Robust Inference in Dynamic Regression Models with Persistent ... Robust Inference in Dynamic Regression Models with Persistent ...

Robust Inference in Dynamic RegressionModels with Persistent RegressorsJörg Breitung and Matei DemetrescuUniversity of Bonn, GermanyConference on Macro and Financial EconomicsBrunel University, May 24, 2011

The inference problemConsider the dynamic system:y t = βx t−1 + u tx t = ϱx t−1 + v t ,whereΣ = E(( )ut ( ) ) ( σ2ut vv t = u σ uvt σ uv σv2).The null hypothesis is that x t does not predict y t+1 orH 0 : β = 0

Review of the literature◮ Elliott, G and JH Stock (1994) Inference in Time Series Regressionwhen the Order of Integration of a Regressor Is Unknown,Econometric Theory 10, 672–700.◮ Cavanagh, CL, G Elliott, JH Stock (1995), Inference in Models withNearly Integrated Regressors, Econometric Theory 11, 1131–1147.◮ Jansson M and MJ Moreira (2006) Optimal Inference in RegressionModels with Nearly Integrated Regressors, Econometrica, 74,681–714.◮ Toda, HY and T Yamamoto (1995) Statistical Inference in VectorAutoregressions with Possibly Integrated Processes, Journal ofEconometrics, 66, 225–250.◮ Dolado, JJ and H Lütkepohl (1996) Making Wald Tests Work forCointegrated VAR Systems, Econometric Reviews, 15, 369–386.◮ Phillips, PCB and A Magdalinos (2009) Econometric Inference inthe Vicinity of Unity, Working paper.

◮ Consider the t-statistic of the hypothesis β = 0 iny t = β∆ξ t−1 + ψξ t−2 + u t◮ Distribution under the local alternative:(i) If α T = ᾱ with |ᾱ| < 1 and β T =˜t vaβ(d→ N bb √T, then)σ√ vσ , 1 u 1 − α2(ii) If α T = 1 − abTwith 0 < η < 1 and β η T = , thenT (1+η)/2˜t vaβ( )d b σv→ N √ , 1 .σ u 2a⇒ The modified VA test has superior power.

Tests based on instrumental variables◮ Idea: Replace the regressor x t by a suitable instrument z t◮ IV t-statistic:T∑z t−1 y ttβ iv = t=2√ , (1)T∑̂σ uzt−12t=2◮ Valid instruments obey the following assumption:The instrument z t satisfy(i) non-stationary instruments:[rT ]1 ∑√TVTzj=2z j⇒ Z (r)for some process Z with integrable paths such thatV Tz :=1T 1+2δT ∑t=2z 2 t−1 ⇒ V z ,

(ii) stationary instruments[rT ]1 ∑√Tj=2ζ j⇒ σ u W ζ (r)1with ζ t = √ z t−1 u t such thatT 2δ V Tz⎛⎜⎝√1∑ [rT ]T j=2 ζ j√1∑ [rT ]T j=2 u j√1∑ [rT ]T j=2 v j⎞⎟⎠ ⇒ ⎝⎛⎞σ u W ζ (r)σ u W u (r) ⎠σ v W v (r):= B (r)where B (r) is a 3-dimensional Brownian motion withcovariance matrix⎛σ 2 ⎞u σ ζu σ ζvΣ B = ⎝ σ ζu σu 2 σ uv⎠σ ζv σ uv σv2

Local power of some IV tests1. Short-memory instrument If z t = ∆x t and β T = b/ √ T , thentβiv d→ b σ √ ( ∫v 1 + ᾱ1)√ B c (1) 2 − B c (r) dB c (r) + N (0, 1)σ u 1 − ᾱ2. Linear trend If z t = t and β T = b/T , thentβiv d→ b σ √ ∫v 3 1rB c (r) dr + N (0, 1)σ u3. Cauchy instrument If z t = sign(x t ) and β T = b/T , thent ivβd→ b σ v00∫ 1|B c (r)| dr + N (0, 1)σ u04. Fractionally integrated instrument If z t = ∆ 1−d ∗+ x t for somed ∗ ∈ (0, 0.5) and β T = b/T 0.5+d ∗ , thent ivβd→ bσ vσ u√Γ(1−2d ∗ )Γ 2 (1−d ∗ )(B d ∗ +1c (1) B c (1) −∫ 10)B d ∗ +1c (r) dB c (r) +N (0, 1)Nwhere B d ∗ +1c (r) is the Ornstein-Uhlenbeck process driven by thefractional Brownian motion W d ∗ +1 .

Combination of instruments◮ The two-stage least-squares (2SLS) IV t-statistic is given by) −1 ( T∑)z t−1 y tt=2t 2Sβ =( T∑) ( T∑x t−1 z t−1′ z t−1 z t−1′t=2t=2√ ( T∑) −1 ( T∑) .̂σ u x t−1 zt−1′ z t−1 x t−1t=2 t=2) ( T∑z t−1 zt−1′t=2◮ The vector of instruments z t obeys a generalized version ofAssumption 1◮ In general, the 2SLS estimator has a non-standard limitingdistribution◮ Exception:Vector of m instruments z t = (z 1t , . . . , z mt ) ′ with associatedconstants δ 1 , . . . , δ m and κ 1 , . . . , κ m . If there exist aninstrument z it such that κ i > κ j for all j ≠ i, the 2SLSt-statistic possess a standard normal limiting distributionunder the null hypothesis β = 0.

The Anderson-Rubin statistic◮ Test of γ = 0 in the regression:y t= γ ′ z t−1 + e t⇒ x t−1 is replaced by z t−1◮ Anderson-Rubin (or Sargan) statistic:AR = 1̂σ 2 u(∑ T) ( T) −1 (∑T)∑y t z t−1′ z t−1 z t−1′ z t−1 y tt=1t=1t=1◮ The limiting distribution under the sequence of localalternativesbβ T =T max(κ i )is noncentral χ 2 with k degrees of freedom.◮ Critical values increase with m ⇒ loss of power

Small sample comparison◮ Data generated according towhere E(x t ) = µy t = α + βx t−1 + u tx t = ϱx t−1 + v t ,◮ correlation between u t and v t is ω = 0.8.◮ Test statistics:OLS: OLS t-statistic (biased)VA(ᾱ = 0): original VA test (Dolado and Ltkepohl 1996)VA mild: modified VA test with ᾱ = 1 − 3/T 0.8IV Cauchy: IV with sign(x t−1 ) as instrumentIV(d ∗ = 0.5): IV tests with fractionally integrated instrumentIV trend: IV test with linear trendIV comb: IV combining trend and fract. instrument

Table: Empirical size and power when the regressor is nearly integratedϱ = 1.00b OLS VA VA mild IV Cauchy IV(d*=0.5) IV trend IV comb0 20.7 5.4 7.2 5.0 5.2 5.0 5.75 19.8 5.4 7.1 11.5 8.6 29.2 27.210 56.7 7.4 15.5 26.4 26.5 54.9 60.315 81.6 12.4 29.7 40.7 46.2 68.4 79.420 93.4 18.4 48.4 50.8 61.6 76.6 89.7ϱ = 0.98b OLS VA VA mild IV Cauchy IV(d*=0.5) IV trend IV comb0 9.7 5.4 5.6 4.8 4.5 5.0 4.85 14.9 5.8 8.1 10.7 10.2 19.7 22.110 45.4 8.1 18.7 22.0 26.2 35.3 46.615 72.7 13.1 34.4 36.2 45.8 48.2 68.120 89.2 21.4 54.6 48.6 64.5 57.6 83.1ϱ = 0.94b OLS VA VA mild IV Cauchy IV(d*=0.5) IV trend IV comb0 7.4 5.2 5.5 5.1 4.7 4.9 5.95 11.4 5.2 7.7 9.6 8.8 14.0 16.210 36.4 8.9 17.8 21.1 21.4 25.1 36.915 63.2 13.4 33.8 36.2 39.4 34.4 57.420 82.8 21.4 52.0 50.4 57.1 43.1 73.6

Generalizations◮ Consider the model with lagged dependent variable:y t = γy t−1 + βx t−1 + u tx t = ϱx t−1 + v t◮ Assume under the null hypothesis:ϱ = 1 − (c/T )γ = 1 − (c ∗ /T )⇒ also y t is near-integrated◮ Asymptotic null distribution of the OLS estimator:whereR(r) = B v c (r) − B u c ∗(r) ( ∫ 1t βd →∫ 10 R(r)dW u(r)√ ∫ 10 R(r)2 dr) −1 ∫ 10 Bu c ∗(r)Bv c (r)dr.0 Bu c ∗(r)2 dr◮ OLS test is biased even if u t and v t are uncorrelated.

Three alternatives hypotheses1. Double integrationIf γ and ϱ are close to unity we havey t( t∑≈ βs=2)s∑v r + u t .r=22. Cointegrated I(1) alternativesAssume that γ = ¯γ < 1 and ϱ T = 1 − c/T .x t and y t are cointegrated with (¯γ − 1)x t + β T y t ∼ I (0)3. Non-cointegrated I(1) alternativesy t = γy t−1 + β 1 x t−1 + β 2 x t−2 + u t (2)x t = ϱx t−1 + v t , (3)where β 2 = −ϱ T β 1 and γ = 1 − c ∗ /T .

Local Power1. If y t is double integrated the OLS t-statistic is asymptoticallydistributed as∫ √1∫dt β →0 U (r) dW u (r)√ ∫ + b σ 1vU1σ 2 (r) dr0 U2 (r) dr u 0( ∫ )where U (r) = Bc v 1−1 ∫(r) − Y (r)0 Y 1(r)2 dr0 Y (r) Bv c (r) dr2. If y t is cointegrated with x t :∫ 1dt β → ω0 B c (r) dW v (r)√ ∫+ √ √ ∫1 − ω 2 N (0, 1)+b σ 1vB c (r)1σ 2 dr.0 B2 c (r) dru 03. If y t and x t are not cointegrated the Wald statistic ofβ 1 = β 2 = 0 is distributed as[(W = W ζ (1) + b σ ) ∫ 12v0 R(r)dW v (r)+ ∫σ 1u0 R(r)2 dr] 2

Further generalizations1. x t is fractionally integrated with d ∈ (0.5, 1)2. Deterministic terms (constant, trend, dummy variables)3. More that 2 variables◮ The (modified) VA test remain valid in all cases◮ Only short-memory instrumentse.g. ∆x t−1 or (1 − L) 1−d∗ x t−1yield a valid IV test statistic

Conclusions◮ Two possible strategies:(a) Pretest (using unit root tests)(b) Robust inferenceHere we focus on (b)◮ Various possibilities to construct robust test statistics:(i) Variable addition test(ii) Tests based on instrumentsBoth tests perform similarly◮ There is some loss in power by using robust tests but the lossis moderate

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