Cointegration - Wikipedia
Econometrics: Cointegration and Autoregressive Conditional . is that you may model both short run and long run relationship jointly. (if variables cointegrated). Cointegration occurs when a relationship ties together nonstationary economic time series such that a combination of those time series is I(0). The concepts of. or cointegration relations, between non-stationary time series variables such This is a sequentially defined model; yt is generated as a function of its own past.
The number of cointegrating vectors is called the cointegrating rank. An interpretation of cointegrated variables is that they share a common stochastic trend.
Consider in which mit is a random walk and eit is white noise. The remaining linear combination of the white noise terms is itself white noise. Given our notions of equilibrium in economics, we must conclude that the time paths of cointegrated variables are determined in part by how far we are from equilibrium.
That is, if the variables wander from each other, there must be some way for them to get back together, as in the Spike and Sal example. The errors are necessarily I 0. Therefore, the two interest rates must cointegrated of order CI 1,1. We could add lagged Drit to the RHS of both equations without changing the interpretation of the model. With the added terms we would have a model similar to a vector autoregression VAR.
However, if we were to estimate an unrestricted VAR then we would introduce a misspecification error. At least one of as and aL must be non-zero. If the roots of Characteristic equation's the fraction polynomial in L the lag operator lie inside the unit circle then both zt and yt are stationary see the first point and cannot be cointegrated. Even if only 1 root of the characteristic equation lies outside the unit circle then both variables are explosive, so cannot be CI 1,1. If both roots are unity then both variables are I 2 and cannot be CI 1,1.
For yt and zt to CI 1,1 one root must be 1 and the other must be less than 1. For this particular example we can show For cointegration either a12 or a21 must be non-zero and the condition that the second root of the polynomial in L be more than 1 in absolute value requires the following condition on the aij: Suppose yt and zt are cointegrated.
Testing for cointegration 1. Test the variables for order of integration. They must both all be I d. Estimate the parameters of the long run relationship. For example, when yt and zt are cointegrated OLS is super consistent.
Unit Roots and Cointegrated Series
That is, the rate of convergence is T2 rather than just T in Chebyshev's inequality. Denote the residuals from step 2 as and fit the model The null and alternate hypotheses are Interpretation: Rejection of the Null implies the residual is stationary. If you reject the null in step 3 then estimate the parameters of the ECM The terms in parentheses are the error correction terms.
In the earlier section on unit roots we observed that this was not the case. Then by the PPP model there is a long run relationship such that ut is stationary.
Now the cointegrating vector between ft and pt is under a strict interpretation of the PPP model. If they were integrate of different orders then we could reject PPP immediately.
For each country the pairs are I 1. Third, the statistical theory of unit-root processes aids inference about the empirical existence of cointegration. Econometric theory historically relied on the assumption of stationary data even though many observed economic time series were trending and nonstationary.
Cointegration explicitly allows for nonstationarity, thus providing a sounder basis for empirical inference. Cointegration also clarifies the problem of nonsense regressions, in which intrinsically unrelated nonstationary time series are highly correlated with each other.
Fourth, cointegration implies, and is implied by, the existence of an equilibrium correction representation of the relevant variables. Cointegration thus solidifies the statistical and economic bases for the empirically successful class of equilibrium correction models, in which past disequilibria in levels have an effect on current changes in the variables. Through ECMs, cointegration provides a systematic framework for jointly analyzing short-run e.
Fifth, optimal forecasts of cointegrated variables are themselves cointegrated.
Hence, the existence of cointegration may improve the long-term forecasting of economic time series. Hendry and Mary Morgan, in The Foundations of Econometric Analysishighlight the following events in that history. InClive Granger and Paul Newbold reemphasized the dangers of nonsense correlations, and Peter Phillips presented a formal analysis inwhich he updated in Denis Sargan established the link between static-equilibrium economic theory and ECMs in In the s, Granger and Robert Engle developed coin-tegration analysis as such.
Cointegration entails, and is entailed by, an ECM, which explicitly embeds a steady-state solution for its variables, while also allowing them to deviate from that steady state in the short run.
In a nonstochastic steady state, an equilibrium relation would typically be motivated by economic theory. Hence, economic hypotheses are testable in a cointegration framework. With suitable tests, asymptotically correct inferences can be obtained. In addition, spurious regressions can be detected and avoided, as can unbalanced regressions involving variables of different orders of integration. Economic theory rarely specifies orders of integration for variables, so a practitioner must analyze the data for both integration and cointegration.
While the presence of unit roots complicates inference because some associated limiting distributions are nonstandard, critical values have been tabulated for many common cases. David Dickey and Wayne Fuller calculated critical values of tests for unit roots in univariate processes, and many robust unit-root tests have subsequently been developed.
Numerous cointegration tests have also been designed. InEngle and Granger proposed a single-equation approach that is intuitive and easy to implement, though it includes nuisance parameters in inference and may lack power see Hendry Several authors have tabulated critical values, which are also embodied in software such as Cats for Rats and PcGive.
For instance, testing the lon-grun homogeneity of money with respect to prices is equivalent to testing whether the logs of money and prices are cointegrated with a unit coefficient. Weak exogeneity is satisfied if the cointegrating vector entering the conditional model does not appear in the marginal model of the conditioning variables.
Under weak exogeneity, inference on those parameters from the conditional model alone is without loss of information relative to inference in the complete system. In summary, cointegration and equilibrium correction help us understand short-run and long-run properties of economic data, and they provide a framework for testing economic hypotheses about growth and fluctuations.
At the outset of an empirical investigation, economic time series should be analyzed for integration and cointegration, and tests are readily available to do so. Such analyses can aid in the interpretation of subsequent results and may suggest possible modeling strategies and specifications that are consistent with the data, while also reducing the risk of spurious regressions.
Galbraith, and David F. Hendry, Frank Srba, and Stephen Yeo. Economic Journal 88 Journal of the American Statistical Association 74 Co-integration and Error Correction: Representation, Estimation, and Testing.
Cointegration, Exogeneity, and Policy Analysis: Journal of Policy Modeling 14 3: