# Illuminating ARIMA model-based seasonal adjustment with three fundamental seasonal models

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## Abstract

Our starting place is the first order seasonal autoregressive model. Its series are shown to have canonical model-based decompositions whose finite-sample estimates, filters, and error covariances have simple revealing formulas from basic linear regression. We obtain analogous formulas for seasonal random walks, extending some of the results of Maravall and Pierce (J Time Series Anal, 8:177–293, 1987). The seasonal decomposition filters of the biannual seasonal random walk have formulas that explicitly reveal which deterministic functions they annihilate and which they reproduce, directly illustrating very general results of Bell (J Off Stat, 28:441–461, 2012; Center for Statistical Research and Methodology, Research Report Series, Statistics #2015-03, U.S. Census Bureau, Washington, D.C. https://www.census.gov/srd/papers/pdf/RRS2015-03, 2015). Other formulas express phenomena heretofore lacking such concrete expression, such as the much discussed negative autocorrelation at the first seasonal lag quite often observed in differenced seasonally adjusted series. An innovation that is also applied to airline model seasonal decompositions is the effective use of signs of lag one and first-seasonal-lag autocorrelations (after differencing) to indicate, in a formal way, where smoothness is increased by seasonal adjustment and where its effect is opposite.

## Keywords

ARIMA models Signal extraction smoothness Timeseries## JEL Classification

C4 C8## 1 Overview

The graph of \(\hat{N}_{t}\) in Fig. 3 appears less smooth than \( Z_{t}\), and this will be established in a formal way in Sect. 13.2. The signal estimate \(\hat{S}_{t}\) is graphed by calendar month in Fig. 4. The \(\hat{S}_{t}\) visibly smooth each of the 12 annual calendar month series of \(Z_{t}\), a property connected to the fact that the lag 12*k*, \(k\ge 1\), autocorrelations of \(\hat{S}_{t}\) are larger than those of \(Z_{t}\), see Sect. 13.

For any stationary \(Z_{t}\) with known autocovariances \(\gamma _{j}\), typically from an ARMA model for \(Z_{t}\), the first step toward obtaining linear estimates of an uncorrelated component decomposition (4) is the determination or specification of an appropriate *autocovariance decomposition*, \(\gamma _{j}=\gamma _{j}^{S}+\gamma _{j}^{N},j=0,1,\ldots \) . The SAR(1) estimated decomposition is detailed in Sect. 3.2.

*MMSE*) linear estimates (or estimators), \( \hat{S}=\beta _{S}Z\) and \(\hat{N}=\beta _{N}Z\), of the unobserved components. Such estimates are also called minimum variance estimates. Another standard formula provides the variance matrix of the estimation errors. Everything is illustrated for the two-component SAR(1) decomposition.

*seasonal random walk*or SRW, obtained by setting \(\Phi =1\) in (1),

*Wiener-Kolmogorov (W-K) filter formulas*presented in Sect. 6. The original W-K formulas provide MMSE component estimates from bi-infinite stationary data \( Z_{t},-\infty <t<\infty \) and immediately reproduce the SAR(1) formulas of Sect. 5.1 for the intermediate times between the first and last years, \(q+1\le t\le n-q\).

In Sect. 7, after formally defining the pseudo-spectral density (pseudo-s.d.) of an ARIMA model, we illustrate the kinds of non-stationary W-K calculations that are done with pseudo-s.d.’s in TRAMO-SEATS (Gómez and Maravall 1996), hereafter T-S, and in its implementations in TSW (Caporello and Maravall 2004), X-13ARIMA-SEATS (U.S. Census Bureau 2015), hereafter X-13A-S, and JDemetra+ (Seasonal Adjustment Centre of Competence 2015), hereafter JD+. We derive the simple formulas of all of the filters associated with the three-component seasonal, trend, and irregular decomposition of the \(q=2\) SRW. We proceed a little more directly than the tutorial article Maravall and Pierce (1987), which develops fundamental properties of this model’s decomposition estimates with somewhat different goals. In Sect. 7.2, we obtain the forecast and backcast results which are required to derive the asymmetric filters for initial and final years of a finite sample as well as the error variances of their estimates. These illustrate in a simple way the fundamental role of Bell’s Assumption A. In Sect. 10, we provide an extended and corrected version of Maravall and Pierce’s Table 1 giving variances and autocorrelations of the stationary transforms of the estimates.

For the Box-Jenkins airline model, Sect. 12 provides graphs of the MMSE filters determined by small, medium and large values of the seasonal moving average parameter \(\Theta \). Graphs also display the quite different visual smoothing effects of filters from such \(\Theta \) on the monthly International Airline Passenger totals series for which the model is named.

This is a prelude to Sect. 13, which has the most experimental material. In a formal way based on autocorrelations of the component estimates of the different types of models considered (fully differenced in the nonstationary case), it shows where *smoothness* is enhanced, and where an opposite result occurs, among the seasonal decomposition components. Same-calendar-month subseries are the main setting and are considered separately from the monthly series. Complete results are presented, first for the two-component SAR(1) decomposition, next for the \( q=2\) SRW’s three-component decomposition. Thereafter, smoothness results are presented for airline model series over an illustrative set of coefficient pairs.

There are many formulas. In most cases, more useful for readers than their derivations or details will be to study how the formulas are used.

## 2 Some conventions and terminology

A generic primary time series \(X_{t}\), stationary or not, will be assumed to have \(q\ge 2\) observations per year, with the *j*-th observation for the *k* -th year having the time index \(t=j+\left( k-1\right) q,1\le j\le q\). For simplicity, the series of *j*-th values from all available years of is called the \(\,\,j\) *-th calendar month subseries* of \(X_{t}\) even when \(q\ne 12\). When \(q=12\), these are the series of January values, the series of February values, etc., 12 series in all. Some seasonal adjustment properties, especially those of seasonal component estimates, are best revealed by the calendar month subseries. When \(X_{t}\) is stationary (mean \(EX_{t}=0\) assumed), the lag *k* autocorrelation of a calendar month subseries is the lag *kq*, or *k*-th seasonal autocorrelation, of \(X_{t}\). Because some formulas simplify when *q* / 2 is an integer, we only consider even *q*. In our examples \(q=2\) or 12. (In practice, \(q=3,4,6\) also occur.) Some basic features of canonical ARIMA-model-based seasonal adjustment (*AMBSA* for short) will be related to smoothing of the calendar month subseries or detrended versions thereof, see Sect. 11. The definition of *canonical* is given in Sect. 3.2.

Features of SEATS referred to are also features of the implementations of SEATS in X-13A-S and JD+.

## 3 The general stationary setting

### 3.1 Autocovariance and spectral density decompositions

*spectral density*function (s.d.),

*j*the autocovariance \(\gamma _{j}\) can be recovered from \(g\left( \lambda \right) \) as

### 3.2 The SAR(1) canonical signal + white noise decomposition

Conceptually attractive and unique decompositions result from the following restriction, introduced by Tiao and Hillmer (1978). An s.d. decomposition with two or more component s.d.’s is called *canonical* if at most one of the components, usually a constant (white noise) s.d., has a non-zero minimum. A nonconstant s.d. (or pseudo-s.d. as defined in Sect. 7) is called canonical if its minimum value is zero. The two-component SAR(1) case provides the simplest seasonal example.

*seasonal frequencies*\(\lambda =k/12,1\le k\le 6\) cycles per year.

*I*the identity matrix of order

*n*,

*white noise free*. \(N_{t}\) has the largest variance possible for a white noise component. This white noise free plus white noise decomposition has filter formulas for the MMSE linear estimates of \(S_{t}\) and \(N_{t}\) that are especially simple and revealing, as will be seen in Sect. 5.1.

## 4 Regression formulas for two-component decompositions

*S*and \(\hat{N}\) of

*N*.

### 4.1 The estimated decomposition

*n*, we have \(\Sigma _{SZ}=ESZ^{\prime }=\Sigma _{SS}\). Similarly \(\Sigma _{NZ}=\Sigma _{NN}\). Thus the usual regression coefficient formulas \(\beta _{S}=\Sigma _{SZ}\Sigma _{ZZ}^{-1}\) (with \(\Sigma _{SZ}=ESZ^{\prime }\)) and \( \beta _{N}=\Sigma _{NZ}\Sigma _{ZZ}^{-1}\) simplify. We have

*Z*, see Wikipedia 2013.

*t*-th row of \(\beta _{S}\) provides the filter coefficients for the estimate \(\hat{S}_{t}\) and correspondingly with \( \beta _{N}\) for \(\hat{N}_{t}\), as will be illustrated in Sect. 5.

In summary, regression based on (20) provides an observable decomposition of *Z* in terms of MMSE linear estimates. Such a decomposition, with \(N_{t}\) specified as white noise, exists for any stationary \(Z_{t}\) whose s.d. has a positive minimum.

### 4.2 Variance and error variance matrix formulas

*the total variance*of \(S,\Sigma _{\hat{S}\hat{S}}\) the

*variance of*

*S*

*explained by*

*Z*, and \(\Sigma _{\epsilon \epsilon }\) is the

*residual variance*. Similarly for

*N*with the same residual variance, from (26), which shows that

Seasonal economic indicator series \(Z_{t}\) are generally modeled with nonstationary models, e.g., ARIMA models rather than ARMA models. Then AMBSA uses pseudo-spectral density decompositions, discussed in Sect. 7. For finite-sample estimates in the ARIMA case, McElroy (2008) provides matrix formulas for \(\hat{S},\hat{N}\) and \(\Sigma _{\epsilon \epsilon }\) which involve matrices implementing differencings and autocovariance matrices of the differenced *S* and *N*. We will be able to easily convert the SAR(1) formulas developed next to obtain the same finite-sample results as McElroy’s formulas for the two-component decomposition of the ARIMA SRW model (6).

## 5 SAR(1) Decomposition estimation formulas

*q*-th diagonals above and below it. The sub- and superdiagonals have the entries \(-\Phi \sigma _{a}^{-2}\). The first and last

*q*entries of the main diagonal are \(\sigma _{a}^{-2}\) and the rest are \( \sigma _{a}^{-2}\left( 1+\Phi ^{2}\right) \).

### 5.1 The general filter formulas

*q*and \(n\ge 2q+1\), the \(\Sigma _{ZZ}^{-1}\) formula of Wise (1955) yields the filter formulas for \(\hat{N}\) and \(\hat{S}=Z-\hat{N}\) shown in (33)–(37) and (38)–(40), generalizing from the special cases (31) and (32). For the intermediate times \(q+1\le t\le n-q\), the noise component estimate \(\hat{N}_{t}\ \)is given by the symmetric filter

^{1}are asymmetric. For \(1\le t\le q,\)

*q*, it follows that \(\hat{S}_{t}\) and \(\hat{S}_{t\pm kq}\) are positively correlated, more strongly than \(Z_{t}\) and \(Z_{t\pm kq}\) it will be shown.

#### 5.1.1 Filter re-expressions and filter terminology

*t*when \(n\ge 2q+2\) (Recall that \(q\ge 2\)). To reveal this better, let

*B*denote the

*backshift*or

*lag operator*defined as usual: for any time series \(X_{t}\) and integer \(j\ge 0, B^{j}X_{t}=X_{t-j} \) and \(B^{-j}X_{t}=X_{t+j}\) (a

*forward shift*if \( j\ne 0\)). Since \(B^{0}X_{t}=X_{t}\), one sets \(B^{0}=1\). A constant-coefficient sum \(\Sigma _{j}c_{j}B^{j}\) is a (linear, time-invariant) filter, a

*symmetric*filter if the same filter results when

*B*is replaced by \(B^{-1}\), as with the intermediate time filters (33) and (38). Their backshift operator formulas reveal factorizations like others that will be useful:

*t*such that \(q+1\le t\le n-q\) and to all

*t*when the conceptually important case of bi-infinite data \(Z_{\tau },-\infty <\tau <\infty \) is considered.

The one-sided filter that produces the MMSE estimate for final time \(t=n\) is called the *concurrent filter*. In our finite-sample context, the concurrent \(\hat{N}_{t}\) and \(\hat{S}_{t}\) filters, \(\left( 1+\Phi \right) ^{-2}\left\{ -\Phi B^{q}+1\right\} \) and \(\Phi \left( 1+\Phi \right) ^{-2}\left\{ B^{q}+\left( \Phi +2\right) \right\} \) respectively, could be applied to all \(Z_{t}\) after the first year, \(q+1\le t\le n\), but are only MMSE in the final year.

### 5.2 The error covariances of the SAR(1) estimates

^{2}of using \(\Phi \left( 1+\Phi \right) ^{-2}\left\{ \Phi Z_{t}\right\} \) to backcast/forecast \(\Phi \left( 1+\Phi \right) ^{-2}Z_{t\pm q}\) in (34) and (37), since from (2) we have

*n*.

The two-component SAR(1) decomposition derived above has exceptional pedagogical value because of the simplicity of its filter and error variance formulas derived above. With the aid of results of Bell (1984), a rederivation of the filter formulas via the Wiener-Kolmogorov formulas, presented next, makes possible a quick transition to the nonstationary SRW model (6) and its canonical MMSE two-component decomposition filter and error autocovariance formulas, all obtained by setting \(\Phi =1\) in the SAR(1) formulas above.

## 6 Wiener–Kolmogorov formulas applied to SAR(1) and SRW models

Initially for the two-component case (4) with bi-infinite data, we consider a fundamental and relatively simple approach to obtaining MMSE decomposition estimates. It also applies to the ARIMA case under a productive assumption of Bell (1984) discussed and applied in Sect. 7.

### 6.1 Filter transfer functions and the input–output spectral density formula

*transfer function*of the filter \(\beta \left( B\right) \) and \(\left| \beta \left( e^{i2\pi \lambda }\right) \right| ^{2}\) is its

*squared gain*. When a filter’s transfer function \(\beta \left( e^{i2\pi \lambda }\right) \) is known, then the filter coefficients can be obtained from it, in general by integration

### 6.2 The W–K formulas

### 6.3 SAR(1) intermediate-time filters again and an alternate model form

*Remark* A formula of the usual backward-time form, \(\hat{N} _{t}=\left( 1+\Phi \right) ^{-2}(c_{t}-\Phi c_{t-2})\), can be obtained for (56) with \(c_{t}=\left( 1-\Phi B^{2}\right) ^{-1}\left( 1-\Phi B^{-2}\right) a_{t}\). Complex conjugation preserves magnitude, so \( \left| 1-\Phi e^{i2\pi 2\lambda }\right| ^{-2}\left| 1-\Phi e^{-i2\pi 2\lambda }\right| ^{2}=1\) for all \(\lambda \). Thus, from (46), the spectral densities satisfy \(g_{c}\left( \lambda \right) =g_{a}\left( \lambda \right) =\sigma _{a}^{2}\) showing that \(c_{t}\) is white noise with variance \(\sigma _{a}^{2}\). The expanded formula \(c_{t}=\left( 1-\Phi B^{-2}\right) \sum _{j=0}^{\infty }\Phi ^{j}a_{t-2j}\), shows that \( c_{t}\) is a function of future and past \(a_{t}\), specifically of \( a_{t+2-2j},0\le j\,<\infty \). Analogues of these results hold for all forward-time moving average models derived below.

### 6.4 Going nonstationary with the seasonal random walk

In Sect. 8, we will detail the seasonal, trend, irregular decompositions of (58) and the SRW with \(q=2\).

## 7 ARIMA component filters from pseudo-spectral density decompositions

*pseudo-spectral density*(pseudo-s.d.) is defined by

In the nonstationary signal plus nonstationary noise case of interest, \( \delta \left( B\right) =\delta _{S}\left( B\right) \delta _{N}\left( B\right) \), and \(\delta _{S}\left( e^{i2\pi \lambda }\right) \) and \(\delta _{N}\left( e^{i2\pi \lambda }\right) \) have no common zero. In the seasonal plus nonseasonal case, \(\delta _{S}\left( e^{i2\pi \lambda }\right) \) has zeroes only at seasonal frequencies \(\lambda =k/q,k=\pm 1,\ldots ,q/2\), and \(\delta _{N}\left( e^{i2\pi \lambda }\right) =0\) only for \(\lambda =0\), as with \(\delta _{S}\left( B\right) =1+B+\cdots +B^{q-1}\) and \(\delta _{N}\left( B\right) =\left( 1-B\right) ^{2}\) for \(\delta \left( B\right) =\left( 1-B\right) \left( 1-B^{q}\right) \) of the airline model. The pseudo-s.d. \(g\left( \lambda \right) \) must be decomposed into a sum of seasonal and nonseasonal pseudo-s.d.’s associated with \(\delta _{S}\left( B\right) \) and \(\delta _{N}\left( B\right) \), respectively.

*inadmissible*case. Stationary components in addition to the irregular occur with s.d.’s for cyclical or other “transitory” components, see Gómez and Maravall (1996) and Kaiser and Maravall (2001). SEATS has options for several.

Generalizing the stationary case definition, a pseudo-s.d. decomposition is *canonical* if, with at most one exception, its component pseudo- s.d.’s and s.d.’s have minimum value zero, as in (59).

### 7.1 Bell’s assumptions

In addition to the requirements on \(\delta _{S}\left( B\right) \) and \(\delta _{N}\left( B\right) \), Bell (1984) also requires that the series \(\delta _{S}\left( B\right) S_{t}\) and \(\delta _{N}\left( B\right) N_{t}\) be uncorrelated. This can be obtained “automatically” from the implied s.d. decomposition \( g_{\delta \left( B\right) Z}\left( \lambda \right) =g_{\delta \left( B\right) S}\left( \lambda \right) +g_{\delta \left( B\right) N}\left( \lambda \right) \), see Findley (2012). From here, Bell’s Assumption A provides MMSE optimality for finite-sample component estimates from the matrix formulas of McElroy (2008), for bi-infinite data estimates from pseudo-spectral W-K formulas, and for semi-infinite data estimates like those considered in Bell and Martin (2004).

*Bell’s*

*Assumption A:*

*For*\(\delta (B)\)

*of degree*

*d*,

*the*

*d*

*initial values, say*\(Z_{1},\ldots ,Z_{d}\),

*are uncorrelated with the bi-infinite ARMA series*\(w_{t}=\delta \left( B\right) Z_{t},-\infty <t<\infty \),

*which generates the bi-infinite*\(Z_{t}\)

*via*

### 7.2 Assumption A yields MMSE forecasts and backcasts

Forecasts of ARIMA \(Z_{t}\) are traditionally obtained as in Box and Jenkins (1976, Ch. 5), namely by generating future \(Z_{t},t>d\) from initial \( Z_{1},\ldots ,Z_{d}\) and subsequent \(w_{t}=\delta \left( B\right) Z_{t}\,, t>d\,\), replacing unobserved \(w_{t}\) by their MMSE ARMA forecasts to obtain desired forecasts of \(Z_{t}\). The MMSE optimality of such forecasts follows somewhat straightforwardly from the forecasting results obtained under Assumption A in the third text paragraph on p. 652 of Bell (1984). For backcasts the reverse recursion in (63) is used.

For backcasts the result that, for all \(k\ge 1,Z_{1}\) and \(Z_{2}\) are the MMSE optimal backcasts of \(Z_{-1-2\left( k-1\right) }\) and \(Z_{-2\left( k-1\right) }\) respectively, follows from Assumption A and the analogues of (65) and (66) yielded by the backward form \( Z_{t-2}=Z_{t}-a_{t}\) of the recursion used above.

## 8 The 3-component \(q=2\) SRW decomposition

*s*instead of

*S*, for components of three-component decompositions.) It is easily verified that

### 8.1 The estimates’ symmetric filters and ARIMA models

### 8.2 The estimates’ asymmetric filter formulas

### 8.3 Symmetric and concurrent SWP filter coefficient features

*monthly*random walk model \(Z_{t}=Z_{t-12}+a_{t}\). All of these filters are very localized, ignoring data more than a year way from the time point of the estimate, which gets the largest weight, always positive, contrasting with the next largest coefficients, negative for the closest same-calendar-month data. They are therefore very adaptive in a way that is appropriate for series with such erratic trend and seasonal movements.

## 9 What seasonal decomposition filters annihilate or preserve

The occurrence of the trend and seasonal differencing factors of \(\delta \left( B\right) \) and \(\delta \left( B^{-1}\right) \), sometimes squared, in the filter formulas of the Sect. 8.1 reveal which fixed seasonal or trend components the filters annihilate and which their complementary filters preserve. Regarding symmetric filters, \(\beta _{s}\left( B\right) \) contains \(\delta _{s}\left( B\right) \delta _{s}\left( B^{-1}\right) =\left( 1-B\right) ^{2}\left( 1-B^{-1}\right) ^{2}=B^{-2}\left( 1-B\right) ^{4}\). Differencing lowers the degree of a polynomial by one, e.g., \(\left( 1-B\right) t^{3}= t^{3}-\left( t-1\right) ^{3}=3t^{2}-3t+1\). Hence this \(\beta _{s}\left( B\right) \) will annihilate a cubic component \(at^{3}\) and the seasonal adjustment filter \(\beta _{sa}\left( B\right) =1-\beta _{s}\left( B\right) \) will reproduce it. \( \beta _{sa}\left( B\right) \) has \(\left( 1+B\right) ^{2}\) as a factor. A \( q=2 \) stable seasonal component \(a\left( -1\right) ^{t}\) has the property that \(\left( 1+B\right) a\left( -1\right) ^{t}=a\left\{ \left( -1\right) ^{t}-\left( -1\right) ^{t-1}\right\} =0\). Hence \(\beta _{u}\left( B\right) \) and \(\beta _{sa}\left( B\right) \) annihilate such a component (and also an unrealistic linearly increasing \(at\left( -1\right) ^{t}\)) and \(\beta _{s}\left( B\right) \) preserves it. This is also true for their concurrent filters. The reader can further observe that \(\beta _{p}\left( B\right) a\left( -1\right) ^{t}=0\) and that the irregular filter \(\beta _{u}\left( B\right) =-\frac{1}{8}B^{-2}\left( 1+B\right) ^{2}\left( 1-B\right) ^{2}\) will eliminate both a linear trend and a stable seasonal component.

These are quite general results, applying to AMBSA filters, finite or infinite, symmetric or asymmetric, from any ARIMA model whose differencing operator has \(1-B^{q}=\left( 1-B\right) U\left( B\right) ,q\ge 2\), with \(U\left( B\right) =1+B+\cdots B^{q-1}\) as a factor. The tables of Bell (2012) cover more general differencing operators and also several generations of symmetric and asymmetric X-11 filters. In many cases, linear functions in *t*, and in exceptional cases, such as (71) and others covered in Bell (2015), even higher degree polynomials in *t* are eliminated by \(\beta _{s}\left( B\right) \) and preserved by \(\beta _{sa}\left( B\right) \).

## 10 Auto- and cross-correlations of stationary transforms of \(q=2\) SRW components

We continue the exploration of properties of estimated components, including how they differ from properties of the unobserved components, now for the ARIMA case, using the \(q=2\) SRW 3-component decomposition. We examine the most accessible properties, those of autocorrelations and cross-correlations of the minimally differenced components estimates, with numerical results in Table 1. These can be used as diagnostics. We begin with an important difference between the ARIMA case and the stationary case after setting the scene.

*stationary transformation*of its estimate, and the same term is used with unobserved unobserved components. The right hand sides of the Sect. 8.1 model formulas provide examples we will use.

The important difference from the stationary case: From (72) and (73), in contrast to \(E\hat{S}_{t}\hat{N}_{t}>0\) for stationary SAR(1) decompositions, we have \(E\big ( \big \{ \left( 1+B\right) \hat{s}_{t}\big \} \left\{ \left( 1-B\right) \widehat{sa}_{t}\right\} \big ) =0\). This result might suggest that estimates have the uncorrelatedness property, required of unobserved components from Sect. 7.1, the property that all stationary transforms with different differencing operators must be uncorrelated, requiring in particular \(E\left( \left\{ \left( 1+B\right) s_{t}\right\} \left\{ \left( 1-B\right) sa_{t-j}\right\} \right) =0\) for all *j*. However, \(E(\left\{ \left( 1+B\right) \hat{s}_{t}\right\} \{\left( 1-B\right) \widehat{sa}_{t-1}\}=13\sigma _{a}^{2}/16^{2}\) and \(E\left( \left\{ \left( 1+B\right) \hat{s}_{t}\right\} \left\{ \left( 1-B\right) \widehat{sa}_{t-2}\right\} \right) =-8\sigma _{a}^{2}/16^{2}\) show otherwise.

Autocorrelations of stationary transforms of the unobserved and estimated components

Transforms | \(\gamma _{0}/\sigma _{a}^{2}\) | \(\rho _{1}\) | \(\rho _{2}\) | \( \rho _{3}\) |
---|---|---|---|---|

\(\left( 1+B\right) s_{t}\) | \(1/8=0.125\) | \(-0.50\) | 0 | 0 |

\(\left( 1+B\right) \hat{s}_{t}\) | \(20/16^{2}\doteq 0.078\) | \(-0.75\) | 0.30 | \(-0.05\) |

\(\left( 1-B\right) p_{t}\) | \(1/8=0.125\) | 0.50 | 0 | 0 |

\(\left( 1-B\right) \hat{p}_{t}\) | \(20/16^{2}\doteq 0.078\) | 0.75 | 0.30 | 0.05 |

\(u_{t}\) | \(1/8=0.125\) | 0 | 0 | 0 |

\(\hat{u}_{t}\) | \(1/32\doteq 0.031\) | 0 | \(-0.50\) | 0 |

\(\left( 1-B\right) sa_{t}\) | \(3/8=0.375\) | \(-1/6\doteq -0.167\) | 0 | 0 |

\(\left( 1-B\right) \widehat{sa}_{t}\) | \(52/16^{2}\doteq 0.203\) | \(15/52\doteq 0.288\) | \(-10/52\doteq -0.192\,\) | \(1/52\doteq 0.019\) |

## 11 Reciprocal smoothing properties of seasonal and trend estimates

Consider a finite detrended series, \(Z-\hat{p}=\hat{s}+\hat{u}\) in vector notation. In the X-11 method, \(\hat{s}\) is estimated from the detrended series, see Ladiray and Quenneville (2001). With AMBSA estimates, Theorem 1 and Remark 4 of McElroy and Sutcliffe (2006) for ARIMA \(Z_{t}\) show that, in addition to being the MMSE linear estimate of *s* from \(Z,\hat{s}\) is also the MMSE linear function of \(Z-\hat{p}\) for estimating *s*. Further, reciprocally, the estimated trend \(\hat{p}\) is the MMSE linear estimate of *p* from the seasonally adjusted series \(Z-\hat{s}\) (a parallel to how the “final” X-11 trend is estimated). Under correct model assumptions, their paper also provides convergence results to the MMSE estimates of trend and seasonal for iterations starting with a non-MMSE estimate of trend (or seasonal).

## 12 Airline model results: the role of \(\Theta \)

We continue with (79) and special cases thereof to demonstrate important aspects of AMBSA. Hillmer and Tiao (1982) show that, when \(\Theta \ge 0\), the airline model is *admissible* (i.e., has a pseudo-s.d. decomposition) for all \(-1\le \theta <1\). (Admissible decompositions exist for negative \(\Theta \ge -0.3\) for a \(\Theta \,\)-dependent interval of \( \theta \) values.) We display the influence of \(\Theta \ge 0\) on the effective length of the filter through the rate of decay of its largest coefficients away from the time point of estimation. In general, for the estimate at time *t*, the observation \(Z_{t}\) gets the largest coefficient. The coefficients of the same-calendar-month values \(Z_{t\pm 12k}\) in the observation interval \(1\le t\le n\), decrease effectively exponentially in *k*. The dominating effect of \(\Theta \) is most clearly observed with the concurrent seasonal adjustment filters.

### 12.1 Seasonal filters from various \(\theta ,\Theta \) and their seasonal factors for the international airline passenger series

*n*.

Since \(\beta _{s}\left( B\right) =1-\beta _{sa}\left( B\right) \), at non-zero lags the magnitude effect of \(\Theta \) on seasonal filter coefficients is the same. Figures 7, 8, 9 show the calendar month seasonal factor estimates for the International Airline Passenger data from the filters determined by small, intermediate and large values of \( \Theta \), always with \(\theta =0.6\). The coefficient values were specified as fixed in X-13ARIMA-SEATS and thus are not data-dependent. (For the Airline Passenger series the estimates are \(\left( \hat{\theta },\hat{\Theta } \right) =(0.4,0.6)\)). The factors from \(\Theta =0.0\) change rapidly, resulting in excessive smoothing of the seasonal adjustment, see Fig. 12, and in large revisions (not shown). For \(\Theta =0.9\) the seasonal factors are effectively fixed and not locally adaptive. They thus have small revisions (whose cumulative effect over time can be large).

## 13 Smoothness properties of the models’ estimates

### 13.1 Simplistic autocorrelation comparison criteria for smoothness and nonsmoothness

We begin our autocorrelation-based consideration of the smoothing properties of estimates. The simplistic definitions used will support a systematic analysis that provides insight regarding seasonal decompositions.

Given two stationary series \(X_{t}\) and \(Y_{t}\), we say that \(X_{t}\) is *smooth* if \(\rho _{1}^{X}>0\). If also \(\rho _{1}^{X}>\rho _{1}^{Y}\), then \(X_{t}\) is *smoother than * \(Y_{t}\). If \(\rho _{1}^{Y}<0\), the series \(Y_{t}\) is *nonsmooth*. A smooth series is therefore smoother than all white noise series and all nonsmooth series. If \(Y_{t}\) is nonsmooth and \(\rho _{1}^{Y}<\rho _{1}^{X}\) holds, then \(Y_{t}\) is * more nonsmooth* than \(X_{t}\). A nonsmooth series is thus more nonsmooth than all white noise series and all smooth series. To examine if visual impressions of smoothness or nonsmoothness align with the conclusions of these formal criteria, differences of scale must be accounted for, see Sect. 13.2 and 13.3 and associated figures for illustrations.

*t*and \(\tau \), immediately yields that \(\delta \left( B\right) \left( \hat{s}_{t}-s_{t}\right) \) is uncorrelated with \(\delta \left( B\right) Z_{\tau }\) for all

*t*and \(\tau \).

When a series considered is a calendar month subseries, the autocorrelations considered are the seasonal autocorrelations in the time scale of the original \(Z_{t}\). We sometimes say that the conclusion of *smoother* is *strengthened* if the relevant autocorrelation at lag 2 (and perhaps consecutive higher lags) is also positive. This calls attention to the stronger smoothness properties of monthly trends and calendar month seasonal factors or their stationary transforms. Other intensifiers are only suggestive and will not be formally defined.

### 13.2 SAR(1): increased nonsmoothness of \(\hat{N}_{t}\)

### 13.3 SAR(1): greater calendar month smoothness of \(\hat{S}_{t}\)

### 13.4 Smoothness properties of \(q=2\) SRW component estimates

Autocorrelations of fully differenced estimates

Estimate | \(\gamma _{0}/\sigma _{a}^{2}\) | \(\rho _{1}\) | \(\rho _{2}\) | \(\rho _{3}\) | \(\rho _{4}\) |
---|---|---|---|---|---|

\(\left( 1-B^{2}\right) \hat{s}_{t}\) | \(70/16^{2}\) | \( -0.80\) | 0.40 | \(-\frac{4}{35}\doteq -0.114\) | \(\frac{1}{70}\doteq 0.014\) |

\(\left( 1-B^{2}\right) \hat{p}_{t}\) | \(70/16^{2}\) | 0.80 | 0.40 | \(\frac{4}{35}\doteq 0.114\) | \(\frac{1}{70}\doteq 0.014\) |

\(\left( 1-B^{2}\right) \widehat{sa}_{t}\) | \(134/16^{2}\) | \(\frac{72}{134}\doteq 0.537\,\) | \(-\frac{2}{67}\doteq -.030\) | 0 | \(\frac{1}{134}\doteq 0.007\) |

\(\left( 1-B^{2}\right) \hat{u}_{t}\) | \(24/16^{2}\) | 0 | \(-\frac{2}{3} \doteq -.667\) | 0 | \(\frac{1}{6}\doteq 0.167\) |

\(\delta \left( B\right) Z_{t}=a_{t}\) is white noise, so \(\rho _{j}^{\delta \left( B\right) Z}=0,j>0\). Thus in Table 2, \(\rho _{2}>0\) indicates a differenced estimate with smoother calendar month series than \(\delta \left( B\right) Z_{t}\), whereas \(\rho _{1}<0\) indicates a differenced estimate whose monthly series is more nonsmooth than monthly \(\delta \left( B\right) Z_{t}\), etc. Regarding the *monthly* series: as determined by \(\rho _{1} ,\delta \left( B\right) \widehat{sa}_{t}\) and \(\delta \left( B\right) \hat{p}_{t}\) are smoother than \(\delta \left( B\right) Z_{t}\) with strengthened smoothness. The seasonal’s \(\delta \left( B\right) \hat{s}_{t}\) is more nonsmooth than \(\delta \left( B\right) Z_{t}\). Regarding * calendar month series*, the seasonal adjustment’s, \(\delta \left( B\right) \widehat{sa}_{t}\) and especially the irregular’s \(\delta \left( B\right) \hat{u}_{t}\)’ s are more nonsmooth than \(\delta \left( B\right) Z_{t}\). The seasonal’s \(\delta \left( B\right) \hat{s}_{t}\) and the trend’s \(\delta \left( B\right) \hat{p}_{t}\) are smoother than \(\delta \left( B\right) Z_{t}\) , with strengthened smoothness.

### 13.5 Airline model component estimates: autocorrelations after full differencing

#### 13.5.1 Empirical lag 12 autocorrelation results of McElroy (2012) for seasonal adjustments

With a set of 88 U.S. Census Bureau economic indicator series for which the airline model was selected over alternatives, McElroy (2012) found that all but one had negative lag 12 sample autocorrelation in the fully differenced seasonally adjusted series, \(\delta \left( B\right) \widehat{sa}_{t}\) in our notation. This negative autocorrelation is statistically significant at the 0.05 level for 46 of the series. From the perspective of detrended calendar month series, which seem always to be visually smoothed by seasonal factor estimates (in logs when appropriate for AMBSA), this should not a be surprising result–removal of a smooth component causes loss of smoothness, which negative autocorrelation can formally identify.

#### 13.5.2 Correct model results for various \(\theta ,\Theta \) and component estimates

#### 13.5.3 Seasonal autocorrelations and calendar month smoothness

Lag 12 autocorrelations \(\rho _{12}^{\delta \left( B\right) Z}\left( \Theta ,\theta \right) =-\Theta \left( 1+\Theta ^{2}\right) ^{-1}\) of \(\delta \left( B\right) Z\)

\(\Theta {\backslash }\theta \) | \(-0.3\) | 0.0 | 0.3 | 0.6 | 0.9 |
---|---|---|---|---|---|

0.0 | 0 | 0 | 0 | 0 | 0 |

0.3 | -0.275 | -0.275 | -0.275 | -0.275 | -0.275 |

0.6 | -0.442 | -0.442 | -0.442 | -0.442 | -0.442 |

0.9 | -0.497 | -0.497 | -0.497 | -0.497 | -0.497 |

Lag 12 Autocorrelations of \(\delta \left( B\right) \hat{s}\)

\(\Theta \backslash \theta \) | \(-\)0.3 | 0.0 | 0.3 | 0.6 | 0.9 |
---|---|---|---|---|---|

0.0 | 0.347 | 0.467 | 0.589 | 0.622 | 0.222 |

0.3 | 0.568 | 0.644 | 0.714 | 0.731 | 0.481 |

0.6 | 0.763 | 0.803 | 0.836 | 0.844 | 0.715 |

0.9 | 0.943 | 0.952 | 0.959 | 0.960 | 0.931 |

Lag 24 Autocorrelations of \(\delta \left( B\right) \hat{s}_{{}}\)

\(\Theta \backslash \theta \) | \(-0.3\) | 0.0 | 0.3 | 0.6 | 0.9 |
---|---|---|---|---|---|

0.0 | 0.035 | 0.072 | 0.121 | 0.131 | 0.013 |

0.3 | 0.197 | 0.244 | 0.294 | 0.305 | 0.154 |

0.6 | 0.474 | 0.510 | 0.545 | 0.552 | 0.435 |

0.9 | 0.852 | 0.864 | 0.873 | 0.875 | 0.840 |

Lag 36 Autocorrelations of \(\delta \left( B\right) \hat{s}\)

\(\Theta \backslash \theta \) | -0.3 | 0.0 | 0.3 | 0.6 | 0.9 |
---|---|---|---|---|---|

0.0 | \(\doteq 0\) | \(\doteq 0\) | \(\doteq 0\) | \(\doteq 0\) | \(\doteq 0\) |

0.3 | 0.059 | 0.073 | 0.088 | 0.092 | 0.046 |

0.6 | 0.284 | 0.306 | 0.327 | 0.331 | 0.261 |

0.9 | 0.767 | 0.777 | 0.786 | 0.788 | 0.756 |

Lag 12 autocorrelations of \(\delta \left( B\right) \widehat{sa}\)

\(\Theta \backslash \theta \) | -0.3 | 0.0 | 0.3 | 0.6 | 0.9 |
---|---|---|---|---|---|

0.0 | \(-\)0.297 | \(-\)0.465 | \(-\)0.590 | \(-\)0.646 | \(-\)0.659 |

0.3 | \(-\)0.520 | \(-\)0.548 | \(-\)0.573 | \(-\)0.586 | \(-\)0.590 |

0.6 | \(-\)0.520 | \(-\)0.525 | \(-\)0.529 | \(-\)0.532 | \(-\)0.533 |

0.9 | \(-\)0.502 | \(-\)0.502 | \(-\)0.502 | \(-\)0.502 | \(-\)0.502 |

Lag 12 autocorrelations of \(\delta \left( B\right) \hat{u}\)

\(\Theta \backslash \theta \) | \(-\)0.3 | 0.0 | 0.3 | 0.6 | 0.9 |
---|---|---|---|---|---|

0.0 | \(-\)0.667 | \(-\)0.667 | \(-\)0.667 | \(-\)0.667 | \(-\)0.667 |

0.3 | \(-\)0.591 | \(-\)0.591 | \(-\)0.591 | \(-\)0.591 | \(-\)0.591 |

0.6 | \(-\)0.533 | \(-\)0.533 | \(-\)0.533 | \(-\)0.533 | \(-\)0.533 |

0.9 | \(-\)0.502 | \(-\)0.502 | \(-\)0.502 | \(-\)0.502 | \(-\)0.502 |

#### 13.5.4 Lag 1 autocorrelation and monthly smoothness results

Familiarly, an estimated trend visually smooths a seasonally adjusted monthly series. We examined the lag 1–12 autocorrelations (not shown) of the differenced trend estimates, \(\delta \left( B\right) \hat{p}_{t}\) for the \( \Theta ,\theta \) under consideration. At lags 1–6 all are positive. At lags 7–11, some or all can have either sign. Thus \(\delta \left( B\right) \hat{p}_{t}\) will have at least a half-year of resistance to oscillation. At lag 12, all are negative. This is in strong contrast to \(\delta \left( B\right) Z_{t}\), which, among lags 1–6, has a non-zero autocorrelation only at lag one, with a negative value indicating nonsmoothness (except when \( \theta <0\)), see (86).

Lag 1 autocorrelations \(\rho _{1}^{\delta \left( B\right) Z}\left( \Theta ,\theta \right) =-\theta \left( 1+\theta ^{2}\right) ^{-1}\) of \(\delta \left( B\right) Z_{{}}\)

\(\Theta \backslash \theta \) | \(-0.3\) | 0.0 | 0.3 | 0.6 | 0.9 |
---|---|---|---|---|---|

\(\text {all} \Theta \) | 0.275 | 0.0 | \({-0.275}\) | \(-0.441\) | \(-0.497\) |

Lag 1 Autocorrelations of \(\delta \left( B\right) \hat{u}\)

\(\Theta \backslash \theta \) | \({-0.3}\) | 0.0 | 0.3 | 0.6 | 0.9 |
---|---|---|---|---|---|

\(\text {all }\Theta \) | \(-0.756\) | -2/3 | \(-0.591\) | \(-0.533\) | \(-0.502 \) |

## 14 Concluding remarks

The simple seasonal models considered have provided very informative formulas for two- and three-component decompositions of seasonal time series. The factored formulas for the seasonal random walk simply display the full range of differencing operators (in biannual form) of ARIMA model-based seasonal decomposition filters identified in Bell (2012, 2015). The formulas for the estimates’ auto- and cross-correlation formulas have led to new insights and results. For example, the finding of negative sample autocorrelations at the seasonal lag of the differenced seasonally adjusted series now appears as an inevitable result of removing a seasonal component whose calendar month subseries are smooth. It is not a defect of the seasonal adjustment procedure, contrary to a view expressed in some of the literature motivating McElroy (2012).

For the irregular component, there are the common empirical findings, with airline and similar models, of negative sample autocorrelations, often at the first lag (see Table 11 in Appendix 1) and at the first seasonal lag of the estimated irregular component \(\hat{u}\) or differenced \(\hat{u}\) as in Tables 1 and 10. These can now be anticipated from the knowledge that \(\hat{u} \) can be regarded both as the detrended version of the seasonally adjusted series \(Z-\hat{s}\), and also as the deseasonalized version of the detrended series \(Z-\hat{p}\), in both cases resulting from removal of a smooth component.

The capacity to provide illuminating precise answers to many questions is a very valuable feature of ARIMA-model-based seasonal adjustment, as is its conceptual simplicity relative to nonparametric procedures, at least for adjusters with sufficient modeling background and experience. (The challenge is always to find an adequate model for the data span to be adjusted, if one exists.) Also valuable are the error variance and autocovariance measures (not accounting for sampling or model error) that AMBSA easily provides (only) for additive direct seasonal adjustments and their period-to-period changes. The latter, with log-additive/multiplicative adjustments, the most common kind, yield approximate uncertainty intervals for growth rates, quantities of special interest for real-time economic analysis.

**Disclaimer** Results of ongoing research are provided to inform interested parties and stimulate discussion. Any opinions expressed are those of the authors and not necessarily those of the U.S. Census Bureau or the Bank of Spain.

## Footnotes

- 1.
A more general perspective, applicable to any finite sample estimate, is that there is just one filter, a filter whose coefficients change with

*t*. The coefficients of \(\hat{S}_{t}=c_{1}\left( t\right) Z_{t-q}+c_{2}\left( t\right) Z_{t}+c_{3}\left( t\right) Z_{t+q}\) are piecewise constant, fixed at different values in the first year, last year, and the interval between. - 2.
With more general models for \(Z_{t}\), more forecasts and backcasts are needed, and their error covariances at nonzero leads and lags occur in the more complex mean square error formulas.

## Notes

### Acknowledgments

The SAR(1) results were presented by the first author at the March 13-14, 2014 Bank of Spain conference “Celebrating 25 Years of TRAMO-SEATS and honoring the 70th Birthday of Agustin Maravall”. The authors are indebted Brian Monsell, Tucker McElroy and Bill Bell for beneficial comments on earlier drafts. Bell’s comments and suggestions were especially extensive and influential. We are also grateful to Domingo Pérez, Brian Monsell and especially Tucker McElroy for computational support.

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