Abstract
In this chapter vector time series models are considered for stationary processes. There is a brief discussion of stationarity, but we leave the reader to refer for further detail to Patterson (2010) and (2011). The models are decomposed into VAR, VMA and mixed models with both characteristics (VARMA). The conventional classical assumptions will be considered and related to the likelihood function and the regression and maximum likelihood estimators. The notion of cointegration is not considered in this chapter, but the reparameterization into error correction form related to persistence is. Often these models are analysed in terms of the causal structure, and it is also possible to consider parameter stability and the related subject of exogeneity. The latter topics will be handled in more detail in Chap. 5 when long-run behaviour and short-run behaviour are considered. VAR in particular is often seen as a tractable reduced form (RF) of a rational expectation model (Sims 1980). In association with VAR, macroeconomic theory is linked to the behaviour of the impulse response function. Here the uniqueness of the impulse response function and the various approaches to handle it are considered. Time series models are also useful for forecasting, which has been a key rationale for their construction. Forecasting stationary VAR and VARMA processes is relatively straightforward when the series are all stationary. However, the problem is open to significant debate when non-stationarity and cointegration are considered; this is considered further in Chap. 5. There is also a more detailed treatment by Clements (2005), while the interested reader is directed to Clements and Hendry (2011).
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Notes
- 1.
In fact the sequence of a random variable underlying the time series observations is referred to as a stochastic process, and it is the stochastic process that is labelled “stationary”.
- 2.
- 3.
All white noise in this book is viewed as zero mean.
- 4.
The detailed theory draws a distinction between two components of a time series: that which is perfectly predictable from its own past, called a deterministic component, and that which cannot be perfectly predicted from its own past. A purely non-deterministic process has no component that can be predicted from its own past, and it is this type of series to which this very abbreviated version of the theorem refers.
- 5.
In addition, ε t is uncorrelated. With future values of the process, x t+j , j > 0.
- 6.
The AR(1) process x t = ϕ x t−1 +ε t will have one root given by λ 1 = ϕ −1.
- 7.
Time series identification is often considered in terms of the autocorrelation and partial autocorrelation function (Box and Jenkins 1976). The latter derives from the parameters of a sequence of AR models. One process arises when the Hannan-Rissanen method is applied where the data are filtered using a high order AR model which can be used to estimate MA and ARMA models. These regression models are amenable to comparison.
- 8.
Preserving the ordering so the inverse operator is the premultiplying factor on the right-hand side of (2.21) is not necessary in the univariate case, but is good practice since in the multivariate case discussed subsequently it is important.
- 9.
There may be other forms of deterministic components, while when the underlying data has been transformed to produce stationarity it is possible the parameter μ may be given the alternative interpretation as a drift coefficient in relation to the original data that may trend upwards over time. Other terms that may be included to render the model stationary would be deterministic trends (t) and polynomial terms in t such as their square. There may also be intervention dummies that capture anomalous observations or shifts in deterministic components such as the intercept or trend.
Such features may also impact on the structure of the variance of a model. For example see the discussion in Diebold (1986).
- 10.
Such polynomials are often used to characterize the dynamic behaviour of economic processes subject to adjustment (Chiang and Wainwright 2005). One may consider these to be fundamental reduced forms.
- 11.
Once it is understood that the series may require differencing to render them stationary, then the next step is to undertake a test of trend stationarity. The Dickey-Fuller test model is augmented by a deterministic trend and, should the series be found to be stationary relative to the appropriate critical value, the dependent variable can be specified without differencing; estimation follows in the usual manner with the inclusion of the trend variable. This is an ordered procedure once non-stationarity is decided on and whether it is captured via differencing the data or by capturing a trend or further components.
- 12.
A similar argument arises in the case of the maximum likelihood (ML) estimators that derive from the solution to an inversion of the MA polynomial \( \tilde{\varTheta}(L)\). The concentrated likelihood in this case is:
$$\displaystyle{ L = c + NT\log (\tilde{\Sigma }) }$$where \(\tilde{\Sigma } =\) \(\tilde{\epsilon }_{t}\) \(\tilde{\epsilon }_{t}^{{\prime}}\) and \(\tilde{\epsilon }_{t} =\tilde{ \Theta }(L)^{-1}\tilde{\Gamma }\left (L\right )x_{t}-\tilde{\mu }\). Phadke and Kedem (1978) for example use a decomposition of \(\tilde{\Theta }(L) = M(L)Q(L)M(L)^{{\prime}}\), and similar results can be derived using unimodular matrices (see Chap. 4 for further details), where the exact form of the estimations relies on the normality assumption, which is not a requirement of what may be seen as a quasi-maximum likelihood or multi-step regression approach.
- 13.
Similar results are obtained under cointegration by Hunter and Dislis (1996).
- 14.
This is implemented in Microfit 5 (Pesaran and Pesaran 2009). While Bahram Pesaran also made the same process operational in the multivariate context in his PhD.
- 15.
Extra terms can be included when the series are cointegrated: such transformations are used to estimate the forwarded looking model in Dunne and Hunter (1998).
- 16.
The interested reader might note the early application of time series methods by Wold and Jureen (1953). In the macro-context dynamic simulated general equilibrium models are popular, though in econometric terms they have not performed well.
- 17.
In this case the VAR in restricted form is estimated as a weighted least squares (WLS) problem, where appropriate, by seemingly unrelated (SUR) methods.
- 18.
- 19.
- 20.
- 21.
It also follows from (2.49) that the shocks are uncorrelated over time, which is implicit in the VAR structure but would not be in the case of a VMA.
- 22.
This issue is a manifestation of the wider property of observational equivalence. Any non-singular linear transformation of the VAR is observationally equivalent. This goes back to the Blaschke matrices that led (Lippi and Reichlin 1994) to emphasize fundamentalness.
- 23.
See Doornik and Hendry (2009) for discussion of what is meant by causality in economics and econometrics in relation to the long and the short run as well as the empirical and in theory.
- 24.
However, it should be noted that the maximum likelihood estimator of structural models may have no finite sample moments and fat tails asymptotically (Sargan 1988).
- 25.
Here for simplicity we will not look at the mean: the series may be zero mean or the data may be prefiltered to account for constants and deterministic variables (Hendry 1995).
- 26.
In the VAR(1) case the inversion is straightforward as \(\Gamma (L) = (I - \Gamma _{1}L)\) and \(\Gamma \left (L\right )^{-1}\Theta ^{\infty }(L) = I.\) In general the terms in \(\Theta _{i}\) follow from recursive substitution at each lag for terms in \(\Gamma (L).\) The VAR(1) leads to a precise solution as \(\Gamma (L)^{-1} = \frac{1} {(I-\Gamma _{1}L)} =\sum _{ i=0}\nolimits^{\infty }\Gamma _{1}^{i}\) and matching terms in i implies that \(\Theta _{i} = \Gamma _{1}^{i}.\) In practice the number of recursions relate to the size of the roots of the VAR. The nearer to unity the larger the number of terms in \(\Theta _{i}\) that need to be included. Therefore \(\Theta ^{\infty }(L) = (I - \Theta _{1}L\ldots - \Theta _{T}L^{T}\ldots.)\) will be truncated at some point such as T, where the sample is used up, or at the point in the VAR(1) case, where the spectral radius of \(\rho (\Gamma _{1}^{i}) \approx 0.\)
- 27.
Similar results flow from the law of iterated predictions and similar results can be found in Whittle (1983).
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Hunter, J., Burke, S.P., Canepa, A. (2017). Multivariate Time Series. In: Multivariate Modelling of Non-Stationary Economic Time Series. Palgrave Texts in Econometrics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-137-31303-4_2
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