Standard Testing Procedures for White Noise and Heteroskedasticity

Transcription

1 Standard Testing Procedures for White Noise and Heteroskedasticity Violetta Dalla 1, Liudas Giraitis 2 and Peter C.B. Phillips 3 1 University of Athens, 2 Queen Mary, UL, 3 Yale University A Celebration of Peter Phillips Forty Years at Yale University of Yale 2th October 218

2 Introduction Major tools in data analysis: Testing for correlation (x t ) Testing for bivariate cross correlation (x t ), (y t ) Testing for ARCH effect: (x t ) White noise, (x 2 t ) correlated Test for i.i.d. Known phenomenon-danger: standard testing procedures produce spurious correlations Yule, Bartlett, Box-Pierce, Ljung-Box procedures often require assumptions of independence. Used in: STATA, EVIEWs

3 Large literature on testing: GU Yule (1924, J. Royal Stat. Soc.): Why do we sometimes get nonsense correlations between time series? Peter CB Phillips (1984, J. of Econometrics): Understanding spurious regressions in econonometrics Peter CB Phillips (1998, Econometrica): New tools for understanding spurious regressions PA Ernst, LA Shepp, AJ Wyner (217, Ann. Statistics): Yule s nonsense correlation solved! Our objective: Simple standard testing procedures that allow to be agnostic about dependence heteroscedasticity structure of (x t ), (y t ) Ready to use R, STATA, EVIEWs applications/packages

4 2/1/28 2/1/29 2/1/21 2/1/211 2/1/212 2/1/213 2/1/214 2/1/215 2/1/216 Daily S&P5 returns, Data in % 15 S&P

5 Daily S&P5 returns x t, Correlogram ˆρ k with standard confidence band (CB) for insignificance.2 ρ Standard CB(95%) Standard CB(99%)

6 6/1/28 6/1/29 6/1/21 6/1/211 6/1/212 6/1/213 6/1/214 6/1/215 6/1/216 Weekly S&P5 and gold returns, Data in % 1 S&P5 Gold

7 Weekly S&P5 and gold returns, Cross-correlogram ˆρ xy,k, x=s&p5, y=gold, lags k =, 1,..., 1.2 ρ (x,y(-k)) Standard CB(95%) Standard CB(99%)

8 Daily S&P5 returns (squared), Correlogram ˆρ k for x 2 t = Ex 2 t + (x 2 t Ex 2 t ).5 ρ Standard CB(95%) Standard CB(99%)

9 Univariate tests for zero autocorrelation (x t ) Tests for H : ρ k = (at lag k) ˆt k = n ˆρ k standard where t k = ρ k adjusted by Taylor (1984) ρ k = n t=k+1 x t x t k n t=1 x, ρ t 2 k = n t=k+1 x t x t k ( n t=k+1 ( x t x t k ) 2 ) 1/2 with x t = x t x.

10 Correct size/ size distortions Let (x t ) be uncorrelated variables: Standard test ˆt k N(, 1): size correct if x t iid, ˆt k size distorted if (a) x t GARCH process (b) x t m.d.; e.g. x t = ε t ε t 1, ε t iid (c) under heteroscedasticity: x t = h t ε t, (d) under changing mean Ex t Modified test t k N(, 1): size correct if x t = h t e t heteroscedastic martingale differences; (e t ) -stationary ergodic martingale differences, h t C, n t=1 (h t h t k ) 4 = o( n t=1 h4 t ) for any k 1.

11 Cumulative test H : ρ 1 =... = ρ m = Ljung-Box LB m = n(n + 2) m k=1 ˆρ 2 k n k Modified test Q m = t (ˆR ) 1 t, t = ( t 1,..., t m ), ˆR = (r jk) R = E[ t t ], r jk = n jki ( n jk /n jk > 1.96) n 1/2 jj n 1/2 kk Thresholding: Cov( t j, t k ) =, then n jk /n jk N(, 1). n jk = n t=k+1 x 2 t x t j x t k, n jk = ( n t=k+1 x 4 t x 2 t j x 2 t k )1/2 (Novelty: c.f. Guo, Phillips (21); Lobato, Nankervis, Savin (22) ) Size correct: LB m χ 2 m if x t iid; Qm χ 2 m if x t heteroscedastic m.d.

12 Monte-Carlo simulations Model: x t iid N(,1), n = 3, 5 replications Rejection frequency in (%) of standard ˆt k = nˆρ k and corrected t k statistics ; standard LB m and corrected Q m cumulative statistics 6 t t 7 LB Q*

13 Monte-Carlo simulations Model: x t GARCH(1,1) α =.2, γ =.7, n = 3, 5 replications Rejection frequency of standard ˆt k = nˆρ k and corrected t k statistics ; standard LB m and corrected Q m cumulative statistics 14 t t 25 LB Q*

14 Monte-Carlo simulations Model: x t = (1 + I (t/n >.5))ε t, ε t iid N(,1), n = 3, 5 replications Rejection frequency of standard ˆt k = nˆρ k and corrected t k statistics ; standard LB m and corrected Q m cumulative statistics 1 t t 35 LB Q*

15 Monte-Carlo simulations Model: x t = ε t ε t 1, ε t iid N(,1), n = 3, 5 replications Rejection frequency of standard ˆt k = nˆρ k and corrected t k statistics ; standard LB m and corrected Q m cumulative statistics 25 t t 25 LB Q*

16 Monte-Carlo simulations One replication from model: x t iid N(,1) Correlogram ˆρ k with standard and corrected confidence bands (CB) for insignificance; standard ˆt k = nˆρ k and corrected t k statistics ; standard LB m and corrected Q m cumulative statistics.2 ρ Standard CB(95%) Corrected CB(95%) 3 Standard t Corrected t Standard LB Corrected Q*

17 Monte-Carlo simulations One replication from model: x t = I (t/n >.5) + ε t, ε t iid N(,1) Correlogram ˆρ k with standard and corrected confidence bands (CB) for insignificance; standard ˆt k = nˆρ k and corrected t k statistics ; standard LB m and corrected Q m cumulative statistics ρ Standard CB(95%) Corrected CB(95%) Standard t Corrected t Standard LB Corrected Q*

18 Daily S&P5 returns, Correlogram ˆρ k with standard and corrected confidence bands (CB) for insignificance at lags k = 1,..., ρ Standard CB(95%) Standard CB(99%) Corrected CB(95%) Corrected CB(99%) Standard LB Corrected Q* cv(5%) cv(1%)

19 Bivariate test for zero cross autocorrelation (x t ), (y t ) Tests for H : ρ xy,k = (at lag k) ˆt xy,k = n ˆρ xy,k standard t xy,k = ρ xy,k adjusted where ρ xy,k = n t=k+1 x tỹ t k ( n t=1 x t 2 ) 1/2 ( n t=1 ỹ t 2 ), ρ 1/2 xy,k = n t=k+1 x tỹ t k ( n t=k+1 x t 2 ỹ t k ỹ t j ) 1/2 with x t = x t x, ỹ t = y t y.

20 Correct size/ size distortions Let (x t ), (x t ) have zero mean /be such that x t y t k = h t e t uncorrelated heteroscedastic m.d. (as in univariate case). Then Modified test t xy,k N(, 1) has correct size. Standard test ˆt xy,k N(, 1) requires (x t ) to be independent of (y t ). Size of ˆt xy,k distorted: if (x t ) and (y t ) are dependent variables (x t ) independent of (y t )

21 Cumulative tests for zero cross-autocorrelation Bivariate case Tests for H : ρ xy,1 = ρ xy,2 =... = ρ xy,m = Standard LB xy,m = n 2 m k=1 ˆρ 2 xy,k n k Modified Qxy,m = t xy(r xy) 1 t xy, t xy = ( t xy,1,..., t xy,m ). Under H, and some additional restrictions: LB xy,m χ 2 m, Q xy,m χ 2 m.

22 Monte-Carlo simulations Model: x t = (1 + I (t/n >.5))ε t, y t = (1 + I (t/n >.5))η t, ε t, η t iid N(,1), x and y independent, n = 3, 5 replications Rejection frequency of standard ˆt xy,k = nˆρ xy,k and corrected t xy,k test; standard LB xy,m and corrected Q xy,m cumulative test 12 t (x,y(-k)) t (x,y(-k)) 35 LB(x,y(-k)) Q*(x,y(-k))

23 Monte-Carlo simulations Model: x t iid N(,1), y t = x t x t 1, n = 3, 5 replications Rejection frequency of standard ˆt xy,k = nˆρ xy,k and corrected t xy,k statistics ; standard LB xy,m and corrected Q xy,m cumulative statistics t (x,y(-k)) t (x,y(-k)) LB(x,y(-k)) Q*(x,y(-k))

24 Monte-Carlo simulations One replication from model: x t iid N(,1), y t = x t x t 1 Cross-correlogram ˆρ xy,k with standard and corrected confidence bands (CB) for insignificance; standard ˆt xy,k = nˆρ xy,k and corrected t xy,k statistics ; standard LB xy,m and corrected Q xy,m cumulative statistics.2 ρ (x,y(-k)) Standard CB(95%) Corrected CB(95%) 3 2 Standard t (x,y(-k)) Corrected t (x,y(-k)) Standard LB(x,y(-k)) Corrected Q*(x,y(-k))

25 Monte-Carlo simulations One replication from model: x t AR(1) φ x =.7, y t AR(1) φ y =.7, x and y independent Cross-correlogram ˆρ xy,k with standard and corrected confidence bands (CB) for insignificance; standard ˆt xy,k = nˆρ xy,k and corrected t xy,k statistics ; standard LB xy,m and corrected Q xy,m cumulative statistics.2 ρ (x,y(-k)) Standard CB(95%) Corrected CB(95%) 4 3 Standard t (x,y(-k)) Corrected t (x,y(-k)) Standard LB(x,y(-k)) Corrected Q*(x,y(-k))

26 Weekly S&P5 and gold returns, Cross-correlogram ˆρ xy,k with standard and corrected confidence bands (CB) for insignificance at lags k =, 1,..., 1, x=s&p5, y=gold ρ (x,y(-k)) Standard CB(95%) Standard CB(99%) Corrected CB(95%) Corrected CB(99%) Standard LB(x,y(-k)) Corrected Q*(x,y(-k)) cv(5%) cv(1%)

27 Testing hypothesis (x t ) iid. Test for i.i.d. at lag k. Notation ρ x,k = Corr(x t, x t k ) 1) H : ρ x,k = and ρ x,k = : based on J k (x, x ) = n2 2 (ˆρ n k x,k + ˆρ x,k) 2 2) H : ρ x,k = and ρ x 2,k = : based on J k (x, x 2 ) = n2 2 (ˆρ n k x,k + ˆρ 2 x,k) 2 Under iid H : J k (x, x ) χ 2 1, J k (x, x 2 ) χ 2 1.

28 Cumulative tests for iid 1) H holds at lags 1,..., m : based on C m (x, x ) = m J k (x, x ) k=1 2) H holds at lags 1,..., m : based on C m (x, x 2 ) = m J k (x, x 2 ) k=1 Under the null: C m (x, x ) χ 2 2m, C m (x, x 2 ) χ 2 2m.

29 Monte-Carlo simulations Model: x t iid N(,1), n = 3, 5 replications Rejection frequency in (%) of J k (x, x ) and J k (x, x 2 ) statistics ; C m(x, x ) and C m(x, x 2 ) cumulative statistics 6 J(x, x ) J(x,x²) 7 C(x, x ) C(x,x²)

30 Monte-Carlo simulations Model: x t GARCH(1,1) α =.2, γ =.7, n = 3, 5 replications Rejection frequency in (%) of J k (x, x ) and J k (x, x 2 ) statistics ; C m(x, x ) and C m(x, x 2 ) cumulative statistics J(x, x ) J(x,x²) C(x, x ) C(x,x²)

31 Monte-Carlo simulations Model: x t = (1+I (t/n >.5))ε t, ε t iid N(,1), n = 3, 5 replications Rejection frequency in (%) of J k (x, x ) and J k (x, x 2 ) statistics ; C m(x, x ) and C m(x, x 2 ) cumulative statistics 7 J(x, x ) J(x,x²) 1 C(x, x ) C(x,x²)

32 Monte-Carlo simulations One replication from model: x t GARCH(1,1) α =.2, γ =.7 J k (x, x ) and J k (x, x 2 ) statistics ; C m(x, x ) and C m(x, x 2 ) cumulative statistics 14 J(x, x ) J(x,x²) 1 C(x, x ) C(x,x²)

33 Monte-Carlo simulations One replication from model: x t = (1 + I (t/n >.5))ε t, ε t iid N(,1) J k (x, x ) and J k (x, x 2 ) statistics ; C m(x, x ) and C m(x, x 2 ) cumulative statistics J(x, x ) J(x,x²) C(x, x ) C(x,x²)

34 Daily S&P5 returns, J k (x, x ) and J k (x, x 2 ) statistics at lags k = 1,..., 1; C m (x, x ) and C m (x, x 2 ) cumulative statistics at lags k = 1,..., 3 4 J(x, x ) J(x,x²) cv(5%) cv(1%) 7 C(x, x ) C(x,x²) cv(5%) cv(1%)

35 Code Daily S&P5 returns, : Correlogram ˆρ k with standard (S-CB) and corrected (C-CB) confidence bands and corrected Q m cumulative statistic at lags k, m = 1,..., 1 EViews add-in Series: SP5 Transformation: Levels Sample: 2/1/28-3/12/ AC 95% S-CB 95% C-CB Q* 5% cv

36 Code Daily S&P5 returns, : Correlogram ˆρ k with standard (S-CB) and corrected (C-CB) confidence bands and corrected Q m cumulative statistic at lags k, m = 1,..., 1 R library AC 95% S CB 95% C CB Q* 5% cv