Many economic variables exhibit persistent upward or downward movement. This feature can be generated by stochastic trends in integrated variables. If the same stochastic trend is driving a set of integrated variables jointly, they are called cointegrated. In this case, certain linear combinations of integrated variables are stationary. Such linear combinations that link the variables to a common trend path are called cointegrating relationships. They sometimes may be interpreted as equilibrium relationships in economic models.

Cointegrating relationships can be imposed by reparameterizing the VAR model as a vector error correction model (VECM). In Section 3.1 cointegrated variables are introduced and VECMs are set up. Sections 3.2 and 3.3 consider the estimation as well as the specification of VECMs. Diagnostic tools are presented in Section 3.4, and the implications of including cointegrated variables in VAR models for forecasting and Granger causality analysis are discussed in Section 3.5. Our focus in this chapter is on reduced-form models. We leave extensions to structural VECMs to later chapters.

The concept of cointegration was introduced in the econometrics literature by Granger (1981) and Engle and Granger (1987). Early work on error correction models goes back to Sargan (1964), Davidson, Hendry, Srba, and Yeo (1978), Hendry and von Ungern-Sternberg (1981), and Salmon (1982). Lütkepohl (1982b) discusses the cointegration feature without using the cointegration terminology. A full analysis of the VECM is presented in Johansen (1995), among others. Parts of the present chapter follow closely Lütkepohl (2005, part II; 2006, 2009).

Cointegrated Variables and Vector Error Correction Models

Common Trends and Cointegration

Cointegrated processes were introduced by Granger (1981) and Engle and Granger (1987). If two integrated variables share a common stochastic trend such that a linear combination of these variables is stationary, they are called cointegrated. For example, the plots of quarterly U.S. log output and investment in the upper panel of Figure 3.1 both exhibit an upward trend. Because both series are driven by the same trend, the log of the GDP-investment ratio is fluctuating about a constant mean. As a result, the difference between the log series in the lower panel of Figure 3.1 has no obvious trend anymore. It is mean reverting and appears stationary.