# Can we use seasonally adjusted variables in dynamic factor models?

Maximo Camacho, Yuliya Lovcha and Gabriel Perez Quiros

# Abstract

We examine the short-term performance of two alternative approaches of forecasting from dynamic factor models. The first approach extracts the seasonal component of the individual variables before estimating the model, while the alternative uses the non seasonally adjusted data in a model that endogenously accounts for seasonal adjustment. Our Monte Carlo analysis reveals that the performance of the former is always comparable to or even better than that of the latter in all the simulated scenarios. Our results have important implications for the factor models literature because they show the that the common practice of using seasonally adjusted data in this type of models is very accurate in terms of forecasting ability. Using five coincident indicators, we illustrate this result for US data.

JEL Classification: E32; C22; E27

Corresponding author: Maximo Camacho, Universidad de Murcia, Facultad de Economia y Empresa, Departamento de Metodos Cuantitativos para la Economia y la Empresa, 30100, Murcia, Spain, e-mail:

### Appendix

Since the empirical data are quarterly, the seasonal component of each time series, si, is the sum of two cyclical components, sit=si1t+si2t, which are evaluated at the seasonal frequencies, λ1=π/2, and λ2=π. According to (4), the dynamics of the first cyclical component is

(A1) ( s i 1 t s i 1 t * ) = ( cos π / 2 sin π / 2 sin π / 2 cos π / 2 ) ( s i 2 t 1 s i 2 t 1 * ) + ( ξ i 1 t ξ i 1 t * ) .  (A1)

Using cosπ/2=0 and sinπ/2=1 and rearranging terms, one can obtains

(A2) ( s i 1 t s i 1 t * ) = 1 1 + L 2 ( 1 L L 1 ) ( ξ i 1 t ξ i 1 t * ) ,  (A2)

which implies that (1+L2)si1t=ζi1t, where ζi1t=ξi1t+ξi1t1*. Ifvar(ξi1t)=var(ξi1t*)=σξi12,t, then var(ζi1t)=2σξi12.

Similarly, the dynamics of the second cyclical component can be obtained from

(A3) ( s i 2 t s i 2 t * ) = ( cos π sin π sin π cos π ) ( s i 2 t 1 s i 2 t 1 * ) + ( ξ i 2 t ξ i 2 t * ) ,  (A3)

which, using cosπ=–1 and sinπ=0, leads to

(A4) ( s i 2 t s i 2 t * ) = 1 ( 1 + L ) 2 ( 1 + L 0 0 1 + L ) ( ξ i 2 t ξ i 2 t * ) .  (A4)

This expression implies that (1+L)si2t=ζi2t, where ζi2t=ξi2t and var(ζi2t)=σξi22. Let us additionally assume that σξi12=σξi22=σξi2.

Accordingly, the seasonal component of each time series can be expressed as

(A5) s i t = ζ i 1 t ( 1 + L ) + ζ i 2 t ( 1 + L 2 ) ( 1 + L 2 ) ( 1 + L ) ,  (A5)

or

(A6) ( 1 + L + L 2 + L 3 ) s i t = ζ i 1 t ( 1 + L ) + ζ i 2 t ( 1 + L 2 ) .  (A6)

Since the greatest polynomial of the two terms from the right-hand side is of power two, the resulting polynomial (the result of summation) is of power two as well

(A7) ( 1 + L + L 2 + L 3 ) s i t = ( 1 + α L + β L 2 ) ζ i t .  (A7)

To find the unknown coefficients α and β, we derive the spectra of right-hand sides of both expressions. On the one hand, the spectrum of right-hand side of (A6) is

(A8) ( 1 + e i ω ) ( 1 + e i ω ) σ ζ i 1 2 + ( 1 + e 2 i ω ) ( 1 + e 2 i ω ) σ ζ i 2 2 = ( 2 + 2 cos ω ) σ ζ i 1 2 + ( 2 + 2 cos 2 ω ) σ ζ i 2 2 = ( 6 + 4 cos ω + 2 cos 2 ω ) σ ξ i 2 .  (A8)

The first equality follows from the fact that e+e=2 cos λ, ∀λ. The last equation uses σζi12=2σξi2 and σζi22=σξi2. On the other hand, the spectrum of right-hand side of (A7) is

(A9) ( 1 + α e i ω + β e 2 i ω ) ( 1 + α e i ω + β e 2 i ω ) σ ζ i 2 = ( ( 1 + α 2 + β 2 ) + 2 ( α + α β ) cos ω + 2 β cos 2 ω ) σ ζ i 2 .  (A9)

Since the two spectra must represent the same dynamics, one can use the system of three equations with three unknowns 6σξi2=(1+α2+β2)σζi2,4σξi2=2(α+αβ)σζi2 and 2σξi2=2βσζi2 to obtain α4–4α3+12α2–16α+4=0. The real solutions of this equation are α1=0.3187 and α2=1.6813, and using again the system of equations, it is easy to obtain that they correspond to values β1=0.1869 and β2=5.2745. The first pair of solutions (α=0.3187, β=0.1869) produces invertible MA polynomial in (A7), opposite, the second pair of solutions results in non-invertible (A7). Using the first pair of real solutions we find σζi2=5.3505σξi2 from the last equation of the system. In this way, the seasonal component for the series i is given by:

( 1 + L + L 2 + L 3 ) s i t = ( 1 + 0.3187 L + 0.1869 L 2 ) ζ i t .

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## Article note

Maximo Camacho thanks CICYT for its support through grants ECO2010-19830 and ECO2013-45698. The views in this paper are those of the authors and do not represent the views of the Bank of Spain or the Eurosystem.

## Supplemental Material

The online version of this article (DOI:https://doi.org/10.1515/snde-2013-0096) offers supplementary material, available to authorized users.

Published Online: 2014-09-23
Published in Print: 2015-06-01