# Linear System Challenges of Dynamic Factor Models

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## Abstract

**:**

## 1. Introduction

- The theoretical underpinning for allowing modelling to focus on AR rather than ARMA modelling, and the simplifications this makes possible for the identification of models.
- Issues arising in practice from numerical problems in attempting to build models; some of these errors can be mitigated through the use of carefully chosen forms for the models.
- The difficulties arising from modelling time series of different but multiply related periodicities, e.g., monthly and quarterly, and existing tools for resolving, at least partly, the difficulties. These build on the ideas developed for single-periodicity modelling.

## 2. Dynamic Factor Models

**Assumption**

**1.**

#### 2.1. The Goal of Modelling

#### The Issue of Errors and Continuity

- If there is a true underlying process with an associated canonical spectral factor, and a canonical spectral factor obtained using approximate values of lagged covariance coefficients, there will be an error between the two.
- We should use a description of the computed canonical spectral factor that is appropriate to the application, in the sense that the error mechanisms in the relevant computations will not cause significant errors in the quantities computed as part of the application.

**Assumption**

**2.**

#### 2.2. Controlling the Number of Parameters to Be Estimated

**Assumption**

**3.**

- The dimension n of a minimal state-space realization of the canonical spectral factor (or its McMillan degree)
- The number of columns q in the minimal state-space factor, or the dimension of an innovations sequence (this being a white noise sequence, known in this context as the dynamic factor sequence)
- The rank r of ${\gamma}^{N}\left(0\right):=\mathbb{E}\left[{\chi}_{t}^{N}{\left({\chi}_{t}^{N}\right)}^{\top}\right]$, which is the dimension of the space spanned by ${\chi}_{1t},{\chi}_{2t},\dots ,{\chi}_{Nt}$.

**Assumption**

**4.**

**Theorem**

**1.**

#### 2.3. Strong and Weak Cross-Dependence

**Assumption**

**5.**

**Assumption**

**6.**

## 3. Obtaining the Common Component Process from the Measurement Process

#### Two Alternative Approaches to Additive Decomposition of the Spectral Matrix

## 4. Tall Stable Miniphase Spectral Factors and Singular Rational Spectra

**Theorem**

**2**

- Suppose that for polynomial matrices ${\overline{A}}^{N}\left(z\right)\in {\mathbb{R}}^{N\times N}\left[z\right],{\overline{B}}^{N}\left(z\right)\in {\mathbb{R}}^{N\times q}\left[z\right]$ with ${\overline{A}}^{N}$ and ${\overline{B}}^{N}$ left coprime8 there holds ${K}^{N}\left(z\right)={\left({\overline{A}}^{N}\right)}^{-1}{\overline{B}}^{N}$. Then zero freeness implies ${\overline{B}}^{N}\in {\mathbb{R}}^{N\times q}$ has full rank for all z. Accordingly, there exists a polynomial unimodular (constant determinant) common premultiplier ${U}^{N}\left(z\right)$ of ${\overline{A}}^{N},{\overline{B}}^{N}$ such that a new polynomial fraction description of ${K}^{N}\left(z\right)$ exists with$${K}^{N}\left(z\right)={\left({U}^{N}\left(z\right){\overline{A}}^{N}\left(z\right)\right)}^{-1}\left({U}^{N}\left(z\right){\overline{B}}^{N}\left(z\right)\right)={\left({A}^{N}\left(z\right)\right)}^{-1}{B}^{N}$$
- The system (3) is left invertible with unknown initial state, that is, there exists an integer $L\le n$ such that from the sequence ${\chi}_{k}^{N},{\chi}_{k+1}^{N},\dots ,{\chi}_{k+L-1}^{M}$ the state ${x}_{k}$ and the sequence ${w}_{k+1},{w}_{k+2},\dots ,{w}_{k+L-1}$ can be determined.

**Theorem**

**3**

- 1.
- The spectrum${\mathrm{\Phi}}_{\chi}^{N}$can be generated by a singular autoregressive process$$[I+{A}_{1}^{N}z+{A}_{2}^{N}{z}^{2}+\cdots +{A}_{m}^{N}{z}^{m}]{\chi}_{t}^{N}={B}^{N}{w}_{t}$$
- 2.
- Suppose the rational spectral matrix${\mathrm{\Phi}}_{\chi}^{N}\left({e}^{j\omega}\right)$is written in the form$${\overline{H}}^{N}{(I-z\overline{F})}^{-1}{\overline{Q}}^{N}+{\left({\overline{Q}}^{N}\right)}^{\top}{(I-{z}^{-1}{\overline{F}}^{\top})}^{-1}{\left({\overline{H}}^{N}\right)}^{\top}-{\overline{H}}^{N}{\overline{Q}}^{N},\phantom{\rule{1.em}{0ex}}z={e}^{j\omega}$$
- (a)
- there exists${\overline{G}}^{N}$such that the stable miniphase (canonical) spectral factor can be written in the form${\overline{H}}^{N}{(I-z\overline{F})}^{-1}{\overline{G}}^{N}$
- (b)
- ${\overline{G}}^{N}$is computable in a finite number of rational calculations from$\overline{F},{\overline{H}}^{N}$and${\overline{Q}}^{N}$, and
- (c)
- the triple$\{\overline{F},{\overline{H}}^{N},{\overline{G}}^{N}\}$is minimal.

- 3.
- In the event that${\overline{G}}^{N}$defined above is independent of N, say$G={\overline{G}}^{N}$, then${\overline{Q}}^{N}=P{\left({\overline{H}}^{N}\right)}^{\top}$for some positive definite P which is independent of N, with P the unique solution of$P-\overline{F}P{\overline{F}}^{\top}=G{G}^{\top}$.

**Theorem**

**4.**

## 5. Obtaining Parameters in the AR and State-Variable Models

## 6. Mixed Frequency Systems

#### 6.1. Blocked Linear Systems

#### 6.2. Systems with Multiple Periodicities in Their Outputs

**Theorem**

**5.**

- Depending on the value of J and the dimensions of ${u}_{t},{y}_{t}$, the dimension of ${U}_{t}$ may end up larger than that of ${Y}_{t}$, i.e., the transfer function of the blocked system may be fat, as noted in Anderson et al. (2016b).
- Depending on the value of J and the dimensions of ${u}_{t},{x}_{t}$, the dimension of ${U}_{t}$ may end up greater than that of ${x}_{t}$ (which is the state dimension of the blocked system).

## 7. Mixed Frequency System Identification

#### 7.1. Mixed Frequency System Identificaiton with an AR Underlying System

#### 7.2. A Simple AR(1) Example

#### 7.3. Mixed Frequency System Identification Exploiting Blocked System Structure

#### 7.4. VARMA and VMA System Identification

**Theorem**

**6.**

## 8. Conclusions

- The zero-free property has been applied to macroeconomic analysis as a means to solve the fundamentalness problem. Further work is ongoing on this and, we hope, the zero-free property will increasingly become familiar to macroeconomists, see the literature cited below Theorem 4.
- An important observation on the zero-free property in statements like Theorem 4 is that its genericity is usually obtained with respect to a parameterization in which each parameter is free to vary independently of all the others, see the comment below Theorem 2. However, in applications to macroeconomic models, structural restrictions may prevent such free variability and can therefore interfere with the genericity of zero-freeness. For an attempt to solve this issue, see Forni et al. (2020).
- Another observation on Theorem 4 is that an AR representation may exist also for non-generic parameters, for which zerolessness does not hold but the zeros of $B\left(z\right)$ lie outside the unit circle. However, in that case the AR would not be of finite length.
- In some cases, we have outlined algorithms which have not yet been tried on real data. It would be worthwhile to investigate them using real data.
- A full treatment (algorithms through to testing on real data) including the decomposition into common components and idiosyncratic processes is needed for systems with multiple frequencies.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AIC | Akaike Information Criterion |

AR | Autoregression |

ARMA | Autoregression Moving Average |

MA | Moving Average |

PCA | Principal Components Analysis |

SVD | Singular Value Decomposition |

VAR | Vector Autoregression |

VARMA | Vector Autoregression Moving Average |

VMA | Vector Moving Average |

## Appendix A. Proof of Theorem 3, Part 3

## Notes

1 | This paper underpins a lecture by the first author presented at the 5th Vienna Workshop on High-dimensional Time Series in Macroeconomics and Finance, and celebrating the 80th birthday of Manfred Deistler. |

2 | State space models have larger equivalences classes and therefore more flexibility when choosing “nice” representatives. Special representatives, in echelon form, can yield state space models with the same parameters as ARMA models. |

3 | This is an easy consequence of the celebrated solution of the partial realization problem for covariance sequences, Byrnes and Lindquist (1997). |

4 | In a practical situation, given that sensors normally are not noise free, the introduction of more sensors is likely to aid the de-noising process. Our remarks here pertain to the process of common components. |

5 | Such a spanning set of entries can be obtained by identifying the rows defining an $r\times r$ nonsingular principal submatrix of $\mathbb{E}\left[{\chi}_{t}^{N}{\left({\chi}_{t}^{N}\right)}^{\top}\right]$ |

6 | As an alternative, PCA can also be used to obtain a minimal static factor. |

7 | For a scalar rational transfer function written as a ratio of coprime polynomials $n\left(z\right)/d\left(z\right)$, the zero free property is equivalent to $n\left(z\right)$ being a nonzero constant. The idea generalizes to coprime matrix fraction representations of a rational transfer function matrix |

8 | Coprimeness means there exists no polynomial ${C}^{N}\left(z\right)\in {\mathbb{R}}^{N\times N}\left[z\right]$ with nonconstant determinant and matrices ${\tilde{A}}^{N}\in {\mathbb{R}}^{N\times N}\left[z\right],{\tilde{B}}^{N}\in {\mathbb{R}}^{N\times q}\left[z\right]$ such that ${\overline{A}}^{N}={C}^{N}{\tilde{A}}^{N}$ and ${\overline{B}}^{N}={C}^{N}{\tilde{B}}^{N}$ |

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**MDPI and ACS Style**

Anderson, B.D.O.; Deistler, M.; Lippi, M.
Linear System Challenges of Dynamic Factor Models. *Econometrics* **2022**, *10*, 35.
https://doi.org/10.3390/econometrics10040035

**AMA Style**

Anderson BDO, Deistler M, Lippi M.
Linear System Challenges of Dynamic Factor Models. *Econometrics*. 2022; 10(4):35.
https://doi.org/10.3390/econometrics10040035

**Chicago/Turabian Style**

Anderson, Brian D. O., Manfred Deistler, and Marco Lippi.
2022. "Linear System Challenges of Dynamic Factor Models" *Econometrics* 10, no. 4: 35.
https://doi.org/10.3390/econometrics10040035