Still puzzling: evaluating the price puzzle in an empirically identified structural vector autoregression

The order is suggested by Fig. 1 of Brissimis and Magginas (2006, p. 1230), but is for a longer time period: 1959:08 to 2007:12. For definitions of the data, see Appendix.

In his classic treatment of the identification problem, Fisher (1966, p. 46) writes:

[in the just-identified case] the equations to be solved have an infinite number of solutions; manyother equations than the true structural one satisfy them. This is true no matter how large a samplewe consider and no matter how we abstract from problems of sampling error. In the overidentifiedcase, on the other hand, no problem would arise if we could so abstract... It is thus ... a practicalproblem ... rather than a problem of principle.

Swanson and Granger (1997), Akleman et al. (1999) , Bessler and Loper (2001), Bessler and Lee (2002), Demiralp and Hoover (2003), Haigh et al. (2004), Awokuse (2005), Hoover et al. (2009), and Reale and Tunnicliffe Wilson (2001), (2002) offer a substantially different, but still graph-theoretically based, method approach to the causal ordering of SVARs. A comparison of these methods with the PC algorithm would be interesting but beyond the scope of this article.

The number of acyclical models is huge. An estimate can be formed by considering that each is nested in a just-identified model. If there are \(n\) variables, then there are \(n\)! such models. Consider the \(\mathbf{A}_{0}\) matrix for each such model with the variables ordered such that there are zeroes above the main diagonal. Then every possible acyclical model nested in that just-identified model places either places a zero or a nonzero value in the cells below the main diagonal. There are \(n(n - 1)/2\) such cells, so there are \(2^{n(n-1)/2}\) combinations for each just-identified model. For the 12-variable systems reported below, an upper bound is given by 12!(\(2^{66}\)) = \(3.53 \times 10^{28}\) models. This is an overestimate of the number of models, since some overidentified models are nested in more than one just-identified model and are counted more than once. But a lower bound is the number of combinations nested in a single just-identified model (\(2^{66} = 7.38 \times 10^{19}\))—still a huge number.

More formal and more detailed expositions of these ideas are found in Spirtes et al. (2001) and Pearl (2000). Cooper (1999), Demiralp and Hoover (2003), and Hoover (2005) provide more concise expositions.

The algorithm operates under the assumption of causal sufficiency—namely that any omitted variables cause at most one of the included variables and so may have their effects absorbed in the error terms and constants without altering the dependencies among other variables in the system. While this is a strong assumption, it is no stronger than what is normally made with any SVAR model. Spirtes et al. (2001) have developed algorithms that relax this assumption and detect the effect of omitted latent variables on the independence relations among the included variables.

Demiralp and Hoover (2003) provide Monte Carlo evidence that shows that the PC algorithm is highly effective at recovering the skeleton of the graph of the data-generating process and moderately effective at recovering the directions of individual links, provided that signal-to-noise ratios are high enough. Demiralp et al. (2008) develop and validate a bootstrap procedure to assess the effectiveness of the closely related SGS algorithm.

See http://​www.​federalreserve.​gov/​monetarypolicy/​taf.​htm.

Estimates were conducted using E-views 5.1 econometric software. The actual VAR and final SVAR estimates, as well as tests for VAR lag length, are available on the Hoover’s website: http://​public.​econ.​duke.​edu/​~kdh9/​.

Implementation of the PC algorithm and the bootstrap evaluations use a Gauss program downloadable from the Hoover’s website: http://​public.​econ.​duke.​edu/​~kdh9/​.

By “piecewise” we mean that the algorithm tests pairs and triples of variables in isolation but does not ever test all the variables jointly. To illustrate how low power could result in a bidirectional edge, consider the following case in which our focus is on the bidirectional edge between FFR and CPI: A subgraph of the graph represented in Table 1 is \({ NBORRES}\, \rightarrow \,{ FFR} \leftrightarrow \, { CPI} \leftrightarrow \,{ COMP}\). Starting from the skeleton in which all of the arrows lack heads, in order to orient the first three variables as \({ NBORRES}\, \rightarrow { FFR} \,\leftarrow \, { CPI}\), we must not reject the null \(\rho _{{ NBORRES}, { CPI}}\) = 0, where \(\rho \) is the correlation coefficient, but we must reject the null \(\rho _{{ NBORRES},\, { CPI|FFR}} = 0\), where the vertical (|) indicates conditioning. Suppose that we have done that successfully. To orient \({ FFR} \rightarrow \, { CPI} \leftarrow \, { COMP}\), which would turn the previous single-headed arrow between FFR and CPI into a two-headed arrow, we must not reject the null \(\rho _{{ FFR},\, { COMP}}= 0\), where \(\rho \) is the correlation coefficient, but we must reject the null \(\rho _{{ FFR},\, { COMP}|{ CPI}} = 0\). If our first orientation was correct, then this should not be possible, since COMP causes CPI and CPI causes FFR. It could, however, happen if we made a type II error: our null \(\rho _{{ FFR},\,{ COMP}} = 0\) was in fact false and yet we accepted it.

Complete impulse-response functions are reported on the Hoover’s website: http://​public.​econ.​duke.​edu/​~kdh9/​.

Many studies have included commodity prices among the activity variables. Some order monetary policy variables after activity variables, assuming that monetary policy affects the activity variables only with a lag (e.g., Strongin 1995; Christiano et al. 1996; Giordani 2004); while others order monetary policy variables ahead of activity variables (Sims 1992; Eichenbaum 1992).

We also do not consider other, and possibly more workable, proxies for expectations such as those based on the Federal funds futures market; but see Faust et al. (2004).

The components of the index are: 1. Average weekly hours, manufacturing; 2. Average weekly initial claims for unemployment insurance; 3. Manufacturers’ new orders, consumer goods and materials; 4. Index of supplier deliveries—vendor performance; 5. Manufacturers’ new orders, nondefense capital goods; 6. Building permits, new private housing units; 7. Stock prices, 500 common stocks; 8. Money supply, M2; 9. Interest rate spread: 10-year Treasury bonds less federal funds; 10. Index of consumer expectations.