Regressions of two independent time senes are considered. The variables are covariance stationary but display time-varying although not trending means. Two prominent examples are mean shifts due to structural breaks and seasonally varying means. If the variation of the means is not taken into account, this induces nonsense correlation. The asymptotic treatment is supplemented by experimental evidence

Hassler, Uwe

Universidad Carlos III de Madrid e-Archivo

Working Paper 99-78 Statistics and Econometrics Series 30 October 1999 Departamento de Estadfstica y Econometrfa Universidad Carlos III de Madrid Calle Madrid, 126 28903 Getafe (Spain) Fax (34-91) 624-9849 NONSENSE REGRESSIONS DUE TO TIME-VARYING MEANS. Uwe Hassler. * Abstract --------------------------------------------------------Regressions of two independent time senes are considered. The variables are covariance stationary but display time-varying although not trending means. Two prominent examples are mean shifts due to structural breaks and seasonally varying means. If the variation of the means is not taken into account, this induces nonsense correlation. The asymptotic treatment is supplemented by experimental evidence. Keywords: Structural breaks; deterministic seasonality; spurious correlation. *Free University of Berlin. Institute of Statistics and Econometrics. Boltzmannstr. 20 D-14195 Berlin Germany, e-mail: uwe@wiwiss.fu-berlin.de.This work was carried out while visiting Universidad Carlos III de Madrid. Financial support from the European Commision through the Training and Mobility of Researchers programme is gratefully acknowledged. 1 Introduction Since the early paper by Yule (1926) statisticians are aware of the danger of nonsense correlation between unrelated random walks. Granger and Newbold (1974) followed his work and established experimentally that stochastically independent random walks give rise to spurious regressions. l They coined the latter term to describe the fact that testing for the true null of no correlation one observes a much higher rejection rate than the nominal level. Phillips (1986) provided an asymptotic treatment of spurious regressions that arise whenever random walks are not cointegrated, see also the discussion e.g. in Banerjee, Dolado, Galbraith and Hendry (1993). His results were extended to the cases of cointegrated regressors and of stationary covariates by Choi (1994) and Hassler (1996), respectively. Similar findings were established in the presence of trending variables integrated of order two or higher orders, see Haldrup (1994) and Marmol (1996), respectively, or in the presence of linear time trends, cf. Hassler (1996a). This note demonstrates that nonsense correlation, or spurious regressions, may arise even if the series are not trending. It is motivated by Perron (1990) who showed that the distinction between trending series integrated of order one and stationary series with a mean shift may be difficult as long as the shift is not taken into account. His finding suggests that independent covariance stationary time series with mean shifts may give rise to nonsense regression. Indeed, this will be proven here. Hence this paper contributes to a field of growing interest recently surveyed in Maddala and Kim (1998). 1 I use the terms nonsense or spurious regressions interchangeably and speak as well of nonsense or spurious correlation. 2 In fact, I consider the more general case that the means of two independent series are time-varying although not trending. This includes e.g. series with deterministic seasonality where the mean varies from season to season. Section 2 becomes precise on the model and establishes the general result which is illustrated by the example of quarterly varying means. Section 3 turns to the case of mean shifts due to structural breaks that is more relevant in practice. The asymptotic formulae are confronted with Monte Carlo evidence. The final section contains a more detailled summary. 2 The general case The simple model considered here consists of two covariance stationary series Ult and U2t that are stochastically independent of each other and that are added to a deterministic mean function, Xit = dit + Uit, t = 1,2, ... ,T, i = 1,2. (1) The stochastic components are assumed to be zero mean processes with vari-ance a; independent of each other. The deterministic components are sup-posed to be not trending. They are bounded and square summable, more precisely, JditJ ~ Di < 00, t = 1,2, ... ,T, i = 1,2, (2) (3) 3 where .!... denotes convergence in probability. For the stochastic series I as-sume in contrast _ 1 T (Ui' u;, U1 U2) = T 2:)Uit? U;t, Ult U2t) .!... (0, al, 0), t=1 and that they satisfy the central limit theorem, which implies T L Uit = Op(TO.5 ). t=1 Given (I), (2) and (3) it is straightforward to prove as T ---* 00 1 T T L(Xit - Xi)2 u; + d; - d/ + Op(T-O.5 ) t=1 .!... a; + "d - 0;, i = 1,2, (4) (5) Please note that (2) and (3) allow for the special case of constant means, dit = di · In this case 0i = di , "/f = d; = 0; and /12 = d1d2 = 0102 so that (4) and (5) reproduce the standard case of independent series with constant means. The limits from (4) and (5) render themselves to determine the probability limit of the ordinary least squares (OLS) estimator from Xlt = 0: + ;3X2t + Et, t = 1,2, ... ,T, (6) where the variable nature of dit is not taken into account. We obtain under the assumptions made so far A p /12 - 0102 -;3 ---* 2 2 02 =: ;3. a2 + /2 - 2 (7) 4 With (7) the behaviour of the standard error of regression (6), becomes obvious. It is needed to construct the t-statistic tf3 testing for f3 = o. To establish the divergence of tf3 it is convenient to normalize as T-O.5tf3 = ~JT-l LA(X2t - X2")2 s Moreover, the limit of the coefficient of determination can be derived from A2 R2 = 1 _ _ --=~s __ _ T-l 2:(Xl - Xl)2 vVith (4), (5) and (7) the following results are easily established. Proposition 1 (Nonsense regression) Let Xlt and X2t from (1) be stochas-tically independent of each other. Under the assumptions of this section it then holds for the OLS statistics from (6) as T ---+ 00: A 2 P 2 2 );"2 f3~ ( );" );" ) ~ 2 S ---+ (Jl + 11 - vI - 112 - vlv2 =: s , 5 Just as in the case of random walks, see Phillips (1986), we observe that in general /3, T-O.5t(3 and R2 have non-zero limits, only that the limits are deterministic in the present setup of deterministic time-varying means. Such nonsense correlation of course only occurs if the righthand side in (5) and hence fJ is different from zero. Consider e.g. the case of additive outliers where dit is zero except for a finite number of times. This clearly entails /12 = Di = 0, and therefore does not lead to spurious regression results. As a first application of Propostion 1 I consider series with seasonally varying means. A beysinghe (1991) observed spurious regression effects if seasonally integrated series are regressed on each other, where the regression includes seasonal dummies. Here the series are not integrated but display deterministic seasonality that is ignored in the regression. I restrict the treatment to quarterly varying means with observations over n years, Sli, t = 4j + 1 dit = S2i, t = 4j + 2 j = 0,1, ... ,n - 1, (8) , S3i, t = 4j + 3 S4i, t = 4j + 4 where T = 4n is assumed. It is again simple to check that Hence Proposition 1 can be applied to the situation of quarterly series with deterministic seasonality. Corollary 2 (Deterministic seasonality) Let Xlt and X2t from (1) and (8) satisfy the assumptions of this section. Then the results from Proposition 6 1 hold with Please note that quarterly varying means like in (8) do not necessarily imply spurious regressions. For instance (su, 0, 0, -SU), yield ')'12 = 0, 151 = ° and hence do not give rise to nonsense correlation, /3 = 0. Moreover, regressing seasonal series on each other empirical researchers typically include seasonal dummies accounting for seasonally varying means. That is why the nonsense regression situation underlying Corollary 2 will not be encountered often in practice. Therefore, I provide a more thorough discussion only for the more relevant example of mean shifts due to structural breaks. 3 Mean shifts As a second example of model (1) I consider mean shifts, (9) where E(Xit) = 0 is assumed without loss of generality for t ::; AiT. Both observed series Xlt and X2t are subject to a break in the mean at possibly different times \T where \ E [0,1] is the proportion when this shift occurs. 7 It is easily verified that where Amax = max(A1' A2) . Propositon 1 hence provides the following corol-lary_ Corollary 3 (Mean shifts) Let Xlt and X2t from (1) and (9) satisfy the assumptions of the previous section. Then the results from Proposition 1 hold with It is not difficult to supplement Corollary 3 with a statement referring to the Durbin-Watson statistic. To simplify matters I assume now that Ult and U2t are white noise series. With /3 and s defined in Proposition 1 it then follows T -1 ""r A A 2( 2 f3~2 2) d = Dt=2 Utt ~ 0"1 + 0"2 W A2 ~2· S S We observe that dw does not tend to zero as with spurious regressions of random walks. However, its probability limit will in general differ from 2 even if the stochastic components are white noise. Please note that nonsense correlation according to Corollary 3 arises if and only if both series are subject to a structural break, bi ~ 0 and 0 < Ai < 1 for i = 1,2. This is true because for Amax = max(A1' A2) implies j3 ~ 0 with j3 = b1b2(1 - Amax) - b1b2(1- A1)(1 - A2) O"~ + b~(l - A2)A2 8 Furthermore, certain symmetries appear. Looking closely at the limits in Proposition 1, Corollary 3 reveals the following symmetries about the line A = 0.5. Corollary 4 (Symmetries) Let Xlt and X2t from (1) and (9) satisfy the assumptions of Corollary 3 with bi =I 0 and 0 < Ai < 1 for i = 1,2. Then it holds for the limits in Proposition 1: • if Al = 1 - A2 then all limits are symmetric about A2 = 0.5; • if Al = 0.5 then all limits are symmetric about A2 = 0.5; • if A2 = 0.5 then all limits are symmetric about Al = 0.5; • if Al = A2 = :\ then all limits are symmetric about :\ = 0.5. Bearing in mind the German unification in 1990 the last case in Corollary 4 of a common breakpoint :\ may be of particular interest. It is straightfor-ward to show that 1;31 has a maximum at :\ = 0.5, i.e. the spurious effect is strongest if the breakpoint is in the middle of the sample. To verify the relevance of the asymptotic results a Monte Carlo experi-ment was performed. Two independent white noise series with means shift were simulated according to (1) and (9) with Uit f"V N(O, 1). The number of observations was T = 100. From 5000 replications done with GAU8832 the mean values of the OL8 statistics were computed. In Tables 1 and 2 they are compared with the probability limits according to Corollary 3. More-over, I report the frequency of rejection when testing with tf3 for the true null hypothesis f3 = 0 at the 5% level, Itf31 > 1.96. As one would expect the nonsense correlation becomes more severe as the breaks bi > 0 are increasing. 9 Throughout we observe that the asymptotic values well explain the experi-mental means already for T = 100. The asymptotic symmetry according to Corollary 4 was also found to be well reproduced experimentally. That's why I do not report tables for symmetric cases. Table 1 is restricted to the situation of different breakpoints. With Al = 0.25, A2 = 0.75 only moderate spurious results arise. As the breakpoints move closer, Al = 0.5, A2 = 0.25 or Al = 0.25, A2 = 0.5, the problem of nonsense regressions becomes more severe. In Table 2 results for common breakpoints are collected. If Al = A2, the feature of nonsense correlation is stronger in comparison with Table 1, and it is strongest if the break occurs in the middle of the sample. Very low values of the coefficient of determination are common to Tables 1 and 2. This suggests to detect spurious regressions due to mean shifts by means of the R2. Alternatively one might of course avoid nonsense correlation from the beginning by testing the univariate series for a break and eventually removing it prior to regression. 4 Summary It is widespread textbook knowledge that nonsense regressions may arise between independent time series that are trending. Moreover, there is a grow-ing literature concerned with the effect of structural breaks. In this paper the two topics are related. It is shown that the danger of nonsense correlation, or spurious regression, is present even if the independent series are covari-ance stationary. This is true if their means are time-varying functions and 10 Table 1: Mean values and limits, )'1 =1= '\2 11 11 s 1 t~/TO.5 1 R2 1 dw 11 5% 11 Al = 0.25, A2 = 0.75 (AI = 0.75, A2 = 0.25) b1 = 1 0.050 0.050 0.013 1.709 8.06 b2 = 1 (0.053) (0.053) (0.003) (1.694) bl = 1 0.071 0.087 0.017 1.722 14.02 b2 = 2 (0.071) (0.087) (0.008) (1.706) b1 = 2 0.107 0.089 0.017 1.194 13.02 b2 = 1 (0.105) (0.087) (0.008) (1.164) b1 = 2 0.146 0.1.46 0.029 1.220 29.30 b2 = 2 (0.143) (0.144) (0.020) (1.190) Al = 0.5, A2 = 0.25 (AI = 0.5, A2 = 0.75) b1 = 1 0.108 0.105 0.020 1.644 18.42 b2 = 1 (0.105) (0.103) (0.011) (1.635) b1 = 1 0.144 0.173 0.037 1.698 39.28 b2 = 2 (0.143) (0.171) (0.029) (1.681) h =2 0.212 0.166 0.035 1.103 37.00 b2 = 1 (0.211) (0.164) (0.026) (1.073) b1 = 2 0.287 0.279 0.079 1.195 82.22 b2 = 2 (0.286) (0.277) (0.071) (1.165) Al = 0.25, A2 = 0.5 (AI = 0.75, A2 = 0.5) h = 1 0.103 0.106 0.021 1.738 17.84 b2 = 1 (0.100) (0.103) (0.011) (1.719) b1 = 1 0.125 0.164 0.035 1.778 38.06 b2 = 2 (0.125) (0.164) (0.026) (1.757) b1 = 2 0.202 0.173 0.038 1.256 41.90 b2 = 1 (0.200) (0.1711 (0.029) (1.224) b1 = 2 0.255 0.283 0.081 1.343 80.78 b2 = 2 (0.250) (0.277) (0.071) (1.308) For notes see Table 2. Table 2: Mean values and limits, Al = A2 = :x 11 dw " 5% 11 ~ = 0.25 (~= 0.75) b1 = 1 0.159 0.162 0.034 1.785 36.22 b2 = 1 (0.158) (0.159) (0.025) (1.770) b1 = 1 0.215 0.270 0.076 1.904 77.72 b2 = 2 (0.214) (0.269) (0.067) (l.889) b1 = 2 0.319 0.272 0.077 l.374 76.62 b2 = 1 (0.316) (0.269) (0.067) (1.348) b1 = 2 0.431 0.479 0.191 l.682 99.78 b2 = 2 (0.429) (0.474) (0.184) (l.657) ~ = 0.5 b1 = 1 0.202 0.206 0.049 l.750 54.76 b2 = 1 (0.200) (0.204) (0.040) (l.733) b1 = 1 0.250 0.335 0.107 l.905 92.40 b2 = 2 (0.250) (0.333) (0.100) (l.888) b1 = 2 0.405 0.337 0.109 l.310 92.68 b2 = 1 (0.400) (0.333) (0.100) (l.289) b1 = 2 0.505 0.583 0.257 1.686 100 b2 = 2 (0.500) (0.577) (0.250) (l.666) Simulated white noise series with mean shifts as in (9). Given are mean values from 5000 replications of regression (6) with T = 100 observations (asymptotic values according to Corollary 3 in brackets). Moreover, the column denoted '5%' contains the frequency of rejecting the true null f3 = 0 when performing a two-sided test based on t{3 at the nominal 5% level. 12 if this is not taken into account when regressing them on each other. The first example are series with deterministic seasonality where the means differ from season to season. Nonsense regressions are avoided by simply including seasonal dummies in the regression, which is of course standard practice. The second example discussed at some length are mean shifts due to structural breaks. The following asymptotic results are well supported by finite sample evidence. The closer the breakpoints of the two series are, the more severe is the problem of nonsense correlation. The strongest spurious regression effects occur for a common breakpoint in the middle of the sam-ple. Depending on the magnitude of the breaks the true null of no correlation will be rejected with very high probability. This clearly highlights the im-portance of testing for structural breaks and removing them before running regressions. References Abeysinghe, T. (1991): Inappropriate use of seasonal dummies in regres-sions; Economics Letters 36, 175-179. Banerjee, A., J.J. Dolado, J.W. Galbraith, and D.F. Hendry (1993): Co-integration, error-correction, and the econometric analysis of non-stationary data; Oxford University Press. Choi, 1. (1994): Spurious regressions and residual-based tests for cointe-gration when regressors are cointegrated; Journal of Econometrics 60, 313-320. 13 Granger, C.W.J., and P. Newbold (1974): Spurious regressions in econo-metrics; Journal of Econometrics 2, 111-120. Haldrup, N. (1994): The asymptotics of single-equation cointegration regressions with 1(1) and 1(2) variables; Journal of Econometrics 63, 153-18l. Hassler, U. (1996): Spurious regressions when stationary regressors are included; Economics Letters 50, 25-3l. Hassler, U. (1996a): Nonsense correlation between time series with linear trends; Allgemeines Statistisches Archiv 80, 227-235. Maddala, G.S., and I.-M. Kim (1998): Unit roots, cointegration, and structural breaks; Cambridge University Press. Marmol, F. (1996): Nonsense regressions between integrated processes of different orders; Oxford Bulletin of Economics and Statistics 58, 525-536. Perron, P. (1990): Testing for a unit root in a time series with a changing mean; Journal of Business & Economic Statistics 8, 153-162. Phillips, P.C.B. (1986): Understanding spurious regressions in economet-rics; Journal of Econometrics 33, 311-340. Yule, G.D. (1926): Why do we sometimes get nonsense correlation be-tween time series? A study in sampling and the nature of time series; Journal of the Royal Statistical Society 89, 1-64. 14 

Nonsense regressions due to time-varying means

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Nonsense regressions due to time-varying means

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