Unbalanced Regressions and Spurious Inference

doi:10.4236/ojs.2012.23035

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Open Journal of Statistics, 2012, 2, 297-299

http://dx.doi.org/10.4236/ojs.2012.23035 Published Online July 2012 (http://www.SciRP.org/journal/ojs)

Unbalanced Regressions and Spurious Inference

Daniel Ventosa-Santaulària

Centro de Investigación y Docencia Económicas, CIDE, Mexico City, Mexico

Email: [email protected]

Received April 23, 2012; revised May 24, 2012; accepted June 5, 2012

ABSTRACT

Spurious regression has been extensively studied in time series econometrics since Granger and Newbold’s [1] seminal

paper. Recently, it has been advanced that this phenomenon is due to a mistreatment of short-range autocorrelation in

the residuals of the regression when at least one of the variables in a bivariate regression is stationary. HAC errors, fea-

sible GLS and Cochrane-Orcutt-type procedures are then proposed to draw correct inference. Such a proposal should be

cautiously considered, since nonsense inference might also be due to deterministic trend mechanisms, structural breaks,

and long range dependence. In these cases, standard autocorrelation correction procedures would not solve the problem

of spurious regression. We aim to make the later argument clear.

Keywords: Spurious Regression; Stationarity; Unbalanced Regression; Unit Root

1. Unbalanced Spurious Regressions

Spurious regression has been extensively studied since

Granger and Newbold [1] seminal paper in which inde-

pendent nonstationary variables are simulated and then

used to estimate a simple bivariate regression. Phillips [2]

provided the theoretical framework to understand the

phenomenon in the simplest case (independent driftless

unit root processes). Since then, the spurious regression

phenomenon has been identified for many data-generat-

ing processes (DGPs), such as unit root with drifts, (bro-

ken-) trend stationary and long range, for example1.

Here, we are concerned with the results presented in

Noriega and Ventosa-Santaulària [4] and Stewart [5]

pertaining to the spurious regression phenomenon under

the following conditions: 1) both variables, t and t

(see Equation (1)), are stationary (i.e., integrated of order

0, I(0)), and 2) at least one of the variables (the regressor

or the regressand) is integrated of order 1, I(1). The later

combinations result in an unbalanced regression and

Noriega and Ventosa-Santaulària [4] found that, in a sim-

ple regression specification,

ttt

yxu





 

(1)

where either , t

(or both) is I(0)2, the t-ratio associ-

ated with ˆ



, ˆ



, does not diverge as the sample size

grows; i.e. . Results in Noriega and Ventosa-

Santaulària [4] imply that the asymptotic spurious re-

gression phenomenon does not occur. Nevertheless, non-

sense inference cannot be fully discarded. In a recent

paper, Stewart [5] argues that, although the t-ratio does

not diverge, it may not necessarily converge to a standard

normal distribution. Furthermore, in the absence of auto-

correlation in the DGP’s innovations, only when both

variables are iid I(0) processes, the t-ratio behaves—

asymptotically—as a standard normal. Other DGP com-

binations, such as







~1 − t

I and vice versa,

do have asymptotic nonstandard distributed t-ratios. Nev-

ertheless, the size distortions are better explained by the

presence of autocorrelation in the DGP innovations. This

point is illustrated by Stewart [5] throughout a number of

finite-sample experiments. The problem comes as no

surprise since the estimated residuals behave as an auto-

correlated process and size distortion should be expected

in that case. Moreover, the use of heteroskedasticity and

autocorrelation consistent (HAC) errors considerably re-

duce size distortions in some cases, as argues Stewart [5].

Table 1 summarizes the relevant DGPs for both the de-

pendent and the explanatory variables, similar to those

used by Noriega and VentosaSantaulària [4] and Stewart

[5], to estimate a simple linear specification.

For simplicity, we assume that innovations,

t, for e

,zx



, are iid white noises. Following Noriega and

Ventosa-Santaulària [4] and using the aforementioned

DGPs, we present the following corollary:



ˆ1





Corollary Let t and

1For a recent survey see Ventosa-Santaulària [3]. yt

, be generated by DGPs i

and j of Table 1. Denote ij as the DGP combination

that generated y and x, respectively, and use them to

2In Noriega and Ventosa-Santaulària [4], the other variable may be-

have as: 1) driftless I(1); 2) I(1) with drift (with a possible drift break);

3) I(2); and 4) (Broken-)Trend stationary.

D. VENTOSA-SANTAULÀRIA

298

Table 1. The DGPs for ,

ttt

.

Case Name Model

1 I(0) tzz





2 )1(I 1tt zt

zz e





estimate specification (1) by ordinary least square. The

asymptotic distribution of the t-ratio associated with ˆ



ˆ, is: t









 



12 1/

()

d1d

xy y

xxr

rrr



 



















 







 





d1d

:yx x

rrr

rr rr



 























,,zxy

21 1/2

where , for , is a standard Brownian mo-





tion and

Proof: See Noriega and Ventosa-Santaulària [4]3.

Results in the corollary reveal that the asymptotic dis-

tribution of the t-ratio is nonstandard when the regression

is unbalanced. However, a simple simulation of the as-

ymptotic distribution shows a striking resemblance of

this distribution with a standard normal (insets (a) and (b)

in Figure 1). Such resemblance fades out in the presence

of autocorrelation (insets (c) and (d) in Figure 1). We,

therefore, confirm that the size distortions pointed out by

Stewart [5] are due to autocorrelation; the latter happens

to be an important source of spurious regression when at

least one of the variables is I(0) and confirms the results

of Granger Hyung and Jeon [6] and Mikosch and Vries

[7] results. Nevertheless, short range autocorrelation

should not be considered as the sole source of spurious

inference. It is well documented that deterministic trends,

structural breaks, and long range dependence, also gen-

erate nonsense inference (see Perron [8] and Tsay and

Chung [9]). It is important to note that the latter cannot

be prevented by using Cochrane-Orcutt or Feasible GLS.

Using standard correction procedures to deal with the



 denotes convergence in distribution.

(a) (b)

Figure 1. t-ratio asymptotic distribution for unbalanced regressions: insets (a) and (c) I(0) vs I(1); insets (b) and (d) I(1) vs

I(0); insets (a) and (b) iid innovations; insets (c) and (d) AR(1) innovations ( and

0.4

x



0.7

y



). Number of re-

plications: 10,000. The blue area corresponds to the standard normal distribution whilst the red dashed line depicts the as-

mptotic distribution of the t-ratio.

e



3Noriega and Ventosa-Santaulària [4] only provide the order in convergence of the t-ratio. However, by following the instructions in the appendix, the

asymptotic expressions can also be obtained, as we demonstrated in this paper. The Mathematica code is available upon request.

D. VENTOSA-SANTAULÀRIA 299

spurious regression phenomenon is tempting, even if such

procedures cannot always provide correct inference (see

Stewart [5] and McCallum [10], for example). Sun [11]

proposed a convergent t-statistic using modified HAC

errors with a bandwidth proportional to the sample size

when the variables are highly persistent. The author ac-

knowledges, however, that such a procedure cannot be

used in empirical applications, since the limit distribution

of the test depends on the memory parameter under the

null hypothesis and critical values cannot, therefore, be

tabulated. McCallum [10] and Kolev [12] also advocate

classical correction procedures to deal with spurious re-

gressions, such as the Cochrane-Orcutt procedure and

Feasible Generalized Least Squares. They argue that us-

ing them reduces size distortions of the t-test. However,

Martínez-Rivera and Ventosa-Santaulària [13] proved that

such methods are not always effective and remain highly

dependent on the DGP of the series4.

2. Concluding Remarks

There is finite-sample evidence showing that spurious

inference in unbalanced regressions mostly occurs when

the innovations of the DGPs are not iid. In that sense,

standard autocorrelation-correction procedures, such as

HAC errors, Feasible GLS and Cochrane-Orcutt esti-

mates, have been advanced to eliminate/reduce the size

distortions and, thus, spurious inference. This approach

should, nevertheless, be reconsidered. First, there is evi-

dence that spurious regression using stationary series

cannot always be interpreted as a short range autocorre-

lation phenomenon: long range dependence and struc-

tural breaks (level shifts, for example) also cause spuri-

ous inference; spurious regression cannot, therefore, be

always corrected using classical procedures. Second, an

unbalanced regression (in which the order of integrations

of the involved series is not the same) is an empirical

situation which remains to be proved relevant. The esti-

mation of an unbalanced regression is not intuitive, al-

though there are cases such as in the predictive equation

in the finance literature, in which the market returns

(usually, found to be stationary) is regressed against

dividend yield (stationary but highly persistent). Spurious

regression cannot be simply considered as a short-mem-

ory autocorrelation phenomenon and cannot, therefore,

be treated using standard procedures. The main conclu-

sion is, therefore, twofold: 1) practitioners should inter-

pret cautiously their results whenever they find evidence

of autocorrelation, since the inference could be spurious;

2) they should, however, be aware that spurious regres-

sions arise for many diverse reasons, autocorrelation be-

ing only one of them; standard autocorrelation correction

procedures are not to be considered as the sole solution

to prevent spurious inference; on the contrary: parameter

stability, long memory and cointegration tests should

always be also considered.

REFERENCES

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[2] P. C. B. Phillips, “Understanding Spurious Regressions in

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[3] D. Ventosa-Santaulària, “Spurious Regression,” Journal

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155-182. doi:10.1155/2009/802975

[4] A. E. Noriega and D. Ventosa-Santaulària, “Spurious Re-

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Economics and Statistics, Vol. 69, No. 3, 2007, pp. 439-

444. doi:10.1111/j.1468-0084.2007.00481.x

[5] C. Stewart, “A Note on Spurious Significance in Regres-

sions Involving I(0) and I(1) Variables,” Empirical Eco-

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doi:10.1007/s00181-010-0404-5

[6] C. W. J. Granger IV, N. Hyung and Y. Jeon, “Spurious

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[7] T. Mikosch and C. G. Vries, “Tail Probabilities for Re-

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[8] P. Perron, “The Great Crash, the Oil Price Shock, and the

Unit Root Hypothesis,” Econometrica, Vol. 57, No. 6,

1989, pp. 1631-1401. doi:10.2307/1913712

[9] W. J. Tsay and C. F. Chung, “The Spurious Regression of

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[10] B. T. McCallum, “Is the Spurious Regression Problem

Spurious?” Economics Letters, Vol. 107, No. 3, 2010, pp.

321-323. doi:10.1016/j.econlet.2010.02.004

[11] Y. Sun, “A Convergent T-Statistic in Spurious Regres-

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943-962. doi:10.1017/S0266466604205072

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Bulletin, Vol. 31, No. 1, 2011, pp. 925-937.

4There are other approaches worth mentioning. In Davidson and Mac-

Kinnon [14], for example, the authors consider that the spurious re-

gression phenomenon is, at least partially due to a misspecification o

the model. The authors argue that instead of Equation (1), practitioners

should estimate yt = α + βxt + γyt-1 + μy. Using this specification makes

the null hypothesis of the t-test valid. Simulations presented in David-

son and MacKinnon [14] reveal, however, that even a correct specifi-

cation is unable to provide an adequate size of the t-test under the null

hypothesis when the variables are nonstationary.

[13] B. Martínez-Rivera and D. Ventosa-Santaulària, “A Com-

ment on ‘Is the Spurious Regression Problem Spurious?’”

Economics Letters, Vol. 115, No. 2, 2012, pp. 229-231.

[14] D. Davidson and J. G. MacKinnon, “Econometric Theory

and Methods,” Oxford University Press, New York, 2004.