58 I U David A. Dickey, Dennis W. Jansen and Daniel L. Thornton I David A. Dickey is a professor of statistics at North Carolina State University. Dennis W Jansen is an associate professor of economics at Texas A&M University and Daniel T Thornton is I an assistant vice president at the Federal Bank of St. Louis. The authors would like to thank Mark Watson, Tom Fomby and David Small for helpful comments on an earlier draft of this article. Lora Holman and Kevin L. Kliesen provided research assistance. I I A Primer On Cointegration I with an Application to Money I and Income I 1 I OR SOME TIME NOW, macroeconomists stems from a common prediction of macroeco- have been aware that many macroeconomic nomic theory that there should be a stable long- time series are not stationary in their levels and run relationship among the levels of certain eco- that many time series are most adequately nomic variables. That is, theory often suggests represented by first differences.’ In the parlance that some set of variables cannot wander too of time-series analysis, such variables are said to far away from each other. If individual time be integrated of order one and are denoted 1(1). series are integrated of order one, however, The level of such variables can become arbitrari- they may be “cointegrated.” Cointegration means ly large or small so there is no tendency for that one or more linear combinations of these them to revert to their mean level. Indeed, variables is stationary even though individually neither the mean nor the variance is a mean- they are not. If these variables are cointegrated, ingful concept for such variables. they cannot move too far” away from each Nonstationarity gives rise to several economet- other. In contrast, a lack of cointegration sug- ric problems.2 One of the most troublesome gests that such variables have no long-run link; 1 2lt can give rise to the possibility of a spurious relationship That is, formal statistical tests often cannot reject the null among the levels of the economic variables. Also, the hypothesis of a unit root. The results of these tests, parameter estimates from a regression of one such however, are sensitive to how the tests are performed— variable on others are inconsistent unless the variables are that is, whether the tests assume a non-zero mean or a cointegrated. time trend, whether an MA or AR data generating pro- cesses is assumed [Schwert (1987)) and whether the test is performed using classical or Bayesian statistical in- ference (Sims (1988), and Sims and Uhlig (1988)). These sensitivities are partly due to the lack of power these tests II have against an alternative hypothesis of a stationary but large root. FEDERAL RESERVE SANK OF ST. LOWS I I~- 59 in principle, they can %vander arbitrarily far When ~= 1, these deviations are permanent. In away from each other.-’ this case, y, is said to follow a random walk— I it can wander arbitrarily far from any given This article illustrates the salient features of 4 cointegration and tests for cointegration. The constant if enough time passes. In fact, when discussion, initially motivated by the simple ex- = 1 the variance of y, approaches infinity as I ample of Irving Fisher’s “equation of exchange,” increases and the mean of y,, y, is not defined. draws an analogy between cointegration and Alternatively, when ~[< 1, the series is said to unit roots on the one hand and tests for cointe- be mean reverting and the variance of y, is F gration among multiple time series and the usual finite. tests for unit roots in univariate time-series Although there is a similarity between tests analysis on the other. The article then addresses for cointegration and tests for unit roots, as we I the broader question of the economic inter- shall see below, they are not identical. Tests for pretation of cointegration by contrasting it with unit roots are performed on univariate time the usual linear, dynamic, simultaneous equa- series. In contrast, cointegration deals with the tion model which is frequently used in relationship among a group of variables, where I macroeconomics. (unconditionally) each has a unit root. The article goes on to compare three recently To be specific, consider Irving Fisher’s impor- proposed tests for cointegration and outlines the 7 tant equation of exchange, M\ = Pq, where Mis Ii procedures for applying these tests. An applica- a measure of nominal money, V is the velocity tion of these tests to U.S. time-series data using of money, Pisthe overall level of prices and q alternative monetary aggregates, income and in- is real output)’ This equation can be rewritten II terest rates suggests that there is a stable long- in natural logarithms as: run relationship among real output, interest rates and several monetary aggregates, including (2) lnM + mV — lnP — lnq = 0. the monetary base. In this form, the equation of exchange is an II identity. The theory of the demand for money, TESTING FOR COINTEGRATION: however, converts this identity into an equation by making velocity a function of a number of I f A GENERAL FRAMEWORK economic variables; both the form of the func- Because of the close correspondence between tion and its arguments change from one for cointegration and standard tests for theoretical specification to another. In the theory I testsunit roots, it is useful to begin the discussion by of money demand, V is unobservable and in ap- considering the univariate time-series model plied work it is proxied with some function of economic variables, V’, mV’ =lnV+E, where E denotes a random error associated with the use (i)y,—ji = Q(y,~ —jA) + II where y, denotes1 some univariate time series, p of the proxy for V.’ The proxy is a function of is the series’ mean and e, is a random error one or more observed variables, other than in- with an expected value of zero and a constant, come and prices, that are hypothesized to deter- ‘I finite variance. The coefficient Q measures the mine the demand for money. Hence, equation 2 degree of persistence of deviations of y, from j.i. is replaced with -‘At the present time, tests for cointegration deal only with pie, money holders might have a money illusion or money looking for stable linear relationships among economic demand might not be homogenous of degree one in real income. ‘For the classic discussion of velocity and a long list of its variables.relationshiptheredoes notis nonecessarilyConsequently,stable, amonglong-run,themeanvariables,a failurethat linearthereittorelationshiponly findis nosuggestscointegrationstable, amongthat long-run potential determinants, see Friedman (1956). Empirical proxies for velocity often contain one or more of these them. determinants. 4 Thatany startingis, for any valuenumbersY, thereC is>-a0time, and T,0 -<suchp <that,1 andfor forall generating>- 1, Pr(flYjY, >-isC)stationary> p. Whenin thatj~it<Idoesthe notprocesswander too far ‘Thefindfromacointegratingconstantits mean,C vectori.e.,> 0 forsuchcouldanythatgivenbePrdifferent(~Y,.-~4probabilityfromC) <p thep.we can hypothesized one for other reasons as well. For exam- I MARCH/APRlL 1991 I 60 I a (2’) lnM + lnV’ — lnP — lnq = E. In multiple comparison tests, an experimenter If the proxy is good, the expected value of E is usually concerned with, say, comparing the should be zero. Furthermore, E should be sta- highest and lowest sample means among several. I tionary, so that, V’ might deviate from its true He wants to find the pair of sample means with value in the short-run, but should converge to it the largest difference to see whether the dif- in the long-run. Failure to find a stationary rela- ference is statistically significant. When the I tionship among these variables—that is, to find means to be compared are chosen ahead of that they are not cointegrated—implies either time, tests for a significant difference between that V’ is a poor proxy for V or that the long- the means can be done using the usual t- I run demand for money does not exist in any statistics. If, however, the means to be com- meaningful sense. pared are chosen simply because they are the largest and smallest from a sample of, say, five In essence, the Fisher relationship embodies a means, the rejection rate under the null I long-run relationship among money, prices, out- hypothesis will be much higher than that im- put and velocity. In particular, it hypothesizes plied by the percentile of the t-distribution. In that the cointegrating vector (1,1,—i,— 1) exists. order to control for the experimentwise error I This vector combines the four series into a rate, as it is called, tables of distributions of univariate series, E. Given this assumption, a highest mean minus lowest mean (standardized) test for cointegration can be performed by ap- have been computed for the case of no true dif- plying any conventional unit x-oot test to E. ferences in the population means. These “tables Using conventional unit root tests to test for of the studentized range” are then used to test cointegration [such as tests prepared by Dickey for significant differences between the highest 7 and Fuller (1979) and Phillips (1987)], however, and lowest means. II requires prior knowledge of the cointegrating vector. And most often, this vector is unknown. The price paid for controlling the experiment- wise error rate is a loss in power. That is, the Therefore, some linear combination of these 11 variables, for example, difference between the means must be much larger than in the case of the standard t-test (3) b,lnM + b,lnv’ + b,lnP + b,lnq, before it can be declared significant at some is hypothesized to be stationary, where the predetermined significance level.