Local Power of Fixed-T Panel Unit Root Tests with Serially Correlated Errors and Incidental Trends

Transcription

1 Local Power of Fixed-T Panel Unit Root Tests with Serially Correlated Errors and Incidental Trends Y. Karavias and E. Tzavalis University of Nottingham and Athens University of Economics and Business Tokyo, July 2014 Y. Karavias (AUEB) Local Power Tokyo, July / 22

2 Motivation Panel data unit root tests are attractive because they are more powerful However, they have their own complications (see e.g. Moon et al (2007)) y it = a i + β i t + ζ it, ζ it = ϕζ it 1 + u it where ϕ = 1 1 T N Y. Karavias (AUEB) Local Power Tokyo, July / 22

3 Research Question In this paper we examine the problem for fixed T tests Short term serial correlation is also a major factor Previous work in the area: Bond et al. (2005), Kruiniger (2008,2009), Madsen (2010), Westerlund(2014) Y. Karavias (AUEB) Local Power Tokyo, July / 22

4 Implementation Two tests are relative in this framework IV test (De Wachter et al. (2007) intercepts trends WG test (Kruiniger and Tzavalis (2002) intercepts trends Derive local power functions of the IV test for serial correlation and incidental trends Y. Karavias (AUEB) Local Power Tokyo, July / 22

5 Contributions 1 The IV test is dominates WG test in terms of power 2 The effects of serial correlation depend on the type of tests and on the deterministic specification 3 The presence of serial correlation does not necessarily mean loss of power 4 The incidental trends problem may not exist in the presence of serial correlation 5 The IV test does not suffer from the incidental trends problem Y. Karavias (AUEB) Local Power Tokyo, July / 22

6 Incidental Intercepts Consider the AR(1) model: y it = a i + ζ it ζ it = ϕζ it 1 + u it, i = 1,..., N and t = 1,..., T Y. Karavias (AUEB) Local Power Tokyo, July / 22

7 Incidental Intercepts Consider the AR(1) model: Stacked over i y it = a i + ζ it ζ it = ϕζ it 1 + u it, i = 1,..., N and t = 1,..., T y i = a i e + ζ i, ζ i = ϕζ i 1 + u i, y i = (y i1,..., y it ), e = (1,..., 1), ζ i = (ζ i1,..., ζ it ), u i = (u i1,..., u it ) ζ i 1 = (ζ i0,..., ζ it 1 ) and also y i 1 = (y i0,..., y it 1 ) Y. Karavias (AUEB) Local Power Tokyo, July / 22

8 Incidental Intercepts Consider the AR(1) model: Stacked over i y it = a i + ζ it ζ it = ϕζ it 1 + u it, i = 1,..., N and t = 1,..., T y i = a i e + ζ i, ζ i = ϕζ i 1 + u i, y i = (y i1,..., y it ), e = (1,..., 1), ζ i = (ζ i1,..., ζ it ), u i = (u i1,..., u it ) ζ i 1 = (ζ i0,..., ζ it 1 ) and also y i 1 = (y i0,..., y it 1 ) Hypothesis of interest ϕ N = 1 c N H 0 : c = 0 H 1 : c > 0 Y. Karavias (AUEB) Local Power Tokyo, July / 22

9 Individual Intercepts - Assumption {u it } have E (u it ) = 0, and are independent and homogeneous across i. {u it } are serially correlated across time but E (u i1 u it ) = 0. The u it, are independent of a i and y i0 and all variables have bounded 4 + ε moments. Y. Karavias (AUEB) Local Power Tokyo, July / 22

10 Individual Intercepts - The Estimators The WG estimator where Q = I T e(e e) 1 e. ˆϕ WG = N N y i 1 Qy i y i 1 Qy i 1 Y. Karavias (AUEB) Local Power Tokyo, July / 22

11 Individual Intercepts - The Estimators The WG estimator ˆϕ WG = N N y i 1 Qy i y i 1 Qy i 1 where Q = I T e(e e) 1 e. The IV estimator [ T p 1 ] E t=1 y it u i,t+p+1 (ϕ) = 0, or thus E (y i 1Π p u i ) = 0 ˆϕ IV = N N y i 1 Π py i y i 1 Π py i 1 Y. Karavias (AUEB) Local Power Tokyo, July / 22

12 Individual Intercepts - Tests Theorem 1: Under Assumption A and as N N ˆV 1 2 ˆδ( WG ˆϕ WG 1 ˆb ˆδ ) d N( ck WG, 1) where ˆb ˆδ = tr(ψ ˆΓ) 1 N N y i, 1 Qy i, 1 and A WG = 1 2 (Λ Q + QΛ Ψ Ψ ), V WG = 2tr((A WG Γ) 2 ), ˆΓ = 1 N N y i y i Y. Karavias (AUEB) Local Power Tokyo, July / 22

13 Individual Intercepts - Tests Theorem 1: Under Assumption A and as N where ˆb ˆδ = N ˆV 1 2 ˆδ( WG ˆϕ WG 1 ˆb ˆδ ) d N( ck WG, 1) tr(ψ ˆΓ) 1 N N y i, 1 Qy i, 1 and A WG = 1 2 (Λ Q + QΛ Ψ Ψ ) Theorem 2: Under Assumption A and as N N( ˆϕIV 1) ˆV 1 2 IV, V WG = 2tr((A WG Γ) 2 ), ˆΓ = 1 N d N( ck IV, 1) N y i y i where V IV = 2tr((A IV Γ) 2 ) tr(λ Π p ΛΓ) 2, A IV = 1 2 (Λ Π p + Π pλ) Y. Karavias (AUEB) Local Power Tokyo, July / 22

14 Individual Intercepts - General Local Power Functions where k WG = tr(λ QΛΓ) + tr(f QΓ) tr(ψλγ) tr(λ ΨΓ) VWG and k IV = 1 VIV Then the asymptotic local power function is Φ(z a + ck) Y. Karavias (AUEB) Local Power Tokyo, July / 22

15 Individual Intercepts - Examples k WG IV p=0 p=1 p=2 p=3 3(T 1) and for MA(1) with parameter θ 1 T 2 2T T (T 2 T ) 3(T 2 3T +2) T T 2 6T 24 T + 12 T (T 2 5T +6) T T 2 10T 80 T + 60 T (T 2 7T +12) T T 2 14T 196 T T T 2 T 2 2 3T T T 2 2 7T k WG = k IV = (T 2)(T θ 2 θ 2 + 3T θ 7θ + T 1) 2T R 1,WG θ 4 + R 2,WG θ 3 + R 3,WG θ 2 + R 2,WG θ + R 1,WG D 1,IV θ 2 + D 2,IV θ + D 1,IV R 1,IV θ 4 + R 2,IV θ 3 + R 3,IV θ 2 + R 2,IV θ + R 1,IV Y. Karavias (AUEB) Local Power Tokyo, July / 22

16 Individual Intercepts - Local Power Functions - IV Y. Karavias (AUEB) Local Power Tokyo, July / 22

17 Individual intercepts - Local Power Functions - WG Y. Karavias (AUEB) Local Power Tokyo, July / 22

18 Individual Intercepts - Local Power Functions - IV Y. Karavias (AUEB) Local Power Tokyo, July / 22

19 Individual Intercepts - Local Power Functions - WG Y. Karavias (AUEB) Local Power Tokyo, July / 22

20 Incidental Trends - Estimators The incidental trends model is y i = a i e + β i τ + ζ i, ζ i = ϕζ i 1 + u i Y. Karavias (AUEB) Local Power Tokyo, July / 22

21 Incidental Trends - Estimators The incidental trends model is The WG estimator ˆϕ WG = ( N y i = a i e + β i τ + ζ i, ζ i = ϕζ i 1 + u i y i 1Qy i 1 ) 1 ( N where Q = I T X (X X ) 1 X with X = [e, τ]. y i 1Qy i ) Y. Karavias (AUEB) Local Power Tokyo, July / 22

22 Incidental Trends - Estimators The incidental trends model is The WG estimator ˆϕ WG = ( N y i = a i e + β i τ + ζ i, ζ i = ϕζ i 1 + u i y i 1Qy i 1 ) 1 ( N where Q = I T X (X X ) 1 X with X = [e, τ]. Taking first differences E (y i 1Π pu i ) = 0 where ui leads to = u i = ( u i2,..., u it ) and y i ˆϕ IV = ( N y i 1Π py y i 1Qy i ) = y i = ( y i2,..., y it ) which i 1) 1 ( N yi 1Π py i ) Y. Karavias (AUEB) Local Power Tokyo, July / 22

23 Incidental Trends - Tests Theorem 3: Under Assumption A and as N where ˆb ˆδ = and ˆΘ = 1 N N N ˆV 1 2 WG ˆδ( ˆϕ WG 1 ˆb ˆδ ) N( ck WG, 1) tr(φ ˆΓ) 1 N N y i, 1 Qy and V WG = vec(λ Q Φ ) Θvec(Λ Q Φ ) i, 1 ( vec( yi y i )vec( y i y i ) ) Y. Karavias (AUEB) Local Power Tokyo, July / 22

24 Incidental Trends - Tests Theorem 3: Under Assumption A and as N where ˆb ˆδ = and ˆΘ = 1 N N N ˆV 1 2 WG ˆδ( ˆϕ WG 1 ˆb ˆδ ) N( ck WG, 1) tr(φ ˆΓ) 1 N N y i, 1 Qy and V WG = vec(λ Q Φ ) Θvec(Λ Q Φ ) i, 1 ( vec( yi y i )vec( y i y i ) ) Theorem 4: Under Assumption A and as N N( ˆϕIV 1) ˆV 1 2 IV N( ck IV, 1) where V IV = 2tr ((A FDIV Θ) 2 ) tr (Λ Π p Λ Θ) 2, A IV = 1 2 (Λ Π p + Π p Λ ) and Θ = 2Γ 1 Γ 2 Γ 2 where Γ 1 = E (u i u i ) and Γ 2 = E (u i u i 1 ). Y. Karavias (AUEB) Local Power Tokyo, July / 22

25 Individual Trends - Local Power Functions where for the WG test k WG = tr(λ QΓ) + tr(f QΓ) tr(φλγ) tr(λ ΦΓ) 2tr((A WGT Γ) 2 ) and for the IV test k IV = 1 VIV Then Table: Values of slope parameter p. T=7 θ k IV k WG T=10 θ k IV k WG Y. Karavias (AUEB) Local Power Tokyo, July / 22

26 Large T If we assume T to be asymptotic the hypothesis of interest becomes ϕ NT = 1 c T N Theorem 5: If T, N jointly and the following condition holds: N/T 0. T N( 2) 1 ( ˆϕ IV 1) N( c 1, 1), 2 d and T N( 3) 1 ˆδ WG ( ˆϕ WG 1 ˆb ) ˆδ d N( c0, 1), Y. Karavias (AUEB) Local Power Tokyo, July / 22

27 Large T If we assume T to be asymptotic the hypothesis of interest becomes ϕ NT = 1 c T N Theorem 5: If T, N jointly and the following condition holds: N/T 0. T N( 2) 1 ( ˆϕ IV 1) N( c 1, 1), 2 d and T N( 3) 1 ˆδ WG ( ˆϕ WG 1 ˆb ) ˆδ d N( c0, 1), Table 2: Slopes of large-t tests. IV MPP LLC/HT SGLS IPS WG 1/ 2 1/ 2 (3/2) (5/51) 1/ Y. Karavias (AUEB) Local Power Tokyo, July / 22

28 Large T If we assume T to be asymptotic the hypothesis of interest becomes ϕ NT = 1 c T N Theorem 5: If T, N jointly and the following condition holds: N/T 0. T N( 2) 1 ( ˆϕ IV 1) N( c 1, 1), 2 d and T N( 3) 1 ˆδ WG ( ˆϕ WG 1 ˆb ) ˆδ d N( c0, 1), Table 2: Slopes of large-t tests. IV MPP LLC/HT SGLS IPS WG 1/ 2 1/ 2 (3/2) (5/51) 1/ In the presence of incidental trends k IV = 0 k WG = 0 Y. Karavias (AUEB) Local Power Tokyo, July / 22

29 Heterogeneous Alternatives For alternatives of the form ϕ Ni = 1 c i N the hypothesis of interest is H 0 : c i = 0, for all i H 1 : c i > 0, for some i where c i are i.i.d. with support in a subset of a bounded interval [0, M c ], for some M c 0. Then, only the mean of c i affects power, i.e. N( ˆϕIV 1) ˆV 1 2 IV N( E (c i )k IV, 1) Y. Karavias (AUEB) Local Power Tokyo, July / 22

30 Monte Carlo Simulations Intercepts N Theory θ = 0.5 IV WG θ = 0 IV WG θ = 0.5 IV WG Trends N Theory θ = 0.5 IV WG θ = 0 IV WG θ = 0.5 IV WG Y. Karavias (AUEB) Local Power Tokyo, July / 22

31 Conclusions The IV test is always better than the WG test and better than the HT test The effect of serial correlation is case specific Local power is possible in a fixed T therefore theoretically solving the incidental trends problem Power comes either from the presence of serial correlation or the use of double differences Non-trivial power vanishes when T is asymptotic Y. Karavias (AUEB) Local Power Tokyo, July / 22