To difference or not to difference: a Monte Carlo investigation of inference in vector autoregression models

International Journal of Data Analysis Techniques and Strategies

Volumn 1, Issue 3, 2009, Pages 242-274

  Ashley, Richard A ^a   Verbrugge, Randal J ^b  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

^b BUREAU OF LABOR STATISTICS   (United States)

Author keywords

differencing; impulse response function; unit root; VAR model; vector autoregressions

Indexed keywords

EID: 77954213774     PISSN: 17558050     EISSN: 17558069     Source Type: Journal    
DOI: 10.1504/IJDATS.2009.024295     Document Type: Article

References (58)

1. Ashley, R. and Patterson, D.M. (2008) ‘Apparent long memory in time series as an artifact of a time-varying mean’, Unpublished manuscript, http://ashleymac.econ.vt.edu/working_papers/long_memory.pdf.
2. Ashley, R. and Verbrugge, R. (2009) ‘Frequency dependence in regression model coefficients: an alternative approach for modeling nonlinear dynamic relationships’, Econometric Reviews, forthcoming, http://ashleymac.econ.vt.edu/working_papers/freq_depend.pdf.
3. Bernanke, B.S., Gertler, M. and Watson, M. (1997) ‘Systematic monetary policy and the effects of oil price shocks’, Brookings Papers on Economic Activity, pp.91–157.
4. Christiano, L.J., Eichenbaum, M. and Evans, C. (1996) ‘The effects of monetary policy shocks: evidence from the flow of funds’, Review of Economics and Statistics, Vol. 78, No. 1, pp.16–34.
5. Cogley, T. and Nason, J.M. (1995) ‘Effects of the Hodrick-Prescott filter on trend and difference stationary time series: implications for business cycle research’, J. Economic Dynamics and Control, Vol. 19, pp.253–278.
6. Diebold, F.X. and Kilian, L. (2000) ‘Unit root tests are useful for selecting forecasting models’, Journal of Business and Economic Statistics, Vol. 18, No. 3, pp.265–273.
7. Granger, C.W.J. and Hyung, N. (2004) ‘Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns’, Journal of Empirical Finance, Vol. 11, pp.399–421.
8. Granger, C.W.J., Hyung, N. and Jeon, Y. (2001) ‘Spurious regressions with stationary series’, Applied Economics, Vol. 33, pp.899–904.
9. Granger, C.W.J. and Newbold, P. (1977) Forecasting Economic Time Series, New York: Academic Press.
10. Harvey, A.C. and Jaeger, A. (1993) ‘Detrending, stylized facts and the business cycle’, J. Applied Econometrics, Vol. 8, pp.231–247.
11. Hodrick, R. and Prescott, E.C. (1980) ‘Post-war U.S. business cycles: an empirical investigation’, Mimeo, Carnegie-Mellon University, Pittsburgh, Pennsylvania.
12. Jensen, M.J. and Liu, M. (2006) ‘Do long swings in the business cycle lead to strong persistence in output’, Journal of Monetary Economics, Vol. 53, No. 3, pp.597–611.
13. Kilian, L. (1998) ‘Small-sample confidence intervals for impulse response functions’, Review of Economics and Statistics, Vol. 80, No. 2, pp.218–230.
14. Kilian, L. and Chang, P. (2000) ‘How accurate are confidence intervals for impulse response functions in large VAR models?’, Economics Letters, Vol. 69, pp.299–307.
15. King, R.G. and Rebelo, S.T. (1993) ‘Low frequency filtering and real business cycles’, J. Economic Dynamics and Control, Vol. 17, pp.207–231.
16. Kurozumi, E. and Yamamoto, T. (2000) ‘Modified lag augmented vector autoregressions’, Econometric Reviews, Vol. 19, No. 2, pp.207–231.
17. Lin, J.L. and Tsay, R. (1996) ‘Cointegration constraints and forecasting: an empirical examination’, Journal of Applied Econometrics, Vol. 11, pp.519–538.
18. Lütkepohl, H. (1990) ‘Asymptotic distributions of impulse response functions and forecast error decompositions of vector autoregressive models’, Review of Economics and Statistics, Vol. 72, pp.116–125.
19. Ohanian, L.E. (1988) ‘The spurious effects of unit roots on vector autoregressions: a Monte Carlo study’, Journal of Econometrics, Vol. 39, pp.251–256.
20. Ohanian, L.E. (1991) ‘A note on spurious inference in a linearly detrended vector autoregression’, Review of Economics and Statistics, Vol. 73, No. 3, pp.568–571.
21. Pesavento, E. and Rossi, B. (2006) ‘Impulse response function confidence intervals for persistent data: what have we learned’, Journal of Economic Dynamics and Control, Vol. 31, No. 7, pp.2398–2412.
22. Phillips, P.C.B. (1986) ‘Understanding spurious regressions in econometrics’, J. Econometrics, Vol. 33, pp.311–340.
23. Phillips, P.C.B. (1995) ‘Fully modified least squares and vector autoregression’, Econometrica, Vol. 63, pp.1023–1078.
24. Plosser, C.I. and Schwert, G.W. (1977) ‘Estimation of a non-invertible moving average process’, Journal of Econometrics, Vol. 6, pp.199–224.
25. Plosser, C.I. and Schwert, G.W. (1978) ‘Money, income, and sunspots: measuring economic relationships and the effects of differencing’, Journal of Monetary Economics, Vol. 4, pp.637–660.
26. Sims, C.A. (1980) ‘Macroeconomics and reality’, Econometrica, Vol. 48, No. 1, pp.1–48.
27. Sims, C.A., Stock, J.H. and Watson, M.W. (1990) ‘Inference in linear times series models with some unit roots’, Econometrica, Vol. 58, pp.113–144.
28. Sims, C.A. and Zha, T. (1999) ‘Error bands for impulse responses’, Econometrica, Vol. 67, pp.1113–1155.
29. Swanson, N.R., Ozyildrim, A. and Pisu, M. (2003) ‘A comparison of alternative causality and predictive accuracy tests in the presence of integrated and co-integrated economic variables’, in D.E.A. Giles (Ed.) Computer-Aided Econometrics, London: Routledge, pp.91–148.
30. Toda, H.Y. and Phillips, P.C.B. (1993) ‘Vector autoregressions and causality’, Econometrica, Vol. 61, pp.1367–1393.
31. Toda, H.Y. and Phillips, P.C.B. (1994) ‘Vector autoregressions and causality: a theoretical overview and simulation study’, Econometric Reviews, Vol. 13, No. 2, pp.259–285.
32. Toda, H.Y. and Yamamoto, T. (1995) ‘Statistical inference in vector autoregressions with possibly integrated processes’, J. Econometrics, Vol. 66, pp.225–250.
33. Yamada, H. and Toda, H.Y. (1998) ‘Inference in possibly integrated vector autoregressive models: some finite sample evidence’, J. Econometrics, Vol. 86, pp.55–95.
34. See also Plosser and Schwert (1977; 1978). Phillips (1986) provides a firm theoretical basis for Granger and Newbold’s results by showing that limiting distributions do not exist for the estimated t ratio in such circumstances so even attempts to adjust the critical values are asymptotically unjustified in this context.
35. However, a higher R2 does not even imply a better fit to the data since the dependent variables are different for the VAR-in-differences and VAR-in-levels models. Note also that R2 does not even necessarily admit of an interpretation as proportion of variance explained in the VAR-in-levels model, since the variance is not finite for an I(1) level series.
36. This is, of course, despite the published work (e.g., Harvey and Jaeger, 1993; King and Rebelo, 1993; Cogley and Nason, 1995) showing that HP filtering can severely distort estimated dynamic relationships.
37. See also Kurozumi and Yamamoto (2000). The related FM-VAR approach suggested by Phillips (1995) and the sequential ECM approach suggested by Toda and Phillips (1994) are not implemented here based on simulation results presented in Yamada and Toda (1998) indicating that the lag-augmented VAR (‘LA-VAR’) approach is better-sized in small samples.
38. Data generating processes with feedback are not explicitly considered because including such terms did not materially affect the results quoted below. Note that equations for generating both x(t) and y(t) are needed, but the focus below is entirely on the estimated equations for y(t).
39. t.
40. HP filtering software written by E. Prescott was obtained from the Quantitative Macroeconomics and Real Business Cycle website (http://ideas.uqam.ca/QM RBC); the smoothing parameter was set to the standard value of 1600.
41. t-3.
42. The modified lag-augmented VAR procedure proposed by Kurozumi and Yamamoto (2000) might provide still better results.
43. This is not to say that unit root testing may not be useful in other contexts. For example, Diebold and Kilian (2000) find such testing helpful in choosing forecasting models. Nor – as becomes apparent from the results of Section 3.3 below – does it imply that differenced-based models are necessarily preferable for producing confidence intervals on impulse response functions in models for which Granger causality is already known to be present.
44. Note, however, that the results in Section 3.1 above rationalise the existence of this belief without validating it, since the apparent power of an over-sized test will be artificially inflated.
45. This fraction is meaningless for the tests based on the models estimated in levels or in HP-filtered levels since, in these cases, the actual size of a nominally 5% test is substantially different from .05. The parameter ψ explicitly appears in DGP I, DGP II, and DGP III in Section 2.
46. All three estimation methods have approximately the same power (ca. .05) for the generating process where xt adjusts to yt (DGP IV of Section 2) since xt does not Granger cause yt in that case.
47. Bayesian interval estimation (as in Sims and Zha, 1999) is beyond the scope of the present paper. We also note that both methods are only asymptotically justified; the relevant issue is the relative and absolute effectiveness of each method using the finite sample lengths considered here.
48. Numerical precision issues also arise in calculating ∂ Ωyεj / ∂Ayy1, say, for large j since terms like Ωyεj involve (Ayy1)j.
49. See Kilian (1998) for details on how this bias correction is performed; for example, the correction is explicitly adjusted so that it never makes an originally stationary model nonstationary.
50. In fairness, however, we should note that this lag limitation in our results may cause us to understate the superiority of the bootstrap confidence intervals over those obtained using the asymptotic standard errors.
51. Estimated models involving differenced variables were re-written (post-estimation) in terms of the levels of the variables. The estimated sampling distributions of Ayy1… Axx2 were obtained based on the estimated sampling distributions of the coefficients in the equations actually estimated. Tables 6 and 7 are based on parameter values of ψ = .15 and ρ = 0.
52. We cannot rule out the possibility that the bootstrap is substantially superior at larger lags. Also, results which we do not report in these tables overwhelmingly support the contention of Kilian (1998) that bias-corrected bootstrap intervals are definitely preferable to ordinary bootstrap intervals.
53. Over differencing is also consequential for long term forecasting – see Lin and Tsay (1996).
54. In results not reported in Tables 6 and 7, we find very similar coverage results for ψ = 0; thus, this result is also robust with respect to whether or not this explanatory variable was erroneously included in the model.
55. Note that the dependent variables in lag-augmented VAR model equations are in levels and that each explanatory variable also appears in level form at the largest lag, so it is not appropriate to think of the lag-augmented VAR as a ‘differenced’ model.
56. Bayesian interval estimation – e.g., Sims and Zha (1999) – is beyond the scope of the present paper.
57. Users of HP filtering sometimes imply that their filtering choice is justified by an interest in relationships ‘only at business cycle frequencies’. If one really believes that the relevant relationships are frequency-dependent, then regression methods which explicitly allow for this – e.g., Ashley and Verbrugge (2009) – should be employed.
58. This conclusion is consistent with the results of Kilian and Chang (2000), who look at impulse response function confidence interval coverage for several higher dimensional VAR models (see also Pesavento and Rossi, 2006).