In this paper we overview the literature on common features analysis of economic time series. Starting from the seminal contributions by
Economic time series could be characterized by several features such as trends, cycles, seasonality, serial correlation, and so on. When a set of series possesses the same type of feature, it could be the case that a linear combination of them does not necessarily possesses the feature: this is the most interesting case, for which Engle and Kozicki (1993) provided the following definition: “A feature, which is present in each of a set of series, is said to be common to those series when there exists a nonzero linear combination of these series that does not have the feature”. A well known example of common features is cointegration (Engle and Granger, 1987; Johansen, 1988): a group of series that possesses stochastic trends is cointegrated when there are some linear combinations of the variables that are stationary, i.e. do not have stochastic trends. Nowadays, there is a huge collection of special cases of common features. A comprehensive, although still partial, list includes: codependence (Gourieroux and Peaucelle, 1988; Vahid and Engle, 1997) and the scalar component model (Tiao and Tsay, 1989), when a linear combination of variables possesses shorter memory than individual series; common serial correlation (Engle and Kozicki, 1993; Vahid and Engle, 1993), when a linear combination of serially correlated series is an innovation w.r.t. the past; cotrending (Chapman and Ogaki, 1993), when a linear combination of trend-stationary time series no longer displays deterministic trend; common volatility (Engle and Kozicki, 1993; Engle and Susmel, 1993), when a linear combination of conditionally heteroskedastic time series eliminates conditional heteroskedasticity; seasonal cointegration (Hylleberg
Common features among economic time series are often predicted by economic theory. For example, in King
The rest of the paper is organized as follows. In Section 2, after introducing the general notion of common features and linking it to the reduced-rank regression model, we focus on the various forms of common cyclical features and their implication in terms of common short-run components. We also illustrate similarities and differences between these approaches and other popular types of multivariate time series modelling. Section 3 takes into account the consequences of the presence of common features for the univariate representation of multiple time series. Section 4 deals with the estimation methods of the models implied by the various form of common features, distinguishing between the cases of small systems and medium-large systems. Finally, Section 5 draws some conclusions.
In this section we first present and discuss the general notion of common features that was originally proposed by Engle and Kozicki (1993). We stress the link between the presence of common features and the multivariate Reduced-Rank Regression model (RRR) that was introduced by Anderson (1951). A detailed survey on this modelling may be found in Reinsel and Velu (1998). Then we focus on the common autocorrelation feature and its interplay with the notion of common cycles in the multivariate Beveridge and Nelson (1981) decomposition. Starting from the seminal work by Vahid and Engle (1993), we illustrate the various forms of common cyclical features that have been proposed in the literature and their implication in terms of common unobserved components. Finally, we illustrate similarities and differences between the common serial correlation approach and other types of multivariate time series modelling that are popular in statistics and econometrics, such the dynamic factor model (see, e.g., Stock and Watson (2011) and the references therein) and the multivariate autoregressive index model (Reinsel, 1983).
Engle and Kozicki (1993) considered features that satisfy the following axioms:
The vector series
If two
If
Then, any dynamic property of the data could be viewed as a special case of feature: for example, series with stochastic trends satisfies all axioms. As pointed out by Engle and Kozicki (1993) a linear combination of two series that both have the feature does not necessarily possess the feature. This is the most interesting case, and to this issue Engle and Kozicki gave particular attention by the following definition:
From a statistical point of view, the existence of such linear combinations can be linked to a common factor representation. Indeed, let
where (
The main idea is that a small number of unobserved components possess a given feature and transmit it to a larger set of time series. It is then possible to combine such time series in order to cancel the influence of these unobserved components, thus removing the Common Feature (CF) from the data.
In order to illustrate the connection between CF’s and RRR, let us assume that
where
Moreover, assume that:
Variables
There exists a
In view of the above assumptions we get
that is equivalent to
where
In view of
The cofeature matrix
Obtain the partial regression model
where
for two generic stationary time series
Solve the following maximization problem
Then we get the solution
Note that R^{2}(
Solve the following maximization problem
for
We get the solution
and
Since
Since
Finally, the loading matrix
It is worth remarking that the eigenvalue problem
is equivalent to finding the roots of the equation:
with the normalization
One easily recognizes that the solution of the problem (
A very well known example of CF is cointegration (see, e.g., Johansen (1996) and the references therein), where linear combinations of series having nonstationary stochastic trends feature are stationary.
Let us assume that the elements of a
where
When variables
where
Since Δ
where
By expanding
where
Since we know from the Engle-Granger representation theorem (Engle and Granger, 1987; Johansen, 1996) that
Note the analogy between (
However, detrended economic time series often display clear evidence of comovements (Lucas, 1977), which cannot be due to cointegration, thus suggesting the presence of common cycles. If this is the case, we expect that there exist linear combinations of cyclical series the are not cyclical. This Common Cyclical Feature (CCF) (Engle and Kozicki, 1993; Vahid and Engle, 1993) can be interpreted as short-run equilibrium relationships, similarly to the interpretation of the cointegration relations as long-run equilibrium.
From the multivariate BN decomposition (
where
While the presence of cointegration implies reduced-rank restrictions on the VECM parameters that are responsible for the long-run behavior of series
However, differently from cointegration, there is not a unique notion of common short-run components. Indeed, also the degree of synchronicity of the common cycle plays a role in the definitions. Alternative notions of CCF impose differing reduced-rank structures to the VAR. Let us briefly review various form of CCF starting with the seminal notion proposed by Engle and Kozicki (1993).
Series Δ
where
Cubadda and Hecq (2001) propose the notion of polynomial serial correlation common features (PSCCF) as a measure of non-contemporaneous cyclical comovements. Non-synchronous common cycles arises, for example, in economic model of consumption with several types of consumer goods as,
By definition, series Δ
, and the VECM (
where
In order to interpret the notion of PSCCF, Cubadda and Hecq (2001) show that there exists a first-order polynomial matrix
Hence, PSCCF requires that there exists a first-order polynomial matrix
In the above definitions of CCF, the number of SCCF’s or PSCCF’s,
where
In order to uncover interesting implications of the WF for the BN cycle, Cubadda (2007) shows that there exists a first-order polynomial matrix
As consequence, since
A limitations of the above methods for cyclical features analysis is that they cannot handle the possible coexistence of differing types of reduced-rank restrictions in the same vector. In order to overcome this limitation, Cubadda (2007) introduced the notion of weak form of PSCCF (WFP), which encompasses most of the existing formulations: series Δ
where
The WFP requires the existence of a second-order polynomial matrix
An important implication of the WFP is that the polynomial matrix
The CCF analysis was extended even to the case of series having different forms of stationarity than I(1)-ness. In particular, Cubadda (1999, 2001) explored the presence of common cycles in seasonal time series that are also integrated at (a subset of) the seasonal frequencies, whereas Paruolo (2006) focused on the case of I(2) systems.
Franchi and Paruolo (2011) offered a comprehensive theoretical analysis of the conditions of existence of the various form of CCF’s and of the characterization of the CCF relations in I(0), I(1) and I(2) systems.
It is interesting to analyze similarities and differences of the CF approach with other popular multivariate time models. For the sake of simplicity, we will refer within this subsection to the basic SCCF model, which can be formulated as
where series
Ahn and Reinsel (1988) proposed a variant of the basic RRR model that is called Nested Reduced- Rank AR model (NRRAR). The main assumption is that the VAR coefficient matrices have reduced ranks, which are nested each other: Rank(
The NRRAR is a very general statistical model since it is easy to see that both the SCCF and its polynomial extensions are particular cases of
A different modelling, which is also endowed with a reduced-rank structure, is the Multivariate Autoregressive Index model (MAI) as originally proposed by Reinsel (1983). The basic version of the MAI reads
where
Notice that the regression coefficient matrix implied by the MAI has the following structure
which implies that
Notwithstanding both SCCF and MAI have a reduced-rank structure, the mathematical properties of these two modelling approaches are only partially similar. Indeed, although the interpretation of the canonical variates
A specific property of the MAI is that the indexes themselves follow a VAR(
whereas linear combinations of series generated by an unrestricted VAR(
A different approach that gained large popularity is the Dynamic Factor Model (DFM), see e.g. Stock and Watson (2011) and the references therein. As shown by Stock and Watson (2005), the DFM can be represented in the VAR form as follows
where
A key difference between the DFM and the previously considered approaches is that the former requires for inferential purposes that the number of series
Apart from the different asymptotic frameworks, the DFM and the MAI have some degree of similarity in their mathematical formulations. Indeed,
It is less obvious how to relate the DFM to the SCCF. One may notice that in the particular case that
It is well known that each series generated by a VAR process admits a univariate ARIMA representation, see e.g. Zellner and Palm (1974). However, the VAR models that are typically used in macroeconomic analysis would imply highly non parsimonious ARIMA models for individual time-series, whereas low order ARIMA models are empirically appropriate. This is the so-called “autoregressivity paradox”. Cubadda
Indeed, let us assume that the
The so-called Final Equations (FEs) of series
where det[
Since det[
Following Cubadda
where
For the above VAR, the FEs are:
such that individual series follow ARMA(3, 2) models.
However, if
which produces the FEs:
This implies that the univariate representations are parsimonious ARMA(1, 1) models with the same autoregressive parameter and cross-correlated VMA errors having a factor structure.
More generally, Table 1 summaries the reduction of the individual ARMA orders due to common features restrictions.
As one can see, the existence of CCF’s provides a possible, economically meaningful, solution of the autoregressivity paradox. Notice that Table 1 provides the
The presence of short-run comovements has also consequences for the VMA part of FEs. Cubadda
Cubadda
Theoretical FEs orders imply an ARMA(2, 1) processes. However, SCCF test statistics is in favor of
As expected, the AR coefficients are very similar. Moreover, since the estimated cofeature vector
In this section we first illustrate how to estimate the RRR models implied by the various form of CCF’s when the system dimension is small. In this case, a Maximum Likelihood (ML) under the Gaussianity assumption is generally employed. However, ML methods break down when the number of regressors becomes large compared to the typical sample size in macroeconomic datasets (100 ≤
Under the assumptions that series
In particular, the LR test statistic on the existence of
where
The ML estimator of the cofeatures matrix
where
The ML estimator of the coefficient matrix
where
The ML estimator of the loading matrix
The degrees of freedom of the test statistic (
Without loss of generality, we assume that
where
In order to conduct inference on the various forms of CCF’s, a two-step procedure is usually employed (see,
Hence, ML inference on the various forms of common features is obtained by solving CanCor{Δ
For each of the models in Table 2, under the null that
Moreover, optimal estimates of both the common features vectors and (partial) RRR coefficients are then obtained as described in Table 4.
Finally, the remaining parameters of the various RRR models are estimated by OLS after fixing the matrices
Notice that the two step procedures previously illustrated do not maximize the Gaussian likelihood, although the estimated parameters have the same distribution as the optimal ones. Centoni
The merits of imposing CCF restrictions in VAR models are investigated in several Monte Carlo studies. Vahid and Issler (2002) documented the advantages of simultaneously choosing the order
The ML approach discussed so far is typically applied to small scale multivariate models, i.e. in situations where the number of series
The main reason why ML inference performs poorly in high-dimensional systems is that the inversion of large variance-covariance matrices is required. Hence, most of methods that have been proposed to use RRR with large
The first attempt in this direction can be dated back to Vinod (1976), who proposed the canonical ridge model. When applied to the SCCF modelling, this approach would require to substitute the eigenvalues and eigenvectors that are used for CCA with those of the matrix
where
Cubadda and Hecq (2011) suggested a Partial Least Squares (PLS) approach to test and impose the SCCF restriction to a VAR even when CCA is not feasible due a lack of degrees of freedom. PLS are a family of multivariate techniques with the aim of maximizing the covariance between linear combinations of two variable sets, see, e.g. Rosipal and Krämer (2006) for a detailed survey. The idea is to consistently estimate the SCCF matrix
where
In order to consistently estimate the factor weights
where
In order to detect the presence of SCCF in large dimensional systems, Cubadda and Hecq (2011) proposed to replace the condition that a linear combination of variables must be orthogonal to the past, namely
with the one of absence of autocorrelation, namely
where
Having controlled for the overall size of the test when different values of
Carriero
Carriero
Bernardini and Cubadda (2015) proposed to regularize the estimate of the autocorrelation matrix prior on performing CCA. In particular, they suggest to use, in place of the natural estimator, a proper shrinkage estimator of the covariance matrix of
where
Notice that when
Bernardini and Cubadda (2015) document, both by simulations and empirical applications, that RCCA improves both forecasting and estimation of structural parameters over traditional medium-size (
The main goal of this survey was to create a “common thread” between various topics, related to each other by the idea of modelling various forms of comovements that are typically observed in economic time series and often predicted by economic theory. From a statistical point of view, common features imply a reduction to more parsimonious structure such as common factor representation: a small number of unobserved components possesses a given feature and transmits it to a larger set of economic time series. As we have seen, RRR is often the solution to the inferential problem. Due to the large amount of literature on common features, we focused the discussion on common cyclical features. However, we have tried to take into account recent developments as the implications of common features for univariate time series models and the statistical issues that arise when the number of the variables is fairly large.
The main drawback of the methods discussed in this paper is that different features are evaluated separately; one important exception is the unifying framework for analyzing common cyclical features that we discussed in Subsection 2.2. Then, a major goal ahead is to develop estimation and testing procedures that allow for the joint identification of several forms of common features using an integrated statistical approach.
Another challenge for future research is the extension of common cycle analysis to nonstationary medium-large dimensional systems. Indeed, all the methods that are presented in Subsection 4.2 require that series are individually stationarized prior to the multivariate analysis. This is clearly a methodological limitation since possible cointegration relations are ignored. Hence, it is still missing a statistical modelling that allows for the simultaneous analysis of common trends and cycles when the number of time series is not small.
Maximum ARIMA orders of univariate series generated by an
Model | AR order | I( | MA order |
---|---|---|---|
I(0)VAR | 0 | ( | |
SCCF | ( | 0 | ( |
PSCCF | ( | 0 | ( |
CI(1, 1) VAR | 1 | ( | |
SCCF | ( | 1 | ( |
PSCCF | ( | 1 | ( |
WF | ( | 1 | ( |
SCCF = serial correlation common feature; PSCCF = polynomial SCCF; WF = weak form.
Canonical correlations and CCF’s
Model | ||
---|---|---|
SCCF | ||
WF | ||
PSCCF | ||
WFP |
SCCF = serial correlation common feature; WF = weak form; PSCCF = polynomial SCCF; WFP = weak form of PSCCF.
Tests for common features
Model | Degress of freedom |
---|---|
SCCF | |
WF | |
PSCCF | |
WFP |
SCCF = serial correlation common feature; WF = weak form; PSCCF = polynomial SCCF; WFP = weak form of PSCCF.
Estimators of the common features vectors and RRR coefficients
Model | ( | ( |
---|---|---|
SCCF | ||
WF | ||
PSCCF | ||
WFP |
SCCF = serial correlation common feature; WF = weak form; PSCCF = polynomial SCCF; WFP = weak form of PSCCF.