10-12 M AY 2006 A Random Walk through Seasonal Adjustment: - PDF document

C ONFERENCE ON S EASONALITY , S EASONAL A DJUSTMENT AND THEIR IMPLICATIONS FOR S HORT -T ERM A NALYSIS AND F ORECASTING 10-12 M AY 2006 A Random Walk through Seasonal Adjustment: Noninvertible Moving Averages and Unit Root Tests Tomas del Barrio Castro Denise Osborn

A Random Walk through Seasonal Adjustment: Noninvertible Moving Averages and Unit Root Tests by Tomas del Barrio Castro University of Barcelona And Denise R Osborn University of Manchester April 2006 Key words: Unit root tests, seasonality, seasonal adjustment, X- 11. JEL codes: C22, C12, C82 1

Wallis (1974) and Sims (1974). del Barrio Castro and Osborn (2004), Ericsson, Hendry and Tran (1994), Franses (1995, 1996), Ghysels (1990), Ghysels and Perron (1993, 1996), Ghysels and Liebermann (1996), Maravall (1993), Matas Mir and Osborn (2004) and Otero and Smith (2002). Seasonal adjustment found to have no asymptotic impact on tests under the null hypothesis of (zero frequency) integration and cointegration; see, in particular, Ghysels and Perron (1993) and Ericsson et al. (1994). Galbraith and Zinde-Walsh (1999) and Gonzalo and Pitarakis (1998) 1.- Introduction 2.- Seasonal adjustment and moving average components 3.- Noninvertible moving averages 4.- Seasonally adjusted random walk 2

2. Seasonal adjustment and moving average components Unit root test are based on: y t = ρ y t -1 + u t (1) H 0 : ρ = 1 or, α = ρ - 1 = 0. u t in (1) may exhibit temporal dependence and/or heteroskedasticity. Limiting distribution of UR tests under H 0 : Phillips (1987, Theorem 3.1). Typical assumption in UR analyses being that the process for u t is stationary and invertible. (for example, Ghysels and Perron, 1993, Elliot, Rothenberg and Stock, 1996, Galbraith and Zinde-Walsh, 1999). However, the invertibility of u t may be questioned when the series under analysis has been seasonally adjusted by conventional procedures.i.e. X-11 or X-12 ARIMA programs. Seasonal adjustment by X-11 can be represented as the application of a sequence of linear filters, see Laroque (1977) quarterly case and Ghysels and Perron (1993) monthly case. As shown by Burridge and Wallis (1984), X-11 assumes that the unadjusted series is generated by: d u − = ( 1 L ) S ( L ) y w (2) t t where d = 1 or 2 (with the best-fitting model implying d = 2) and w t is a moving average (MA) process. 3

Empirical studies of the properties of seasonal time series find, in general, little evidence for the presence of the full set of seasonal unit roots implied S ( L ) in (2); see, among others Beaulieu and Miron (1993), Osborn (1990), or the discussion in Ghysels and Osborn (2001) Application of seasonal adjustment based on an assumption of a DGP of the form (2) when the true DGP has no seasonal unit roots will induce the full set of (seasonal) unit roots implied by S ( L ) into the MA component. Consequently, the stylized fact that macroeconomic time series are not seasonally integrated implies that the disturbance u t in the unit root test regression of (1), when y t is seasonally adjusted, may be anticipated to be a noninvertible moving average process. In Schwert (1989) it was shown that UR tests are poorly sized in the presence of moving average components in (1). Galbraith and Zinde-Walsh (1999) and Gonzalo and Pitarakis (1998) analytically show why such distortions occur. However, Galbraith and Zinde-Walsh (1999) and Gonzalo and Pitarakis (1998) focus on invertible MA components in (1). Nevertheless, Galbraith and Zinde-Walsh hint at the importance of this assumption, 4

3. Noninvertible moving averages y t = y t -1 + u t t = 1, 2, …, T (3) where u t = ε t + ε t -1 and ε t ~ iid (0, σ 2 ). y 0 = ε 0 = ε -1 = 0. The moving average unit root of -1 in (3) implies a zero in the spectral density of ∆ y t at a frequency of π . 3.1 No correction for autocorrelation ∆ y t = α y t -1 + u t , t = 1, 2, …, T (4) α = 0 and u t = ε t + ε t -1 . Proposition 1. Let y t follow (3) with u t = ε t + ε t -1 and ε t ~ iid (0, σ 2 ). The asymptotic distribution of the normalized bias test statistic in (4) is then given by: 2 − + 1 [ W ( 1 ) 1 ] 0 . 5 α ⇒ T . (5) ˆ 2 2 ∫ [ W ( r )] dr and that for the t-ratio test statistic is: 2 − + 1 [ W ( 1 ) 1 ] 0 . 5 ⇒ t α . (6) { } ˆ 1 / 2 2 2 ∫ [ W ( r )] dr 2 2 − − 1 [ W ( 1 ) 1 ] 1 [ W ( 1 ) 1 ] α ⇒ ⇒ T t α (7) ˆ { } ˆ 2 1 / 2 2 ∫ 2 [ W ( r )] dr 2 ∫ [ W ( r )] dr 5

3.2 Autoregressive augmentation Now consider the usual ADF regression p ∑ ∆ = α + φ ∆ + y y y v (8) − − t t 1 i t i t = i 1 where the DGP is again given by (3). under the H 0 : α = 0 the autoregressive augmentation of (8) results in an MA( p +1) disturbance process, with coefficients p ∑ p p ∆ = φ ∆ + y y e − t i t i t = i 1 p p ∑ ∑ p p p = ∆ − φ ∆ = ε + ε − φ ε + ε e y y ( ) ( ) − − − − − t t i t i t t 1 i t i t i 1 = = i 1 i 1 p p p p p p = ε + − φ ε − φ + φ ε − − φ + φ ε − φ ε ( 1 ) ( ) ... ( ) . − − − − − − t t 1 t 2 p t p p t p 1 1 1 2 p 1 + p 1 ∑ p = ε + θ ε . − t i t i = i 1 ⎛ ⎞ 1 + p i 1 θ = − ⎜ ⎟ = + (9) ( 1 ) , i 1 , ...., p 1 ⎜ ⎟ i + p 1 ⎝ ⎠ Notice that, the coefficients do not decline towards zero as i increases. The AR approximation does not account for the noninvertible MA seasonal unit root unit root -1. 6

Proposition 2. Let y t follow (3) with u t = ε t + ε t -1 and ε t ~ iid (0, σ 2 ). The normalized bias and t-ratio test statistic test statistics in regression (8) then satisfy: [ ] ⎧ 1 2 − − W ( 1 ) 1 ⎪ + 1 p 1 ⎪ p odd 2 ⎪ 4 ∫ W ( r ) dr ⎪ ⎪ α ⇒ T (10) ˆ ⎨ ⎪ [ ] ⎛ + ⎞ p 2 1 ⎪ 2 ⎜ ⎟ − + W ( 1 ) 1 + + ⎪ 1 p 1 p 1 ⎝ ⎠ p even ⎪ 2 4 ∫ ⎪ W ( r ) dr ⎩ and [ ] ⎧ 1 2 − − W ( 1 ) 1 ⎪ + 1 p 1 ⎪ p odd [ ] 1 / 2 ⎪ 2 ⎡ + ⎤ 1 / 2 p 2 2 ∫ ⎪ W ( r ) dr ⎢ ⎥ + p 1 ⎣ ⎦ ⎪ ⎪ ⇒ t (11) ⎨ α ˆ ⎪ [ ] ⎛ + ⎞ p 2 1 2 ⎪ ⎜ ⎟ − + W ( 1 ) 1 ⎪ + + 1 p 1 p 1 ⎝ ⎠ p even ⎪ [ ] 1 / 2 2 ⎡ ⎤ + 1 / 2 p 2 ⎪ 2 ∫ W ( r ) dr ⎢ ⎥ ⎪ + p 1 ⎣ ⎦ ⎩ 7

Table 1. The Null Distribution of the DF t -Statistic in the Presence of a Noninvertible MA. Quantile 0.010 0.025 0.050 0.100 0.500 0.900 0.950 0.975 0.990 Size DF dist. -2.565 -2.248 -1.958 -1.626 -0.510 0.860 1.260 1.635 2.030 0.051 Noninvertible MA Process with Augmentation P = 0 -1.770 -1.518 -1.300 -1.055 -0.059 1.646 2.151 2.607 3.119 0.005 P = 1 -3.210 -2.762 -2.409 -2.037 -0.765 0.495 0.832 1.128 1.444 0.115 p = 2 -2.223 -1.902 -1.647 -1.359 -0.313 1.177 1.612 1.994 2.411 0.021 p = 3 -2.905 -2.500 -2.186 -1.828 -0.640 0.661 1.026 1.335 1.665 0.081 p = 4 -2.348 -2.009 -1.743 -1.443 -0.380 1.064 1.483 1.837 2.245 0.029 p = 8 -2.440 -2.098 -1.819 -1.508 -0.428 0.979 1.391 1.737 2.122 0.037 p = 12 -2.486 -2.133 -1.849 -1.535 -0.447 0.944 1.358 1.691 2.077 0.039 p = 16 -2.474 -2.143 -1.878 -1.563 -0.456 0.944 1.324 1.662 2.090 0.042 p = 20 -2.487 -2.163 -1.890 -1.568 -0.472 0.932 1.316 1.663 2.060 0.044 p = 24 -2.470 -2.177 -1.888 -1.575 -0.472 0.927 1.299 1.637 2.048 0.043 Notes: The quantiles of the empirical distribution of the ADF test t -ratio test are based 15,000 replications and a sample size of 4,000 observations. The DF distribution is obtained from a random walk where the innovation is the white noise process ε t ~ N (0, 1). The noninvertible MA is an I (1) process where the innovation is given by u t = ε t + ε t -1 , ε t ~ N (0, 1), and the ADF regression is estimated with no augmentation and augmentation of orders 1, 2, 3, 4, 8, 12, 16, 20, 24. 8

3.3 PP approach ( ) 2 2 − 1 s s ( ) l u α = α − Z T . (12) ˆ ˆ T 2 − ∑ 2 2 T y − t 1 = t 1 and ( ) 2 2 ⎛ ⎞ − s s s 1 ( ) ⎜ l ⎟ l u = − Z t t (13) ⎜ ⎟ α α ˆ ˆ s 2 T ⎝ ⎠ u − ∑ 2 2 s T y − l t 1 = t 1 where T − ∑ 2 1 2 = s T u ˆ u t = t 1 (14) p T T ) ∑ ( − ∑ − ∑ 2 1 2 1 = + s T u 2 T w i , p u u ˆ ˆ ˆ − l t t t i = = = + t 1 i 1 t i 1 9

Proposition 3. Let y t follow (3) with u t = ε t + ε t -1 and ε t ~ iid (0, σ 2 ). Then the asymptotic distributions of the PP unit root test statistics of (12) and (13) are given by: ( ( ) ) 2 − − 1 [ W ( 1 ) 1 ] 0 . 25 1 w 1 , p ( ) α ⇒ + Z (15) ˆ [ ] ( ) ∫ 2 2 2 ∫ [ W ( r )] dr 4 w r dr 2 − 1 [ W ( 1 ) 1 ] ( ) ⇒ + Z t [ ] α ( ( ) ) ˆ 1 / 2 1 / 2 + 2 2 w 1 , p ∫ 2 [ W ( r )] dr . (16) ( ( ) ) − 0 . 5 1 w 1 , p + ) [ ] 1 / 2 ( ( ) 1 / 2 2 ∫ + 2 2 w 1 , p [ W ( r )] dr With the Bartlett window ( ) ( ) ] = − + w i , p 1 [ i / p 1 . ⎧ ⎫ 1 / 2 2 ⎡ + ⎤ − + + ⎪ ⎪ 1 p 1 [ W ( 1 ) 1 ] 0 . 5 [ p /( p 1 )] ( ) ⇒ Z t . (17) ⎨ ⎬ ⎢ ⎥ [ ] α ˆ + 1 / 2 2 p 0 . 5 ⎣ ⎦ 2 ⎪ ∫ ⎪ [ W ( r )] dr ⎩ ⎭ 10