Infinite-variance, alpha-stable shocks in monetary SVAR

Abstract

This paper outlines a theory of what might be going wrong in the monetary SVAR (structural vector autoregression) literature and provides supporting empirical evidence. The theory is that macroeconomists may be attempting to identify structural forms that do not exist, given the true distribution of the innovations in the reduced-form VAR. This paper shows that this problem occurs whenever (1) some innovation in the VAR has an infinite-variance distribution and (2) the matrix of coefficients on the contemporaneous terms in the VAR's structural form is nonsingular. Since (2) is almost always required for SVAR analysis, it is germane to test hypothesis (1). Hence, in this paper, we fit α-stable distributions to the residuals from 3-lag and 12-lag monetary VARs, and, using a parametric-bootstrap method, we reject the null hypotheses of finite variance (or equivalently, α = 2) for all 12 error terms in the two VARs. These results are mostly robust to a sample break at the February 1984 observations. Moreover, ARCH tests suggest that the shocks from the subperiod VARs are homoskedastic in seven of 24 instances. Next, we compare the fits of the α-stable distributions with those of t distributions and a GARCH(1,1) shock model. This analysis suggests that the time-invariant α-stable distributions provide the best fits for two of six shocks in the VAR(12) specification and three of six shocks in the VAR(3). Finally, we use the GARCH model as a filter to obtain homoskedastic shocks, which also prove to have α < 2, according to ML estimates.

Keywords:

• vector autoregression
• Lévy-stable distribution
• infinite variance
• monetary policy shocks
• structural identification

JEL Classifications:

• C320
• C460
• C500
• E300
• E520

Acknowledgements

The author wishes to thank the managing editor of this journal, along with Olivier Blanchard, James Stock, and anonymous referees for comments that led to many substantial improvements in the paper. In addition, he thanks John Nolan for answers and advice, and those who attended a seminar at the Levy Institute for their useful comments and suggestions. The author is also grateful to Michael Woodroofe for guidance on Woodroofe's research. Discussions with Greg Colman and Kenneth Hannsgen led to improvements in some of the explanations and arguments presented in this and the earlier papers. Although grateful to each of the scholars mentioned above, the author has, of course, not followed all of their advice throughout this paper. The author thanks the Levy Institute for the opportunity to issue a version of this paper as number 682 in its working paper series. An earlier version of the paper appeared as Working Paper no. 596, which was in turn a greatly revised and expanded version of Working Paper no. 546 (Hannsgen 2008).

Notes

1. Two key reference works that cover SVAR are Lütkepohl (2006, especially 357–86) and Hamilton (1994, especially 324–40). Watson (1994) is an early handbook article on VARs, and SVARs in particular, while Christiano, Eichenbaum, and Evans (1999) and Stock and Watson (2001) are surveys that emphasize applied SVAR work in macroeconomics. Qin (2010) surveys VAR research since the late 1970s, providing an historical account of the ‘rise of VAR modeling approach,’ and Sims (2010) provides a retrospective on the SVAR literature.

2. The bias of some tail-index estimators is often very large for stable distributions with α > 1.5.

3. The fact that the parameters lie on the boundary of the parameter space does not preclude a valid bootstrap, because we test only an equality restriction (Andrews 2000).

4. Stable distributions were largely discovered by Paul Lévy (1925). Two references on stable distributions and processes are Samorodnitsky and Taqqu (1994). More applied introductions can be found in Adler, Feldman, and Taqqu (1998), Borak, Misiorek, and Weron (2011), Embrechts, Klüppelberg, and Mikosch (1997), Nolan (forthcoming), and Rachev and Mittnik (2000). Econometric results and issues involving stably distributed variables are discussed in Rachev, Kim, and Mittnik (1999a, 1999b).

5. Many studies make more specific distributional assumptions about the disturbance term η_t , especially for maximum likelihood estimation (Hamilton 1994, 291–302). Also, E(η_tη_t ′) is sometimes assumed to be an arbitrary diagonal matrix D with strictly positive diagonal elements, rather than the identity matrix (Bernanke 1986; Sims 1986).

6. The stability condition requires that the characteristic roots of the system (1) lie within the complex unit circle.

7. An instrumental-variables estimator for SVARs with long-run restrictions is presented in Shapiro and Watson (1989). Proposition 1 below applies to this case as well.

8. In addition, if more than one innovation has infinite variance, some off-diagonal entries in the variance-covariance matrix will be infinite.

9. Another implication of infinite variance time series is that standard estimators will generally be inefficient. Robust estimation for stable models is a complex subject; see Note 4 for some references. Moreover, bootstraps for impulse response functions can fail when the shocks have thick tails (Kilian 1998). Athreya (1987) also discusses problems with the bootstrap under infinite variance.

10. This period does not precisely correspond to the sample period, because of the use of presamples for all VAR estimates reported in this paper. See notes below Table 3.

11. This commodity price index is generally included in monetary VARs for the reasons discussed in Sims (1992) and elsewhere in the subsequent literature.

12. The NBR variable, described below, fell to negative levels in January 2008, making the log transformation impossible. The decline began with a sharp fall in the previous month. A somewhat arbitrary decision was made to truncate the sample so as to omit the entire episode, rather than including one part of it but not another.

13. Differencing all of the variables in equation (2) or transforming equation (2) to a VECM would not affect the α of a VAR error term with a stable distribution, because a linear combination of α-stable variables is α-stable (Samorodnitsky and Taqqu 1994, 2).

14. A typical element of this latter matrix is where f is the likelihood function and θ_i is an element of the stable parameter vector θ = (α, β, γ, δ) (Nolan 2001, 384).

15. For this use of the term ‘pointwise consistent,’ see Greene (1993, 309). The superconsistency of the ML stable-parameter estimator is covered in DuMouchel (1983). Lanne and Lütkepohl (2008b, 7) use a similar two-step ‘quasi-ML’ procedure in the context of a similar problem.

16. We found that using the McCulloch (1997) Monte Carlo critical values for the LR test would have resulted in a large number of additional rejections of our null hypotheses, compared with our actual bootstrap LR test.

17. The stability condition was not met by some of our subsample VARs. Some had one or two roots just outside the unit circle: for the three-lag specification, the 1959:1–1984:1 and for the 12-lag specifications, the 1959:1–1984:1 subperiod. Standard diagnostics were satisfactory in all cases.

18. Hall and Yao (2003) is a recent study of GARCH estimation in the presence of heavy tails.

19. The Anderson-Darling and Kolmogorov-Smirnov measures of goodness-of-fit are somewhat standard. The formulas for these criteria can be found in Rachev and Mittnik (2000, 163).

20. The formula for the abscissa in Michael’s stabilized P-P plots is t_i = (2/π)arcsin(((i–0.5)/n)^0.5), and the ordinate can be found using s_i = (2/π)arcsin(((F(x_i ))^0.5) where x_i is the ith highest observation and F(.) is the estimated cumulative distribution function (Michael 1983, 12).