Threshold models Endogenous threshold models in cross section data Panel threshold model with endogenous threshold variable Monte Carlo Simulations Future studies Panel Threshold regression models with endogeneous threshold variable Chien-Ho Wang 1 Eric S. Lin 2 1 National Taipei University 2 National Tsinghua University Panel data meeting, 2010 Chien-Ho Wang Panel Threshold regression models with endogeneous thresh
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Panel Threshold regression models with endogeneous threshold …€¦ · Panel Threshold regression models with endogeneous threshold variable Chien-Ho Wang 1 Eric S. Lin 2 1National
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Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
1National Taipei University 2National Tsinghua University
Panel data meeting, 2010
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Outline
1 Threshold models
2 Endogenous threshold models in cross section data
3 Panel threshold model with endogenous threshold variable
4 Monte Carlo Simulations
5 Future studies
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Threshold models under different types of data
Cross section data
yi =
{β1xi + ui if zi ≤ γβ2xi + ui if zi > γ
Time series data
yt =
{β1yt−1 + ut if yt−1 ≤ γβ2yt−1 + ut if yt−1 > γ
Panel data
yit =
{β1xit + uit if zit ≤ γβ2xit + uit if zit > γ
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Threshold models under different types of data
Cross section data
yi =
{β1xi + ui if zi ≤ γβ2xi + ui if zi > γ
Time series data
yt =
{β1yt−1 + ut if yt−1 ≤ γβ2yt−1 + ut if yt−1 > γ
Panel data
yit =
{β1xit + uit if zit ≤ γβ2xit + uit if zit > γ
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Threshold models under different types of data
Cross section data
yi =
{β1xi + ui if zi ≤ γβ2xi + ui if zi > γ
Time series data
yt =
{β1yt−1 + ut if yt−1 ≤ γβ2yt−1 + ut if yt−1 > γ
Panel data
yit =
{β1xit + uit if zit ≤ γβ2xit + uit if zit > γ
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Possible estimation problems
If we want to obtain consistent estimators of β1 and β2, thethreshold value γ must be estimated consistency in advance.
Under zit is correlated with uit , We cannot obtain consistentestimator for γ.
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Possible estimation problems
If we want to obtain consistent estimators of β1 and β2, thethreshold value γ must be estimated consistency in advance.
Under zit is correlated with uit , We cannot obtain consistentestimator for γ.
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Threshold model with endogenous threshold variable
Consider the threshold model
yi =
{β1xi + ui if qi ≤ γβ2xi + ui if qi > γ
(1)
If we want to obtain consistent estimators of β1 and β2, qicannot be correlated with ui . If qi is correlated with ui , we neednew method to estimate γ, β1 and β2.
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Bias-Correction Estimator for endogenous thresholdmodels
Kourtellos, Stengos and Tan (2007) consider a threshold modelwith endogenous threshold variables like Equation(2).
yi =
{xiβ1 + ui if qi ≤ γxiβ2 + ui if qi > γ
(2)
zi are exogenous variables. qi is an endogenous variable. Thethreshold equation is
qi = ziπ + υi . (3)
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Define the indicator variable
Ii =
{1 iff υi ≤ γ − ziπ0 iff υi > γ − ziπ
The joint distribution between ui and υi is defined as(uiυi
)|xi , zi ∼ N
(0
(σ2
u σuυ
σuυ 1
) )Use the relationship between ui and εi .(
εiυi
)=
(1 −σuυ
0 1
) (uiυi
)
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
When two regimes have the same error structure, THRETmodel can be estimate by
yi = xiβ2 + xi(γ)ϕ + κλi(γ − ziπ) + ei , (4)
where xi(γ) = xi I(qi ≤ γ) and ϕ = β2 − β1.This estimation method looks like sample-selection model. Themain difference about these two model is THRET model usingall data.
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Estimation of Threshold model with endogenousthreshold variable
Kourtellos, Stengos and Tan (2007) use three steps to obtainconsistent estimator.
1 Estimate the parameter vector π in Equation (3) by leastsquare.
2 Estimate the threshold value γ by minimizing aconcentrated two stage least square criterion using π fromfirst stage.
Sn(γ) =n∑
i=1
(yi − xiβ1 − xi(γ)ϕ− κλi(γ − zi π))2.
3 Estimate the lease square estimates of the slopparameters based on the split samples implied by γ.
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
PTHET models
We consider three kinds of panel threhold models withendogeneous threshold variable
2 Assumption 2: {yit , xit , qit , eit : 1 ≤ i ≤ n, 1 ≤ t ≤ T} arefrom balanced panel data
3 Assumption 3: uit |zit ∼ N (0, 1)4 Assumption 4’: The joint distribution between ∆eit and uit
is defined as:[∆eituit
]|xit , zit ∼ N
(0,
[σ2
e γjγj 1
] ),
where γj is covariance between ∆eit and uit , γj = γ1 whenqit ≤ θ and γj = γ2 when qit > θ.
5 Assumption 5: n →∞ and T is fixed.
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Estimation of panel threshold model with endogenousthreshold variable
Using first difference transformation to eliminate fixedeffect ηi in Equation (11).Estimate the parameter vector π in Equation (12) by leastsquare.Estimate the threshold value γ by minimizing aconcentrated two stage least square criterion using π fromfirst stage.
we use the first difference transformation to cancel out fixedeffect ηi in main equation.
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
For the threshold equation, we cannot use first differencetransformation to elminiate fixed effect. The least squaredummy variable method is used to obtain π and ci . Thebias-correction estimator γ can be estimated under π and ci
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Estimation of panel threshold model with endogenousthreshold variable
Using first difference transformation to eliminate fixedeffect ηi in Equation (13).Estimate the parameter vector π and ci in Equation (14) byleast square dummy variable method.Estimate the threshold value γ by minimizing aconcentrated two stage least square criterion using π andci from first stage.
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
Estimate the lease square estimates of the slopparameters based on the split samples implied by γ.
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
The data generation process considered is based on Equation(11), where the threshold value γ = 2. We fixed β2 = 1 andβ1 − β2 = (0.05, 0.5). For threshold equation, we considerwithout fixed effect case:
qit = 2 + 3z1it + 3z2it + uit . (15)
where xit , z1it , and z2it are identical and mutually independentstandard normal distributed. The fixed effects ci and ηi followi .i .d . N (0, 1).
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Threshold modelsEndogenous threshold models in cross section data
Panel threshold model with endogenous threshold variableMonte Carlo Simulations
Future studies
For controlling the endogeneity, we generate (εit , uit) ∼ i .i .d .
N (0, 1), and eit = σ2e(κoεit + (1− κo)uit)/
√κ2
o + (1− κo)2. Sothat the correlation between eit and threshold variable qit is
given by ρ = κo/√
κ2o + (1− κo)2. We set κo = (0.05, 0.5, 0.9)
to simulate strong and weak correlations between eit and uit .
Chien-Ho Wang Panel Threshold regression models with endogeneous threshold variable
Table: Finite-sample Performance for Threshold Value Estimation(β1 − β2 = 0.05; γ = 2)
Method T = 10 T = 20
κ0 = 0.05 mean median MSE mean median MSEN = 50 pthet -0.665 -0.036 24961.78 -0.628 0.265 25832.25