{smcl}
{* *! version 0.5.2  18apr2026}{...}
{cmd:help xtdpthresh}
{hline}

{title:Title}

{pstd}
{hi:xtdpthresh} {hline 2} Dynamic Panel Threshold Model with Endogeneity,
supporting Unbalanced Panels via Forward Orthogonal Deviations, and Grid
Bootstrap Inference (Gong-Seo 2026)


{title:Syntax}

{p 8 17 2}
{cmd:xtdpthresh} {it:depvar} [{it:indepvars}] {ifin}
{cmd:,} {cmdab:qx(}{it:varname}{cmd:)} [{it:options}]

{pstd}
where:

{pmore}
{it:depvar}    — dependent variable y{p_end}

{pmore}
{it:indepvars} — exogenous regressors (enter β and δ parts of the model){p_end}

{pmore}
{cmd:qx(}{it:varname}{cmd:)} — {bf:threshold variable} (REQUIRED; self-documenting style following xthreg2){p_end}


{synoptset 30 tabbed}{...}
{synopthdr}
{synoptline}
{syntab:Model specification}
{synopt:{opt qx(varname)}}threshold variable (REQUIRED){p_end}
{synopt:{opt endo:genous(varlist)}}endogenous regressors; lagged levels used as IV{p_end}
{synopt:{opt exo:genous(varlist)}}extra exogenous regressors (same treatment as {it:indepvars}){p_end}
{synopt:{opt iv(varlist[, sub-opts])}}additional user-supplied instruments.
Concise xtabond2-style sub-options allowed: {cmd:iv({it:varlist}, maxlag(# [#]) collapse)}.
Sub-options override the top-level {cmd:maxlag()}/{cmd:collapse} when both are given.{p_end}
{synopt:{opt kink}}enforce continuity (kink) restriction{p_end}
{synopt:{opt static}}static model; do NOT auto-add L.{it:depvar} as regressor{p_end}
{synopt:{opt td}}purge common time shocks by within-time demeaning of y and regressors{p_end}
{synopt:{opt collapse}}collapse block-diagonal instruments across time (Roodman 2009); drastically reduces #instruments when T is large{p_end}
{synopt:{opt maxlag(# [#])}}restrict lag depth for instruments in the TRANSFORMED equation; {cmd:maxlag(L)} caps at lag L; {cmd:maxlag(a b)} uses lags a through b{p_end}
{synopt:{opt levmaxlag(# [#])}}lag range for level-equation IVs in {cmd:method(system)}; default {cmd:(1 1)} (Blundell-Bond convention); {cmd:levmaxlag(1 2)} uses Δy_{t-1} and Δy_{t-2}{p_end}

{syntab:Transformation and estimation}
{synopt:{opt method(fd|fod|system)}}panel transformation; default is {cmd:fd}{p_end}
{synopt:{opt grid(#)}}# grid points for γ search; default {cmd:30}{p_end}
{synopt:{opt trim(#)}}trim rate for γ grid (xthenreg convention); default {cmd:0.10}{p_end}

{syntab:Inference (Grid Bootstrap, Gong-Seo 2026)}
{synopt:{opt citype(grid|none)}}CI type for threshold; default {cmd:grid}.
Asymptotic CI not provided — the threshold estimator has a non-standard
limiting distribution (n^(1/4)-rate under continuity; Gong-Seo 2026 show
asymptotic CIs are invalid in that case). Use grid bootstrap instead.{p_end}
{synopt:{opt gridci(#)}}# grid points for CI construction; default {cmd:25}{p_end}
{synopt:{opt boot(#)}}bootstrap replications; default {cmd:299}{p_end}
{synopt:{opt nosearch}}skip grid bootstrap (point estimate only){p_end}

{syntab:Reporting}
{synopt:{opt l:evel(#)}}confidence level; default {cmd:95}{p_end}
{synopt:{opt verbose}}print per-γ bootstrap progress (25+ lines); default is a compact one-dot-per-γ progress bar{p_end}
{synoptline}

{p 4 6 2}
Use {help xtset} before running {cmd:xtdpthresh}.


{title:Description}

{pstd}
{cmd:xtdpthresh} estimates the dynamic panel threshold model of
Seo and Shin (2016), extended to unbalanced panels via Forward Orthogonal
Deviations (Arellano-Bover 1995) and System GMM (Blundell-Bond 1998).
Inference for the threshold location uses the grid bootstrap of
Gong and Seo (2026), which is uniformly valid regardless of whether the
threshold model is continuous (kink) or discontinuous (jump).

{pstd}
The estimated dynamic model is:

{p 8 8 2}
{it:y_it = β1·y_{i,t-1} + x_it'β2 + (1, y_{i,t-1}, x_it')δ · 1(q_it > γ) + η_i + ε_it}

{pstd}
where {it:x_it} stacks the regressors listed in {it:indepvars} plus any
variables specified via {opt endogenous()} / {opt exogenous()}. The lagged
dependent variable {it:y_{i,t-1}} is appended automatically (the command is
dynamic by default; see {opt static} to disable). The coefficient vector δ
contains both the intercept jump (δ_1) and the regime-specific slope changes
for {it:y_{i,t-1}} and each element of {it:x_it}. In the static case
({opt static}), the term β1·{it:y_{i,t-1}} drops out of both the β and δ
blocks.

{pstd}
When the true model is continuous (satisfies δ_1 + δ_p·γ = 0 and δ_2:p-1 = 0),
the GMM threshold estimator converges at {it:n^(1/4)}-rate with a non-normal
distribution; the standard nonparametric bootstrap is inconsistent. The grid
bootstrap in {cmd:xtdpthresh} imposes the null at each candidate γ and is
valid regardless of continuity.


{title:Syntax conventions relative to xthenreg}

{pstd}
{cmd:xtdpthresh} is designed to feel familiar to users of {cmd:xthenreg}
(Kim-Kim-Seo 2019) while extending it along several dimensions. Key
differences:

{phang}
{cmd:*} Positional arguments are {bf:depvar [indepvars]}. The threshold
variable is passed via the REQUIRED option {cmd:qx(}{it:varname}{cmd:)},
not as a positional argument. This follows the self-documenting
convention of {cmd:xthreg2} (Wang-Lian 2019).

{phang}
{cmd:*} Added: {cmd:method()} for FD/FOD/System choice. {cmd:method(fd)}
is the default and reproduces {cmd:xthenreg}; {cmd:method(fod)} enables
unbalanced panel support; {cmd:method(system)} adds Blundell-Bond level
moments.

{phang}
{cmd:*} Added: {cmd:iv(}{it:varlist}[, sub-opts]{cmd:)} for external
instruments, with xtabond2-style sub-options {cmd:maxlag()} and
{cmd:collapse} that override the top-level settings. Functionally
similar to {cmd:xthenreg}'s {cmd:inst()} but with finer control.

{phang}
{cmd:*} Added: {cmd:collapse}, {cmd:maxlag()}, {cmd:levmaxlag()} for
xtabond2-style instrument count management — essential when T is
moderate and the Hansen J diagnostic would otherwise be unreliable.

{phang}
{cmd:*} Added: {cmd:citype()}, {cmd:gridci()}, {cmd:boot()}, {cmd:nosearch}
for grid bootstrap threshold CI (not in {cmd:xthenreg}).

{phang}
{cmd:*} Added: {cmd:td} to purge common time shocks by within-time
demeaning.

{phang}
{cmd:*} Added: {cmd:exogenous()} for explicit extra exogenous vars. In
practice, putting these in {it:indepvars} achieves the same effect.


{title:Options}

{dlgtab:Model specification}

{phang}
{opt endogenous(varlist)} specifies endogenous regressors. These are
appended to the regressor matrix (in addition to {it:indepvars}), and the
command automatically uses their lagged levels as instruments. {it:endogenous()}
variables must NOT appear in {it:indepvars}.

{phang}
{opt exogenous(varlist)} specifies additional exogenous regressors. These
are treated identically to {it:indepvars} and included in Δx as IV. This
option is redundant with {it:indepvars} but provided for explicitness.

{it:Default instrument structure (iv() is OPTIONAL).} When {cmd:iv()} is
NOT specified, the command still builds a valid instrument matrix Z
automatically, per Arellano-Bond convention. Per time block t, Z contains:

{phang2}
(i) a constant column;

{phang2}
(ii) lags of the dependent variable y_{t-2}, y_{t-3}, ... (dynamic models);

{phang2}
(iii) for each exogenous regressor (listed in {it:indepvars} or
{opt exogenous()}): the contemporaneous Δx_t or x_t itself — exog vars
instrument themselves under E[x · ε]=0;

{phang2}
(iv) for each variable in {opt endogenous()}: its own lags x_{t-1},
x_{t-2}, ... as Arellano-Bond moment conditions.

{pstd}
So a minimal call like {cmd:xtdpthresh y x1 x2, qx(q)} already uses
x1, x2, and the lagged y as instruments — no need to specify {cmd:iv()}.

{phang}
{opt iv(varlist[, sub-opts])} specifies {it:additional} user-supplied
instruments that enter Z without being added as regressors. Useful when
an {it:external} IV exists (e.g., industry-level average that moves with
a firm endogenous variable). Each {it:iv} variable contributes one extra
column per time block. Under {cmd:method(system)}, user IVs are added to
BOTH transformed- and level-equation moment blocks — valid when the user
IV is exogenous in levels. User is responsible for supplying only IVs
satisfying E[z · ε] = 0 in levels.
Sub-options {cmd:maxlag(# [#])} and {cmd:collapse} inside {cmd:iv()}
override the corresponding top-level options (xtabond2-style).

{phang}
{opt kink} requests the continuity-restricted (kink) model of Seo-Shin
(2016). Under this restriction, the model is continuous at γ:
δ_2 = 0_{p-1} and δ_1 + δ_3·γ = 0. The threshold variable's slope changes
at γ, but no jump occurs. The coefficient vector has p+2 elements
(β_1, ..., β_p, kink_slope) instead of 2p+2 in the unrestricted (jump)
model.

{phang}
{opt static} specifies a static model. The default is dynamic, which
automatically includes L.{it:depvar} as a regressor. With {cmd:static},
L.{it:depvar} is not auto-added; users must include any lag explicitly
in {it:indepvars}.

{phang}
{opt td} purges common time shocks by subtracting the cross-sectional
mean at each time period from {it:depvar}, {it:indepvars}, and variables
listed in {opt endogenous()}, {opt exogenous()}, and {opt iv()}. The
threshold variable {it:q} is left untouched so γ retains its original
scale and interpretation. Demeaning is applied after internal {cmd:preserve},
so user data is not modified permanently. This is the standard panel
econometrics approach to controlling for aggregate time effects (e.g.,
macroeconomic shocks); it is equivalent to including year dummies whose
coefficients are restricted to be equal across regimes.

{phang}
{opt collapse} collapses block-diagonal instruments across time periods
(Roodman 2009). By default, each moment (lag of y, Δx, endog lag, user
inst) gets a separate column per time block, causing the instrument
count to grow quadratically with T. With {cmd:collapse}, instruments
are stacked into a single shared column per lag depth, reducing the
total count by a factor of (T−2). Use this for long panels (T ≥ 10) to
mitigate instrument proliferation and produce reliable Hansen J tests.
Downside: weaker identification at each γ due to fewer moments.

{phang}
{opt maxlag(# [#])} restricts which lags are used as instruments,
analogous to xtabond2's {cmd:lag()} suboption.

{pmore}
{cmd:maxlag(L)} caps instrument lag depth at L (uses lags 1 through L
for endogenous variables; lags 2 through L+1 for the auto-added L.y).

{pmore}
{cmd:maxlag(a b)} uses lags a through b inclusive. For example,
{cmd:maxlag(2 4)} uses lags 2, 3, 4. Useful when short lags are
"too recent" to be valid IVs (possibly still correlated with ε).

{pmore}
Combine with {cmd:collapse} for aggressive instrument reduction:
{cmd:maxlag(2 4) collapse} yields few moments and reliable Hansen J
at the cost of identification strength. Monitor {cmd:e(N_iv)} to
ensure the model remains identified (N_iv ≥ # regressors).

{phang}
{opt levmaxlag(# [#])} controls lag depth for LEVEL-equation instruments
under {cmd:method(system)}. Default is {cmd:levmaxlag(1)} = single
lag (Blundell-Bond 1998 convention): Δy_{t-1} as IV for y_{t-1},
Δx_t for exog x, Δx_{t-1} for endog x. Specify {cmd:levmaxlag(1 2)}
to add Δy_{t-2} and corresponding second-lag differences for exog/endog.
Multiple level lags can improve efficiency under stationarity but are
typically redundant with transformed-equation moments. Use cautiously.

{dlgtab:Transformation and estimation}

{phang}
{opt method(fd|fod|system)} selects the panel transformation used for
moment conditions:

{pmore}
{cmd:fd} — first-difference (Arellano-Bond 1991). Requires strongly
balanced panel or near-balanced (gaps drop obs). Matches {cmd:xthenreg}
results.

{pmore}
{cmd:fod} — forward orthogonal deviations (Arellano-Bover 1995).
Recommended for unbalanced panels. Transforms each observation as a
weighted difference from the mean of future observations; preserves
observations whenever at least one future value exists.

{pmore}
{cmd:system} — system GMM (Blundell-Bond 1998). Stacks FOD equations
with level equations using Δ-lag instruments.

{phang}
{opt grid(#)} sets the number of grid points for the γ search. Default 30.
For Monte Carlo or final estimation, use a denser grid (50-100).

{phang}
{opt trim(#)} sets the trimming rate for the γ grid, using xthenreg's
convention: {cmd:trim(0.2)} means grid spans the 10th to 90th percentile
of {it:q_var}. Must be in [0.01, 0.45]. Default 0.10.

{dlgtab:Inference}

{phang}
{opt citype(grid|none)} chooses CI construction method for the threshold.

{pmore}
{cmd:grid} — grid bootstrap CI via test inversion (Gong-Seo 2026 Algorithm 1),
uniformly valid regardless of model continuity. Default.

{pmore}
{cmd:none} — skip threshold CI; report point estimate and coefficient SE
only.

{pmore}
Asymptotic CI is NOT provided as an option. Under the continuous (kink)
model, γ̂ is n^(1/4)-consistent with a non-normal limit distribution
(Theorem 2 of Gong-Seo 2026), making standard asymptotic intervals
invalid. The grid bootstrap is uniformly valid across continuous and
discontinuous cases.

{phang}
{opt gridci(#)} sets grid points for CI construction. Finer grid gives
more precise CI but slower runtime. Default 25.

{phang}
{opt boot(#)} sets bootstrap replications. Minimum 10 (enforced for debugging
only; a warning is issued if {it:#} < 99). For valid 5% inference follow
Davidson and MacKinnon (2000, {it:Econometric Reviews}): 99+ is the minimum,
299-499 is recommended for production, 99-199 for exploratory runs.
Default 299.

{phang}
{opt nosearch} skips the grid bootstrap CI entirely. Equivalent to
{cmd:citype(none)}.

{dlgtab:Reporting}

{phang}
{opt level(#)} sets the confidence level. Default 95.

{phang}
{opt verbose} prints per-γ progress during the grid bootstrap (one line
per γ point, 25+ lines of output). Useful for monitoring long-running
bootstraps on large panels or for debugging convergence. Without
{opt verbose}, the command prints a compact one-dot-per-γ progress bar.


{title:Examples}

{pstd}
Firm investment model with debt threshold, all regressors exogenous:

{phang2}{cmd:. xtset firm year}{p_end}
{phang2}{cmd:. xtdpthresh invest tobin_q cashflow, qx(debt) method(fd)}{p_end}

{pstd}
Same but with cashflow as endogenous (auto-lag instruments):

{phang2}{cmd:. xtdpthresh invest tobin_q, qx(debt)}{break}
{cmd:      endogenous(cashflow) method(fd)}{p_end}

{pstd}
Unbalanced banking panel, FOD transformation, with user-supplied IV:

{phang2}{cmd:. xtdpthresh roa gdp_growth credit_growth, qx(car)}{break}
{cmd:      endogenous(L.roa) iv(industry_avg_roa) method(fod)}{p_end}

{pstd}
Concise xtabond2-style: lag range and collapse set inside {cmd:iv()}:

{phang2}{cmd:. xtdpthresh invest tobin_q cashflow, qx(debt)}{break}
{cmd:      iv(L2_tobin, maxlag(2 4) collapse) method(fd)}{p_end}

{pstd}
Point estimate only (skip bootstrap, useful for quick checks):

{phang2}{cmd:. xtdpthresh invest tobin_q cashflow, qx(debt)}{break}
{cmd:      method(fd) nosearch}{p_end}

{pstd}
Full grid bootstrap with fine settings:

{phang2}{cmd:. xtdpthresh invest tobin_q cashflow, qx(debt)}{break}
{cmd:      method(fd) grid(50) gridci(40) boot(499) trim(0.10)}{p_end}

{pstd}
System GMM with dense instruments:

{phang2}{cmd:. xtdpthresh y x1 x2, qx(q)}{break}
{cmd:      endogenous(L.y) method(system) grid(40)}{p_end}

{pstd}
Kink (continuity-restricted) model; continuity test compares kink vs
jump:

{phang2}{cmd:. xtdpthresh invest tobin_q cashflow, qx(debt) method(fd) kink}{p_end}
{phang2}{cmd:. xtdpthresh invest tobin_q cashflow, qx(debt) method(fd)}{p_end}
{phang2}{cmd:. display "Continuity test p-value: " e(pval_cont)}{p_end}

{pstd}
The second call (unrestricted) reports {cmd:e(pval_cont)}; if less than
0.05, the jump model is preferred; otherwise kink is not rejected.


{title:Stored results}

{synoptset 22 tabbed}{...}
{p2col 5 22 26 2: Scalars}{p_end}
{synopt:{cmd:e(N)}}number of observations used in GMM (= N_trans + N_level){p_end}
{synopt:{cmd:e(N_raw)}}raw sample size (rows after {it:touse}, before transformation){p_end}
{synopt:{cmd:e(N_trans)}}transformed rows (FD or FOD equations){p_end}
{synopt:{cmd:e(N_level)}}level equation rows (nonzero only for {cmd:method(system)}){p_end}
{synopt:{cmd:e(N_iv)}}number of instruments (columns of Z after dropping zero cols){p_end}
{synopt:{cmd:e(N_units)}}number of panel units used{p_end}
{synopt:{cmd:e(hansen)}}Hansen J over-identification statistic{p_end}
{synopt:{cmd:e(hansen_df)}}Hansen J degrees of freedom (N_iv − #params){p_end}
{synopt:{cmd:e(hansen_p)}}Hansen J p-value ({it:H}_0: moments valid){p_end}
{synopt:{cmd:e(ar1)}}Arellano-Bond AR(1) m-statistic{p_end}
{synopt:{cmd:e(ar1_p)}}AR(1) p-value (typically rejects for FD due to MA(1)){p_end}
{synopt:{cmd:e(ar2)}}Arellano-Bond AR(2) m-statistic{p_end}
{synopt:{cmd:e(ar2_p)}}AR(2) p-value (should NOT reject if moments valid){p_end}
{synopt:{cmd:e(gamma)}}threshold point estimate γ̂{p_end}
{synopt:{cmd:e(gamma_lo)}}grid bootstrap CI lower bound{p_end}
{synopt:{cmd:e(gamma_hi)}}grid bootstrap CI upper bound{p_end}
{synopt:{cmd:e(pval_lin)}}linearity test p-value (Seo-Shin 2016 style){p_end}
{synopt:{cmd:e(pval_cont)}}continuity test p-value (Gong-Seo 2026, Section 4.3 and Theorem 7); missing when {cmd:kink} is set{p_end}
{synopt:{cmd:e(obj)}}GMM objective at γ̂{p_end}
{synopt:{cmd:e(k_exog)}}# exogenous regressors{p_end}
{synopt:{cmd:e(k_endog)}}# endogenous regressors{p_end}
{synopt:{cmd:e(k_inst)}}# user-supplied instruments{p_end}
{synopt:{cmd:e(flag_kink)}}1 if kink option specified{p_end}
{synopt:{cmd:e(flag_static)}}1 if static option specified{p_end}
{synopt:{cmd:e(flag_td)}}1 if td option specified{p_end}
{synopt:{cmd:e(balanced)}}1 if panel is strongly balanced{p_end}

{synoptset 22 tabbed}{...}
{p2col 5 22 26 2: Macros}{p_end}
{synopt:{cmd:e(cmd)}}{cmd:xtdpthresh}{p_end}
{synopt:{cmd:e(depvar)}}name of dependent variable{p_end}
{synopt:{cmd:e(q_var)}}name of threshold variable{p_end}
{synopt:{cmd:e(indepvars)}}names of exogenous regressors{p_end}
{synopt:{cmd:e(endog)}}names of endogenous regressors{p_end}
{synopt:{cmd:e(exog_extra)}}names of extra exogenous regressors (from {opt exogenous()}){p_end}
{synopt:{cmd:e(inst)}}names of user-supplied instruments{p_end}
{synopt:{cmd:e(method)}}transformation method used{p_end}
{synopt:{cmd:e(panelvar)}}panel variable name{p_end}
{synopt:{cmd:e(timevar)}}time variable name{p_end}

{synoptset 22 tabbed}{...}
{p2col 5 22 26 2: Matrices}{p_end}
{synopt:{cmd:e(b)}}coefficient vector (β, δ){p_end}
{synopt:{cmd:e(V)}}variance-covariance matrix{p_end}


{title:Coefficient labels}

{pstd}
The coefficient vector {cmd:e(b)} is labeled following {cmd:xthenreg}
convention.

{pstd}
{bf:Unrestricted (jump) model} — 2K+1 coefficients:

{phang2}{cmd:Lag_y_b}       coefficient on L.{it:depvar} in β (if dynamic){p_end}
{phang2}{cmd:<var>_b}       coefficient on {it:var} in β (β part){p_end}
{phang2}{cmd:cons_d}        intercept jump δ_1{p_end}
{phang2}{cmd:Lag_y_d}       coefficient on L.{it:depvar} in δ (if dynamic){p_end}
{phang2}{cmd:<var>_d}       coefficient on {it:var} in δ (δ part){p_end}

{pstd}
{bf:Kink (continuity-restricted) model} — K+1 coefficients:

{phang2}{cmd:Lag_y_b}       coefficient on L.{it:depvar} (if dynamic){p_end}
{phang2}{cmd:<var>_b}       coefficient on {it:var}{p_end}
{phang2}{cmd:kink_slope}    change in slope of threshold variable at γ (δ_3){p_end}


{title:Specification tests (reported automatically)}

{pstd}
{cmd:xtdpthresh} reports three standard dynamic panel GMM specification
tests after every estimation:

{phang}
{bf:Hansen J over-identification test}: {it:H}_0 = moment conditions are
valid. Statistic {cmd:e(hansen)} = n·ḡ'Ω̂⁻¹ḡ at θ̂ ~ χ²(df), where
df = N_iv − #params. Reject → moment misspecification. {it:Caveat:} under
instrument proliferation the test over-rejects (Roodman 2009). Apply
{cmd:collapse} or {cmd:maxlag()} and monitor whether rejection persists.

{phang}
{bf:Arellano-Bond AR(1) test}: {it:H}_0 = no first-order autocorrelation
in the transformed residuals. Statistic {cmd:e(ar1)} follows {it:N}(0,1)
asymptotically. For FD, AR(1) is expected to REJECT (by construction,
Δε has MA(1) structure); for FOD, both directions are possible.

{phang}
{bf:Arellano-Bond AR(2) test}: {it:H}_0 = no second-order autocorrelation.
{cmd:e(ar2)}. {bf:Should NOT reject at conventional levels.} Rejection
signals that moment conditions are invalid (ε_{t-2} correlated with ε_t),
and the GMM estimator is inconsistent.

{title:Hypothesis tests (bootstrap-based, require citype(grid))}

{pstd}
{cmd:xtdpthresh} reports three specification tests when grid bootstrap
inference is enabled (default, {cmd:citype(grid)}):

{phang}
{bf:Threshold CI} (Gong-Seo 2026 §4.1): 100(1-α)% confidence interval
for γ constructed by test inversion. For each γ_ℓ in a grid, the null
γ = γ_ℓ is tested via the GMM distance statistic D_n(γ_ℓ); γ_ℓ is
accepted if D_n(γ_ℓ) ≤ (1-α)-quantile of bootstrap distribution D*_n.
The CI is the convex hull of accepted γ_ℓ. Uniformly valid regardless
of whether the model is continuous or discontinuous. Stored in
{cmd:e(gamma_lo)}, {cmd:e(gamma_hi)}.

{phang}
{bf:Linearity test} (Seo-Shin 2016): tests H0: δ = 0 (no threshold
effect). Sup-Wald statistic with wild bootstrap under H0. Unit-level
Mammen weights. Low p-value rejects linearity → threshold effect is
significant. Stored in {cmd:e(pval_lin)}.

{phang}
{bf:Continuity test} (Gong-Seo 2026, Section 4.3 and Theorem 7): tests H0: model is
continuous (kink) vs H1: discontinuous (jump). Test statistic
T_n = n·(Q̂_kink(θ̃) - Q̂_jump(θ̂)). Bootstrap p-value under kink DGP.
Low p-value rejects continuity → use the unrestricted (jump) model.
High p-value (> 0.10) supports the kink specification. Stored in
{cmd:e(pval_cont)}. Only computed when {cmd:kink} is NOT set (the test
compares unrestricted vs kink; it makes no sense under kink estimation).


{title:Notes and known limitations}

{pstd}
{cmd:*} {bf:Sample size}: the framework requires large n for reliable
inference. Gong-Seo (2026) Table 1 simulations use n ∈ {400, 800, 1600}.
With n < 100, both point estimation and CI show substantial finite-sample
variance (we verified this on invest.dta subsets: at n=50, both
{cmd:xtdpthresh} and {cmd:xthenreg} produce unstable γ̂ across different
50-firm subsets).

{pstd}
{cmd:*} {bf:FD vs FOD}: on balanced data, FD gives the same point estimate
as {cmd:xthenreg}. FOD identifies a potentially different threshold
because the moment structure differs; FOD's advantage is robustness to
unbalanced patterns (late entry, early exit, random missing).

{pstd}
{cmd:*} {bf:Grid bootstrap runtime}: scales roughly as
{it:gridci × boot × grid × n}. With the per-γ cache and the fixed
first-stage weight (Gong-Seo 2026 Algorithm 1), each bootstrap
replication is a single matrix-vector product. With default settings
{cmd:grid(30) gridci(25) boot(299)}, expect 3-10 minutes per call
for n = 500 (System GMM is slowest, FD fastest). For exploratory work
use {cmd:grid(15) gridci(10) boot(99)} or {cmd:nosearch}; for published
results use the defaults.

{pstd}
{cmd:*} {bf:Linearity test}: the p-value from {cmd:e(pval_lin)} uses a
wild bootstrap implementation of the sup-Wald test of Seo and Shin
(2016). Both {cmd:xtdpthresh} and {cmd:xthenreg} use bootstrap-based
critical values; finite-sample power is comparable when bootstrap reps
are set equal.

{pstd}
{cmd:*} {bf:Continuity test interpretation}: reject H0 (continuous) at
5% → evidence for jump model with discontinuous threshold. Fail to
reject → either model is truly continuous, OR sample is underpowered.
The test is based on Gong-Seo (2026, Section 4.3 and Theorem 7) and uses bootstrap under
the kink DGP; critical values depend on a non-standard limiting
distribution V_1 - V_2 + V_3.


{title:References}

{phang}
Arellano, M., and O. Bover. 1995. Another look at the instrumental
variable estimation of error-components models. {it:Journal of
Econometrics} 68: 29-51.

{phang}
Arellano, M., and S. Bond. 1991. Some tests of specification for panel
data: Monte Carlo evidence and an application to employment equations.
{it:Review of Economic Studies} 58: 277-297.

{phang}
Blundell, R., and S. Bond. 1998. Initial conditions and moment
restrictions in dynamic panel data models. {it:Journal of Econometrics}
87: 115-143.

{phang}
Gong, W., and M. H. Seo. 2026. Bootstraps for dynamic panel threshold
models. {it:Journal of Econometrics} 253: 106153.

{phang}
Hansen, B. E. 1999. The grid bootstrap and the autoregressive model.
{it:Review of Economics and Statistics} 81: 594-607.

{phang}
Kim, S., Y. J. Kim, and M. H. Seo. 2019. Estimation of dynamic panel
threshold model using Stata. {it:Stata Journal} 19: 685-697.

{phang}
Seo, M. H., and Y. Shin. 2016. Dynamic panels with threshold effect and
endogeneity. {it:Journal of Econometrics} 195: 169-186.


{title:Authors}

{pstd}
Duy Chinh Nguyen{break}
School of Business, International University, Ho Chi Minh City, Vietnam{break}
Vietnam National University Ho Chi Minh City, Vietnam{break}
Email: {browse "mailto:ndchinh@hcmiu.edu.vn":ndchinh@hcmiu.edu.vn}{break}
ORCID: {browse "https://orcid.org/0000-0002-9157-9358":0000-0002-9157-9358}

{pstd}
Nhat Duy Lai (corresponding author){break}
Faculty of Finance and Accounting, Saigon University,{break}
273 An Duong Vuong St, Cho Quan Ward, Ho Chi Minh City, Vietnam{break}
Email: {browse "mailto:lnduy@sgu.edu.vn":lnduy@sgu.edu.vn}{break}
ORCID: {browse "https://orcid.org/0009-0008-5365-2893":0009-0008-5365-2893}


{title:Also see}

{psee}
Online: {help xtset}, {help xthenreg} (Kim-Kim-Seo 2019), {help xtabond},
{help xtabond2} (Roodman 2009){p_end}