The central objective in structural VAR analysis is to quantify causal relationships in the data. Before discussing the identification of causal relationships in structural VAR models, it is useful to review the precursors to structural VAR analysis. Our discussion traces how the focus of the literature has evolved from documenting lead-lag patterns in the data, as discussed in Sections 7.2–7.4, to quantifying unanticipated shifts in the data reflecting exogenous events, as discussed in Section 7.5. There are several approaches to constructing such exogenous shocks. We review the narrative approach to measuring exogenous policy shocks, the derivation of exogenous shocks from data-based counterfactuals, the construction of news shocks from macroeconomic announcements, and the measurement of shocks to financial market expectations. The definition of exogenous shocks was generalized with the introduction of the structural VAR framework, as discussed in Section 7.6. The latter approach is based on decomposing fluctuations in the data that cannot be predicted based on past data into mutually uncorrelated exogenous shocks with economic interpretation that need not be directly observable. As we trace the evolution of this literature, we also formally introduce the concepts of predeterminedness, strict exogeneity, and Granger causality, highlighting the extent to which each approach relies on these concepts.

A Motivating Example

The need for structural models in studying causal relationships between economic time series is best illustrated by the debate about causality from monetary aggregates to national income in the 1960s and 1970s. It had long been observed that money growth and income growth in the United States were positively correlated. Based on a careful review of the historical evidence, Friedman and Schwartz (1963) in their Monetary History of the United States concluded that changes in money growth are causing changes in income growth (an obvious implication being that the Federal Reserve should pursue a constant money growth rule to stabilize the business cycle). This position evolved into a school of thought known as monetarism. Monetarism emphasizes the relation of the level of the money stock to the level of aggregate real economic activity (see Sims 1980b).

The monetarist position contrasted with the prevailing Keynesian wisdom that monetary policy was not nearly as important as fiscal policy in explaining economic fluctuations.

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.

Typical VAR models used for policy analysis include only small numbers of variables. One motivation for considering larger VAR models is that policy institutions such as central banks and government organizations consider large panels of time series variables in making policy decisions. If important variables are not included in a VAR model, that model becomes informationally deficient and estimates of the responses to policy shocks will be distorted by omitted-variable bias. Thus, unless a variable is known to be irrelevant, one should ideally include it in the structural VAR model.

Deciding on the relevance of a particular variable for an empirical model is a difficult task because the variables for which data are available may not correspond exactly to the variables used in theoretical models. For example, consider a monetary policy reaction function that includes inflation and the output gap as explanatory variables. It is well known that the output gap is difficult to measure. Hence, one can make the case for including all variables that contain information about the output gap because they could all be important for the analysis of the impact of monetary policy.

Another motivation for considering larger VAR models is that we may wish to examine the impact of monetary policy shocks at a more disaggregate level. For example, one may be interested not only in the response of the overall price level to a monetary policy shock, but also in the response of sub-indices corresponding to specific expenditure components. Such an analysis necessitates the inclusion of disaggregate price level data in the model. Likewise, one may be interested in the output response in specific sectors of the economy. In that case, again, these additional variables have to be included in the analysis.

Conventional unrestricted VAR models do not allow for the situations described above because the inclusion of many additional variables undermines the precision of the model estimates in small samples. Moreover, the extent to which a VAR model can be enlarged is limited by the fact that the number of regressors cannot exceed the number of observations. This restriction can easily become binding when working with large-dimensional VAR models because the number of parameters in a VAR model increases with the square of the number of variables included.

Introduction

As we have seen in Chapters 8 and 10, the identification of structural VAR models typically relies on economically motivated identifying restrictions. Another strand of the literature exploits certain statistical properties of the data for identification. In particular, changes in the conditional or unconditional volatility of the VAR errors (and hence of the observed variables) can be used to assist in the identification of structural shocks. For example, Rigobon (2003), Rigobon and Sack (2003), and Lanne and Lütkepohl (2008) rely on unconditional heteroskedasticity, whereas Normandin and Phaneuf (2004), Bouakez and Normandin (2010), and Lanne, Lütkepohl, and Maciejowska (2010) exploit conditional heteroskedasticity.

In this chapter, we explain the principle of identification by heteroskedasticity. In Section 14.2, the general modeling strategy is presented and its advantages and limitations are discussed. The central idea is that in a conventional structural VAR analysis the structural shocks are recovered by transforming the reduced-form residuals. As we have seen in previous chapters, this is typically done through exclusion restrictions. The current chapter considers the question of how changes in the volatility of the model errors can be used for this purpose. We show that the assumption that the structural impulse responses are time invariant, as the volatility of the reduced-form shocks changes, provides additional restrictions that can be used to uniquely pin down mutually uncorrelated shocks. There is nothing in this purely statistical identification procedure, however, that ensures that these shocks are also economically meaningful, making it difficult to interpret them as structural VAR shocks.

One way to assess whether all or some of the shocks identified by heteroskedasticity correspond to economic shocks and, hence, can be interpreted as proper structural shocks is to treat conventional identifying restrictions as overidentifying restrictions within the heteroskedastic model, facilitating formal tests of these restrictions.

In some cases it may also be possible to infer the economic interpretation of these shocks informally from comparisons of the implied impulse responses with impulse response estimates based on conventional structural VAR models, as illustrated in Lütkepohl and Netšunajev (2014). A necessary condition for such comparisons is that the structural impulse responses in the VAR model based on conventional identifying restrictions can be estimated consistently under the assumptions maintained in the explicitly heteroskedastic VAR model.