Nonsense Regressions due to Neglected Time-varying Means Uwe Hassler Free University of Berlin Institute of Statistics and Econometrics Boltzmannstr. 20 D-14195 Berlin Germany email: uwe@wiwiss.fu-berlin.de 3 September 2001 Abstract Regressions of two independent time series are considered. The variables are covariance stationary but display time-varying although not trending means. Two prominent examples are level 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 Level shifts; deterministic seasonality; spurious correlation This work was carried out while visiting Universidad Carlos III de Madrid. Financial support from the European Commission through the Training and Mobility of Researchers programme is gratefully acknowledged. An earlier version was circulated as Universidad Carlos III working paper 99-78. I am grateful to Artur da Silva Lopes for comments that improved the presentation.
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. 1 They coined the latter term to describe the fact that, when testing for the true null of no correlation one observes a much higher rejection rate than the nominal level. This is not a finite sample problem. 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, and to seasonally integrated series by Abeysinghe 1991, 1994). Similar findings were established in the presence of trending variables integrated of order two or higher orders, see Haldrup 1994) and Marmol 1996), respectively, in case of nonstationary fractionally integrated processes, see Cappuccio and Lubian 1997) and Marmol 1998), or in the presence of linear time trends, cf. Hendry 1980) and Hassler 1996a). All those results rely on nonstationary series, a recent exception being Tsay and Chung 2000) who find nonsense regressions with only mildly trending, namely stationary fractionally integrated processes. This note demonstrates that nonsense correlation, or spurious regressions, may arise even if the series are not trending. It is motivated by Perron 1989, 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, see also Montañés and Reyes 1998) and Montañés and Reyes 1999). These findings suggest that independent covariance stationary time series with mean shifts give rise to nonsense regressions, as long as the shift is neglected. Indeed, this will be proven here. Hence, those results on structural breaks contribute to a field of growing interest recently surveyed in Maddala and Kim 1998). In fact, I consider 1 I use the terms nonsense or spurious regressions interchangeably and speak as well of nonsense or spurious correlation.
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 level shifts due to structural breaks that is more relevant in practice. The asymptotic formulae are well supported by Monte Carlo evidence. The final section contains a more detailed summary. 2 The general case The simple model considered here consists of two zero mean series u 1t and u 2t that are stochastically independent of each other and that are added to a deterministic mean function, x it = d it + u it, t = 1, 2,..., T, i = 1, 2. 1) The stochastic components are assumed to be covariance stationary processes with variance σi 2. Hence the observed series are also covariance stationary with E[x it d it )x is d is )] = Eu it u is ). The deterministic components are supposed to be not trending. They are bounded and square summable, more precisely it is assumed that d it D i <, t = 1, 2,..., T, i = 1, 2, 2) d i, d 2 i, d 1d 2 ) = 1 T T ) dit, d 2 ) it, d 1t d 2t δi, γi 2, γ 12, 3) t=1 where the limits in 3) are finite. For the stochastic series it is assumed that u i, u 2 i, u 1u 2 ) = 1 T T ) uit, u 2 p it, u 1t u 2t 0, σ 2 i, 0 ), t=1
where p denotes convergence in probability. Given 1), 2) and 3) it is straightforward to prove that, as T, 1 T x it x i ) 2 T = u 2 i + d2 i d i2 + op 1) t=1 p σ 2 i + γ 2 i δ 2 i, i = 1, 2, 4) 1 T T x 1t x 1 )x 2t x 2 ) = d 1 d 2 d 1 d 2 + o p 1) t=1 p γ 12 δ 1 δ 2. 5) Notice that 2) and 3) allow for the special case of constant means, d it = d i. In this case δ i = d i, γ 2 i = d 2 i = δ 2 i and γ 12 = d 1 d 2 = δ 1 δ 2 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 x 1t = ˆα + ˆβx 2t + ˆɛ t, t = 1, 2,..., T, 6) where the variable nature of d it is not taken into account. We obtain under the assumptions made so far ˆβ p γ 12 δ 1 δ 2 σ 2 2 + γ 2 2 δ 2 2 With 7) the behaviour of the standard error of regression 6), ŝ 2 = 1 T T ˆɛ 2 t, ˆɛ t = x 1t x 1 ˆβx 2t x 2 ), t=1 =: β. 7) becomes obvious. It is needed to construct the t-statistic t β testing for β = 0. To establish the divergence of t β it is convenient to normalize as T 0.5 t β = ˆβ T 1 T t=1 x 2t x 2 ) 2. ŝ Moreover, the limit of the coefficient of determination, which equals the squared correlation coefficient, can be derived from R 2 = ˆβ 2 x2t x 2 ) 2 x1t x 1 ) 2. Using 4), 5) and 7) the following results are easily established.
Proposition 1 Nonsense regression) Let x 1t and x 2t from 1) be stochastically independent of each other. Under the assumptions of this section it then holds for the OLS statistics from 6) as T : ˆβ p γ 12 δ 1 δ 2 σ 2 2 + γ 2 2 δ 2 2 =: β, ŝ 2 p σ 2 1 + γ 2 1 δ 2 1 β γ 12 δ 1 δ 2 ) =: s 2, T 0.5 p σ 2 t β β 2 + γ2 2 δ2 2, s R 2 p β 2 σ2 2 + γ2 2 δ2 2. σ1 2 + γ1 2 δ1 2 Just as in the case of random walks, see Phillips 1986), we observe that ˆβ, T 0.5 t β and R 2 have non-zero limits if and only if β 0), which means that t statistics diverge. However, those limits are deterministic in the present setting of deterministic time-varying means. Such nonsense correlation of course only occurs if the righthand side in 5) and hence β is different from zero. Consider e.g. the case of additive outliers where d it is zero except for a finite number of times. This clearly entails γ 12 = δ i = 0, and therefore does not lead to spurious regression results. As a first application of Propostion 1 I consider series with seasonally varying means. The treatment is restricted to quarterly means with observations over n years, s 1i, t = 4j + 1 s 2i, t = 4j + 2 d it = s 3i, t = 4j + 3 s 4i, t = 4j + 4 where T = 4n. It is again simple to check that ) 4 d i, d 2 i, d p 1 4 1d 2 s ki, 4, j = 0, 1,..., n 1, i = 1, 2, 8) k=1 k=1 s 2 ki, ) 4 s k1 s k2. Hence Proposition 1 can be applied to the situation of quarterly series with deterministic seasonality. k=1
Corollary 2 Deterministic seasonality) Let x 1t and x 2t from 1) and 8) satisfy the assumptions of this section. Then the results from Proposition 1 hold with ) 4 δi, γi 2 1, γ 12 = s ki, 4 k=1 4 k=1 s 2 ki, ) 4 s k1 s k2, i = 1, 2. k=1 From the work by Wooldridge 1991) it is not surprising that nonsense regressions may arise. Wooldridge 1991) considered the case of monthly means underlying the true processes while the regressions are run without seasonal dummies. He suggests a new coefficient of determination adjusted for the deterministic seasonality to avoid spuriously high values of the R 2. However, note that quarterly varying means like in 8) do not necessarily imply spurious regressions. For instance s 11, s 21, s 31, s 41 ) = s 11, 0, 0, s 11 ), s 12, s 22, s 32, s 42 ) = 0, s 22, s 32, 0) yield γ 12 = 0, δ 1 = 0 and hence do not give rise to nonsense correlation, β = 0. Moreover, when regressing seasonal series on each other empirical researchers hopefully 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, a more thorough discussion is provided 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 level shifts, 0, t λ i T d it =, b i 0, i = 1, 2, 9) b i, t > λ i T where Ex it ) = 0 is assumed without loss of generality for t λ i T. Both observed series x 1t and x 2t are subject to a break in the mean at possibly different times λ i T
where λ i [0, 1] is the proportion when this shift occurs. Notice that the series from 1) with 9) have a so-called common feature in the sense of Engle and Kozicki 1993) only if the break takes place at the same time, λ 1 = λ 2 = λ. In the general case it is easily verified that d i, d 2 i, d 1d 2 ) p bi 1 λ i ), b 2 i 1 λ i ), b 1 b 2 1 λ max ) ), where λ max = maxλ 1, λ 2 ). Propositon 1 hence provides the following corollary. Corollary 3 Mean shifts) Let x 1t and x 2t from 1) and 9) satisfy the assumptions of the previous section. Then the results from Proposition 1 hold with δi, γ 2 i, γ 12 ) = bi 1 λ i ), b 2 i 1 λ i ), b 1 b 2 1 λ max ) ), i = 1, 2. If u 1t and u 2t are in particular white noise, u it iid0, σ 2 i ), then it holds for the Durbin-Watson statistic dw = T 1 T t=2 ˆɛ t) 2 ŝ 2 p 2σ 2 1 + β 2 σ 2 2) s 2. We observe that the Durbin-Watson statistic 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 not serially correlated. Note that nonsense correlation according to Corollary 3 arises whenever both series are subject to a structural break, b i 0 and 0 < λ i < 1 for i = 1, 2. This is true because for λ max = maxλ 1, λ 2 ) it holds that which implies β 0 with 1 λ 1 )1 λ 2 ) 1 λ max ), β = b 1b 2 1 λ max ) b 1 b 2 1 λ 1 )1 λ 2 ) σ 2 2 + b 2 21 λ 2 )λ 2. Furthermore, certain symmetries appear. Looking closely at the limits in Proposition 1, Corollary 3 reveals the following symmetries about the line λ = 0.5.
Corollary 4 Symmetries) Let x 1t and x 2t from 1) and 9) satisfy the assumptions of Corollary 3 with b i 0 and 0 < λ i < 1 for i = 1, 2. Then it holds for the limits in Proposition 1: if λ 1 = 1 λ 2, then all limits are symmetric about λ i = 0.5; if λ 1 = 0.5, then all limits are symmetric about λ 2 = 0.5; if λ 2 = 0.5, then all limits are symmetric about λ 1 = 0.5; if λ 1 = λ 2 = λ, then all limits are symmetric about λ = 0.5. Bearing in mind the German unification in 1990 the last case in Corollary 4 of a common breakpoint λ may be of particular interest. It is straightforward to show that β has a maximum at λ = 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 experiment was performed. Two independent white noise series with mean shifts were simulated according to 1) and 9) with u it N0, 1). The level shifts vary between 2 twice the standard deviation) and 4. The number of observations was T = 100. From 5000 replications done with GAUSS the mean values of the OLS statistics were computed. In Tables 1 and 2 they are compared with the probability limits according to Corollary 3. Moreover, I report the frequency of rejection when testing with t β for the true null hypothesis β = 0 at the 5% level, t β > 1.96. Throughout we observe that the asymptotic values very well explain the experimental means already for T = 100. The asymptotic symmetry according to Corollary 4 was also found to be well reproduced experimentally. That s why tables for symmetric cases are not reported. Table 1 is restricted to the situation of different breakpoints. Already for λ 1 = 0.25, λ 2 = 0.75 and breaks that are only twice the standard deviation, b i = 2, clearly
spurious results arise in that the experimental level is a multiple of the nominal one, and the means of ˆβ are far away from 0. As one would expect the nonsense correlation becomes more severe as the breaks b i > 0 are increasing. In the lower panel of Table 1 the breakpoints are closer, λ 1 = 0.25, λ 2 = 0.5. This aggravates the nonsense correlation in terms of ˆβ and its t statistic considerably. Throughout Table 1, however, it is observed that the means and limits of R 2 hardly exceed 0.2. Hence, the coefficient of determination seems to be a useful device to detect those kinds of spurious regressions. In Table 2 results for common breakpoints are collected. In this situation x 1t and x 2t have a common feature in the sense of Engle and Kozicki 1993). If λ 1 = λ 2, the nonsense correlation is therefore much stronger in comparison with Table 1, and it is strongest if the break occurs in the middle of the sample. The latter feature is clear from the comment following Corollary 4. It is rather disquieting that coefficients of determination of a magnitude of 0.5 may easily occur. Working with covariance stationary series, values of this magnitude are often considered as satisfactory in practice. Therefore, it will be difficult to reveal nonsense regressions from statistics that ignore level shifts taking place at the same time. And this is probably the case most often encountered in practice. 4 Summary It is widespread textbook knowledge that nonsense regressions may arise between independent time series that are trending. Moreover, there is a growing 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 covariance stationary. This is true if their means are time-varying functions and 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 can be avoided by simply including seasonal dummies in the regression, which should be 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 sample. Depending on the magnitude of the breaks the true null of no correlation will be rejected with probability one. Of course, just as differencing removes the problem of spurious regressions with integrated series, the removal of mean shifts will remedy nonsense correlations in the presence of level breaks. This clearly highlights the importance of testing for level shifts and removing them before running regressions.
Table 1: Mean values and limits, λ 1 λ 2 ˆβ t β /T 0.5 R 2 dw 5% λ 1 = 0.25, λ 2 = 0.75 b 1 = 2 0.146 0.146 0.029 1.220 29.30 b 2 = 2 0.143) 0.144) 0.020) 1.190) b 1 = 3 0.218 0.178 0.037 0.844 40.98 b 2 = 2 0.214) 0.176) 0.030) 0.802) b 1 = 2 0.141 0.177 0.037 1.236 41.22 b 2 = 3 0.140) 0.176) 0.030) 1.201) b 1 = 3 0.211 0.215 0.050 0.851 58.30 b 2 = 3 0.209) 0.214) 0.044) 0.812) b 1 = 4 0.253 0.260 0.068 0.614 81.92 b 2 = 4 0.250) 0.258) 0.063) 0.567) λ 1 = 0.25, λ 2 = 0.5 b 1 = 2 0.255 0.283 0.081 1.343 80.78 b 2 = 2 0.250) 0.277) 0.071) 1.308) b 1 = 3 0.380 0.345 0.112 0.944 95.50 b 2 = 2 0.375) 0.342) 0.105) 0.948) b 1 = 2 0.232 0.333 0.105 1.370 94.44 b 2 = 3 0.231) 0.331) 0.099) 1.336) b 1 = 3 0.348 0.414 0.151 1.024 99.64 b 2 = 3 0.346) 0.412) 0.145) 0.975) b 1 = 4 0.402 0.500 0.204 0.787 100 b 2 = 4 0.400) 0.500) 0.200) 0.725) Simulated standard normal 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 β = 0 when performing a two-sided test based on t β at the nominal 5% level.
Table 2: Mean values and limits, λ 1 = λ 2 = λ ˆβ t β /T 0.5 R 2 dw 5% λ = 0.25 b 1 = 2 0.431 0.479 0.191 1.682 99.78 b 2 = 2 0.429) 0.474) 0.184) 1.657) b 1 = 3 0.649 0.613 0.276 1.463 100 b 2 = 2 0.643) 0.607) 0.269) 1.439) b 1 = 2 0.421 0.612 0.276 1.854 100 b 2 = 3 0.419) 0.607) 0.269) 1.838) b 1 = 3 0.634 0.814 0.401 1.733 100 b 2 = 3 0.628) 0.807) 0.394) 1.713) b 1 = 4 0.755 1.144 0.568 1.809 100 b 2 = 4 0.750) 1.134) 0.563) 1.786) λ = 0.5 b 1 = 2 0.505 0.583 0.257 1.686 100 b 2 = 2 0.500) 0.577) 0.250) 1.666) b 1 = 3 0.760 0.737 0.355 1.495 100 b 2 = 2 0.750) 0.728) 0.346) 1.471) b 1 = 2 0.465 0.733 0.352 1.870 100 b 2 = 3 0.462) 0.728) 0.346) 1.855) b 1 = 3 0.700 0.969 0.486 1.772 100 b 2 = 3 0.692) 0.959) 0.479) 1.748) b 1 = 4 0.803 1.345 0.645 1.846 100 b 2 = 4 0.800) 1.333) 0.640) 1.822) For notes see Table 1.
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