NONSTATIONARY DISCRETE CHOICE: A CORRIGENDUM AND ADDENDUM. PETER C. B. PHILIPS, SAINAN JIN and LING HU COWLES FOUNDATION PAPER NO.

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1 NONSTATIONARY DISCRETE CHOICE: A CORRIGENDUM AND ADDENDUM BY PETER C. B. PHILIPS, SAINAN JIN and LING HU COWLES FOUNDATION PAPER NO. 24 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 2828 New Haven, Connecticut

2 Journal of Econometrics 4 (27) Nonstationary discrete choice: A corrigendum and addendum Peter C.B. Phillips a,b,c,, Sainan Jin d, Ling Hu e a Cowles Foundation, Yale University, USA b University of Auckland, New Zealand c University of York, UK d Guanghua School of Management, Peking University, China e Department of Economics, Ohio State University, USA Available online 26 March 27 Abstract We correct the limit theory presented in an earlier paper by Hu and Phillips [24a. Nonstationary discrete choice. Journal of Econometrics 2, 3 38] for nonstationary time series discrete choice models with multiple choices and thresholds. The new limit theory shows that, in contrast to the binary choice model with nonstationary regressors and a zero threshold where there are dual rates of convergence (n =4 and n 3=4 ), all parameters including the thresholds converge at the rate n 3=4. The presence of nonzero thresholds therefore materially affects rates of convergence. Dual rates of convergence reappear when stationary variables are present in the system. Some simulation evidence is provided, showing how the magnitude of the thresholds affects finite sample performance. A new finding is that predicted probabilities and marginal effect estimates have finite sample distributions that manifest a pile-up, or increasing density, towards the limits of the domain of definition. r 27 Elsevier B.V. All rights reserved. JEL classification: C23; C25 Keywords: Brownian motion; Brownian local time; Discrete choices; Integrated processes; Pile-up problem; Threshold parameters Corresponding author. Cowles Foundation, Yale University, USA. address: peter.phillips@yale.edu (P.C.B. Phillips) /$ - see front matter r 27 Elsevier B.V. All rights reserved. doi:.6/.econom.27..7

3 6 P.C.B. Phillips et al. / Journal of Econometrics 4 (27) 5 3. Introduction This note corrects the limit theory given in Hu and Phillips (24a, hereafter HP) for discrete choice models with integrated covariates and nonzero thresholds that determine an ordered set of choices. The error occurs in Lemma and Theorem of HP. Those results sought to provide the asymptotic theory for sample moment expressions that appear in the score function and hessian (Eqs. (7) (9), in HP); and they gave dual rates of convergence (n =4 and n 3=4 ) and limit expressions involving the local time of Brownian motion at the origin. Those results turn out to apply only when the threshold parameters are unscaled or zero, and in these cases the results correspond to those in the binary choice model considered in Park and Phillips (2, hereafter PP). When the threshold parameters p are ffiffi nonzero and are scaled to have the same order of magnitude as the covariates (i.e., by n for integrated regressors), a single convergence rate of n 3=4 applies to both parameters and thresholds and the limit theory involves expressions with local time evaluated at the thresholds rather than the origin. The limit theory for the parameter estimates is still mixed normal and usual procedures for statistical inference remain valid, as do the expressions for the arc sine laws and extended arc sine laws given in PP and HP. As discussed in Hu and Phillips (24b, hereafter HP 2 ), practical empirical work on ordered discrete choice models frequently involves explanatory variables that display random wandering characteristics. For instance, HP 2 construct a discrete choice model of the empirical behavior of the Federal Reserve in making discrete adustments to the federal funds target rate, where the explanatory variables involve economic fundamentals monitored by the Fed such as the inflation rate and unemployment as well as leading indicators like consumer and business confidence. In modeling such intervention decisions where some of the explanatory variables behave like stochastic trends, it seems appropriate for the thresholds in the decision choices to be scaled to have the same order as the regressors so that there are nontrivial effects. This scaling is a theoretical device for developing a more meaningful asymptotic theory. Otherwise, the limit distribution will be degenerate and trivial. When the latent variable y t in the choice model is nonstationary and p ffiffiffi converges to a continuous stochastic process like Brownian motion after scaling by n, the choices ultimately depend on the behavior of the limiting stochastic process. For example, the observed dependent variable y t may take on a discrete value such as unity (corresponding pffiffi to a p certain ffiffi choice) when y t falls in the interval between the scaled thresholds n m and n m 2, and for such realizations the limit Brownian motion lies in the interval between m and m2, and the associated probability will be nonzero when m am2. However, if the thresholds were unscaled, the limiting probability of y t falling in the fixed interval between m and m2 would be zero (since m =n=2 ; m 2 =n=2 ) and would therefore be trivial. The thresholds could, in fact, be determined by other variables, although this is not explored in HP or the present paper. In the development that follows, we use the same model and notation as HP. In the interests of brevity, the set-up of HP will not be repeated in detail here and this paper provides a revised version of Lemma and Theorem of HP (given in Lemma R and Theorem R2) and the results which depend on them. We also need some supplementary results on convergence to functionals of Brownian local time at spatial points away from the origin, which are of independent interest. These are provided, together with proofs of the main results, in Appendices I and II. Readers are referred to the full length version of this paper (Phillips et al., 25, hereafter PJH) available on the authors websites for

4 complete details. Further empirical illustrations of Brownian local time are given in Phillips (998/25, 2). The results of some simulation experiments are summarized, again with details in PJH. These reveal that the finite sample distributions of the regression coefficient and threshold estimates are generally well approximated by the mixture normal limit theory. A new finding is that predicted probabilities and marginal effect estimates have finite sample distributions in which the density increases towards the limits of the domain of definition. This pile-up problem is shown to occur also in the stationary discrete choice model. 2. Revised notation and assumptions The set-up here follows HP and PP with some differences and extensions. In particular, we consider the regression model given by y t ¼ x t b t for t ¼ ;...; n, () where x t is a (m ) vector of explanatory variables and t is an error with cdf F. The dependent variable y t is unobserved. Instead, what is observed is the indicator y t, which takes the following possible ðj þ Þ values: y t ¼ if y t 2 ð ; pffiffi n m Š ¼ if y t 2ð ffiffiffi n p pffiffi m ; n m 2 Š.. ¼ J if y t 2ð pffiffi n m J ¼ J if y t 2ð pffiffi n m J ; Þ. ; pffiffi n m J Š In () x t is predetermined and is an integrated process satisfying Assumption of HP and for which n =2 x ½nŠ ) VðÞ, Brownian motion with variance matrix S. The conditions are also sufficient to ensure that Skorohod embedding arguments may be used. The parameters are assembled in the vector y, whose true value y ¼ðb ; m Þ is an interior point of a subset of R mþj which is compact and convex. As in HP, the regressor space is rotated using an orthogonal matrix H ¼ðh ; H 2 Þ with h ¼ b =ðb b Þ =2 to isolate the effects of the nonlinearities. The process V is correspondingly transformed as V ¼ h V, V 2 ¼ H 2 V, L V ðt; sþ is the local time of V at the spatial point s over the time interval ½; tš, and L ðt; sþ ¼ð=s ÞL V ðt; sþ, where s is the variance of V. Under rotation by H, () becomes y t ¼ x t b t ¼ x t HH b t ¼ x t a þ x 2t a2 t, where x t ¼ h x t, x 2t ¼ H 2 x t, a ¼ h b ¼ðb b Þ =2, a 2 ¼ H 2 b ¼, and a ¼ H b with a ¼ða ; a2 Þ. Denote y ¼ða ; m Þ. The conditional probabilities of y t are written as Pðy t ¼ F t Þ¼P ðx t ; y Þ. The log likelihood function is log L n ðyþ ¼ P n P J t¼ ¼ Lðt; Þ log P ðx t ; yþ, where Lðt; Þ ¼fy t ¼ g, and the score function is S n ðyþ ¼ðS n ðbþ ; S n ðmþ Þ ¼ðqlog L n =qb ; q log L n =qm Þ with elements q log L n qb ¼ Xn P.C.B. Phillips et al. / Journal of Econometrics 4 (27) t¼ X J ¼ Lðt; Þ P ðx t ; yþ p ðx t ; yþx t, (3) ð2þ

5 8 P.C.B. Phillips et al. / Journal of Econometrics 4 (27) 5 3 q log L n p qm ¼ ffiffiffi X n Lðt; Þ Lðt; Þ n P t¼ ðx t ; yþ P ðx t ; yþ where ffiffi p ðx t ; yþ ¼ fðx t b p n m Þ, p ðx t ; yþ ¼f ðx t b p ffiffi n m Þ fðx t b p J ðx t ; yþ ¼f ðx t b p ffiffiffi n m J Þ. f ðx t b p ffiffi n m Þ, (4) p ffiffi n m þ Þ for ¼ ;...; J, The first and second derivatives of F are written as f and _ f. The following assumption about the distribution function F and density f of t extends Assumption 2 of HP by placing some additional explicit component functions in the classes and placing uniform tail conditions on F and f. Both probit and logit functions satisfy conditions (a) (c) of Assumption R2 (as discussed in PP and HP) and (5), as is easily verified. As in HP, we use the following classifications for nonlinear functions: g : R R is regular if it is bounded, integrable, and differentiable with bounded derivative; F R denotes the class of regular functions; F I is the class of bounded and integrable functions; and F the class of functions that are bounded and vanish at infinity. The notation _g and g is used to denote the first and second derivatives of g. The definitions for Z kl ; A k ; B l ; C k ; t klpq are given in PJH. Assumption R2. (Updates Assumption 2 of HP). F is three times differentiable with bounded derivatives and satisfies Fðx M þz mþ sup ¼ oðþ; xom FðxÞ Fðx þ M þz mþþ sup ¼ oðþ, xpm FðxÞ f ðx M þz mþ sup ¼ oðþ, xom f ðxþ as M for any Z; m4. Further, for k; l ¼ ;...; J: (a) Z kl A k B l ; Z kl A k A l ; Z kl B k B l 2 F R ; (b) Z kk A k ; Z kk B k ; ðz _ kl Ak B l Þ; ðz _ kl Ak A l Þ; ðz kl _B k B l Þ; Z =2 _ kk C k 2 F I ; (c) t klpq A k A l A p A q ; t klpq A k A l B p B q ; t klpq B k B l B p B q ; C k C l Z kl 2 F. ð5þ 3. Correction to Lemma of HP Lemma R gives some limit results for partial sum expressions that are needed in analyzing the asymptotic behavior of the score and hessian functions. Lemma R corrects Lemma of HP. Proofs and complementary results are given in the Appendix and PJH. Lemma R. Let f and P be the density and probability distribution functions defined above, let Assumption in HP and Assumption R2 hold, and let m a and k X. Then, as n, (a) X n n =2ðþk Þ t¼ f 2 ðx t a p ffiffi n m Þ x k t P ) ðm Þk Z ða Þk þ L ; m a FðsÞ ds,

6 (b) X n n =2ðþk Þ t¼ (c) n 3=2 X n t¼ (d) n 3=2 X n t¼ (e) n 3=2 X n t¼ (f) n 3=2 X n t¼ P.C.B. Phillips et al. / Journal of Econometrics 4 (27) f 2 ðx t a p ffiffiffi n m Þ x k t P ) ðm Þk Z ða Þk þ L ; m f 2 ðx t a p ffiffi n m Þ x t x 2t ) m P ða Þ2 f 2 ðx t a p ffiffi n m Þ x t x 2t ) m P ða Þ2 f 2 ðx t a p ffiffi n m Þ x 2t x 2t P ) a f 2 ðx t a p ffiffi n m Þ x 2t x 2t P ) a Z Z Z Z a FðsÞ ds, Z V 2 ðrþ dl r; m a FðsÞ ds, V 2 ðrþ dl r; m a Z FðsÞ ds, Z V 2 ðrþv 2 ðrþ dl r; m a FðsÞ ds, V 2 ðrþv 2 ðrþ dl r; m a Z FðsÞ ds, (g) (h) Xn =2 n t¼ Xn =2 n t¼ f ðx t a f ðx t a p ffiffiffi n m Þf ðx ta pffiffiffi n m Þ p, P p ffiffiffi n m Þf ðx ta pffiffiffi n m þ Þ p. P Remark. In a similar fashion to part (a) when k ¼ 2 (as occurs in the hessian expression considered below), we obtain the limit X n f ðx n 3=2 t a ffiffiffi Z p n m Þx2 t ) ðm Þ2 ða L Þ3 ; m a f ðsþ ds, (6) t¼ whereas, when m ¼, we have (e.g. from Lemma 2 part (a) of PP) X n n =2 t¼ f ðx t a Þx2 t ) ða Þ3 L ð; Þ Z f ðsþs 2 ds. (7) Thus, a maor effect of the nonzero p threshold m a is to change the rate of convergence (or standardization) from = ffiffi n in (7) to =n 3=2. Another effect is that the limit random variable involves Brownian local time at m =a instead of the origin. Finally, the scale effect arising from the spatial integral changes from R f R ðsþs2 ds in (7) to m 2 f ðsþ ds in (6). Each of these effects arises from pffiffi the fact that the principal contribution to the partial sum comes when x t is around n m =a. These are the changes in the limit theory for the nonzero threshold case that lead to the corrections needed for HP. Lemma R. (Corrects Lemma of HP). Let Assumption in HP hold, and write A k ðx t ; y Þ¼A k, B k ðx t ; ; y Þ¼B k. Assume for k; l; ¼ ;...; J, that A k A l Z kl, A k B l Z kl,

7 2 P.C.B. Phillips et al. / Journal of Econometrics 4 (27) 5 3 B k B l Z kl 2 F R, A k Z kk, B k Z kk 2 F I, and t kkkk A 4 k, t kkkkb 4 k 2 F for A k, B k : R R. Then n 3=4P n P J t¼ k¼ A kz kt x t n 3=4P n t¼ P J k¼ A kz kt x 2t n =4P n t¼ P J k¼ B kz kt C A ) M=2 WðÞ, (8) where M ¼ð½M i ŠÞ is partitioned conformably with component submatrices ( M ¼ XJ ðm Z ) Þ2 ¼ ða L Þ3 ; m a FðsÞð FðsÞÞ ds, ð9þ ( M 2 ¼ XJ m Z Z ) V ¼ ða 2 ðrþ dl r; m Þ2 a FðsÞð FðsÞÞ ds, ðþ ( Z M 22 ¼ XJ Z ) a V 2 ðrþv 2 ðrþ dl r; m ¼ a FðsÞð FðsÞÞ ds, ðþ Z M 3 ¼ m ða L Þ2 ; m FðsÞð FðsÞÞ ds, ð2þ M 23 ¼ a Z M 33 ¼ a L ; m a a Z dl r; m V 2 ðrþ a Z FðsÞð FðsÞÞ ds, FðsÞð FðsÞÞ ds, and W is m-dimensional Brownian motion with covariance matrix I, which is independent of V. Remarks. () The main correction that Lemma R makes to Lemma of HP is to include the component n 3=4P n P J t¼ k¼ A kz kt x t, which has the same rate of convergence (n 3=4 ) as the element n 3=4P n P J t¼ k¼ A kz kt x 2t involving the factor x 2t. The corrections, notably that the limit functional involves Brownian local time at spatial points fm =a : ¼ ;...; Jg away from the origin, are discussed in the Remark above. (2) It is pointed out in PP that if x 2t were replaced by a stationary variate (as it would in some directions were x 2t to be cointegrated), then the norming would be different. Thus, suppose x 3t is a stationary (m 3 ) vector with coefficient g, satisfies the same conditions as v t in Assumption of HP and is independent of u t. Then we have: X n f ðx n =2 3t g þ x t a ffiffi Z p n m Þx 3tx 3t ) a L ; m a f ðsþ dss 33, t¼ where S 33 ¼ Eðx 3t x 3tÞ, and X n n =4 t¼ X J k¼ A k z kt x 3t ) MN ; XJ ¼ ( Z a L ; m )S a FðsÞð FðsÞÞ ds 33. ð3þ ð4þ

8 4. Correction to the main results P.C.B. Phillips et al. / Journal of Econometrics 4 (27) The maximum likelihood estimator b y n ¼ð b b n ; bm n Þ of y ¼ðb ; m Þ satisfies the expansion ¼ S n ð b y n Þ¼S n ðy ÞþJ n ð e yþð b y n y Þ, (5) where e y is on the line segment between b y n and y, which differs from row to row of the hessian matrix J n ð e yþ. Corresponding to the rotation in the regressor space, define G ¼ H, I J and let y ¼ða ; m Þ. Then, the score function and hessian matrix for the new parameters are based on S n ðyþ ¼G S n ðyþ and J n ðyþ ¼G J n ðyþg, and ¼ S n ð^y n Þ¼S n ðy ÞþJ n ð y ~ n Þð^y n y Þ. (6) Using Lemma R, we obtain the following limit theory for the score function and the hessian, which corrects Theorem of HP. Theorem R2. Let Assumption in HP and Assumption R2 hold. Then n 3=4 S n ðy Þ)Q =2 WðÞ and n 3=2 J n ðy Þ) Q ointly, where Q is the symmetric matrix partitioned as q q 2 q 3 B Q q 2 q 22 q C 23 A q 3 q 32 q 33 (7) conformably with ða ; a2 ; m Þ, and where ( q ¼ XJ ðm Z ) Þ2 ¼ ða L Þ3 ; m a FðsÞð FðsÞÞ ds, ( q 2 ¼ XJ m Z Z ) dl ¼ ða r; m Þ2 a V 2 ðrþ FðsÞð FðsÞÞ ds, Z q 3 ðþ ¼ m ða L Þ2 ; m a FðsÞð FðsÞÞ ds, ( Z q 22 ¼ XJ Z ) a V 2 ðrþv 2 ðrþ dl r; m ¼ a FðsÞð FðsÞÞ ds, q 23 ðþ ¼ Z Z a dl r; m a V 2 ðrþ FðsÞð FðsÞÞ ds, Z q 33 ð; Þ ¼ a L ; m a FðsÞð FðsÞÞ ds, q 33 ð; iþ ¼ for ia. and W is defined as in Lemma R.

9 22 P.C.B. Phillips et al. / Journal of Econometrics 4 (27) 5 3 Remarks. () Notice that with threshold parameters in the model, even if t has a symmetric distribution, as in the probit and logit models, q 2 ; q 3 ; q 2 and q 3 are not zero and Q does not reduce to a block diagonal matrix, which differs from the result in PP. (2) When stationary m 3 -dimensional variables x 3t are present in the model, we get multiple convergence rates. Suppose x 3t is an m 3 -vector of zero mean, stationary time series with coefficient g defined as above. Let r ¼ðg ; y Þ, r ¼ðg ; y Þ, and I m3 B G 2 ¼ H A, I J D n ¼ Diagðn =4 I m3 ; n 3=4 I mþj Þ. Following similar steps as those in the proof of Theorem R2, and using Remark 2 after Lemma R, we obtain the following limit theory: D n S n ðr Þ)X =2 WðÞ and D n J nðr ÞD n ) X, where X ¼ X, Q with ( Z ) X ¼ XJ a L ; m ¼ a FðsÞð FðsÞÞ ds S 33, and Q is defined as in Theorem R2, and S 33 ¼ Eðx 3t x 3t Þ. The asymptotic results for S n ðy Þ and J n ðy Þ in Theorem R2 lead to the limit distribution of b y n. From expansion (6), the normed and centered estimator satisfies n 3=4 ð b y n y Þ¼ ðn 3=2 J n ðy ÞÞ n 3=4 S n ðy Þþo p ðþ, (8) a result that is established in the proof of Theorem R3, which corrects Theorem 2 of HP. Theorem R3. Let Assumption in HP and Assumption R2 hold. Then there exists a sequence of ML estimators for which as b y n p y, and n 3=4 ð b y n y Þ)Q =2 WðÞ, in the notation introduced in Theorem R2. Remarks. () From the above, we get n 3=4 G ð b y n y Þ)Q =2 WðÞ, (9) and therefore n 3=4 ð b y n y Þ)GQ =2 WðÞ ¼MNð; GQ G Þ, (2) Following arguments similar to those in Theorem 3 and using Remark 2, when there are stationary variables in the model, we have and D n ðbr n r Þ)X =2 WðÞ D n G 2 ðbr n r Þ)X =2 WðÞ

10 P.C.B. Phillips et al. / Journal of Econometrics 4 (27) or Thus n =4 ðbg n g Þ)X =2 WðÞ, n 3=4 G ð b y n y Þ)Q =2 WðÞ. n 3=4 ð b y n y Þ)GQ =2 WðÞ ¼MNð; GQ G Þ, n =4 ðbg n g Þ)X =2 WðÞ, which we formalize in the Corollary that follows, which replaces Corollary of HP. Corollary R4. Under Assumption in HP and Assumption R2, as n n 3=4 ð b n b Þ n 3=4 ) MNð; GQ G Þ. (2) ðbm n m Þ When there are stationary variables in the system with coefficients g, n =4 ðbg n g Þ) MNð; X Þ and is independent of (2), so that n =4 ðbg n g Þ n 3=4 ð b B n b Þ A ) MNð; G 2X G 2 Þ, n 3=4 ðbm n m Þ in which case the convergence rates for the parameter estimates differ, with a slower n =4 rate for the parameters of stationary variables, and a faster n 3=4 convergence rate for the other parameter estimates. The conditional covariance matrix of ^y n can be estimated by the hessian inverse J n ð^y n Þ, or the more commonly used alternative J n ð^y n Þ, where J n;i excludes the terms in J n;i that involve martingale differences (see HP and PJH for details). The following result replaces Theorem 3 of HP. Theorem R5. Under Assumption in HP and Assumption R2, ½n 3=2 J n ð b y n ÞŠ ) GQ G as n, with the same limit holding for ½n 3=2 J n ð b y n ÞŠ. Again, when we have stationary variables, ½n =2 J n ðbg n ÞŠ ) X, and ½n 3=2 J n ð b y n ÞŠ ) GQ G as n. 5. Predicted probabilities and marginal effects 5.. Predicted probability Next consider ^P ;x ¼ ^P ðx t ; ^y n Þ, the predicted probability of the choice y t ¼, and bu ;x ¼ ^p ðx t ; ^y n Þ^b n, the estimated marginal effect of x t on ^P ðx t ; ^y n Þ both evaluated for some x t ¼ x. To achieve comparability between x b and the thresholds, and thereby assist in simulating the finite sample and asymptotic distributions of the predicted probabilities, we write the scaled thresholds in the comparable form z n m pffiffi (in place of n m ) and p suppose ffiffi z n 4 is a realization of some (independent) unit root time series so that z n ¼ O p ð n Þ, and the ordering on the p thresholds ffiffiffi is positively scaled and therefore not reversed. This scaling is analogous to the n m scaling of the thresholds used in previous sections and serves as a

11 24 P.C.B. Phillips et al. / Journal of Econometrics 4 (27) 5 3 device for developing the asymptotic theory in a convenient way. The probabilities P are then evaluated at x t ¼ x and z n ¼ z for some specific values x and z. The probabilities satisfy P ðx t ; y Þ¼ Fðx b zm Þ, P ðx t ; y Þ¼Fðx b zm Þ Fðx b zm þ Þ for ¼ ;...; J, P J ðx t ; y Þ¼Fðx b zm J Þ. To analyze these quantities, we define a matrix RðÞ ¼DiagðI m ; i Þ where i is a vector of length J with the th element and other elements zero. Similarly, RðJÞ ¼DiagðI m ; i J Þ and for ppj, RðÞ ¼DiagðI m ; ði ; i þ Þ Þ. Accordingly, we may write ^b n b ^b n b ^b n b ^b n b ^m n ¼ RðÞ m ^m n m ^m J n ¼ RðJÞ, mj ^m n m and for ppj, ^b n b ^m n m m þ ^m þ n C A ¼ RðÞ ^bn b ^m n m. Corollary R6. Let Assumption in HP and Assumption R2 hold. Given x t ¼ x, z n ¼ z, for ¼ ;...; J, the predicted probabilities of y t ¼ ð ¼ ;...; JÞ satisfy n 3=4 ð bp ;x P ;x Þ)MNð; UðÞGQ G UðÞ Þ. The above expressions use the following notation: P ;x ¼ P ðx; y Þ for ¼ ; ;...; J, UðÞ ¼f ðx b zm Þ x RðÞ, z UðJÞ ¼f ðx b zm J Þ x RðJÞ, z ½f ðx b zm Þ fðx b zm þ ÞŠx UðÞ ¼ f ðx b zm Þz B C RðÞ for ¼ ;...; A f ðx b zm þ Þz When we have stationary variables, given x t ¼ x, z n ¼ z, x 3t ¼ x 3 becomes n =4 ð bp ;x P ;x Þ)MNð; f ðx 3 g þ x b zm Þ2 x 3 X x 3Þ for ¼ ; J, n =4 ð bp ;x P ;x Þ)MNð; ½f ðx b zm Þ fðx b zm þ ÞŠ 2 x 3 X x 3Þ for ¼ J;...; J. the limit theory Therefore, the limit theory when stationary variables are present is dominated by the stationary coefficients and the convergence rate is n =4, ust as in PP.

12 5.2. Marginal effects P.C.B. Phillips et al. / Journal of Econometrics 4 (27) For the marginal effects, we have the following limit theory. Corollary R7. Let Assumption in HP and Assumption R2 hold. Given x t ¼ x, z n ¼ z, for ¼ ;...; J, the estimated marginal effects bu ;x have the following asymptotic distributions as n n 3=4 ðbu ;x u ;x Þ)MNð; PðÞGQ G PðÞ Þ. These expressions use the notation: u ;x ¼ u ðx; y Þ¼p ðx; y Þb for ¼ ; ;...; J, f _ ððx b zm ÞÞxb f ððx b zm ÞÞI m PðÞ ¼@ A f_ ððx b zm ÞÞzb RðÞ, f_ ððx b zm J ÞÞxb þ f ððx b zm J ÞÞI m PðJÞ ¼@ f _ A ððx b zm J ÞÞzb RðJÞ, ½ f _ ððx b zm ÞÞ f_ ððx b zm þ ÞÞŠxb þ p ðx; y ÞI m PðÞ ¼ f _ ððx b zm ÞÞzb B C A f_ ððx b zm þ ÞÞzb for ¼ ;...; J. When stationary variables are present, given x t ¼ x, z n ¼ z; x 3t ¼ x 3, the estimated marginal effects bu ;x have the following asymptotic distributions as n n =4 ðbu ;x u ;x Þ)MNð; LðÞX LðÞ Þ, where LðÞ ¼ f _ ððx 3 g þ x b zm ÞÞrx 3 f ððx 3 g þ x b zm ÞÞI m 3, LðÞ ¼½ f _ ððx 3 g þ x b zm ÞÞ f_ ððx 3 g þ x b zm þ ÞÞŠrx 3 þ p ðx; x 3 ; r ÞI m3, LðJÞ ¼ f _ ððx 3 g þ x b zm J ÞÞrx 3 þ f ððx 3 g þ x b zm ÞÞI m 3. Therefore, the limit theory when stationary variables are present is dominated by the stationary coefficients and the convergence rate is n =4, ust as in PP. 6. Simulation experiments Some extensive simulations were conducted to examine the finite sample performance of ML estimation, predicted probabilities, and marginal effects in a polychotomous choice model under nonstationarity. This section briefly summarizes some of the findings and readers are referred to PJH for details and further discussion. The experimental design was based on a model with m ¼ 2 explanatory variables and J ¼ 2, giving a triple-choice dependent variable y t. The DGP for the exogeneous data

13 26 P.C.B. Phillips et al. / Journal of Econometrics 4 (27) 5 3 is the system x t ¼ r xt þ x 2t r 2 x 2t v t v 2t, (2) with v t ¼ðv t ; v 2t Þ ¼ iid Nð; I 2 Þ, and r ¼ r 2 ¼ r ¼. The coefficient parameter vector was set at b ¼ð; Þ,sox t b ¼ b x t ¼ x t and the direction orthogonal to b is ð; Þ, giving the coefficient b 2 ¼ ofx 2t. This set-up is analogous to that of the simulation study of PP. The number of replications was 5, and sample sizes ranging from n ¼ to were used. The main conclusions are as follows:. As the magnitude of the threshold parameters increased (from : to :5Þ, the convergence rates of the coefficient estimates in the two directions showed evidence of equalizing, thereby corroborating the limit theory of Theorem R3 in contrast to the zero threshold case of PP, where the convergence rates differ. 2. The distributions of both parameters and threshold estimates generally appear to approach symmetric distributions corresponding to the mixed normal limit theory. However, there is some evidence that, as the magnitude of the thresholds m increase, the distributions of the estimates become biased. The reason for the bias appears to be related to the behavior of the choice probabilities in such cases, which quickly go to zero or unity when the arguments are large. This bias is also found to occur in the stationary case (for values of r i X:95 in (2)) when the thresholds are large. 3. Fig. shows kernel estimates of the sampling distributions of the (scaled and centered) choice probability when ¼ for sample sizes n ¼ ; 25; 5;. Different choices of m, b, z, and x do not change the results in a material way provided the parameter.25 density.2.5. choice, limit choice, n= choice, n=25 choice, n=5 choice, n= Fig.. Density of choice probability for ¼.

14 P.C.B. Phillips et al. / Journal of Econometrics 4 (27) settings are small, but when they are large the choice probabilities can quickly go to zero or unity and this appears to bias the distributions, as mentioned above. 4. The finite sample distribution of the choice probability has finite support and reveals a pile-up problem where the density increases towards the limits of the domain of definition, as is apparent in Fig.. This pile-up problem, which to our knowledge has not before been noticed in the discrete choice literature, also occurs in the stationary case see Fig. 2, where r ¼ r 2 ¼ r ¼ :95, and Fig. 3 where r ¼ r 2 ¼ r ¼ :99, with sample sizes n ¼ ; 5; ; 5. The figures show that as n passes to infinity the pile-up problem steadily dissipates. For n ¼ 5 the upper and lower bounds are close to the extremes of the support where the limit distribution is nonnegligible. Thus, the problem of pile-up is not confined to the nonstationary discrete choice problem but is a more generic problem. In effect, the asymptotic approximations (such as those given in Corollary R6) are valid in an immediate interval around the true values. Outside that interval, behavior is rather different because of the fact that bp ;x goes to zero or unity depending on the sign of its argument, resulting in a pile-up of the distribution in finite samples. It might therefore be argued that the true finite sample distribution would be better approximated by a mixture of three distributions, one of which is the local asymptotic result given above and the other two are based on pile-ups around bp ;x, and bp ;x. Developing such a mixture approximation clearly involves further complications and is left for the future research. 5. Figs. 4 and 5, show kernel estimates of the sampling distributions of the marginal effects bu ;x ¼ bp ðx t ; b y n Þ b n ¼ bu ;x ¼ bp ðx t ; b y n Þð b n ; b 2 n Þ when ¼ for sample sizes n ¼ ; 25; 5;. In the graphs, we use ME to denote bp ðx t ; b y n Þ b b n, and ME2 to denote bp ðx t ; b y n Þ b b 2 n : The graphs show that in large samples the distributions of scaled marginal effects appear to approach the asymptotic distributions derived in the paper. Again, there.7.6 density choice, n= choice, n=5 choice, n= choice, n= Fig. 2. Stationary models with r ¼ :95.

15 28 P.C.B. Phillips et al. / Journal of Econometrics 4 (27) choice, n= choice, n=5 choice, n= choice, n=5.25 density Fig. 3. Stationary models with r ¼ :99. density. ME,limit ME, n= ME, n=25 ME, n=5 ME, n= Fig. 4. Density of marginal effects when ¼. appears to be a pile-up problem towards the limits of the domain of definition particularly in the case of ME. Investigation shows that this problem also occurs in the stationary case for large values of the autoregressive coefficient. As for the predicted probabilities, this phenomenon deserves further study.

16 P.C.B. Phillips et al. / Journal of Econometrics 4 (27) density ME2, limit ME2, n= ME2, n=25 ME2, n=5 ME2, n= Fig. 5. Density of marginal effects when ¼. Acknowledgments We thank the referees and Co-Editor for helpful comments and advice. A full length version of this paper (Phillips et al., 25) is available on the authors websites. Phillips gratefully acknowledges research support from a Kelly Fellowship at the School of Business, University of Auckland, and the NSF under Grant No. SES Jin thanks the Cowles Foundation for support under a Cowles Fellowship. Appendix. Useful lemmas and proofs The corresponding appendix of PJH contains some useful lemmas, proofs of those lemmas, and proofs of the main results in the paper, which update and extend those in HP, PP and Park and Phillips (999, 2). The updating takes into account the explicit form of the dependence of functions on the threshold. The reader is referred to PJH for details. We provide here only the statement of the key lemma that is used directly in the derivation of the main results, showing the effects of nonzero thresholds. Some related work, which gives a version of part (a) of the lemma, is contained in the recent paper by Jeganathan (24). Lemma E (Extends Lemma 2 of PP to local time away from the origin). Let Assumption in HP hold, f : R R be regular, and ma. Then we have: p (a) ð= ffiffiffi P n Þ n t¼ f ðx p t ffiffi n mþ)l ð; mþ R f ðsþ ds, (b) ð=nþ P n t¼ f ðx p t ffiffiffi n mþx2t ) R V 2ðrÞ dl ðr; mþ R f ðsþ ds, (c) ð=n 3=2 Þ P n t¼ f ðx p t ffiffi n mþx2t x 2t ) R V 2ðrÞV 2 ðrþ dl ðr; mþ R f ðsþ ds.

17 3 P.C.B. Phillips et al. / Journal of Econometrics 4 (27) 5 3 References Hu, L., Phillips, P.C.B., 24a. Nonstationary discrete choice. Journal of Econometrics 2, Hu, L., Phillips, P.C.B., 24b. Dynamics of the federal funds target rate: a nonstationary discrete choice approach. Journal of Applied Econometrics 9, Jeganathan, J., 24. Convergence of functionals of sums of random variables to local times of fractional stable motions. Annals of Probability 32, Park, J.Y., Phillips, P.C.B., 999. Asymptotics for nonlinear transformations of integrated time series. Econometric Theory 5, Park, J.Y., Phillips, P.C.B., 2. Nonstationary binary choice. Econometrica 68, Park, J.Y., Phillips, P.C.B., 2. Nonlinear regression with integrated processes. Econometrica 69, 7 6. Phillips, P.C.B., 998/25. Econometric analysis of the Fisher equation. Cowles Foundation Discussion Paper No. 8. Published in American Journal of Economics and Sociology 64(). Phillips, P.C.B., 2. Descriptive econometrics for non-stationary time series with empirical applications. Journal of Applied Econometrics 6, Phillips, P.C.B., Jin, S., Hu, L., 25. Nonstationary discrete choice: a corrigendum and addendum. Cowles Foundation Discussion Paper No. 56, Yale University.