Department of Economics and Business Economics

Aarhus University

Fuglesangs Allé 4

DK-8210 Aarhus V

Denmark

Email: oekonomi@au.dk

Tel: +45 8716 5515

Wavelet Estimation for Dynamic Factor Models with Time-

Varying Loadings

Duván Humberto Cataño, Carlos Vladimir Rodríguez-

Caballero and Daniel Peña

CREATES Research Paper 2019-23

Wavelet Estimation for Dynamic Factor Modelswith Time-Varying Loadings

Duván Humberto Cataño∗†, Carlos Vladimir Rodríguez-Caballero‡, andDaniel Peña §,

Abstract

We introduce a non-stationary high-dimensional factor model with time-varying loadings. We propose an estimation procedure based on two stages.First, we estimate common factors by principal components. Afterwards,in the second step, considering the factors estimates as observed, the time-varying loadings are estimated by an iterative procedure of generalized leastsquares using wavelet functions. We investigate the finite sample features ofthe proposed methodology by some Monte Carlo simulations. Finally, weuse this methodology to study the electricity prices and loads of the NordPool power market.

Keywords: Factor models, wavelet functions, generalized least squares, elec-tricity prices and loads.

1 Introduction

Factor models have been widely used in the last decade due to their ability to ex-plain the structure of a common variability among time series by a small numberof unobservable common factors. In this sense, these models are used to reduce di-mensionality on complex systems. The studies of factor models have encompassedthe stationary and nonstationary frameworks by different estimation methods, seeamong many others, Pena and Box (1987), Forni et al. (2000), Stock and Wat-son (2002), Bai and Ng (2002), Bai (2003), Forni et al. (2004), and Forni et al.(2005) for the stationary cases, and Bai and Ng (2004), Peña and Poncela (2006),∗The author would like to thank financial supports from CAPES, CNPq, and University of São

Paulo, Brasil.†University of Antioquia, Colombia.‡Corresponding author. Department of Statistics, ITAM, Mexico & CREATES, Aarhus Univer-

sity, Denmark. Address: Río Hondo No.1, Col. Progreso Tizapán, Álvaro Obregón, CDMX. 01080.Mexico. E-mail: vladimir.rodriguez@itam.mx

§Department of Statistics and Institute UC3M-BS of Financial Big Data. Universidad Carlos IIIde Madrid, Getafe, Spain.

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Barigozzi et al. (2016), and Rodríguez-Caballero and Ergemen (2017) for the non-stationary cases.

One non stationary framework of special interest is the factor model withloadings that vary over time. Motta et al. (2011) introduced deterministic smoothvariations in factor loadings and proposed estimation procedures based on locallyweighted generalized least squares using kernel functions under a time-domain ap-proach. Similarly, Eichler et al. (2011) allowed for a non stationary dynamic struc-ture in a factor model and considered deterministic time-dependent functions in thefactor loadings. Their estimation procedure can be seen as a time-varying spectraldensity matrix of the underlying process. Furthermore, Mikkelsen et al. (2018)proposed a factor model with time-varying loadings that evolve as stationary VARprocesses. They employed Kalman filter procedures to obtain the maximum likeli-hood estimators of the parameters of the factor loadings.

In this paper, we use wavelet functions to define smooth variations of load-ings in high-dimensional factor models. Our model can be useful in some eco-nomic applications when the dynamics is driven by smooth progressive variations,whose cumulative effects cannot be simply ignored, see Su and Wang (2017), andBai and Han (2016), for details. In our framework, to capture the common smoothvariations in the vector of time series, the parameters in the loading matrix areassumed to be well aproximated by deterministic functions over time.

The estimation procedure consists of two steps. First, the common factorsare estimated by standard principal component analysis (PCA). Then, in the secondstep, considering the factors as known, factor loadings are estimated by an iterativeprocedure that combines generalized least squares (GLS) using wavelet functions.We show that factors estimated by principal components are consistent after con-trolling the magnitude of the loadings instabilities. We highlight that a necessaryrequirement for such consistency is that common factors need to be independent ofthe functions of factor loadings. This model is extended for the case when variablesare non-stationary where the factor are first estimated by the approach of Peña andPoncela (2006). We use Monte Carlo simulations to show that the method cor-rectly identify the factors and loadings. Finally, we use the methodology proposedwith non-stationary variables to analyze an electricity market: the Nord Pool powermarket. We use the electricity system prices and loads throughout two years and ahalf to show that factor loadings are not invariant along time, illustrating the usefulof our model in energy markets. We find that some features of electricity prices andloads, see e.g. Weron (2007), and Weron (2014), are well extracted by commonfactors estimates, and by time-varying loadings estimates.

The remainder of the paper is organized as follows. Section 2 introduces themodel, shows the consistency of principal components with stationary and non-stationary variables, and introduces the wavelet functions. Section 3 discusses theproposed estimation procedure. Monte Carlo simulation studies are presented inSection 4, and an empirical illustration is provided in Section 5. Section 6 con-cludes. Throughout the paper we use bold-unslanted letters for matrices, bold-slanted letters for vectors and unbold (normal) letters for scalars. We denote by

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tr(·) the trace operator, by rank(A) the rank of a matrix A, by In the identitymatrix of dimension n, by ⊗ the Kronecker product and by ‖ · ‖ the Frobenius(Euclidean) norm, i.e., ‖A‖ =

√tr(A′A).

2 The model

The model we propose is as follows

Yt = Xt + et, (1)

Xt = [Λ0 + Λ(t/T )]Ft, (2)

where the common component, Xt, is aN−dimensional locally stationary process,in sense of Dahlhaus et al. (1997b) and the loadings are defined in the re-scaledtime, u = t/T ∈ [0, 1]. Ft are the unobservable common factors and et, theidiosyncratic component, is a sequence of weakly dependent variables. Λ(u) ={λij(u), i = 1, . . . , N ; j = 1, . . . , r, is the time-dependent factor loading matrix.In this respect, the influence of Ft on the observed process varies over time andΛ(u) capture the smooth variations of the loadings with respect Λ0, a constantloading matrix.

When Λ(u) = 0, ∀u ∈ [0, 1], the standard factor model is considered,therefore the model proposed in (1) can be seen as a generalization of the standardfactor model of Bai (2003).

Definition 1. The sequence Yt in (1) follows a factor model with time-varyingloadings if:

a. For N ∈ N, there is a function with

Λ(·) : [0, 1] → RN×ru 7→ Λ(u)

such that ∀ T ∈ N,

ΓY (u) = [Λ0 + Λ(u)]ΓF [Λ0 + Λ(u)]′ + Γe,

where rank[Λ0 + Λ(u)] = r and ΓF = Var(Ft), is a positive definitediagonal matrix.

b. Γe = var(et) is a positive definite matrix.

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2.1 Assumptions

With an arbitrary constant Q ∈ R+, the assumptions of the model in (1) followingBai and Ng (2002) are as follows,

Assumption A. Factors:

A1 E‖Ft‖4 < Q,

A2 T−1∑T

t=1 FtF′tp→ ΓF when T → ∞, where ΓF is a positive definite diag-

onal matrix.

Assumption B. Factor loadings:

B1 ‖λi0‖ ≤ λ and ‖Λ′0Λ0/N − D‖ → 0, when N → ∞, with r × r positivedefinite matrix D, where λi0 is the i−th row of Λ0,

B2 supu∈(0,1) ‖λi(u)‖ ≤ λ <∞, where λi(u) is the i−th row of Λ(u),

B3 λij(u) ∈ L2[0, 1], for i = 1, . . . , N and j = 1, . . . , r.

Assumption C. Idiosyncratic terms:

C1 E(eit) = 0,E|eit|4 < Q,

C2 E[eitejt] = τij,t,with |τij,t| < |τij |, for a constant |τij | andN−1∑N

i,j=1 |τij | <Q, for all t,

C3 E[N−1∑N

i=1 eiseit] = γN (s, t), |γN (s, s)| < Q, for all (s, t); andT−1

∑Ts=1

∑Tt=1 |γN (s, t)| < Q.

Assumption D. Time-varying factor loadings and factors:K1NT , K2NT and K3NT are functions such that the following conditions are ful-filled for any n,m, k, q = 1, . . . , r and λij(s/T ) ≡ λij(s).

D1 sups,t∑N

i,j |λin(s)λjm(t)||E[FnsFmt]| < K1NT ,

D2∑T

s,t=1

∑Ni,j=1 |λin(s)λjm(s)||E[FnsFmsFktFqt]| < K2NT ,

D3∑T

s=1

∑Ni,j=1 |λin(s)λjm(s)λik(t)λjq(t)||E[FnsFmsFktFqt]| < K3NT .

Assumption E. Independence:The process eit and Fjs are independent of each other for any (i, j, s, t).

Assumption A imposes standard moment conditions, that is, the unobserv-able factors have finite fourth moments and their covariance converges in probabil-ity to a positive definite matrix. AssumptionsB1 andB2 ensure that each factor hasa nontrivial contribution to the variance of Yt. Assumption B3 ensures existenceof the expansion in the wavelet function for the factor loadings. Assumptions Callows dependence in the idiosyncratic process. Assumption D is required to guar-antee the consistency of the principal components. Finally, independence betweenfactors and the idiosyncratic term is provided in the Assumption E.

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2.2 Principal component estimators

We use PCA to estimate the factors Ft. It is well-known that principal componentsare obtained solving the optimization problem

minF,Λ

(NT )−1N∑i=1

T∑t=1

(Yit − λ′iFt)2, (3)

where F = (F1,F2, . . . ,FT )′ is a T × r matrix and Λ is a N × r matrix. We needto impose some restrictions to guarantee the identification of the parameters asusual. Solving for Λ, the normalization F′F = Ir provides the necessary numberof restrictions. With this, minimize (3) is equivalent to maximize tr[F′(YY′)F],where Y = (Y1, . . . ,YT )′. Then, the estimated factor matrix, F, is

√T times the

eigenvectors corresponding to the r largest eigenvalues of the T × T matrix YY′.It is well-known that this solution is not unique, that is, any orthogonal rotation ofF is also a solution. See Bai et al. (2008) for more details.

The following theorem is a modified version of Theorem 1 in Bates et al.(2013) and shows that, under the assumptions previously stated, it is possible toconsistently estimate any rotation of the factors by principal components even ifthe loadings are time-varying.

Theorem 1. Under Assumptions A-E, there exists an r × r matrix H such that

T−1T∑t=1

‖Ft −H ′Ft‖2 = Op(RNT ), (4)

whenN,T →∞, whereRNT = max{

1N ,

1NT ,

K1NTN2 , K2NT

N2T 2 ,K3NTN2T 2

}withK1NT ,

K2NT , and K3NT defined in the Assumption D. Furthermore,H = (Λ

′0Λ0/N)(F′F/T )V −1

NT , where VNT is a diagonal matrix of the r largesteigenvalues of the matrix (NT )−1YY′.

Proof. See the appendix A.1.Theorem 1 points out that the average squared deviation between the esti-

mated factors and the space spanned by a rotation of the true factors will vanishat rate RNT , which is similar to that in Bai and Ng (2002). Note that from (4),the estimated common factors, Ft, are identified through a rotation, then, principalcomponents converge to a rotation of the true common factors H ′Ft.

2.3 Nonstationary factors

In many areas, as in economics and finance, there is strong evidence in favor of thepresence of non-stationarity processes, which have repeatedly found in many em-pirical studies with economic or financial time series. Consequently, it is natural tothink that many panel data may include non-stationary economic or financial vari-ables. There has been some debate concerning the use of differenced variables, the

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main argument being discussed is whether differencing the series causes a severeloss of information. In this paper, to treat with non-stationary variables, we use themethodology proposed by Peña and Poncela (2006) who, assuming that Yt ∼ I(d)with a positive integer d, use generalized covariance matrices, which is defined as

Cy(k) =1

T 2d+d′

T∑t=k+1

(Yt−k − Yt)(Yt − Yt), (5)

where Y = 1T

∑Tt=1 Yt and d′ can be either 0 or 1. In this framework, a consistent

estimate for the factors are:

F = YΛ, (6)

where F = (F1, F2, . . . , FT )′ is a T × r matrix, Y = (Y1, . . . ,YT )′ is a T ×Nmatrix and Λ is a N × r matrix composed by the first r eigenvectors of Cy(k).Following the same reasoning as in Theorem 1, it can be shown that the averagesquared deviation between the estimated factors (6) and the space spanned by arotation of the true factors will vanish when (N,T )→∞.

2.4 Wavelets

The basic idea of a wavelet is to construct infinite collections of translated andscaled versions of the scale function φ(t) and the wavelet ψ(t) such as φjk(t) =2j/2φ(2jt−k), andψjk(t) = 2j/2ψ(2jt−k) for j, k ∈ Z. Suppose that {φlk(·)}k∈Z∪{ψjk(·)}j≥l;k∈Z forms an orthonormal basis ofL2(R), for any coarse scale l. A keypoint is to construct φ and ψ with a compact support that generates an orthonormalsystem, which has location in time-frequency. From this, we can get parsimoniousrepresentations for a wide class of wavelet functions, see Chiann and Morettin(2005) and Porto et al. (2008) for details. In some applications, these functions aredefined in a compact set such as [0, 1]. We consider this compact set for functionsλij(u), for i = 1, . . . , N and j = 1, . . . , r defined in (2). Then, it is necessaryto consider an orthonormal system that generates L2[0, 1]. For the construction ofthese orthonormal systems we follow the procedure by Cohen and Ryan (1995),that generates multiresolution levels V0 ⊂ V1 ⊂ · · · , where the spaces Vj are gen-erated by ψjk. Negative values of j are not necessary since φ = φ00 = 1, and ifj ≤ 0, ψjk(u) = 2−j/2, see Vidakovic (2009) for more details and different ap-proaches. Therefore, for any function λ(u) ∈ L2[0, 1], can be expand in series oforthogonal functions

λ(u) = α00φ(u) +∑j≥0

∑k∈Ij

βjkψjk(u), (7)

where we take l = 0 and Ij = {k : k = 0, . . . , 2j − 1}. For each j, the set Ijgenerates values of k such that βjk belongs to the scale 2j . For example, for j = 3,

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there are eight wavelet coefficients in the scale 23, whereas for j = 2, only fourcoefficients in the scale 22.

Some applications consider the equation in (7) for a maximum resolutionlevel J , through

λ(u) ≈ α00φ(u) +

J−1∑j=0

∑k∈Ij

βjkψjk(u). (8)

In this way, the function λ(u) approximates to the space VJ . In this paper, we useordinary wavelets as in Dahlhaus et al. (1997a) due to their performance is suitablein the case of smooth functions. Particularly, Daubechies D8 and Haar wavelets ofcompact supports are employed.

3 Estimation of time-varying loadings by wavelets

We consider the process in (1) with r common factors (r < N) to discuss theestimation procedure of the time-varying loadings. From now on, we consider theloading matrix, [Λ(u) + Λ0], as a unique function over time Λ(u), given by

Yt = Λ(u)Ft + et, (9)

with t = 1, 2, . . . , T , and u = t/T ∈ [0, 1]. In matrix form, we have

Y1t

Y2t...Yrt

...YNt

=

λ11(u) λ12(u) . . . λ1r(u)λ21(u) λ22(u) . . . λ2r(u)

......

. . ....

λr1(u) λr2(u) . . . λrr(u)...

.... . .

...λN1(u) λN2(u) . . . λNr(u)

F1t

F2t...Frt

+

e1t

e2t...ert...eNt

, (10)

where Yt is an N−dimensional vector of time series, Λ(u) is the time-varyingloading matrix with λij(u) ∈ L2[0, 1], for i = 1, 2, . . . , N , and j = 1, 2, . . . , r.Ft is the common factor, and et is the idisyncratic process. From (9), and theAssumption E, the structure of the covariance matrix of the process Yt is writtenas

ΓY (u) = Λ(u)ΓFΛ′(u) + Γe, ∀u ∈ [0, 1],

that means, the variance of the common component is ΓX(u) = Λ(u)ΓFΛ′(u).For the construction of the time-varying loadings estimates, we first as-

sume that the estimator of the r common factors are obtained by PCA of theN−dimensional time series Yt. Then, functions of time-varying loadings are ap-proximated in series of orthogonal wavelets as in (8), for a fixed resolution level

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J < T ,

λmn(u) = α(mn)00 φ(u) +

J−1∑j=0

∑k∈Ij

β(mn)jk ψjk(u). (11)

The values of j, k vary depending on the resolution level in the wavelet de-composition. We choose the maximum resolution J , such that 2J−1 ≤

√T ≤ 2J ,

see Dahlhaus et al. (1997a) for details of this selection. In practice, the coeffi-cients α(mn)

00 , β(mn)00 , β

(mn)10 , . . . , β

(mn)

J−1,2J−1are obtained for a particular estimation

method. In this paper, we use GLS to estimate these coefficients and to reconstructthe loadings functions.

Let Yt = (Y1t, Y2t, . . . , YNt) be N time series with t = 1, 2, . . . , T whichare generated by

Yt = Λ(u)Ft + et, (12)

where the r common factors, Ft, are estimated by principal components. Eachloading function λmn(u) is written as in (11), then when plugging each λmn(u)into (10), we have

Y11...

Y1T...Yr1

...YrT

...YN1

...YNT

︸ ︷︷ ︸vec(Y)

=

Ψ(1)

FΨ

(2)

F. . . Ψ

(r)

F. . . O O . . . O O O . . . O

O O O O . . . O O . . . O O O . . . O...

......

.... . .

......

......

......

O O O O . . . Ψ(1)

FΨ

(2)

F. . . Ψ

(r)

FO O . . . O

......

......

......

......

......

O O O O . . . O O . . . O Ψ(1)

FΨ

(2)

F. . . Ψ

(r)

F

︸ ︷︷ ︸

Θ

β(1)

β(2)

β(3)

...β(r)

...β(N)

︸ ︷︷ ︸

β

+

e11...e1T

...er1

...erT

...eN1

...eNT

,

︸ ︷︷ ︸vec(e)

(13)

where

Ψ(i)

F=

φ(1/T )Fi1 ψ00(1/T )Fi1 . . . ψJ−1,2J−1(1/T )Fi1φ(2/T )Fi2 ψ00(2/T )Fi2 . . . ψJ−1,2J−1(2/T )Fi2

......

. . ....

φ(T/T )FiT ψ00(T/T )FiT . . . ψJ−1,2J−1(T/T )FiT

,

are T × 2J matrices for i = 1, 2, . . . , r and O is T × 2J null matrix.Let ΨrF = [Ψ

(1)

F, . . . ,Ψ

(r)

F] be a T × r2J matrix, then

Θ = IN ⊗ΨrF

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isNT×Nr2J matrix that depends on the estimated factors, Ft, the wavelets ψ(u),and the resolution level J , with vector of parameters β(m) =

(β(m1),β(m2), . . . ,

β(mr))′

of dimension r2J × 1 for m = 1, 2, . . . , N , where β(mn) =(α

(mn)00 ,

β(mn)00 , β

(mn)10 , . . . , β

(mn)

J−1,2J−1

)′.

Each β(m) is composed by the wavelets coefficients of the m−th row of thematrix Λ(u). Therefore, the total number of wavelets parameters to be estimatedis 2JNr.

Hence, the model in (13) can be represented in a linear model form as

vec(Y) = Θβ + vec(e),

where vec(Y) is the response vector and Θ is the usual design matrix in regressionanalysis. Assuming that the covariance matrix of the idiosyncratic errors, Γe, isknown, then the GLS estimator of the coefficients β is given by

β = (Θ′Σ−1e Θ)−1Θ′Σ−1

e vec(Y),

where Σe is a NT ×NT matrix defined as

Σe = Γe ⊗ IT =

ITγe,11 ITγe,12 . . . ITγe,1NITγe,21 ITγe,22 . . . ITγe,2N

......

. . ....

ITγe,N1 ITγe,N2 . . . ITγe,NN

These results provide linear estimators of wavelets coefficients for the time-

varying loadings assuming that the covariance matrix of the idiosyncratic error isknown. Some procedures, such as maximum likelihood methods, are not computa-tionally efficient for estimating such model, since the number of parameters tendsto be very large. In this light, we use GLS to simplify the implementation.

Note that we can use different basis of wavelet functions φ(u), and ψjk(u)for each λmn(u). In the simulation section, we use similar basis to simplify theexposition.

3.1 Estimation

The estimation procedure is based on a general two-step procedure, which can beexecuted by the following algorithm:

Step 1. Use principal components to estimate Ft as

F =√T [v1, v2, . . . , vr],

where vi is the eigenvector corresponding to the i−th largest eigenvalue, λifor i = 1, . . . , r, of the matrix (NT )−1YY′. Here, Y and F denote T ×N

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and T × r matrices, respectively. Next, the loading function matrix, Λ(t), isapproximated by the wavelets

λmn(t) = α(mn)00 φ(t) +

J−1∑j=0

∑k∈Ij

β(mn)jk ψjk(t),

with Ij = {k : k = 0, 1, . . . , 2j − 1}, m = 1, . . . , N , and n = 1, . . . , r.Then, we write the equation (12) as

vec(Y) = Θ(F, ψ, φ)β + vec(e),

where vec(Y) and vec(e) areNT×1 vectors, and the dimensions of Θ(F, ψ, φ)and β are (NT × 2JNr) and (2JNr × 1), respectively, where J indicatesthe resolution level chosen in the wavelet expansions.

Step 2. Estimate by GLS the wavelet coefficients as

β = (Θ′Σ−1e Θ)−1Θ′Σ−1

e Z,

where Θ(F, ψ, φ) ≡ Θ, Z = vec(Y), and Σe = INT is used as initial value.

Step 3. Using the estimated coefficient in the Step 2, the loadings are obtained as

Λ(t)(0) = {λ(0)mn(t)}n=1,...,r

m=1,...,N ,

where

λ(0)mn(t) = α

(mn)00 φ(t) +

J−1∑j=0

∑k∈Ij

β(mn)jk ψjk(t).

Step 4. With Λ(t)(0), obtain residuals, Yt − Λ(t)(0)Ft = e(0)t . Then, compute

Γ(0)e =

T∑t=1

e(0)t e(0)′

t /T.

Step 5. Back to step 2 with Σe = Γ(0)e . Iterate n-times the procedure to obtain the

sequences {Λ(t)(i), Γ(i)e }i=1,...,n. Stop the iteration when

‖Λ(t)(i−1) − Λ(t)(i)‖ < δ,

for any small δ > 0, where ‖·‖ denotes the Frobenius norm for t = 1, . . . , T.

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4 Monte Carlo simulation

We examine the finite-sample properties of the estimation procedure proposedabove using a Monte Carlo study. The model in (1) is generated as

Yit = λ′i(t)Ft + eit, i = 1, . . . , N and t = 1, . . . , T,

Fkt(1− θkB) = ηkt, k = 1, . . . , r. ηt ∼ Nr(0, diag{1− θ21, . . . , 1− θ2

r}),et ∼ NN (0,Γe),

where the matrix Γe is generated by a couple of different structures: i) Γe ={γ|i−j|}i,j=1,...,N , that is a Toeplitz matrix, and ii) a diagonal matrix. Furthermore,at instant t, Yit denotes the i−th time series, λ′i(t) = (λi1(t), . . . , λir(t)) is thevector of loadings, which are generated alternately by some smooth functions. Wediscuss a couple of functions used below. Ft = (F1t, . . . , Frt)

′ is the vector of fac-tors, and ηt = (η1t, . . . , ηrt)

′ and et = (e1t, . . . , eNt)′ are vectors of idiosyncratic

terms which are independent to each other.In our Monte Carlo study, the model is generated with N ∈ {20, 30} cross-

sectional units and T ∈ {512, 1024, 2048} sample sizes. We consider for sim-plicity only two common factors, that is r = 2. Furthermore, three values forθ are considered; θ ∈ {0, 0.5, 1}. For the case with θ = 1, ηt is simulated asNr(0, diag{θ2

1, . . . , θ2r}). Two values for Γe; a Toeplitz matrix Γe = Toep with

γ = 0, 7 for correlated noise and Γe = Diag for uncorrelated, where the entries ofthe diagonal matrix are generated by an uniform distribution U(0.5, 1.5). Note thatsimulated data are standardized before extracting the principal components. Com-mon factors are estimated by principal components in cases with θ < 1, and by theprocedure of Peña and Poncela (2006) in the case with θ = 1. All simulations arebased on 1000 replications of the model.

We rotate the obtained factors in order to compare proposed estimations withthe actual factors. The optimal rotation A∗ is obtained by maximizingtr[corr(F, FA)]. The solution is given by A∗ = V U where V and U are orthog-onal matrices of the decomposition corr(F, F ) = USV ′. When the number ofk principal components is not equal to the number of factors r, we rotate the firstl = min{k, r} principal components, see Eickmeier et al. (2015). Both estimatedand simulated factors are re-scaled to keep the same standard deviation, then

F∗k =σ(Fk)

σ(Fk)Fk, k = 1, . . . , r, (14)

where Fk is the k−th column of the matrix of the rotated principal componentsFA∗.

As explained before, these rotated factors are now treated as observed vari-ables in the regression model

vec(Y) = Θ(F∗, ψ, φ)β + vec(e),

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to estimate the wavelet coefficients, β, where F∗ = (F∗1, . . . , F∗r).

To investigate the performance of the estimation procedure, estimated andsimulated factors are compared as following:

i) The precision of the estimation factors is measured by the R2F ,F

statistics asin Bates et al. (2013), given by

R2F ,F

=tr[F′F(F′F)−1F′F]

tr[F′F], (15)

where F is the T × r matrix of estimated factors as in (14) and F is theT × r matrix of the actual factors, that is the simulated ones. This statisticsis a multivariate R2 in a regression of the actual factors on the principalcomponents. When the canonical correlation of the actual and estimatedfactors tends to one, then R2

F ,F→ 1 as well.

ii) The precision of the estimation loadings is measured by the mean square er-rors (MSE) between estimated and actual loadings, as in Motta et al. (2011).The MSE is computed as follows

MSE(v) = (NT )−1T∑t=1

‖Λ(v)(t)−Λ(t)‖,

for v = 1, . . . , 1000. The estimator of the factor loadings matrix, Λ(t), ischosen by a path such that

Λ(t) = {Λ(m)

(t) : MSEm = median{MSE(1), . . . ,MSE(1000)}}. (16)

Table 1 shows the results of estimations of (15) and (16). As can be seen, themethodology proposed in this paper performs very well in relatively small samplesregardless of size distortion between N and T . As seen in Table 1, R2

F ,Fis rela-

tively high indicating a good performance of the estimator, although the precision isa bit reduced when increasing the value of θ. These findings are maintained for theboth type of wavelets used and even when we allow for cross-correlation betweenidiosyncratic errors. Furthermore, inspecting the MSE in Table 1, we find that theMSEs decrease as T increases in all cases, even if the common factors are seri-ally correlated and even if idiosyncratic errors are cross-correlated. Furthermore,another finding indicates that in general, factor loadings using the wavelet D8 per-form better than the wavelet Haar. We think that such findings are reasonable dueto the smoothness of the wavelet D8 in contrast with the other one. The waveletHaar should be implemented when dynamics of the factor loadings have breaks orperhaps some aggressive jumps. These structures are not considered in the currentpaper, but these possible features of the wavelet Haar is part of an another researchand is out of the present scope.

12

Finally, in Figures 1 and 2, we display the good performance of the method-ology to estimate the factor loadings. We choose a couple of different smoothfunctions for each value of θ and in both types of wavelets for comparison pur-poses. The following functions are considered: i) λ1,12(t) = 0.4 cos−3πt, and ii)λ2,8(t) = 0.6(0.7

√t− 0.5 sin 1.2πt), where λ1,12 indicates the loading of the first

factor of the cross-sectional unit i = 12, and λ2,8, the loadings of the second factorof the unit i = 8. Figures display the actual and estimated time-varying loadings aswell as their bootstrap confidence interval at 95% with B = 100 replications, fol-lowing de A. Moura et al. (2012). In such figures, we can see that the methodologyworks well independently of the value taken in θ.

5 Application

In this section, we provide an application of the model proposed to study comove-ments in loads and prices of the Nord Pool power market.

Nord Pool runs the leading power market in Europe and operates in the day-ahead and intraday markets. Elspot is the day-ahead auction market, where par-ticipants act in a double auction and submit their supply and demand orders (spotprices and loads) for each individual hour of the next day. The market is in equilib-rium when demand and supply curves intersect to each other at the system pricesand loads for each hour. The hourly system prices and loads series are announcedas 24 dimensional vectors which are determined simultaneously. See Bredesen andNilsen (2013) for more information about the operation and history of the market.

Electricity markets have particular features that do not exist in another typeof commodity market. Particularly, the non-storability of electricity provokes thatthe price behavior shows an excessive volatility, possible negative prices and manyspikes along time. These characteristics are intrinsically more linked to prices thanto loads which make them very different, however both series include strong intra-day, weekly, and yearly seasonality, see Weron (2007).

Hourly electricity prices and loads have been mostly studied by univariatetime series methods, see Weron (2014) for a rich review. Until the last years, someauthors have explored these series by multivariate techniques as high dimensionalfactor models in different power markets. Using data from the Iberian Electric-ity Market (MIBEL), seasonal factor have been extracted in the works of Alonsoet al. (2011) and Garcıa-Martos et al. (2012), whereas Alonso et al. (2016) pro-pose to employ model averaging of factor models to improve the performance offorecasting. The Pennsylvania - New Jersey - Maryland (PJM) interconnectionmarket is studied by Maciejowska and Weron (2015), who estimate factor modelsfor forecasting evaluation using hourly and zonal prices. Furthermore, the NordPool power market is studied in the works of Ergemen et al. (2016), who study thelong-term relationship between system prices and loads, and Rodríguez-Caballeroand Ergemen (2017) who use regional prices in a multi-level setting. However, allthese empirical studies consider that factor loadings do not vary along time. In this

13

sense, to the best of our knowledge, we are the first to consider a factor model withtime-varying loadings for the power market.

We consider a balanced panel data set consisting of N = 24 hourly pricesand loads for each day from 13th March 2016 to 31th December 2018, yieldinga total of T = 1024 daily observations in each hour. The series are downloadedfrom the Nord Pool ftp server and prices are denominated in Euros per Mwh ofload. Figures 3 and 4 display six time series in logs from which we can observesome characteristics in both time series. First, electricity system prices and loadsvary differently over the months with a common pattern in the evolution of hourlyseries. Second, the price series show many spikes which are related to the ownfeatures of the commodity as explained above. Third, the seasonal variation isstronger in loads than in prices series even if this component is present in bothas discussed in Ergemen et al. (2016). Fourth, electricity prices and loads havenonstationary performances, see e.g. Haldrup et al. (2010), Haldrup and Nielsen(2006), and Koopman et al. (2007). These papers focus on the feature of electricityprices and loads have autocorrelation functions that decay at a hyperbolic rate,suggesting the use of fractionally integrated processes. In this paper, we keep inthe I(1) case to focus on the main ideas. Then, we set our estimation on the non-stationary approach discussed in the section 2.3. The possibility of long-memorydynamics is beyond the scope of the present paper and is not further explored.

We estimate the model in (12) with r = 2 for modeling the commonality ofhourly system prices and loads. The literature has consistently used two commonfactors in both prices and loads. Since we work with heteroscedastic time series,we use (in first differences) the procedure proposed by Alessi et al. (2010) whichintroduce a tuning multiplicative constant in the penalty function to improve thecriteria of Bai and Ng (2002). We also find two common factors in prices andloads which are in line with the literature. Using these factors, we explain 95% ofthe variation in the panel of electricity prices and 97% of electricity loads. Thesepercentages are slightly higher than that found in Ergemen et al. (2016) but ourperiod of time is much shorter than the one covered in their study. In this respect,what may be provoking such differences is that we do not cover the years in whichthe Nord Pool power market experienced a number of events in the delineation andmarket infrastructure.

Besides the estimated model, our study differs of Ergemen et al. (2016) andRodríguez-Caballero and Ergemen (2017) in two aspects: first, we do not sea-sonally adjust and detrend each panel element of prices and loads as they do intheir study. This is because we find that the seasonality is strongly extracted notonly in common factors but also in factor loadings which evolve along time inour approach. Second, we do not follow their estimation procedure based on afractionally differencing and integrating back strategy. Instead, we follow the pro-cedure proposed by Peña and Poncela (2006) using the common eigenstructure ofthe generalized covariance matrices as discussed before. Using the methodlogyproposed in that paper, we set that the first common factor in loads and pricesare nonstationary, I(1), whereas that the second common factors in both series

14

are stationary, I(0). These findings are in line with the non-stationary literatureon electricity modeling, see e.g. Ergemen et al. (2016), Rodríguez-Caballero andErgemen (2017), for instance.

Figures 5 and 8 display estimates of common factors of hourly system loadsand prices, respectively. In addition, Figures 6 and 7 show time-varying loadingsof the first and second common factors of hourly system loads, respectively, whileFigures 9 and 10 show the time-varying loadings estimates of hourly system prices.We only display the results for six hours for exposition purposes: 04:00, 08:00,12:00, 16:00, 20:00, and 24:00 hrs. Note that with these hours, we can observe dif-ferent performances of loadings between working and non-working hours, whichexemplify the intrinsic nature of the market. We also estimate the standard factormodel by PCA for comparison purposes. Static factor loadings are representedin Figures 6, 7, 9 and 10 by horizontal red lines. Common factors by static andtime-varying approaches overlap, consequently static factors are not displayed.

As seen in Figure 5, the first factor captures the strong seasonal componentshowing possible weekly, and monthly periodicity in the hourly system loads. Em-pirical studies have also documented this regular behavior in such frequencies bothin the Nord pool power market and in other energy markets. The second factorseems to capture mainly a kind of weekly variability, which occurs mostly duringworking hours as seen in Figure 3.

A further inspection on Figure 6 indicates that factor loadings correspond-ing to the first common factor of system loads are all positive showing smoothvariations along time. Loadings have positive peaks more frequent during the sum-mer. In this regard, time-varying loadings capture a kind of readjustment of systemloads when the demand reaches a maximum level (in winter) and a minimum level(in summer). This makes sense with the nature of this commodity. Moreover,time-varying loadings in the remaining months remain stable, oscillating smoothlyaround the static factor loading. In turn, factor loadings corresponding to the sec-ond common factor of system loads do not have a regular behavior in contrast tothose of the first factor, see Figure 7. However, these loadings are positive duringnight hours and negative during working hours as also discussed in Ergemen et al.(2016). This indicates that the second common factor plays a positive role duringnon-working hours and have negative contributions along the working hours. Inthis sense, such changes in the demand of electricity around the day may provokethe strong variability of the second common factor as discussed before.

With respect to hourly system prices, at a first glance, Figure 8 shows thatestimated common factors exhibit some stylized facts of electricity prices. Volatil-ity clustering is being captured by the first factor, while, excessive price spikesoccurring in 2018 by March, July, and August are extracted by both common fac-tors, but mostly by the second one. Similarly to the case of system loads, the factorloadings of the first common factor of system prices are all positive (Figure 9),whereas those of the second factor are positive during the night and negative dur-ing working hours (Figure 10). However, unlike system loads, we do not find thatfactor loadings in the case of prices have regular periodic movements, although in

15

general, loadings in some consecutive hours are very similar (5-7, 15-18 hrs, forinstance). The latter indicates that the impact of common factors to explain pricevariability will be very similar as the hours approach.

In conclusion, this empirical study leaves open many possibilities for futureresearch. First, testing whether loadings vary or if they are constant over timewould improve the specification of factor models and could help to understandmarket behavior. Second, in the context of the liberalization of energy markets,short-term forecasting of prices and loads are essential, then, in a subsequent pa-per, we are investigating if loads/prices forecasts obtained by time-varying factormodels are better than those provided by standard or dynamic factor models. Third,as discussed in the Monte Carlo study, the factor loadings using the wavelet D8 per-form better than Haar in situation where time series evolve smoothly as in the caseof system loads. In this sense, a deeper study on electricity prices could help usunderstand if the wavelet Haar represents a better choice for modeling the perfor-mance of factor loadings, since electricity prices are more volatile than loads.

6 Concluding remarks

In the last years, factor models has been widely used in many branches due tothe flexibility of treating high dimensional panels. However, most of the standardapproaches have some limitations when assuming that factor loadings are invariantalong time. In this respect, we relaxed such an assumption and focused on thestudy of factor models with time-varying loadings.

In this paper, we proposed a two-step procedure based on GLS with waveletfunctions to estimate factor models, whose loadings are defined as smooth andcontinuous functions of time. We consider stationary as well as non-stationaryvariables to cover more possibilities of application. The finite-sample propertiesof our estimator were supported by some Monte Carlo simulations. Our findingsindicate that the methodology proposed performed very well regardless of sizedistortion between N and T and even in relatively small samples. We also foundthat Wavelet D8 estimates were more attractive than wavelet Haar estimates due toits smoothness. This indicates that Wavelet D8 should provide better estimationswhen factor loadings do not have sudden changes.

Finally, we motivated the empirical relevance of allowing for time-varyingloadings in factor models by studying the electricity loads and prices of the NordPool power market. At a first glance, we found that factor loadings seem to varyover time, which has not been discussed in the empirical literature on electricitymarkets. Furthermore, we also found that the strong seasonality, which have beendocumented in many other studies, is extracted not only by common factors, butalso by factor loadings, which may improve the forecasting of electricity prices andloads.

16

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19

Appendix

A Technical appendix

A.1 Proof of Theorem 1

Proof. Let Y = (Y1, . . . , YT )′ be a T × N matrix, and let VNT be a r × r diago-nal matrix composed by the r largest eigenvalues of the matrix (NT )−1YY′. Bydefinition of eigenvectors and eigenvalues, we have

1

NTYY′F = FVNT ⇐⇒ 1

NTYY′FV −1

NT = F, (17)

where F′F = Ir. The model defined in (1) and (2) can be re-written as

Yt = [Λ0 + Λ(t)]Ft + et= Λ0Ft + Λ(t)Ft + et= Λ0Ft + wt + et,

where wt = Λ(t)Ft.Now, we define the following T×N matrices, e = (e1, e2, . . . , eT )′

and w = (w1,w2, . . . ,wT )′. Note that the model in (1), can also be re-written inmatrix form as

Y = FΛ′0 + w + e.

Consequently, after taking products we get

YY′ = FΛ′0Λ0F′ + FΛ

′0(e + w)′ + (e + w)Λ0F′ + (e + w)(e + w)′.

Then, from definitions of Ft in (17) and the matrix rotation H , we can writefor a fixed t

Ft −H ′Ft = (NT )−1V −1NT F′YY

′t − V −1

NT (F′F/T )(Λ′0Λ0/N)Ft

=V −1NTNT [F′(FΛ

′0Λ0Ft + FΛ

′0(wt + et) + (w + e)Λ0Ft + (e+ w)(et + wt))

−(F′F)(Λ′0Λ0)Ft]

=V −1NTNT [F′FΛ

′0et︸ ︷︷ ︸

D1t

+ F′eΛ0Ft︸ ︷︷ ︸D2t

+ F′eet︸ ︷︷ ︸D3t

+ F′FΛ′0wt︸ ︷︷ ︸

D4t

+ F′wΛ0Ft︸ ︷︷ ︸D5t

+ F′wwt︸ ︷︷ ︸D6t

+

F′ewt︸ ︷︷ ︸D7t

+ F′wet︸ ︷︷ ︸D8t

]

= V −1NT

∑8i=1Dit.

Then, after applying squared norms in both sides, adding on t, and dividingby T in V −1

NT

∑8i=1Dit, we get by the Loèv inequality

1

T

T∑t=1

‖Ft −H ′Ft‖2 ≤ ‖V −1NT ‖

28

8∑i=1

(1

T

T∑t=1

‖Dit‖2)

(18)

20

whereD1t = F′FΛ

′0et/NT D2t = F′eΛ0Ft/NT

D3t = F′eet/NT D4t = F′FΛ′0wt/NT

D5t = F′wΛ0Ft/NT D6t = F′wwt/NTD7t = F′ewt/NT D8t = F′wet/NT

Then, from Mikkelsen et al. (2018) (Lemma A.1), VNT converges to a def-inite positive matrix, therefore ‖V −1

NT ‖ = Op(1). Considering Assumptions A-Eand properties of principal components, for each term Dit, i = 1, . . . , 8, we have

T−1T∑t=1

‖D1t‖2 = Op(N−1),

T−1T∑t=1

‖D2t‖2 = Op(N−1T−1),

T−1T∑t=1

‖D3t‖2 = Op(N−1T−1),

T−1T∑t=1

‖D4t‖2 = Op(N−2K1NT ),

T−1T∑t=1

‖D5t‖2 = Op(N−2T−2K2NT ),

T−1T∑t=1

‖D6t‖2 = Op(N−2T−2K3NT ),

T−1T∑t=1

‖D7t‖2 = Op(N−2K1NT ),

T−1T∑t=1

‖D8t‖2 = Op(N−2K1NT ).

Finally, the right-hand side of equation (18) is a sum of variables with orders{1N ,

1NT ,

K1NTN2 , K2NT

N2T 2 ,K3NTN2T 2

}, respectively, and the proof is now completed.

B Tables

21

Tabl

e1:T∈{5

12,1

024,2

048}

,N∈{2

0,30}

,andr

=2.

The

mea

sure

ofco

nsis

tenc

yof

the

estim

ated

unob

serv

able

fact

ors

asw

ella

sth

ees

timat

edfa

ctor

load

ings

are

pres

ente

din

the

repo

rt.

Wav

elet

func

tions

Haa

rD

8

θ=

0θ

=0.

5θ

=1

θ=

0θ

=0.

5θ

=1

NT

Γe

R2 F,F

MSEm

R2 F,F

MSEm

R2 F,F

MSEm

R2 F,F

MSEm

R2 F,F

MSEm

R2 F,F

MSEm

2051

2Diag

0.93

410.

0103

0.85

420.

1970

0.74

500.

1562

0.94

470.

0011

0.86

550.

0023

0.77

140.

1191

1024

0.94

210.

0076

0.81

350.

0900

0.75

010.

1202

0.93

470.

0082

0.87

640.

0011

0.74

390.

0876

2048

0.91

270.

0020

0.82

360.

0053

0.73

120.

0931

0.94

700.

0005

0.84

550.

0009

0.75

460.

0045

3051

2Diag

0.92

360.

0098

0.84

560.

0128

0.76

110.

1091

0.95

600.

0071

0.85

210.

0127

0.81

310.

1093

1024

0.95

470.

0051

0.81

250.

0090

0.71

120.

0859

0.94

870.

0047

0.86

710.

0079

0.79

630.

0759

2048

0.93

410.

0011

0.82

680.

0027

0.73

240.

0738

0.94

980.

0020

0.86

010.

0060

0.82

130.

0028

2051

2Toep

0.87

310.

0672

0.69

320.

0891

0.67

110.

2201

0.89

110.

0128

0.68

710.

0593

0.72

420.

1223

1024

0.86

340.

0501

0.68

530.

0702

0.70

120.

1901

0.90

260.

0100

0.70

010.

0337

0.68

320.

0693

2048

0.89

110.

0137

0.77

230.

0433

0.73

210.

1642

0.89

710.

0091

0.73

070.

0108

0.68

410.

0031

3051

2Toep

0.85

310.

0472

0.79

840.

0621

0.68

120.

0912

0.90

210.

0117

0.78

150.

0311

0.74

830.

1114

1024

0.86

710.

0231

0.77

300.

0539

0.71

220.

0734

0.91

260.

0090

0.72

700.

0276

0.71

260.

0554

2048

0.84

510.

0111

0.82

000.

0311

0.72

120.

0819

0.87

320.

0068

0.73

050.

0109

0.75

310.

0037

Not

es:

The

DG

PisYit

=λ′ i(t)

F t+e i

t,w

herei∈{2

0,3

0}

andT∈{5

12,1

024,2

048},Fkt(1−θ kB

)=ηkt,w

ithk∈{1,2}.

Idio

sync

ratic

term

sar

ein

depe

nden

tlyge

nera

ted

as

ηt∼

N1(0,diag{1−θ2 1,1−θ2 2})

withθ∈{0,0.5,1}

defin

ing

the

degr

eeof

seri

alco

rrel

atio

nam

ong

fact

ors,

and

e t∼

NN

(0,Γ

e)

with

Γe

defin

edas

diag

onal

and

Toep

litz

mat

rice

s.

Com

mon

fact

ors

are

estim

ated

bypr

inci

palc

ompo

nent

sin

case

sw

ithθ<

1,w

hile

byth

epr

oced

ure

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C Figures

Figure 1: Comparison between the actual factor loadings in solid red line, theestimated factor loadings in solid black line, and Bootstrap confidence interval at95% in dashed black line. By column, from left to right: λ12, and λ28. By row,from top to bottom: θ = 0, θ = 0.5, and θ = 1. Γe = Toep, N = 20, T = 1024and Wavelet Haar.

23

Figure 2: Comparison between the actual factor loadings in solid red line, theestimated factor loadings in solid black line, and Bootstrap confidence interval at95% in dashed black line. By column, from left to right: λ12, and λ28. By row,from top to bottom: θ = 0, θ = 0.5, and θ = 1. Γe = Toep, N = 20, T = 1024and Wavelet D8.

24

04:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

10.2

10.6

08:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

10.2

10.6

11.0

12:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

10.4

10.8

16:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

10.4

10.8

20:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

10.4

10.8

24:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

10.3

10.6

10.9

Figure 3: Hourly system loads in logs for six different hours showing working andnon-working hours performances, 12 March 2016 to 31 December 2018.

04:00

2016.5 2017.5 2018.5

1.0

2.5

4.0

08:00

2016.5 2017.5 2018.5

2.5

3.5

4.5

12:00

2016.5 2017.5 2018.5

3.0

4.0

16:00

2016.5 2017.5 2018.5

2.5

3.5

20:00

2016.5 2017.5 2018.5

3.0

4.0

24:00

2016.5 2017.5 2018.5

1.5

2.5

3.5

Figure 4: Hourly system prices in logsfor six different hours showing working andnon-working hours performances, 12 March 2016 to 31 December 2018.

25

First factor

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−10

05

Second factor

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−1

12

Figure 5: Common factors of hourly system loads.

04:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.19

0.21

0.23

08:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.15

0.25

12:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.19

0.21

16:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.12

0.18

0.24

20:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.18

0.24

24:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.18

0.22

Figure 6: Time-varying loadings of the first factor of hourly system loads. Thehorizontal red line represents the level of loadings by the standard method.

26

04:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.24

0.30

08:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−0.

5−

0.3

12:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−0.

22−

0.14

16:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−0.

25−

0.10

20:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−0.

150.

000.

15

24:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.15

0.25

Figure 7: Time-varying loadings of the second factor of hourly system loads. Thehorizontal red line represents the level of loadings by the standard method.

First factor

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−15

−5

5

Second factor

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−12

−6

0

Figure 8: Common factors of hourly system prices.

27

04:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.20

0.30

08:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.15

0.25

12:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.18

0.22

16:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.10

0.20

20:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.15

0.25

24:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.10

0.20

Figure 9: Time-varying loadings of the first factor of hourly system prices. Thehorizontal red line represents the level of loadings by the standard method.

04:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

0.2

0.4

0.6

08:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−0.

4−

0.1

12:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−0.

25−

0.10

16:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−0.

4−

0.1

20:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−0.

30.

00.

2

24:00

2016.5 2017.0 2017.5 2018.0 2018.5 2019.0

−0.

20.

2

Figure 10: Time-varying loadings of the second factor of hourly system prices.The horizontal red line represents the level of loadings by the standard method.

28

Research Papers 2019

2019-08: Søren Kjærgaard, Yunus Emre Ergemen, Marie-Pier Bergeron Boucher, Jim Oeppen and Malene Kallestrup-Lamb: Longevity forecasting by socio-economic groups using compositional data analysis

2019-09: Debopam Bhattacharya, Pascaline Dupas and Shin Kanaya: Demand and Welfare Analysis in Discrete Choice Models with Social Interactions

2019-10: Martin Møller Andreasen, Kasper Jørgensen and Andrew Meldrum: Bond Risk Premiums at the Zero Lower Bound

2019-11: Martin Møller Andrasen: Explaining Bond Return Predictability in an Estimated New Keynesian Model

2019-12: Vanessa Berenguer-Rico, Søren Johansen and Bent Nielsen: Uniform Consistency of Marked and Weighted Empirical Distributions of Residuals

2019-13: Daniel Borup and Erik Christian Montes Schütte: In search of a job: Forecasting employment growth using Google Trends

2019-14 Kim Christensen, Charlotte Christiansen and Anders M. Posselt: The Economic Value of VIX ETPs

2019-15 Vanessa Berenguer-Rico, Søren Johansen and Bent Nielsen: Models where the Least Trimmed Squares and Least Median of Squares estimators are maximum likelihood

2019-16 Kristoffer Pons Bertelsen: Comparing Tests for Identification of Bubbles

2019-17 Dakyung Seong, Jin Seo Cho and Timo Teräsvirta: Comprehensive Testing of Linearity against the Smooth Transition Autoregressive Model

2019-18 Changli He, Jian Kang, Timo Teräsvirta and Shuhua Zhang: Long monthly temperature series and the Vector Seasonal Shifting Mean and Covariance Autoregressive model

2019-19 Changli He, Jian Kang, Timo Teräsvirta and Shuhua Zhang: Comparing long monthly Chinese and selected European temperature series using the Vector Seasonal Shifting Mean and Covariance Autoregressive model

2019-20 Malene Kallestrup-Lamb, Søren Kjærgaard and Carsten P. T. Rosenskjold: Insight into Stagnating Life Expectancy: Analysing Cause of Death Patterns across Socio-economic Groups

2019-21 Mikkel Bennedsen, Eric Hillebrand and Siem Jan Koopman: Modeling, Forecasting, and Nowcasting U.S. CO2 Emissions Using Many Macroeconomic Predictors

2019-22 Anne G. Balter, Malene Kallestrup-Lamb and Jesper Rangvid: The move towards riskier pensions: The importance of mortality

2019-23 Duván Humberto Cataño, Carlos Vladimir Rodríguez-Caballero and Daniel Peña: Wavelet Estimation for Dynamic Factor Models with Time-Varying Loadings