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VAR models with an index structure: A survey with new results
The Vector Auto-Regressive model [VAR] and the Dynamic Factor Model [DFM] are arguably among the most popular tools for analyzing economic and financial variables over time. Since the seminal contribution of sims1980macroeconomics, VARs have been theoretically extended and practically implemented to forecast, perform structural analysis, and detect comovements in multivariate time series. DFMs were introduced more recently (forni2000generalized, Forni_Lippi2001, stock2002forecasting, stock2002macroeconomic, bai_ng2003, and bai2003), but rapidly contested the role of the workhorse in empirical macroeconomics.
The main reason for the success of the DFM is two-fold. First, it allows handling a much larger number of variables than those that are generally employed in traditional small-scale VARs, thus potentially boosting forecasting accuracy and solving the informational deficiency problems that arise in structural analyses when the agent's information set is richer than the econometrician's information set (see, e.g., Forni2014). Second, DFM allows one to disentagle the shocks that drive the common components of several time series and recover the structural shocks from those common shocks. Hence, in structural DFMs, the number of shocks is smaller than the number of variables (ForniET), which is in line with dynamic stochastic general equilibrium models (see DSGE2016 and the references therein) and, more generally, with the standard macroeconomic view that a small number of shocks drives aggregate fluctuations.
Efforts have recently been made to endow the VAR with the above-mentioned features of the DFM. On the one hand, shrinkage estimators have been proposed for medium-large VARs, both from a Bayesian perspective (e.g., banbura2010large, koop2013forecasting, and carriero2015bayesian) and from a classical standpoint (e.g., Hsu2008, KC2005, and HMSJoFE). On the other hand, the Multivariate Autoregressive Index (MAI) model -- originally proposed by reinsel1983some as a convenient approach to dimension reduction in stationary VARs -- has recently gained renewed attention.\footnote{At the end of 2024, the annual citation rate of reinsel1983some in Scopus has increased by about 54% in the last 9 years, with the majority of recent citations coming from econometric journals.}.
Late advances have shown that MAI and its variants allow for both forecasting variables and identifying shocks analogously to the DFM but without encountering some issues in model identification and statistical inference that characterize the latter, such as the requirement that the number of variables diverges at a given rate and the need of specific assumptions on both the correlation structure of the idiosyncratic components and the factor loadings (see, i.a., bai2003, and bai2006confidence). Moreover, VARs with index structures have been shown to be able to accommodate features such as stochastic volatility (carriero2022global) and time-varying parameters (CGG2025), which are not easy to handle within the DFM framework.
The MAI falls within reduced-rank VARs, a general class of models that include as special cases both the cointegrated VAR (see johansen1995likelihood and the references therin) and the Common Serial Correlation [CSC] models (see CH2022ORE and the references therin). Although CSC models and MAI have similar mathematical formulations, their respective goals and properties are rather different; whereas the former are based on the existence of (possibly dynamic) linear combinations of autocorrelated time series that are white noise (EK1993, Vahid1993, Cubadda2007, carriero2011forecasting, cubadda2011, Bernardini2015), VARs with an index structure assume that there is a limited number of channels through which information from the past is transmitted to the variables of interest.
The main aim of this paper is twofold. First, recent developments in MAI are reviewed, such as the structural MAI (carriero2016structural), the vector heterogeneous index model for realized volatilities (cubadda2017vector), as well as augmentations of the original model with individual autoregressive structures (cubadda2019representation), stochastic volatility (carriero2022global), time-varying parameters (CGG2025), high-dimensionality (cubadda2022), and cointegration (CM2024). Second, new results are provided in terms of representation theory for the various models, and a novel modeling is proposed, namely the cointegrated index-augmented autoregressive model, which combines and extends the results in cubadda2019representation and CM2024.
This paper is organized as follows. Focusing on representation theory, Section 2 reviews previous contributions and provides new insights into some of them. Section 3 presents the new model and deals with its estimation, whereas some details of the estimation procedure are relegated to the Appendix. Finally, Section 4 provides some conclusions.
In this Section we review the models that are rooted from the original MAI formulation and provide new results on representation theory of some of them. Analogies and differences with the DFM are discussed in detail. Estimation and identification issues are also covered.
Let us assume that the $n-$vector time series $Y_{t}=(y_{1t},...,y_{nt} )^{^{\prime}}$ is generated by the following stationary VAR$(p)$ model:
where $L$ is the lag operator, $\Phi(L)=$ $I_{n}-\sum_{j=1}^{p}\Phi_{j}L^{j}$, and $\varepsilon_{t}$ is a vector or $n$ errors with $\mathrm{E} (\varepsilon_{t}\varepsilon_{t}^{\prime})=\Sigma$ (positive definite) and finite fourth moments, $\mathrm{E}(\varepsilon_{t}|\digamma_{t-1})=0$, and $\digamma_{t}$ is the natural filtration of the process $Y_{t}$. For simplicity, deterministic elements are ignored.
The key assumption of MAI (reinsel1983some) is the following:
Under Assumption 1, Model ((ref)) can be rewritten as
where linear combinations $f_{t}=\omega^{\prime}Y_{t}$ are called the indexes. The MAI has at most $nq(p+1)-q^{2}$ mean parameters, which implies a significant dimension reduction when $p$ is small w.r.t. $n$.\footnote{Indeed, the matrix $\omega$, once identified through normalizing restrictions, has $q(n-q)$ free parameters.}
By premultiplying with $\omega^{\prime}$ both sides of Equation ((ref)) we get
which shows that the indexes follow a VAR$(p)$ process and not a VARMA, as is generally the case for linear combinations of elements of a VAR (see Cubadda2009 and the references therein).
Putting emphasis on the analogies between MAI and EDFM, carriero2016structural propose identifying structural shocks as linear transformations of the index shocks only. Starting from the Wold representation of series $Y_{t}$ \[ Y_{t}=\Psi(L)\varepsilon_{t}, \] and inserting between $\Psi(L)$ and $\varepsilon_{t}$ the decomposition of the identity matrix as in centoni2003measuring
one gets the following decomposition of series $Y_{t}:$
where
$\underline{\Sigma}=\omega^{\prime}\Sigma\omega$, $\varepsilon_{t}^{\chi }=\omega^{\prime}\varepsilon_{t}$, $\varepsilon_{t}^{\iota}=\omega_{\perp }^{\prime}\Sigma^{-1}\varepsilon_{t}$, $\mathrm{E}(\varepsilon_{t}^{\chi }\varepsilon_{t}^{\iota\prime})=0$, and $\mathrm{E}(\chi_{t}\iota _{t-j}^{\prime})=0$ for $\forall j$.
Since the shocks $\varepsilon_{t}^{\chi}$ are those of the indexes, $\varepsilon_{t}^{\chi}$ may be interpreted as the common shocks and $\chi _{t}$ as the common components of the series $Y_{t}$. Similarly, $\varepsilon_{t}^{\iota}$ and $\iota_{t}$ can be labeled, respectively, as uncommon shocks and a uncommon component.
Interestingly, post-multiplying with $\omega_{\perp}$\ both sides of the relation \[ \Psi(L)(I_{n}-\sum_{j=1}^{p-1}\alpha_{j}\omega^{\prime}L^{j})=I_{n} \] we get $\Psi(L)\omega_{\perp}=\omega_{\perp}$, which in turn implies that the Wold polynomial matrix of the MAI has the form
where $\theta_{j}$ is an $n\times q-$matrix for $j>0$.
Having substituted $\Psi(L)$ in Equations ((ref)) and ((ref)) with the RHS of Equation ((ref)), we can finally prove the following proposition:
carriero2016structural suggest to recover the structural shocks as linear transformations of the common shocks $\varepsilon_{t}^{\chi}$ only. Hence, at most $q<n$ structural shocks can be recovered, as occurs in DFMs and in dynamic stochastic general equilibrium models. In principle, all the identification strategies that are available for structural VARs or structural DFMs (see SW2016chapter and the references therein) can be adopted.
On the estimation side, carriero2016structural prove that the iterative maximum likelihood procedure proposed by reinsel1983some is consistent when $n=o(\sqrt{T})$. Moreover, they provide an MCMC algorithm for Bayesian estimation and show by simulations that the Bayesian approach outperforms the classical one when $n=15,20$. Finally, they document the practical value of the structural MAI by two empirical applications, on the transmission mechanism of monetary policy and on the propagation of demand and supply shocks.
The univariate Heterogeneous AR model [HAR], originally proposed by Corsi2009, is a popular tool to analyze and forecast daily realized volatility [RV] measures without resorting to more involved long-memory models. Technically speaking, the HAR is a constrained AR$(22)$ model where the predictors are the first lags of: (i) the daily RV; (ii) the weekly (5 days) average of the daily RV; (iii) the monthly (22 days) average of the daily RV.
cubadda2017vector propose a multivariate HAR for a set of $n$ daily RV measures $Y_{t}^{(d)}\equiv\left( Y_{1,t}^{(d)},\ldots,Y_{n,t}^{(d)}\right) ^{\prime}$ that is endowed with an index structure. In particular, the Vector Heterogeneous Autoregressive Index model [VHARI] reads as follows \[ Y_{t}^{(d)}=\alpha^{(d)}\omega^{\prime}Y_{t-1d}^{(d)}+\alpha^{(w)} \omega^{\prime}Y_{t-1d}^{(w)}+\alpha^{(m)}\omega^{\prime}Y_{t-1d} ^{(m)}+\varepsilon_{t}, \] where $(d)$, $(w)$, and $(m)$ denote, respectively, time horizons of one day, one week, and one month such that \[ Y_{t}^{(w)}=\frac{1}{5}\sum_{j=0}^{4}Y_{t-jd}^{(d)},\text{ \ \ }Y_{t} ^{(m)}=\frac{1}{22}\sum_{j=0}^{21}Y_{t-jd}^{(d)} \]
The VHARI enjoys two important properties that are not shared by alternative approaches to inducing dimension reduction in the Vector HAR\footnote{The most obvious alternatives to the VHARI likely are multivariate principal component regression and reduced-rank regression.}: First, the indexes $f_{t} ^{(d)}=\omega^{\prime}Y_{t-1d}^{(d)}$ preserve the temporal cascade structure of the HAR model since \[ f_{t}^{(w)}=\frac{1}{5}\sum_{j=0}^{4}f_{t-jd}^{(d)},\text{ \ \ }f_{t} ^{(m)}=\frac{1}{22}\sum_{j=0}^{21}f_{t-jd}^{(d)}, \] Second, pre-multiplying both sides of the VHARI by $\omega^{\prime}$ yields the following: \[ f_{t}^{(d)}=\omega^{\prime}\alpha+\omega^{\prime}\alpha^{(d)}f_{t-1d} ^{(d)}+\omega^{\prime}\alpha^{(w)}f_{t-1d}^{(w)}+\omega^{\prime}\alpha ^{(m)}f_{t-1d}^{(m)}+\omega^{\prime}\varepsilon_{t}, \] which shows that the indexes follow a multivariate HAR model. In particular, when $q=1$ a univariate HAR model generated all the dynamics of the $n$ RVs.
On the estimation side, cubadda2017vector suggest using a switching algorithm [SA], an iterative method for numerical maximization of the log-likelihood of complex models that has a long tradition in time-series analysis (see Boswijk2004 and the references therein). In particular, the proposed SA requires the following steps:
A key point of the above SA is that both Steps 1 and 2 require running OLS regressions only. This feature provides the SA with several advantages over Newton-type optimization methods, such as computational simplicity, no need for normalization conditions in $\omega$, explicit optimization at each step, and ease of application of regularization schemes or linear restrictions on parameters (see cubadda2019representation for additional discussion). Furthermore, when the SA is initialized with consistent estimates and is iterated sufficiently often, the resulting estimator is asymptotically equivalent to the ML one (HautschJoFE). cubadda2017vector show by simulation that the suggested SA performs well even when elements of $\varepsilon_{t}$ have a log-normal error distribution with GARCH variances.
Following Patton2009, cubadda2017vector use the VHARI to build the optimal linear combination of ten different estimators of the volatility of the same market and to evaluate its merits through an out-of-sample forecasting exercise. The VHARI model proves to work well, often outperforming previously existing methods.
A possible limitation of MAI as a forecasting tool is that the only predictors of the series $y_{i,t}$, for $i=1,\ldots,n$, are the lagged indexes, whereas the forecasts obtained through the DFM exploit information coming from the past of both factors and the series $y_{i,t}$ itself (see the seminal contributions by stock2002forecasting and stock2002macroeconomic ). Although the indexes may be interpreted as 'supervised' factors that are constructed for emphasizing the comovements between the present and the past of the system, it may occur that some variables are better predicted by their own lags rather than by any linear combination of all variables only.
In order to overcome such limitation, cubadda2019representation extended the basic MAI model by allowing individual AR structures for each element of $Y_{t}$. Their key assumption is the following.
In words, Assumption 2 states that there is a reduced number of channels $p$ through which each variable is influenced by the past of other variables in the system, which is consistent with the common view that few shocks are responsible for most macroeconomic fluctuations.
Under Assumption 2 and using the reparametrization $\delta_{ii}^{(j)} =\phi_{ii}^{(j)}-\sum_{m=1}^{q}\alpha_{im}^{(j)}\omega_{im}$, Model ((ref)) can be rewritten into the following {Index-Augmented Auto-Regressive model} [IAAR]:
where $D_{j}$\ is a $n\times n$ diagonal matrix with $\delta_{ii}^{(j)}$ as generic diagonal element, and, for greater generality, $s\leq p$.
cubadda2019representation proposed a two-step SA for the estimation of the IAAR, along with a variant where a $\ell_{2}$ regularization scheme is applied in both steps. They show by simulations that the regularized version of the SA outperforms the standard one with $n=20$. Regarding model specification, they opt for the use of Information Criteria [IC], in line with previous contributions showing that IC outperform likelihood ratio tests in selection of reduced-rank VAR models (see, e.g., Gonzalo1999chapter, CavaliereOBES, CavaliereET). Finally, the IAAR proves to outperform well-known macroeconomic forecasting methods when applied to systems with $n$ ranging from $4$ to $40$.
carriero2022global endowed the IAAR with Stochastic Volatility [IAAR-SV] in the errors $\varepsilon_{t}$ and offered Bayesian estimation using Markov Chain Monte Carlo [MCMC] techniques. Furthermore, they use ((ref)) to decompose the time-varying volatility $\mathrm{E}(\varepsilon _{t}\varepsilon_{t}^{\prime})=\Sigma_{t}$\ as follows: \[ \Sigma_{t}=\underset{\mathrm{common}}{\underbrace{\Sigma_{t}\omega (\omega^{\prime}\Sigma_{t}\omega)^{-1}\omega^{\prime}\Sigma_{t}} }+\underset{\mathrm{uncommon}}{\underbrace{\omega_{\perp}(\omega_{\perp }^{\prime}\Sigma_{t}^{-1}\omega_{\perp})^{-1}\omega_{\perp}^{\prime}}} \]
carriero2022global apply the IAAR-SV to analyze the commonality in both levels and volatilities of inflation rates in several countries, and their main finding is that a substantial fraction of inflation volatility can be attributed to a global factor that also drives inflation levels and their persistence.
A further step towards taking into account parameter instabilities over time was taken by CGG2025, who proposed the following MAI with Time Varying Parameters and Time-Varying Volatility [MAI-TVP-TVV]:
where $\boldsymbol{\alpha}_{t}=\mathrm{Vec}(\alpha_{1,t}^{\prime} ,\ldots,\alpha_{n,t}^{\prime})^{\prime}$, $\varepsilon_{t}\sim N(0,\Sigma _{t})$, $\kappa_{t}\sim N(0,Q_{t})$, $\varepsilon_{t}$ and $\kappa_{t}$ are independent at any lag and lead. Notice that it is assumed that the index loadings evolve over time as random walks, while the index weights $\omega$ remain stable.
In order to overcome the computational limitation related to MCMC procedures, CGG2025 offers a hybrid estimation method that combines the SA, Kalman filter with forgetting factors (KK2014), and exponentially weighted moving average techniques (JOPSB2023) for the time-varying volatility.
An empirical application, where 25 US quarterly time series are used to forecast three key macroeconomic variables, shows that the MAI-TVP-TVV is one of the best models in a large set of competitors for all targets, improving upon its counterparts especially at short horizons. Other interesting findings are that, once the MAI is endowed with time-varying volatility [MAI-TVV], there are no clear improvements in adding time-varying parameters for point forecasting, but the MAI-TVP-TVV always outperforms the MAI-TVV in density forecasting.
cubadda2022 studied the conditions under which the dynamics in a large-dimensional VAR are entirely generated by a small-scale VAR. They show that such conditions are met when the coefficient matrices of the large VAR have the same common right space and a common left null space. This entails combing Assumption 1 with the following:
Assumption 3 is popularly known in time series econometrics as the CSC (see CH2022ORE and the reference therein) given that \[ \omega_{\perp}^{\prime}Y_{t}=\omega_{\perp}^{\prime}\varepsilon_{t}, \] that is, there exist $(n-q)$ linear combinations of variables $Y_{t}$ that are white noise and, as such, cannot exhibit cyclical behavior.
Taking Assumptions 1 and 3 together leads to the Dimension-Reducible VAR model [DRVAR]:
where $\phi_{j}$ is a $q\times q$ matrix for $j=1,...,p$.
Assuming, without loss of generality, that $\omega^{\prime}\omega=I_{q}$ and $\omega_{\bot}^{\prime}\omega_{\bot}=I_{n-q}$, we can decompose series $Y_{t}$ as follows
where $f_{t}$ is the dynamic component and $\eta_{t}=\omega_{\bot}^{\prime }\varepsilon_{t}$ is the static one. Premultiplying both sides of DRVAR by $\omega^{\prime}$\ one gets \[ f_{t}=\sum_{j=1}^{p}\phi_{j}f_{t-j}+\varepsilon_{t}^{\chi}, \] where $\varepsilon_{t}^{\chi}=\omega^{\prime}\varepsilon_{t}$, which shows that $f_{t}$\ is generated by a $q-$dimensional VAR($p$) process.
By inserting the Wold representation of the dynamic components $f_{t}$\ in Equation ((ref)) it follows:
where $\gamma(L)^{-1}=I_{n}-\sum_{j=1}^{p}\phi_{j}L^{j}$. Finally, by linearly projecting $\omega_{\bot}\eta_{t}$ on $\varepsilon_{t}^{\chi}$, we obtain $\omega_{\bot}\eta_{t}=\rho\varepsilon_{t}^{\chi}+\nu_{t}$ with $\mathrm{E} (\varepsilon_{t}^{\chi}v_{t}^{\prime})=0$, which can be inserted into Equation ((ref)) to get
where $C_{0}=\omega+\rho$\ and $C_{j}=\omega\gamma_{j}$ for $j>0$.
Representation ((ref)) highlights that system dynamics are completely generated by common reduced form errors $\varepsilon_{t}^{\chi}$. Consequently, cubadda2022 label $\nu_{t}$ as the ignorable errors, as they are noise without structural interpretation. Since the errors $\varepsilon_{t}^{\chi}$\ and $\nu_{t}$ are uncorrelated at any lead and lag, it is then possible to recover the structural shocks solely from the reduced form errors $\varepsilon_{t}^{\chi}$ of the common component $\chi_{t}$ using any of the procedures that are commonly employed in structural VARs or structural DFMs (see SW2016chapter and the references therein).
In order to estimate the matrix $\omega$, one may rely on a nonparametric estimator proposed by Lam2011. The underlying intuition is that the matrix $\omega$ lies in the space generated by the eigenvectors associated with the $q$ nonzero eigenvalues of the symmetric and semipositive definite matrix. \[ M=\sum_{j=1}^{p_{0}}\Sigma_{y}(j)\Sigma_{y}(j)^{\prime}, \] where $p_{0}\geq p$ and $\Sigma_{y}(j)$ is the autocovariance matrix of series $Y_{t}$ in lag $j$. Under some regularity conditions, the matrix formed by the eigenvectors associated with the $q$ largest eigenvalues of the sample estimate of $M$ is a $\sqrt{T}$-consistent estimator of $\omega$ (up to an orthonormal transformation) when $q$ is fixed, $n,T\rightarrow\infty$, and $\omega_{i}^{\prime}\omega_{i}=O(n)$ for $i=1,...,q$, where $\omega =[\omega_{1},...,\omega_{q}]$. Remarkably, the speed of convergence of the estimator, namely $\sqrt{T}$, is the same as when the dimension $n$ is finite.
Moreover, cubadda2022 provide both the OLS and the GLS estimators of the coefficients $\phi$' s in equation ((ref)) and consistent information criteria for the selection of $q$, and show by simulations that the proposed methodology works well with the temporal and cross-sectional sizes that are typical in macroeconomics. Finally, the approach is applied to analyze a large set of US economic time series and to identify the shock that is responsible for most of the common volatility in the business cycle frequency band.
The models considered so far do not explicitly deal with the possible presence of unit roots. Given that most macroeconomic and financial time series are characterized by stochastic trends, it is important to understand how a cointegrated VAR model can be augmented with an index structure.
Let us assume that series $Y_{t}$ follow the Vector Error-Correction Model [VECM]
where $\alpha_{0}$ and $\beta$ are full-rank $n\times r$ ($r<n$) matrices such that $\alpha_{0}\beta^{\prime}=-\Phi(1)$, $\Pi_{j}=- {\textstyle\sum\limits_{i>j}} \Phi_{i}$ for $j=1,\ldots,p-1$, $\alpha_{0}{}_{\perp}^{\prime}\bar{\Pi} \beta_{\perp}$ is non-singular, and $\bar{\Pi}=I_{n}-\sum_{j=1}^{p-1}\Pi_{j}$. Under such assumptions, it is well known that the elements of $Y_{t}$ are individually, at most, $I(1)$ and that they are jointly cointegrated of order $1$, in the sense that $\beta^{\prime}Y_{t-1}$\ is $I(0)$ (see johansen1995likelihood and the references therein).
To possibly reduce the number of parameters in the VECM, CM2024 take the following assumptions:
Under Assumptions 4 and 5, Model ((ref)) can be rewritten in the following {Vector-Error Correction Index Model [VECIM]:} \[ \Delta Y_{t}=\alpha_{0}\gamma^{\prime}f_{t-1}+\sum_{j=1}^{p-1}\alpha_{j}\Delta f_{t-j}+\varepsilon_{t}, \] where $\gamma$ is a full-rank $q\times r$ matrix ($q\geq r$), and $\alpha_{j}$ is an $n\times q$ matrix for $j=1,...,p-1$ such that $\mathrm{rank} ([\alpha_{1}^{\prime},...,\alpha_{p-1}^{\prime}]^{\prime})=q$. Notice that the cointegration matrix is given by $\beta=\omega\gamma$.
Interestingly, the indexes $f_{t}$ themselves are generated by a $q-$dimensional VECM: \[ \Delta f_{t}=\underline{\alpha}_{0}\gamma^{\prime}f_{t-1}+\sum_{j=1} ^{p-1}\underline{\alpha}_{j}\Delta f_{t-j}+\varepsilon_{t}^{\chi}, \] where $\underline{\alpha}_{j}=\omega^{\prime}\alpha_{j}$, for $j=0,1\ldots ,p-1$.
By first inserting the decomposition ((ref)) between $\Psi(L)$ and $\varepsilon_{t}$ into the Wold representation of the first differences $\Delta Y_{t}$: \[ \Delta Y_{t}=\Psi(L)\varepsilon_{t}, \] and then further decomposing the common component $\chi_{t}$ into permanent and transitory subcomponents as in centoni2003measuring, we get the following:
where
Since errors $\varepsilon_{t}^{\pi}$ are the innovations of the common trends of the indexes $f_{t}$ (see e.g. johansen1995likelihood) and errors $\varepsilon_{t}^{\tau}$ are such that $\mathrm{E}(\varepsilon_{t}^{\pi }\varepsilon_{t}^{\tau\prime})=0$, CM2024 label $\pi_{t}$\ as the common permanent component, $\tau_{t}$\ as the common transitory component, whereas $\iota_{t}$\ is the uncommon component given that $\mathrm{E} (\varepsilon_{t}^{\iota}\varepsilon_{t}^{\pi\prime})=0$ and $\mathrm{E} (\varepsilon_{t}^{\iota}\varepsilon_{t}^{\tau\prime})=0$.
Following a similar reasoning as the one leading to Proposition 1, post-multiplying with $\omega_{\perp}$\ both sides of the relation \[ \Psi(L)(\Delta I_{n}-\sum_{j=1}^{p-1}\alpha_{j}\omega^{\prime}\Delta L^{j}-\alpha_{0}\gamma^{\prime}\omega^{\prime}L)=\Delta I_{n} \] we again get $\Psi(L)\omega_{\perp}=\omega_{\perp}$, which in turn implies that the Wold polynomial matrix of the VECIM has the same form as ((ref)). Finally, inserting ((ref)) in Equations ((ref) ), ((ref)), and ((ref)) we can prove the following proposition:
Notice that Proposition 2 implies that Corollary 1 applies to the VECIM as well.
CM2024 offer a three-step SA for the estimation of the VECIM and propose to select the triple $(p,q,r)$ in a unique search by IC. An extensive Monte Carlo study shows that the proposed methodology works reasonably well for $n$ ranging from $6$ to $18$ when the model is identified by the Hannan-Quinn IC. Moreover, in an empirical application, they identify a shock that maximizes the variability of the common transitory component of unemployment at the business cycle frequencies, and another one that does the same, but for the common permanent component of unemployment. These two shocks are endowed with a neater economic interpretation than a unique main business cycle shock identified according to angeletos2020business.
A possible limitation of the VECIM is that the uncommon component $\iota_{t}$ is necessarily a random walk, which may be considered restrictive for some applications. For example, barigozzi2021large propose a DFM where the idiosyncratic components may be I(0) or I(1).
In order to overcome this issue, one can combine the VECIM with the IAAR. Formally, this involves taking Assumption 5 along with the following one:
Taking Assumptions 5 and 6, the model ((ref)) can be rewritten into the following Cointegrated Index-Augmented Auto-Regressive model [CIAAR]
where $D_{j}$\ is a $n\times n$ diagonal matrix with $\delta_{ii}^{(j)} =\pi_{ii}^{(j)}-\sum_{m=1}^{q}\alpha_{im}^{(j)}\omega_{im}$\ as generic diagonal element.
When the elements of the series $Y_{t}$ are I(1), Model ((ref)) includes several earlier models as special cases. In fact, we have a MAI for series $\Delta Y_{t}$ if $p,r=0$, an IAAR for series $\Delta Y_{t}$ is obtained if $r=0$, and a VECIM if $p=0$.
Following cubadda2019representation and CM2024, the estimation procedure is based on an SA where each step is designed to increase the Gaussian likelihood of Model ((ref)). In detail, when $0<r<q$, the procedure goes as follows:
When $r=0$, step 3 is clearly not needed, and steps 1 and 2 must be modified as follows:
Finally, when $r=q$, we can assume without loss of generality that $\gamma=I_{q}$. Then Step 3 is again not needed, whereas Steps 1 and 2 must be modified as follows.
The choice of initial values for the above procedures is discussed in the Appendix, whereas the selection of the quadruple $(p,s,q,r)$ can be done by IC, sequentially or in a unique search as suggested by CM2024.
The DFM and the VAR are, arguably, among the most popular tools in macroeconometrics and financial econometrics. The two approaches should be considered complementary rather than substitutive, since each of them has its own merits. The MAI represents a link between these two methodologies: On the one hand, it is a VAR with a specific reduced-rank structure that alleviates the dimensionality problem; on the other hand, the MAI and its variants have several analogies with the DFM; in particular, they allow for identifying a small number of common reduced-form errors and for recovering structural shocks from those errors only.
However, the MAI is not affected by some theoretical limitations of the DFM such as the requirement that the cross-sectional dimension diverges to infinity and the need for specific assumptions on the dynamic correlation structure of the idiosyncratic component and on the factor loadings. In a more practical perspective, VARs with an index structure can also handle features such as stochastic volatility (carriero2022global) and time-varying parameters (CGG2025), which are not easily accommodated in DFMs.
Recent developments in VAR models with index structures have considerably extended the original MAI formulation, endowing the model with individual autoregressive structures, stochastic volatility, time-varying parameters, high dimensionality, and cointegration. These extensions have proven to be useful tools for detecting common components, obtaining efficiency gains through the imposition of parameter restrictions, performing structural analysis, and boosting forecast accuracy.
Having reviewed most of the recent advances on the MAI and provided new insights on the representation theory underlying the various formulations, a new model, namely the CIAAR, has been proposed along with an estimation procedure. The hope is that this paper will contribute to providing room for future research on VAR models with index structures.