chapter 3. panel threshold regression models · 2. the panel threshold regression model example...

Chapter 3. Panel Threshold Regression ModelsSchool of Economics and Management - University of Geneva

Christophe Hurlin (Université of Orléans)

University of Orléans

May 2018

C. Hurlin (University of Orléans) Advanced Econometrics II May 2018 1 / 86

1. Introduction

In econometrics, threshold regression models are a category ofregime-switching models in which

The slope parameters vary according to a "regime" switchingmecanism that depends on a threshold variable,

The regime is observable ex-post, contrary to the Markovian regimeswitching models.


1. Introduction

De�nition (threshold regression model)A typical threshold regression model is given by

yt = α+ β00xt + β01xth (qt ; θ) + εt

where β0 and β1 are K � 1 vectors, qt is a threshold variable, θ a vectorof parameters and h (qt ; θ) a transition function.


1. Introduction

Example (threshold regression model)If the transition function is a binary function such that

h (qt ; c) =

(1 if qt � c0 if qt < c

,

then the model is simply de�ned by

yt = α+ β00xt + β01xtI(qt�c ) + εt

where I(.) is the indicator function and c is a location parameter.


1. Introduction

Example (threshold regression model)If the transition function is a binary function, we have a regime-switchingmechanism for the slope parameters that depends on the thresholdvariable and a location parameter c

yt =

(α+

�β00 + β01

�xt + εt if qt � c

α+ β00xt + εt if qt < c,

In this simple case, we have two "regimes" for the slope parameters, i.e.β0 and β0 + β1.


1. Introduction

Remarks

The transition function h (qt ; θ) may be smoother than a binaryfunction.

In general, the transition function h (qt ; θ) is assumed to veri�ed

0 � h (qt ; θ) � 1

Sometimes, we have

limqt!+∞

h (qt ; θ) = 1 limqt!�∞

h (qt ; θ) = 0


1. Introduction

Example (Logistic transition function)A logistic transition function is de�ned as:

h (qt ;γ, c) =1

1+ exp (�γ (qt � c)), γ > 0


1. Introduction

Figure: Logistic transition function with c = 0

6 4 2 0 2 4 6threshold variable q

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1tra

nsiti

on fu

nctio

n h(

q,)

=0.5=1=5


1. Introduction

Panel data

The threshold regressions in panel data models allow to model theheterogeneity of the slope parameters.

These models give a parametric approach of the heterogeneitywhich is associated to an economic "story" (interpretation).


Introduction

The outline of this chapter is the following:

Section 1: Introduction

Section 2: Panel Threshold Regression (PTR) Model

Section 3: Panel Smooth Threshold Regression (PSTR) Model


Section 2

The Panel Threshold Regression (PTR) Model


2. The panel threshold regression model

Objectives

1 Introduce the panel threshold regression (PTR) model.

2 Understand the link with heterogeneous panel models.

3 Understand the link with time varying parameters panel models.

4 Understand the di¤erences with random coe¢ cient models.



The Panel Threshold Regression (PTR) model has beenintroduced by Hansen (1999).

In this paper, threshold regression methods are developed fornon-dynamic panels with individual �xed e¤ects.

Hansen, B. E. 1999. Threshold e¤ects in non-dynamic panels: estimation,testing, and inference, Journal of Econometrics, 93, 334-368



De�nition (panel threshold regression model)

The panel threshold regression (PTR) model is de�ned as

yit = αi + β01xitI(qit�c ) + β02xitI(qit>c ) + εit

where the dependent variable yit is scalar, αi is a �xed e¤ect, the thresholdvariable qit is scalar, the regressor xit is a k vector, and I(.) is the indicatorfunction and c is a threshold parameter.



Assumptions

The threshold variable is exogeneous or at least predetermined(qit = yi ,t�d with d � 1).

For the identi�cation of β1 and β3, it is required that the elements ofxit are not time invariant.

The threshold variable qit is not time invariant.

The error εit is assumed to be i.i.d. with E (εit ) = 0 and V (εit ) = σ2ε .



Panel Threshold Regression model

An alternative speci�cation of the PTR is

yit = αi + β00xit + β03xitI(qit�c ) + εit

or equivalently

yit = αi + β00xit + β03xith (qit ; c) + εit

whith h (qit ; c) = I(qit�c ) the binary transition function

yit =

(αi +

�β00 + β03

�xit + εit if qit � c

αi + β03xit + εit if qit > c,



Figure: Transition function h (qit ; c) with c = 0


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q,c)



De�nition (heterogeneous panel data model )The PTR model can be viewed as an heterogeneous and time-varyingparameters panel data model

yit = αi + β0itxit + εit =

(αi + β01xit + εit if qit � cαi + β02xit + εit if qit > c

,

where the marginal e¤ect (slope parameters) satisfy

∂yit∂xit

= βit =

(β1 if qit � cβ2 if qit > c

,



Heterogeneous panel data model

yit =


,

At a given time t, two cross-section units i and j may have two di¤erentslope parameters

∂yit∂xit

= β1 6=∂yjt∂xjt

= β2 if qit � c and qjt > c .



Time-varying parameter panel data model

yit =


,

A given cross-section unit i may have di¤erent slope parameters atdi¤erent dates t and s

∂yit∂xit

= β1 6=∂yis∂xis

= β2 if qit � c and qis > c .



Example (Marginal e¤ect and PTR)Consider a PTR model with K = 1 regressor such that

yit = αi + 0.5xitI(qit�3) � 0.8xitI(qit>3) + εit

Then, the marginal e¤ect of the regressor xit on yit is equal to

∂yit∂xit

= 0.5I(qit�3) � 0.8I(qit>3) =(

0.5 if qit � 3�0.8 if qit > 3

,



Figure: Marginal e¤ect with K = 1, β1 = 0.5, β2 = �0.8 and c = 3

0 10 20 30 40 50Time

2

1

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resh

old v

ariab

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Mar

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effe

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yit/d

x it



Panel Threshold Regression model

The PTR model is an alternative to the random coe¢ cient model

Panel Threshold Regression βit = f (qit ; c) βit is a constant term

Economic interpretation (through qit)

Random Coe¢ cient Model βi i.i.d.�

β,∆�

βi is a random variable

No economic interpretation



Estimation by NLS

The PTR model can be rewritten as:

yit = αi + β0xit (c) + εit

xit (c)(2K ,1)

=

�xitI(qit�c )xitI(qit>c )

�β

(2K ,1)=

�β1β2

�



Estimation by NLS

In order to eliminate the individual e¤ects, we apply the Withintransformation

y �it = β0x�it (c) + ε�it

y �it = yit � y i y i = T�1

T

∑t=1yit

x�it (c) = xit (c)� x i (c) x i (c) = T�1T

∑t=1xit (c)

ε�it = εit � εi εi = T�1T

∑t=1

εit



Let us de�ne

y �i(T ,1)

=

0BB@y �i ,1y �i ,2...y �it

1CCA X �i (c)(T ,2K )

=

0BB@x1,i ,1 (c)

0

x1,i ,2 (c)0

...

x1,it (c)0

1CCA ε�i(T ,1)

=

0BB@ε�i ,1ε�i ,2...ε�it

1CCA

Y �(Tn,1)

=

0BB@y �1y �2...y �n

1CCA X � (c)(Tn,2K )

=

0BB@X �1 (c)X �2 (c)...X �n (c)

1CCA ε�(Tn,1)

=

0BB@ε�1ε�2...ε�n

1CCA



De�nition (given threshold)

For any given threshold c , the slope coe¤cient β can be estimated byordinary least squares (OLS).

bβ (c) = �X � (c)0 X � (c)��1 X � (c)0 YThe vector or residuals is given by

bε� (c) = Y � � X � (c) bβ (c)and the sum of squared errors is

SSR (c) = bε� (c)0bε� (c)C. Hurlin (University of Orléans) Advanced Econometrics II May 2018 27 / 86


De�nition (Threshold estimation)The estimation if the threshold parameter c is obtained by minimization ofthe concentrated sum of squared

bc = argminc2Θ

SSR (c)



Remarks

It is undesirable for a threshold c to be selected which sorts too fewobservations into one or the other regime.

This possibility can be excluded by restricting the search in to valuesof c such that a minimal percentage of the observations (say, 1% or5%) lie in each regime.

c 2 Θ =hquantile

�fqitgN ,Ti=1,t=1 , 0.05

�, quantile

�fqitgN ,Ti=1,t=1 , 0.95

�iGiven bc , we can compute the estimates for β as

bβ = β (bc) = � β1 (bc)β2 (bc)

�



De�nition (extension to r + 1 regimes)A general speci�cation of the PTR can be proposed with r + 1 > 2regimes or r threshold parameters c1, ..cr .

yit = αi +r

∑j=1

β0jxitI(cj�1<qit�cj ) + εit

with c0 = �∞ and cr+1 = +∞.



Example (PTR with three regimes)

For instance, a three-regimes model (r = 3) with 2 threshold parametersc1 and c2 is given by

yit = αi + β01xitI(qit�c1) + β02xitI(c1<qit�c2)+β03xitI(qit>c2) + εit



Example (PTR with four regimes)

For instance, a three-regimes model (r = 4) with 3 threshold parametersc1, c2 and c3 is given by

yit = αi + β01xitI(qit�c1) + β02xitI(c1<qit�c2)+β03xitI(c2<qit�c3) + β04xitI(qit>c3) + εit



Inference

If one comes to test whether the threshold e¤ect is staticallysigni�cant in the model with two regimes, the null hypothesis is

H0 : β1 = β2

This null hypothesis corresponds to the hypothesis of no thresholde¤ect.

Under H0 the model is then equivalent to a linear model.

yit = αi + β0xit + εit

with β = β1 = β2.



Inference

The null hypothesis H0 : β1 = β2 can be tested by a standard test.

If we note S0 the sum of squared of the linear model, the approximatelikelihood ratio test of H0 is based on:

F1 =SSR0 � SRR1 (bc)bσ2

where bσ2 denotes a convergent estimate of σ2.



Inference

The main problem is that under the null, the threshold parameter c isnot identi�ed (nuisance parameter).

Consequently, the asymptotic distribution of F1 is not standard and,in particular, does not correspond to a chi-squared distribution.

This issue has been largely studied in literature devoted to thresholdmodels, notably since the seminal paper by Davies (1977, 1987).

One solution is to use a bootstrap procedure to determine theasymptotic distribution of the statistic F1.



Number of regimes (thresholds)

The same kind of inference procedure can be applied in order todetermining the number of thresholds.

A likelihood ratio test of one threshold versus two thresholds is basedon the statistic

F2 =SSR1 (bc)� SSR2 (bc1,bc2)bσ2

where bc1 and bc2 denote the threshold estimates of the model withthree regimes , and SSR2 (bc1,bc2) denotes the corresponding residualsum of squares.




The hypothesis of one threshold is rejected in favor of two thresholdsif F2 is larger than its critical value.

The corresponding asymptotic p-value can be approximated bybootstrap simulations (Hansen, 1999).

If the model with two thresholds (three regimes) is not rejectedaccepted, we to test the hypothesis of two thresholds (three regimes)against the alternative of three thresholds (four regimes).




The corresponding likelihood ratio statistic, denoted F3., is de�ned as:

F3 =SSR2 (bc1,bc2)� SSR3 (bc1,bc2,bc3)bσ2

where SSR3 (bc1,bc2,bc3) denotes the residual sum of squares of themodel with four regimes and three threshold parameters.

Thus, a sequential procedure based on F1, F2, F3, etc. allows todetermining the number of regimes



ExampleCandelon, Colletaz, Hurlin (2012) investigate the threshold e¤ects in theproductivity of infrastructure investment in developing countries.

yit =

(ai + α1kit + β1hit + γ1xit + εit

ai + α2kit + β2hit + γ2xit + εit

if qit � λ

if qit > λ.

where yit is the aggregate added value, kit is physical capital, hit is humancapital, xit is infrastructure stock.

Candelon B., Colletaz G., Hurlin C. (2013), Network E¤ects andInfrastructure Productivity in Developing Countries, Oxford Bulletin ofEconomics and Statistics, 75(6), 887-913.



Key Concepts Section 2

1 Panel Threshold Regression Model

2 Heterogeneous and time-varying parameters

3 Non Linear Least Squares

4 Inference for the number of regimes

5 Davies problem


Section 2

The Panel Smooth Threshold Regression

(PSTR) Model


3. The panel smooth threshold regression model

Objectives

1 Introduce the panel smooth threshold regression (PSTR) model.

2 Understand the link with heterogeneous panel models.

3 Understand the link with time varying parameters panel models.



The Panel Smooth Threshold Regression (PSTR) model has beenintroduced by Gonzalez, Teräsvirta and van Dijk (2005).

This model can be viewed as a generalization of the PTR model.

González, A., Teräsvirta, T., van Dijk, D., 2005. Panel smooth transitionregression model. Working Paper Series in Economics and Finance, vol. 604.



De�nition (PSTR model)The PSTR with two extreme regimes can be de�ned as

yit = αi + β00xit + β01xitg (qit ;γ, c) + εit

where g (qit ;γ, c) is a transition function, qit a threshold variable, c alocation parameter and γ a slope parameter.



Transition function

The transition function g (qit ;γ, c) is a continuous function of theobserved variable qit

The transition function is normalized to be bounded between 0 and 1.

0 � g (qit ;γ, c) � 1



De�nitionGonzalez, Teräsvirta and van Dijk (2005) consider a logistic transitionfunction

g (qit ;γ, c) =1

1+ exp (�γ (qit � c)), γ > 0



Figure: Logistic transition function with c = 0


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Remark 1

If γ ! ∞, the logistic function tends to an indicator function and thePSTR corresponds to a PTR model

limγ!∞

g (qit ;γ, c) = I(qit�c ) =

(1 if qit � c0 if qit < c

,

yit = αi + β00xit + β01xitI(qit�c ) + εit



Remark 2

If γ ! 0, the logistic function tends to an indicator function and thePSTR corresponds to a linear panel model

limγ!∞

g (qit ;γ, c) =12

8qit

yit = αi + β0xit + εit β =β0 + β12



De�nition (heterogeneous panel data model )The PSTR model can be viewed as an heterogeneous and time-varyingparameters panel data mode since the marginal e¤ect (slope parameters)satisfy

∂yit∂xit

= βit = β0 + β1g (qit ;γ, c)

as soon as qit does not belong to xit , with by convention

β0 � βit � β0 + β1



Remark 3

Even there are two "extreme" regimes/values for the slope parameters, infact there is an in�nity of regimes (possible values) for the slopeparameters βit given the observed values for qit :

βit = β0 + β1g (qit ;γ, c)

β0 � βit � β0 + β1



De�nition (generalization to r+1 regimes)The PSTR model can be generalised to r + 1 extreme regimes as follows:

yit = αi + β00xit +r

∑j=1

β0jxit gj (qit ;γj , cj ) + εit

where the r transition functions gj (qit ;γj , cj ) depend on the slopeparameters γj and on the location parameters cj .



Marginal e¤ects

If the transition variable does not belong to the set of regressors, the slopeparameters become:

βit =δyitδxit

= β0 +r

∑j=1

βj gj (qit ;γj , cj )



Marginal e¤ects

If the transition variable does not belong to the set of regressors, the slopeparameters become:

βit =δyitδxit

= β0 +r

∑j=1

βj gj (qit ;γj , cj ) +r

∑j=1

βjδgj (qit ;γj , cj )

δqitqit



Estimation

The estimation of the parameters of the PSTR model consists ofeliminating the individual e¤ects αi and then in applying NLS to thetransformed model

See Gonzàlez et al. (2005) or Colletaz and Hurlin (2006), for moredetails.

González, A., Teräsvirta, T., van Dijk, D., 2005. Panel smooth transitionregression model. Working Paper Series in Economics and Finance, vol. 604.

Colletaz, G., Hurlin, C., 2006. Threshold e¤ects in the public capitalproductivity: an international panel smooth transition approach. Documentde Recherche du Laboratoire d�Economie d�Orléans. 2006-1.



Inference

Gonzàlez et al. (2005) propose a testing procedure in order (i) to testthe linearity against the PSTR model and (ii) to determine thenumber, r , of transition functions, i.e. the number of extreme regimeswhich is equal to r + 1.

Testing the linearity in a PSTR model can be done by testing

H0 : γ = 0 or H0 : β0 = β1

But in both cases, the test will be non standard since under H0 thePSTR model contains unidenti�ed nuisance parameters.



Inference

A possible solution is to replace the transition function gj (qit ,γj , cj )by its �rst-order Taylor expansion around γ = 0 and to test anequivalent hypothesis in the auxiliary regression

yit = αi + θ0 xit + θ1 xit qit + εit

The parameters θi are proportional to the slope parameter γ of thetransition function.

Thus, testing the linearity of the model against the PSTR simplyconsists of testing

H0 : θ1 = 0



De�nition (linearity/homogeneity test)

The F-statistic for the homogeneity assumption H0 : θ1 = 0 is thende�ned by:

LMF = (SSR0 � SSR1)/ [SSR0/(TN �N � 1)]

with SSR0 the panel sum of squared residuals under H0 (linear panelmodel with individual e¤ects) and SSR1 the panel sum of squared residualsunder H1 (PSTR model with two regimes). Under the null hypothesis, theF-statistic has an approximate F (1,TN �N � 1) distribution.



Choice of number of transitions

The logic is similar when it comes to testing the number of transitionfunctions in the model or equivalently the number of extreme regimes.

We use a sequential approach by testing the null hypothesis of noremaining nonlinearity in the transition function.




We want to test whether there is one transition function (H0 : r = 1)or whether there are at least two transition functions (H0 : r = 2).Let us assume that the model with r = 2 is de�ned as:

yit = αi + β0xit + β1 xit g1(qit ;γ1, c1) + β2 xit g2(qit ;γ2, c2) + εit

The logic of the test consists in replacing the second transitionfunction by its �rst-order Taylor expansion around γ2 = 0 and then intesting linear constraints on the parameters in

yit = αi + β0xit + β1 xit g1(qit ;γ1, c1) + θ1 xit qit + ε�it

and the test of no remaining nonlinearity is simply de�ned byH0 : θ1 = 0.




Let us denote SSR0 the panel sum of squared residuals under H0, i.e.in a PSTR model with one transition function.

yit = αi + β0xit + β1 xit g1(qit ;γ1, c1) + εit

Let us denote SSR1 the sum of squared residuals of the transformedmodel.

yit = αi + β0xit + β1 xit g1(qit ;γ1, c1) + θ1 xit qit + ε�it

The F-statistic LMf can be calculated in the same way by adjustingthe number of degrees of freedom.



Testing procedure

1 Given a PSTR model with r = r �, we test the null H0 : r = r �

against H1 : r = r � + 1.

2 If H0 is not rejected the procedure ends.

3 Otherwise, the null hypothesis H0 : r = r � + 1 is tested againstH1 : r = r � + 2.

4 The testing procedure continues until the �rst acceptance of H0.

5 Given the sequential aspect of this testing procedure, at each step ofthe procedure the signi�cance level must be reduced by a constantfactor 0 < ρ < 1 in order to avoid excessively large models. Gonzàlezet al. (2005) suggest ρ = 0.5.



ExampleHurlin, Rabaud and Fouquau (2008) considers a PSTR model fordetermining the relative in�uence of �ve factors on the Feldstein andHorioka result for OECD countries.

Iit = αi + β0Sit + β1Sit g(qit ; c) + εit

They consider �ve main factors (as potential threshold variable) generallyconsidered in this literature: (i) economic growth, (ii) demography, (iii)degree of openness, (iv) country size and (v) current account balance.

Hurlin C., Rabaud I., and Fouquau J. (2008) The Feldstein-Horioka Puzzle:a PSTR Approach. Economic Modelling, 25, 284-299.



Key Concepts Section 3

1 Panel Smooth Threshold Regression Model

2 Marginal e¤ects (slope parameters)

3 Non Linear Least Squares

4 Inference for the number of regimes


End of Chapter 3

Christophe Hurlin (University of Orléans)


chapter 3. panel threshold regression models · 2. the panel threshold regression model example...

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