On the Power of Tests for Regime Switching

Transcription

1 On the Power of Tests for Regime Switching joint work with Drew Carter and Ben Hansen Douglas G. Steigerwald UC Santa Barbara May 2015 D. Steigerwald (UCSB) Regime Switching May / 42

2 Motivating Example Cecchetti, Lam and Mark 1990 Y t - GNP annual growth rate (real, per capita) U t i.i.d.n 0, σ 2 S t latent Markov process Y t = µ 0 + δs t + U t S t = 0 contraction mean µ0 variance σ 2 1 expansion mean µ 1 = µ 0 + δ variance σ 2 P (S t = 1jS t 1 = 0) = p 0 P (S t = 0jS t 1 = 1) = p 1 Goal: test for the presence of two regimes D. Steigerwald (UCSB) Regime Switching May / 42

3 Testing ssues null hypothesis H 0 : δ = 0 (implicit 0 < p 0 < 1) expanded null includes boundary of parameter space F p 0 = 0 (and p 1 = 0) quasi likelihood stationary probability π = P (S t = 1) replaces p 0 and p 1 test statistic: quasi-likelihood ratio Q n asymptotic null distribution (Cho and White 2007) F maximum of a dependent Gaussian process F maximize over δ 2 must select F a ects size and power asymptotic distribution under alternatives unknown D. Steigerwald (UCSB) Regime Switching May / 42

4 Contributions asymptotic distribution of Q n under a sequence of local alternatives expression for limit process that relates to moments of the data Hermite polynomial expansion provide guidance to researcher on selection of guidance based on sample size link between Q n and moment-based tests D. Steigerwald (UCSB) Regime Switching May / 42

5 Background ssues: General Model Y t = µ + X T t β + δs t + Ũ t Ũ t N 0, σ 2 P (S t = 1) = π QMLE inconsistent for autoregressions Carter-Steigerwald 2012 Y t = µ + αy t 1 + δs t + Ũ t ˆα P 9 α dependence in S t is confounded with dependence in Y t F if S t is independent over time, QMLE is MLE X t does not contain lagged values of Y t Q n depends only on scaled regime separation Carter-Steigerwald 2013 Y t = µ + X T t β + σδs t + σu t U t N (0, 1) P (S t = 1) = π parameters of interest (π, δ) nuisance parameters γ = µ, β, σ 2 D. Steigerwald (UCSB) Regime Switching May / 42

6 Background ssues: Framing the Null Hypothesis Conditional Density Representation to form the likelihood, work with conditional densities h f (Y t, µ 0 ) = (2π) 2 1 exp 1 Y 2σ 2 t µ 0 Xt Tβ 2 i if S t = 0 h f (Y t, µ 1 ) = (2π) 2 1 exp 1 Y 2σ 2 t µ 1 Xt Tβ 2 i if S t = 1 under the null hypothesis of only a single regime, the density is D. Steigerwald (UCSB) Regime Switching May / 42

7 Background ssues: Framing the Null Hypothesis Conditional Density Representation to form the likelihood, work with conditional densities h f (Y t, µ 0 ) = (2π) 2 1 exp 1 Y 2σ 2 t µ 0 Xt Tβ 2 i if S t = 0 h f (Y t, µ 1 ) = (2π) 2 1 exp 1 Y 2σ 2 t µ 1 Xt Tβ 2 i if S t = 1 under the null hypothesis of only a single regime, the density is h f (Y t, µ ) = (2π) 2 1 exp 1 Y 2σ 2 t µ Xt Tβ 2 i for all t D. Steigerwald (UCSB) Regime Switching May / 42

8 Background ssues: Framing the Null Hypothesis Conditional Density Representation to form the likelihood, work with conditional densities h f (Y t, µ 0 ) = (2π) 2 1 exp 1 Y 2σ 2 t µ 0 Xt Tβ 2 i if S t = 0 h f (Y t, µ 1 ) = (2π) 2 1 exp 1 Y 2σ 2 t µ 1 Xt Tβ 2 i if S t = 1 under the null hypothesis of only a single regime, the density is h f (Y t, µ ) = (2π) 2 1 exp 1 Y 2σ 2 t µ Xt Tβ 2 i for all t more generally, one parameter shift: intercept, slope or variance D. Steigerwald (UCSB) Regime Switching May / 42

9 Expanded Null Hypothesis Ghosh and Sen 1985 Under H 0 : equivalently represented by 3 curves standard identi cation f (Y t, µ ) δ = 0 (µ 0 = µ 1 = µ ) 0 < π < 1 (curve 1) boundary of parameter space π = 0 µ 0 = µ (curve 2) π = 1 µ 1 = µ (curve 3) D. Steigerwald (UCSB) Regime Switching May / 42

10 Null Space Null Space δ=0 π=0 D. Steigerwald (UCSB) Regime Switching May / 42

11 Local Alternatives parameters of interest π P (S t = 1) δ number of σ separating regime means θ = π δ null hypothesis local alternatives H 0 : θ = 0 H 1,n : θ n = π n δ π n = n 1 2 h δ = δ interesting features of the limit distribution in a neighborhood of π = 0 F mixture with infrequent, widely separated second regime D. Steigerwald (UCSB) Regime Switching May / 42

12 Background ssues: Quasi-Likelihood quasi-likelihood for observation t (1 π) f (y t, µ 0 ) + πf (y t, µ 1 ) π = P (S t = 1) quasi-likelihood ratio statistic Q n = 2 (L n ( ˆω) L n ( ω)) ˆω - unrestricted estimates L n ˆθ, ˆδ, ˆγ ω - restricted estimates L n (0,, γ) (at π = 0, δ not identi ed) D. Steigerwald (UCSB) Regime Switching May / 42

13 Background ssues: Dependent Gaussian Process µ = 0 σ 2 = 1 x δ and construct the QLR x δ = δ 0 q n δ 0 χ 2 (1) δ 0 x δ = δ 00 q n δ 00 χ 2 (1) δ 00 F χ 2 (1) δ 0 = G δ 0 2 for xed δ, L n is maximized at ˆθ δ 0, δ 0, ˆγ δ 0 for xed δ 0 ˆθ δ 00, δ 00, ˆγ δ 00 for xed δ 00 G δ 0 N (0, 1) dependence - two estimates are derived from the same sample F Cov G δ 0, G δ 00 6= 0 D. Steigerwald (UCSB) Regime Switching May / 42

14 Background ssues: Compact Parameter Space Hartigan 1985 for any xed δ 0 lim Cov G δ 0, G δ 00 = 0 δ 00! G () contains a countably in nite number of asymptotically independent components sup δ2r jg (δ)j diverges to in nity must specify compact parameter space δ 2 D. Steigerwald (UCSB) Regime Switching May / 42

15 Asymptotic Distribution: Under Null Cho and White 2007 under H 0 : π = 0 Q n max (max (0, G )) 2, sup (min [0, G (δ)]) 2 δ2 Zero-Mean Gaussian Process G (δ) N (0, 1) correlated process G N (0, 1) correlated with G (δ) D. Steigerwald (UCSB) Regime Switching May / 42

16 Asymptotic Distribution: Under Local Alternatives under H 1,n : π = n 2 1 h δ = δ Q n max (max (0, G )) 2, sup (min [0, G (δ)]) 2 δ2 Nonzero-Mean Gaussian Process 0 G (δ) G N hδ 3 p6, 1 e δδ 1 δδ (δδ) 2 2 h δ 2 e δ2 1 δ 2 δ4 2 1 i 1, 1A 2 correlated process correlated with G (δ) D. Steigerwald (UCSB) Regime Switching May / 42

17 Quantiles of the Asymptotic Distribution need to know joint distribution of G (δ), G δ 0 under H 0, G (δ) is a zero-mean Gaussian process F Cov G (δ), G δ 0 alone determines joint distribution under H 1,n, G (δ) is a non-zero mean Gaussian process F covariance and mean together determine joint distribution Cho and White do not represent G (δ) directly simulate G A (δ) with Cov does not yield mean, cannot yield power we represent G (δ) with Hermite polynomials direct calculations of mean and covariance yields power G A (δ), G A δ 0 = Cov G (δ), G δ 0 D. Steigerwald (UCSB) Regime Switching May / 42

18 Gaussian Process Joint Distribution: Framework to obtain joint distribution of G (δ), G δ 0 x δ and consider q n (δ) = 2 L n ˆθ, δ, ˆγ L n (0,, γ) we will focus on (uniform) convergence of q n (δ) convergence of Q n = sup δ2 q n (δ) follows q n (δ) depends on representation of the likelihood D. Steigerwald (UCSB) Regime Switching May / 42

19 Likelihood Representation Chen and Chen 2001 L n (θ, δ, γ) = n log φ (v t ) + t=1 n log (1 + θz δ (v t )) t=1 "su cient" statistic for δ Z δ (v t ) = 1 δ e v t δ 1 2 δ 2 1 v t is the null hypothesis error v t = 1 σ y t µ x T t β represent Z δ (v t ) with Hermite polynomials D. Steigerwald (UCSB) Regime Switching May / 42

20 Hermite Polynomial Expansion e v t δ 1 2 δ 2 = j=0 δj j! H j (v t ) fh j g Hermite Polynomials H 0 (v t ) = 1 H 1 (v t ) = v t H 2 (v t ) = v 2 t 1 H 3 (v t ) = v 3 t 3v t if v t N (0, 1) F H j (v t ) is the j th centered moment F Cov H j (v t ), H k (v t ) = 0 thus Z δ (v t ) := 1 δ e v t δ 1 2 δ 2 1 = j=3 δ j 1 j! H j (v t ) asymptotic behavior of Z depends on asymptotic behavior of H D. Steigerwald (UCSB) Regime Switching May / 42

21 Hermite Polynomials: Asymptotic Behavior as n! (under normality) 1 p n n t=1 H j (v t ) H0 N (0, j!) n 1 p n H j (v t ) H 1,n N t=1 hδ j, j! for each J, multivariate CLT yields (H 1 (v t ),..., H J (v t )) N J (m, V ) V = diag (1!,..., J!) under H 0 m =0 under H 1,n m = hδ,..., hδ J D. Steigerwald (UCSB) Regime Switching May / 42

22 Asymptotic Behavior of Z as n!, J n!, J n /n! 0 1 p n n t=1 Z δ (v t ) = J n j=3 δ j 1 j!! n 1 p n H j (v t ) t=1 + o P (1) multivariate CLT for triangular arrays yields n 1 p n Z δ (v t ) N (m, V ) V = δ 2 e δ2 1 δ 2 δ 4 t=1 2! under H 0 m = 0 under H 1,n m = h e δδ (δδ 1 δδ ) 2 2 D. Steigerwald (UCSB) Regime Switching May / 42

23 Asymptotic Behavior of QLR: Fixed Delta let ṽ t be the OLS residuals q n (δ) = 0 1 p n n t=1 Z δ (ṽ t ) h i δ 2 e δ2 1 δ 2 1 δ C A 2 + o P (1) in contrast, Cho and White study q n (δ) = S T π 1 π S π + o P (1) D. Steigerwald (UCSB) Regime Switching May / 42

24 Asymptotic Distribution of QLR for all δ outside a local neighborhood of 0 q n (δ) (min [0, G (δ)]) 2 uniformly in δ if δ is local to 0 and π 6= 1 2 q n (δ) min G 0, G if δ is local to 0 and π = 1 2 q n (δ) max (max (0, G )) 2, min G 0, G therefore Q n max (max (0, G )) 2, sup (min [0, G (δ)]) 2 δ2 D. Steigerwald (UCSB) Regime Switching May / 42

25 Null Space Null Space δ=0 (max[0,g]) 2 sup δ (min[0,g(δ)]) 2 π=0 D. Steigerwald (UCSB) Regime Switching May / 42

26 Parameter Space Selection: Behavior Near Origin Gaussian Process G (δ) = ν δ j=3 δ j p η j! j ν δ = e δ2 1 δ 2 δ 4 2! 1 2 η j corresponds to H j H j is the j th central moment of ṽ t H η 0 H 1,n j N (0, 1) η j N hδ j, 1 local to δ = 0 G (δ) ν δ δ 3 p 3! η 3 if π 6= 1 2 G (δ) ν δ δ 4 p 4! η 4 if π = 1 2 D. Steigerwald (UCSB) Regime Switching May / 42

27 Parameter Space Selection: Behavior Near Origin Test Statistic local to δ = 0 q n (δ) is equivalent to the skewness in ṽ if π 6= 1 2 q n (δ) is equivalent to the kurtosis in ṽ if π = 1 2 local to δ = 0, Q n is similar to a Jarque-Bera test as δ increases in magnitude, higher-order moments grow in importance graph ν δ δ j p j! D. Steigerwald (UCSB) Regime Switching May / 42

28 Components of Hermite Polynomial Expansion D. Steigerwald (UCSB) Regime Switching May / 42

29 Link Between Parameter Space and Sample Size with larger, ˆδ n may be larger as δ grows, q n (δ) increasingly depends on higher-order moments higher-order moments "require" larger samples with larger, adequacy of asymptotic approximation of q n (δ) over requires larger sample sizes smaller can still have substantial power to reject because of shift in G (δ) D. Steigerwald (UCSB) Regime Switching May / 42

30 mplementation: Obtaining Critical Values generate fg (δ)g (for a set from together with G) as ν δ J j=3 δ j p j! η j η j N (0, 1) 2 J 2 max grid ( ) jδj Carter and Steigerwald (2013) obtain max h(max (0, G )) 2, sup (min [0, G (δ)]) 2i repeat; 5% size!.95 quantile Tabulated Critical Values (5 percent) χ 2 (1) [ 1, 1] [ 2, 2] [ 3, 3] [ 4, 4] [ 5, 5] [ 10, 10] critical value is 3.84 D. Steigerwald (UCSB) Regime Switching May / 42

31 Laboratory Performance Comparison Tests Skewness & kurtosis! evidence of regime switching Jarque-Bera JB = n s 2 n6 + (k n 3) 2 24 Neyman (modi ed for zero second derivative) sn C (α) = n max 2 6, min h i 2 0, k n /2 D. Steigerwald (UCSB) Regime Switching May / 42

32 Laboratory Performance QLR Test: Relative Performance location example n = 100 δ = 2 π =.7 = ( 3, 3) Test JB C (α) Q Size 2.9% 4.4% 5.1% Power 9.4% 5.7% 28.6% 3000 replications D. Steigerwald (UCSB) Regime Switching May / 42

33 Field Performance: Testing for Multiple Equilibria Bloom, Canning and Sevilla 2003 explain large di erence in income levels across countries 1 di erences in intrinsic geography 2 multiple equilibria (regimes) with poverty traps Y - income (log GDP) Z - latitude (catchall for geography) poverty trap (regime 1) occurs with probability π (Z ) regime 2 Y = µ 1 + β 1 Z + U 1 Var (U 1 ) = σ 2 1 Y = µ 2 + β 2 Z + U 2 Var (U 2 ) = σ 2 2 D. Steigerwald (UCSB) Regime Switching May / 42

34 Field Performance: Testing for Multiple Equilibria Reported Results 152 countries, 1985 GDP/capita bq n = 26.0 critical value from Monte Carlo Simulations sensitive to assumptions about data generating process 2 sensitive to numeric procedures (grid search, bounds on σ 2 1 and σ2 2 ) evidence supports multiple equilibria with poverty traps valid inference 1 construct Q n di erently 2 use tabulated critical values from the asymptotic null distribution D. Steigerwald (UCSB) Regime Switching May / 42

35 Field Performance: Testing for Multiple Equilibria Test Statistic Construction cross section ) MLE not QMLE for valid inference - changes to alternative model null model Y = µ + βz + U Var (U) = σ 2 alternative model changes 1 regime probability not a function of covariates 2 one parameter shift (µ or β or σ 2 ) regime 1, poverty trap, occurs with probability π Y = µ 1 + βz + U 1 Var (U 1 ) = σ 2 regime 2 Y = µ 2 + βz + U 2 Var (U 2 ) = σ 2 D. Steigerwald (UCSB) Regime Switching May / 42

36 Field Performance: Testing for Multiple Equilibria Valid nference 152 countries, 1985 GDP/capita bq n = 26.0 becomes bq n = 4.03 critical value based on = [ 3, 3] 6.18 becomes 7.0 evidence no longer supports multiple equilibria with poverty traps valid inference di erence key is replacing π (Z ) with π potential power loss from ignoring added nonlinearity empirics really do support intrinsic geography D. Steigerwald (UCSB) Regime Switching May / 42

37 Remarks asymptotic distribution under a local alternative expansion for G (δ) that relates to moments of the data connection between and n for larger δ, q n (δ) relies more heavily on higher-order moments link between Q n and moment-based tests provides powerful alternative to JB and C (α) D. Steigerwald (UCSB) Regime Switching May / 42

38 Asymptotic Null Distribution: Point Mass at Zero Arises from Boundary Requirement ntuition: Ĥ 3 = 3 (indicating a standard normal with another normal shifted to the left) x δ at a positive value, forcing a shift to the right H 3 = π (1 π) δ 3 forcing π negative hence if you allow both positive and negative shifts, the maximizing value for π is not negative Boundary Requirement: π 0 if ˆπ < 0, then set the estimate ˆθ = 0: when is ˆπ < 0? q n δ 0 min 0, G δ 0 2 q n δ 0 = 2 L ˆθ, L (0, ) = 0 ˆπ δ 0 a = 0 cg δ 0 c > 0 D. Steigerwald (UCSB) Regime Switching May / 42

39 Asymptotic Distribution: Simpli cation Q n max (max (0, G )) 2, sup (min [0, G (δ)]) 2 δ2 if = [ d, d] G (δ) is locally an odd function, which implies P (min [0, G (δ)] = 0) = 0 D. Steigerwald (UCSB) Regime Switching May / 42

40 Gaussian Process Behavior Odd Function Near the Origin Behavior of G in a Neighborhood of δ=0 δ D. Steigerwald (UCSB) Regime Switching May / 42

41 Asymptotic Null Distribution Gaussian Density Derivatives if the error density is Gaussian U t i.i.d.n 0, σ 2 then f (2) = c 0 f (1) setting the rst derivative equal to 0 automatically sets the second derivative to 0 if δ < ɛ, then asymptotically L n might not be maximized near π = 0 D. Steigerwald (UCSB) Regime Switching May / 42

42 Asymptotic Null Distribution Distribution and Error Density U t i.i.d.f f not Gaussian f is Gaussian Q n 0, σ 2 max Q n sup min [0, G (δ)] 2 (max (0, G )) 2, sup min [0, G (δ)] 2 D. Steigerwald (UCSB) Regime Switching May / 42

43 Consistency Result QMLE inconsistent for autoregressions Consistency of QMLE under H 1 ) consistency of QLR test Cho and White (Theorem 1.b): QMLE is consistent under H 1 for class of processes that includes h f (Y t, θ 0 ) = c exp 1 2 (Y t θ i αy t 1 ) 2i if S t = i i = 0, 1 Y t 1 impacts Y t directly through αy t 1 indirectly through S t 1 issue: QMLE ignores dependence in S t dependence in S t is confounded with dependence in Y t QMLE inconsistent for α D. Steigerwald (UCSB) Regime Switching May / 42

44 Consistency QMLE vs. QLRT QMLE inconsistent for autoregression Y t = θ i + αy t 1 + U t U t white noise if S t = i moving average Y t = θ i + U t U t = ε t + δε t 1 if S t = i QLR test may still be consistent consistency of QLRT requires only E [l t (π, θ)] maximized outside null space for AR(1), gradient is zero in every coordinate except α F F indicates maximum occurs outside null space de nitive treatment - bound likelihood under null and demonstrate there is always a point in the alternative space for which the likelihood exceeds the bound power likely a ected by ˆπ n, ˆα n, ˆθ n 9 (π, α, θ ) under H 1 D. Steigerwald (UCSB) Regime Switching May / 42