Semiparametric Cointegrating Rank Selection Xu Cheng Peter C. B. Phillips Department of Economics Yale University Workshop on Current Trends and Challenges in Model Selection and Related Areas Vienna, July 2008 Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Papers and Outline Cheng & Phillips (2008a) “Semiparametric cointegrating rank selection” – consistent cointegrating rank estimation by information criteria – asymptotics for weakly dependent innovations – simulation Cheng & Phillips (2008b) “Cointegrating rank selection in models with time varying variance” – robust to unconditional heterogeneity of unknown form – asymptotics under time varying variances – empirical application on exchange rate dynamics and simulation Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Papers and Outline Cheng & Phillips (2008a) “Semiparametric cointegrating rank selection” – consistent cointegrating rank estimation by information criteria – asymptotics for weakly dependent innovations – simulation Cheng & Phillips (2008b) “Cointegrating rank selection in models with time varying variance” – robust to unconditional heterogeneity of unknown form – asymptotics under time varying variances – empirical application on exchange rate dynamics and simulation Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Papers and Outline Cheng & Phillips (2008a) “Semiparametric cointegrating rank selection” – consistent cointegrating rank estimation by information criteria – asymptotics for weakly dependent innovations – simulation Cheng & Phillips (2008b) “Cointegrating rank selection in models with time varying variance” – robust to unconditional heterogeneity of unknown form – asymptotics under time varying variances – empirical application on exchange rate dynamics and simulation Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
SP ECM Model semiparametric ECM � X t = �� 0 X t � 1 + u t � and � are m � r full rank matrices – u t is weakly dependent with mean zero general short memory component u t – no speci…cation of VAR lags as in � X t = �� 0 X t � 1 + P p i =1 � i � X t � i + u t – no speci…cation of the distribution of u t – allow for unconditional unknown heterogeneity in u t permit near integration as well as strict unit roots Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
SP ECM Model semiparametric ECM � X t = �� 0 X t � 1 + u t � and � are m � r full rank matrices – u t is weakly dependent with mean zero general short memory component u t – no speci…cation of VAR lags as in � X t = �� 0 X t � 1 + P p i =1 � i � X t � i + u t – no speci…cation of the distribution of u t – allow for unconditional unknown heterogeneity in u t permit near integration as well as strict unit roots Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Cointegrating rank determination …tted model � X t = �� 0 X t � 1 + u t information criteria � ( r ) j + C n n � 1 � 2 mr � r 2 � IC ( r ) = log j b ; 0 � r � m – b � ( r ) is the residual covariance matrix from reduced rank regression – penalty C n : 2 (AIC), log ( n ) (BIC), c log log ( n ) (HQ) – degrees of freedom: 2 mr � r 2 b r = arg min 0 � r � m IC ( r ) Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Cointegrating rank determination …tted model � X t = �� 0 X t � 1 + u t information criteria � ( r ) j + C n n � 1 � 2 mr � r 2 � IC ( r ) = log j b ; 0 � r � m – b � ( r ) is the residual covariance matrix from reduced rank regression – penalty C n : 2 (AIC), log ( n ) (BIC), c log log ( n ) (HQ) – degrees of freedom: 2 mr � r 2 b r = arg min 0 � r � m IC ( r ) Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Cointegrating rank determination …tted model � X t = �� 0 X t � 1 + u t information criteria � ( r ) j + C n n � 1 � 2 mr � r 2 � IC ( r ) = log j b ; 0 � r � m – b � ( r ) is the residual covariance matrix from reduced rank regression – penalty C n : 2 (AIC), log ( n ) (BIC), c log log ( n ) (HQ) – degrees of freedom: 2 mr � r 2 b r = arg min 0 � r � m IC ( r ) Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Cointegrating rank determination …tted model � X t = �� 0 X t � 1 + u t information criteria � ( r ) j + C n n � 1 � 2 mr � r 2 � IC ( r ) = log j b ; 0 � r � m – b � ( r ) is the residual covariance matrix from reduced rank regression – penalty C n : 2 (AIC), log ( n ) (BIC), c log log ( n ) (HQ) – degrees of freedom: 2 mr � r 2 b r = arg min 0 � r � m IC ( r ) Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Outline of Basic Results � X t = �� 0 X t � 1 + u t � ( r ) j + C n n � 1 � 2 mr � r 2 � IC ( r ) = log j b IC ( r ) is weakly consistent provided C n ! 1 and C n =n ! 0 AIC inconsistent, limit distribution given Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Literature — Order Estimation Semiparametric approaches – Phillips (2008) “Unit root model selection” Parametric approaches & joint order estimation – Johansen (1988, 1991) – Phillips and Ploberger (1996), Phillips (1996), Chao and Phillips (1999) – Phillips & McFarland (1997) Order selection & nonstationarity – Tsay (1984), Potscher (1989), Wei (1992), Nielsen (2006), Kapetanios (2004), Wang & Bessler (2005), Poskitt (2000), Harris and Poskitt (2004) Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Literature — Time-varying Variance literature – classical unit root testing Hamori and Tokihisa, 1997; Kim et al , 2002; Cavaliere, 2004; Cavaliere and Taylor, 2007; Beare, 2007 – autoregressive models Phillips and Xu, 2006; Xu and Phillips, 2008 SP model choice method – robust to time-varying variance – no change in implementation Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Contribution � X t = �� 0 X t � 1 + u t SP ECM: u t standard method our method valid valid stationary specify lag length avoid misspeci…cation easy to implement time-varying var invalid valid same implementation Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Reduced Rank Regression � X t = �� 0 X t � 1 + u t suppose the cointegrating rank is r for given � � ( � ) = S 01 � ( � 0 S 11 � ) � 1 – b � � � 1 � 0 S 10 – b � 0 S 11 � � ( � ) = S 00 � S 01 � – notation S 00 = n � 1 P n S 11 = n � 1 P n t =1 � X t � X 0 t =1 X t � 1 X 0 t ; t � 1 ; S 10 = n � 1 P n t =1 � X t X 0 S 10 = S 0 t � 1 ; 01 0 S 11 b b � ( � ) j ; subject to b � = arg min � j b � � = I r � ( b � ) and b � ( r ) = b �( b � ( � ) = b b � ) Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
RRR Estimation – as if model correctly speci…ed Johansen (1988,1995) � � � �S 11 � S 10 S � 1 � = 0 determinantal equation 00 S 01 – ordered eigenvalues 1 > b � 1 > � � � > b � m > 0 – corresponding eigenvectors b V = [ b v 1 ; � � � ; b v m ] ; normalized by V 0 S 11 b b V = I m b i =1 (1 � b v r ] and j b � ( r ) j = j S 00 j � r � = [ b v 1 ; � � � ; b � i ) Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Information criteria components criterion has the form i =1 (1 � b – IC ( r ) = log( j S 00 j � r � i )) + C n n � 1 (2 mr � r 2 ) � � – b � �S 11 � S 10 S � 1 � = 0 � i are ordered solutions of 00 S 01 …nd SP limits of b � i; for i = 1 ; :::; m; using limit theory for S 11 ; S 10 ; S 00 Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Heuristics j �S 11 � S 10 S � 1 00 S 01 j = 0 when X t is stationary, i.e. r = m – S 11 ; S 10 ; and S 00 are all O p (1) ) 0 < � i < 1 for all i when X t is full rank integrated, i.e. r = 0 – S 11 = O p ( n ) ) � i decreases to 0 at rate n � 1 for all i Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Heuristics (cont.) when 0 < r < m – � 0 X t is stationary ) 0 < � i < 1 for all 1 � i � r – � 0 ? X t is an m � r vector of unit root time series ) � i decreases to 0 at rate n � 1 for r + 1 � i � m same asymptotic orders apply when u t is – weakly dependent – with time varying variance KEY: for weak consistency of IC ( r ) , only asymptotic order matters Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
Heuristics (cont.) when 0 < r < m – � 0 X t is stationary ) 0 < � i < 1 for all 1 � i � r – � 0 ? X t is an m � r vector of unit root time series ) � i decreases to 0 at rate n � 1 for r + 1 � i � m same asymptotic orders apply when u t is – weakly dependent – with time varying variance KEY: for weak consistency of IC ( r ) , only asymptotic order matters Xu Cheng & Peter C.B. Phillips Semiparametric Cointegrating Rank Selection
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