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10.2.4 Extracting Factors from the Frequency Domain

Forni et al. (2000) propose an alternative method for factor extraction that does not require a rational lag distribution or autoregressive representation for the dynamics in the factors. Their model starts from the moving average representation

$x_{i t} = b_{i 1} (L) u_{1 t} + b_{i 2} (L) u_{2 t} + \cdot \cdot \cdot + b_{i q} (L) u_{q t} + ξ_{i t}, i = 1, \dots, N, (10.20)$

where each of the common shock processes $u_{j t} (i = 1, \dots, q)$ are uncorrelated white noise with zero mean and unit variance, $(ξ_{1 t}, \dots, ξ_{N t})$ is a stationary process that is orthogonal to the $u_{j t}$ processes, and the coefficients of the lag polynomials $b_{ij} (L)$

are square-summable. Moreover, the idiosyncratic components $ξ_{i t}$ may be crosssectionally correlated. These assumptions imply that $(x_{1 t}, \dots, x_{n t})^{'}$ is a stationary process. In this representation, the contribution of the common component to the ith observable variable, $x_{i t},$ is captured through the linear combination of the $q$ factors $χ_{i t} = b_{i 1} (L) u_{1 t} + b_{i 2} (L) u_{2 t} + \cdot \cdot \cdot + b_{i q} (L) u_{q t}$

Let $Σ_{n} (θ)$ be the spectral density matrix of $x_{t} .$ , where $θ \in [- π, π]$ . Define $λ_{nj}$ as the real nonnegative j th eigenvalue of $Σ_{n} (θ)$ organized in descending order of magnitude or, equivalently, the j th dynamic eigenvalue of $Σ_{n} (θ)$ . Similarly, let $λ_{nj}^{χ}$ be the dynamic eigenvalues associated with the covariance matrix of the common components, $Σ_{n}^{χ}$ , and let $λ_{nj}^{ξ}$ be the eigenvalues of the covariance matrix of the idiosyncratic components, $Σ_{n}^{ξ}$ . A key assumption in Forni et al. (2000) is that the first idiosyncratic dynamic eigenvalue $ψ_{n 1}^{ξ}$ is uniformly bounded, i.e., $λ_{n 1}^{ξ} (θ) \leq \overset{ˉ}{λ}$ for any $θ \in [0, 2 π]$ , while conversely the first q common dynamic eigenvalues diverge, $i.e., lim_{n \infty} λ_{nj}^{χ} (θ) = \infty for j \leq q$ and $θ \in [0, 2 π]$ almost everywhere.

The common components can be recovered from a sequence of filters $K_{ni} (L) (x_{1 t}, \dots, x_{n t})^{'}$ which are functions of the unknown spectral density matrices $Σ_{n} (θ)$ . Estimation of these can proceed using a discrete Fourier transform of a truncated two-sided sequence of covariance matrices of $x_{n t}$

$Σ_{n}^{T} (θ_{m}) = k = - M \sum M Γ_{nk}^{T} ω_{k} e^{- ik θ_{m}},$

where $i = - 1, θ_{m} = 2 π m / (2 M + 1), m = 0, 1, \dots, 2 M,$ and $Γ_{nk}^{T}$ is the sample covariance matrix of $(x_{1 t}, \dots, x_{n t})^{'}$ and $(x_{1 t - k}, \dots, x_{n t - k})^{'}$ , and the weights, $ω_{k}$ follow a Bartlett window of size M, i.e., $ω_{k} = 1 - [∣ k ∣/ (M + 1)]$ . Denote the first q eigenvectors of $Σ_{n}^{T} (θ_{m})$ by $p_{nj}^{T} (θ_{m})$ , for $j = 1, \dots, q$ , and $m = 0, 1, \dots, 2 M .$ while $\tilde{P}_{nj}^{T} (θ_{m})$ is its adjoint (transposed, complex conjugate). Then we can compute

$K_{ni}^{T} (θ_{m}) = \tilde{p}_{n 1, i}^{T} (θ_{m}) p_{n 1}^{T} (θ_{m}) + \cdot \cdot \cdot + \tilde{p}_{n q, i}^{T} (θ_{m}) p_{n q}^{T} (θ_{m}) .$

The filter $\underline{K}_{nj} (L)$ can be estimated from the inverse Fourier transform of the vector $(K_{ni}^{T} (θ_{0}), \dots, K_{ni}^{T} (θ_{2 M}))^{'}$ as follows:

$\underline{K}_{ni, k}^{T} = \frac{1}{2 M + 1} m = 0 \sum 2 M K_{ni}^{T} (θ_{m}) e^{ik θ_{m}} .$

Finally, the common components can be obtained from the relation $ϰ_{i t, n} =$ $\underline{K}_{ni}^{T} (\dot{L}) (x_{1 t}, \dots, x_{n t})^{'}$ , where the estimated two-sided filter is given by

$\underline{K}_{ni, k}^{T} (L) = k = - M \sum M \underline{K}_{ni, k}^{T} L^{k} .$

To the extent that the estimation of the factors relies on two-sided filters, this approach is not well suited for forecasting since estimates of the current values of the factors are not available and so cannot be used to condition on when forecasting future values of the dependent variable.

练习题

In Forni et al. (2000) factor extraction model, what is the nature of the common shock processes $u_{j t}$ ?

A. Correlated white noise with non-zero mean

B. Uncorrelated white noise with zero mean and unit variance

C. Correlated white noise with zero mean and non-unit variance

D. Uncorrelated white noise with non-zero mean and non-unit variance

What is the role of the spectral density matrix $Σ_{n} (θ)$ in Forni et al. (2000) model?

A. It represents the covariance matrix of the common components

B. It represents the covariance matrix of the idiosyncratic components

C. It is the spectral density matrix of

x_{t}

D. It is the moving average representation of

x_{i t}

Which of the following are assumptions about eigenvalues in Forni et al. (2000) model? (Select all that apply)

A. The first idiosyncratic dynamic eigenvalue

λ_{n 1}^{ξ} (θ)

is uniformly bounded

B. The first q common dynamic eigenvalues diverge

C. All eigenvalues are uniformly bounded

D. The idiosyncratic components have diverging eigenvalues

In Forni et al. (2000) model, the common components can be recovered using filters that are functions of the unknown spectral density matrices $Σ_{n} (θ)$ .

The estimation of the spectral density matrices in Forni et al. (2000) model involves the use of a discrete Fourier transform of a truncated two-sided sequence of covariance matrices of $x_{n t}$ .

The weights, $ω_{k}$ , in the estimation of spectral density matrices follow a Bartlett window of size M, i.e., $ω_{k} = 1 - \frac{∣ k ∣}{M + 1}$ for $k = - M, \dots, M$ . The value of $ω_{k}$ when $k = 0$ is ___.

In Forni et al. (2000) model, the first q eigenvectors of $Σ_{n}^{T} (θ_{m})$ are denoted by $p_{nj}^{T} (θ_{m})$ for $j = 1, \dots, q$ . The adjoint (transposed, complex conjugate) of $p_{nj}^{T} (θ_{m})$ is denoted by ___.

Explain the process of computing the filters $K_{ni}^{T} (θ_{m})$ in Forni et al. (2000) model.

What is the limitation of using two-sided filters for forecasting in Forni et al. (2000) model?

Which of the following statements are true regarding the estimation of the filter $\underline{K}_{nj} (L)$ in Forni et al. (2000) model? (Select all that apply)

A. It is estimated from the inverse Fourier transform of the vector

(K_{ni}^{T} (θ_{0}), \dots, K_{ni}^{T} (θ_{2 M}))^{'}

B. The estimation involves the use of a discrete Fourier transform

C. The formula for

\underline{K}_{ni, k}^{T}

\frac{1}{2 M + 1} \sum_{m = 0}^{2 M} K_{ni}^{T} (θ_{m}) e^{ik θ_{m}}

D. The estimation process does not involve any Fourier transform

In the context of Forni et al. (2000) model, which of the following is true about the moving average representation of $x_{i t}$ ?

A. It includes only the common shock processes

u_{j t}

B. It includes only the idiosyncratic components

ξ_{i t}

C. It includes both the common shock processes

u_{j t}

and the idiosyncratic components

ξ_{i t}

D. It does not include any lag polynomials

b_{ij} (L)

Which of the following are true about the common components $χ_{i t}$ in Forni et al. (2000) model? (Select all that apply)

A. They are captured through the linear combination of the

q

factors

B. They are orthogonal to the idiosyncratic components

ξ_{i t}

C. They are represented by the equation

χ_{i t} = b_{i 1} (L) u_{1 t} + b_{i 2} (L) u_{2 t} + \dots + b_{i q} (L) u_{q t}

D. They are uncorrelated with the common shock processes

u_{j t}

Which of the following are key assumptions in Forni et al. (2000) model regarding the eigenvalues? (Select all that apply)

A. The first idiosyncratic dynamic eigenvalue

λ_{n 1}^{ξ} (θ)

is uniformly bounded for any

θ \in [0, 2 π]

B. The first q common dynamic eigenvalues diverge for

j \leq q

and

θ \in [0, 2 π]

almost everywhere

C. All eigenvalues are positive

D. The idiosyncratic components have eigenvalues that are zero

Which of the following statements are true regarding the estimation of the common components in Forni et al. (2000) model? (Select all that apply)

A. The common components can be obtained from the relation

ϰ_{i t, n} = \underline{K}_{ni}^{T} (\dot{L}) (x_{1 t}, \dots, x_{n t})^{'}

B. The estimated two-sided filter is given by

\underline{K}_{ni, k}^{T} (L) = \sum_{k = - M}^{M} \underline{K}_{ni, k}^{T} L^{k}

C. The common components are estimated using only the idiosyncratic components

ξ_{i t}

D. The estimation process involves the use of filters that are functions of the unknown spectral density matrices

Σ_{n} (θ)

In the Forni et al. (2000) factor extraction model, which of the following statements about the eigenvalues of the spectral density matrix is correct?

A. The first idiosyncratic dynamic eigenvalue

λ_{n 1}^{ξ} (θ)

is unbounded for any

θ \in [0, 2 π]

B. The first q common dynamic eigenvalues

λ_{nj}^{χ} (θ)

converge to zero as

n \to \infty

for

j \leq q

and

θ \in [0, 2 π]

C. The first idiosyncratic dynamic eigenvalue

λ_{n 1}^{ξ} (θ)

is uniformly bounded for any

θ \in [0, 2 π]

, while the first q common dynamic eigenvalues

λ_{nj}^{χ} (θ)

diverge as

n \to \infty

for

j \leq q

and

θ \in [0, 2 π]

D. The first q common dynamic eigenvalues

λ_{nj}^{χ} (θ)

are uniformly bounded for any

θ \in [0, 2 π]

, while the first idiosyncratic dynamic eigenvalue

λ_{n 1}^{ξ} (θ)

diverges as

n \to \infty

Which of the following statements are correct regarding the estimation of spectral density matrices in the Forni et al. (2000) model?

A. The spectral density matrix

Σ_{n} (θ)

is estimated using a discrete Fourier transform of a truncated two-sided sequence of covariance matrices of

x_{n t}

B. The weights

ω_{k}

in the estimation process follow a uniform window of size M.

C. The sample covariance matrix

Γ_{nk}^{T}

is computed using

(x_{1 t}, \dots, x_{n t})^{'}

and

(x_{1 t - k}, \dots, x_{n t - k})^{'}

D. The estimation process involves the use of a Bartlett window of size M for the weights

ω_{k}

E. The spectral density matrix

Σ_{n} (θ)

is estimated using a continuous Fourier transform of the covariance matrices of

x_{n t}

The common components in the Forni et al. (2000) model can be recovered using a sequence of filters that are functions of the unknown spectral density matrices $Σ_{n} (θ)$ , and these filters can be estimated using the inverse Fourier transform of the vector $(K_{ni}^{T} (θ_{0}), \dots, K_{ni}^{T} (θ_{2 M}))^{'}$ .

In the Forni et al. (2000) model, the first q eigenvectors of $Σ_{n}^{T} (θ_{m})$ are denoted by $p_{nj}^{T} (θ_{m})$ for $j = 1, \dots, q$ and $m = 0, 1, \dots, 2 M$ , and the adjoint (transposed, complex conjugate) is denoted by $\tilde{P}_{nj}^{T} (θ_{m})$ . The filters $K_{ni}^{T} (θ_{m})$ can be computed as $K_{ni}^{T} (θ_{m}) = \tilde{p}_{n 1, i}^{T} (θ_{m}) p_{n 1}^{T} (θ_{m}) + \cdot \cdot \cdot + \tilde{p}_{n q, i}^{T} (θ_{m}) p_{n q}^{T} (θ_{m})$ . The common components are then obtained using the relation $ϰ_{i t, n} =$ ___ $(L) (x_{1 t}, \dots, x_{n t})^{'}$ , where the estimated two-sided filter is given by $\underline{K}_{ni, k}^{T} (L) = \sum_{k = - M}^{M} \underline{K}_{ni, k}^{T} L^{k}$ .

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10.2.4 Extracting Factors from the Frequency Domain