Series Decomposition

Default series decomposition

The usual seasonal adjustment methods consist in a pre-processing step, which models deterministic effects by means of a linear regression and in a decomposition step, which splits the so-called linearized series in its trend, seasonal and irregular component.
We describe below the decomposition considered in JD+. The different equations explain the relationships between the components and the meaning of the codes used in JD+ to identify them.

We consider separately the additive and the multiplicative cases

Regression model

The regression model contains calendar variables (trading days, including leap year, Easter and other moving holidays effects), outliers and other regression variables.

Additive decomposition

y_c=α∙cal+β∙out+γ∙reg+μ

Multiplicative decomposition

ln(y_c)-ln(lp)=α∙cal+β∙out+γ∙reg+μ

The different regression effects are split as follows

cal=[tde ee omhe]
out=[out_t out_s out_i]
reg=[reg_t reg_s reg_i reg_sa reg_y reg_u]

Code	Description	y_lin	t	s	i	sa
tde	Trading days (default, holidays, user-defined)	x		x
ee	Easter	x		x
omhe	Other moving holidays	x		x
AO	Additive outlier	x			x	x
TC	Transitory change	x			x	x
LS	Level shift	x	x			x
SO, SLS	Seasonal outlier / seasonal level shift	x		x
Reg_i	Regression variables allocated to irregular, IV (p)	x		x		x
Reg_t	Regression variables allocated to trend, ramps, IV (p)	x	x			x
Reg_s	Regression variables allocated to seasonal, IV (p)	x		x
Reg_sa	Regression variables allocated to seas. adjusted	x				x
Reg_y	Regression variables removed before decomp.
Reg_u	Regression variables unallocated to a component	x	p	p	p	p

p stands for partially, IV(p) for intervention variables (partially)

Decomposition of the stochastic component

In the following equations, the coefficients have been omitted

Additive decomposition

reg_u+μ=y_cmp=t_cmp+s_cmp+i_cmp
t_cmp=t_lin, s_cmp=s_lin, i_cmp=i_lin

Multiplicative decomposition

ln(reg_u+μ)=y_lin=t_lin+s_lin+i_lin
reg_u∙μ=y_cmp=t_cmp∙s_cmp∙i_cmp

Final decomposition

Additive decomposition

t=out_t+reg_t+t_cmp
s=cal+out_s+reg_s+s_cmp
cal=tde+lp+ee+omhe
i=out_i+reg_i+i_cmp
sa=t+i+reg_sa=y_c-reg_y-s
y_c=t+s+i+reg_y

Multiplicative decomposition

t=out_t∙reg_t∙t_cmp
cal=tde∙lp∙ee∙omhe
s=cal∙out_s∙reg_st∙s_cmp
i=out_i∙reg_i∙i_cmp
sa=t∙i∙reg_sa=y_c/(reg_y∙s)
y_c/reg_y=t∙s∙i

Besides the codes defined above, JD+ uses the suffixes _f to indicate forecasts and when it makes sense, _e, (_ef) for standard deviations (on forecasts)

Exhaustive list of the codes

code	definition
y	original series
y_c	original series completed for missing values
y_f	forecasts of the original series
y_ef	forecasts errors for the original series
t	final trend
t_f	forecasts of t
s	final seasonal
s_f	forecasts of s
sa	final seasonally adjusted series
sa_f	forecasts of sa (usually identical to t_f)
i	final irregular
i_f	forecasts of i (usually 0 or 1)
cal	calendar effects
cal_f	forecasts of cal
ycal	original series corrected for calendar effects
ycal_f	forecasts of ycal
tde	trading days effects (including leap year)
tde_f	forecasts of tde
ee	Easter effect
ee_f	forecasts of ee
omhe	other moving holidays effects
omhe_f	forecasts of omhe
out	outliers effects
out_f	forecasts of out
out_t	outliers associated to t (LS)
out_t_f	forecasts of out_t
out_s	outliers associated to s (SO, SLS)
out_s_f	forecasts of out_s
out_i	outliers associated to i (TC, AO)
out_i_f	forecasts of out_i
reg	all other regression effects
reg_f	forecasts of reg
reg_u	regression effects not associated to a component
reg_u_f	forecasts of reg_u
reg_y	regression effects associated to y
reg_y_f	forecasts of reg_y
reg_sa	regression effects associated to sa
reg_sa_f	forecasts of reg_sa
reg_t	regression effects associated to t
reg_t_f	forecasts of reg_t
reg_s	regression effects associated to s
reg_s_f	forecasts of reg_s
reg_i	regression effects associated to i
reg_i_f	forecasts of reg_i
y_lin	linearized series
y_lin_f	forecasts of y_lin
t_lin	linearized trend
t_lin_f	forecasts of t_lin
t_lin_ef	forecast errors of t_lin
s_lin	linearized seasonal
s_lin_f	forecasts of s_lin
s_lin_ef	forecast errors of s_lin
i_lin	linearized irregular
i_lin_f	forecasts of i_lin
i_lin_ef	forecast errors of i_lin
sa_lin	linearized seasonally adjusted series
sa_lin_f	forecasts of sa_lin
sa_lin_ef	forecast errors of sa_lin
y_cmp	linearized component (level)
y_cmp_f	forecasts of y_cmp
t_cmp	linearized trend component
t_cmp_f	forecasts of t_cmp
t_cmp_ef	forecast errors of t_cmp
s_cmp	linearized seasonal component
s_cmp_f	forecasts of s_cmp
s_cmp_ef	forecast errors of s_cmp
i_cmp	linearized irregular component
i_cmp_f	forecasts of i_lin
i_cmp_ef	forecast errors of i_lin
sa_cmp	linearized seasonally adjusted component
sa_cmp_f	forecasts of sa_cmp
sa_cmp_ef	forecast errors of sa_cmp