Abstract
If we take the notion of general equilibrium seriously, then everything in the economy is related to everything else. For this reason, it is impossible to say which variable is exogenous. It is possible that all variables are endogenous: they can all be caused by, and simultaneously be the cause of, some other variable.
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Notes
- 1.
It was found not to be the case. There are many hidden assumptions in VARs; the researcher cannot stand outside the research process, even in the case of VARs.
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As of June 2017, it has been cited over 11,700 times.
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The book by Amisano and Giannini (2012) is considered a definitive guide to SVARs.
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Runkle (1987) emphasizes the importance of including and properly calculating confidence intervals when reporting IRFs and FEVDs.
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As examples, Hatemi-J (2003) proposes a lag-selection criterion that simply averages the SIC and HQIC; the theory is that this is a straight-forward approach that is good enough for general use. Other approaches are tailored to more specific uses. Schorfheide (2005), for example, proposes a final prediction error approach to lag selection when the goal is multi-step forecasting.
There is no requirement that the number of lags in a VAR needs to be constant across equations or variables. Ignoring irrelevant parameters means that the remaining parameters can be estimated more efficiently. Precisely estimated coefficients produce better IRFs and better forecasts. Hsiao (1979, 1981) developed a fully asymmetric VAR model to specifically address Sims’ money/income causality question. Keating (2000) explores this concept within a class of VARs where each variable takes on different lags, but the lag structure is the same across equations. The AIC is known to select the correct symmetric lag lengths better than the other commonly used alternatives. Keating develops an alternative to the AIC for asymmetric lag-length selection. In a Monte Carlo simulation, Ozcicek and McMillin (1999) examine the small-sample performance of the standard IC and Keating’s versions of these. They find that the KAIC more frequently identified the correct number of asymmetric lags than did the other information criteria, and had good forecasting properties. Ozcicek and McMillin (1999) conclude that the AIC and KAIC should be used over SIC when forecasting; their results are reversed when the IRFs are the focus of the study.
Ivanov et al. (2005) review much of the literature and conduct extensive Monte Carlo tests of lag-order’s effect on IRFs. Their findings are sensitive to the observation frequency, with monthly data preferring AIC and quarterly data preferring SBIC and HQIC.
The most obvious conclusion that we can draw from all this is that the field has not yet reached a conclusion. But if we were to offer advice, it would be the following: if you wish to forecast, use AIC or KAIC. If you wish to construct IRFs, then BIC or SICs are preferred. IRFs will fit the data very well when they are fit with lots of lags.
- 7.
It is np × np because each component in matrix (10.11) is, itself, an n × n matrix.
- 8.
We use “stability” and “stationarity” interchangeably. They are not the same thing. However, stability implies stationarity if the error process is stationary. Stability applies to the coefficients affecting the mean; stationarity is a broader concept that also demands that the autocovariances and the error variances do not change over time. Given that we do not address GARCH errors in this chapter, stability is enough to ensure stationarity.
- 9.
Equivalently, the length of a complex vector is equal to the square root of the product of the vector and its complex conjugate: \(\sqrt {\left (r+ci\right )\left (r-ci\right )} = \sqrt {\left (r^2 +rci -rci - {ci}^2\right )} = \sqrt {\left (r^2 + c^2 \right )}\).
- 10.
Granger (1980) provides an interesting discussion on the philosophical nature and various definitions of causality. In that paper, he also generalizes his own definition of causality to include non-linear models, providing a broader operational definition of causality.
- 11.
Sims argued that if X causes Y (and not vice versa), then this should be evident in zero-coefficients in future values of X whenever regressing Y on past, present and future values of X. Granger’s approach was that if X causes Y, then there should be non-zero coefficients when regressing Y on past X and past Y. Ultimately, a string of research proved that the two approaches are identical.
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To follow Sims’ specific technique, we would estimate reg Y L( 0/8) .MB F( 1/4) .MB time quarter and then test F1.MB F2.MB F3.MB F4.MB to see whether MB causes Y. Then, we would estimate reg MB L( 0/8) .Y F( 1/4) .Y time quarter and test F1.Y F2.Y F3.Y F4.Y.
- 14.
Sims arbitrarily chose his lag length. Hsiao (1979, 1981) proposed using Akaike’s FPE to select different lag lengths for each variable and each equation. He estimated such a fully asymmetric-lab VAR model to explore Sims’ money/income causality results. He found bi-directional Granger causality for the US and Canada. Thornton and Batten (1985) replicated Sims’ paper on “Money, Income and Causality,” using different lag lengths chosen by several different selection procedures. Different models give different results, so you should not choose a lag length arbitrarily. Thornton and Batten suggest relying on a lag selection method such as Akaike’s FPE. You should never choose one simply because it gives you the results you were hoping for.
In their review of the money/income causality literature, Stock and Watson (1989) report that adding a deterministic time trend strengthens money’s estimated effect on output. Further, the sample data can affect the results (ex: Eichenbaum and Singleton 1986). (Structural breaks can often be confused with unit roots, leading to inappropriate detrending in the money/income regressions.) Stock and Watson’s (1989) main econometric finding is that the initial method of detrending is responsible for the diverging results; detrending can cause the test statistics to have non-standard distributions. Their main economic finding is that shocks to the money growth rate that are greater than those predicted by the trend do have an effect on output.
Hall (1978) is also notable, reminding researchers that permanent rather than transitory income matters, so simple regressions of consumption on past income conflate two different effects. Dickey et al. (1991) revisited the money/income question, with interest rates added, in the context of cointegration analysis.
- 15.
This presumes that the lag-structure is correctly specified in the VAR.
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Levendis, J.D. (2018). Vector Autoregressions I: Basics. In: Time Series Econometrics. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-98282-3_10
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