Publication: Essays on Identification of Structural VARMA models
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2023-06
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2023-06-23
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Abstract
Structural vector autoregressive models are widely employed in macro-econometrics
since they provide a simple, practical representation of structural theoretical models.
For decades, the most preferred framework has been the structural vector
autoregressive models, under the belief that structural economics models are
fundamental, which intuitively means that information generated from the history
(current and past values) of observable, endogenous variables is equivalent to the
information constructed from the history of unobservable economic shocks.
Under such an assumption, most empirical studies only focus on the identification
challenges of the contemporaneous effects matrix or the static component of the
model¿the strategies for overcoming this identification problem resort to using external
information. Early proposals impose zero or linear restrictions over the matrix of
contemporaneous effects. Another type of restriction is imposed on the long-run effects
of economic shocks over some outcomes of interests, also known as long-run
identification restrictions, such as the one employed. An alternative strategy is based
on sign restrictions, which only restrain the direction (sign) of the response of some
endogenous variable to changes in a single or a subset of economic shocks. For
instance, it is a consensus to assume that a positive demand shock increases output
and inflation in the short run.
On the other hand, a more economic-based identification strategy is the narrative
approach. This scheme constructs a measure for an unobserved economic shock by
analyzing extensive policy change information. For instance, Romer and Romer (1989)
construct a measure of monetary policy shock by analyzing historical records of
Federal Reserve meetings. Recently, an identification strategy that uses external
measures as a proxy or instrumental variables for identifying a single shock or a subset
of structural shocks has been introduced.
Despite the usefulness of these strategies, they possess essential drawbacks. In the
case of zero, linear or long-run identification restrictions, most of these conditions are
not subject to empirical evaluation unless they are overidentification restrictions. Their
support must be done by appealing to economic intuitions, some of which may be
arguable. Besides, some of the identification restrictions may be challenging to
implement. For instance, zero long-run restrictions are commonly approximated by
restricting the cumulative response of a shock over a finite time horizon.
Regarding identification based on sign restrictions, these can only identify a set of
models. Consequently, this strategy entails two sources of variability: sample and
model uncertainty. Isolating each of these sources of variance is challenging. Finally,
regarding the strategy using proxy or instrumental variables, its identification power
depends on the exogeneity and relevance conditions. Besides, as it is documented by
Angelini et al. (2002) when proxy SVARs feature multiple target shocks, it requires
additional point or sign restrictions. Last but not least, most of these strategies have in
common that they were developed for identifying shocks in a causal (or fundamental)
structural VAR model.
Besides, the fundamental representation assumption of the structural model may need
to be revised, mainly when the econometrician's information set is smaller than the
information set of the theoretical model. Moreover, it has been documented that this
case may not be quite an unrealistic situation. Empirically, to deal with possibly non-
fundamental structural models is necessary not to impose the location of roots outside
the complex unit circle but only rule out unit roots in dynamic lag polynomials.
Therefore, it is critical to develop methods for estimating structural vector
autoregressive moving-average models, where the location of dynamic lag polynomial
roots is unknown and must be determined from data.
Nevertheless, distinguishing fundamental from non-fundamental models cannot be
accomplished if the researcher only exploits second-order information, i.e., requiring
that structural shocks are serially and mutually uncorrelated. Because, as has been
documented in the literature, it is possible to find at least two models, fundamental and
non-fundamental, with serially uncorrelated errors. This issue is what the literature calls
the dynamic identification problem (see Velasco (2022)).
Consequently, to overcome this identification problem, higher-order information on the
errors must be exploited. This fact leads to abandoning the Gaussian behavior
assumption regarding the structural shocks and assuming that structural shocks are
non-Gaussian distributed. Otherwise, it is unfeasible to distinguish fundamental from
non-fundamental models. In recent years, several methodological works have
proposed alternatives for identifying and estimating structural vector autoregressive
moving-average models without imposing the location of roots by exploiting the non-
Gaussian behavior of structural shocks. Most of these investigations assumed that
shocks are serial and mutually independent and identified the structural parameters
following two approaches. The first strategy is to characterize non-Gaussian behavior
by assuming non-zero third or fourth-order cumulants. This strategy is like a method of
moments approach. Therefore, for constructing standard errors of the estimates, it is
necessary to impose the finiteness of sixth or eighth-order moments. This assumption
rules out essential types of non-Gaussian behavior.
On the other hand, a second approach is to characterize the non-Gaussian behavior
assuming a particular joint density function for the structural shocks and imposing, at
least, non-zero third-order cumulants. This research attempts to fill a gap in the
previous literature by proposing an estimation method that exploits the joint
characteristic function of pairwise errors. This approach does not require imposing the
finiteness of many higher-order moments or the knowledge of a specific joint density
distribution of structural shocks. Moreover, the proposed method may be adapted for
situations where structural shocks exhibit higher-order serial dependence while mutual
independence is still assumed.
Specifically, the first chapter explains that causal (or fundamental) structural vector
autoregressions models may deliver biased estimates of causal effects of economic
shocks when data comes from a non-fundamental structural model. As discussed
above, this paper exploits the joint characteristic function of pairwise model errors
instead of using specific moments or cumulants or imposing a particular density
function for the structural shocks. Thus, our measure dependence is a generalized
autocovariance function of the complex exponential model errors. We summarize the
information on the autocovariance function using the cumulative spectral distribution,
which is the integral of the spectral density function of model errors. We opt for the
cumulative spectral function instead of the spectral density because its sample
counterpart does not require a smoothing kernel. Moreover, this feature makes that
weak convergence of estimators occurs at the standard rate.
In this line, the contributions of this chapter are twofold. On the one hand, we extend a
well-known identification result for SVARMA models with independent shocks to a
situation in which disturbances can exhibit a more general serial dependence structure;
on the other hand, we propose an estimation procedure that minimizes the distance of
the generalized cumulative spectral distribution of unrestricted and restricted model
errors. This procedure does not require imposing a particular joint non-Gaussian
distribution of structural errors, as well as it does not need the existence of a large
number of higher-order moments. Furthermore, in the case of having serially
dependent shocks, identification (up to sign-permutation) only requires the existence of
second-order moments of structural shocks and peculiar characterization of non-
Gaussian behavior, consisting of non-linearity of expectations of structural errors
conditional to non-constant linear filters. Regarding the asymptotic properties of our
estimators, in the case of fully independent shocks, the finiteness of third-order
moments is the only requisite to find the asymptotic distribution of the structural
parameter estimator; in the case of serially dependent structural errors, it is required
the finiteness of at least fifth-order moments for finding the asymptotic distribution of
the estimator.
The simulation evidence shows that our method satisfactorily identifies the location of
the roots with a relatively small sample size. In the case of the bivariate structural
model, the rate of correct root location is above 90 percent when structural shocks are
asymmetric. In contrast, this rate decays when the errors follow a symmetric non-
Gaussian distribution, especially for VARMA models with non-zero AR and MA
components. To analyze our method's robustness, we study how well the proposal
identifies the location of roots when structural shocks behave like a multivariate
Gaussian random element. The rate of correct root location obtained with the
Montecarlo experiment is quite close to the theoretical rate when only errors are
required to be white noise.
Finally, we applied our proposal to the dataset analyzed by Blanchard and Quah
(1989). This dataset contains only two endogenous variables, GDP growth, and
unemployment rate. We select two different specifications. The first case assumes that
the roots of the AR polynomial lie outside the unit circle, but the location of the roots of
the MA polynomial is not determined. The second specification is an unrestricted
structural model, which does not restrict the root location of both the AR and MA
polynomials. In both scenarios, the polynomials' overall degree equals one,
respectively. The estimation results are the following: for the first restricted
specification, our proposed method identifies that the roots of the MA polynomial lie
outside the unit circle, i.e., our method identifies that the MA polynomial is invertible.
Based on the estimation of the structural model, we compute the estimated causal
effects of both identified shocks (i.e., impulse response functions of each shock). The
analysis of these causal effects shows that we identify a shock with a non-zero long-
run effect and another with zero long-run effect over GDP growth. It is worth mentioning
that identifying these shocks did not require imposing any external identification
restriction. In the scenario of not restricting the SVARMA model's root location, our
method identifies MA roots outside the unit circle but AR roots inside the unit circle. In
other words, our proposal identifies a non-causal but invertible SVARMA model.
In the second chapter, we investigate the feasibility of empirically verifying one of the
critical identification assumptions of the statistical strategy for identifying structural
vector autoregressive moving-average models, i.e., the requirement about the non-
Gaussian behavior of all shocks in the system. In particular, the statistical identification
of possibly non-fundamental structural vector autoregressive moving-average models
requires that shocks satisfy: (i) to be an i.i.d process, (ii) to be mutually independent
across components, and (iii) each shock must be non-Gaussian distributed. Therefore,
provided the first two requirements, it is crucial to be able to evaluate the third condition
empirically without requiring estimation of the structural model because this latter may
not be identified, and with a non-zero probability, an attempt to estimate the structural
model may deliver an inconsistent one. However, in the literature, few efforts have
been posed to determine the non-Gaussian dimension in a structural system. Most of
these investigations assume that the structural model is fundamental, which makes it
unfeasible to apply such proposals to more general contexts.
Consequently, this research attempts to fill this gap in the literature and proposes a
method for determining the non-Gaussian dimension when the location of polynomial
roots is still being determined. In that sense, our proposal is robust to the location of
the roots of the dynamic polynomials in the model. Therefore, it can be applied to
fundamental or non-fundamental contexts.
We address the unknown location of roots by relating the non-Gaussian dimension of
structural errors to the rank of a matrix built from the higher-order spectrum of reduced-
form errors. The estimation of the spectral density of the reduced form errors is done
non-parametrically, making the convergence rates of the asymptotic distribution of the
estimators to be slower than the standard one. Our estimation approach for the non-
Gaussian dimension is a sequential testing procedure. Precisely, we follow the
Kleibergen and Paap (hereafter KP) approach. The KP -statistic is built from the
singular value decomposition of the interest matrix. The asymptotic distribution of the
statistic is a standard chi-square whose degrees of freedom change depending on the
null hypothesis. Unlike the KP article, our context changes the asymptotic distribution
of the test statistic for some particular rank values, specifically under joint Gaussianity.
For other null hypotheses, the asymptotic distribution is chi-square, but the degrees of
freedom are generally unknown.
For this reason, we propose a bootstrap test strategy. This path implies another
significant challenge: to impose the null hypothesis in the resampled data. The way for
creating bootstrap samples follows the idea of Nordhausen et al. (2017).
The idea is to employ the characteristic vectors associated with zero singular values
under each specific null hypothesis. These vectors span the Gaussian dimension.
These vectors create a projection matrix over the non-Gaussian dimension, while the
Gaussian dimension is completed with values drawn from a multivariate standard
Normal distribution. The Montecarlo exercises were performed to analyze the size and
power of the bootstrap test to show that our strategy estimates satisfactorily the non-
Gaussian dimension. We apply our procedure to two well-known macroeconomic
datasets, such as Blanchard and Quah (1989) and Blanchard and Perotti (2002).
Unlike the work of Guay (2021), we can detect at least two non-Gaussian shocks in the
Blanchard-Perotti dataset. Meaning that a statistical identification strategy can be
applied to this database, at least assuming that polynomial roots are outside the
complex unit circle.
Finally, the third chapter has explored the conditions under which the combination of
higher order cumulants identification and sign restrictions may be combined to obtain
global identification. The statistical identification strategy of causal structural vector
autoregressive models has gained attention recently because it permits the
identification of the causal effects of structural shocks without resorting to external
identification restrictions. This strategy exploits higher-order information provided
structural errors are mutually independent and non-Gaussian distributed. This
approach only identifies the structural model up to the signed permutation, which
implies a finite set of admissible models. The solution proposed in the literature for
selecting a model relies on applying a mechanical procedure. This paper aims to study
an alternative scheme based on imposing economic-motivated sign restrictions over
the causal effects of structural shocks. Only the permutations matter when a single,
strict sign restriction is applied to each shock. A sufficient condition for achieving global
identification is that the matrix of contemporaneous effects has a generalized recursive
structure. Furthermore, when all shocks are either asymmetric or non-mesokurtic, we
can find a relationship between the recursive structure of the contemporaneous effects
matrix and the rank of the blocks of the matrix version of third or fourth-order cumulants
of reduced-form errors. This result can be seen as an alternative path for determining
the recursive structure to the standard procedure based on applying Granger's
causality test to each endogenous variable.
A weaker sufficient condition for accomplishing point identification of a single shock
requires imposing as many sign restrictions as the number of endogenous variables in
the model and that the system of sign restrictions is sign-solvable. It also explores the
necessary conditions for having a non-empty identified set when only second-order
cumulants and sign restrictions are employed. Although evident, this condition is not
considered and taken as granted in several empirical works that employ sign
restrictions for identifying economic shocks. The condition states that when the Gram-
Schmidt algorithm is applied to each tuple of the vectors in the solution set of the sign
restrictions intersects the Cartesian product, the identified set is non-empty. This
condition holds immediately when only a small set of sign restrictions are imposed over
the shocks and the restrictions are set over the same objects. On the other hand, when
the model contains more than one Gaussian structural error, identification using higher-
order cumulants ensures that the non-Gaussian block remains identified, although the
order is unknown.
Structural vector autoregressive models are widely employed in macro-econometrics since they provide a simple, practical representation of structural theoretical models. For decades, the most preferred framework has been the structural vector autoregressive models, under the belief that structural economics models are fundamental, which intuitively means that information generated from the history (current and past values) of observable, endogenous variables is equivalent to the information constructed from the history of unobservable economic shocks. Under such an assumption, most empirical studies only focus on the identification challenges of the contemporaneous effects matrix or the static component of the model¿the strategies for overcoming this identification problem resort to using external information. Early proposals impose zero or linear restrictions over the matrix of contemporaneous effects. Another type of restriction is imposed on the long-run effects of economic shocks over some outcomes of interests, also known as long-run identification restrictions, such as the one employed. An alternative strategy is based on sign restrictions, which only restrain the direction (sign) of the response of some endogenous variable to changes in a single or a subset of economic shocks. For instance, it is a consensus to assume that a positive demand shock increases output and inflation in the short run. On the other hand, a more economic-based identification strategy is the narrative approach. This scheme constructs a measure for an unobserved economic shock by analyzing extensive policy change information. For instance, Romer and Romer (1989) construct a measure of monetary policy shock by analyzing historical records of Federal Reserve meetings. Recently, an identification strategy that uses external measures as a proxy or instrumental variables for identifying a single shock or a subset of structural shocks has been introduced. Despite the usefulness of these strategies, they possess essential drawbacks. In the case of zero, linear or long-run identification restrictions, most of these conditions are not subject to empirical evaluation unless they are overidentification restrictions. Their support must be done by appealing to economic intuitions, some of which may be arguable. Besides, some of the identification restrictions may be challenging to implement. For instance, zero long-run restrictions are commonly approximated by restricting the cumulative response of a shock over a finite time horizon. Regarding identification based on sign restrictions, these can only identify a set of models. Consequently, this strategy entails two sources of variability: sample and model uncertainty. Isolating each of these sources of variance is challenging. Finally, regarding the strategy using proxy or instrumental variables, its identification power depends on the exogeneity and relevance conditions. Besides, as it is documented by Angelini et al. (2002) when proxy SVARs feature multiple target shocks, it requires additional point or sign restrictions. Last but not least, most of these strategies have in common that they were developed for identifying shocks in a causal (or fundamental) structural VAR model. Besides, the fundamental representation assumption of the structural model may need to be revised, mainly when the econometrician's information set is smaller than the information set of the theoretical model. Moreover, it has been documented that this case may not be quite an unrealistic situation. Empirically, to deal with possibly non- fundamental structural models is necessary not to impose the location of roots outside the complex unit circle but only rule out unit roots in dynamic lag polynomials. Therefore, it is critical to develop methods for estimating structural vector autoregressive moving-average models, where the location of dynamic lag polynomial roots is unknown and must be determined from data. Nevertheless, distinguishing fundamental from non-fundamental models cannot be accomplished if the researcher only exploits second-order information, i.e., requiring that structural shocks are serially and mutually uncorrelated. Because, as has been documented in the literature, it is possible to find at least two models, fundamental and non-fundamental, with serially uncorrelated errors. This issue is what the literature calls the dynamic identification problem (see Velasco (2022)). Consequently, to overcome this identification problem, higher-order information on the errors must be exploited. This fact leads to abandoning the Gaussian behavior assumption regarding the structural shocks and assuming that structural shocks are non-Gaussian distributed. Otherwise, it is unfeasible to distinguish fundamental from non-fundamental models. In recent years, several methodological works have proposed alternatives for identifying and estimating structural vector autoregressive moving-average models without imposing the location of roots by exploiting the non- Gaussian behavior of structural shocks. Most of these investigations assumed that shocks are serial and mutually independent and identified the structural parameters following two approaches. The first strategy is to characterize non-Gaussian behavior by assuming non-zero third or fourth-order cumulants. This strategy is like a method of moments approach. Therefore, for constructing standard errors of the estimates, it is necessary to impose the finiteness of sixth or eighth-order moments. This assumption rules out essential types of non-Gaussian behavior. On the other hand, a second approach is to characterize the non-Gaussian behavior assuming a particular joint density function for the structural shocks and imposing, at least, non-zero third-order cumulants. This research attempts to fill a gap in the previous literature by proposing an estimation method that exploits the joint characteristic function of pairwise errors. This approach does not require imposing the finiteness of many higher-order moments or the knowledge of a specific joint density distribution of structural shocks. Moreover, the proposed method may be adapted for situations where structural shocks exhibit higher-order serial dependence while mutual independence is still assumed. Specifically, the first chapter explains that causal (or fundamental) structural vector autoregressions models may deliver biased estimates of causal effects of economic shocks when data comes from a non-fundamental structural model. As discussed above, this paper exploits the joint characteristic function of pairwise model errors instead of using specific moments or cumulants or imposing a particular density function for the structural shocks. Thus, our measure dependence is a generalized autocovariance function of the complex exponential model errors. We summarize the information on the autocovariance function using the cumulative spectral distribution, which is the integral of the spectral density function of model errors. We opt for the cumulative spectral function instead of the spectral density because its sample counterpart does not require a smoothing kernel. Moreover, this feature makes that weak convergence of estimators occurs at the standard rate. In this line, the contributions of this chapter are twofold. On the one hand, we extend a well-known identification result for SVARMA models with independent shocks to a situation in which disturbances can exhibit a more general serial dependence structure; on the other hand, we propose an estimation procedure that minimizes the distance of the generalized cumulative spectral distribution of unrestricted and restricted model errors. This procedure does not require imposing a particular joint non-Gaussian distribution of structural errors, as well as it does not need the existence of a large number of higher-order moments. Furthermore, in the case of having serially dependent shocks, identification (up to sign-permutation) only requires the existence of second-order moments of structural shocks and peculiar characterization of non- Gaussian behavior, consisting of non-linearity of expectations of structural errors conditional to non-constant linear filters. Regarding the asymptotic properties of our estimators, in the case of fully independent shocks, the finiteness of third-order moments is the only requisite to find the asymptotic distribution of the structural parameter estimator; in the case of serially dependent structural errors, it is required the finiteness of at least fifth-order moments for finding the asymptotic distribution of the estimator. The simulation evidence shows that our method satisfactorily identifies the location of the roots with a relatively small sample size. In the case of the bivariate structural model, the rate of correct root location is above 90 percent when structural shocks are asymmetric. In contrast, this rate decays when the errors follow a symmetric non- Gaussian distribution, especially for VARMA models with non-zero AR and MA components. To analyze our method's robustness, we study how well the proposal identifies the location of roots when structural shocks behave like a multivariate Gaussian random element. The rate of correct root location obtained with the Montecarlo experiment is quite close to the theoretical rate when only errors are required to be white noise. Finally, we applied our proposal to the dataset analyzed by Blanchard and Quah (1989). This dataset contains only two endogenous variables, GDP growth, and unemployment rate. We select two different specifications. The first case assumes that the roots of the AR polynomial lie outside the unit circle, but the location of the roots of the MA polynomial is not determined. The second specification is an unrestricted structural model, which does not restrict the root location of both the AR and MA polynomials. In both scenarios, the polynomials' overall degree equals one, respectively. The estimation results are the following: for the first restricted specification, our proposed method identifies that the roots of the MA polynomial lie outside the unit circle, i.e., our method identifies that the MA polynomial is invertible. Based on the estimation of the structural model, we compute the estimated causal effects of both identified shocks (i.e., impulse response functions of each shock). The analysis of these causal effects shows that we identify a shock with a non-zero long- run effect and another with zero long-run effect over GDP growth. It is worth mentioning that identifying these shocks did not require imposing any external identification restriction. In the scenario of not restricting the SVARMA model's root location, our method identifies MA roots outside the unit circle but AR roots inside the unit circle. In other words, our proposal identifies a non-causal but invertible SVARMA model. In the second chapter, we investigate the feasibility of empirically verifying one of the critical identification assumptions of the statistical strategy for identifying structural vector autoregressive moving-average models, i.e., the requirement about the non- Gaussian behavior of all shocks in the system. In particular, the statistical identification of possibly non-fundamental structural vector autoregressive moving-average models requires that shocks satisfy: (i) to be an i.i.d process, (ii) to be mutually independent across components, and (iii) each shock must be non-Gaussian distributed. Therefore, provided the first two requirements, it is crucial to be able to evaluate the third condition empirically without requiring estimation of the structural model because this latter may not be identified, and with a non-zero probability, an attempt to estimate the structural model may deliver an inconsistent one. However, in the literature, few efforts have been posed to determine the non-Gaussian dimension in a structural system. Most of these investigations assume that the structural model is fundamental, which makes it unfeasible to apply such proposals to more general contexts. Consequently, this research attempts to fill this gap in the literature and proposes a method for determining the non-Gaussian dimension when the location of polynomial roots is still being determined. In that sense, our proposal is robust to the location of the roots of the dynamic polynomials in the model. Therefore, it can be applied to fundamental or non-fundamental contexts. We address the unknown location of roots by relating the non-Gaussian dimension of structural errors to the rank of a matrix built from the higher-order spectrum of reduced- form errors. The estimation of the spectral density of the reduced form errors is done non-parametrically, making the convergence rates of the asymptotic distribution of the estimators to be slower than the standard one. Our estimation approach for the non- Gaussian dimension is a sequential testing procedure. Precisely, we follow the Kleibergen and Paap (hereafter KP) approach. The KP -statistic is built from the singular value decomposition of the interest matrix. The asymptotic distribution of the statistic is a standard chi-square whose degrees of freedom change depending on the null hypothesis. Unlike the KP article, our context changes the asymptotic distribution of the test statistic for some particular rank values, specifically under joint Gaussianity. For other null hypotheses, the asymptotic distribution is chi-square, but the degrees of freedom are generally unknown. For this reason, we propose a bootstrap test strategy. This path implies another significant challenge: to impose the null hypothesis in the resampled data. The way for creating bootstrap samples follows the idea of Nordhausen et al. (2017). The idea is to employ the characteristic vectors associated with zero singular values under each specific null hypothesis. These vectors span the Gaussian dimension. These vectors create a projection matrix over the non-Gaussian dimension, while the Gaussian dimension is completed with values drawn from a multivariate standard Normal distribution. The Montecarlo exercises were performed to analyze the size and power of the bootstrap test to show that our strategy estimates satisfactorily the non- Gaussian dimension. We apply our procedure to two well-known macroeconomic datasets, such as Blanchard and Quah (1989) and Blanchard and Perotti (2002). Unlike the work of Guay (2021), we can detect at least two non-Gaussian shocks in the Blanchard-Perotti dataset. Meaning that a statistical identification strategy can be applied to this database, at least assuming that polynomial roots are outside the complex unit circle. Finally, the third chapter has explored the conditions under which the combination of higher order cumulants identification and sign restrictions may be combined to obtain global identification. The statistical identification strategy of causal structural vector autoregressive models has gained attention recently because it permits the identification of the causal effects of structural shocks without resorting to external identification restrictions. This strategy exploits higher-order information provided structural errors are mutually independent and non-Gaussian distributed. This approach only identifies the structural model up to the signed permutation, which implies a finite set of admissible models. The solution proposed in the literature for selecting a model relies on applying a mechanical procedure. This paper aims to study an alternative scheme based on imposing economic-motivated sign restrictions over the causal effects of structural shocks. Only the permutations matter when a single, strict sign restriction is applied to each shock. A sufficient condition for achieving global identification is that the matrix of contemporaneous effects has a generalized recursive structure. Furthermore, when all shocks are either asymmetric or non-mesokurtic, we can find a relationship between the recursive structure of the contemporaneous effects matrix and the rank of the blocks of the matrix version of third or fourth-order cumulants of reduced-form errors. This result can be seen as an alternative path for determining the recursive structure to the standard procedure based on applying Granger's causality test to each endogenous variable. A weaker sufficient condition for accomplishing point identification of a single shock requires imposing as many sign restrictions as the number of endogenous variables in the model and that the system of sign restrictions is sign-solvable. It also explores the necessary conditions for having a non-empty identified set when only second-order cumulants and sign restrictions are employed. Although evident, this condition is not considered and taken as granted in several empirical works that employ sign restrictions for identifying economic shocks. The condition states that when the Gram- Schmidt algorithm is applied to each tuple of the vectors in the solution set of the sign restrictions intersects the Cartesian product, the identified set is non-empty. This condition holds immediately when only a small set of sign restrictions are imposed over the shocks and the restrictions are set over the same objects. On the other hand, when the model contains more than one Gaussian structural error, identification using higher- order cumulants ensures that the non-Gaussian block remains identified, although the order is unknown.
Structural vector autoregressive models are widely employed in macro-econometrics since they provide a simple, practical representation of structural theoretical models. For decades, the most preferred framework has been the structural vector autoregressive models, under the belief that structural economics models are fundamental, which intuitively means that information generated from the history (current and past values) of observable, endogenous variables is equivalent to the information constructed from the history of unobservable economic shocks. Under such an assumption, most empirical studies only focus on the identification challenges of the contemporaneous effects matrix or the static component of the model¿the strategies for overcoming this identification problem resort to using external information. Early proposals impose zero or linear restrictions over the matrix of contemporaneous effects. Another type of restriction is imposed on the long-run effects of economic shocks over some outcomes of interests, also known as long-run identification restrictions, such as the one employed. An alternative strategy is based on sign restrictions, which only restrain the direction (sign) of the response of some endogenous variable to changes in a single or a subset of economic shocks. For instance, it is a consensus to assume that a positive demand shock increases output and inflation in the short run. On the other hand, a more economic-based identification strategy is the narrative approach. This scheme constructs a measure for an unobserved economic shock by analyzing extensive policy change information. For instance, Romer and Romer (1989) construct a measure of monetary policy shock by analyzing historical records of Federal Reserve meetings. Recently, an identification strategy that uses external measures as a proxy or instrumental variables for identifying a single shock or a subset of structural shocks has been introduced. Despite the usefulness of these strategies, they possess essential drawbacks. In the case of zero, linear or long-run identification restrictions, most of these conditions are not subject to empirical evaluation unless they are overidentification restrictions. Their support must be done by appealing to economic intuitions, some of which may be arguable. Besides, some of the identification restrictions may be challenging to implement. For instance, zero long-run restrictions are commonly approximated by restricting the cumulative response of a shock over a finite time horizon. Regarding identification based on sign restrictions, these can only identify a set of models. Consequently, this strategy entails two sources of variability: sample and model uncertainty. Isolating each of these sources of variance is challenging. Finally, regarding the strategy using proxy or instrumental variables, its identification power depends on the exogeneity and relevance conditions. Besides, as it is documented by Angelini et al. (2002) when proxy SVARs feature multiple target shocks, it requires additional point or sign restrictions. Last but not least, most of these strategies have in common that they were developed for identifying shocks in a causal (or fundamental) structural VAR model. Besides, the fundamental representation assumption of the structural model may need to be revised, mainly when the econometrician's information set is smaller than the information set of the theoretical model. Moreover, it has been documented that this case may not be quite an unrealistic situation. Empirically, to deal with possibly non- fundamental structural models is necessary not to impose the location of roots outside the complex unit circle but only rule out unit roots in dynamic lag polynomials. Therefore, it is critical to develop methods for estimating structural vector autoregressive moving-average models, where the location of dynamic lag polynomial roots is unknown and must be determined from data. Nevertheless, distinguishing fundamental from non-fundamental models cannot be accomplished if the researcher only exploits second-order information, i.e., requiring that structural shocks are serially and mutually uncorrelated. Because, as has been documented in the literature, it is possible to find at least two models, fundamental and non-fundamental, with serially uncorrelated errors. This issue is what the literature calls the dynamic identification problem (see Velasco (2022)). Consequently, to overcome this identification problem, higher-order information on the errors must be exploited. This fact leads to abandoning the Gaussian behavior assumption regarding the structural shocks and assuming that structural shocks are non-Gaussian distributed. Otherwise, it is unfeasible to distinguish fundamental from non-fundamental models. In recent years, several methodological works have proposed alternatives for identifying and estimating structural vector autoregressive moving-average models without imposing the location of roots by exploiting the non- Gaussian behavior of structural shocks. Most of these investigations assumed that shocks are serial and mutually independent and identified the structural parameters following two approaches. The first strategy is to characterize non-Gaussian behavior by assuming non-zero third or fourth-order cumulants. This strategy is like a method of moments approach. Therefore, for constructing standard errors of the estimates, it is necessary to impose the finiteness of sixth or eighth-order moments. This assumption rules out essential types of non-Gaussian behavior. On the other hand, a second approach is to characterize the non-Gaussian behavior assuming a particular joint density function for the structural shocks and imposing, at least, non-zero third-order cumulants. This research attempts to fill a gap in the previous literature by proposing an estimation method that exploits the joint characteristic function of pairwise errors. This approach does not require imposing the finiteness of many higher-order moments or the knowledge of a specific joint density distribution of structural shocks. Moreover, the proposed method may be adapted for situations where structural shocks exhibit higher-order serial dependence while mutual independence is still assumed. Specifically, the first chapter explains that causal (or fundamental) structural vector autoregressions models may deliver biased estimates of causal effects of economic shocks when data comes from a non-fundamental structural model. As discussed above, this paper exploits the joint characteristic function of pairwise model errors instead of using specific moments or cumulants or imposing a particular density function for the structural shocks. Thus, our measure dependence is a generalized autocovariance function of the complex exponential model errors. We summarize the information on the autocovariance function using the cumulative spectral distribution, which is the integral of the spectral density function of model errors. We opt for the cumulative spectral function instead of the spectral density because its sample counterpart does not require a smoothing kernel. Moreover, this feature makes that weak convergence of estimators occurs at the standard rate. In this line, the contributions of this chapter are twofold. On the one hand, we extend a well-known identification result for SVARMA models with independent shocks to a situation in which disturbances can exhibit a more general serial dependence structure; on the other hand, we propose an estimation procedure that minimizes the distance of the generalized cumulative spectral distribution of unrestricted and restricted model errors. This procedure does not require imposing a particular joint non-Gaussian distribution of structural errors, as well as it does not need the existence of a large number of higher-order moments. Furthermore, in the case of having serially dependent shocks, identification (up to sign-permutation) only requires the existence of second-order moments of structural shocks and peculiar characterization of non- Gaussian behavior, consisting of non-linearity of expectations of structural errors conditional to non-constant linear filters. Regarding the asymptotic properties of our estimators, in the case of fully independent shocks, the finiteness of third-order moments is the only requisite to find the asymptotic distribution of the structural parameter estimator; in the case of serially dependent structural errors, it is required the finiteness of at least fifth-order moments for finding the asymptotic distribution of the estimator. The simulation evidence shows that our method satisfactorily identifies the location of the roots with a relatively small sample size. In the case of the bivariate structural model, the rate of correct root location is above 90 percent when structural shocks are asymmetric. In contrast, this rate decays when the errors follow a symmetric non- Gaussian distribution, especially for VARMA models with non-zero AR and MA components. To analyze our method's robustness, we study how well the proposal identifies the location of roots when structural shocks behave like a multivariate Gaussian random element. The rate of correct root location obtained with the Montecarlo experiment is quite close to the theoretical rate when only errors are required to be white noise. Finally, we applied our proposal to the dataset analyzed by Blanchard and Quah (1989). This dataset contains only two endogenous variables, GDP growth, and unemployment rate. We select two different specifications. The first case assumes that the roots of the AR polynomial lie outside the unit circle, but the location of the roots of the MA polynomial is not determined. The second specification is an unrestricted structural model, which does not restrict the root location of both the AR and MA polynomials. In both scenarios, the polynomials' overall degree equals one, respectively. The estimation results are the following: for the first restricted specification, our proposed method identifies that the roots of the MA polynomial lie outside the unit circle, i.e., our method identifies that the MA polynomial is invertible. Based on the estimation of the structural model, we compute the estimated causal effects of both identified shocks (i.e., impulse response functions of each shock). The analysis of these causal effects shows that we identify a shock with a non-zero long- run effect and another with zero long-run effect over GDP growth. It is worth mentioning that identifying these shocks did not require imposing any external identification restriction. In the scenario of not restricting the SVARMA model's root location, our method identifies MA roots outside the unit circle but AR roots inside the unit circle. In other words, our proposal identifies a non-causal but invertible SVARMA model. In the second chapter, we investigate the feasibility of empirically verifying one of the critical identification assumptions of the statistical strategy for identifying structural vector autoregressive moving-average models, i.e., the requirement about the non- Gaussian behavior of all shocks in the system. In particular, the statistical identification of possibly non-fundamental structural vector autoregressive moving-average models requires that shocks satisfy: (i) to be an i.i.d process, (ii) to be mutually independent across components, and (iii) each shock must be non-Gaussian distributed. Therefore, provided the first two requirements, it is crucial to be able to evaluate the third condition empirically without requiring estimation of the structural model because this latter may not be identified, and with a non-zero probability, an attempt to estimate the structural model may deliver an inconsistent one. However, in the literature, few efforts have been posed to determine the non-Gaussian dimension in a structural system. Most of these investigations assume that the structural model is fundamental, which makes it unfeasible to apply such proposals to more general contexts. Consequently, this research attempts to fill this gap in the literature and proposes a method for determining the non-Gaussian dimension when the location of polynomial roots is still being determined. In that sense, our proposal is robust to the location of the roots of the dynamic polynomials in the model. Therefore, it can be applied to fundamental or non-fundamental contexts. We address the unknown location of roots by relating the non-Gaussian dimension of structural errors to the rank of a matrix built from the higher-order spectrum of reduced- form errors. The estimation of the spectral density of the reduced form errors is done non-parametrically, making the convergence rates of the asymptotic distribution of the estimators to be slower than the standard one. Our estimation approach for the non- Gaussian dimension is a sequential testing procedure. Precisely, we follow the Kleibergen and Paap (hereafter KP) approach. The KP -statistic is built from the singular value decomposition of the interest matrix. The asymptotic distribution of the statistic is a standard chi-square whose degrees of freedom change depending on the null hypothesis. Unlike the KP article, our context changes the asymptotic distribution of the test statistic for some particular rank values, specifically under joint Gaussianity. For other null hypotheses, the asymptotic distribution is chi-square, but the degrees of freedom are generally unknown. For this reason, we propose a bootstrap test strategy. This path implies another significant challenge: to impose the null hypothesis in the resampled data. The way for creating bootstrap samples follows the idea of Nordhausen et al. (2017). The idea is to employ the characteristic vectors associated with zero singular values under each specific null hypothesis. These vectors span the Gaussian dimension. These vectors create a projection matrix over the non-Gaussian dimension, while the Gaussian dimension is completed with values drawn from a multivariate standard Normal distribution. The Montecarlo exercises were performed to analyze the size and power of the bootstrap test to show that our strategy estimates satisfactorily the non- Gaussian dimension. We apply our procedure to two well-known macroeconomic datasets, such as Blanchard and Quah (1989) and Blanchard and Perotti (2002). Unlike the work of Guay (2021), we can detect at least two non-Gaussian shocks in the Blanchard-Perotti dataset. Meaning that a statistical identification strategy can be applied to this database, at least assuming that polynomial roots are outside the complex unit circle. Finally, the third chapter has explored the conditions under which the combination of higher order cumulants identification and sign restrictions may be combined to obtain global identification. The statistical identification strategy of causal structural vector autoregressive models has gained attention recently because it permits the identification of the causal effects of structural shocks without resorting to external identification restrictions. This strategy exploits higher-order information provided structural errors are mutually independent and non-Gaussian distributed. This approach only identifies the structural model up to the signed permutation, which implies a finite set of admissible models. The solution proposed in the literature for selecting a model relies on applying a mechanical procedure. This paper aims to study an alternative scheme based on imposing economic-motivated sign restrictions over the causal effects of structural shocks. Only the permutations matter when a single, strict sign restriction is applied to each shock. A sufficient condition for achieving global identification is that the matrix of contemporaneous effects has a generalized recursive structure. Furthermore, when all shocks are either asymmetric or non-mesokurtic, we can find a relationship between the recursive structure of the contemporaneous effects matrix and the rank of the blocks of the matrix version of third or fourth-order cumulants of reduced-form errors. This result can be seen as an alternative path for determining the recursive structure to the standard procedure based on applying Granger's causality test to each endogenous variable. A weaker sufficient condition for accomplishing point identification of a single shock requires imposing as many sign restrictions as the number of endogenous variables in the model and that the system of sign restrictions is sign-solvable. It also explores the necessary conditions for having a non-empty identified set when only second-order cumulants and sign restrictions are employed. Although evident, this condition is not considered and taken as granted in several empirical works that employ sign restrictions for identifying economic shocks. The condition states that when the Gram- Schmidt algorithm is applied to each tuple of the vectors in the solution set of the sign restrictions intersects the Cartesian product, the identified set is non-empty. This condition holds immediately when only a small set of sign restrictions are imposed over the shocks and the restrictions are set over the same objects. On the other hand, when the model contains more than one Gaussian structural error, identification using higher- order cumulants ensures that the non-Gaussian block remains identified, although the order is unknown.
Description
Keywords
Series temporales económicas, Series temporales económicas, Econometría