator vector, and as a result the same minimal residual sum of squares.

  • $r_search, a vector with the \(r\)-values that were considered.

  • $tar, Logical: TRUE if a TAR model was fitted.

Implemented generics for the hystar_fit class:

  • plot() plots the z variable and the y variable above one another. Shading of the background visualizes the regimes. Thresholds are drawn as horizontal lines in the z plot. You can provide regime_names (char vector of 2), main (char vector of 1), xlab (char vector of 1) and ylab (char vector of 2).

  • summary(), this also provides the p-values and standard errors for the estimates of the coefficients.

  • print() prints the estimates within the mathematical representation of the model. Note that the scalar multiplied with e[t] is the standard deviation of the residuals, not the variance. See also the model definition above.

  • coef()

  • confint()

  • residuals()

  • fitted()

  • nobs()

Details

In regime 0, \(y_{t}\) is predicted by values up to \(y_{t - p_0}\). This implies that the first \(p_0\) time points can not be predicted. E.g., if \(p_0 = 2\), \(y_1\) would miss a value from \(y_{-1}\). Similarly, the value of the delay parameter implies that the regime is unknown for the first \(d\) time points. To ensure that the same data are used on all options for d, p0 and p1, the first max(d, p0, p1) observations are discarded for estimation of the parameters.

The HysTAR model

The HysTAR model is defined as:

\( y_t = \begin{cases} \phi_{00} + \phi_{01} y_{t-1} + \cdots + \phi_{0 p_0} y_{t-p_0} + \sigma_{0} \epsilon_{t} \quad \mathrm{if}~R_{t} = 0 \\ \phi_{10} + \phi_{11} y_{t-1} + \cdots + \phi_{1 p_1} y_{t-p_1} + \sigma_{1} \epsilon_{t} \quad \mathrm{if}~R_{t} = 1, \\ \end{cases} \)

with \( R_t = \begin{cases} 0 \quad \quad \mathrm{if} \, z_{t-d} \in (-\infty, r_{0}] \\ R_{t-1} \quad \mathrm{if} \, z_{t-d} \in (r_0, r_1] \\ 1 \quad \quad \mathrm{if} \, z_{t-d} \in (r_1, \infty), \\ \end{cases} \)

where \(p_j\) denotes the order of regime \(j \in \{0,1\}\) with coefficients \(\phi_{j0}, \dots, \phi_{j p_j \in (-1, 1)}\), \(\sigma_{j}\) is the standard deviation of the residuals, and \(d \in \{0, 1, 2, \dots\}\) is a delay parameter. The parameters of primary interest are the thresholds \(r_0 \le r_1\). We let \(t = 0, 1, 2, ..., T\), where \(T\) is the number of observations.

References

Li, Guodong, Bo Guan, Wai Keung Li, en Philip L. H. Yu. ‘Hysteretic Autoregressive Time Series Models’. Biometrika 102, nr. 3 (september 2015): 717–23.

Zhu, Ke, Philip L H Yu, en Wai Keung Li. ‘Testing for the Buffered Autoregressive Process’. Munich Personal RePEc Archive, (november 2013).

Author

Daan de Jong.

Examples

simulated_control_variable <- z_sim()
simulated_hystar_model <- hystar_sim(simulated_control_variable)
fitted_hystar_model <- hystar_fit(simulated_hystar_model$data)