DPTS: Dynamic panel multiple threshold model with fixed effects.

In DPTM: Dynamic Panel Multiple Threshold Model with Fixed Effects

Dynamic panel multiple threshold model with fixed effects.

Description

Use a MCMC-MLE based on two-step procedure to estimate the dynamic panel multiple threshold model with fixed effects.

Usage

DPTS(formula = NULL, formula_cv = NULL, data, index=NULL, Th = 1, q, timeFE = FALSE, 
NoY = FALSE, y1 = NULL, iterations = 2000, sro = 0.1, r0x = NULL, r1x = NULL,
...)

## S6 method for class 'DPTM' 
#print(...)

Arguments

formula	formula of the covariates with threshold effects; If a setting is not provided, defaults (no covariates with threshold effects) will be used. Defaults to 'NULL'.
formula_cv	formula of the covariates without threshold effects; If a setting is not provided, defaults (no covariates without threshold effects) will be used. Defaults to 'NULL'.
data	data frame of the observed data.
index	variable names of individuals and period; If a setting is not provided, defaults (the first variables in data will be as "id", while the second will be "year") will be used.Defaults to 'NULL'.
Th	number of thresholds; Defaults to '1'.
q	threshold variable.
timeFE	logicals. If TRUE the time fixed effects will be allowed. Defaults to 'FALSE'.
NoY	logicals. If TRUE the lags of dependent variables will be without threshold effects. Defaults to 'FALSE'.
y1	lags of dependent variables; If a setting is not provided, defaults (the first-order lag) will be used. Defaults to 'NULL'.
iterations	MCMC iterations (50% used for burnining). Defaults to '2000'.
sro	regime (subsample) proportion; If a setting is not provided, defaults (10%) will be used. Defaults to '0.1'.
r0x	lower bound of threshold parameter space; If a setting is not provided, defaults (15% quantile of threshold variable) will be used.
r1x	upper bound of threshold parameter space; If a setting is not provided, defaults (85% quantile of threshold variable) will be used.
...	additional arguments to be passed to the settings of MCMC (see BayesianTools::applySettingsDefault)

Details

DPTS can fit the dynamic panel threshold model with fixed effects proposed by Ramírez-Rondán (2020), and also allow a multiple threshold model by setting Th > 1.

Given the diverse forms and versatile applications of threshold models, we advocate for aligning model selection with specific research objectives, thereby granting users autonomy in specifying the model structure.

Take the model with one threshold (Ramírez-Rondán, 2020) as example.

For a standard threshold model

\begin{aligned}y_{i t} &=\left(\rho_1 y_{i t-1}+\beta_1 x_{i t}\right) I\left(q_{i t}\leq \gamma\right)+\left(\rho_2 y_{i t-1}+\beta_2 x_{i t}\right) I\left(q_{i t}> \gamma\right)\\&+\alpha_i+u_{i t},\end{aligned}

, can use DPTS(y~x,data = data, q = q, Th = 1).

For a threshold model who has regressors with threshold effects (x) and regressors without threshold effects (z)

\begin{aligned}y_{i t} &=\left(\rho_1 y_{i t-1}+\beta_1 x_{i t}\right) I\left(q_{i t}\leq \gamma\right)+\left(\rho_2 y_{i t-1}+\beta_2 x_{i t}\right) I\left(q_{i t}> \gamma\right)\\&+\theta z_{i t}+\alpha_i+u_{i t},\end{aligned}

, can use DPTS(y~x,y~z,data = data, q = q, Th = 1).

If user only cares about the regressors with threshold effects (thus hopes there is no threshold effects in the lag of dependent variable y_1), like

\begin{aligned}y_{i t} &= \rho y_{i t-1}+ \beta_1 x_{i t} I\left(q_{i t}\leq \gamma\right)+\beta_2 x_{i t} I\left(q_{i t}> \gamma\right)\\&+\theta z_{i t}+\alpha_i+u_{i t},\end{aligned}

, can use DPTS(y~x,y~z,data = data, q = q, Th = 1, NoY = TRUE).

And, the threshold model with the following form

\begin{aligned}y_{i t} &=\rho_1 y_{i t-1}I\left(q_{i t}\leq \gamma\right)+\rho_2 y_{i t-1}I\left(q_{i t}> \gamma\right)+\beta x_{i t}\\&+\theta z_{i t}+\alpha_i+u_{i t},\end{aligned}

, is also allowed by DPTS(NULL,y~x+z,data = data, q = q, Th = 1).

In addition, a special threshold model having the following form

\begin{aligned}\Delta y_{i t} &=\left(\rho_1 y_{i t-1}+\beta_1 x_{i t}\right) I\left(q_{i t}\leq \gamma\right)+\left(\rho_2 y_{i t-1}+\beta_2 x_{i t}\right) I\left(q_{i t}> \gamma\right)\\&+\theta z_{i t}+\alpha_i+u_{i t},\end{aligned}

, can use DPTS(dy~x,dy~z,data = data, q = q, Th = 1) with y1= y_{it-1}.

The MCMC we used is based on BayesianTools, and the default method is "DREAMzs" (see Vrugt et al., 2009). If user wants to use other MCMC, can use ... (see BayesianTools::applySettingsDefault). As for the length of iterations, it can be set by iterations (50% used for burnining) and default length is 2000. The trace plot and the Gelman and Rubin's convergence diagnostic are supplied by DPTS (print) to test the convergence of MCMC sample.

Additionally, we assume the exogenous regressor x is weakly exogenous, and thus the first period after difference is given by

\Delta y_{i1}=\delta_0 + {\boldsymbol\delta}'_1 \Delta {\bf x}_{i1}+ v_{i1},

where E(v_{i1}| \Delta {\bf x}_{i1} )=0. E(v_{i1}^2)=\sigma^2_v, E(v_{i1}\Delta u_{i2})=-\sigma^2_u and E(v_{i1} \Delta u_{it})=0 for t=3,4,...,T and i=1,...,N. For more details, see Hsiao et al. (2002).

Finally, we solve the log-likelihood function by stats::nlm who uses iterlim to set the maximum number of iterations, and thus iterlim is allowed by ... in DPTS.

Value

DPTS returns an object of class "DPTM". The function print are used to obtain and print a print of the results. An object of class "DPTM" is a list containing at least the following components:

coefficients	a named vector of coefficients
NNLL	the negative log-likelihood function value
Zvalues	a vector of t statistics
Ses	a vector of standard errors
covariance_matrix	a covariance matrix
Th	the number of thresholds
thresholds	a named vector of thresholds

Author(s)

Hujie Bai

References

Ramírez-Rondán, N. R. (2020). Maximum likelihood estimation of dynamic panel threshold models. Econometric Reviews, 39(3), 260-276.

Vrugt, Jasper A., et al. (2009)."Accelerating Markov chain Monte Carlo simulation by differential evolution with self-adaptive randomized subspace sampling." International Journal of Nonlinear Sciences and Numerical Simulation 10.3: 273-290.

Hsiao, C., Pesaran, M. H., & Tahmiscioglu, A. K. (2002). Maximum likelihood estimation of fixed effects dynamic panel data models covering short time periods. Journal of econometrics, 109(1), 107-150.

Examples

data(d1)

# single threshold

## standard form 
#Model1_1 <- DPTS(y~x,data = d1, index = c('id','year'), q = d1$q, Th = 1, 
#iterations = 1000)
#print(Model1_1)


### Examples elapsed time > 15s
## with x \& z
#Model2_1 <- DPTS(y~x,y~z,data = d1, index = c('id','year'), q = d1$q, Th = 1, 
#iterations = 1000)
#print(Model2_1)

## with x \& z (y1 no threshold effects)
#Model3_1 <- DPTS(y~x,y~z,data = d1, index = c('id','year'), q = d1$q, Th = 1,
#NoY = TRUE, iterations = 1000)
#print(Model3_1)

## only y1 with threshold effects
#Model4_1 <- DPTS(NULL,y~x+z,data = d1, index = c('id','year'), q = d1$q, Th = 1, 
#iterations = 1000)
#print(Model4_1)

# two thresholds (Th = 2)
## with x \& z
#Model2_2 <- DPTS(y~x,y~z,data = d1, index = c('id','year'), q = d1$q, Th = 2, 
#iterations = 1000)
#print(Model2_2)

# Adding time fixed effects (timeFE = TRUE)
#Model2_2T <- DPTS(y~x,y~z,data = d1, index = c('id','year'), q = d1$q, Th = 2,
#timeFE = TRUE, iterations = 1000)
#print(Model2_2T)

DPTM documentation built on April 3, 2025, 6:28 p.m.