Motivation Literature Methodology Empirical Results Conclusions The Portuguese Business Cycle: Chronology and Duration Dependence Vtor Castro University of Coimbra and NIPE, Portugal Presentation at the 5th Annual Meeting of the Portuguese Economic Journal University of Aveiro, Portugal 8-9 July, 2011 Vtor Castro The Portuguese Business Cycle
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MotivationLiterature
MethodologyEmpirical Results
Conclusions
The Portuguese Business Cycle:Chronology and Duration Dependence
Vítor Castro
University of Coimbra and NIPE, Portugal
Presentation at the5th Annual Meeting of the Portuguese Economic Journal
University of Aveiro, Portugal8-9 July, 2011
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Outline
1 Motivation
2 Literature
3 Methodology
4 Empirical Results
5 Conclusions
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Motivation
The NBER has been publishing a business cycle chronologyfor the United States since 1929.
The ECRI and the CEPR have recently started to producesimilar chronologies for some market oriented economies andthe Euro Area, respectively.
However, to our knowledge, no national or internationalorganization or scienti�c study has produced, so far, abusiness cycle chronology for Portugal.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Motivation
The NBER has been publishing a business cycle chronologyfor the United States since 1929.
The ECRI and the CEPR have recently started to producesimilar chronologies for some market oriented economies andthe Euro Area, respectively.
However, to our knowledge, no national or internationalorganization or scienti�c study has produced, so far, abusiness cycle chronology for Portugal.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Motivation
The NBER has been publishing a business cycle chronologyfor the United States since 1929.
The ECRI and the CEPR have recently started to producesimilar chronologies for some market oriented economies andthe Euro Area, respectively.
However, to our knowledge, no national or internationalorganization or scienti�c study has produced, so far, abusiness cycle chronology for Portugal.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Motivation
This study intends to identify a chronology for thePortuguese business cycle, and...
At the same time, to test for the presence of durationdependence in expansions and contractions.
Employing a duration dependent Markov-switching vectorautoregressive model (DDMSVAR).
This model alows: (i) to date the business cycle; (ii) andcontrol for the presence of duration dependence in its phases.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Motivation
This study intends to identify a chronology for thePortuguese business cycle, and...
At the same time, to test for the presence of durationdependence in expansions and contractions.
Employing a duration dependent Markov-switching vectorautoregressive model (DDMSVAR).
This model alows: (i) to date the business cycle; (ii) andcontrol for the presence of duration dependence in its phases.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Motivation
This study intends to identify a chronology for thePortuguese business cycle, and...
At the same time, to test for the presence of durationdependence in expansions and contractions.
Employing a duration dependent Markov-switching vectorautoregressive model (DDMSVAR).
This model alows: (i) to date the business cycle; (ii) andcontrol for the presence of duration dependence in its phases.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Motivation
This study intends to identify a chronology for thePortuguese business cycle, and...
At the same time, to test for the presence of durationdependence in expansions and contractions.
Employing a duration dependent Markov-switching vectorautoregressive model (DDMSVAR).
This model alows: (i) to date the business cycle; (ii) andcontrol for the presence of duration dependence in its phases.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Outline
1 Motivation
2 Literature
3 Methodology
4 Empirical Results
5 Conclusions
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (i) business cycle chronology available
No need to date the business cycle => traditional durationanalysis can be employed to test for duration dependence.
Great attention to the US business cycle because its turningpoint dates are well documented by the NBER.
ECRI and CEPR chronologies => other countries under thescope of other studies.
Do business cycles exhibit positive duration dependence?
Are expansions and contractions in economic activity morelikely to end as they become older?
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (i) business cycle chronology available
No need to date the business cycle => traditional durationanalysis can be employed to test for duration dependence.
Great attention to the US business cycle because its turningpoint dates are well documented by the NBER.
ECRI and CEPR chronologies => other countries under thescope of other studies.
Do business cycles exhibit positive duration dependence?
Are expansions and contractions in economic activity morelikely to end as they become older?
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (i) business cycle chronology available
No need to date the business cycle => traditional durationanalysis can be employed to test for duration dependence.
Great attention to the US business cycle because its turningpoint dates are well documented by the NBER.
ECRI and CEPR chronologies => other countries under thescope of other studies.
Do business cycles exhibit positive duration dependence?
Are expansions and contractions in economic activity morelikely to end as they become older?
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (i) business cycle chronology available
No need to date the business cycle => traditional durationanalysis can be employed to test for duration dependence.
Great attention to the US business cycle because its turningpoint dates are well documented by the NBER.
ECRI and CEPR chronologies => other countries under thescope of other studies.
Do business cycles exhibit positive duration dependence?
Are expansions and contractions in economic activity morelikely to end as they become older?
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (i) business cycle chronology available
Parametric duration models:Sichel (1991), Diebold et al. (1993), Zuehlke (2003) andDavig (2007): evidence of positive duration dependence forpre-WWII expansions and post-WWII contractions in the US.
Diebold et al. (1990): reach a similar conclusion for France,Germany and the UK in the pre-WWII period.
Abderrezak (1998): evidence of positive duration dependencein the whole growth cycles and growth phases (upswings anddownswings) for a group of 11 countries.
Castro (2010): evidence of positive duration dependence forboth expansions and contractions (using a discrete-timeduration model over a panel of 13 industrial countries). Healso notices that the probability of a contraction endingincreases more quickly with its age than an expansion.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (ii) chronology not needed or unavailable
Other approaches regard the business cycle as an unobservedstochastic process.
The business cycle is seen as the outcome of a Markov processthat switches between the states of expansion and contraction(Hamilton, 1989).
Turning point dates identi�ed by the NBER, ECRI or CEPRare not necessary.
Very useful to identify the business cycle chronology when it isnot available at all.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (ii) chronology not needed or unavailable
Other approaches regard the business cycle as an unobservedstochastic process.
The business cycle is seen as the outcome of a Markov processthat switches between the states of expansion and contraction(Hamilton, 1989).
Turning point dates identi�ed by the NBER, ECRI or CEPRare not necessary.
Very useful to identify the business cycle chronology when it isnot available at all.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (ii) chronology not needed or unavailable
Other approaches regard the business cycle as an unobservedstochastic process.
The business cycle is seen as the outcome of a Markov processthat switches between the states of expansion and contraction(Hamilton, 1989).
Turning point dates identi�ed by the NBER, ECRI or CEPRare not necessary.
Very useful to identify the business cycle chronology when it isnot available at all.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (ii) chronology not needed or unavailable
Other approaches regard the business cycle as an unobservedstochastic process.
The business cycle is seen as the outcome of a Markov processthat switches between the states of expansion and contraction(Hamilton, 1989).
Turning point dates identi�ed by the NBER, ECRI or CEPRare not necessary.
Very useful to identify the business cycle chronology when it isnot available at all.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (ii) chronology not needed or unavailable
Markov-switching models:
Hamilton (1989): �rst to implement this kind of analysis tothe US business cycle.
Krolzig (1997), Krolzig (2001), Artis et al. (2004) andKrolzig & Toro (2005): apply some variants of these modelsto identify the European business cycle.
Schirwitz (2009): also employs them to identify a chronologyfor the German business cycle.
Ignore that the switching process can be a¤ected by theduration of each state.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (ii) chronology not needed or unavailable
Markov-switching models:
Hamilton (1989): �rst to implement this kind of analysis tothe US business cycle.
Krolzig (1997), Krolzig (2001), Artis et al. (2004) andKrolzig & Toro (2005): apply some variants of these modelsto identify the European business cycle.
Schirwitz (2009): also employs them to identify a chronologyfor the German business cycle.
Ignore that the switching process can be a¤ected by theduration of each state.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (ii) chronology not needed or unavailable
Duration dependent Markov-switching models:
Allow for state transition probabilities to be durationdependent.
Durland & McCurdy (1994), Kim & Nelson (1998) andPelagatti (2001, 2002): evidence of duration dependence forthe US contractions but not for expansions after WWII.
Perruchoud (2008): similar conclusion for the Swiss businesscycle.
Kim (1996) and Iiboshi (2007): evidence of positiveduration dependence for, respectively, Korean and Japaneseexpansions and contractions.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (ii) chronology not needed or unavailable
Duration dependent Markov-switching models:
Allow for state transition probabilities to be durationdependent.
Durland & McCurdy (1994), Kim & Nelson (1998) andPelagatti (2001, 2002): evidence of duration dependence forthe US contractions but not for expansions after WWII.
Perruchoud (2008): similar conclusion for the Swiss businesscycle.
Kim (1996) and Iiboshi (2007): evidence of positiveduration dependence for, respectively, Korean and Japaneseexpansions and contractions.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (iii) additional remarks
Advantages of Pelagatti�s (2001, 2002) DDMSVAR:
Applies a Bayesian approach => inference does not depend onthe sample size and it is not conditional on the estimatedparameters.
Considers a multivariate speci�cation => inference using amulti-move Gibbs sampler => faster convergence to theinvariant distribution.
Chen & Shen (2006) and Ozun & Turk (2009) employ thismodel to identify the Taiwanese and Turkish business cycles,respectively, and �nd evidence of positive duration dependence.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Literature: (iii) additional remarks
Advantages of Pelagatti�s (2001, 2002) DDMSVAR:
Applies a Bayesian approach => inference does not depend onthe sample size and it is not conditional on the estimatedparameters.
Considers a multivariate speci�cation => inference using amulti-move Gibbs sampler => faster convergence to theinvariant distribution.
Chen & Shen (2006) and Ozun & Turk (2009) employ thismodel to identify the Taiwanese and Turkish business cycles,respectively, and �nd evidence of positive duration dependence.
� A1, ...,Ap � coe¢ cient matrices of a stable VAR process;
� εt � gaussian white noise vector with covariance matrix Σ;� St � binary unobservable random variable following a Markov chain
with varying transition probabilities (1=expans.; 0=contr.);
� µ0, µ0 + µ1 �average growth rates of yt in state 0 (contraction)and state 1 (expansion), respectively.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The DDMSVAR model
A Markov chain is built for the pair (St ;Dt)
The probability of St being in a particular state is assumed tobe dependent on the previous state St�1 and on the durationdependent variable Dt�1, where:
Dt =
(Dt�1 + 1
1
if
if
St = St�1St 6= St�1
Duration dependent transition probabilities:
pi jj (d) = Pr(St = i jSt�1 = j ;Dt�1 = d), i , j = 0, 1
� probability of the economy moving to or staying in state i giventhat in the previous period it was in state j with duration d .
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The DDMSVAR model
A Markov chain is built for the pair (St ;Dt)
The probability of St being in a particular state is assumed tobe dependent on the previous state St�1 and on the durationdependent variable Dt�1, where:
Dt =
(Dt�1 + 1
1
if
if
St = St�1St 6= St�1
Duration dependent transition probabilities:
pi jj (d) = Pr(St = i jSt�1 = j ;Dt�1 = d), i , j = 0, 1
� probability of the economy moving to or staying in state i giventhat in the previous period it was in state j with duration d .
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The DDMSVAR model
Hence, the vector of observable variables yt is dependentupon the unobserved states from St to St�p , and the durationdependent variable Dt�1.
Implies an extended state variable
S�t = (Dt ,St ,St�1, ...,St�p)
that includes all combinations of states in the last p periods.
A probit speci�cation is used to model the transitionprobabilities and to characterize the duration dependence inthe business cycle.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The DDMSVAR model
Hence, the vector of observable variables yt is dependentupon the unobserved states from St to St�p , and the durationdependent variable Dt�1.
Implies an extended state variable
S�t = (Dt ,St ,St�1, ...,St�p)
that includes all combinations of states in the last p periods.
A probit speci�cation is used to model the transitionprobabilities and to characterize the duration dependence inthe business cycle.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The DDMSVAR model
Hence, the vector of observable variables yt is dependentupon the unobserved states from St to St�p , and the durationdependent variable Dt�1.
Implies an extended state variable
S�t = (Dt ,St ,St�1, ...,St�p)
that includes all combinations of states in the last p periods.
A probit speci�cation is used to model the transitionprobabilities and to characterize the duration dependence inthe business cycle.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The DDMSVAR model
# A latent variable, S�t , can then be expressed as:
where d = 1, ..., τ, and Φ(�) is the standard normal CDF.- Transition probabilities de�ned by β = (β1, β2, β3, β4).- If β2 = β4 = 0) no business cycle duration dependence.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The DDMSVAR model
# A latent variable, S�t , can then be expressed as:
where d = 1, ..., τ, and Φ(�) is the standard normal CDF.- Transition probabilities de�ned by β = (β1, β2, β3, β4).- If β2 = β4 = 0) no business cycle duration dependence.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The DDMSVAR model: Gibbs sampling
To obtain parameter estimates from the DDMSVAR model,we employ a Bayesian (Gibbs sampler) approach.
� Parameters: θ =�θ01, θ
02, θ
03, θ
04
�0θ01=(A
0,Σ0)0, θ02=(µ00, µ
01)0, θ03=β0, θ04=(f(St ,Dt )gTt=1)
0
� Prior distribution:
p (µ,A,Σ, β,S�0 ) = p(µ).p(A).p(Σ).p(β).p(S0,D0)
� Conditional distribution of θk :
p�θk jY, θ�k
�, k=1,...,4; Y=(y1,...,yT ); θ-k : set without θk .
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The DDMSVAR model: Gibbs sampling
Bayesian inference using Markov chain Monte Carlo:
� From arbitrary initial values θ(0) =�
θ(0)01 , θ
(0)02 , θ
(0)03 , θ
(0)04
�0,
(1) draw θ(i )1 from p
�θ1jY, θ(i�1)2 , θ
(i�1)3 , θ
(i�1)4
�(2) draw θ
(i )2 from p
�θ2jY, θ(i )1 , θ
(i�1)3 , θ
(i�1)4
�(3) draw θ
(i )3 from p
�θ3jY, θ(i )1 , θ
(i )2 , θ
(i�1)4
�(4) draw θ
(i )4 from p
�θ4jY, θ(i )1 , θ
(i )2 , θ
(i )3
�� The i th realization of θ is then θ(i ) =
�θ(i )01 , θ
(i )02 , θ
(i )03 , θ
(i )04
�0.
� Repeating steps (1) to (4) I times, we get the Gibbs sequenceθ(1), θ(2), θ(3), ..., θ(I ) that will converge to the distribution of θ,
� Shorter time period: June 1984 - October 2010.� Stong evidence of positive duration dependence for contractions (β4).
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Figure 6a - Transition probabilities for contractions: y-o-y with dlEmp12Pr(S t = 1 |S t 1= 0,D t 1= d )
0.00.10.20.30.40.50.60.70.80.91.0
0 5 10 15 20 25 30 35 40 45 50 55 60
mean 2.5% median 97.5%
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Figure 6b - Transition probabilities for expansions: y-o-y with dlEmp12Pr(S t = 0 |S t 1= 1,D t 1= d )
0.00.10.20.30.40.50.60.70.80.91.0
0 5 10 15 20 25 30 35 40 45 50 55 60
mean 2.5% median 97.5%
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Figure 7 - Probability of expansion: y-o-y with dlEmp12
Pr(St =1)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1984 1988 1992 1996 2000 2004 2008
� "Weak" contraction in 1995-1996 (very close to the 0.5-rule).� Early 2000s contraction seems to die out sooner.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Outline
1 Motivation
2 Literature
3 Methodology
4 Empirical Results
5 Conclusions
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Conclusions
The DDMSVAR model with year-on-year (monthly) growthrates of IP, CLI and Emp (Jun1983-Oct2010) together withthe model without Emp (Jan1978-Oct2010) provide:
a reasonable picture of the Portuguese business cyclechronology over the last 33 years,
and indicate the presence of positive duration dependencein contractions but not in expansions.
Four important periods of contraction are identi�ed:
October 1983 - June 1984;March 1991 - November 1993;May 2001 - February 2006;May 2007 - November 2009.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Conclusions
The DDMSVAR model with year-on-year (monthly) growthrates of IP, CLI and Emp (Jun1983-Oct2010) together withthe model without Emp (Jan1978-Oct2010) provide:
a reasonable picture of the Portuguese business cyclechronology over the last 33 years,
and indicate the presence of positive duration dependencein contractions but not in expansions.
Four important periods of contraction are identi�ed:
October 1983 - June 1984;March 1991 - November 1993;May 2001 - February 2006;May 2007 - November 2009.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The model is quite successful in identifying the contraction inthe Portuguese economy caused by the recent �nancial crisis.
The model shows that the 2000s can be seen as a "lostdecade" for Portugal in terms of economic expansion.
Big concern for the next years since economic expansion isneeded to serve private and public debt that was accumulatedover the last decade.
The likelihood of a contraction ending increases over time,but for expansions it remains constant.
These results are similar to others obtained for the US:contractions are duration dependent, while expansions are not.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The model is quite successful in identifying the contraction inthe Portuguese economy caused by the recent �nancial crisis.
The model shows that the 2000s can be seen as a "lostdecade" for Portugal in terms of economic expansion.
Big concern for the next years since economic expansion isneeded to serve private and public debt that was accumulatedover the last decade.
The likelihood of a contraction ending increases over time,but for expansions it remains constant.
These results are similar to others obtained for the US:contractions are duration dependent, while expansions are not.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The model is quite successful in identifying the contraction inthe Portuguese economy caused by the recent �nancial crisis.
The model shows that the 2000s can be seen as a "lostdecade" for Portugal in terms of economic expansion.
Big concern for the next years since economic expansion isneeded to serve private and public debt that was accumulatedover the last decade.
The likelihood of a contraction ending increases over time,but for expansions it remains constant.
These results are similar to others obtained for the US:contractions are duration dependent, while expansions are not.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The model is quite successful in identifying the contraction inthe Portuguese economy caused by the recent �nancial crisis.
The model shows that the 2000s can be seen as a "lostdecade" for Portugal in terms of economic expansion.
Big concern for the next years since economic expansion isneeded to serve private and public debt that was accumulatedover the last decade.
The likelihood of a contraction ending increases over time,but for expansions it remains constant.
These results are similar to others obtained for the US:contractions are duration dependent, while expansions are not.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
The model is quite successful in identifying the contraction inthe Portuguese economy caused by the recent �nancial crisis.
The model shows that the 2000s can be seen as a "lostdecade" for Portugal in terms of economic expansion.
Big concern for the next years since economic expansion isneeded to serve private and public debt that was accumulatedover the last decade.
The likelihood of a contraction ending increases over time,but for expansions it remains constant.
These results are similar to others obtained for the US:contractions are duration dependent, while expansions are not.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Extensions
This analysis can be extended to the study of the businesscycle in other countries for which no organization is datingtheir business cycle turning points.
The model employed here may also be useful to test for thepresence of duration dependence in the phases of their cycles.
The DDMSVAR model can also be used to detect thepresence of duration dependence in other areas where cyclesmight be present, like in the stock or housing markets.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Extensions
This analysis can be extended to the study of the businesscycle in other countries for which no organization is datingtheir business cycle turning points.
The model employed here may also be useful to test for thepresence of duration dependence in the phases of their cycles.
The DDMSVAR model can also be used to detect thepresence of duration dependence in other areas where cyclesmight be present, like in the stock or housing markets.
Vítor Castro The Portuguese Business Cycle
MotivationLiterature
MethodologyEmpirical Results
Conclusions
Extensions
This analysis can be extended to the study of the businesscycle in other countries for which no organization is datingtheir business cycle turning points.
The model employed here may also be useful to test for thepresence of duration dependence in the phases of their cycles.
The DDMSVAR model can also be used to detect thepresence of duration dependence in other areas where cyclesmight be present, like in the stock or housing markets.