Allowing the Data to Speak Freely: The Macroeconometrics of the Cointegrated Vector AutoregressionI. Between Data and TheoryII. Models of Theory, Models of DataIII. The Cointegrated Vector Autoregression ModelIV. Case Study: Purchasing Power ParityREFERENCES251American Economic Review: Papers & Proceedings 2008, 98:2, 251–255http://www.aeaweb.org/articles.php?doi=10.1257/aer.98.2.251All economists agree that reality is complex and that the tools with which we confront it are far simpler. Theorists sometimes deal with this gap by asking very little of the data. Start with the “stylized” facts and develop relatively sim-ply theories to account for them. Unfortunately, stylized facts are often too stylized to discrimi-nate among plausible candidate theories or to provide a basis for accurate quantification. Alternative approaches start from the other end and ask much of the data. One European tra-dition, which derives from Trygve Haavelmo’s “The Probability Approach in Econometrics” 119442 , focuses on obtaining good character-izations of data before testing and on drawing out the implications of data that ought to con-strain economic theorizing. The application of the cointegrated vector autoregression 1CVAR2 recounted, for example, in Katarina Juselius’s 120062 textbook and facilitated by the CATS in RATS econometric software 1Jonathan G. Dennis et al. 20062 is a special macroeconometric case of the Probability Approach. The message of the Probability Approach and the CVAR approach can be summarized in the slogan: “facts, not stylized facts.”I. Between Data and TheoryAll econometrics aims ultimately to confront theory and data. Different approaches differ in how they conceive the relationship and the prob-lems that it poses. To start, think of an ideal case such as one might find in a physics textbook. The law of gravity is applied to the dropping of a ball from a tower. The law, together with Allowing the Data to Speak Freely: The Macroeconometrics of the Cointegrated Vector AutoregressionBy Kevin D. Hoover, Søren Johansen, and Katarina Juselius*an initial condition 1the height of the tower2, determines the distance the ball falls for each time … in theory. Of course, no object conforms perfectly to the gravity law. If one had a gener-ous enough notion of approximation, if the ball were steel and the initial height were not too high, then tight bounds of approximation would work; but not if the ball were styrofoam. Now there are three choices: (A2 declare that theory is no good; (B2 modify the original theory to account for the factors such as air resistance; or (C2 attempt to assess empirically the combined forces that must be used to adjust the gravity law to its application in the particular case.In economics, as in physics, the difficulty is that our theory holds ceteris paribus. When other things are not equal, there is always some residual left unexplained which, if large, may render the theory empirically irrelevant. As scientists, we can either attempt to elaborate the theory in such a way that fewer and fewer ceteris paribus conditions are invoked (B2 or we can attempt to provide an adequate empirical characterization of the factors that determine the initial gap between theory and data 1C2. Strategy C can be seen as the passive analogue to a controlled experiment 1cf. Haavelmo 1944, esp. chaps. 1 and 22. It has an advantage over B, in the sense of providing clues as to how the theory needs to be developed—clues that would be helpful in strategy B, but for which strategy B itself offers no internal resources. What is more, strategy C gives some hope of actually isolating the action of the gravity law and, therefore, in fact testing whether it is a contributing factor to a successful account of the data. It is only when we can control for enough of the complicating factors that the underlying quadratic relation-ship of the gravity law can be detected.The extreme limit of strategy B is what Milton Friedman has called the “Walrasian” methodology, by which he refers not to general-equilibrium theory per se, but to the idea that one must have a complete, detailed theoretical * Hoover: Department of Economics, Duke University, Box 90097, Durham, NC 27278 (e-mail: [email protected]); Johansen, Department of Economics, University of Copenhagen, Studiestræde 6, 1455, København K, Denmark (e-mail: [email protected]); Juselius: Department of Economics, University of Copenhagen, Studiestræde 6, 1455 København K, Denmark (e-mail: [email protected]).MAY 2008252AEA PAPERS AND PROCEEDINGSaccount in order to say anything useful about the economy at all 1see Hoover 20062. The extreme limit of strategy C is a completely atheoretical analysis of data. Both extremes are hopeless: strategy B because we lack the cog-nitive capacity to elaborate a complete theory from first principles; strategy C because without some prior conceptual notion we would never find a starting place for any investigation. Still, the Probability Approach leans toward strategy C: the weight of the analysis is on characteriz-ing the data and on using the data to criticize and guide theorizing. In Friedman’s terms, the approach is “Marshallian” or, as Hoover 120062 puts it, “archaeological”: we learn about the economy a piece at a time by removing the overlay of detritus to uncover the underlying structure, guided by our theoretical conception of what we are looking for, which is tested and enriched by each new discovery.II. Models of Theory, Models of DataThe CVAR approach builds on Haavelmo’s 119442 great insight that the gap between theory and data need not be treated as an unstructured residual of approximation, but could itself be modeled statistically using the theory of proba-bility. The cost is that we now need another level of modeling in addition to theory—a statistical model constructed in such a way that (a2 theory has implications interpretable in its terms, and (b 2 data are described fully enough that its only residuals are identically independent random errors—that is, unsystematic noise. The pay-off is that such a