Coherent Trends, Turning Points, and Forecasts for ACS Data

Tucker McElroy

U.S. Census Bureau

Abstract

The American Community Survey (ACS) provides one-year (1y), three-year (3y), and fiveyear (5y) multi-year estimates (MYEs) of various demographic and economic variables for each

“community,” although for small communities the 1y and 3y may not be available. These survey

estimates are not truly measuring the same quantities, since they each cover different time spans.

We present a metric to measure the compatibility of different MYEs; for those that are deemed

to be sufficiently compatible, we describe methods for generating trends, turning points, and

forecasts of ACS data at 1y, 3y, and 5y intervals, in such a way that the different estimates can

be compared with one another. The filters utilized are non-model-based, require only a short

span of data, and are designed to preserve the appropriate linear characteristics of the time

series that are relevant for trends, turning points, and forecasts respectively. The basic method,

which only requires polynomial algebra, is outlined and applied on ACS data. The resulting

filters are analyzed in the frequency domain.

Keywords. Filtering, Frequency Domain, Nonstationary time series.

Disclaimer This paper is released to inform interested parties of ongoing research and to encourage discussion of work in progress. Any views expressed are those of the author and not necessarily

those of the U.S. Census Bureau.

1 Introduction

The American Community Survey (ACS) replaces the former Census Long Form, providing timely

estimates available throughout the decade. The ACS sample size is comparable to that of the Census

Long Form; variability in the sampling error component of the ACS is partially reduced through

a rolling sample (Kish, 1981). The rolling sample refers to the pooling of sample respondents over

time – in some cases this may be viewed as an approximate temporal moving average of single

period estimates. In particular, estimates from regions with at least 65, 000 people are produced

with a single year of data, whereas if the population is between 20, 000 and 65, 000 then three years

of data are combined, and if the population is less than 20, 000 then five years of data are pooled.

1
A somewhat dated overview of the ACS can be found in Alexander (1998). More current details

can be found in Census Bureau (2006).

The ACS database for the Multi-Year Estimates Study (MYES) is now publicly available at

www.census.gov/acs/www/AdvMeth/Multi Y ear Estimates/online data year.html.

The MYES produces one, three, and five year estimates for counties included in the 1999 − 2001

demonstration period and their constituent geographies, using data from 1999 through 2005. One

selects a Multi-Year Estimate (MYE) category – either one-year (1y), three-year (3y), or five-year

(5y) – then a time period, then a county, and then a geographic type within the county (e.g., school

district). Then the user can select from four types of information: demographic, economic, social,

and housing. Within each of these categories are dozens of variables, which for the most part are

simple totals or averages, but some of which are more complicated statistics (e.g., medians and

percentiles).

Because some counties have a low population, it was deemed desirable by the U.S. Census Bureau

to decrease sampling error for smaller geographies and subpopulations by using a rolling sample;

this methodology was confirmed by the National Academy of Sciences Panel on the Functionality

and Usability of Data from the American Community Survey (Citro and Kalton, 2007). In essence,

responses over a 3y or even a 5y span are gathered together into one database, and a statistic of

interest is computed over the temporally enlarged sample. In many cases, this is approximately

equal to computing a simple moving average of 1y estimates (see Section 3 below). This is known

as a rolling sample – see Kish (1981, 1998) and Alexander (2001) for a discussion. For larger

counties, the 1y MYE would be available as well. The question of whether each year should be

equally weighted was addressed by Breidt in a report to NAS (Citro and Kalton, 2007); since

all the responses are pooled in the 3y and 5y cases, it was judged to be impractical to use some

alternative weighting scheme (such as weighting the most recent year of data more highly). Hence,

the MYEs are formed from contributions over multiple years that are equally weighted. Although

this approach is simple, one repercussion is that some lag (or time delay) is induced by the use of

rolling samples.

The effect of this time delay is that the 3y estimate is roughly speaking an estimate of the previous

year’s value, and the 5y estimate gives the value corresponding to two years back. For example,

putting a straight line through a 3-year and 5-year moving average viewed as a concurrent filter will

result in a line lagged by 1 and 2 time points respectively. Because for many regions and statistics

the ACS data will be trending upwards in a local linear fashion (this is dictated by macroeconomic

theory and demographic principles at work), the 3y and 5y estimates will tend to be lagging

2
behind the 1y values. Admittedly this effect is much less pronounced for characteristics reported as

percentages, versus the actual levels. This has repercussions for cross-county comparisons, which

is seen as follows. Suppose that county A has a small population, so we only have a 5y estimate.

But county B has a large population, so we have 1y, 3y, and 5y available. Since the 1y is at hand

for county B, attention may well focus on it and the 5y will be largely ignored. It will then be

natural for a user to compare the 1y MYE for county B with the 5y MYE for county A. But this

is comparing apples and oranges; it would be valid, roughly speaking, to compare the 1y MYE for

county B two years back to the 5y MYE for county A (or alternatively, to take a five year moving

average of the 1y MYE for county B). Alexander (2001) provides additional discussion.

These subtleties, although easily illustrated through a few pictures, cannot be assumed to be

comprehensible to the average consumer of ACS data. The dangers of deliberate or unintentional

abuse seem to be great. Therefore, a method that produces trends (as well as turning points

and forecasts) from available data should be developed, such that the trends (respectively, turning

points and forecasts) all generate the same value for 1y, 3y, and 5y estimates. That is, we have

three separate trend filters, for each of the 1y, 3y, and 5y estimates, such that the output of all three

operations are approximately identical. It will then be mathematically valid to compare a county

B 1y trend with a county A 5y trend, for example. Such trends, turning points, and forecasts could

be either published along with the ACS data, or could be available in a small portable program

with a GUI front-end (e.g., R with JAVA interface) that the average consumer could use with a few

clicks of the mouse. The first option has the advantage of pre-empting the ability of the public user

to misuse the data. A similar thing is done in the arena of seasonal adjustment, where the U.S.

Census Bureau strongly encourages consumers to base all analysis upon the seasonally adjusted

data rather than the raw (see the discussion in Bell and Hillmer, 1984).

In the scheme of trend filter design, there are two general approaches: the “window trend” and

the “on-line trend.” The former takes a span of data X1, · · · ,XN and produces a trend value

S1, · · · , SN for each time point. The latter may utilize less data, and produces a trend only at time

N. The window trend method will typically produce trends that depend on the size of the window

(since the filters that are used depend on this length). This can be a practical drawback, which we

will illustrate. If we obtain an additional datum XN+1, then we can obtain new trend estimates,

based on either the window X1 · · · XN+1 or X2 · · · XN+1 (depending on whether a fixed window

or expanding window is desired). Either way, the new trend estimate at time N will generally be

different from the previous value SN . This is known as the problem of revisions in the seasonal

adjustment literature. A publishing agency would then have the onerous task of updating the trend

estimate at time N (and the updating can continue indefinitely, although they generally tend to

stabilize). This is onerous, because the public invariably complains that the agency is “changing

3
the numbers again.”

In contrast, the on-line approach requires no updating, since a trend is only produced at the

current time point (using one fixed concurrent filter). Although the trend values will then be consistent, there is invariable phase delay (i.e., lag in the trend estimates). Generally there is a trade-off

between smoothness and lag, which represents an inherent scientific limitation; the acquisition of

a zero-lagging smoother would be an incredibly powerful tool in economic and demographic analysis. Nevertheless, careful design of concurrent filters may help to strike a proper balance between

smoothness and lag. Due to the greater possibilities of user confusion – and the fact that the

greater lag inherent in concurrent filtering can be mitigated – we advocate the on-line approach.

From this discussion, we have gathered several criteria for ACS “signals”; by signals, we refer to

trends, turning points, forecasts, and potential additional applications.

1. Coherency between signals of 1y, 3y, and 5y estimates

2. Proper treatment of linear dynamics

3. Concurrent filtering with minimal phase-delay

4. Filter length is short

Item 1 has been discussed above, and will be formulated mathematically below. As for the linear

dynamics in 2, we mean that underlying local linear behavior in the time series data should be appropriately preserved. For trends, this means that the exact linear polynomial should be unaltered

by the filtering; for turning points, the sign of the slope coefficient of the line should be transmitted

(so that a negative slope indicates a down-turn, whereas a positive slope indicates an up-turn). In

forecasting, the line should actually be advanced one step in time. Each of these three concepts can

be given a precise mathematical formulation. Item 3 indicates that we adopt the on-line filtering

approach, and seek to mitigate the effects of lag. This is related to item 2, since in producing trends

we seek a filter that is the identity operator on linear polynomials, and thus the lag at “frequency

zero” will be automatically reduced. A full discussion of this important topic requires a frequency

domain analysis of the signal filters. Finally, point 4 is of practical importance; since we use on-line

filtering, we must have the signals produced from a minimal span of data. If the filter consists of

d weights, then d data points are required to produce the signal, i.e., data XN−d+1, · · · ,XN are

required for the signal SN . Thus no signal will ever be available for X1, · · · ,Xd−1; we therefore

seek to minimize d.

The remainder of this paper is structured as follows. In Section 2 we discuss the basic approach

to constructing the signal filters through using polynomial algebra; the specific cases of trends,

4
forecasts, and turning points are considered in Sections 2.1 through 2.3. Section 3 provides a way of

assessing compatibility between various MYEs, essentially providing the conditions for the validity

of Section Two’s methodology. Then Section 4 illustrates the methods on some actual publicly

available ACS data. Finally in Section 5 the properties of the various filters are explored in the

frequency domain through squared gain and phase delay plots. A discussion of the applications is

given in Section 6, and some derivations are contained in the Appendix.

2 Mathematical Formulation and Solution

In this section we assume that the 3y and 5y estimates are exactly equal to simple moving

averages of the 1y data. We write the k-year estimate at year t as Y

(k)

, where t

Yt

(k) = Θ(k)

(B)Xt (1)

and k = 1, 3, 5 (B is the backshift operator). Here Xt

is the 1y estimate (which is not directly observable for small geographical areas), and Θ(k)

(z) is the Simple Moving Average (SMA) polynomial

of order k given by

1

Θ

(k)

(z) = 3

1 + z + · · · + z

k−1

 ́

.

k

Now (1) may be only approximately true in reality, but in many cases we may assume that (1)

holds with a small amount of error; for more discussion of this assumption see Section 3. Now for

a signal of interest St

, we look for three filters Ψ(k)(B) for k = 1, 3, 5 such that

St = Ψ(k)

(B)Y

(k)

k = 1, 3, 5. (2) t

This ensures condition 1, namely that we can filter the 1y, 3y, and 5y estimates with different filters

in order to obtain a common signal St

. This signal must have the characteristics dictated by its

definition, and this is where conditions 2 and 3 enter. Because we are doing concurrent filtering,

we can write

(k) j Ψ

(k)

(z) = Xψ z . j

j≥0

(k)

In practice only a finite number of the coefficients ψjare nonzero. The specifics of condition 2

are discussed in the following subsections. Now combining (2) and (1) yields

Ψ

(1)(z) = Ψ(3)(z)Θ(3)(z) = Ψ(5)(z)Θ(5)(z). (3)

Since Θ(3)(z) and Θ(5)(z) are known ahead of time, we only need determine Ψ(3)(z) and Ψ(5)(z);

then Ψ(1)(z) is determined.

5
Now focusing on trends, we seek concurrent filters Ψ(k)

such that the composite with the SMA

passes lines, i.e.,

Ψ

(k)

(B)Θ(k)(B) [at + b] = at + b (4)

for any a, b and all integers t, and k = 1, 3, 5. This property, together with those discussed above,

define the optimal trend filter for our purposes. The following theorem provides explicit formulas

for the optimal trend filters.

Theorem 1 The minimal length concurrent filters Ψ(k)

satisfying (3) and (4) are given by

Ψ

(5)(z) = ¡

4 + z + z

2 − 3z3

¢

/3

Ψ

(3)(z) = ¡

4 + z + z

2 + z3 + z4 − 3z5

¢

/5

Ψ

(1)(z) = ¡

4 + 5z + 6z

2 + 3z3 + 3z4 − z5 − 2z6 − 3z7

¢

/15.

Next focusing on 1-step ahead forecasting, we seek concurrent filters Ψ(k)

such that the composite

with the SMA advances lines by one time step, i.e.,

Ψ

(k)

(B)Θ(k)(B) [at + b] = a(t + 1) + b (5)

for any a, b and all integers t, and k = 1, 3, 5. This property defines the optimal forecast filter, and

the following theorem provides explicit formulas:

Theorem 2 The minimal length concurrent filters Ψ(k)

satisfying (3) and (5) are given by

Ψ

(5)(z) = ¡

5 + z + z

2 − 4z3

¢

/3

Ψ

(3)(z) = ¡

5 + z + z

2 + z3 + z4 − 4z5

¢

/5

Ψ

(1)(z) = ¡

5 + 6z + 7z

2 + 3z3 + 3z4 − 2z5 − 3z6 − 4z7

¢

/15.

Finally, consider turning point (t.p.) filters Ψ(j)

for j = 1, 3, 5. A t.p. estimate should indicate

a change in the overall direction of the data; a crude t.p. signal can be generated by differencing

consecutive trend values. Since we wish to anticipate turning points, we take the difference between

the forecast filter and the trend filter, which yields the following for the 5y, 3y, and 1y t.p.’s

respectively:

Ψ

(5)(z) = ¡

1 − z

3

¢

/3 (6)

Ψ

(3)(z) = ¡

1 − z

5

¢

/5

Ψ

(1)(z) = ¡

1 + z + z

2 − z5 − z6 − z7

¢

/15.

This last t.p. filter is just (1 − z)Θ(3)(z)Θ(5)(z). These filters satisfy conditions 1, 2, and 4.

6
3 A Compatibility Measure

The above development relies on (1), which in reality is only approximately true. For a 3y MYE,

a sample of people is gathered over all three years and their responses are pooled together. For the

5y MYE the sample is gathered over five years. Then for whatever characteristic is being estimated

from the sample, it will be rescaled by the sampling frequency. Thus in comparing a 3y MYE and

5y MYE for the same time period and area, we might expect (1) to hold true. However, for some

characteristics we found large discrepancies between the different MYEs, and it was observed that

(1) fails. The validity of (1) depends most upon what type of characteristic is being measured: for

totals and averages (e.g., total population) the approach of Section 2 was largely valid, whereas

for other estimates (e.g., median income) we saw that (1) was false. Thus it is necessary to devise

a “compatibility measure” for each characteristic (in each region) that assesses the validity of our

main approach. Consider the following modification of (1):

(k) (k)

Yt = Θ(k)(B)Xt + 2t(7)

(k) (1) for k = 1, 3, 5. Here 2 represents an error process, but 2 = 0 by assumption (there is no loss of t t

generality in assuming this). Now we will develop the exposition under the supposition that MYE’s

are available for k = 1, 3, 5; if only 3y and 5y MYE’s are available, then set 2

(3) = 0 and ignore the t

k = 1 equation. Likewise, if only a 5y MYE is available, then there is no compatibility issue, since

there is only one type of estimate. Now applying a trend, t.p., or forecasting filter Ψ(k)

(B) yields

(k) (k) Ψ

(k)

(B)Y = Ψ(1)(B)Xt + Ψ(k)(B)2 . (8) t t

On the left is the estimate, whereas Ψ(1)(B)Xt should be viewed as our target, or signal. The

(k) (k)

associated error process is Ψ(k)(B)2 . Our objective is to quantify the size of 2 in order to t t

(k) (k)

understand its impact on the error process Ψ(k)(B)2t t . The closer 2 is to zero, the more valid

the methodology of Section 2 will be. The basic idea is to consider a “noise-signal” ratio

(k) (k)

2t Yt = − 1.

Θ(k)(B)Xt Θ(k)(B)Xt

This is only well-defined when Θ(k)

(B)Xtis nonzero, and we generally suppose that it is positive

at all times. Since it is more convenient we will instead work with the first-order Taylor series

approximation of the “noise-signal” ratio, which is given by

(k) (k) NSR = log Y − log Θ

(k)

(B)Xt.

t t

The NSR notation stands for Noise-Signal Ratio. For k = 1 it is trivially equal to zero, but for

k = 3, 5 it may be positive or negative. In order to compute it, we substitute Xt = Y

(1)

. Now this t

7
4

quantity shall be calculated over all times t for which it is possible to do so, and then we let the

compatibility measure be defined by

(k) C

(k) = max |NSR |.

t

t

Clearly C

(k) will get updated as more data is available. Now it follows from the definitions that

−C(k) (k) C(k)

3

e − 1

 ́

Θ

(k)

(B)Xt ≤ 2 ≤

3

e − 1

 ́

Θ

(k)

(B)Xt.

t

Applying this to (8) yields

−C(k)

Ψ

(1)(B)Xt ≤ Ψ(k)

(B)Y

(k) Ψ

(1)(B)Xt e .

t ≤ e

C(k)

This inequality shows us how to set tolerance thresholds for the compatibility measure, e.g., C

(k) =

log(1.01) (approximately one percent discrepancy). When the compatibility measure (for both

k = 3 and k = 5) is below the threshold, then we can apply the methods of Section 2, confident in

the knowledge that the error process is very small.

Illustrations on ACS Data

As of the time of writing of this article, the following MYEs are available: 00–05 for 1y, 01–05

for 3y, and 03–05 for 5y. The year index here refers to the last year that entered into the sample,

and so is consistent with our notation for Y

(k)

. Letting t range between 00 and 05 (referring to the t

(1) (1) (3) (3) (5) (5) year), the available database is Y , · · · ,Y ,Y , · · · , Y ,Y , · · · ,Y . In order to apply our 00 05 01 05 03 05

methods, we need 8 years of 1y data, 6 years of 3y data, and 4 years of 5y data. So we need two

more 1y estimates, and one more of the 3y and 5y. So in order to apply our method, we need to

extend our database.

We first obtain a backcast estimate for Y

(1) using the supposed compatibility of the MYEs. If 99

(1) were true, then

(1) (3) (1) (1) (5) (1) (1) (1) (1) ́

Y = 3Y −

3

Y + Y

 ́

= 5Y −

3

Y + Y + Y + Y . 99 01 01 00 03 03 02 01 00

So these provide two estimates for Y

(1)

, which can be averaged for better precision. Thus 99

(1) 3 (3) 5 (5) 1 (1) (1) (1) (1) 3

Y

 ́

Y = Y + Y − + Y + 2Y + 2Y . 99 01 03 03 02 01 00

b

2 2 2

(1) (3) (5) Next, in order to obtain estimates for Y , Y , and Y (we choose to forecast rather than 06 06 06

backcast, because our projections can be verified in a year’s time), we apply a very simple random

8
(1) (1) 1 (1) (1) ́

Y = Y +

3

b

06 05 Y05 00 − Y

5

(3) (3) 1

Y

(3) (3)

06 = Y05 + 05 01 − Y

4

3  ́

(5) (5) (5) (5) = Y + Y − Y . 06 05 05 03 2

walk model to obtain

3  ́

Yb

1

Yb

Note that we cannot use the forecast filter to get these future estimates, since that supposes that 8

1y data points are available. So with these crude forecasts, we can now apply our filters for trends,

forecasts, and t.p.’s.

We consider five examples out of a plethora, which were chosen to illustrate various aspects of this

research. First consider the Mean Travel Time variable for Bronx, NYC, NY, referred to as Travel.

We first note the compatibility measures for this variable were C

(3) = .071% and C(5) = .043%.

This indicates a very small error in (7), so that we can expect our filtering methodology to be

highly valid. The extended data-set for Travel is given in Table 1; we can see the mismatch and

lag between the various years just by eye-balling the chart along the various sub-diagonals (i.e.,

consider 1y 01, 3y 02, 5y 03, and 1y 02, 3y 03, 5y 04, and so forth).

In Table 2 all of the compatibility measures are given (in percentages) for the five examples,

along with trend, forecast, and t.p. estimates for year 2006 (though the 2006 forecast should be

interpreted as a trend estimate for year 2007). These examples are: Divorced Males for Lake

County, IL (Divorce); Mean Travel Time for Bronx, NYC, NY (Travel); Median Income for Pima,

AZ (Income); Median Age for Hampden, MA (Age); Rental Vacancy Rate for Madison, MI (Rent).

The 1y, 3y, and 5y estimates of trend, forecast, and t.p. respectively are all quite similar when the

compatibility measure is low, as in Divorce, Travel, and Age. The forecasts in these cases seem

reasonable, but ultimately cannot be checked until next year. For Divorce, Rent, and Income the

compatibility is progressively worse, and the signal estimates have corresponding discrepancies. The

3y trend, forecast, and t.p. for Income is especially noteworthy – the t.p.’s sign is even different! We

observe that in this case the series is based off of a median of incomes, and thus it is understandable

that the compatibility breaks down.

9

5 Frequency Domain Properties

The definitions of the Gain and Phase and Phase-Delay functions of a filter Ψ(B) can be found

in Findley and Martin (2006); this reference explains how a continuous Phase function can be

constructed by allowing the Gain function to be negative. We follow this convention here. That is,
we have the following decomposition:

Ψ(e

−iλ) = G(λ) exp{iΦ(λ)}.

Here G and Φ are continuous real functions, and are called the Gain and Phase functions, respectively. We also have the Phase-Delay function Υ(λ) = Φ(λ)/λ when Φ(λ)/λ|λ=0 is finite; its

interpretation is the amount (in time units) of delay or lag that each frequency of the data inherits

from application of the filter. Note that this interpretation is valid for stationary time series, but

must be adjusted somewhat when considering random walks or other nonstationary time series.

Likewise, the Gain function shows how much attenuation of stochastic variance each frequency in

the data inherits from application of the filter. For example, a stationary process can be written

as an orthogonal increments integral

Z π

iλt X dZ(λ), t = e

−π

where dZ is an orthogonal increments random measure. Application of the filter Ψ(B) produces

Z π

Z π

iλtΨ(e

iλ(t−Υ(λ)) Ψ(B)X G(λ)dZ(λ). t = e−iλ)dZ(λ) = e

−π −π

From this, we see the effect of the Phase Delay at time t (high values of Υ(λ) create a lag) and the

Gain function, which scales the original random measure dZ. For a nonstationary time series, a

modified orthogonal increments representation is required and the above formulation does not hold

true. Supposing that the process Xt

is given by a Generalized Random Walk (GRW), we can write

iλt − 1

Z π

e

Xt = X0 + dZ(λ)

−π 1 − e−iλ

for t = 1, 2, · · · . This is a very general once-integrated stochastic process, and is plausible as a

crude model for much of the ACS demographic and economic data. Application of a filter Ψ(B)

produces

Z π

e

iλ(t−Υ(λ))G(λ) − Ψ(1) Ψ(B)Xt = Ψ(1)X0 + dZ(λ).

−π 1 − e−iλ

So for the GRW, Υ and G have a somewhat different effect, although it is still measurable. Observe

that for trend and forecast filters, Ψ(1) = 1. Thus the integrand in the orthogonal increments

integral essentially undergoes the transformation

iλt − 1 e e

iλ(t−Υ(λ))G(λ) − 1

7→ 1 − e−iλ −iλ 1 − e

under application of the filter. Thus the interpretation is that the time index t gets lagged by Υ(λ),

and so in the GRW case this can still be interpreted as Phase Delay. However, G does not multiply

the whole fraction (e

iλt−1)/(1−e−iλ), so it cannot really be interpreted as the Gain for the process.

iλt If we take first temporal differences of the above integrands, we obtain e 7→ eiλ(t−Υ(λ))G(λ); this

shows that G is interpreted as the Gain function of the differenced GRW process. Now for the t.p.

filter we have Ψ(1) = 0, so the Υ and G functions can only really be interpreted as Phase Delay

and Gain for the differenced GRW process.

10
With these observations in mind, we now proceed to the Gain, Phase, and Phase Delay plots

for trend, t.p.’s, and forecast filters. Figure 1 shows the Gain function for the trend filter Ψ(1)(B).

We see it passes frequency zero, accentuates some low frequencies, and tends to attenuate higher

frequencies, albeit in a nonuniform (monotonic) fashion. This behavior is due to the enforced

constraints, ensuring coherency across 1y, 3y, and 5y estimates. The zeroes are at λ = 2π/5, 4π/5,

and 2π/3, which can also be deduced directly from the formulas for the filter. The Phase Delay

plot in Figure 2 shows there is no lag at frequency zero (this is ensured by the condition that a line

be preserved), and at the first zero λ = 2π/5 the delay is roughly 2.5 time units. The delay does

not increase beyond 3 time units, which is entirely reasonable for a concurrent trend filter.

The t.p. filter is nonzero at frequency zero, which prohibits us from formulating a well-defined

Phase Delay function. The Gain function (Figure 3) shows this complete attenuation at frequency

zero; as designed, the action of this filter on a line is that only the constant slope is reproduced.

Basically all frequencies are attenuated, since the Gain is always less than one in magnitude. In

terms of the stochastic nature of the data, this filter greatly reduces the variance; this is fine,

because a t.p. filter only reveals changes in directionality, and is not concerned with preserving the

overall scale of the original data. The other zeroes of the t.p. are the same as the trend filter. The

Phase function (Figure 4) gives a constant increase, with a slope of 3. Note that because the Phase

is negative for λ ≤ π/2, there is actually a “phase advance” of these frequencies. Dividing by λ to

obtain the Phase Delay function, the phase advance is actually quite large at these low frequencies

(the function tends to −∞ at λ = 0). But Υ tends asymptotically to 3 for larger λ, indicating a

time lag of three units at higher frequencies. Because of the unbounded behavior at λ = 0, we do

not show the Phase Delay plot.

Finally, the forecast filter is displayed in Figures 5 and 6. The Gain function is almost identical

to that of the trend filter, which is not surprising given their similar derivation. The forecast Gain

has a bit more accentuation of low frequencies. The main difference lies in the Phase Delay plots;

although the shape of this function is similar to that of the trend, note that Phase Delay is negative

for frequencies up to roughly .1. Thus the low frequency range has a phase advance property, or

in other words it is projected forward in time. So the forecast filter does indeed forecast these low

frequencies. At higher frequencies, phase delay takes over and gets as high as three time units.

11

6 Discussion and Conclusion

We began with a discussion about key properties for any signal filter, and arrived at conditions 1

through 4. Guided by these principles, we were able to construct filters for trends, turning points,

and forecasts in a straightforward fashion. One underlying assumption that should be emphasized
is that much of the analysis depends upon a local linear structure in the data. The more formal

analysis of Section 5 considers the data to be given (at least locally) by a GRW or stationary process.

However, the methods presented in Section 2 can, in principle, be generalized to twice-integrated

stochastic processes, where interest focuses on quadratic trends. The needed calculations would be

much more involved. But macro-economic and demographic theory tend to support the idea of a

single order of integration in the stochastic dynamics (at least locally, i.e., for a short span of data).

What about other signals? The typical econometric analysis of an economic time series considers

seasonal and cyclical dynamics, as well as trading day and holiday effects (Findley, Monsell, Bell,

Otto, and Chen, 1998). Since the data is measured annually, the only relevant dynamic is the cycle.

The design of a cycle filter is far beyond the scope of this work, since it is unclear about how to

relate such a filter to linear dynamics (the cycle is concerned with truly periodic effects somewhat

removed from frequency zero). But other signals of interest can arise from the desire to forecast

components. We note that the forecast filter we consider essentially forecasts the trend, since the

phase advance is concentrated in the low frequency band. One possibility is to consider a highfrequency forecast filter, though in general the high-frequency components contain less interest for

typical users.

If we view the evolution of an economic or demographic process from a kinetic perspective, it is

often interesting to consider the “velocity” and “acceleration” of such a series, in addition to other

kinematic characteristics (e.g., curvature, zero-crossings of the velocity, etc.). The turning point

filter can be viewed as a low-frequency velocity filter, since it has the effect of the first derivative

on the line mx + b. Change of sign of the turning point filter indicates a change in the overall

trending behavior of the data. The ability to forecast such a zero-crossing amounts to prediction of

change in an economic or demographic trend. An acceleration filter would give information about

the convexity of the low-frequency portions of the series. The Newtonian characterization of local

peaks and troughs through velocity and acceleration seems to be a useful concept for analysis of

trending data. To this end, filters to measure velocity and acceleration, and forecasts of such, would

be of value and interest, and could be designed along the lines of the methodology of Section 2.

Whatever the choice of signal and filter, the main consideration is that the 1y, 3y, and 5y results

be made compatible. At the heart of the issue is the non-invertibility of Θ(3)(B) and Θ(5)(B); these

filters essentially become components of any signal filter for the 1y estimate that we design (hence,

we expect that only low-frequency behavior can be successfully gauged through the methods of

this work). For the average user, this means that a few basic signals (such as the trend and t.p.)

should be provided and published so that inapt comparisons are avoided. For the professional user

(i.e., internal analyst), they should first define what characteristic or signals they are interested in,

12
and develop the corresponding filters along the lines that have been delineated here. Presumably,

interest will focus on cross-county comparisons of characteristics such as trends, forecasts, and

so forth. It is essential that condition 1 be satisfied before any analysis is conducted. Thus,

this work will be of value to two diverse groups: the statistically unsophisticated public user (for

which the internal details of this methodology must remain opaque) and the professionally trained

statistician (who is encouraged to learn and expand this methodology so that their analyses will

obtain a heightened validity).

Appendix

A.1 Action of Filters on Polynomials

Here we discuss the action of filters on polynomials; for a related reference, see Brockwell and

Davis (1996). Consider a generic filter G(z) acting on a line at + b:

G(B)[at + b] = aG(B)t + G(1)b.

For a trend filter, we wish G(z) to pass the line, which requires that G(B)t = t and G(1) = 1. The

first condition is equivalent to (1 − G(B))t = 0, or

1 − G(z) = H(z)(1 − z)

2

for some polynomial H. This is true, because 1 − B reduces t to unity, and another difference is

needed to produce zero. It follows from this condition that G(1) = 1. Hence, it is necessary and

sufficient for G(z) to pass lines that 1 − G(z) is divisible by (1 − z)

2

. If we wish for a line to be

passed with a lag of j time units (where if j is negative there is an advance), we have G(B)[t] = t−j,

or

(1 − G(B))[t] = j.

This implies that 1 − G(z) must accomplish differencing with a change in level, or

1 − G(z) = H(z)(1 − z),

where H(1) = j. For the trend filter j = 0, whereas j = −1 for the forecast filter.

A.2 Proofs of Theorems 1 and 2.

We seek trend/forecast filters that will “pass” lines, and Appendix A.1 gives conditions that

guarantee this. Now for k = 3, 5 we have

k−3

1 − Ψ + ·

(k)

(z)Θ(k)(z) = 1 − Ψ

(k)

(z) + Ψ(k)(z)(1 − z)

3

z

k−2 + 2z · · + (k − 2)z + (k − 1) ́

/k,

13
which uses (1−Θ(k)

(z))/(1−z) = (z

k−2 +2zk−3 +· · ·+(k−2)z +(k−1))/k. Choosing Ψ(k)

(z) such

that Ψ(k)

(1) = 1, we know that 1 − z divides 1 − Ψ(k)(z). Letting Φ

(k)

(z) denote the polynomial

quotient, we have

1 − Ψ(k)

(z)Θ(k)(z) k−3 = Φ + ·

(k)

(z) + Ψ(k)(z)

3

z

k−2 + 2z · · + kz + (k − 1) ́

/k.

1 − z

We wish to construct Φ

(k)

(z) such that the right hand side evaluated at z = 1 is j, where j = 0 for

trends and j = −1 for forecasts (see Appendix A.1). Thus we obtain the condition that

k − 1

Φ

(k)

(1) = j − . (A.1)

2

So, given a choice of Φ

(k)

(z) such that (A.1) holds, we simply let

Ψ

(k)

(z) = 1 − (1 − z)Φ(k)(z).

This will ensure that 1 − Ψ(k)

(z)Θ(k)(z) is divisible by (1 − z)

2

, so that it passes lines. The other

constraints of our solution depend on (3); if Φ

(5)(z) is degree d, then there are d + 8 constraints

and 2d + 4 degrees of freedom offered by the coefficients of Φ

(3)(z) and Φ(5)(z). Although this

implies that the minimal degree polynomial solution is d = 4, there are redundancies among the

constraints that can be exploited, such that we can take d = 2. We next derive the solution.

If Φ

(5)(z) has degree 2, it follows from (3) that Φ(3)(z) has degree 4. We write

2 Φ(5)(z) = a0 + a1z + a2z

from which it follows that

3 Ψ(5)(z) = (1 − a0) + (a0 − a1)z + (a1 − a2)z2 + a2z

Ψ

(5)(z)Θ(5)(z) = ¡

(1 − a0) + (1 − a1)z + (1 − a2)z

2 + z3 + z4 + a0z5 + a1z6 + a2z7

¢

/5.

Similarly we have

4 Φ(3)(z) = b0 + b1z + b2z2 + b3z3 + b4z

from which it follows that

5 Ψ(3)(z) = (1 − b0) + (b0 − b1)z + (b1 − b2)z2 + (b2 − b3)z3 + (b3 − b4)z4 + b4z

Ψ

(3)(z)Θ(3)(z) = ¡

(1 − b0) + (1 − b1)z + (1 − b2)z

2 + (b0 − b3)z3 + (b1 − b4)z4 + b2z5 + b3z6 + b4z7

¢

/3.

At this point we use (3) and match coefficients. Consider the trend (j = 0) and forecast (j = −1)

filters together. From matching coefficients, we obtain the following equations for bk and ak:

2 3 2 3 3 3 3

b0 = + a0 b1 = + a1 b2 = a0 b3 = a1 b4 = a2 (A.2)

5 5 5 5 5 5 5

14
We also obtain several constraints on the ak’s:

3 2 3 3 2 3 2 3

= + (a1 − a2) = + (a0 − a1) = (a0 − a2) (A.3)

5 5 5 5 5 5 5 5

The third equation is implied by the other two. Finally, applying (8) gives the additional constraints

j − 2 = a0 + a1 + a2

j − 1 = b0 + b1 + b2 + b3 + b4.

The second equation, together with (A.2) yields

4 3

j − 1 = + (2a0 + 2a1 + a2).

5 5

This results in the following matrix system

⎡

1

⎤ ⎡

0 3 −3

⎤ ⎡ ⎤

a0

1

⎢

⎢

⎣

5j − 9

=

⎥

⎥

⎦

3

⎢

⎢

⎣

6

−3

6

0

3

a1

⎥

⎥

⎢

⎢

⎦ ⎣

a2

.

⎥

⎥

⎦

(A.4)

The inverse matrix is

⎡

1 1 1

⎤

1

15

⎢

⎢

⎣

1

−4

−2

−2

2

1

.

⎥

⎥

⎦

(A.5)

Now for j = 0, this yields the solution

Φ

(5)(z) = −

¡

1 + 2z + 3z

2

¢

/3

Φ

(3)(z) = ¡

1 − z

2 − 2z3 − 3z4

¢

/5.

Then the 1y, 3y and 5y trend filters are

Ψ

(5)(z) = ¡

4 + z + z

2 − 3z3

¢

/3

Ψ

(3)(z) = ¡

4 + z + z

2 + z3 + z4 − 3z5

¢

/5

Ψ

(1)(z) = ¡

4 + 5z + 6z

2 + 3z3 + 3z4 − z5 − 2z6 − 3z7

¢

/15.

For j = −1 we obtain

Φ

(5)(z) = −

¡

2 + 3z + 4z

2

¢

/3

Φ

(3)(z) = −

¡

z + 2z

2 + 3z3 + 4z4

¢

/5.

Then the 1y, 3y and 5y forecast filters are

Ψ

(5)(z) = ¡

5 + z + z

2 − 4z3

¢

/3

Ψ

(3)(z) = ¡

5 + z + z

2 + z3 + z4 − 4z5

¢

/5

Ψ

(1)(z) = ¡

5 + 6z + 7z

2 + 3z3 + 3z4 − 2z5 − 3z6 − 4z7

¢

/15.

It was demonstrated above that these filters have the shortest length possible with the stated

properties. This completes the proof. 2

15
References

[1] Alexander, C. (1998) Recent developments in the American Community Survey. Proceedings

of the Survey Research Methods Section, ASA, 92–100.

[2] Alexander, C. (2001) Still rolling: Leslie Kish’s “rolling samples” and the American Community Survey. Proceedings of Statistics Canada Symposium 2001, Achieving Data Quality in a

Statistical Agency: A Methodological Perspective.

[3] Bell, W. and Hillmer, S. (1984) Issues involved with the Seasonal Adjustment of Economic

Time Series. Journal of Business and Economic Statistics 2, 291–320.

[4] Brockwell, P. and Davis, R. (1996) Introduction to Time Series and Forecasting. New York:

Springer.

[5] Census Bureau, U.S. (2006) Technical Paper 67. Design and Methodology, American Community Survey.

http://www.census.gov/acs/www/Downloads/tp67.pdf

[6] Citro, C. and Kalton, G. (2007) Using the American Community Survey: benefits and challenges. The National Academies Press: Washington, D.C.

[7] Findley, D. and Martin, D. (2006) Frequency Domain Analyses of SEATS and X-12-ARIMA

Seasonal Adjustment Filters for Short and Moderate-Length Time Series. Journal of Official

Statistics 22, 1–34.

[8] Findley, D. F., Monsell, B. C., Bell, W. R., Otto, M. C. and Chen, B. C. (1998) New Capabilities and Methods of the X-12-ARIMA Seasonal Adjustment Program. Journal of Business

and Economic Statistics 16, 127–177 (with discussion).

[9] Kish, L. (1981) “Using cumulated rolling samples to integrate census and survey operations of

the Census Bureau,” Washington, D.C., U.S. Government Printing Office.

[10] Kish, L. (1998) Space/Time variations and rolling samples. Journal of Official Statistics 14,

31–46.

16
MYEs

Time 1y 3y 5y

99 40.05

00 40.00

01 41.00 40.00

02 41.80 41.00

03 40.80 41.20 40.70

04 40.60 41.00 40.80

05 41.70 41.10 41.20

06 42.04 41.28 41.45

Table 1: MYEs for Mean Travel Time of Bronx, NYC, New York in minutes. Estimates have been

backcast and forecast extended to the years 99 and 06, written in bold.

Characteristics

Divorce Travel Income Age Rent

C

(3) .59 .07 10.10 .16 5.65

C

(5) .11 .04 9.4 .12 2.83

1y Trend 16404 41.79 36125 37.61 4.38

3y Trend 17410 41.64 53003 37.50 5.12

5y Trend 16718 41.90 32370 37.93 3.40

1y Forecast 16552 42.01 35240 37.77 4.13

3y Forecast 17775 41.82 56229 37.63 5.02

5y Forecast 16954 42.15 29173 38.18 2.85

1y T.p. 149 .22 -885 .15 -.25

3y T.p. 365 .18 3226 .13 -.10

5y T.p. 236 .25 -3197 .25 -.55

Table 2: Compatibility measures for seven series (in percentages), as well as trends, forecasts and

t.p.s for 2006.

17
Frequency

0.0 0.2 0.4 0.6 0.8 1.0

−0.5 0.0 0.5 1.0

Figure 1: Gain Function for the Trend Filter

Frequency

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Figure 2: Phase Delay Function for the Trend Filter

18
Frequency

0.0 0.2 0.4 0.6 0.8 1.0

−0.1 0.0 0.1 0.2 0.3

Figure 3: Gain Function for the Turning Point Filter

Frequency

0.0 0.2 0.4 0.6 0.8 1.0

0 2 4 6 8

Figure 4: Phase Function for the Turning Point Filter

19
Frequency

0.0 0.2 0.4 0.6 0.8 1.0

−0.5 0.0 0.5 1.0 1.5

Figure 5: Gain Function for the Forecast Filter

Frequency

0.0 0.2 0.4 0.6 0.8 1.0

−1 0 1 2 3

Figure 6: Phase Delay Function for the Forecast Filter

20