Package 'DPTM'

Example Dataset d1

Description

A simulated dataset for demonstrating the package

Usage

d1

Format

## 'd1' A data.frame with 1000 rows and 7 columns:

id

individuals

year

periods

y

dependent variable

y1

the first lag of y

q

threshold variable

x

regressor with threshold effects

z

regressor without threshold effects

Source

Simulated data with two thresholds

---

Dynamic panel model with fixed effects.

Description

Use a MLE procedure to estimate the dynamic panel model with fixed effects.

Usage

DPML(formula, data, index=NULL, timeFE = FALSE, y1 = NULL,...)

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

Arguments

formula	formula of the covariates with threshold effects.
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'.
timeFE	logicals. If TRUE the time fixed effects will be allowed. 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'.
...	additional arguments, seestats::nlm.

Details

DPML can fit the dynamic panel model with fixed effects proposed by Hsiao et al. (2002), which is based on the first difference and the maximum likelihood (MLE) method.

For a classical dynamic panel model with fixed effects having following form:

$y_{it}=\rho y_{it-1}+\beta_1x_{1,it}+\beta_2x_{2,it}+\alpha_i+u_{it}$

, can use y~x1+x2.

For a special dynamic panel model with fixed effects having the following form:

$\Delta y_{it}=\rho y_{it-1}+\beta_1x_{1,it}+\beta_2x_{2,it}+\alpha_i+u_{it}$

, can use dy~x1+x2 with y1$= y_{it-1}$.

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).

In addition, 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 DPML.

Value

DPML 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

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)

# No time fixed effects
model1 <- DPML(y~x+z, data = d1)
print(model1)

# With time fixed effects
model2 <- DPML(y~x+z, data = d1, timeFE = TRUE)
print(model2)

---

Dynamic panel multiple threshold model with fixed effects.

Description

Use a MCMC-MLE or grid-search-MLE based 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,
grid_search = FALSE, grids = 100, grid_search_type = c("jointly", "sequential"),
grid_search_iter = 1,
...)

## 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.
grid_search	logicals. If TRUE, thresholds are searched by deterministic grid search instead of MCMC. Defaults to 'FALSE'.
grids	number of grid points (threshold vectors) searched when grid_search = TRUE. Defaults to '100'.
grid_search_type	type of grid search for multiple thresholds. "jointly" searches the full Cartesian grid (grids^Th); "sequential" adds thresholds one at a time (grids * Th). Defaults to "jointly".
grid_search_iter	number of coordinate-refinement cycles after the initial grid search when grid_search = TRUE. For two thresholds, one cycle re-estimates the first threshold conditional on the second and then re-estimates the second conditional on the first. Set to 0 to skip this refinement. Defaults to 1.
...	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.

Alternatively, set grid_search = TRUE to use deterministic grid search for the unknown thresholds. In that case, grids controls the number of threshold vectors evaluated; for example, grids = 100 searches 100 candidate points in the ordered threshold-parameter space. For Th > 1 and grid_search_type = "jointly", the full Cartesian grid is searched, so Th = 2 and grids = 100 evaluates 100 * 100 candidate threshold pairs. For grid_search_type = "sequential", thresholds are added one at a time, so Th = 2 and grids = 100 evaluates about 200 candidates in total. When Th > 1, grid_search_iter controls an additional coordinate-refinement step after the initial grid search. With two thresholds and grid_search_iter = 1, DPTS re-estimates the first threshold conditional on the second, then re-estimates the second conditional on the first, and sorts the final threshold estimates. Grid search does not produce MCMC threshold confidence intervals or Gelman-Rubin diagnostics.

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
threshold_search	threshold-search method, either "MCMC" or "grid"
grid_points	number of searched grid points when grid_search = TRUE
grid_search_type	grid-search type when grid_search = TRUE
grid_search_iter	number of coordinate-refinement cycles when grid_search = TRUE

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)

## deterministic grid search
#Model1_grid <- DPTS(y~x,data = d1, index = c('id','year'), q = d1$q, Th = 1,
#grid_search = TRUE, grids = 100)
#print(Model1_grid)


### 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)

---

Example Dataset Growth_Inflation

Description

A dataset for economic growth of 74 countries from 1961 to 2015 (five-year average).

Usage

Growth_Inflation

Format

## 'Growth_Inflation' A data.frame with 814 rows and 15 columns:

ncountry

country id

countryname

country name

code

country code

Period

Period

years

years

GDP per capita growth

the difference of ln(GDP per capita)

Inflation rate-semilog

the semi-log of Inflation rate

Transitional convergence

the lag of ln(GDP per capita)

Human capital

Human capital

Financial depth

Financial depth

Governance

Governance

Public infrastructure

Public infrastructure

Trade openness

Trade openness

Economic instability

Economic instability

Inflation rate

Inflation rate

Source

https://doi.org/10.1080/07474938.2019.1624401

---

Tests for multiple thresholds.

Description

Tests for models with different thresholds, using bootstrap method.

Usage

Threshold_Test(formula = NULL, formula_cv = NULL, data, index=NULL, Th = 1, q, 
timeFE = FALSE, bt = 100,NoY = FALSE, y1 = NULL, iterations = 2000, sro = 0.1,
r0x = NULL, r1x = NULL, grid_search = FALSE, grids = 100,
grid_search_type = c("jointly", "sequential"), grid_search_iter = 1,
parallel=TRUE, seed = NULL,...)

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'.
q	threshold variable.
timeFE	logicals. If TRUE the time fixed effects will be allowed. Defaults to 'FALSE'.
bt	the number of bootstrap; If a setting is not provided, defaults (bt = 100) will be used. Defaults to '100'.
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.
grid_search	logicals. If TRUE, thresholds in DPTS calls are searched by deterministic grid search instead of MCMC. Defaults to 'FALSE'.
grids	number of grid points (threshold vectors) searched when grid_search = TRUE. Defaults to '100'.
grid_search_type	type of grid search for multiple thresholds. "jointly" searches the full Cartesian grid (grids^Th); "sequential" adds thresholds one at a time (grids * Th). Defaults to "jointly".
grid_search_iter	number of coordinate-refinement cycles after the initial grid search when grid_search = TRUE. Defaults to 1.
parallel	logicals. If TRUE test will run in parallel for saving time. Defaults to 'TRUE'.
seed	set seeds to guarantee the replication the test (see set.seed);
...	additional arguments to be passed to the settings of MCMC (see BayesianTools::applySettingsDefault)
Th	number of thresholds; Defaults to '1'.

Details

Threshold_Test can run the Test for multiple thresholds (Th is H1). The statistic is

$F_s=\frac{S\left(\hat{\gamma}_{s-1}\right)-S\left(\hat{\gamma}_s\right)}{S\left(\hat{\gamma}_s\right) / N(T-1)},$

where $s$ is the number of thresholds in H1, $S\left(\hat{\gamma}_{s-1}\right)=-\ln L\left(\hat{\gamma}_{s-1}\right)$ and $S\left(\hat{\gamma}_s\right)=-\ln L\left(\left(\hat{\gamma}_{s-1}^{\prime}, \hat{\gamma}_s\right)^{\prime}\right)$. And the p-value is computed by bootstrap method (see Ramírez-Rondán, 2020).

Take the two threshold model as example. User must set Th = 1 firstly to reject the null hypothesis of no threshold effects; Then he should set Th = 2 to reject the null hypothesis of only one threshold; Lastly, set Th = 3 to accept the null hypothesis of two thresholds. In other words, p-values of the first test (Th = 1) and the second test (Th = 1) should be less than significant level while the third test (Th = 3) is not.

Threshold_Test contains all augments in DPTS, but with three new augments: bt, parallel and seed. bt is the number of bootstrap (by default is 100); parallel can allow user to run test in parallel to save time; seed is used to guarantee the replication of tests. When grid_search = TRUE, the null and alternative threshold models in the test use the same deterministic grid-search threshold routine as DPTS. Use grid_search_type = "jointly" for the full multi-dimensional grid or grid_search_type = "sequential" for stepwise threshold addition. For multiple thresholds, grid_search_iter adds coordinate-refinement cycles after the initial grid search; set it to 0 to skip refinement.

It is worthy noting that the test shrinks to the so-called threshold existence test when Th = 1.

Value

A list with class "htest" containing the following components:

statistic	the value of the F-statistic.
parameter	the degrees of freedom for the F-statistic.
p.value	the p-value for the test.
null.value	the specified hypothesized value of the null hypothesis.
alternative	a character string describing the alternative hypothesis.
method	a character string indicating what type of test was performed.
data.name	a character string giving the name(s) of the data.
estimate	the critical value of the statistic (5% significance level).
LRs	a vector of statistics from bootstrap.

Author(s)

Hujie Bai

References

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

Examples

### Examples elapsed time > 15s

#data(d1)

# H0: no threshold effects (no threshold)
#test0 <- Threshold_Test(y~x,y~z,data = d1, index = c('id','year'), q = d1$q, Th = 1,
#bt = 50, iterations = 500)
#test0

# Grid-search version of the threshold existence test
#test0_grid <- Threshold_Test(y~x,y~z,data = d1, index = c('id','year'), q = d1$q,
#Th = 1, bt = 50, grid_search = TRUE, grids = 100)
#test0_grid

# H0: one threshold 
#test1 <- Threshold_Test(y~x,y~z,data = d1, index = c('id','year'), q = d1$q, Th = 2,
#bt = 50, iterations = 500)
#test1

# H0: two threshold 
#test2 <- Threshold_Test(y~x,y~z,data = d1, index = c('id','year'), q = d1$q, Th = 3,
#bt = 50, iterations = 500)
#test2

---