Summary
Causal inference in multivariate time series is challenging because the sampling rate may not be as fast as the time scale of the causal interactions, so the observed series is a subsampled version of the desired series. Furthermore, series may be observed at different sampling rates, yielding mixed-frequency series. To determine instantaneous and lagged effects between series at the causal scale, we take a model-based approach that relies on structural vector autoregressive models. We present a unifying framework for parameter identifiability and estimation under subsampling and mixed frequencies when the noise, or shocks, is non-Gaussian. By studying the structural case, we develop identifiability and estimation methods for the causal structure of lagged and instantaneous effects at the desired time scale. We further derive an exact expectation-maximization algorithm for inference in both subsampled and mixed-frequency settings. We validate our approach in simulated scenarios and on a climate and an econometric dataset.
Keywords: Mixed frequency, Non-Gaussian error, Structural vector autoregressive model, Subsampling, Time series
1. Introduction
Classical approaches to multivariate time series and Granger causality assume that all time series are sampled at the same rate. However, due to data integration across heterogeneous sources, many datasets in econometrics, health care, environment monitoring, and neuroscience comprise multiple series sampled at different rates, referred to as mixed-frequency time series. Furthermore, due to the cost or technological challenge of data collection, many series may be sampled at a rate lower than the true causal scale of the underlying physical process. For example, many econometric indicators, such as gross domestic product, GDP, or housing price data, are recorded at quarterly and monthly scales (Moauro & Savio, 2005), though there may be important interactions between these indicators at the weekly or biweekly scale (Boot et al., 1967; Stram & Wei, 1986; Moauro & Savio, 2005). In neuroscience, imaging technologies with high spatial resolution, such as functional magnetic resonance imaging or fluorescent calcium imaging, have relatively low temporal resolutions, but many important neuronal processes and interactions happen at finer time scales (Zhou et al., 2014). A causal analysis rooted at a slower time scale than the true causal time scale may miss true interactions and add spurious ones (Boot et al., 1967; Breitung & Swanson, 2002; Silvestrini & Veredas, 2008; Zhou et al., 2014). A comprehensive approach to Granger causality in multivariate time series should be able to simultaneously accommodate both mixed-frequency and subsampled data.
Recently, causal discovery in subsampled time series has been studied with methods in causal structure learning using graphical models (Danks & Plis, 2013; Plis et al., 2015; Hyttinen et al., 2016). These methods are model-free and automatically infer a sampling rate for causal relations most consistent with the data. We maintain a similar goal, but take a model-based approach and examine the identifiability of structural vector autoregressive models under both subsampling and mixed-frequency settings. Structural models are an important tool in time series analysis (Harvey, 1990; Lütkepohl, 2005) and are a mainstay in econometrics and macro-economic policy analysis. These models combine classical linear autoregressive models with structural equation modelling (Bowen & Guo, 2011) to allow analysis of both instantaneous and lagged causal effects between time series. However, structural models are commonly applied to regularly sampled data, where each series is observed at the same regular intervals; moreover, the time scale of such an analysis is typically restricted to this shared sampling scale.
Gong et al. (2015) recently explored identifiability and estimation of vector autoregressive models under subsampling with independent innovations, i.e., no instantaneous causal effects or error correlations. They showed that with non-Gaussian errors, the transition matrix is identifiable under subsampling, implying that Granger causality estimation is possible. Unfortunately, their results do not cover correlated errors, a common and important aspect of many real-world time series (Lütkepohl, 2005). Interestingly, non-Gaussian errors have also been shown to aid model identifiability in structural autoregressive models with standard sampling assumptions (Hyvärinen et al., 2008; Zhang & Hyvärinen, 2009; Hyvärinen et al., 2010; Peters et al., 2013; Lanne et al., 2017). This line of work applies techniques developed for structural equation modelling with non-Gaussian errors and independent component analysis (Hyvärinen et al., 2004) to the structural time series context. Non-Gaussian errors allow identification of the structural model without other identifying restrictions (Lanne et al., 2017) and also allow identification of the causal ordering of the instantaneous effects if these are known to follow a directed acyclic graph (Hyvärinen et al., 2010). These models have been successfully applied to many non-Gaussian time series in econometrics (Lanne & Lütkepohl, 2010; Lanne et al., 2010; Lanne et al., 2017; Herwartz & Plödt, 2016).
Our approach to subsampling unifies existing approaches to identifiability along two complementary directions. First, our work connects the non-Gaussian subsampled autoregressive model to the independent innovations method (Gong et al., 2015) in the non-Gaussian structural autoregressive framework (Hyvärinen et al., 2008, 2010; Zhang & Hyvärinen, 2009; Peters et al., 2013; Lanne et al., 2017) by proving identifiability of a structural vector autoregressive model of order one under arbitrary subsampling. As a result, we find that one can identify not only the causal structure of lagged effects from subsampled data with correlated errors, but also the directed acyclic graph of the instantaneous effects, without prior knowledge of the causal ordering.
Second, we generalize our results to the mixed-frequency setting with arbitrary subsampling, where the subsampling level may be different for each time series. In doing so, we provide a unified theoretical approach and estimation method for subsampled and mixed-frequency cases. Deriving identifiability conditions on the model parameters in the mixed-frequency case is difficult (Anderson et al., 2016) and has only been studied based on the first two moments of the mixed-frequency process. Our work follows a complementary direction by leveraging higher-order moments and provides the first set of specific model conditions for mixed-frequency structural models needed for identifiability. Furthermore, previous mixed-frequency approaches have assumed a causal ordering, while our results indicate that this may be estimated by leveraging non-Gaussianity. Finally, our approach to identifiability allows us to move beyond the classical mixed-frequency setting, where the time scale is fixed at the most finely sampled series (Anderson et al., 2016); we instead consider identifiability and estimation in more general mixed-frequency cases. The four sampling types covered by our approach are shown in Fig. 1. To simplify the presentation, we first introduce our theoretical results for subsampled series of case (a) in § 3. We then generalize the results to the mixed-frequency cases (b), (c) and (d) in § 4.
Fig. 1.
Four types of structured sampling, where black lines indicate observed data and dotted lines indicate missing data: (a) both series are subsampled; (b) the standard mixed-frequency case, where only the second series is subsampled; (c) a subsampled version of case (b) where each series is subsampled at a different rate; (d) a subsampled mixed-frequency series that has no common factor across sampling rates and thus is not a subsampled version of case (b).
We introduce an exact expectation-maximization algorithm for inference in both subsampled and mixed-frequency cases. Gong et al. (2015) also use such an algorithm, but because they formulate inference directly on the subsampled process by marginalizing the missing data, their approach requires an extra approximation. Our approach instead casts inference as a missing-data problem and uses a Kalman filter to exactly compute the E-step for both subsampled and mixed-frequency cases. We validate our estimation and identifiability results via extensive simulations and apply our method to evaluate causal relations in a subsampled climate dataset and a mixed-frequency econometric dataset. Taken together, we present a unified theoretical analysis and estimation methodology for subsampled and mixed-frequency cases, which have previously been studied separately. A summary of our contributions is presented in Table 1.
Table 1.
Summary of contributions of our work to identifiability and estimation in mixed-frequency sampling for structural autoregressive models: the subsampling types are as in Fig. 1; citations refer to previous work and check marks indicate our contributions
| Sampling type | None | A | B | D | ||
|---|---|---|---|---|---|---|
|
Identifiability | Lut05 | Gong15 | ✓ | ✓ | ✓ |
| Estimation | Lut05 | Gong15 (approx.) ✓ | ✓ | ✓ (ce) | ✓ (ce) | |
 free |
Identifiability | Hyv08 | ✓ | ✓ | ✓ | ✓ |
| Estimation | Hyv08 | ✓ | ✓ | ✓ (ce) | ✓ (ce) |
(ce), computationally expensive; Hyv08, Hyvärinen et al. (2008); Gong15, Gong et al. (2015); Lut05, Lütkepohl (2005).
2. Background
Let 
be a 
-dimensional multivariate time series generated at a fixed sampling rate. We collect all the 
into the matrix 
. We assume that the dynamics of 
follows a combination of instantaneous effects, autoregressive effects and independent noise:
 |
(1) |
where 
is the structural matrix that determines the instantaneous-time linear effects, 
is an autoregressive matrix that specifies the lag-one effects conditional on the instantaneous effects, and 
is a white noise process such that 
for all 
and 
is independent of 
for all 
such that 
. We assume that 
is distributed as 
. Solving (1) in terms of 
gives the following lag-one structural vector autoregressive process for the evolution of 
:
 |
(2) |
Under the representation in (2), each element 
denotes the lag-one linear effect of series 
on series 
and 
is the structural matrix. The error 
is referred to as the shock to series 
at time 
, and the element 
is the linear instantaneous effect of 
on 
.
Conditions on 
, or equivalently 
, for model identifiability and estimation have been explored (Harvey, 1990; Kilian & Lütkepohl, 2016). The most typical condition is that 
is a lower triangular matrix with ones on the diagonal, implying a known causal ordering of the instantaneous effects. In this case, one may interpret the instantaneous effects as a directed acyclic graph (Lauritzen, 1996), i.e., a graph 
with vertices 
and directed edge set 
that has no directed cycles. A causal ordering is an ordering of the vertices into a sequence, 
, such that if 
comes before 
in 
then 
does not contain a path of edges from 
to 
; see, e.g., Shojaie & Michailidis (2010) for details. In the structural context, for 
there exists a directed edge 
from 
to 
in 
if and only if 
is nonzero. Classical estimation for structural models with known causal ordering typically proceeds by simultaneously fitting 
and 
with the identifiability constraint that 
be lower triangular. When there are no unobserved confounders, as we assume throughout this paper, we may refer to the entries in 
as instantaneous causal effects.
A recent line of work (Zhang & Hyvärinen, 2009; Hyvärinen et al., 2010; Lanne et al., 2017) focuses on estimating 
and 
when 
is unknown. When the errors 
are non-Gaussian, both the causal ordering and the instantaneous effects 
may be inferred directly from the data using techniques from independent component analysis (Hyvärinen et al., 2010). Alternatively, one may dispense with orderings and lower triangular restrictions and directly estimate 
under non-Gaussian errors (Lanne et al., 2017). Our analysis continues along these directions, leveraging non-Gaussianity of the structural model with subsampling or mixed frequencies.
3. Subsampled structural vector autoregressive models
3.1. The subsampled process
Subsampling occurs when, due to low temporal resolution, we only observe 
every 
time-steps, as displayed graphically in case (a) of Fig. 1. In this situation, we only have access to observations 
, where 
is the number of subsampled observations. By marginalizing out the unobserved 
, we obtain the evolution equations
 |
(3) |
 |
(4) |
where 
is the stacked vector of errors for time 
and the unobserved time-points between times 
and 
, and 
. Equation (3) states that the subsampled process is a linear transformation of the past subsampled observations with transition matrix 
and a weighted sum of the shocks across all unobserved time-points. Each shock is weighted by 
raised to the power of the time lag. We provide an example of (3) in the Supplementary Material.
Equation (4) appears to take a similar form to the structural process in (1), but now the vector of shocks, 
, is of dimension 
, with special structure on both the structural matrix 
and the distributions of the elements in 
. Unfortunately, this representation does not have the interpretation of instantaneous causal effects described in § 2, as there are now multiple shocks per individual time series. We will refer to the full parameterization of the subsampled structural model in (4) as 
. Identifiability of this model means that there is a unique pair of matrices 
and 
consistent with the joint distribution of 
at subsampling rate 
.
3.2. Lagged and instantaneous causality confounds of subsampling
A classical analysis based on 
that does not account for subsampling would incorrectly estimate lagged Granger causal effects in 
, because 
does not imply 
, and vice versa (Gong et al., 2015). Similarly, estimation of structural interactions may also be biased if subsampling is ignored. Classical structural estimation methods that assume a known causal ordering of the instantaneous shocks simply estimate the covariance of the error process, 
, and let the estimated structural matrix be the Cholesky decomposition of 
. Under subsampling, the covariance of the error process is
 |
(5) |
where 
is the Kronecker product and 
is the identity matrix of size 
. The causal structure given by zeros in the Cholesky decomposition of (5) need not be the same as that implied by 
.
Example 1.
Consider the process (Gong et al., 2015)
so that
. Then, for a subsampling of
,
This implies no lagged causal effect between
and
but a relatively large instantaneous interaction, contrary to the true data-generating model; see Fig. 2.
Fig. 2.
Graphical depiction of how subsampling confounds causal analysis of both lagged and instantaneous effects: (a) the true causal diagram for the regularly sampled data; (b) the estimated causal structure of the subsampled process when the effects of subsampling are ignored.
3.3. Identifiability of L under subsampling
While both lagged Granger causality and instantaneous structural interactions are confounded by subsampling, we show here that by accounting for subsampling we may, under some conditions, still estimate the 
and 
matrices of the underlying process directly from the subsampled data. As a first step towards proving the identifiability of 
and 
, we show that the matrix 
in (4) is identifiable up to permutation and scaling of columns when the 
, the distributions of the 
, are all non-Gaussian.
Proposition 1.
Suppose that all the
are non-Gaussian. Given a known subsampling factor
and subsampled data
generated according to (4),
may be determined up to permutation and scaling of columns.
The proof closely follows that of Proposition 1 in Gong et al. (2015) and depends on the following fundamental result in independent component analysis (Eriksson & Koivunen, 2004).
Lemma 1.
Let
and
be two representations of the
-dimensional random vector
, where
and
are constant matrices of orders
and
, respectively, and
and
are random vectors with independent components.
If the
th column of
is not proportional to any column of
, then
is Gaussian. Moreover, if the
th column of
is proportional to the
th column of
, then the logarithms of the characteristic functions of
and
differ by a polynomial in a neighbourhood of the origin.
This result states that if 
is non-Gaussian with independent elements and if 
, then 
and 
must be equal up to permutation and scaling of columns. This implies that one may estimate 
from only observations of 
and that the estimate of 
should be equal, up to permutations and scalings, to the true 
.
To prove Proposition 1 using Lemma 1, note that 
is identifiable by linear regression. Hence, the error component 
satisfies the conditions of Lemma 1 and 
is identifiable up to permutations and scalings since the 
are non-Gaussian.
3.4. Complete identifiability of the structural autoregressive model when 
Using the identifiability result for 
in Proposition 1, we can derive identifiability statements and conditions for 
and 
in the subsampled case. We require a few mild assumptions.
Assumption 1.
Let
be stationary so that all singular values of
have modulus less than
.
Assumption 2.
The distributions
are distinct for each
after rescaling
by any nonzero scale factor; their characteristic functions are all analytic, or are all nonvanishing, and none of them has an exponent factor with polynomial of degree at least two.
Assumption 3.
All the
are asymmetric.
Assumption 1 is standard in time series modelling (Lütkepohl, 2005), and Assumption 2 is common in independent component analysis. While many of our identifiability results for 
only require that the 
distributions be non-Gaussian, our identifiability results for 
in part (ii) of Theorems 1 and 2 and part (iii) of Theorems 3 and 4 further require Assumption 3, namely that the 
be asymmetric. In practice, assuming fully Gaussian errors may be unrealistic, as aspects of non-Gaussianity, such as asymmetry (Harvey & Siddique, 2000; Walls, 2005; Lanne & Pentti, 2007), heavy tails (Cont, 2001; Rachev, 2003) or stochastic volatility (Justiniano & Primiceri, 2008), are often observed. Not only are non-Gaussian errors empirically appealing but, furthermore, theoretical and modelling approaches that harness the higher-order moments of non-Gaussian distributions aid in identifying model parameters that are unidentifiable from the first two moments alone.
Gong et al. (2015) give identifiability results under Assumptions 1 and 2 for the subsampled autoregression with no error correlations, 
. We restate their result in our framework.
Theorem 1
(Gong et al., 2015) Suppose that
is non-Gaussian for all
and
, and that the data
are generated by (2) with
. Assume that the process admits another representation
. If Assumptions 1 and 2 hold, then we have the following:
(i)
can be represented as
, where
is a diagonal matrix with
or
on the diagonal; if we constrain the self influences to be positive, represented by the diagonal entries, then
.
(ii) If Assumption 3 also holds, then
.
3.5. Complete identifiability of general structural autoregressive models
For identifiability of the full model under subsampling, we require two more assumptions.
Assumption 4.
The variance of each
is equal to
, i.e.,
.
Assumption 5.
The matrix
is of full rank.
Assumption 4 is common in structural modelling and removes the nonidentifiability between scaling the 
and scaling the columns of 
. Assumption 5 is mild, and covers the more restrictive assumption in non-Gaussian structural models that 
may be row- and column-permuted to a lower triangular matrix (Shimizu et al., 2006). Under these assumptions, we have the following identifiability result for general subsampled structural models.
Theorem 2.
Suppose that the
are all non-Gaussian and independent, and that the data
are generated by (2) with representation
. Assume that the process also admits another subsampling representation
. If Assumptions 1, 2 and 4 hold, then we have the following:
(i)
is equal to
up to permutation of columns and scaling of columns by
or
; that is,
where
is a scaled permutation matrix with elements being
or
; this implies that
.
(ii) If Assumptions 3 and 5 also hold, then
.
The requirement that 
be of full rank stems from the structure of 
. Since one may identify 
as the first 
columns of 
, to obtain 
we must premultiply the second set of 
columns of 
by 
. The asymmetry assumption is needed since the scaling of the columns of 
and 
by factors of 
or 
is ambiguous if the distributions are symmetric; the asymmetry assumption ensures that the unit scalings are identifiable; see the Supplementary Material.
If the instantaneous causal effects follow a directed acyclic graph, we may identify the structure without any prior information about causal ordering of the variables.
Corollary 1.
Suppose that Assumptions 1, 2 and 4 hold. If the true structural process corresponds to a directed acyclic graph
, i.e., it has a lower triangular structural matrix
with positive diagonals, and if it admits another representation with structural matrix
, then
. Hence the structure of
is identifiable without prior specification of the causal ordering of
.
This result follows because 
may be identified up to a column permutation. Based on the identifiability results of Shimizu et al. (2006), if 
corresponds to an acyclic graph, it may be row- and column-permuted to a unique lower triangular matrix. The row permutations identify the causal ordering, and the nonzero elements below the diagonal identify the edges in 
. See Shimizu et al. (2006) for more details on identifiability and estimation of the graph from 
.
Taken together, the results of Theorem 2 and Corollary 1 imply that when the shocks 
are independent and asymmetric, a complete causal diagram of the lagged effects and the instantaneous effects is fully identifiable from the subsampled time series, 
.
4. Mixed-frequency structural autoregressive models
4.1. Background and motivation
Estimation and forecasting of mixed-frequency time series are commonly approached using autoregressive models (Schorfheide & Song, 2015). Typically, the model is fitted at the same scale as the fastest sampled time series, which is depicted in Fig. 1(c). The primary motivating example of Fig. 1(c) in the literature is GDP (Anderson et al., 2016). The subsampled structure of Fig. 1(c) simplifies the temporal aggregation of GDP and is used to simplify analysis. Due to costly data collection, especially for macro-economic indicators such as GDP, the scale of the fastest sampled series is generally arbitrary and may not reflect the true causal dynamics, leading to confounded Granger and instantaneous causality judgements (Breitung & Swanson, 2002; Zhou et al., 2014). If the true causal scale, or one of interest to an analyst, is at a lower rate, as in case (d) of Fig. 1, then an analysis at the observed rate will run into the same problems as those for the single-frequency subsampling case discussed in § 3.2. We provide an example at the end of § 4.2.
Finding identifiability conditions for mixed-frequency autoregressive models with no subsampling at the fastest scale, Fig. 1(b), was an open problem for many years (Chen & Zadrozny, 1998). Recently Anderson et al. (2016) showed that the mixed-frequency nonstructural autoregressive model of Fig. 1(b) is generically identifiable from the first two observed moments, so unidentifiable models make up a set of measure zero in the parameter space. Explicit identifiability conditions for the lag-one, bivariate case from the first two moments have also been established (Anderson et al., 2012). However, no explicit identifiability conditions for structural models or models in higher dimensions have been explored.
In this section, we generalize our identifiability results from § 3 to the mixed-frequency case with arbitrary levels of subsampling for each time series. Our analysis indicates that Granger and instantaneous causal effects can be accurately estimated from mixed-frequency time series. Specifically, we use the results from § 3 to provide explicit identifiability conditions for mixed-frequency structural models under arbitrary subsampling, namely cases (b), (c) and (d) in Fig. 1, with non-Gaussian error assumptions. Altogether, our framework provides a unified way of deriving explicit identifiability conditions for both subsampling and mixed-frequency cases. While case (c) in Fig. 1 is a subsampled version of the standard mixed-frequency case, our results also cover mixed-frequency subsampling as in case (d). To the best of our knowledge, these results are the first identifiability results for subsampled mixed-frequency cases like (c) and (d).
4.2. Mixed-frequency structural autoregressive models
We assume that each time series in 
is sampled at one of two sampling rates, a slow subsampling rate 
and a fast subsampling rate 
. We write 
, where 
are those series subsampled at 
and 
are those subsampled at 
. Let 
be the list of subsampling rates for each time series. In Fig. 1(b), 
and 
, whereas in Fig. 1(c), 
and 
. Analogously to the subsampled case, we refer to a parameterization of a mixed-frequency structural model as 
, where 
is now a 
-vector. Let 
be the smallest multiple of both 
and 
; for example, in Fig. 1(c) we have 
.
We may derive a similar representation to (4) for mixed-frequency series. Fix a time-point 
such that all series are observed. Let 
be a modified 
identity matrix where all rows 
such that 
is not observed at time 
are set to zero. Further, let 
, 
and 
. Then
 |
(6) |
where
 |
Equation (6) has the same form as (4), suggesting that similar identifiability results will hold. We provide an example of (6) for a mixed-frequency series in the Supplementary Material.
In a subsampled mixed-frequency setting where the fastest rate is greater than unity, Fig. 1(c), not accounting for subsampling leads not only to the kind of mistaken inferences discussed in § 3.2 but also to further mistakes unique to the mixed-frequency case.
Example 2.
Consider a subsampled mixed-frequency structural process generated by (6) with the
parameters given by Example 1. Suppose subsampling is not taken into account and that
is analysed instead as a classical mixed-frequency series, case (b), based on the first two moments (Anderson et al., 2016). We consider two cases.
Case 1: the sampling rate is
. In this case, if
is analysed at the rate
using the first two moments, then
and
are not identifiable at this rate since both off-diagonal elements of
are zero (Anderson et al., 2016). Thus, no inference of both the instantaneous correlations and the lagged effects is possible.
Case 2: the sampling rate is
. In this case, if
is analysed at the rate
using the first two moments, the estimated
and covariance
will be the same as in Example 1 (Anderson et al., 2016), leading to an incorrect inference that there is an instantaneous effect but no directed lagged effect.
4.3. Identifiability of mixed-frequency structural autoregressive models
We provide generalizations of Theorems 1 and 2 to the mixed-frequency case.
Theorem 3.
Suppose the
are non-Gaussian and independent for all
and
, and that the data
are generated by (2) with
. Assume that the process also admits another mixed-frequency representation
. If Assumptions 1 and 2 hold, then we have the following:
(i)
can be represented as
, where
is a diagonal matrix with
or
on the diagonal.
(ii) If any multiple of
is
smaller than some multiple of
, then
. If
, this implies
, i.e., the
th columns of
and
are equal,
.
(iii) If Assumption 3 also holds, then
.
Proof.
Statements (i) and (iii) follow since we may further subsample all series in
to a subsampling rate of
. This gives a subsampled
with representation
. Applying Theorem 1 gives the result. Furthermore, if some multiple of
is equal to some multiple of
minus 1, then there exists a set of times
for (6) such that series
is observed at time
and series
is observed at time
. By identifiability of linear regression,
. This resolves the sign ambiguity of the columns in (iii) so that
. □
Theorem 4.
Suppose the
are non-Gaussian and independent for all
and
, and that the data
are generated by (2) with representation
. Assume that the process also admits another mixed-frequency subsampling representation
. If Assumptions 1, 2 and 4 hold, then we have the following:
(i)
is equal to
up to permutation of columns and scaling of columns by
or
, i.e.,
where
is a scaled permutation matrix with elements being
or
; this implies that
.
(ii) If
is lower triangular with positive diagonals, i.e., the instantaneous interactions follow a directed acyclic graph, and if for all
there exists a
such that any multiple of
is
smaller than some multiple of
with
, then
.
(iii) If Assumptions 3 and 5 also hold, then
.
The proofs of statements (i) and (iii) follow the same subsampling argument as in the proof of Theorem 3. The proof of (ii) is given in the Supplementary Material.
Theorems 3 and 4 demonstrate that identifiability of structural models still holds for mixed-frequency series with subsampling under non-Gaussian errors. Statements (i) and (iii) of Theorems 3 and 4 are the same as their subsampled counterparts; statement (ii) in both theorems shows how the mixed-frequency setting provides additional information for resolving parameter ambiguities in the non-Gaussian setting. Specifically, when there is a one-time-step difference between when series 
and 
are sampled, then 
is identifiable. We can then use this information to resolve sign ambiguities in columns of 
, which leads to statement (ii) in both of Theorems 3 and 4. This result applies directly to the standard mixed-frequency setting (Schorfheide & Song, 2015; Anderson et al., 2016), where one series is observed at every time-step in Fig. 1(b). It also applies to case (d), since there exist certain time-steps where one series is observed one time-step before another series.
5. Estimation
5.1. Modelling non-Gaussian errors
We model the non-Gaussian errors as a mixture of Gaussian distributions with 
components. This approach has been adopted widely in econometrics and other fields as a flexible and tractable way of modelling non-Gaussian innovations (Gong et al., 2015; Lanne et al., 2017). Formally, we assume that 
is drawn from the mixture distribution
 |
where 
and 
are 
-vectors specifying the mean, variance and mixing weight of each mixture component. The 
component indicators are auxiliary variables introduced to facilitate inference. The mixture model for the errors implies that conditional on the assignment indicators 
, the mean and variance of the error distribution for each series 
are time-dependent. This mixture model can capture the types of non-Guassianity required for identifiability and also those observed in real-world time series. Asymmetric errors may be formed when the mixture centres are nonzero and the variances or mixture weights are different. A non-Gaussian symmetric distribution with kurtosis greater than 1 may be formed by setting the mixture centres to zero but allowing the mixture variances to have different values. The full set of parameters for the structural model is 
where 
and 
concatenate the mixture parameters of the errors across series. For example, 
is the mean of the 
th mixture component for the 
th error distribution, and likewise for 
and 
.
5.2. Expectation-maximization algorithm
We develop an expectation-maximization algorithm for joint maximum likelihood estimation of the full set of parameters 
based only on the observed subsampled and mixed-frequency data 
. Unlike the method of Gong et al. (2015), which is tailored to the subsampled case, our method is the same for both types of data. Furthermore, the non-structural-specific, i.e., 
, algorithm of Gong et al. (2015) introduces auxiliary noise terms to facilitate inference, rendering the resulting algorithm inexact, whereas our algorithm introduces no such approximations. Since the loglikelihood is nonconvex, we employ multiple random restarts to avoid poor local optima. For the subsampled case, the local optimum problem is particularly severe due to the nonidentifiability under the first two moments; many values of 
can give a good fit to the data. The basic algorithm also suffers from slow convergence due to the large amount of missing data. To speed up the algorithm, we deploy the adaptive over-relaxed method of Salakhutdinov & Roweis (2003).
Let 
, and let 
if error 
was generated by mixture component 
and 
otherwise. The complete loglikelihood, 
, of our structural model is
 |
where 
is the 
th row vector of 
. The algorithm alternates between the E-step, where we compute the conditional expectation 
, and the M-step, where that expectation is maximized with respect to the parameters 
. We first describe the M-step updates, and then explain how the conditional expectations are computed using a Kalman filter.
5.3. The M-step
In the M-step, we maximize the expected complete loglikelihood conditional on the observed data, 
, with respect to 
via coordinate ascent, cycling through 
, 
and 
until convergence. The specific updates are as follows.
Updating 
: each row of 
, 
, may be updated independently according to
 |
Updating 
, 
and 
: these may be optimized jointly in one step using
 |
Updating 
: since the maximization is not given in closed form, we use the Newton-Raphson method. Let 
be the 
vectorization. At each step, the next 
iterate is
 |
where 
and 
is the Hessian of 
with respect to 
. Expressions for the gradient and Hessian are given in the Supplementary Material.
5.4. The E-step
All conditional expectations in the M-step above are computed using the Kalman filtering-smoothing algorithm. For simplicity, consider one block of data, so that 
, where 
and 
are fully observed but the 
for 
have some missing data and hence are not included in 
. Any subsampled or mixed-frequency series can be broken into blocks of this type. The conditional expectation 
can be computed by noticing that
 |
For a fixed 
, 
is computed using the Kalman filtering-smoothing algorithm, since for fixed 
, 
is a linear Gaussian state-space model with latent observations 
. We compute 
for each 
combination and then add these together weighted by 
. The probability 
may be computed as
 |
where 
is the set of prior mixture component weights, 
, and 
is the likelihood of the observed data, which may also be computed by one Kalman pass. This process is repeated for all expectations in the E-step. The computational complexity of this exact algorithm scales as 
, since the Kalman filter must be run for all combinations of 
for each block. The approximate algorithm of Gong et al. (2015) has the same complexity. Like Gong et al. (2015), we have explored approximate inference methods based on variational and Markov chain Monte Carlo methods but found their performance to be poor; see § 8.
6. Simulations
6.1. Estimation dependence on the subsampling factor and number of observations
We first investigate the performance of the expectation-maximization algorithm under subsampling. We simulate data with 
time series and 
mixture components. The asymmetric error distributions are given by 
, 
and 
for 
, and 
, 
and 
for 
. We consider two cases for 
and 
:
 |
Simulations are performed for two subsampling factors, 
, and three sample sizes, 
. Due to subsampling, the actual sample sizes are reduced. Data from each parameter configuration are generated 10 times, and the estimation algorithm is run on each realization using 1000 random restarts. Boxplots of the error estimates for two of the scenarios are shown in Figs. 3 and 4; see the Supplementary Material for plots in the other two settings.
Fig. 3.
Boxplots of errors in 
and 
parameter estimates over 10 random data samplings. The original series is of length 203 (top), 403 (middle) or 805 (bottom) and is then subsampled at 
(left) and 
(right).
Fig. 4.
As Fig. 3 but for 
and 
.
We perform a similar experiment for 
. We simulate data with parameters
 |
The mixture of normal error distributions for 
and 
is the same as that for the 
case. The parameters for 
are 
, 
and 
. The average error rates are displayed in Table 2 and indicate increasingly accurate estimation in trivariate structural systems as the sample size increases.
Table 2.
Average log mean squared error of 
and 
in a 
structural system over ten random samples for 
and three sample sizes
|
|
|||||
|---|---|---|---|---|---|---|
|
203 | 403 | 805 | 203 | 403 | 805 |
|
 2.4 |
 7.0 |
 7.5 |
 0.9 |
 1.6 |
 6.8 |
|
 3.6 |
 4.8 |
 5.8 |
 1.8 |
 1.8 |
 3.9 |
6.2. Estimation dependence on the asymmetry of errors
We analyse estimation performance as a function of the skewness of the error distribution, 
, which is a measure of asymmetry. We simulate data from the same 
parameter configurations as in § 6.1 for 
and 
. While keeping the variance fixed, we vary the error distributions across a range of 
, 
, so that 
and 
have the same magnitude of skewness but opposite signs. The skewness values are obtained by gradually modifying the 
, 
and 
values in a bivariate mixture of normals. See the Supplementary Material for the exact parameter values and plots of the simulated error distributions.
The results for estimation of 
are shown in Table 3. First, for 
estimation remains accurate across all skewness settings for 
, while for 
the error stays low for 
but spikes for 
. For 
, estimation is stable until 
for 
, but for 
estimation is only stable at 
. Taken together, these results suggest that the strength of identifiability depends on a combination of factors, 
, 
and 
, and the level of asymmetry of the error distributions. Similar results for 
are reported in the Supplementary Material.
Table 3.
Average log mean squared error of 
over ten random samplings for both 
and 
estimates across multiple settings of the parameters, number of observations, and subsampling factors
|
|
|||||||||
|---|---|---|---|---|---|---|---|---|---|---|
|
1.8 | 1.2 | 0.8 | 0.4 | 0 | 1.8 | 1.2 | 0.8 | 0.4 | 0 |
|
 9.0 |
 7.7 |
−7.3 | −7.0 | −0.018 | −8.1 | −7.0 | −7.1 | −7.0 | −7.4 |
|
 9.0 |
 7.9 |
−7.7 | −7.4 | 0.16 | −7.9 | −7.2 | −7.4 | −7.2 | −7.5 |
|
 9.1 |
 7.9 |
−0.94 | −0.26 | 1.2 | −8.0 | −0.33 | 0.71 | 1.6 | 1.6 |
|
 9.1 |
 8.0 |
−0.94 | 0.15 | 1.3 | −8.0 | −0.32 | 1.0 | 1.4 | 1.2 |
6.3. Estimation dependence on the signal-to-noise ratio
We next investigate estimation performance in subsampling and mixed-frequency sampling as a function of the signal-to-noise ratio. In these experiments we use 
and 
. We scale 
by a factor to set its maximum eigenvalue to the desired level. We perform these experiments both for full subsampling of 
and 
and for mixed-frequency subsampling where one series is observed at every time-point and the other is subsampled. Data from each parameter configuration are generated 40 times. In Fig. 5 we plot the average absolute error for estimating the 
and 
matrices as a function of the maximum eigenvalue of 
. Estimation under subsampling is stable until the maximum eigenvalue falls to about 
, and thereafter estimation becomes dramatically worse, indicating unstable estimation in this regime. The increasing error in the estimation of 
as a function of the signal-to-noise ratio is also observed in the mixed-frequency case. However, the estimation error increases less dramatically than in the subsampled case, partly due to the presence of fewer local optima in the mixed-frequency case. In the mixed-frequency case, the error in 
estimation appears to be constant across the maximum eigenvalue range we considered.
Fig. 5.
Average mean squared error (MSE) in estimation of (a) 
and (b) 
as a function of the maximum eigenvalue of 
. Results are shown for subsamplings of 
(red solid), 
(red dashed), 
(blue solid), and 
(blue dashed). Error bars indicate one standard error from 40 simulation runs.
Unstable estimation arises from a combination of two factors. First, under subsampling, the transition matrix of the subsampled process is 
, indicating that the signal strength between observations scales exponentially as a function of 
. Furthermore, the likelihood surface is multimodal, such that multiple high probability modes all have approximately the same 
value. As the signal-to-noise ratio falls, 
estimation becomes more difficult due to subsampling, and so the multimodal estimation becomes more severe, and modes far from the true 
occasionally have higher likelihood. Overall, these simulations indicate that in the subsampling case, there appears to be a threshold on the maximum eigenvalue, below which inference becomes unstable.
The simulations cover cases (a) and (b) in Fig. 1. Unfortunately, the complexity of the E-step forbids performing simulations in a reasonable time for cases (c) and (d). Future work will explore tractable inference in these cases; see the discussion at the end of § 7.
7. Real data
7.1. Subsampled ozone data
We use the subsampled structural model to analyse the causal scale and pathways in an ozone and temperature dataset. The temperature-ozone data are the 50th causal effect pair from the website https://webdav.tuebingen.mpg.de/cause-effect/, and were also considered by Gong et al. (2015). The dataset consists of temperature and ozone concentration values, sampled daily. First we standardize each time series to zero mean and unit variance. We fit the subsampled structural model to the pre-processed series for 
subsampling regimes under both independent errors, 
, and structural covariance in the instantaneous errors, 
free. To ensure that good optima are found, we perform 30 000 restarts and run the adaptive over-relaxed algorithm until the relative change in loglikelihood is less than 
.
The estimated 
for 
is
 |
with a maximum eigenvalue of 
, suggesting that accurate estimation of subsampled parameters is possible. The Bayesian information criterion scores for all models are displayed in Table 4. Across all subsampling rates, the structural model, 
free, has lower score, indicating that the two extra parameters of the structural model, the off-diagonal elements of 
, provide necessary flexibility. Furthermore, the best-performing model is the structural model with subsampling rate 
. The estimated transition matrix for 
is
 |
similar to that given by Gong et al. (2015) for 
. After normalizing the columns, we obtain
 |
Table 4.
Bayesian information criterion scores of subsampling and covariance types on the atmospheric dataset; an asterisk indicates the lowest value
Model / 
|
1 | 2 | 3 | 4 |
|---|---|---|---|---|
|
901.96 | 791.02 | 839.56 | 797.00 |
 free |
784.53 | 777.78
|
790.46 | 791.23 |
These results indicate weak lagged effects at the subsampled scale, but stronger instantaneous effects between temperature and ozone. Furthermore, the temperature series derives most of its power from a strong error variance, while the ozone series is driven more by the autoregressive component. See the Supplementary Material for quantile-quantile plots of the inferred mixture of error distributions.
7.2. Mixed-frequency data: GDP and treasury bonds
We perform a structural autoregressive analysis on the mixed-frequency dataset of quarterly GDP and monthly price of treasury bonds. The dataset has previously been compiled and analysed in the mixed-frequency setting by Schorfheide & Song (2015) and is available in the Supplementary Material. We follow Schorfheide & Song (2015) and compute the logarithm of both series. Furthermore, as is common in mixed-frequency analysis (Chen & Zadrozny, 1998), we compute first differences to remove first-order nonstationarities.
There are multiple approaches to modelling mixed-frequency GDP data in the literature. Recently, many authors have treated GDP as a flow variable and used state-space models to directly model the aggregation over months in a quarter (Schorfheide & Song, 2015; Ghysels, 2016). Others ignore the generative subsampling structure and instead jointly model the high- and low-frequency variables in a quarter using mixed data sampling methods (Ghysels, 2016). We follow another line of work that simplifies the analysis by ignoring aggregation (Chen & Zadrozny, 1998; Seong, 2012; Eraker et al., 2014; Anderson et al., 2016; Zadrozny, 2016), thus treating GDP as a purely subsampled series, and apply our mixed-frequency structural autoregressive model at the monthly rate. Indeed, recent theoretical work on mixed-frequency autoregressive models for GDP also focuses on the purely subsampled, rather than aggregated, case (Anderson et al., 2016). Since our subsampled approach to modelling GDP is a simplifying assumption that ignores aggregation, extending our framework to handle aggregated variables is an important direction of future research.
In the traditional approaches to mixed-frequency analysis, 
and the instantaneous covariance 
are generically identifiable from the first two moments (Anderson et al., 2016). What sets our non-Gaussian approach apart in this mixed-frequency domain is its ability to uniquely identify the ordering of the instantaneous causal effects in the structural matrix 
. To highlight this ability, we perform model selection on the zero entries in 
to determine the causal ordering of the instantaneous effects. Specifically, we calculate the Bayesian information criterion for the nested models 
, 
, 
, and 
. Models 
, 
and 
represent acyclic structures on the instantaneous effects, while the unrestricted model 
does not. The scores for all models shown in Table 5 indicate that 
performs best. The estimated matrices of
 |
suggest a slight negative lagged interaction from GDP to treasury bonds and an instantaneous interaction at the monthly scale from treasury bonds to GDP. See the Supplementary Material for quantile-quantile plots of the inferred mixture of error distributions.
Table 5.
Bayesian information criterion scores of different instantaneous causality structures on the GDP dataset; an asterisk indicates the lowest value
| Model |
|
|
|
|
|---|---|---|---|---|
| 1984.00 | 1983.41 | 1981.08
|
1987.55 |
The above analysis fits a structural model at the time scale of months, the same sampling rate as the treasury bond time series. The results from § 4 indicate that we could uniquely identify models at bimonthly, or even more granular, time scales. However, even at the bimonthly rate, the computational complexity of the E-step of our algorithm becomes prohibitive due to the large number of combinations of error mixture components in a data block, as discussed in § 5.4. Since the E-step requires running the forward-backward algorithm many times, a considerable computational speed-up could be achieved from a parallel implementation.
8. Discussion
Our results provide sufficient conditions for identifiability of structural autoregressive models for both subsampled and mixed-frequency series. The causal diagram of both lagged and instantaneous effects is identifiable under arbitrary subsampling and non-Gaussian errors.
We have developed an exact expectation-maximization algorithm for estimation and analysed its performance via simulations. Our algorithm has two drawbacks: high complexity due to a Kalman filter evaluation for all mixture error assignments within a time block; and many local optima due to weak identifiability. Our simulations show that the latter problem is more severe under even subsampling factors and low signal-to-noise regimes.
An ongoing line of work is to develop approximate inference for these models using Markov chain Monte Carlo or variational methods. Unfortunately, we have found that the local optima make sampling difficult. A Gibbs sampler we have explored gets stuck in one local mode and requires the same number of random restarts as our algorithm to find a good solution. Perhaps incorporating recent advances in sampling (Ma et al., 2016) may prove beneficial. We have also found the performance of a variational algorithm to be poor. Similarly, Gong et al. (2015) reported significantly worse results for a variational approach than for their approximate expectation-maximization algorithm. By breaking the dependence between the unobserved, subsampled 
and the auxiliary 
, the variational approach avoids the combinatorial evaluation of a Kalman filter; however, this dependence is critical for correctly evaluating the probable trajectories of the latent 
, without which inference of 
suffers.
While our work has focused on point estimation, future research aims to adapt the time series bootstrap to the mixed-frequency and subsampled settings for constructing confidence intervals. It would be interesting to explore method-of-moments estimation for this problem, which may side-step the local optima difficulty and the combinatorial complexity of our algorithm.
Supplementary Material
Acknowledgement
This research was partially funded by the U.S. National Science Foundation, National Institutes of Health, Air Force Office of Scientific Research, and Office of Naval Research. We thank the referees for helpful comments and suggestions.
Supplementary material
Supplementary material available at Biometrika online includes an example of the subsampled and mixed-frequency structural processes, detailed proofs of Theorems 2 and 4, details on the 
update in the expectation-maximization algorithm, additional simulation results, both of the real datasets that we analysed, and the code for the expectation-maximization algorithm.
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