What Drives Inflation? Testing Non-nested Specifications of the New - PDF document

What Drives Inflation? Testing Non-nested Specifications of the New Keynesian Phillips Curve Luis F. Martins Department of Quantitative Methods, ISCTE, Portugal luis.martins@iscte.pt Vasco J. Gabriel Department of Economics, University of Surrey, UK and NIPE-UM v.gabriel@surrey.ac.uk This version: July 2008 1 Basic De…nitions � Test TWO competing non-nested models, H g and H h � � �� � Each model implies a conditional moment condition, E g [ g ( Y t j Z g Y t j Z h t ; � 0 )] = 0 and E h h t ; � 0 = 0 ; where Y t is the variable of interest to model; E g [ � ] is the expectation taken w.r.t. the unknown conditional density under H g ; Z g t is the information set avaliable at t ; and � 0 is the unique unkwown value such that E g [ g ( Y t j Z g t ; � 0 )] = 0 a.s.. � In the usual (unconditional) moment conditional form, E g [ g ( X t ; � 0 )] = 0 and E h [ h ( X t ; � 0 )] = 0 ; where X t ; t = 1 ; :::; T is the entire observable data (variable of interest, regressors and instruments), g ( X t ; � ) is < p g ! < m g and h ( X t ; � ) is < p h ! < m h ; m g � p g ; m h � p h : P T P T g ( � ) = 1 t =1 g t ( � ) = 1 � Sample m g � 1 and m h � 1 counterpart functions, b t =1 g ( X t ; � ) T T P T and b t =1 h t ( � ) where � 2 < p g and � 2 < p h : h ( � ) = 1 T 1

h p i p g ( � 0 ) d T b T b � Let under H g ; ! N m g (0 ; V g ) ; where the m g � m g PD matrix V g = lim T !1 V ar g g ( � 0 ) is estimated by b V g ( � ) evaluated at � = b � (White (1984), Newey and West (1987), Andrews (1991), ...) h i P T @g ( X t ; � ) be estimated by b @g ( X t ; � ) G ( � ) = 1 � Let the m g � p g matrix function G ( � ) = E g 0 0 T t =1 @� @� h i @g ( X t ; � ) and where G = E g evaluated at � = � 0 : Similar for H ( � ) 0 @� 2 Estimation � � � 0 b b e V � 1 � TSGMM = arg min � 2< pg b g ( � ) � b g ( � ) ; (1) g where e � is a …rst-step GMM estimator. For any metric W g ; the objective function is given � � 0 W g b e by b g ( � ) and the TSGMM is the e¢cient one if W g = b V � 1 O g ( �; W g ) = b g ( � ) � : g � 0 b b V � � 2< pg b g ( � ) b � CUE = arg min g ( � ) g ( � ) ; (2) where b g ( � ) is a generalized inverse of b V � V g ( � ) : � Using Newey and Smith (2004) typology, for a concave function � ( v ) and a m g � 1 parameter vector � g 2 � T ( � ) , the GEL estimator solves the following saddle point problem T X 1 b � [ � 0 � GEL = arg min � 2< pg sup g g t ( � )] : (3) T � g 2 � T t =1 The class of estimators de…ned by b � GEL is generally ine¢cient when g t ( � ) is serially correlated. However, Anatolyev (2005) demonstrates that, in the presence of correlation in g t ( � ) , the smoothed GEL estimator of Kitamura and Stutzer (1997) is e¢cient, obtained by smoothing the moment function with the truncated kernel, so that " # T X Q g ( �;� g ) = 1 b b � [ � 0 � SGEL = arg min � 2< pg sup g g tT ( � )] ; (4) T � g 2 � T t =1 P K T k = � K T g t � k ( � ) : Denoted the GEL probabilities as � g 1 with g tT ( � ) � t : 2 K T +1 � CUE: � ( v ) = � (1 + v ) 2 = 2; EL: � ( v ) = ln(1 � v ); ET: � ( v ) = � exp( v ) : 2

� � � 1 � � � d � p b 0 V � 1 � Under H g ; T � � � 0 ! N p g 0 ; G G g � � � 1 � � � d � p b 0 V � 1 � Under H h ; T � � � 0 ! N p h 0 ; H H h 3 Tests 3.1 Cox-Type Tests � Smith (1992) and Ramalho and Smith (2002) (in Singleton, 1985, m g = m h for the GMM case). � Test H g against H h by evaluating, under H g ; the (scaled) di¤erence of the estimated GMM criterion functions. � Smith (1992), page 975, GMM Cox C � � 0 � � � � p T b b b b b b V � 1 C T ( H g j H h ) = h � � A g b g � ; h � � � � 0 � � P T � � : Under H g ; C T ( H g j H h ) d where b b b b b A g = 1 V � 1 0 ; ! 2 t =1 h t � g t � � ! N ; where g g T � � 0 � � � � � � � � 0 0 g = b b b b A g c b M g b b c g b g b b b b ! 2 V � 1 V � 1 b h � � V g � M A � h � ; h h � � �� � 1 � � � 0 � � � � � 0 � � with c M g = I m g � b b b b b b b b b b b b V � 1 V � 1 G � G � � G � G � � : g g � Ramalho and Smith (2002), page 105, general form of Cox-type GC � � � � � 1 = 2 � � p 0 c b 0 0 b b M g b c b g b b GC = T c V g � M c c g � ; � � � � � � � � � � �� b b b b b b b c = b g b 0 c = b 0 g b � b V � 1 V � 1 where b or b A g b A � h � A � h � g � ; the latter with (pos- h h d sibly) improved power. Under H g ; GC ! N (0 ; 1) : � Ramalho and Smith (2002), page 108, GEL Cox-type C � � � 1 = 2 � � � � 0 c p 0 b g b b M g b c Q h ( b b �; b � h ) � b h ( e �; e V � 1 Q � C = T � M � � � h ) ; g P T where b h h tT ( � )] : Here, b � = 1 t =1 g tT ( � ) � [ � 0 Q � h ( �;� h ) is the objective SGEL function with T d t with the corresponding estimator being e �; e � h 1 =T replaced by b � h : Under H g ; C ! N (0 ; 1) : 3

� Ramalho and Smith (2002), page 108, GEL linearized Cox-type LC � � � 1 = 2 � � 0 c p 0 b d b g b 0 b M g b c b V � 1 LC = � T � M � � � � g ! N (0 ; 1) ; under H g : g � Ramalho and Smith (2002), page 108, GEL simpli…ed Cox-type SC � � � 1 = 2 � � � � d 0 c p b g b 0 b M g b c Q h ( b b �; b � h ) � b h ( b �; b V � 1 Q � SC = � T � M � � � h ) ! N (0 ; 1) ; under H g : g 3.2 Encompassing Tests � Smith (1992) and Ramalho and Smith (2002) � The null model H g is capable of predicting the features of the alternative model H h � Parametric Encompassing (behavior of b � under H g ) and Moment Encompassing (behavior of b h ( � ) ; i.e., behavior of a functional of b h under H g ) : � Smith (1992), page 976, GMM Parametric Encompassing E � � 0 � � � � � � 0 � � � � b b 0 g b b b b b g b b b b b b V � 1 Q � V � 1 E T ( H g j H h ) = T b A g b g � A � H � H � � g � ; h h where b Q � g is the generalized inverse of � � 0 � � � � � � � � Q g = b b b b b A g c b M g b b c g b 0 g b 0 b b b V � 1 V � 1 H � � V g � M A � H � : h h Under H g ; E T ( H g j H h ) d rank ( Q g ) where Q g is consistently estimated by b ! � 2 Q g : � Ramalho and Smith (2002), page 105, general form of Encompassing-type GE � � 0 � � d b c b 0 b b � � ! � 2 GE = T b b g b g � c g � rank (� g ) ; � � 0 c under H g ; where b M g b b c 0 � g = b c V g � M g b c: � Ramalho and Smith (2002), page 110, GEL Parametric Encompassing PE 0 1 0 1 0 b � � e b � � e � � B C B C 0 d K g b b g b � � ! � 2 PE = T @ A K @ A rank (� g ) ; g b � h � e b � h � e � h � h under H g ; where the de…nition of b K g can be found in the appendix of Ramalho and Smith (2002), pages 123 and 124. 4

� Ramalho and Smith (2002), pages 111 and 112, GEL Moment Encompassing ME 0 b h ) d m � � � m � ! � 2 ME = T ( b m h � b g ( b m h � b h ) rank (� g ) ; under H g ; where the moment indicator function m h ( X ; � g ; � ; � h ; � ) can be choosen to � � � � h = P T m h = b b � g b m � equal h ( X ; � ) : In this case, b and b t =1 b h � t h t � : If one chooses � � b m � m h ( X ; � g ; � ; � h ; � ) to equal g ( X ; � ) ; then b m h = b and b g � h = 0 which implies PE = J; Hansen’s (1982) statistic for over-identifying moment restrictions. � Ramalho and Smith (2002), pages 111 and 112, GEL linearized Moment Encompassing LME 0 0 b LME = T b g b � b g b d � � ! � 2 � � � g rank (� g ) ; under H g : 3.3 Model Selection Tests � Rivers and Vuong (2002) and Hall and Pelletier (2007) asympt:equiv: � The null hypothesis is H g = H h ; comparing measures of goodness of …t and distance between H g and H h : � If both models are correctly speci…ed or locally misspeci…ed the limiting distribution of the test statistic is not standard and depends on nuisance parameters under the null. � Suppose that both models are misspeci…ed (there is no parameter value at which the moment condition can be set equal to zero). Hall and Pelletier (2007), page 8, N T statistic � � � � �� b b � b b O g �; W g O h �; W h p d N T = T ! N (0 ; 1) ; � 2 b � 2 is a consistent estimator of � 2 under the null hypothesis, where b 0 ; the limiting vari- � � � � �� p � 2 for b b � b b : Hall and Pelletier (2007) derives the formulas of b ance of T O g � O h � � � � 1 P T 0 t =1 Z g t Z g 1 two cases: (1) W g = I m g and W h = I m h ; (2) W g = and W h = t T � � � 1 P T 0 1 t =1 Z h t Z h : t T 5