A noe on spurious regressions beween saionary series Auhor Su, Jen-Je Published 008 Journal Tile Applied Economics Leers DOI hps://doi.org/10.1080/13504850601018106 Copyrigh Saemen 008 Rouledge. This is an elecronic version of an aricle published in Applied Economics Leers [Vol. 15(15), pp. 15-1330]. Applied Economics Leers is available online a: hp:// www.informaworld.com Downloaded from hp://hdl.handle.ne/1007/653 Griffih Research Online hps://research-reposiory.griffih.edu.au
A Noe on Spurious Regressions beween Saionary Series Jen-Je Su Deparmen of Applied and Inernaional Economics Massey Universiy This version: February 8, 006 Running Tile: Spurious Regression, Convergen -Saisic ABSTRACT This paper examines if he convergen -es suggesed by Sun (004) is able o solve spurious regressions wih saionary series. In brief, we find ha he convergen -es does provide beer conrol over size compared o he usual -es and is Newey-Wes modificaion and, in mos cases implemening a pre-whiing procedure size is furher conrolled. Jen-Je Su Applied and Inernaional Economics Deparmen Massey Universiy Privae Bag 11 Palmerson Norh, New Zealand Email: j.j.su@massey.ac.nz Tel: 64-6-3505799 ex 666 Fax: 64-6-3505660
I Inroducion I is a well-known fac ha he use of nonsaionary daa can lead o spurious regressions. Granger and Newbold (1974) firs found via Mone Carlo simulaions ha, given wo independen random walks, i is likely ha a regression of one on he oher will produce a significan slope coefficien according o he usual -es. Since he wo walks are acually unrelaed he saisical significance is spurious and misleading. Laer, Phillips (1986) developed an asympoic heory for regression beween wo unrelaed I(1) processes, showing ha he usual -saisic does no have a limiing disribuion bu diverges as he sample size goes o infiniy. Recenly, Sun (004) found ha he seemingly ineviable divergen behaviour of he usual -es is due o he use of a sandard error ha underesimaes he rue variaion of he slope esimaor. He showed ha once an appropriae sandard error esimae, such as he one suggesed in Kiefer and Vogelsang (00; hereafer, KV), is used, he resuling -saisic is no longer divergen. There is also ineres in sudying spurious regressions in a saionary environmen. As a maer of fac, Granger, Hyung and Jeon (001; hereafer, GHJ) have found ha a spurious regression can even occur in an occasion ha he usual -saisic is acually convergen. Using exensive finie-sample simulaions, GHJ gave evidence ha he null hypohesis of zero-slope in a regression wih wo 1
independen saionary auoregressive series or medium-o-long moving averages is severely over-rejeced. In his paper, we are ineresed in he performance of he convergen - saisic of KV (00) and Sun (004) in he siuaions considered in GHJ (001) hrough Mone-Carlo simulaions. In brief, we find ha he convergen -es provides much beer conrol over size han he usual -es (and is Newey-Wes modificaion). In mos cases, alhough he convergen -es does no produce correc size, he siuaion is no oo bad paricularly, when he sample size is large. However, here are occasions ha he likelihood of over-rejecion is sill high. We also find ha implemening an AR(1) pre-whiening procedure may produce furher conrol over size, bu wih some excepions. The paper is organised as follows. Secion provides a shor review regarding HAC robus ess and spurious regressions. Secion 3 repors and discusses our simulaion resuls. Finally, Secion 4 concludes. II HAC Robus Tes Saisics Consider he regression Y X u, =1,,, T. (1) Le ˆ be he OLS esimae for and be he corresponding -raio. I is well known ha when he error erm in (1) is heeroskedasic or auocorrelaed is no asympoically valid. Alernaively, a kernel-based HAC robus -saisic can be defined as follows:
where ˆ ˆ ˆ T ˆ T ˆ / M X X 1 ˆ M ˆ T 1 Tˆ X X 1 T 1 ˆ (/ ) ˆ ( ) M j M j, T j( T1), 1/, () j1 T ˆ( j) X X uu ˆˆ for j0, ˆ( j) ˆ( j) for j0. j Here, M is he bandwidh parameer and () x is a kernel funcion. Following Newey and Wes (1987), we consider he Barle kernel, () x 1 x 1 if x 1 and 0 oherwise, hroughou paper. To ensure ha ˆ ˆ is a consisen esimae for he variance of ˆ, he convenional approach requires ha MT, and M ot ( ). See Newey and Wes (1987) for an early developmen and den Haan and Levin (1997) for a recen review. Alernaively, Kiefer and Vogelsang (00) sugges using M T (he full bandwidh) in he calculaion of ˆ (denoed by ˆ ) 1. This rae for M M M T clearly violaes he M ot ( ) rule and ˆ is no consisen. However, as shown M T in Kiefer and Vogelsang (00), he es saisic using ˆ is asympoically M T pivoal and exhibis a beer finie-sample size conrol han he convenional 1 Recenly, Kiefer and Vogelsang (005) sugges using M=bT, where b(0,1] is a fixed number. An obvious rade-off has been found: a small bandwidh (small b) leads o ess wih higher power bu greaer size disorions and a large bandwidh (large b) leads o ess wih lower power bu less size disorions. Since conrolling for size disorions is our main concern, we sick o he choice of M=T (i.e. b=1). 3
approach. In his paper, we will refer he HAC robus -es assuming MT, and M ot ( ) as and he es using M T as. NW KV III Mone Carlo Simulaions In his secion we repor he resuls of Mone Carlo experimens designed o invesigae finie-sample size of he hree previously discussed ess namely,, and NW KV. The bandwidh parameer (M) in is se equal o he ineger par NW of 1/ 4 T /100 4, as in GHJ (001). Also, an AR(1) pre-whiening procedure of Andrew and Monahan (199) is implemened for and NW KV denoed by, NW PW NW, PW respecively. All simulaions are preformed in GAUSS and he resuls (he rejecion raes) are calculaed by comparing criical values a 5% (ha is, 1.96 for he usual -es and he NW es and 3.76 for he KV es) via 5,000 ieraions. In he following, simulaion resuls of wo cases ha spurious regressions beween wo saionary series found by GHJ (001). CASE I: Spurious regression beween AR processes The firs experimen is done by considering independen AR(1) processes: Y and X in (1) defined by and X X, x 1 x, Y Y y 1 y, 4
where, are each i.i.d. N(0,1), independen o one anoher. Simulaions y, x, resuls a sample size T=100, 50, 500, 1000, and 000 are repored in Tables 1 for and Table for x y x y. These resuls correspond o hose repored in Tables 1 and of GHJ (001). The resuls in Table 1 can be summarized as follows. Spurious rejecions appear frequenly if he usual -es is used and he es is evidenly divergen as T increases for 0.99 or larger. The Newey-Wes mehod is helpful; bu is advanage is less impressive if is close o 1.0 and T is large. Also, is divergen behaviour becomes apparen for 1.0 bu a a slower rae, as prediced in Phillips (1998). These resuls are generally in agreemen wih hose found in GHJ (001). Performance of he KV es is quie impressive compared o he oher wo ess he size is accurae for 0.90 or less and sill under conrol even for 0.99 if he sample size is large. Moreover, he es is convergen for random walks ( 1.0 ), which is in accordance o he asympoic heory of Sun (004). However, he frequency of rejecion is sill much higher han he nominal size when 0.95. In he random walk case, he size of he KV es reaches 0.33 regardless he sample size. Implemening a pre-whiing procedure does help for he NW es in aming size. Taking 0.9 and T=500 for example, he rejecion rae is reduced dramaically from 0.4 o 0.091. Even so, he procedure fails o make he NW es convergen when is close o 1. When he KV es is considered, a prewhiing procedure can successfully reduce size disorion, provided ha he 5
auoregression parameer is no larger han 0.95. Taking 0.95 and T=500 for example, he rejecion rae is reduced from 0.098 o 0.076, which is close o he nominal size. However, for 0.99 or larger, he procedure acually inflaes he rejecion rae and makes he es divergen when T is large. For example, a T=000, he rejecion rae increases from 0.151 o 0.85 and from 0.36 o 0.667 for equals 0.995 and 1.0, respecively. Therefore, a warning should be addressed in implemening a pre-whiening procedure. Table generally agrees wih Table 1. The KV es is obviously he bes. Performing a pre-whiing procedure is a plus in all cases. Also, as shown in GHJ (001), rejecion raes are similar for ( x a, y b) and ( xb, y a) if ab according o he usual -es and he NW es. Such a symmery does no appear for he KV es: rejecion raes for ( a, b) end o be larger han hose for ( a, b) if ab. x y x y CASE II: Spurious regression beween MA processes The second experimen assumes ha Y and X are generaed by wo independen MA(K) processes: Y K y, j j0 and X x, j, K j0 where x, y are each i.i.d. N(0,1). Table 3 repors he simulaion resuls for K aking he values 5, 10, 0, 50, or 75 and sample size varying from 100 o 000. The resuls correspond o hose repored in Tables 4 of GHJ (001). 6
The resuls are generally consisen wih hose of he previous case. The usual -es over-rejecs frequenly, almos one-half for K=10; he rejecion frequency increases seadily wih K, bu is relaively sable over T. Frequency of over-rejecion is grealy reduced using he NW es, and is reduced furher when he KV es is used. Taking T=50 and K=10 for example, rejecion raes of he usual -es, he NW es and he KV es are 0.491, 0.00 and 0.08, respecively. For boh HAC robus ess, rejecion frequency increases seadily wih K when T is fixed bu decreases wih T when K is fixed. Also, in all cases sudied, size disorion is reduced furher if a pre-whiening procedure is performed. Consider he KV es wih pre-whiening, he rejecion rae of K=75 is around 0.1 a T=500 and less han 0.1 a T>500. The pre-whiened NW es seems o be very conservaive when K is small and T is large he null hypohesis ends o be under-rejeced. IV Conclusion This paper assesses wheher he convergen -es of KV (00) and Sun (004) is able o solve spurious regressions found in GHJ (001). To do so, inensive Mone-Carlo experimens have been conduced. In brief, i has been found ha he convergen -es delivers much beer conrol over size comparing o he usual -saisic and is Newey-Wes modificaion. Also, for mos cases sudied, implemening he convergen -es wih an AR(1) pre-whiing procedure produced furher conrol over size. Ye, in many occasions, he rae of bias 7
rejecion is sill much higher han he nominal size when he convergen es is used. References Andrews, D.W.K. and J.C. Monahan (199) An improved heeroskedasiciy and auocorrelaion consisen covariance esimaor, Economerica, 59, 953-966. den Haan W.J. and A. Levin (1997) A praciioner s guide o robus covariance marix esimaion, in G. Maddala and C. Rao (eds), Handbook of Saisics: Robus Inference, Volume 15, Elsevier, New York, 91-341. Granger, C.W.J. and P. Newbold (1974) Spurious regressions in economerics, Journal of Economerics,, 111-10. Granger, C.W.J., N. Hyung and Y, Jeon (001) Spurious regressions wih saionary series, Applied Economics, 33, 899-904. Kiefer, N.M. and T.J. Vogelsang (00) Heeroskedasiciy-auocorrelaion robus esing using bandwidh equal o sample size, Economeric Theory, 18, 1350-1366. Kiefer, N.M. and T.J. Vogelsang (005) A new asympoic heory for heeroskedasiciy-auocorrelaion robus ess, Economeric Theory, 1, 1130-1164. Newey, W.K. and K.D. Wes (1987) A simple, posiive semi-definie, heeroskedasiciy and auocorrelaion consisen covariance marix, Economerica, 55, 703-708. Phillips, P.C.B. (1986) Undersanding spurious regressions in economerics, Journal of Economerics, 33, 311-340. Phillips, P.C.B. (1998) New oos for undersanding spurious regressions, Economerica, 66, 199-135. Sun Y. (004) A convergen -saisic in spurious regressions, Economeric Theory, 0, 943-96 8
Table 1: Spurious regression beween AR processes ( x = y = T NW NW, PW KV 100.138.110.096.069.063 50.135.079.068.051.047 0.50 500.135.073.056.051.050 1000.16.065.055.050.049 000.131.059.05.050.048 100.9.174.15.096.077 50.89.13.079.064.057 0.75 500.308.114.073.061.058 1000.300.086.054.058.056 000.300.083.053.05.050 100.5.31.178.140.106 50.53.43.115.094.077 0.9 500.55.4.091.073.063 1000.55.168.07.060.056 000.54.151.058.058.056 100.616.41.30.199.149 50.653.381.16.136.103 0.95 500.659.36.118.098.076 1000.66.305.091.077.067 000.663.67.073.066.059 100.739.557.334.99.07 50.804.61.88.5.170 0.99 500.8.65.36.197.145 1000.83.597.188.138.114 000.839.578.174.116.11 100.741.573.343.308.13 50.87.648.35.95.00 0.995 500.856.697.301.50.181 1000.879.695.336.05.00 000.880.687.407.151.85 100.769.613.37.339.31 50.856.699.38.37. 1.0 500.887.77.439.38.53 1000.93.810.64.334.435 000.94.849.793.36.667 KV, PW 9
Table : Spurious regression beween wo AR processes ( x y y, x ) T NW NW, PW KV 100.387.17.13.107.088 (0.75, 0.90) 500.38.137.069.065.059 1000.399.11.060.057.05 100.379.18.140.10.083 (0.90, 0.75) 500.391.149.075.063.057 1000.387.111.061.055.051 100.401.19.141.10.091 (0.75, 0.95) 500.47.159.073.077.069 1000.417.17.067.069.064 100.406.30.147.100.078 (0.95, 0.75) 500.43.153.070.064.056 1000.418.11.058.053.048 100.443.37.14.133.09 (0.75, 1.0) 500.448.174.075.099.081 1000.464.140.064.089.080 100.444.43.156.100.076 (1.0, 0.75) 500.45.167.077.053.040 1000.467.139.066.066.060 100.553.337.194.171.1 (0.90, 0.95) 500.576.63.090.084.069 1000.576.6.074.066.060 100.558.344.19.165.118 (0.95, 0.90) 500.585.76.094.081.066 1000.591.3.080.063.057 100.611.394.06.189.18 (0.9, 1.0) 500.648.346.107.1.089 1000.651.78.080.098.079 100.608.406.45.178.134 (1.0, 0.9) 500.647.348.117.08.063 1000.649.80.086.060.048 100.687.495.66.43.157 (0.95, 1.0) 500.731.469.144.146.099 1000.747.44.099.113.084 100.670.484.95.41.17 (1.0, 0.95) 500.735.481.17.10.089 1000.747.418.11.085.063 KV, PW 10
Table 3: Spurious Regression beween MA(k) processes K T NW NW, PW KV 100.176.103.073.06.051 50.178.081.056.058.051 500.179.07.048.054.048 1000.178.066.048.051.050 000.179.063.045.050.049 100.339.165.087.084.060 50.330.118.046.063.050 5 500.338.113.037.059.051 1000.35.085.034.05.047 000.333.083.035.05.051 100.490.69.19.1.083 50.491.00.070.08.06 10 500.478.166.037.060.048 1000.46.13.09.060.053 000.470.104.0.048.045 100.579.359.17.156.104 50.569.67.081.094.066 15 500.561.7.046.069.054 1000.559.174.036.060.05 000.557.150.08.060.053 100.637.40.17.193.16 50.60.35.107.108.074 0 500.607.85.061.083.061 1000.604.6.041.064.049 000.604.181.03.054.049 100.75.585.35.37.3 50.743.533.06.191.18 50 500.747.483.16.11.080 1000.747.41.070.087.060 000.735.348.037.061.049 100.771.6.367.337.37 50.79.610.7.50.173 75 500.791.571.164.153.101 1000.784.490.096.096.070 000.779.456.060.07.060 KV, PW 11