"P. Rappoport and L. Reichlin, "Segmented Trends and Non-stationary Time Series," Economic Journal, Vol. 99, conference 1989, pp. 168-177.

T. Perron, "The Great Crash, the Oil Price Shock and the Unit Root Hypothesis," Econometrica, Vol. 57, 1989, pp. 1361-1401.

"In-Moo Kim, "Structural Change and Unit Roots," unpublished Ph.D. dissertation. University of Florida, 1990. Kim studies the Nelson-Plosser and Friedman-Schwartz real per capita GNP series for U.S. (annual) and quarterly OECD data. Another study that argues that many of the U.S. economic time series that were considered to be DS type are not necessarily so and that the evidence on the DS versus TS issue is mixed is G. D. Rudebusch, "Trends and Random Walks in Macroeconomic Time Series: A Re-examination," Federal Reserve Paper 1139, Washington, D.C., December 1990.

Test 3 (the Nelson-Plosser test)

: a = 1, 3 = 0 in y, = JL + 3/ + ay,, + u, ,: a < 1

In each case the dimension of the model is the same under both and

Structural Change and Unit Roots

In all the studies on unit roots, the issue of whether a time series is of the DS or TS type was decided by analyzing the series for the entire time period during which many major events took place. The Nelson-Plosser series, for instance, covered the period 1909-1970. which includes the two world wars and the Depression of the 1930s. If there have been any changes in the trend because of these events, the results obtained by assuming a constant parameter structure during the entire period will be suspect. Many studies done using the traditional multiple regression methods have included dummy variables (see Sections 8.2 and 8.3) to allow for different intercepts (and slopes). Rappoport and Richlin (1989) show that a segmented trend model is a feasible alternative to the DS model.2

Perron (1989) argues that standard tests for unit root hypothesis against the trend-stationary (TS) alternatives cannot reject the unit root hypothesis if the time series has a structural break.- Of course, one can also construct examples where, for instance,

y,, 2, . . . , y„ is a random walk with drift

y„+,, . . . , y„+„ is another random walk with a different drift

and the combined series is not the DS type. Perrons study was criticized on the argument that he "peeked at the data" before analysis-that after looking at the graph, he decided that there was a break. But Kim (1990), using Bayesian methods, finds that even allowing for an unknown breakpoint, the standard tests of the unit root hypothesis were biased in favor of accepting the unit root hypothesis if the series had a structural break at some intermediate date.

When using long time series, as many of these studies have done, it is important to take account of structural changes. Parameter constancy tests have

14.6 Cointegration

An important issue in econometrics is the need to integrate short-run dynamics with long-run equilibrium. The traditional approach to the modeling of short-run disequilibrium is the partial adjustment model discussed in Section 10.6. An extension of this is the ECM (error correction model), which also incorporates past periods disequilibrium [see equation (10.17)]. The analysis of short-run dynamics is often done by first eliminating trends in the variables, usually by differencing. This procedure, however, throws away potential valuable information about long-run relationships about which economic theories have a lot to say. The theory of cointegration developed in Granger (1981) and elaborated in Engle and Granger (1987) addresses this issue of integrating short-run dynamics with long-run equilibrium. We discussed this briefly in Section 6.10. We now go through it in greater detail.

We start with a few definitions. A time series y, is said to be integrated of order 1 or 1(1) if , is a stationary time series. A stationary time series is said to be 1(0). A random walk is a special case of an 1(1) series, because, if y, is a random walk, , is a random series or white noise. White noise is a special case of a stationary series. A time series y, is said to be integrated of order 2 or is 1(2) if , is 1(1), and so on. If y, ~ 1(1) and u, ~ 1(0), then their sum Z, = y, + u,~ 1(1).

Suppose that y, ~ 1(1) and x, ~ 1(1). Then y, and x, are said to be cointegrated if there exists a p such that y, - , is 1(0). This is denoted by saying y, and , are CI(], 1). What this means is that the regression equation

y, = Px, -f- M,

makes sense because y, and x, do not drift too far apart from each other over time. Thus there is a long-run equilibrium relationship between them. If y, and JC, are not cointegrated, that is, y, - Px, = u, is also 1(1), they can drift apart from each other more and more as time goes on. Thus there is no long-run equilibrium relationship between them. In this case the relationship between y, and , that we obtain by regressing y, on x, is "spurious" (see Section 6.3).

•C. W. J. Granger, "Some Properties of Time Series Data and Their Use in Econometric Model Specification," Journal of Econometrics, Vol. 16, No. I, 1981, pp. I2I-I30; R. F. Engle and C. W. J. Granger, "Cointegration and Error Correction: Representation, Estimation and Testing," Econometrica, Vol. 55, No. 2, 1987, pp. 251-276.

More generally, ify, ~ 1(d) and x, ~ 1(d), then y, and x, ~ Cl(d, b) it y, - x, ~ l(d - b) with b>0.

frequently been used in traditional regression analysis. But somehow, all the traditional diagnostic tests are ignored when it comes to the issue of the DS versus TS analysis. The main issue with economic time series is how best to model dynamic economic models. The testing for unit roots has received a lot more attention than the estimation aspect.

In the Box-Jenkins method, if the time series is nonstationary (as evidenced by the correlogram not damping), we difference the series to achieve stationarity and then use elaborate ARMA models to fit the stationary series. When we are considering two time series, y, and x, say, we do the same thing. This differencing operation eliminates the trend or long-term movement in the series. However, what we may be interested in is explaining the relationship between the trends in y, and x,. We can do this by running a regression of y, on x„ but this regression will not make sense if a long-run relationship does not exist. By asking the question of whether y, and x, are cointegrated, we are asking whether there is any long-run relationship between the trends in y, and Xf.

The case with seasonal adjustment is similar. Instead of eUminating the seasonal components from and x and then analyzing the deseasonalized data, we might also be asking whether there is a relationship between the seasonais in and X. This is the idea behind "seasonal cointegration." Note that in this case we do not consider first differences or 1(1) processes. For instance, with monthly data we consider 12th differences y, - y, i2. Similarly, for x, we consider x, - X, ,2.

When we talk of common trends, we have to distinguish between what are commonly called deterministic and stochastic trends. In Section 6.10 we talked about these as the TSP (trend stationary process) and DSP (difference stationary process). Detrending (by running a regression on time) assumes the presence of a deterministic trend, and differencing assumes the presence of a stochastic trend. The concept of cointegration refers to the idea of common stochastic trends. But this is not the only kind of common trends. One can also have common deterministic trends. The same concept extends to seasonais as well. Seasonal adjustment using dummy variables assumes a deterministic seasonal, and seasonal adjustment by differencing as discussed in the Box-Jenkins approach in Section 13.6 assumes a stochastic seasonal. The concept of seasonal cointegration appUes to stochastic seasonal. In practice, both deterministic and stochastic components could be present in a time series, so that we can write the time series as

X, = T, + S, + \x., + - , + ,

where T, represents deterministic trends (e.g., a polynomial in t), S, represents a deterministic seasonal (e.g., seasonal dummy variables), x, represents a stochastic trend [e.g., an 1(1) process], and ], represents a stochastic seasonal [e.g., with quarterly data, (1 - V) is stationary].

Ignoring the presence of deterministic components leads to some misleading inferences on cointegration. But to simplify our analysis and to concentrate on the issues of cointegration, we shall assume that there are no deterministic elements in the time series we consider.

S. Hylleberg, R. F. Engle, C. W. J. Granger and S. Yoo, "Seasonal Integration and Co-integration," Working Paper. University of California at San Diego. 1988.

"See H. Kang, "Common Deterministic Trends, Common Factors, and Co-integration," in Fomby and Rhodes (eds.). Advances in Econometrics, pp. 249-269.