Factor Vector Autoregressive Estimation of Heteroskedastic Persistent and Non Persistent Processes Subject to Structural Breaks ()

Claudio Morana

Department of Economics, Management and Statistics, University of Milan-Bicocca, Milan, Italy; Center for Research on Pensions and Welfare Policies, Collegio Carlo Alberto, Moncalieri, Italy.

**DOI: **10.4236/ojs.2014.44030
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Department of Economics, Management and Statistics, University of Milan-Bicocca, Milan, Italy; Center for Research on Pensions and Welfare Policies, Collegio Carlo Alberto, Moncalieri, Italy.

In the paper, a general framework for large scale modeling of macroeconomic and financial time series is introduced. The proposed approach is characterized by simplicity of implementation, performing well independently of persistence and heteroskedasticity properties, accounting for common deterministic and stochastic factors. Monte Carlo results strongly support the proposed methodology, validating its use also for relatively small cross-sectional and temporal samples.

Keywords

Long and Short Memory, Structural Breaks, Common Factors, Principal Components Analysis, Fractionally Integrated Heteroskedastic Factor Vector Autoregressive Model

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Morana, C. (2014) Factor Vector Autoregressive Estimation of Heteroskedastic Persistent and Non Persistent Processes Subject to Structural Breaks. *Open Journal of Statistics*, **4**, 292-312. doi: 10.4236/ojs.2014.44030.

1. Introduction

In the paper, a general strategy for large-scale modeling of macroeconomic and financial data, set within the factor vector autoregressive model (F-VAR) framework, is proposed.^{1}

Following the lead of dynamic factor model analysis proposed in [2] , it is assumed that a small number of structural shocks are responsible for the observed comovement in economic data; it is, however, also assumed that commonalities across series are described by deterministic factors, i.e., common break processes. Comovement across series is then accounted by both deterministic and stochastic factors; moreover, common factors are allowed in both mean and variance, covering the I(0) and I(1) persistence cases, as well as the intermediate case of long memory, i.e., I(d),. As the common factors are unobserved, accurate estimation may fail in the framework of small scale vector autoregressive (VAR) models, but succeed when cross-sectional information is employed to disentangle common and idiosyncratic features.

The proposed fractionally integrated heteroskedastic factor vector autoregressive model (FI-HF-VAR) bridges the F-VAR and (the most recent) G-VAR literature, as, similarly to [3] , a weakly stationary cyclical representation is employed; yet, similarly to [4] , principal components analysis (PCA) is employed for the estimation of the latent factors. Consistent and asymptotically normal estimation is performed by means of QML, also implemented through an iterative multi-step estimation procedure. Monte Carlo results strongly support the proposed methodology.

Overall, the FI-HF-VAR model can be understood as a unified framework for large-scale econometric modeling, allowing for accurate investigation of cross-sectional and time series features, independent of persistence and heteroskedasticity properties of the data, from comovement to impulse responses, forecast error variance and historical decomposition analysis.

After this introduction, the paper is organized as follows. In Section 2, the econometric model is presented; in Section 3, estimation is discussed, while Monte Carlo analysis is performed in Section 4; finally, conclusions are drawn in Section 5.

2. The FI-HF-VAR Model

Consider the following fractionally integrated heteroskedastic factor vector autoregressive (FI-HF-VAR) model

(1)

(2)

where is a vector of real valued integrated I(d) () and heteroskedastic processes subject to structural breaks, in deviation from the unobserved common deterministic () and stochastic (f_{t})

factors; is a finite order matrix of polynomials in the lag operator with all the roots outside the unit circle, , , is a square matrix of coefficients of order N; is a

vector of zero mean idiosyncratic i.i.d. shocks, with contemporaneous covariance matrix, assumed to be coherent with the condition of weak cross-sectional correlation of the idiosyncratic components (Assumption E) stated in [5] p. 143. The model in (1) actually admits the same static representation of [5] , as it can be rewritten as.

2.1. The Common Break Process Component

The vector of common break processes is, with, and matrix of loadings; the latter are assumed to be orthogonal to the common stochastic factors, and of unknown form, measuring recurrent or non recurrent changes in mean, with smooth or abrupt transition across regimes; the generic element in μt is μi,t≡zμ,i(t), where is a function of the time index t,.

The idiosyncratic break process can take different forms. For instance, [6] use a discontinuous function,

(3)

where is the indicator function, such that if and is 0 otherwise; in [6] the break points

are determined through testing; a Markov switching mechanism, as in [7] , could however also be employed to this purpose.

Differently, [8] [9] and [10] model the break process as a continuous and bounded function of time, by means of a Fourier expansion, i.e.,

(4)

Similarly [11] , using a logistic specification

(5)

where the logistic function is, , and

are parameters, , and is the estimated standard deviation of. In particular, as,

becomes the indicator function, yielding therefore a generalization of the specification in [6] .

Also similarly [12] and [13] , using a spline function

, (6)

where is a spline function of order p, are unknown regression coefficients and the functions are spline basis functions defined as, , , , and

, with.

A semiparametric approach has also been suggested by [14] , using a kernel function, i.e.,

, (7)

where b is the bandwidth and is the kernel function, specified as for and

for; and the coefficient are such that.

Finally, a random level shift model has been proposed by [15] -[18] ; for instance, [18] define the break process as

(8)

where and for.

In the case, there are no common break processes, i.e., each series is characterized by its own idiosyncratic break process and the factor loading matrix is square, diagonal and of full rank; when, there exist M common break processes and the factor loading matrix is of reduced rank (M). Hence, in the latter case the series are cotrending, according to [19] , nonlinear cotrending, according to [20] , or cobreaking, according to [21] and [22] . The representation in (1) emphasizes however the driving role of the common break processes, rather than the break-free linear combinations (cobreaking/cotrending relationships) relating the series.

2.2. The Common Break-Free Component

The vector of (zero-mean) integrated heteroskedastic common factors is, with, and

matrix of loadings. The order of integration is in mean, and in variance, , ,

.

The polynomial matrix is of finite order, with all the roots outside the unit circle;, , is a square matrix of coefficients of order; is a vector of common zero mean i.i.d. shocks, with identity covariance matrix, all, respectively.

The matrix is a square diagonal matrix in the lag operator of order, specified according to the integration order (in mean) of the common stochastic factors, i.e., for the case of integration ();for the or no integration (short memory) case ();

for the case of fractional integration (, long memory)

(), where is the fractional differencing operator; the latter admits a binomial expansionwhich can be compactly written in terms of the Hypergeometric function, i.e.,

, where is the Gamma function.

In the case there are no common stochastic processes, i.e., each series is characterized by its own idiosyncratic persistent stochastic component, and the factor loading matrix is square, diagonal and of full rank; when, then there exist R common stochastic processes and the factor loading matrix is of reduced rank (R). Hence, in the latter case the series show common stochastic features, according to [23] . The concept of common feature is broad, encompassing the notion of cointegration ([24] ), holding for the case. The representation in (1) emphasizes however the driving role of the common stochastic factors rather than the feature-free linear combinations (cofeature relationships) relating the series.

2.3. The Conditional Variance Process

The conditional variance-covariance matrix for the unconditionally and conditionally orthogonal common factors is, where is the information set available at time period. Consistent with the constant conditional correlation model of [25] , the ith generic element along the main diagonal of is

(9)

where for the case of fractional integration (long memory) in variance

();for the case of integration in variance ();

for the or no integration (short memory) in variance case (). In all cases, , ,

and all the roots of the and polynomials are outside the unit circle.

The conditional variance process, , is therefore of the

type [26] , with, or the type, for the fractionally integrated and integrated case, respectively; of the type for the non integrated case. The model is however not standard as the intercept component is time-varying, allowing for structural breaks in variance; similarly to the mean part of the model, structural breaks in variance are assumed to be of unknown form, measuring recurrent or non recurrent regimes, with smooth or abrupt transition; then, , where is a continuos or discontinuous bounded function of the time index, , which can be parameterized as in (3), (4), (5), (6), (7), or (8).

The following ARCH (∞) representation can be obtained from each of the three above models

(10)

where and.

The term then bears the interpretation of break in variance process, or time-varying unconditional variance process (no integration case), or long-term conditional variance level (unit root and fractional integration cases).

To guarantee the non negativity of the conditional variance process at eachpoint in time all the coefficients in the representation must be non-negative, i.e., for all and for any. Sufficient conditions, for various parameterization, can be found in [26] and [27] .

2.4. The Reduced Fractional VAR form

By substituting (2) into (1) and rearranging, the vector autoregressive representation for the factors and the gap series can be written as

(11)

where and is differently defined according to persistence properties of the data. In particular, for the case of fractional integration (long memory), by means of the binomial expansion, it follows, , where, , is a square matrix of coefficients of dimension, and; since the infinite order representation cannot be handled in estimation, a truncation to a suitable large lag for the polynomial matrix is required.^{2} Hence,. For the case of no integration (short memory) (), recalling that , and therefore, then; for the case of integration (), it should be firstly recalled that

, with; the latter may be rewritten in the equivalent polynomial matrix form, where, , is a square matrix of coefficients of dimension, and,

; then,.

Reduced Form and Structural Vector Moving Average Representation of the FI-HF-VAR Model

In the presence of unconditional heteroskedasticity, the computation of the impulse response functions and the forecast error variance decomposition (FEVD) should be made dependent on the estimated unconditional variance for each regime. In the case of (continuously) time-varying unconditional variance, policy analysis may then be computed at each point in time. For some of the conditional variance models considered in the paper, i.e., the FIGARCH and IGARCH processes, the population unconditional variance does not actually exist; in the latter cases the component might bear the interpretation of long term level for the conditional variance; policy analysis is still feasible, yet subject to a different interpretation, FEVD referring, for instance, not to the proportion of forecast error (unconditional) variance accounted by each structural shock, but to the proportion of forecast error (conditional) long term variance accounted by each structural shock. With this caveat in mind, the actual computation of the above quantities is achieved in the same way as in the case of well defined population unconditional variance.

Hence, the computation of the vector moving average (VMA) representation for the FI-HF-VAR model depends on the persistence properties of the data. The following distinctions should then be made.

For the short memory case, i.e., the zero integration order case, the VMA representation for the factors and gap series can be written as

, (12)

where, and.

For the long memory case () and the case of non stationarity (), the VMA representation should be computed for the differenced process, yielding

, (13)

where, and. Impulse responses can then be computed as for and and for,

The identification of the structural shocks in the FI-HF-VAR model can be implemented in two steps. Firstly, denoting by the vector of the structural common factor shocks, the relation between reduced and structural form common shocks can be written as, where is square and invertible. Therefore, the identification of the structural common factor shocks amounts to the estimation of the elements of the matrix. It is assumed that, and hence. As the number of free parameters in is

, at most parameters in can be uniquely identified through the

system of nonlinear equations in the unknown parameters of. Additional restrictions need then to be imposed for exact identification of, by constraining the contemporaneous or long-run responses to structural shocks; for instance, recursive (Choleski) or non recursive structures can be imposed on the VAR model for the common factors through exclusion or linear/non-liner restrictions, as well as sign restrictions, on the contemporaneous impact matrix.^{3}

Secondly, by denoting the vector of structural idiosyncratic disturbances, the relation between reduced form and structural form idiosyncratic shocks can be written as, where K is square and invertible. Hence, the identification of the structural idiosyncratic shocks amounts to the estimation of the elements of the K matrix. It is assumed that, and hence. Then, in addition to the equations provided by, restrictions need to be imposed for exact identification ofsimilarly to what required for the common structural shocks.

Note that preliminary to the estimation of the matrix, should be obtained from the residuals of an OLS regression of on; the latter operation would grant orthogonality between common and idiosyncratic residuals.

The structural VMA representation can then be written as

, (14)

where, , , or

, (15)

where, , , according to persistence properties, and any.

3. Estimation

Estimation of the model can be implemented following a multi-step procedure, consisting of persistence analysis, estimation of the common factors and VAR parameters in (1), estimation of the conditional mean model in (2) and the reduced form model in (11), estimation of the conditional variance covariance matrix in (2).

3.1. Step 1: Persistence Analysis

Each component, , in the vector time series is firstly decomposed into its purely deterministic (trend/break process;) and purely stochastic (break-free,) parts.

It is then assumed that the data obey the model

, (16)

where and are orthogonal, , with a bounded function of the time index, evolving according to discontinuous changes (step function) or showing smooth transitions across regimes.

Depending on the specification of, a joint estimate of the two components can be obtained following [7] [10] [11] [13] [14] [30] , by setting up an augmented fractionally integrated ARIMA model

, (17)

where is the integer differencing parameter, is the fractional differencing parameter

(),is a stationary polynomial in the lag operator and is a white noise disturbance.

Heteroskedastic innovations can also be considered, by specifying, with and the conditional variance process according to a model of the GARCH family.

Consistent and asymptotically normal estimation by means of, also implemented through iterative algorithms, is discussed in [10] [13] [14] [18] [31] . Extensions of the Markov switching [7] , logistic [11] and random level shift [15] -[18] models to the long memory case have also been contributed by [32] [33] and [34] , respectively.

Alternatively, following [6] , a two-step procedure can be implemented: firstly, structural break tests are carried out and break points estimated; then, dummy variables are constructed according to their dating and the break process is estimated by running an OLS regression of the actual series on the latter dummies, as in (3); this yields computed as the fitted process and the stochastic part as the estimated residual, i.e.,;

and are then orthogonal by construction.^{4}

As neglected structural breaks may lead to processes which appear to show persistence of the long memory or unit root type, as well as spurious breaks may be detected in the data when persistence in the error component is neglected, testing procedures robust to persistence properties are clearly desirable. In this respect, the RSS-based testing framework in [6] yields consistent detection of multiple breaks at unknown dates for processes, as well as under long range dependence [35] ;^{5} moreover, under long range dependence, the validity of an estimated break process (obtained, for instance, by means of [6] ) may also be assessed by testing the null hypothesis of long memory in the estimated break-free series (), as antipersistence is expected from the removal of a spurious break process [36] [37] . Structural break tests valid for both and series have also recently contributed in the literature.

3.2. Step 2: Estimation of the Conditional Mean Model

estimation of the reduced form model in (11) is performed by first estimating the latent factors and VAR parameters in (1); then, by estimating the conditional mean process in (2); finally, by substituting (2) into (1) in order to obtain a restricted estimate of the polynomial matrix.

3.2.1. Estimation of the Common Factors and VAR Parameters

Estimation of the common factors is performed by, writing the (misspecified) approximating model as

(18)

with log-likelihood function given by

. (19)

QML estimation of the latent factors and their loadings then requires the minimization of the objective function

(20)

which can be rewritten as

, (21)

where, as and are orthogonal vectors, as well as and.

The solution to the minimization problem, subject to the constraints and, is given by firstly minimizing with respect to and, given and, yielding

and then concentrating the objective function to obtain

, (22)

which can be mimized with respect to and. This is equivalent to maximizing

, (23)

which in turn is equivalent to maximizing

(24)

subject to, and

(25)

subject to.

The solution is then found by setting:

equal to the scaled eigenvectors of, i.e., the sample variance covariance matrix of the break processes, associated with its largest eigenvalues; this yields, i.e., the scaled first

principal components of;

equal to the scaled eigenvectors of, i.e., the sample variance covariance matrix of the break-free processes, corresponding to its largest eigenvalues; this yields, i.e., the scaled first

principal components of.

Note that PCA uniquely estimates the space spanned by the unobserved factors; hence, and (

and) are not separately identified, as the common factors and factor loading matrix are uniquely estimated up to a suitable invertible rotation matrix, i.e., PCA delivers estimates of

and, and therefore a unique estimate of the common components only, which is however all what is required for the computation of the gap vector.

As shown by [38] , exact identification of the common factors can also be implemented, by appropriately constraining the factor loading matrix while performing PCA or after estimation. In particular, three identification structures are discussed, involving a block diagonal factor loading matrix, yield by a statistical restriction imposed in estimation, and two rotation strategies, yielding a lower triangular factor loading matrix in the former case and a two-block partitioned factor loading matrix in the latter case, with identity matrix in the upper block and an unrestricted structure in the lower block.

Moreover, the number of common factors is unknown and needs to be determined; several criteria are available in the literature, ranging from heuristic or statistical eigenvalue-based approaches [39] [40] to the more recent information criteria [41] and “primitive” shock ([42] ) based procedures.

Finally, in order to enforce orthogonality between the estimated common break processes and stochastic factors, the above procedure may be modified by computing the stochastic component as the residuals from the OLS regression of on; then PCA is implemented on (the break-free residuals) to yield.

Estimation of the VAR parameters. Conditional on the estimated (rotated) latent factors, the polynomial matrix and the and (rotated) factor loading matrices are obtained by means of OLS estimation of the equation system in (1). This can be obtained by first (OLS) regressing the actual series x_{t} on the estimated common break processes and stochastic factors to obtain and;

alternatively, and can be estimated as yield by PCA, i.e., from the scaled eigenvectors of the matrices and, respectively; then, the gap vector is computed as, as and

, and is obtained by means of OLS estimation of the VAR model in (1).

3.2.2. Iterative Estimation of the Common Factors and VAR Parameters

The above estimation strategy may be embedded within an iterative procedure, yielding a (relatively more efficient) estimate of the latent factors and the VAR parameters in the equation system in (1).

The objective function to be minimized is then written as

(26)

where.

Initialization. The iterative estimation procedure requires an initial estimate of the common deterministic and stochastic factors and the polynomial matrix, i.e., an initial estimate of the equation system in (1). The latter can be obtained as described in Section 3.2.1.

Updating. An updated estimate of the equation system in (1) is obtained as follows.

° First, a new estimate of the (rotated) common deterministic factors, and their factor loading matrix, is obtained by the application of PCA to the (new) stochastic factor-free series

, yielding and.^{6}

° Next, conditional on the new common break processes and their factor loading matrix, the new estimate of the common long memory factors is obtained from the application of PCA to (new) break-free processes

^{7} yielding and.^{8}

° Finally, conditional on the new estimated common break processes and long memory factors, the new estimate of the gap vector is obtained, and the new estimate can be computed by means of OLS estimation of the VAR model in (1).

° The above procedure is iterated until convergence, yielding the final estimates, and. Convergence may be assessed in various ways. For instance, the procedure may be stopped when the change in the value of the objective function is below a given threshold.^{9}

3.2.3. Restricted Estimation of the Reduced Form Model

Once the final estimate of the equation system in (1) is available, the reduced VAR form in (11) is estimated as follows:

1) For the case of fractional integration (long memory) (), the fractional differencing parameter is

(consistently) estimated first, for each component of the (rotated) common factors vector, yielding the estimates, , collected in matrix.

Considering then the generic element, consistent and asymptotically normal estimation of the ith fractional differencing parameter can be obtained, for instance, by means of estimation of the fractionally integrated ARIMA model in (17); alternatively, consistent and asymptotically normal estimation can be obtained by means of the log-periodogram regression or the Whittle-likelihood function.^{10}

Then, conditionally to the estimated fractionally differencing parameter, is obtained by means of OLS estimation of the model for the fractionally differenced common factors in (2); hence,

, where is the diagonal polynomial matrix in the lag operator of order R, containing the th order truncated binomial expansion of the elements in. Then,

and.

Alternatively, rather than by means of the two-step Box-Jenkins type of approach detailed above, VARFIMA estimation of the R-variate version of the model in (17) can be performed by means of Conditional-Sum-ofSquares [45] , exact Maximum Likelihood [46] or Indirect [47] estimation, still yielding consistent and asymptotically normal estimates.^{11} OLS estimation of a VAR approximation for the VARFIMA model has also been recently proposed in [48] , which would even avoid the estimation of the fractional differencing parameter for the common stochastic factors.

For the case of no integration (short memory) () and integration (), we also have:

2) For the case of no integration (short memory) (),is obtained by means of OLS estimation of the VAR(u) model for the (rotated) common stochastic factors () in (2); then;

3) For the case (),is obtained by means of OLS estimation of the model in levels for the (rotated) common stochastic factors implied by (2); then,

.

Consistent with [49] and [50] , in all of the above cases VAR estimation can be performed as the estimated common factors were actually observed.

Following the thick modelling strategy in [51] , median estimates of the parameters of interest, impulse responses and forecast error variance decomposition, as well as their confidence intervals, can be computed through simulation.

3.3. Step 3: Estimation of the Conditional Variance-Covariance Matrix

The estimation of the conditional variance-covariance matrix for the factors in (2) can be carried out using a procedure similar to the O-GARCH model of [52] :

1) Firstly, conditional variance estimation is carried out factor by factor, using the estimated factor residualsyielding,; qml estimation can be performed in a variety of settings, ranging from standard

and models to their “adaptive” generalizations [9] [12] [53] [54] , in order to allow for different sources of persistence in variance;

2) Secondly, consistent with the assumption of conditional and unconditional orthogonality of the factors, the conditional variance-covariance and correlation matrices for the actual series may be estimated as

(27)

(28)

where, and.

Relaxing the assumption of conditional orthogonality of the factors is also feasible in the proposed framework, as the dynamic conditional covariances, i.e., the off-diagonal elements in, can be obtained, after step i) above, by means of the second step in the estimation of the Dynamic Conditional Correlation model [55] or the Dynamic Equicorrelation model [56] .

3.4. Asymptotic Properties

The proposed iterative procedure for the system of equations in (1) bears the interpretation of QML estimation, using a Gaussian likelihood function, performed by means of the EM algorithm. In the E-step, the unobserved factors are estimated, given the observed data and the current estimate of model parameters, by means of PCA; in the M-step the likelihood function is maximized (OLS estimation of the matrix is performed) under the assumption that the unobserved factors are known, conditioning on their E-step estimate. Convergence to the one-step QML estimate is ensured, as the value of the likelihood function is increased at each step [57] [58] . The latter implementation of the EM algorithm follows from considering the estimated factors by PCA as they were actually observed. In fact, the E-step would also require the computation of the conditional expectation of the estimated factors, which might be obtained, for instance, by means of Kalman smoothing [59] [60] . As shown by [49] and [50] , however, when the unobserved factors are estimated by means of PCA in the e-step, the generated regressors problem is not an issue for consistent estimation in the M-step, due to faster vanishing of the estimation error, provided for linear models, and for (some classes of) non linear models, i.e., the factors estimated by means of

can be considered as they where actually observed, therefore not requiring a Kalman smoothing step.

Note also that the Expectation step of the algorithm relies on consistent estimation of the unobserved components. In this respect, under general conditions, consistency and asymptotic normality of, at each point in time, for the unobserved common components, has been established by [5] and [61] for and the case of I(0) and I(1) unobserved components;^{12} this implies the consistent estimation of the gap vector at the same rate, for

, as well. Based on the results for and processes, the same properties can be conjectured also for the intermediate cases of long memory and (linear/nonlinear) trend stationarity; supporting Monte Carlo evidence is actually provided by [63] and in this study.^{13}

Moreover, likewise in the Maximization step of the algorithm, consistent and asymptotically normal estimation of the polynomial matrix is yield by OLS estimation of the VAR model for the

gap vector, which, according to the results in [49] and [50] , can be taken as it were actually observed in the implementation of the iterative estimation procedure.

Similarly, consistent and asymptotically normal estimation of the block of equations in (2) is obtained by means of OLS estimation of the conditional mean process first, holding the estimated latent factors as they were observed, still relying on the results in [49] and [50] and on a consistent estimate of the fractional differrencing parameter if needed, and then performing estimation of the conditional variance-covariance matrix.

4. Monte Carlo Analysis

Consider the following data generation process (DGP) for the vector process

(29)

where is a matrix of coefficients, and are vectors of loadings, and and are the common deterministic and long memory factors, respectively, at time period, with

. (30)

Then, for the conditionally heteroskedastic case it is assumed

while

for the conditionally homoskedastic case.

Different values for the autoregressive idiosyncratic parameter ρ , common across the N cross-sectional units, have been considered, i.e., , as well as for the fractionally differencing parameter and the common factor autoregressive parameter, setting

for the non integrated case and for the fractionally integrated and integrated cases; is always assumed in the experiment. For the conditional variance equation it is assumed and for the short memory case, and, and for the long memory case. The inverse signal to noise ratio is given by, taking values

. Finally, and are set equal to unitary vectors.

Moreover, in addition to the structural stability case, i.e., , two designs with breaks have been considered for the component, i.e.1) Single step change in the intercept at the midpoint of the sample case, i.e.,

2) The two step changes equally spaced throughout the sample case, with the intercept increasing at one third of the way through the sample and then decreasing at a point two thirds of the length of the sample, i.e.,

The sample size investigated is, and the number of cross-sectional units is. For the no breaks case also other cross-sectional sample sizes have been employed, i.e.,. The number of replications has been set to 2,000 for each case.

The performance of the proposed multi-step procedure has then been assessed with reference to the estimation of the unobserved common stochastic and deterministic factors, and the and ρ autoregressive parameters. Concerning the estimation of the common factors, the Theil’s inequality coefficient () and the correlation coefficient () have been employed in the evaluation, i.e.,

where is the population unobserved component and its estimate. The above statistics have been computed for each Monte Carlo replication and then averaged.

In the Monte Carlo analysis, the location of the break points and the value of the fractional differencing parameter are taken as known, in order to focus on the assessment on the estimation procedure contributed by the paper; the break process is then estimated by means of the OLS regression approach in [6] . The Monte Carlo evidence provided is then comprehensive concerning the no-breaks and cases, as well as the no-break case, concerning the estimation of the common stochastic factor. A relative assessment of the various methodologies which can be employed for the decomposition into break and break-free components is however of interest and left for further research.

4.1. Results

The results for the non integration case are reported in Figure 1, Figure 2 (and 5, columns 1 and 3), while Figure 3, Figure 4 (and 5, columns 2 and 4) refer to the fractionally integrated and integrated cases (the integrated case, independent of the type of integration, thereafter). In all cases, results refer to the estimated parameters for the first equation in the model. Since the results are virtually unaffected by the presence of condi-

Figure 1. In the figure, Monte Carlo bias and RMSE statistics for the autoregressive parameter (f) are plotted for the case of no breaks (top and center plots) and one (break 1) and two (break 2) breaks (bottom plots), and a conditionally heteroskedastic common I(0) factor. Results are reported for various values of the persistence spread f ‒ ρ (0.2, 0.4, 0.6, 0.8) against various values of the (inverse) signal to noise ratio (s/n)^{−1} (4, 2, 1, 05, 0.25). The sample size T is 100 and 500 observations, the number of cross-sectional units N is 30, and the number of replications for each case is 2000. For the no breaks case, Monte Carlo bias statistics are also reported for other sample sizes N (5, 10, 15, 50) (center plots).

tional heteroskedasticity, for reasons of space, only the heteroskedastic case is discussed. Moreover, only the results for the case are reported for the integrated case, as similar results have been obtained for the case.^{14}

4.1.1. The Structural Stability Case

As shown in Figure 5 (top plots 1-4), for a cross-sectional sample size N = 30 units, a negligible downward bias for the parameter (on average across (inverse) signal to noise ratio values) can be noted (−0.02 and −0.03, for the non integrated and integrated case, respectively, and (top plots 1-2); −0.01 and −0.006, respectively, and (top plots 3-4)), decreasing as the serial correlation spread, or, or the sample size increase.

On the contrary, as shown in Figure 1 and Figure 3 (top plots 1 and 3), the downward bias in f is increasing with the degree of persistence of the common factor, the (inverse) signal to noise ratio s/n-1 , and the serial correlation spread, or, yet decreasing with the sample size.

For the non integrated case (Figure 1, plots 1 and 3), there are only few cases () when a 10%, or larger, bias in is found, occurring when the series are particularly noisy; for the stationary long memory case a 10% bias, or smaller, is found for, while for the non stationary long memory case for and a (relatively) large sample () (Figure 3, plots 1 and 3). Increasing the cross-sectional dimension yields improvements (see the next section).

Also, as shown in Figure 2 and Figure 4 (top plots 1-4), very satisfactory is the estimation of the unobserved common stochastic factor, as the statistic is always below 0.2 (0.14 (0.10)), on average, for () for the non integrated case (Figure 2, top plots 2 and 4); 0.06 (0.03), on average, for () for the integrated case (Figure 4, top plots 2 and 4). Moreover, the correlation coefficient between

Figure 2. In the figure, Monte Carlo Carlo Theil’s index (IC) and correlation coefficient (Corr) statistics, concerning the estimation of the conditionally heteroskedastic common I(0) factor, are plotted for the case of no breaks (top and center plots) and one (break 1) and two (break 2) breaks (bottom plots). Results are reported for various values of the persistence spread f ‒ ρ (0.2, 0.4, 0.6, 0.8) against various values of the (inverse) signal to noise ratio (s/n)^{−1} (4, 2, 1, 05, 0.25). The sample size T is 100 and 500 observations, the number of cross-sectional units N is 30, and the number of replications for each case is 2000. For the no breaks case, Monte Carlo correlation coefficient statistics are also reported for other sample sizes N (5, 10, 15, 50) (center plots).

the actual and estimated common factors is always very high, 0.98 and 0.99, on average, respectively, for both sample sizes (Figure 2 and Figure 4, top plots 1 and 3).

Results for smaller and larger cross-sectional samples. In Figures 1-4 (center plots, i.e., rows 2 and 3), the bias for the parameter and the correlation coefficient between the actual and estimated common factors are also plotted for different cross-sectional dimensions, i.e., , for the non integrated and integrated cases, respectively; statistics for the parameter are not reported, as the latter is always unbiasedly estimated, independently of the cross-sectional dimension.

As is shown in the plots, the performance of the estimator crucially depends on, , and.

For the non integrated case (Figure 1), when the (inverse) signal to noise ratio is low, i.e., the downward bias is already mitigated by using a cross-sectional sample size as small as, for

; as increases, similar results are obtained for higher, i.e., and, or

and (center plots, column 1-2). For a larger sample size, i.e., (center plotscolumn 3-4), similar conclusions hold, albeit for the the (inverse) signal to noise ratio can be higheri.e.,; similarly for with.

For the integrated case (Figure 3) conditions are slightly more restrictive; in particular, for the stationary long memory case, when the (inverse) signal to noise ratio is low, i.e., , the downward bias is already mitigated by setting and; similar results are obtained for higher and N, i.e., and, or and (center plots, column 1-2). Similar conclusions can be drawn for (center plots, column 3-4), albeit, holding constant, accurate estimation is

Figure 3. In the figure, Monte Carlo bias statistics for the autoregressive parameter (f) are plotted for the case of no breaks (top and center plots) and one (break 1) and two (break 2) breaks (center and bottom plots), and a conditionally heteroskedastic common I(d) factor (). Results are reported for various values of the persistence spread d − ρ (0.2, 0.4, 0.6, 0.8, 1) against various values of the (inverse) signal to noise ratio (s/n)^{−1} (4, 2, 1, 05, 0.25). The sample size T is 100 and 500 observations, the number of cross-sectional units N is 30, and the number of replications for each case is 2000. For the no breaks case, Monte Carlo bias statistics are also reported for other sample sizes N (5, 10, 15, 50) (center plots).

obtained also for higher. Similarly also for the non stationary case (long memory or); yet, holding constant, either larger, or lower, would be required for accurate estimation.

Coherently, the correlation coefficients between the actual and estimated common factors (Figure 2 and Figure 4, center plots) point to satisfactory estimation (a correlation coefficient higher than 0.9) also in the case of a small temporal sample size, provided the (inverse) signal to noise ratio is not too high, and/or the cross-sectional dimension is not too low and; and; and).

4.1.2. The Structural Change Case

While concerning the estimation of the parameter no sizable differences can be found for the designs with structural change, relatively to the case of structural stability^{15}, the complexity of the break process may on the other hand affect estimation accuracy for the parameter, worsening as the number of break points increases, particularly when the temporal sample size is small ().

Yet, for the no integration case (Figure 1, bottom plots), already for the performance is very satisfactory for both designs, independently of the (inverse) signal to noise ratio (bottom plots, columns 3 and 4); on the contrary, for the performance is satisfactory (at most a 10% bias) only when the series are not too noisy () (bottom plots, columns 1 and 2). Also, similar to the structural stability case, the (downward) bias in the parameter is increasing with the degree of persistence of the common factor d, the (inverse) signal to noise ratio, and or, yet decreasing with the sample size.

Coherent with the above results, satisfactory estimation of the unobserved common stochastic factor (Figure 2, bottom plots) and break process can also be noted (Figure 5, bottom plots, columns 1 and 3); for

Figure 4. In the figure, Monte Carlo Carlo correlation coefficient (Corr) statistics, concerning the estimation of the conditionally heteroskedastic common I(d) factor (), are plotted for the case of no breaks (top and center plots) and one (break 1) and two (break 2) breaks (bottom plots). Results are reported for various values of the persistence spread d − ρ (0.2, 0.4, 0.6, 0.8, 1) against various values of the (inverse) signal to noise ratio (s/n)^{−1} (4, 2, 1, 05, 0.25). The sample size T is 100 and 500 observations, the number of cross-sectional units N is 30, and the number of replications for each case is 2000. For the no breaks case, Monte Carlo correlation coefficient statistics are also reported for other sample sizes N (5, 10, 15, 50) (center plots).

the common stochastic factor, the ic statistic (not reported) is in fact always below 0.2 for (0.11 and 0.13, on average, for the single break point and two-break points case, respectively) and below 0.3 for (0.17 and 0.20, on average; column 1), while the actual and estimated common stochastic factors are strongly correlated: for (), on average, the correlation coefficient is 0.96 (0.98) for the single breakpoint case and 0.93 (0.97) for the two-break points case (column 3).

Very accurate is also the estimation of the common break process: the IC statistic is never larger than 0.15 for and 0.075 for (Figure 5, bottom plots, columns 1 and 3), while the correlation coefficient is virtually 1 for the single break case and never below 0.96 for and 0.99 for for the two-break points case (not reported). Given the assumption of known break points, the performance in terms of correlation coefficient is not surprising; yet, the very small Theil’s index is indicative of success in recovering the changing level of the unobserved common break process.

Concerning the integrated case, some differences relatively to the nonintegrated case can be noted; as shown in Figure 5 (bottom plots, columns 2 and 4), albeit the overall recovery of the common break process is always very satisfactory across the various designs, independently of the sample size (the IC statistic is never larger than 0.14; bottom plots), performance slightly worsens as the complexity of the break process and persistence intensity (d ) increase: the average correlation coefficient between the estimated and actual break processes (center plots) falls from 1 when (single break point case) to 0.93 when (two-break points case).

Moreover, concerning the estimation of the common stochastic factor (Figure 4, center and bottom plots, columns 1-4), for the covariance stationary case () results are very close to the non integrated case, i.e., an IC statistic (not reported) always below 0.2 for (0.12 and 0.14, on average, for the single break point and two-break points case, respectively) and below 0.3 for (0.21 and 0.24, on average, respectively); the correlation coefficient is also very high: 0.94 and 0.91, on average, (columns 1

Figure 5. In the figure, average Monte Carlo statistics (across values for the inverse signal to noise ratio) for the bias in the autoregressive idiosyncratic parameter (ρ) (top plots) and Theil’s index (IC) statistic for the common break process (bottom plots) are plotted for the non integrated (I(0)) and integrated (I(d),) cases. Results are reported for various values of the persistence spreads f − ρ (0.2, 0.4, 0.6, 0.8) and d − ρ (0.2, 0.4, 0.6, 0.8, 1). The sample size T is 100 and 500 observations, the number of cross-sectional units N is 30, and the number of replications for each case is 2000.

and 2); 0.97 and 0.96, on average, (columns 3 and 4).

On the contrary, for the non stationary case performance is worse, showing average IC statistics (not reported) of 0.32 (0.32) and 0.42 (0.44), respectively, for the single break point (center plots) and two-break points (bottom plots) case and (); the average correlation coefficient is 0.79 (0.78) and 0.68 (0.66), respectively. Coherently, a worsening in the estimation of the common factor autoregressive parameter, for the and case, can be noted (Figure 3, center and bottom plots), while comparable results to the short memory case can be found for. The latter findings are however not surprising, as the stronger the degree of persistence of the stochastic component (and of the series, therefore) and the less accurate the disentangling of the common break and break-free parts can be expected; overall, Monte Carlo results point to accurate decompositions also for the case of moderate nonstationary long memory, albeit deterioration in performance becomes noticeable.

5. Conclusion

In the paper, a general strategy for large-scale modeling of macroeconomic and financial data, set within the factor vector autoregressive model (F-VAR) framework is introduced. The proposed approach shows minimal pretesting requirements, performing well independently of integration properties of the data and sources of persistence, i.e., deterministic or stochastic, accounting for common features of different kinds, i.e., common integrated (of the fractional or integer type) or non integrated stochastic factors, also heteroskedastic, and common deterministic break processes. Consistent and asymptotically normal estimation is performed by means of QML, implemented through an iterative multi-step algorithm. Monte Carlo results strongly support the proposed approach. Empirical implementations can be found in [37] [67] -[69] , showing the approach being easy to implement and effective also in the case of very large systems of dynamic equations.

Acknowledgements

A previous version of the paper was presented at the 19th and 21st Annual Symposium of the Society for Non Linear Dynamics and Econometrics, the 4th and 6th Annual Conference of the Society for Financial Econometrics, the 65th European Meeting of the Econometric Society (ESEM), the 2011 NBER-NSF Time Series Conference, the 5th CSDA International Conference on Computational and Financial Econometrics. The author is grateful to conference participants, N. Cassola, F. C. Bagliano, C. Conrad, R. T. Baillie, J. Bai, for constructive comments. This project has received funding from the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement no. 3202782013-2015 and PRINMIUR 2009.

As Mitsuo Aida wrote in one of his poems, somewhere in life/there is a path/that must be taken regardless of how hard we try to avoid it/at that time, all one can do is remain silent and walk the path/neither complaining nor whining/saying nothing and walking on/just saying nothing and showing no tears/it is then/as human beings, /that the roots of our souls grow deeper. This paper is dedicated to the loving memory of A.

NOTES

^{*}The literature on F-VAR models is large. See for a survey.

^{2}Monte Carlo evidence reported in suggests that the truncation lag should increase with the sample size and the complexity of the ARFIMA representation of the long memory process, still remaining very small relatively to the sample size. For instance, for the covariance stationary fractional white noise case and a sample of 100 observations truncation can be set as low as 6 lags, while for a sample of 10,000 observations it should be increased to 14 lags; for the case of a covariance stationary ARFIMA (1,d,1) process and a sample of 1000 observations truncation may be set to 30 lags. See for further details.

^{3}See for a recent survey.

^{4}The orthogonality of and can however also be imposed when jointly estimating the deterministic and stochastic components by means of augmented ARFIMA models.

^{5}The strong consistency of the estimator of the break fraction, independently of the rate of decay of the autocovariance function of the error process, has been proved in when the number of break points is known; a modified Bayes-Schwarz selection criterion for the number of break points is also proposed.

^{6}Alternatively, can be obtained by regressing on (and the initial estimate), using OLS.

^{7}Alternatively, the new break-free process can be computed as.

^{8}Alternatively, can be obtained by regressing on and the updated estimate, using OLS. This would also yield a new estimate to be used in the computation of the updated gap vector.

^{9}For instance, the procedure can be stopped when, where the objective function is written as in (26).

^{10}See and for a survey of alternative estimators of the fractional differencing parameter.

^{11}Depending on the parametric structure, system estimation may however become unfeasible when the number of factors is too large.

^{12}In particular, under some general conditions, given any invertible matrix, consistency and asymptotic normality of PCA for, at each point in time, is established for and and the case of unobserved factors and idiosyncratic components, the latter also displaying limited heteroskedasticity in both their time-series and cross-sectional dimensions ; for 𝑁,𝑇→∞ and 𝑁𝑇

3

→0 and the case of (non cointegrated) unobserved factors and idiosyncratic components, similarly showing limited heteroskedasticity in both the time-series and cross-sectional dimensions (). The latter result is actually obtained by applying PCA to the level of the series, rather than their first differences. Moreover, for both the and case, consistency and asymptotic normality of PCA for is established under the same conditions, as well as consistency and asymptotic normality of PCA for the unobserved common components, at each point in time, for.

The conditions for consistency and asymptotic normality reported in and implicitly cover also the case in which PCA is implemented using the estimated break and break-free components, rather than the observed

𝑥

𝑡

series; in fact, by assuming and, then and, which are static factor structures as assumed in and . It appears that assumption E in , page 143, i.e., weak dependence and limited cross-sectional correlation, holding for both noise (estimation error) components and, augmented with the assumption of their contemporaneous orthogonality, i.e., , is then sufficient for the validity of PCA also when implemented on noisy data. In this respect PCA acts as noise suppressor: intuitively, PCs associated with the smallest eigenvalues are noise, which should be neglected when estimating the common factors. PCA estimation of the signal component can actually be shown to be optimal in terms of minimum mean square error ^{[6}2].

13The use of PCA for the estimation of common deterministic trends has previously been advocated by . See also for applications to nonstationary data.

^{14}Detailed results are available in the working paper version of this paper or upon request from the author.

^{15}The average bias is −0.04 and −0.01, independent of the break process design and integration properties, when and, respectively. Moreover, similar to the structural stability case the bias is decreasing as, , or the sample size increase, independent of the (inverse) signal to noise ratio.

Conflicts of Interest

The authors declare no conflicts of interest.

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