On the Robustness of Mixture Models in the Presence of Hidden Markov Regimes with Covariate-Dependent Transition Probabilities

On the Robustness of Mixture Mo dels in the Presence of Hidden Mark o v Regimes with Co v ariate-Dep enden t T ransition Probabilities ∗ Demian P ouzo Departmen t of Ec onomics, Univers ity of California, Berke ley , U.S.A. Email: dp ouzo@berkele y .edu Zac harias Psaradakis Birkb e ck Business Sc ho ol, Birkb ec k, Univ ersit y of London, U.K. Email: z.psaradakis@bbk.ac.uk Martin Sola Departmen t of Economics, Univ ersidad T o rcuato di T ella, Argentina Email: msola@utdt.edu Abstract This pap er studies the robus tness of quasi-maximum-lik elihoo d (QML) esti- mation in hid d en Mark o v mo dels (HMMs) when the regime-switc hing struc- ture is missp ecified. Sp ecifically , we examine the case wh ere the true data- generating pro cess features a hidden Mark o v regime sequence with co v ariate - dep end en t transition probabilities, but estimation pro ceeds u nder a simpli- fied mixture mo del that assumes regimes are in dep end en t and iden tically dis- tributed. W e sh o w that the parameters go v erning the conditional d istribution of the observ ables can still b e consistently estimated u nder this missp ecifi- cation, p ro vided certain regularit y conditions h old. Our results highlight a practical b en efit of us ing computationally simpler mixtur e mo dels in settings where r egime dep end ence is complex or difficu lt to mo del d irectly . Key wor ds and phr ases : Consistency; co v ariate-dep endent transition probabil- ities; id en tifiabilit y; hidden Marko v mo del; mixtu r e mo del; quasi-maxim um- lik eliho o d; missp ecified mo d el. ∗ The authors wish to thank Patrik Guggenberg er, Peter Phillips and tw o referees for their helpful comments and sug gestions. F or the purpo ses of op en a ccess, the cor resp onding author has a pplied a CC-BY public copyright licence to any accepted manuscript version ar ising from this submission. Address corresp ondence to: Za charias Psar adakis, Birk be ck Business School, Birkb eck, Univ er sity of Lo ndon, Malet Street, Lo ndon WC1E 7HX, UK; e-mail: z.psar adakis@ bbk.ac.uk. 1 1 In tro ductio n Consistency and asymptotic normality of least-squares estimators in regression mo d- els in the presence o f p oten tial mo del missp ecification — e.g., missp ecification of the resp onse function or misspecification of the dynamic structure of the errors — are w ell-established fa cts (see, e.g., Domow itz and White ( 19 82 )). Suc h fundamen tal results, together with the related classical work of Hub er ( 1967 ), underpin a large b o dy of literature exploring the feasibilit y of dra wing v alid and meaningful inferences from para metric mo dels that need not necessarily con tain the true data- generating pro cess (DGP). Numerous results of this kind hav e b een established for a wide v a- riet y o f mo dels and estimators, b oth in static and dynamic settings, ranging from inference pro cedures based on estimating equations and momen t conditions (e.g., Bates and White ( 1985 )) to quasi-maxim um-likelihoo d (QML) pro cedures for condi- tional mean, conditional v ariance and conditional quan tile mo dels (e.g., White ( 19 82 , 1994 ), Levine ( 19 8 3 ), Gourieroux et al. ( 1984 ), New ey and Steigerw ald ( 1997 ), Komunje r ( 2005 )). This pap er a dds to the literature by presen ting another example o f robustness with r esp ect to missp ecification. Sp ecifically , we consider the case of Hidden Marko v Mo dels (HMMs), where observ able v ariables exhibit conditional indep endence giv en an underlying unobserv able regime s equence (and, p ossibly , exogenous co v ariate sequence s), fo cusing on situatio ns where the dep endence structure of the regime sequence is missp ecified. In our set-up, the DGP is take n to b e a generalized HMM that ma y include co v ariates and has a finite n um b er of Mark o v regimes, but the p ostulated probability mo del is a finite mixture mo del, that is, an HMM with in- dep enden t, iden tically distributed (i.i.d.) regimes. By considering t he pseudo-true parameter set for the QML estimator in the (misspecified) mixture mo del, it is sho wn that the parameters of the conditional distribution of the observ able r esp o nse v ari- ables are consisten tly estimable ev en if the dep endence of the unobserv able regime sequence is not ta ken into accoun t. A condition on the tail b ehavior of the c har- acteristic function of the (standardized) conditiona l distribution of the observ able 2 resp onses is also pr ovided under whic h the pseudo-true parameter for the Q ML es- timator is a singleton set. An important distinguish ing feature of our a nalysis is that the true regime sequence is allow ed to b e a temp or ally inhomog eneous Mark ov c hain whose transition probabilities a re functions o f observ able v ariables. This case holds pr a ctical significance giv en the widespread use of b oth HMM s and mixture mo dels. HMMs with temp orally inhomogeneous regime sequence s hav e found applications in div erse areas suc h as biolo gy (e.g., Ghavidel et al. ( 2 015 )), eco- nomics (e.g., Dieb old et al. ( 19 94 ), Engel and Hakkio ( 1996 ) ), earth sciences (e.g., Hughes et al. ( 1999 )), and engineering (e.g., Ramesh and Wilp on ( 199 2 )). T emp o- rally homogenous v arian ts of HMMs and of Mar ko v-switc hing regression mo dels are also used extensiv ely in economics and finance (e.g., Engel and Hamilton ( 1990 ), Ryd ´ en et a l. ( 1998 ), Jeanne and Masson ( 2000 ), Bo llen et a l. ( 2008 )), as w ell as in biology , computing, engineering and statistics (see Ephraim and Merhav ( 2002 ) and references therein). Statistical inference in suc h mo dels is typ ically like liho o d-based and the prop erties of QML pro cedures ar e, naturally , of m uc h in terest. Nev ertheless, HMMs a re inherently in tricate and computationally demanding due t o the need to accoun t for the underlying correlated regime sequence a nd for the dep endence of the conditional distribution on the curren t hidden regime. By demonstrating that it is feasible t o use a mixture mo del — a simpler and computationally less demanding framew ork — while still estimating consisten tly the parameters of the conditional distribution of the observ at io ns, this pap er off ers a more accessible av enue f o r prac- titioners to follow without sacrificing the accuracy of parameter estimates. In related recen t w ork, P ouzo et al. ( 202 2 ) considered the asymptotic prop er- ties o f t he QML estimator in a ric h class o f mo dels with Marko v regimes under general conditions whic h allow for autoregressiv e dynamics in the o bserv ation se- quence, co v ariate-dep endence in t he transition probabilities of the hidden regime sequence , and p oten tial mo del missp ecification. The QML estimator w as shown to b e consisten t fo r the pseudo-true parameter (set) that minimizes the Kullbac k– Leibler information measure . Unsu rprisingly , iden tifying the p ossible limit o f the 3 QML estimator when the true pro babilit y structure of the data do es not necessar- ily lie within the para metric f a mily of distributions sp ecified b y the mo del is not alw a ys a feasible task in suc h a general set-up. This pap er provides an answ er in the simpler case of switc hing-regression mo dels, HMMs and related mixture mo dels. Consistency results for missp ecified pure HMMs (with no co v ariates in the o ut come equation) can also b e found in Mev el and Finesso ( 2004 ) and Douc and Moulines ( 2012 ). Unlike our ana lysis, whic h allow s the regime transition probabilities to b e time-dep enden t and driv en b y observ able v ariables, these pap ers restrict attention to the case of time-inv arian t transition mec hanisms. In the next section, w e in tro duce the D GP and statistical mo del of interest, and consider Q ML estimation of the parameters of the outcome equation of a missp ecified generalized HMM. Section 3 discusses numeric al results from a sim ulation study . Section 4 summarizes and concludes. 2 F ramew ork, Results and Di s cussio n 2.1 DGP and Mo del Consider a discrete-time sto chastic pro cess { ( X t , S t ) } t ≥ 0 suc h that X t = ( Y t , Z t , W t ) is an observ able v ariable with v alues in X ⊂ R 3 and S t is a laten t v ariable with v alues in S := { 1 , 2 , . . . , d } ⊂ N for some d ≥ 2. The v aria ble S t is view ed as the hidden regime (or state) asso ciated with index t , whic h is “observ a ble” only indirectly through its effect on X t . The following assumptions are made ab o ut the DGP: 1. F or each t ≥ 1, t he conditional distribution of S t giv en X t − 1 0 := ( X 0 , . . . , X t − 1 ) and S t − 1 0 := ( S 0 , . . . , S t − 1 ), de noted b y Q ∗ ( ·| Z t − 1 , S t − 1 ), de p ends only on ( Z t − 1 , S t − 1 ) and is suc h that Q ∗ ( s | z , s ′ ) > 0 for all ( s, s ′ , z ) ∈ S 2 × Z , where Z ⊂ R is the state space of Z t . 2. F or eac h t ≥ 1, the conditional distribution of Z t giv en ( X t − 1 0 , S t 0 ) dep ends only on Z t − 1 ; furthermore, { ( Z t , S t ) } t ≥ 0 is strictly statio nary with in v ariant 4 distribution ν Z S . 3. F or eac h t ≥ 1, the conditional distribution o f Y t giv en ( X t − 1 0 , S t 0 , W t ) dep ends only on ( W t , S t ) and is sp ecified via the equation Y t = µ ∗ 1 ( S t ) + γ ∗ ( S t ) W t + σ ∗ 1 ( S t ) U 1 ,t , (1) where µ ∗ 1 , γ ∗ and σ ∗ 1 > 0 are known real functions on S ; the noise v ariables { U 1 ,t } t ≥ 0 are i.i.d., indep enden t of { S t } t ≥ 0 , with mean zero, v ariance one, and densit y f . 4. F or each t ≥ 1, the conditional distribution of W t giv en ( X t − 1 0 , S t 0 ) dep ends only on W t − 1 , and W t is indep enden t of ( Z t − 1 , S t − 1 ); furthermore, { W t } ≥ 0 is strictly exogenous in ( 1 ) and strictly stationary with inv a rian t distribution ν W . Instead of the Mark ov-sw itching structure of the DG P , the researc her’s p ostu- lated para metric mo del is a family of finite mixture mo dels (without Marko v dep en- dence). Sp ecifically , the mo del is specified by assuming that the regime v ariables { S t } t ≥ 1 are i.i.d. with distribution Q ¯ ϑ ( s ) = ¯ ϑ s ∈ (0 , 1) , s ∈ S . (2) In addition, the observ able v ariables { Y t } t ≥ 1 are assumed to satisfy the equations Y t = µ ( S t ) + γ ( S t ) W t + σ ( S t ) ε t , t ≥ 1 , (3) where µ , γ and σ > 0 are know n real functions on S a nd { ε t } t ≥ 1 are i.i.d. random v aria bles, indep enden t of { ( S t , W t ) } t ≥ 1 , suc h that ε 1 has the same densit y f as U 1 , 1 . The mixture model defined b y ( 2 ) and ( 3 ) is parameterized b y θ := ( π ( s ) , ¯ ϑ s ) s ∈ S , with π ( s ) := ( µ ( s ) , γ ( s ) , σ ( s )), whic h is assum ed to tak e v alues in a compact set Θ ⊂ R q , q > 1. W e denote b y P π ( ·| W t , S t ) the conditional distribution of Y t giv en ( W t , S t ) that is implied by ( 3 ); the corresp onding conditio na l densit y is denoted by p π ( ·| W t , S t ). 5 Key asp ects of our set-up: First, the DGP has a (generalized) HMM structure in whic h { Y t } t ≥ 0 are indep enden t, conditionally on the regime sequence { S t } t ≥ 0 and an exogenous cov a r ia te sequence { W t } t ≥ 0 (ha ving the Marko v prop ert y), so that the conditional distribution of Y t giv en t he regime and cov ariate sequences dep ends only on ( S t , W t ). The inclusion o f the exogenous cov ariate W t in ( 1 ) and ( 3 ) allow s the study of the causal effect o f W on Y under differen t regimes; this caus al effect is captured b y γ ∗ and is estimable via the mixture sp ecification ( 2 )–( 3 ). Exogeneit y of W (assumption 4 ab o ve ) is essen tial fo r the results discussed in Section 2.2 to hold and, hence, fo r consisten t estimation of the causal effect γ ∗ under the (erroneous) assumption o f indep enden t regimes. 1 Second, the true hidden regimes { S t } t ≥ 0 are a temp orally inhomogeneous Mark ov c hain whose transition probabilities dep end on the lagged v alue of the obse rv able v ariable Z t . The sequence { Z t } t ≥ 0 has the Mark o v prop ert y and is not required to b e exogenous, in the sense that Z t ma y b e con temp oraneously correlated with U 1 ,t . Third, the statistical mo del is missp ecified, in the sense that the D GP is not a mem b er of the fa mily { ( P π , Q ¯ ϑ ) : ( π , ¯ ϑ ) ∈ Θ } ; this is b ecause the dynamic structure of the regimes is missp ecified. As already discussed in Section 1 , this relativ ely simple set-up is o f m uc h pra ctical interes t since HMMs with tempo r a lly inhomogeneous regime sequences ha v e found man y applications. Mixture mo dels with i.i.d. regimes are also widely used in many differen t fields (see McLac hlan and Pe el ( 2000 ) and F r ¨ uh wirth-Sc hnatter ( 2006 )), including economics and econometrics (see Compiani and Kitamura ( 2016 )). It is w orth noting that, although w e fo cus on scalar responses and co v ariates for the sake of simplicit y , all our results can b e extended straigh tfor w ardly to cases where X t ∈ X ⊂ R h with h > 3. F or example, W t ma y b e a v ector of co v ariates, whic h ma y include la gged v alues of W t in case s where dynamic causal effects are of in terest. Similarly , Z t ma y b e a v ector o f information v ariables that affect the dynamic profile of the regime transition probabilities, whose generating mec hanism has a finite-order autoregressiv e structure. 1 The standard HMM formulation is a sp ecial cas e in which W t is absent from the outco me equation ( 1 ). 6 2.2 QML Estimation Giv en observ ations ( X 1 , . . . , X T ), T ≥ 1 , the quasi-log -lik eliho o d function for the parameter θ is θ 7→ ℓ T ( θ ) := T − 1 T X t =1 ln X s ∈ S ¯ ϑ s p π ( Y t | W t , s ) ! . (4) The QML estimator ˆ θ T of θ is defined as an approximate maximizer of ℓ T ( θ ) o ve r Θ, so that ℓ T ( ˆ θ T ) ≥ sup θ ∈ Θ ℓ T ( θ ) − η T , for some sequence { η T } T ≥ 1 ⊂ R + con v erging to zero. It is not t o o onerous to verify that, under assumptions that are common in the literature (e.g., G aussianit y o f U 1 , 1 and Q ∗ ( s | z , s ′ ) = G ( α s,s ′ + β s,s ′ z ) for some con tin uous distribution function G on R whose supp ort is all of R ), the conditions of P ouzo et al. ( 20 22 ) required for con v ergence of the QML estimator of θ to a w ell-defined limit are satisfied. Sp ecifically , let θ 7→ H ∗ ( θ ) := E ¯ P ∗  ln  p ∗ ( Y 1 | W 1 ) p θ ( Y 1 | W 1 )  b e the Kullback–Leibler information function, where p θ ( Y 1 | W 1 ) := P s ∈ S ¯ ϑ s p π ( Y 1 | W 1 , s ) denotes the conditional densit y of Y 1 giv en W 1 induced by ( P π , Q ¯ ϑ ) fo r eac h ( π , ¯ ϑ ) ∈ Θ, p ∗ ( Y 1 | W 1 ) denotes the conditional densit y of Y 1 giv en W 1 induced b y the (t r ue) DGP , a nd the exp ectation E ¯ P ∗ ( · ) is with resp ect to the distribution ¯ P ∗ of { ( X t , S t ) } t ≥ 0 induced by the (true) DGP . Then, w e ha v e inf θ ∈ Θ ∗ || ˆ θ T − θ | | → 0 as T → ∞ , (5) in ¯ P ∗ -probability , where Θ ∗ := arg min θ ∈ Θ H ∗ ( θ ) (6) is the pseudo-true parameter (set) and k·k denotes the Euclidean norm on R q (cf. Theorem 1 of P ouzo et al. ( 2022 )). A sharper result can b e establishe d b y considering the pseudo-true parameter Θ ∗ under the sp ecified DG P . T o g ether with ( 5 ) and ( 6 ), the follow ing theorem 7 sho ws that, despite the erroneous treatmen t of hidden r egimes as indep enden t, QML based on the (missp ecified) mixture mo del provide s consisten t estimators of the true parameters of the outcome equation. Theorem 1. The choic e µ = µ ∗ 1 , σ = σ ∗ 1 , γ = γ ∗ , and ( ¯ ϑ ∗ s ) s ∈ S such that ¯ ϑ ∗ s = E ν Z S [ Q ∗ ( s | Z , S )] for al l s ∈ S is a pseudo-true p ar a m eter, that is, it m a ximizes the function θ 7→ E ¯ P ∗ " ln X s ∈ S ¯ ϑ s σ ( s ) f  Y 1 − µ ( s ) − γ ( s ) W 1 σ ( s )  !# . Pr o of. Obs erv e that the Kullback –Leibler info rmation function H ∗ is prop ortio na l to θ 7→ − Z R 2 ln  P s ∈ S ¯ ϑ s σ ( s ) − 1 f (( y − µ ( s ) − γ ( s ) w ) /σ ( s )) p ∗ ( y | w )  p ∗ ( y | w ) d y ν W ( dw ) , (7) where ( y , w ) 7→ p ∗ ( y | w ) = X s ∈ S Pr ∗ ( S 1 = s | W 1 = w ) σ ∗ 1 ( s ) − 1 f (( y − µ ∗ 1 ( s ) − γ ∗ ( s ) w ) /σ ∗ 1 ( s )) . Under our assumptions ab out ( W t , Z t − 1 ), Pr ∗ ( S 1 = · | W 1 = w ) = Pr ∗ ( S 1 = · ), where Pr ∗ stands for the true probabilit y ov er the hidden regimes, giv en b y s 7→ Pr ∗ ( S 1 = s ) := Z R × S X s ′ ∈ S Q ∗ ( s | z , s ′ ) ν Z S ( dz , ds ′ ) . The minimizers of the function in ( 7 ) are all θ suc h that X s ∈ S ¯ ϑ s σ ( s ) − 1 f (( · − µ ( s ) − γ ( s ) · ) /σ ( s )) = p ∗ ( ·|· ) . It is straigh tforw ard to v erify that the equality abov e holds for µ = µ ∗ 1 , σ = σ ∗ 1 , γ = γ ∗ , and ¯ ϑ ∗ suc h t ha t ¯ ϑ ∗ s = Pr ∗ ( S 1 = s ). Theorem 1 establishes that the true parameters π ∗ ( s ) := ( µ ∗ 1 ( s ) , σ ∗ 1 ( s ) , γ ∗ ( s )), s ∈ S , asso ciated with the observ ation equation ( 1 ), to g ether with ( ¯ ϑ ∗ s ) s ∈ S , minimize the Kullback–Leible r information H ∗ ( θ ). The corollary that follo ws sho ws that this minimizer is unique, a s long as θ ∗ := ( π ∗ ( s ) , ¯ ϑ ∗ s ) s ∈ S is iden tified in Θ. In the presen t 8 con text, θ ∗ is said to b e iden tified in Θ if, f o r an y θ ∈ Θ suc h that p θ ( ·|· ) = p θ ∗ ( ·|· ) ¯ P ∗ -almost surely , θ = θ ∗ up to p erm utations. 2 Corollary 1. If θ ∗ is identifie d in Θ , then it is the unique maximizer (up to p erm u- tations) of θ 7→ E ¯ P ∗ " ln X s ∈ S ¯ ϑ s σ ( s ) f  Y 1 − µ ( s ) − γ ( s ) W 1 σ ( s )  !# . Pr o of. B y Theorem 1 , it suffices to show that Z R 2 ln ( p ∗ ( y | w )) p ∗ ( y | w ) d y ν W ( dw ) > Z R 2 ln ( p θ ( y | w )) p ∗ ( y | w ) d y ν W ( dw ) , for any θ ∈ Θ whic h is not equal to a p erm utation of θ ∗ . Since θ ∗ is identifie d, for an y suc h θ , p θ ( ·|· ) 6 = p θ ∗ ( ·|· ) with p ositiv e probability under ¯ P ∗ . Therefore, the strict inequalit y ab ov e follow s from the (strict) Jensen inequalit y . T o provide some (non- tec hnical) in tuition behind Theorem 1 and its corollary , recall that when the missp ecified mixture mo del is fitted to data using maximu m like- liho o d techniq ues, the ob jectiv e function whic h is maximized is a quasi-lik eliho o d. The resulting QML estimator is a n estimator for the parameters whic h make the mo del’s implied conditional distribution of the resp o nse as close as p ossible — mea- sured in Kullbac k–Leibler div ergence — to the true conditional distribution, ev en when the mo del is missp ecified. Despite the fact that the researc her ignores de- p endence of t he regimes, the conditional distribution of the resp o nse v a riable giv en the co v ariates remains a mixture of distributions under b ot h the true a nd missp ec- ified mo dels. This structural similarity allows the mixture mo del to match the k ey features of the conditional distribution of intere st correctly and, thu s, the QML es- timator consisten tly reco v ers the parameters of the outcome equation. T his result relies heavily on t w o ke y conditions: s ta t io narit y of the regimes (Assumption 2) and exogeneit y of the co v ariates (Assumption 4). The former stationarit y condition is crucial because it ensures stabilit y of the mar ginal distribution of the r egimes, 2 F ormally , ( π ( p [ s ]) , ¯ ϑ p [ s ] ) = ( π ∗ ( s ) , ¯ ϑ ∗ s ) for all s ∈ S and any p ermutation p : S → S . The qualifier ‘up to per m utatio ns ’ reflects the fac t that the problem r emains unc hang ed if the indices of the regimes a re p ermuted. 9 whic h the mixture mo del tries to fit . Without stationa r it y , the limiting ob ject of the QML estimation pro cedure could v ary ov er time, and consistency w ould break do wn. The latter exogeneit y condition is fundamen tal b ecause it guaran tees that the cov ariates do no t “carry” infor mat ion ab out future states or future sho c ks into the noise of t he o utcome equation. If exogeneit y failed, the conditional distribution of the resp onse w ould not b e prop erly captured b y the simple mixture mo del and the QML estimator w ould b e biased and inconsisten t f o r the parameters of in terest. W e conclude b y remarking that when the minimizer θ ∗ of the Kullback –Leibler information function is iden tified in Θ (and b elongs to the in terior o f Θ), asymptotic normalit y of √ T ( ˆ θ T − θ ∗ ) may b e deduced from the results of Pouzo et al. ( 2022 ) under suitable differen tiability and momen t conditions. These conditions a re satis- fied, for example, in the case where f is Ga ussian and Q ∗ ( s | z , s ′ ) = G ( α s,s ′ + β s,s ′ z ) for some con tin uous distribution f unction G on R whose supp o r t is all of R . In the next subsection, w e discuss a sufficien t condition for the high-lev el iden tifiability requiremen t o f Corollary 1 a nd show that this condition holds in some commonly used mo dels. 2.3 Discussion on the Iden tifiabilit y Condition Iden tifiabilit y of mixture mo dels has b een studied extensiv ely in the literat ur e fol- lo wing t he o r iginal con tribution of T eic her ( 1963 ), who established identifiabilit y of finite mixtures o f distributions suc h as the one-dimensional Gaussian and gamma. Y ako witz and Spra gins ( 1968 ) gav e a neces sary and sufficien t condition for iden- tifiabilit y , whic h holds, for example, in the case of finite mixtures of multiv ariate Gaussian distributions. This condition w as exploited by Holzmann et a l. ( 2004 ) and Holzmann et a l. ( 2006 ) to pro vide sufficien t lo w-lev el iden tifiability conditions based on the tail b ehav ior of the c haracteristic function of the comp onen t distributions. 3 Using the approac h of Holzmann et al. ( 2004 ) and Holzmann et al. ( 2006 ), we no w presen t a low-lev el condition, based only on features of f , whic h is sufficien t for 3 A review of related results for pa r ametric and nonpara meter ic mo dels that incorp ora te mixtur e distributions can b e found in Compiani a nd Kitamura ( 2016 ). 10 the iden tifiability of θ ∗ required in Corollary 1 . Lemma 1. L et ϕ b e the c har acteristic function ass o ciate d with f . If, for any a 1 > a 2 , lim τ →∞ ϕ ( a 1 τ ) ϕ ( a 2 τ ) = 0 , then θ ∗ is identifie d in Θ . Pr o of. B y the classical result of Y ako witz and Spra gins ( 1968 ), to establish unique iden tifiabilit y , it suffices to sh ow tha t n 1 σ ( s ) f  y − m ( s,w ) σ ( s ) o s ∈ S are linearly indepen- den t, where m ( s, w ) := µ ( s ) + γ ( s ) w . T o this end, observ e that Z R e i τ y 1 σ ( s ) f  y − m ( s, w ) σ ( s )  dy = e i τ m ( s,w ) Z R e i τ uσ ( s ) f ( u ) d u = e i τ m ( s,w ) ϕ ( τ σ ( s )) . Hence, for a ny λ 1 , . . . , λ d in R , P s ∈ S λ s 1 σ ( s ) f  y − m ( s,w ) σ ( s )  = 0 implies X s ∈ S λ s e i τ m ( s,w ) ϕ ( τ σ ( s )) = 0 , (8) for an y τ ∈ R and any w ∈ R . Without loss of generalit y , let σ (1) ≤ σ (2) ≤ · · · ≤ σ ( d ) , with m ∈ S b eing suc h that σ (1 ) = · · · = σ ( m ) < σ ( m + 1). Then, ( 8 ) is equiv alen t to λ 1 + m X s =2 λ s e i τ [ m ( s,w ) − m (1 ,w )] + d X s = m +1 λ s e i τ [ m ( s,w ) − m (1 ,w )] ϕ ( τ σ ( s )) ϕ ( τ σ (1)) = 0 . (9) Since σ ( s ) > σ (1) for any s ∈ { m + 1 , . . . , d } a nd e i τ [ m ( s,w ) − m (1 ,w )] is uniformly b ounded in τ , it follows b y our assumption that P d s = m +1 λ s e i τ [ m ( s,w ) − m (1 ,w )] ϕ ( τ σ ( s )) ϕ ( τ σ (1)) → 0 as τ → ∞ . This result readily implies that n − 1 P n l =1 P d s = m +1 λ s e i lu 0 [ m ( s,w ) − m (1 ,w )] ϕ ( lu 0 σ ( s )) ϕ ( lu 0 σ (1)) → 0 as n → ∞ , for a n y u 0 > 0. Regarding the term P m s =2 λ s e i τ [ m ( s,w ) − m (1 ,w )] , observ e that m ( s, w ) 6 = m (1 , w ) for an y s ∈ { 1 , . . . , m } — otherwise, since σ ( s ) = σ (1), the regimes asso ciated with S t = s and S t = 1 w ould b e identical rather t ha n distinct. Th us, b y choosing τ = l u 0 , where l ∈ N and u 0 ∈ R + is suc h t ha t u 0 [ m ( s, w ) − m (1 , w )] ∈ ( − π , π ) for a ll s ∈ { 1 , . . . , m } , it follows by Lemma 2.1 in Holzmann et a l. ( 200 4 ) that n − 1 P n l =1 P m s =2 λ s e i lu 0 [ m ( s,w ) − m (1 ,w )] → 0 as n → ∞ . By 11 these tw o results and ( 9 ), λ 1 = − lim n →∞ n − 1 n X l =1 m X s =2 λ s e i lu 0 [ m ( s,w ) − m (1 ,w )] + d X s = m +1 λ s e i lu 0 [ m ( s,w ) − m (1 ,w )] ϕ ( l u 0 σ ( s )) ϕ ( l u 0 σ (1)) ! = 0 . By iterating on this pro cedure, it follo ws that λ 1 = λ 2 = · · · = λ d = 0, thereby establishing the desired result. As an example, consider what is, arguably , the most widely used class of mixture mo dels, namely those in whic h f is Gaussian. In this case, ϕ ( τ ) = e − τ 2 / 2 , τ ∈ R , and ϕ ( a 1 τ ) /ϕ ( a 2 τ ) = e − ( a 2 1 − a 2 2 ) τ 2 / 2 , a 1 > a 2 , so the condition of Lemma 1 is satisfied. Th us, in the Gaussian case, the QML estimator of the parameters of the mixture mo del ( 2 )–( 3 ) con v erges, in ¯ P ∗ -probability , to θ ∗ . This result remains v alid for non- Gaussian distributions, including distributions with heavy tails (and finite v ariance). F or instance, the result holds if f is t he densit y of a (rescaled) Student- t distribution with degrees o f freedom υ > 2 (see Example 1 in Holzmann et al. ( 2 006 )). 2.4 Discussion on the Main Theorem The consistency results in ( 5 )– ( 6 ) and in Theorem 1 are quite general, in the sense that they co v er missp ecified generalized HMMs with temp orally inhomogeneous regime sequences and arbitrary observ ation conditional densities. They imply that dep endence of the r egimes in such HMMs may b e safely igno red a s long as the pa- rameters of in terest a re those of the conditional densit y o f the o bserv ations giv en the regimes and the co v ariates. It is imp ortan t to note, ho w ev er, that care should b e tak en in estimating t he asymptotic co v ariance matrix of the QML estimator since the inv erse of the observ ed information matrix is not necessarily a consisten t esti- mator in a missp ecified mo del. Consisten t estimation in this case typically requires the use of an empirical sandwic h estimator tha t does not rely on the information matrix equalit y (cf. Theorem 5 of P ouzo et al. ( 2022 )). T reating the regimes as an indep enden t sequence simplifies lik eliho o d-based in- ference compared to the case o f correlated Mark ov regimes. In the latter case, an 12 added difficult y , as demonstrated b y Pouzo et al. ( 2022 ), is that consisten t QML estimation of the true para meter v alues in a mo del with Mark o v regimes hav ing co v ariate-dep enden t transition functions t ypically requires join t analysis of equa- tions suc h a s ( 1 ) and the generating mec hanism of { Z t } , eve n if the parameters of in terest are only those a sso ciated with ( 1 ). F urthermore, a s p ointed out by Hamilton ( 2016 ), ric h parameterizations o f the transition mec hanism of the regime sequence ma y no t necessarily b e desirable when w orking with relativ ely short time series b ecause of legitimate conc erns relating to p oten tial o v er-fitt ing and inaccu- rate statistical inference. In suc h cases, parsimonious sp ecifications whic h provide go o d approximations to k ey features of t he data — a nd, in our setting, consisten t estimates of the parameters of in terest — can b e at tractiv e and useful. Note t ha t, for a class of r egime-switc hing mo dels in whic h the regime sequence { S t } is a temp o rally homogeneous, t w o- stat e Mark o v c hain, an observ ation anal- ogous to that implied b y Theorem 1 w as made b y Cho and White ( 2007 ). They argued that the parameters of a mo del fo r the conditional distribution of the ob- serv able v ar ia ble X t , giv en ( X t − 1 0 , S t 0 ), can b e consisten tly estimated b y QML based on a missp ecified v ersion of the mo del with i.i.d. regimes — and exploited this result to construct a quasi-lik elihoo d-ratio test of the n ull h yp othesis of a single regime against the alternative hypothesis of t w o regimes. Ho w ev er, Carter a nd Steigerw ald ( 2012 ) demonstrated that consistency of the QML estimator f o r t he true parame- ters in suc h a setting do es not, in fact, hold if the mo del and the DGP con tain a n autoregressiv e comp onen t. T his observ atio n remains true in o ur more general set- up with temp orally inhomogeneous hidden r egime sequences. Specifically , a result analogous to that in Theorem 1 do es not hold when la gged v alues of Y t are presen t as co v ariates in the outcome equations ( 1 ) and ( 2 ) (e.g., as in Mark o v-switc hing autoregressiv e mo dels). In this case, misspecification of the de p endence structure of the regimes will affect estimation of all the parameters, not just those asso ciated with the tra nsition functions of the regime sequence. 13 3 Numerical Examples As a nu merical illustration of the results discussed in Section 2 , w e rep ort here findings from a small Mon te Carlo sim ulation study in whic h the effect on QML estimators of ignoring Marko v dep endence of hidden regimes is assessed. In the exp eriments , artificial data are generated according to the generalized HMM defined by ( 1 ), with regimes { S t } whic h form a Mark ov c hain on S = { 1 , 2 } suc h t ha t Pr( S t = s | S t − 1 = s, Z t − 1 = z ) = [1 + exp( − α ∗ s − β ∗ s z )] − 1 , s ∈ { 1 , 2 } , z ∈ R , and with { Z t } and { W t } satisfying the autoregressiv e equations Z t = µ ∗ 2 + ψ ∗ Z t − 1 + σ ∗ 2 U 2 ,t , W t = µ ∗ 3 + δ ∗ W t − 1 + σ ∗ 3 U 3 ,t . The noise v ariables { ( U 1 ,t , U 2 ,t , U 3 ,t ) } are i.i.d, Gaussian, indep enden t of { S t } , with mean zero a nd co v ariance matrix   1 ρ ∗ ω ∗ ρ ∗ 1 0 ω ∗ 0 1   . The parameter v alues are α ∗ 1 = α ∗ 2 = 2 , β ∗ 1 = − β ∗ 2 = 0 . 5 , µ ∗ 1 (1) = − µ ∗ 1 (2) = 1, γ ∗ (1) = 0 . 5, γ ∗ (2) = 1, σ ∗ 1 (1) = σ ∗ 1 (2) = 1 , µ ∗ 2 = µ ∗ 3 = 0 . 2, ψ ∗ = δ ∗ = 0 . 8, σ ∗ 2 = σ ∗ 3 = 1, and ρ ∗ , ω ∗ ∈ { 0 , 0 . 65 } . F or each o f 1 000 samples of size T ∈ { 20 0 , 80 0 , 16 00 , 3 200 } fr o m this DGP , estimates of the parameters of the outcome equation are obta ined b y maximizing the quasi-log-likelihoo d function ( 4 ) asso ciated with the mixture mo del ( 2 )–( 3 ), with Pr( S t = 1) = ¯ ϑ and ε t ∼ N (0 , 1). M onte Carlo estimates of the bia s of the QML estimators of µ (1), µ (2), γ (1), γ (2), σ (1) and σ (2) are rep orted in T able 1 . W e also rep ort the ratio of the sampling standard deviation of the estimators to estimated standard errors (a v eraged across replications for eac h design p oin t). The latter ar e computed using a sandwic h estimator based on the Hessian and the gradient of 14 the quasi-log-lik eliho o d function (cf. Pouzo et al. ( 202 2 , Theorem 5)) , with w eigh ts obtained from t he P arzen kerne l and a data-dep endent bandwidth selected by the plug-in metho d of Andrews ( 1991 ). The results fo r ω ∗ = 0 sho wn in the top panel o f T able 1 reve al that, although the estimators of µ ( 1) and µ (2) are somewhat biased in the smallest of the sample sizes considered, finite-sample bias b ecomes insignifican t in the rest of the cases (regardless o f the v alue of the correlation para meter ρ ∗ ), a s is to b e exp ected in light of the result in Theorem 1 . F urthermore, unless the sample size is small, estimated standard errors are v ery a ccurate a s approximations to the standard deviation of the QML estimators. The b otto m panel of T able 1 con tains results for a DGP with ω ∗ = 0 . 6 5. A non-zero v alue for the correlatio n parameter ω ∗ violates the exogeneit y assumption ab out W t that is main tained throughout Section 2 (and it is not ob vious what the limit p oin t of the Q ML estimator based on ( 4 ) might b e in this case). The sim ulation results show that estimators of the parameters of the o utcome equation are significantly biased, ev en for the largest sample size considered in the sim ulations. Biases in this case are clearly a consequence o f the mixture mo del b eing missp ecified b ey ond the assumption of i.i.d. regimes, the additional source of missp ecification b eing the incorrect a ssumption of uncorrelatedness of the cov ariate W t and the noise v aria ble U 1 ,t . The results relating to the accuracy of t he estimated standard error s are not substantially differen t from those obtained with ω ∗ = 0. As p oin ted out in Section 2.4 , another situation in whic h ignoring Mark o v dep en- dence of the regimes is costly inv olves outcome equations that con tain autoregressiv e dynamics. T o demonstrate numerically the difficulties in suc h a case, 100 0 artificial samples of v arious sizes are generated according to the Mark o v-switc hing autore- gression Y t = µ ∗ 1 ( S t ) + φ ∗ Y t − 1 + σ ∗ 1 ( S t ) U 1 ,t , (10) with φ ∗ = 0 . 9; the remaining parameter v alues and the generating mec hanisms of { Z t } , { S t } and { ( U 1 ,t , U 2 ,t ) } are the same as in earlier sim ulation exp erimen ts. F or 15 T able 1: Bias and Standard Deviation o f QML Estimators (HMM) T µ (1) µ (2) γ (1) γ (2) σ (1) σ (2) µ (1) µ (2) γ (1) γ (2) σ (1) σ (2) ρ ∗ = 0 , ω ∗ = 0 ρ ∗ = 0 . 65 , ω ∗ = 0 Bias 200 0.093 - 0.022 -0.0 3 3 0.018 -0.125 -0.045 0.090 - 0.032 -0.0 2 9 0.013 -0.111 -0.041 800 0.017 - 0.006 -0.0 0 3 0.002 -0.028 -0.013 0.021 - 0.006 -0.0 0 3 0.004 -0.023 -0.010 1600 0.009 -0 .001 -0.001 0.000 -0 .014 -0.006 -0.001 -0.008 0.002 0 .003 -0.008 -0.00 8 3200 -0.001 -0.004 -0 .001 0.001 -0.005 -0.003 0.010 0.000 -0.0 03 0.000 -0.006 -0.002 Standard Deviation / Standard Error 200 1.361 1.129 1.338 1.142 1 .452 1.1 7 9 1.365 1.234 1.520 1.168 1 .4 84 1.157 800 1.049 0.954 1.031 1.022 1 .040 0.9 8 4 1.035 0.979 1.036 1.010 1 .0 33 0.990 1600 1.054 1.031 1.022 1.008 1.02 4 1.006 0.998 0.966 1.005 0.972 1.0 16 0.975 3200 1.031 0.973 0.974 0.997 1.04 0 0.992 1.045 1.041 1.015 0.948 0.9 76 1.014 ρ ∗ = 0 , ω ∗ = 0 . 65 ρ ∗ = 0 . 65 , ω ∗ = 0 . 65 Bias 200 -0.190 -0.262 0.228 0 .254 -0.171 -0.12 0 -0.205 -0.280 0 .2 25 0.2 6 1 -0.162 -0.116 800 -0.226 -0.241 0.238 0 .238 -0.098 -0.09 0 -0.233 -0.243 0 .2 35 0.2 4 0 -0.096 -0.090 1600 -0.231 -0.238 0.236 0 .235 -0.089 -0.08 4 -0.2 2 7 -0.237 0.2 3 3 0.23 5 - 0.089 -0.084 3200 -0.235 -0.237 0.236 0 .235 -0.083 -0.08 2 -0.2 3 0 -0.238 0.2 3 4 0.23 6 - 0.085 -0.082 Standard Deviation / Standard Error 200 1.183 1.151 1.328 1.129 1 .328 1.0 7 5 1.182 1.189 1.322 1.111 1 .3 17 1.183 800 0.812 0.888 1.008 0.819 0 .826 0.8 6 1 0.979 1.012 1.035 0.991 1 .0 38 1.025 1600 0.997 1.042 0.998 0.988 1.00 0 0.975 0.989 1.071 0.982 0.965 1.0 11 1.003 3200 1.021 1.080 1.002 0.995 0.97 1 1.022 1.084 1.162 1.024 1.030 1.0 18 1.032 16 T able 2: Bias and Standard Deviation of Q ML Estimators (Mark ov -Switching Au- toregressiv e Mo del) T µ (1) µ (2) σ (1) σ (2) φ µ (1) µ (2) σ (1) σ (2) φ ρ ∗ = 0 ρ ∗ = 0 . 8 Bias 200 -0.500 0.22 2 -0.093 -0.141 -0.012 -0.765 0.468 -0.114 -0.13 5 -0.049 800 -0.400 0.03 9 0.0 42 -0.067 0 .002 -0.626 0.28 8 0.0 27 -0.060 -0.034 1600 -0.440 0.02 3 0.086 -0.049 0.004 -0.753 0.276 0.096 - 0.049 -0.031 3200 -0.462 0.01 3 0.115 -0.036 0.005 -0.699 0.249 0.103 - 0.035 -0.028 Standard Devia tion / Standard Er ror 200 1.546 1.214 1.764 1.563 1.140 0.438 1.261 0.542 1.008 1.023 800 1.282 0.907 1.335 1.177 0.997 1.228 0.928 1.347 1.121 0.954 1600 1.396 0.916 1.335 0.708 0.988 1.542 1.011 1.504 1.066 1.008 3200 1.424 0.897 1.398 0.913 0.968 1.078 0.803 1.102 0.775 0.976 eac h artificial sample, the parameters of the regime-switc hing autoregressiv e mo del Y t = µ ( S t ) + φ Y t − 1 + σ ( S t ) ε t , (1 1) are estimated b y maximizing the quasi-log- lik eliho o d function associated with it under the assumption that t he regime v a riables { S t } are i.i.d., with Pr( S t = 1 ) = ¯ ϑ , and the noise v ariables { ε t } are i.i.d., indep enden t of { S t } , with ε t ∼ N (0 , 1). The Monte Carlo results rep orted in T able 2 rev eal substan tial finite-sample bias in the case of the QML estimators o f the in tercepts µ (1) and µ (2). The QML estimators of σ (1), σ (2) and φ generally exhibit little bias, whic h ma y b e part ly due to the fact that the sim ulation design is such that the v alues of φ ∗ and σ ∗ 1 are the same regardless of the realized regime. Unlik e the HMM case considered b efore, estimated standard erro rs are not a lwa ys a ccurate as approx imatio ns to the finite- sample standard deviation of the Q ML estimators in the autoregressiv e mo del, ev en for a parameter such as σ (1) , whic h is estimated with little bias. W e note tha t qualitativ ely similar results are obtained whe n, in addition to Y t − 1 , an exogenous co v ariate W t , generated as in the previous experiments , is included in the right-hand sides of ( 10 ) and ( 11 ). 17 4 Conclus ion In this pap er, w e ha ve considered QML estimation of the parameters of a generalized HMM with exogenous cov a riates and a finite hidden state space. A distinguishing feature of our approac h is that it allow s the regime sequence to b e a tempo rally inhomogeneous Mark ov chain with cov ariate-dep enden t transition pro babilities. It has b een sho wn that a mixture mo del with independen t regimes is robust in the presence of correlated Mark o v regimes, in the sense that the parameters of the outcome equation can b e estimated consisten tly b y maximizing the quasi-lik eliho o d function asso ciated with the missp ecified mixture mo del. 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