**Changes to Econometric Demand Equations for**

**Market Dominant Products since R2006-1**

Prepared for Postal Regulatory Commission

**[Changes to Econometric Demand Equations since R2006-1]{.underline}**

This document summarizes changes made to the Postal Service's
econometric demand equations for market-dominant products since R2006-1.
Section I describes changes to the underlying econometric methodology
used to estimate the demand equations. Section II identifies some
general changes to the explanatory variables used in the econometric
demand equations. Section III provides an overview of the econometric
equation output, which has been re-worked and expanded somewhat since
R2006-1. Finally, section IV identifies equation-specific changes to the
explanatory variables used to estimate the demand for market-dominant
products.

I.  [Econometric Methodology]{.underline}

There have been five substantive changes to the basic econometric
methodology used to estimate demand equations in R2006-1. Three of these
five changes were in direct response to the Rebuttal Testimony of Dr.
Harry Kelejian on behalf of the Greeting Card Association (GCA-T-5).
These five changes are described briefly below.

A.  [Simultaneous Estimation of Non-Linear Parameters]{.underline}

In some cases, the demand for mail is affected by some variables in a
non-linear way. The one example of this in R2006-1 was the use of
non-linear "Box-Cox" transformations of Internet variables in some
equations.

For these cases, the Postal Service's usual econometric demand equation
specification is not amenable to estimation via Ordinary or Generalized
Least Squares. In these cases, then, the demand equations are estimated
via Non-Linear Least Squares. Non-linear least squares is a technique
which involves taking a linear approximation of the non-linear
specification, solving the linear approximation using a Generalized
Least Squares technique, and iterating this procedure until the
parameter estimates converge. This technique was described in the
Rebuttal Testimony of Thomas Thress in R2006-1 (USPS-RT-2) at pages 51
-- 53.

B.  [Instrumental Variable Estimation]{.underline}

Shifts between First-Class single-piece and workshared mail due to
changes in price are modeled through the inclusion of the average
First-Class worksharing discount in the demand equations for both
single-piece and workshared mail (both letters and cards). Because the
total volume leaving First-Class single-piece mail due solely to changes
in worksharing discounts should be exactly equal to the volume entering
First-Class workshared mail by definition (ceteris paribus), the average
worksharing discount enters the First-Class workshared (letters and
cards) equations as follows:

Ln(V~ws~) = a - β~sp~·\[Ln(d~ws~) / (V~ws~/V~sp~)\] + \... (Equation 1)

where d~WS~ is the average worksharing discount, V~WS~ is the volume of
First-Class workshared mail, and V~SP~ is the volume of First-Class
single-piece mail.

In the above equation, the volume of workshared mail (V~ws~) is
expressed, in part, as a function of the volume of workshared mail. This
creates serious econometric difficulties (and makes it virtually
impossible to forecast the volume of workshared mail).

To solve this econometric problem, the \[Ln(d~ws~) / (V~ws~/V~sp~)\]
term (call it D') in the above equation is replaced with an Instrumental
variable which is not a function of V~ws~. The coefficient on the
Instrumental variable (β~sp~ in the above equation) is then estimated
with the other parameters of the equation in the usual way, using the
Generalized Least Squares framework underlying the Postal Service's
econometric demand equations. This procedure -- replacing the
problematic variable with an Instrumental variable and estimating the
equation which includes the Instrumental variable using a least squares
procedure -- is called two-stage least squares (2SLS).

The Instrumental variable that is used to replace D' in the above
equation is a fitted value for D' from a regression in which D' is the
dependent variable, i.e.,

\[Ln(d~ws~) / (V~ws~/V~sp~)\] = a + b·X~I~ + e (Equation 2)

To ensure that the Instrumental variable is orthogonal to regression
residuals from the volume equation (i.e., the instrumental variable
should be uncorrelated with the regression residuals), the X~I~ matrix
of explanatory variables in the Instrumental equation includes all of
the other explanatory variables from Equation 1 above. Equation 2 may
also include additional explanatory variables as appropriate. For
example, for First-Class workshared letters, the X~I~ matrix in Equation
2 includes the natural logarithm of the average First-Class worksharing
discount - Ln(d~ws~), a time trend (t), and a time trend squared (t^2^).

The \[Ln(d~ws~) / (V~ws~/V~sp~)\] term in Equation 1 is then replaced by
a fitted value for this term from Equation 2. That is,

Ln(V~ws~) = a - β~sp~·\[a + b·X~I~\] + \... (Equation 1~I~)

There is one further minor complication with respect to Instrumental
Variable estimation. For statistical consistency, the
variance-covariance matrix associated with Equation 1~I~
\[σ^2^(X'X)^-1^\] should be calculated based on residuals using the
[actual]{.underline} value of the variable -- D' -- instead of the
Instrumental variable. That is, σ^2^=(e'e)/n, where e is the regression
residual, Ln(V~ws~) - X~WS~·b~WS~, where X~WS~ is the matrix of
explanatory variables used in Equation 1~I~, including, of course, the
Instrumental variable, \[a + b·X~I~\].

For purposes of calculating the variance-covariance matrix, the
Instrumental variable, \[a + b·X~I~\], should be replaced by the
original variable, \[Ln(d~ws~) / (V~ws~/V~sp~)\], in calculating the
residuals used to estimate σ^2^. The X matrix used in calculating the
variance-covariance matrix includes the Instrumental variable, however.

For presentational purposes, the regression residuals presented as part
of the Postal Service's econometric output, as well as regression
diagnostics which rely upon these residuals (e.g., mean-squared errors)
are the residuals which result directly from Equation 1~I~, i.e., these
residuals are calculated using the [Instrumental]{.underline} variable.

This procedure was implemented at the suggestion of Dr. Harry Kelejian
in GCA-T-5.

C.  [Estimation of Autocorrelation Coefficients]{.underline}

At the suggestion of Dr. Kelejian, the autocorrelation coefficients,
when judged necessary, are estimated simultaneously with the other
econometric paramters within a non-linear least squares framework.

D.  [Tests and Corrections for Heteroskedasticity]{.underline}

The restriction on the OLS estimator that var(ε~t~) = σ^2^ implies that
the variance of the disturbance is the same for all the observations.
If, instead, the variance of the disturbance term differs across
observations, then the residual is said to exhibit heteroskedasticity.
Heteroskedasticity could arise in various situations. For example, if
the variance is a function of some independent variable x, such that
var(ε~t~) = σ^2^x~t~^2^, there could be a problem of
heteroskedasticity. In general, this type of heteroskedasticity is
usually a problem when dealing with cross-sectional data, however.

In a time series setting, such as the demand for Postal volumes over
time, another kind of heteroskedasticity called Auto-Regressive
Conditional Heteroskedasticity (ARCH) is more commonly found. As its
name suggests, ARCH means the errors\' conditional variance depends on
its own previous values, e.g., var(ε~t~\| ε~t-1~)=a~0~+a~1~ε~t-1~^2^.
In this example, the period t variance changes over time because it
depends on the realization of period t-1 error.

In the presence of heteroskedasticity, the OLS estimator is unbiased,
but is not efficient, so that the usual inference procedures could be
inappropriate. If one is only interested in the point estimate of
coefficient β~i~, an OLS estimator that is not adjusted for
heteroskedasticity could be good enough. However, if one is also
interested in constructing appropriate confidence intervals and doing
hypotheses tests, then such an estimator would need to be adjusted to
correct for the presence of heteroskedasticity. In the presence of ARCH,
the problem is similar. An OLS estimator is unbiased but a more
efficient estimator could be obtained by correcting for ARCH, which
could provide better inference.

ARCH is tested and corrected by a feasible GLS procedure proposed by
Jack Johnston and John DiNardo ([Econometric Methods]{.underline}, 4th
edition. McGraw-Hill. 1997). A (possibly non-linear) GLS regression, as
outlined above (with outside restrictions and autocorrelation
corrections[^1]), is initially run. The residuals from this regression
are then inspected to assess the presence of ARCH in two steps.

First, the squared residuals are regressed on its squared residuals
lagged one, two, and four quarters. The squared residuals are used as a
proxy of the variance of the residuals. The regression equation for this
is:

> e~t~^2^ = a~0~ + a~1~e~t-1~^2^ + a~2~e~t-2~^2^ + a~4~e~t-4~^2^
> (Equation 5.8)

After this regression is run, the significance of each of a~1~, a~2~,
and a~4~ are tested using a simple t-test. This is exactly parallel to
the technique used to establish the appropriate autocorrelation
technique used in this and previous rate cases.

ARCH corrections are then made using a two-step iterative procedure
whereby the residuals are adjusted for ARCH as estimated via Equation
5.8 and the other parameters are then estimated via Generalized Least
Squares taking the ARCH correction as given. This process is repeated
until both the ARCH estimates as well as the other parameter estimates
converge. This is mathematically equivalent to estimating all of these
parameters simultaneously.

E.  [Non-Linear Intervention Analysis]{.underline}

Intervention analysis is a time series technique which allows one to
identify the effects of an event over time. An "intervention" is an
event which affects the demand for a given product. There are
essentially three different types of impact of intervention events: step
functions, pulse functions, and trends. A generalized Intervention
Analysis technique allows for a functional form which is flexible enough
to accommodate all of these possibilities as dictated by the underlying
data. This function is called the *transfer function.*

The role of the transfer function is to allow the input variable to
affect the volume in different ways and rates over time. Therefore, the
impact of an intervention over the volume is the product of a particular
transfer function and an input variable. The general form of the
transfer function is given by:

![](media/image1.wmf) **(1)**

where *B* is the lag operator: ![](media/image2.wmf). For the stability
of the model, the roots of the equations ![](media/image3.wmf) and
![](media/image4.wmf) must lie outside the unit circle. Of course, a
more generalized form of equation (1) is necessary to limit the number
of ω and δ parameters so that the equation can be uniquely estimated.

The ω(B) terms represent the level impact of the intervention event. For
example, in equation (1), if ω~i~=0, for i\>0, then the intervention
will only affect volume in the current period, and equation (1) will
simplify to a simple dummy variable equal to one in the quarter of
interest and zero elsewhere with coefficient ω~0~. If, on the other
hand, ω~i~ = ω~j~, for all i,j, with δ~i~ = 0 for all i, then equation
(1) simplifies to a simple dummy variable equal to one from the quarter
of interest forward with coefficient ω~0~ (=ω~i~ for all i). Finally, if
ω~i~ is an increasing (or decreasing) function of i, then the transfer
equation identified above will posit a trend response to the
intervention event of interest.

The δ(B) terms represent the rate of increase or decrease of the
intervention events, e.g., the rate of change from a short-run to a
long-run impact. For simplicity, δ~i~ is typically assumed to be
constant across all i. That is, the rate of adoption of an intervention
event is typically assumed to be constant over time.

A simple transfer function that allows for each of the three
possibilities outlined above - pulse, step, or trend response to an
intervention -- is shown in equation (2) below:

I~t~ = {ω~0~ + ω~1~B / (1 -- δB) + (ω~2~ + ω~3~t)B / (1 -- B)}P~t~ (2)

where P~t~ is a pulse function -- i.e., P~t~ = 1 for the period of the
intervention, zero elsewhere.

A step function (=1 for the period of the intervention and all
subsequent periods), S~t~, can be expressed as a function of Pt using
lag notation so that S~t~ = P~t~ / (1-B).

In equation (2), ω~0~ is equal to the initial response to the
Intervention event. If ω~1~=ω~2~=ω~3~=0, then the response to the
Intervention will be equal to zero in all subsequent periods, and the
transfer function will be a pure pulse function (P~t~). If ω~0~=ω~1~ and
δ=ω~2~=ω~3~=0, then the transfer function will be a pure step function
(S~t~ = P~t~ / (1-B)). If ω~1~=ω~2~=0 and ω~0~= ω~3~, then the transfer
function will be a pure linear trend. If, on the other hand, none of
these equalities are realized, then equation (2) will explain a more
flexible transfer function as dictated by the observed data.

The functional form of equation (2), which expresses the transfer
function as a function of the lag operators may not be intuitively
obvious. Re-expressing the lag operator notation here into more
conventional notation yields equation (3):

> I~t~ = ω~0~·P~t~ + ω~1~·(P~t-1~+δ^1^P~t-2~+δ^2^P~t-3~+...) +
> ω~2~·S~t~ + ω~3~·T~t~·S~t~ (3)

where, as noted above, P~t~ is equal to one during the period of the
intervention, zero elsewhere (both before and after), S~t~ is equal to
zero prior to the intervention event being modeled, and equal to one
thereafter, and T is a time trend equal to zero at the point of the
intervention event, increasing by one each quarter thereafter.

While equation (3) is a function of only 5 parameters -- δ and ω~i~ for
i = 0 to 3 -- it nonetheless technically requires the inclusion of an
infinite number of terms in the demand equation of interest. It turns
out, however, that, at any given point in time, all of the P~t-i~ terms
is equal to zero except for, at most, one. To see this, one can re-write
equation (3) as follows:

> I~t~ = ω~0~·P~t~ + ω~1~·Σ~i=1~^∞^(δ^i-1^P~t-i~) + ω~2~·S~t~ +
> ω~3~·T~t~·S~t~

When T~t~ = 1, the value of P~t-1~ = 1, P~t-i~ = 0, for all i≠1.
Similarly, when T~t~ = 2, the value of P~t-2~ = 1, P~t-i~ = 0, for all
i≠2. So, instead of a sum over all values of P~t-i~ one can instead
replace i with T~t-1~ in the above equation. That is,

> I~t~ = ω~0~·P~t~ + ω~1~·S~t~·(δ^T^t-1) + ω~2~·S~t~ + ω~3~·T~t~·S~t~
> (4)

Intervention variables of the form in equation (4) are then added to the
Postal Service's econometric demand equations as necessary. The
Intervention parameters - ω~0~, ω~1~, ω~2~, ω~3~, and δ -- are estimated
simultaneous with the other econometric parameters using non-linear
least squares.

II. [Explanatory Variables]{.underline}

    A.  [Income Variables]{.underline}

The impact of general macro-economic conditions on mail volume was
re-investigated for all of the Postal Service's econometric demand
equations. As part of this investigation, several economic variables
were investigated as candidate variables that were not used in any of
the R2006-1 demand equations. These included personal disposable income,
consumption expenditures, gross domestic product, and the difference
between actual and potential GDP, also called the Output Gap.

B.  [Seasonality]{.underline}

The basic approach of the Postal Service to seasonality in the demand
for mail volumes is the same as it was in R2006-1: a series of seasonal
variables which are tied to specific periods of the Gregorian calendar
as well as quarterly dummy variables for the time period for which
Postal volume data are expressed by Gregorian quarter.

The specific seasonal variables used now differ in two respects from the
seasonal variables used in R2006-1. First, the number of seasonal
variables which are tied to specific periods of the Gregorian calendar
has been reduced to 13. This was done by treating December 11 -- 17 as a
single time period as well as December 18 -- 24. The 13 relevant
variables correspond to the following periods of the Gregorian calendar.
The variable names used in the econometric output associated with these
variables are shown in parentheses:

> September 1 -- 15 (SEP1_15)
>
> September 16 -- 30 (SEP16_30)
>
> October (OCT)
>
> November 1 - December 10 (NOV1_DEC10)
>
> December 11 -- 17 (DEC11_17)
>
> December 18 --24 (DEC18_24)
>
> December 25 -- 31 (DEC25_JAN1)[^2]
>
> January -- February (JAN2_FEB)
>
> March (MARCH)
>
> April 1 -- 15 (APR1_15)
>
> April 16 - May 31 (APR16_MAY)
>
> June (JUNE)
>
> July -- August (JUL_AUG)

Twelve of the 13 seasonal variables are included in each econometric
equation. The excluded seasonal variable is the variable covering the
period from July 1 through August 31, the effect of which is captured
implicitly within the constant term. The coefficients on the 12 included
seasonal variables are estimated along with the other econometric
parameters.

In an effort to maximize the explanatory power of the seasonal
variables, taking into account the cost of including these variables, in
terms of degrees of freedom, the adjoining seasons for which the
coefficients are similar in sign and magnitude are combined in some
cases.[^3] These constraints across seasons are made on an
equation-by-equation basis. The criterion used for this constraining
process is generally to minimize the mean-squared error of the equation
(which is equal to the sum of squared residuals divided by degrees of
freedom). For equations estimated over sufficiently short sample periods
(starting in 1989 or later), DEC18_24 and DEC25_JAN1 were perfectly
correlated (each variable spanned exactly one week of the second Postal
quarter), so that these two variables had to be combined in these
equations.

While Postal volume data are available by Gregorian quarter going back
to 2000, official RPW volume data by Gregorian quarter are only
available starting in 2004. The data for 2000 -- 2003 were reconstructed
after the fact. For some mail categories, there are noticeable
differences in the observed seasonal pattern of the 2000 -- 2003
Gregorian data and the Gregorian data since 2004. Because of this, two
sets of Gregorian seasonal dummies was constructed. The first set of
these variables are equal to zero for all Postal quarters and for three
of the four Gregorian quarters, and equal to one for Gregorian quarters
during the quarter of interest. The second set of quarterly dummies
exactly parallel the first set of quarterly dummies but are equal to
zero through 2003Q4.

By making the months of July and August the excluded variable, the dummy
variables for all four Gregorian quarters are thereby included in the
demand equations. This results in there being an implicit dummy variable
included in the Postal Service's demand equations equal to one
concurrent with the change from Postal to Gregorian quarters (between
1999 and 2000) reflected in the sum of the four Gregorian quarterly
dummies. Implicitly including such a dummy variable may be problematic,
however. To eliminate this potential difficulty, therefore, the sum of
the coefficients on the four Gregorian dummy variables is constrained to
be equal to zero.

For programming simplicity, this restriction is actually implemented
econometrically by freely estimating three variables:

> GQTR1_4 = (Quarter 1 Dummy since 2000) -- (Quarter 4 Dummy since 2000)
>
> GQTR2_4 = (Quarter 2 Dummy since 2000) -- (Quarter 4 Dummy since 2000)
>
> GQTR3_4 = (Quarter 3 Dummy since 2000) -- (Quarter 4 Dummy since 2000)

The coefficients on these three variables are then equal to the
coefficients on the Quarter 1, 2, and 3 dummies, respectively, while the
coefficient on the Quarter 4 dummy is equal to the negative of the sum
of these three coefficients.

The issue mentioned above about the potential for introducing an
implicit dummy variable would hold true for this second set of dummy
variables as well. Hence, a similar restriction that the coefficients on
the four quarterly dummies should sum to zero was imposed here, so that
a second set of three seasonal variables that parallel GQTR1_4, GQTR2_4,
and GQTR3_4 were constructed:

> GQTR04_1_4 = (Quarter 1 Dummy since 2004) -- (Quarter 4 Dummy since
> 2004)
>
> GQTR04_2_4 = (Quarter 2 Dummy since 2004) -- (Quarter 4 Dummy since
> 2004)
>
> GQTR04_3_4 = (Quarter 3 Dummy since 2004) -- (Quarter 4 Dummy since
> 2004)

Both sets of quarterly dummy variables -- those since 2000 as well as
those since 2004 -- were investigated in all of the Postal Service's
demand equations. In those cases where the coefficients on the GQTR04\*
variables were meaningful (not necessarily significant in a strict
statistical sense), these variables were included in the final demand
specification.

C.  [Price Lags]{.underline}

Price variables which enter the Postal Service's econometric demand
equations with lags were re-optimized, so that some lags of prices may
have been added or subtracted for some demand equations.

III. [Econometric Output]{.underline}

Results from the Postal Service's domestic econometric demand equations
are presented over 6 -- 8 pages of output. This report describes the
results presented there and their usefulness in analyzing and
understanding these demand equations.

The econometric output for the First-Class workshared letters demand
equation is used as an example here. In some cases, additional features
of the output are not applicable to the First-Class workshared letters
equation. In these cases, other examples are shown below.

Econometric demand equation output covers six to eight pages, depending
on the specific equation.

[Page 1]{.underline}

The first page of the econometric output presents the most important
results from the econometric equations: the elasticities that are used
to make volume forecasts. There are as many as five tables on Page 1,
depending on the exact specification of the demand equation being
estimated.

The first table has three lines. The first line identifies the mail
volume for which the demand equation is being estimated. The second line
is written by the person who estimated the regression to identify the
specific equation (this line needs to be updated manually and so may be
obsolete or uninformative if the person who estimated the regression did
not make it a point to change it). The final line identifies the sample
period.

> Mail Category: Workshared First-Class Letters
>
> Internet Competition
>
> Sample Period: 1991:1 TO 2007:4

The second table lists the estimated coefficients, standard errors, and
t-ratios for all of the non-seasonal variables included in the demand
equation.

> Non-Seasonal Variables
>
> Coefficients Std. Error T-Ratio
>
> CONSTANT -0.058353 0.114635 -0.509034
>
> EMPLOY 0.423903 0.075609 5.606498
>
> BROADBAND(-4)\*D02Q4 -0.441460 0.030913 -14.280539
>
> D3R_NCR_L -0.055414 0.033143 -1.671981
>
> INSTR_FIT 0.133596 0.003350 39.880656
>
> PX1_3WS -0.115828 0.032698 -3.542407
>
> lag 1 -0.076624 0.020431 -3.750357
>
> lag 2 -0.040265 0.015017 -2.681194
>
> lag 3 -0.012988 0.010111 -1.284474
>
> lag 4 -0.000000 0.000000 -0.041501

The variables in this list are generally listed in the following order:
(1) non-seasonal, non-price variables which enter the equation linearly,
(2) any variables that are adjusted non-linearly (see below), (3)
instrumental variables (see below), and, finally, (4) prices.

A number in parentheses -- e.g., BROADBAND(-4) -- indicates that the
variable is lagged (if the number is negative; a positive number would
indicate a leading variable). If a mathematical operation is indicated
-- e.g., BROADBAND(-4)\*T02Q4 -- then the explanatory variable in the
equation is the result of performing this operation (i.e., in this case
the Broadband variable is multiplied by a time trend starting in 2002Q4;
that product is then used as an explanatory variable in the First-Class
workshared letters equation).

The variable name INSTR_FIT refers to the fitted value of the
Instrumental variable, if one is estimated. This is described in more
detail below.

The terms "lag 1", "lag 2", etc. refer to lags of the variable named
immediately above. In this case, these are lags of PX1_3WS, which is the
price index for First-Class workshared letters.

As a general rule, Postal econometricians are not overly concerned with
strict statistical significance as a necessary condition for including
explanatory variables in demand equations. The principal determinant of
whether to include a variable is the reasonableness of its coefficient.
Reasonableness is primarily determined by having the correct sign,
although the magnitude of the coefficient is also considered.
Statistical significance, while not an absolute criterion, is still
nevertheless taken into consideration, particularly when comparing
competing candidate variables.

The Postal Service's demand equation methodology allows for the
possibility that the relationship between mail volume and one or more
explanatory variables may be non-linear. In such cases, the relationship
between mail volume and the variable of interest is modeled as follows:

Ln(Volume) = a + b·X^λ^ + ...

The character, λ, is the Greek letter Lambda. Lambda is estimated
simultaneous with all of the other econometric parameters. The estimated
value for Lambda, along with its standard error and t-ratio are
presented in the next table. The X variable is identified in the left
column.

> Non-Linear Lambda Coefficients
>
> Lambda Std Error T-Ratio
>
> CS_ISP 0.133675 0.047312 2.825368

The example above is from the First-Class single-piece letters equation.
The variable CS_ISP (consumption expenditures on Internet Service
Providers) enters the equation non-linearly.

The coefficient on the transformed variable (i.e., 'b' in the above
equation) is shown in the second table, with all of the other
non-seasonal variables. If a mathematical operation is identified on
this variable, it is performed on the transformed variable, X^λ^.

For example, the following variables are included in the First-Class
single-piece letters equation:

> Non-Seasonal Variables
>
> Coefficients Std. Error T-Ratio
>
> ...
>
> CS_ISP 0.647279 0.068759 9.413799
>
> CS_ISP\*TREND -0.009266 0.000467 -19.823930
>
> CS_ISP\*T02Q4 -0.006647 0.000986 -6.742352
>
> ...

For all three of these variables, the term "CS_ISP" is understood to
mean CS_ISP^λ^. So, in this case, the First-Class single-piece letters
equation takes the following form:

Ln(Volume) = a + b~1~·CS_ISP^λ^ + b~2~·CS_ISP^λ^·TREND +
b~3~·CS_ISP^λ^·T02Q4 + ...

or, re-combining things in a way that I think makes the results a little
bit more logical:

Ln(Volume) = a + (b~1~+b~2~·TREND+b~3~·T02Q4)·CS_ISP^λ^ + ...

In some cases (currently First-Class workshared letters and cards), a
potential explanatory variable is not included directly in the demand
equation. Instead, an "Instrumental Variable" is included instead.

For those equations where an Instrumental variable is used, a table like
this one will be printed here.

> Instrumental Variable Estimation
>
> Instrumental Variable D1_3WS_RAT
>
> \'Instrument\' Std Error T-Ratio
>
> TREND 0.123854 0.040274 3.075312
>
> TSQ -31.720370 18.892787 -1.678967
>
> D1_3WS 0.514492 0.392674 1.310226
>
> note: All other explanatory variables also used as instruments

The variable which is being replaced with the Instrumental variable is
identified on the second line. In this case, it is the variable
D1_3WS_RAT, which is the average First-Class worksharing discount
(D1_3WS) divided by the ratio of First-Class workshared and single-piece
letters volumes.

This variable (D1_3WS_RAT in this case) is the dependent variable in a
regression which includes, as explanatory variables, all of the other
explanatory variables from the mail category demand equation (i.e., the
First-Class workshared letters equation, in this case) as well as the
other variables shown in the table above. So, in this case, the
Instrumental equation includes three additional variables: TREND (a
linear time trend), TSQ (a linear time trend squared), and D1_3WS. The
coefficient, standard error, and t-ratio values shown here are for these
variables in the equation with the Instrumental variable as the
dependent variable.

The variable used in the final demand equation instead of this variable
is the fitted value from this equation. That variable is identified as
INSTR_FIT in the earlier table which showed all of the non-seasonal
regression coefficients for the First-Class workshared letters equation.

The final table on Page 1 shows results for any variables, typically
prices, which are included with multiple lags and constrained by a
Shiller smoothing technique.

The "SUM" is the sum of the current and lagged price elasticities.
Typically, when the Postal Service refers to "price elasticities" they
mean these long-run price elasticities. As its name suggests, the
T-Statistic on the Sum is the long-run price elasticity divided by its
standard error.

> Long-Run Price Elasticities
>
> PX1_3WS
>
> Current -0.115828
>
> Lag 1 -0.076624
>
> Lag 2 -0.040265
>
> Lag 3 -0.012988
>
> Lag 4 -0.000000
>
> Sum -0.245705
>
> T-Statistic on Sum -3.855345

If an equation includes more than one price variable for which Shiller
smoothing techniques are applied, this table will also show the sum of
[all]{.underline} of these price elasticities (for which Shiller
smoothing is applied) along with a t-statistic for this sum. An example
of this is shown below for Bound Printed Matter, which includes the
prices of Media Mail (PX29) and Bound Printed Matter (PX28).

> Long-Run Price Elasticities
>
> PX29 PX28 Sum of Prices
>
> Current 0.198554 -0.178779
>
> Lag 1 0.120700 -0.176815
>
> Lag 2 0.049597 -0.167978
>
> Lag 3 0.000000 -0.142282
>
> Lag 4 0.000000 -0.085163
>
> Sum 0.368850 -0.751017 -0.382167
>
> T-Statistic on Sum 2.123576 -4.426350 -2.310048

[Page 2 (if needed)]{.underline}

Page 2 presents the results of any Intervention Analysis that may be
included in the demand equation.

As explained above, Intervention Analysis is a means of modeling
non-linear effects of specific events. In the case of First-Class
workshared letters, there is one non-linear Intervention currently
modeled. This Intervention begins in 1996Q4, coinciding with the
implementation of Classification Reform, MC95-1.

A non-linear Intervention is modeled as affecting volume in the
following way:

Ln(Volume) ~t~ = a + ... + ω~0~·P~t~ +
ω~1~·(P~t~+δP~t-1~+δ^2^P~t-2~+δ^3^P~t-3~+...) + ω~2~·S~t~ + ω~3~t·S~t~ +
...

where P~t~ = 1 if t=1996Q4 and 0 otherwise; S~t~ = 1 if t \>=1996Q4 and
0 otherwise; and t is a linear time trend equal to zero in 1996Q4,
increasing by one each quarter thereafter. This equation has an initial
value in 1996Q4 of ω~0~, a value in 1997Q1 of ω~1,~ and decays toward a
long-run trend of the form (ω~2~ + ω~3~·t).

The first table on Page 2 presents coefficients, standard errors, and
t-ratios for the Intervention parameters: ω~0~, ω~1~, ω~2~, ω~3~, and δ.
Any parameters that are not listed in the table are constrained to have
a value of zero. In this case, ω~2~ and ω~3~ are constrained to have
values of zero. This means that the MC95-1 Intervention variable in the
First-Class workshared letters equation is modeled as decaying toward a
long-run value of zero.

Intervention Analysis

Coefficient Std Error T-Ratio

W0: Initial Pulse -0.061259 0.018843 -3.251025

W1: Lag Pulse -0.071288 0.011091 -6.427389

Delta 0.992912 0.011745 84.53745

The value of the Intervention variable for each quarter of the
regression sample period for which the Intervention is in effect is
shown in the next table. The quarter for which the first value is shown
in this table is the quarter in which the Intervention is assumed to
have occurred.

Intervention Value

Quarter 1 Quarter 2 Quarter 3 Quarter 4

1996 -0.061259

1997 -0.071288 -0.070783 -0.070281 -0.069783

1998 -0.069289 -0.068798 -0.068310 -0.067826

1999 -0.067345 -0.066868 -0.066394 -0.065923

2000 -0.065456 -0.064992 -0.064531 -0.064074

2001 -0.063620 -0.063169 -0.062721 -0.062277

2002 -0.061835 -0.061397 -0.060962 -0.060530

2003 -0.060101 -0.059675 -0.059252 -0.058832

2004 -0.058415 -0.058001 -0.057590 -0.057182

2005 -0.056777 -0.056374 -0.055975 -0.055578

2006 -0.055184 -0.054793 -0.054405 -0.054019

2007 -0.053636 -0.053256 -0.052878 -0.052504

Note that the Intervention value in the quarter in which the
Intervention occurs -- 1996Q4 in this case -- is equal to ω~0~, and the
value of the Intervention variable in the next quarter -- 1997Q1 -- is
equal to ω~1~.

The rate at which the Intervention variable decays toward its long-run
value (zero in this case) is determined by δ. A value of δ equal to one
would indicate no decay at all. As δ approaches zero, the Intervention
variable will approach its long-run value more quickly. In this case, δ
has a value of 0.993, so that the Intervention variable decays very
slowly toward its long-run value.

[Page 3]{.underline}

Page 3 shows coefficients, standard errors, and t-ratios for the
seasonal variables included in the demand specification. The treatment
of seasonality in the Postal Service's demand equations was described
above.

Seasonal Variables

Coefficients Std. Error T-Ratio

SEPT 0.012185 0.128709 0.094670

OCT -0.156187 0.139589 -1.118907

NOV1_DEC10 0.326361 0.108668 3.003289

DEC11_17 0.598524 0.205157 2.917399

DEC18_31 -0.906448 0.518609 -1.747844

JAN_MAR 0.314904 0.117059 2.690123

APR1_15 -0.603024 0.360068 -1.674753

APR16_MAY 0.066929 0.098252 0.681198

GQTR1_4 0.113466 0.082093 1.382153

GQTR2_4 -0.212646 0.100576 -2.114284

GQTR3_4 0.091894 0.033761 2.721877

The seasonal coefficients are combined into a single "Seasonal Index"
which shows how the seasonal pattern of the mail has changed over time.

SEASONAL INDEX

Quarter 1 Quarter 2 Quarter 3 Quarter 4

1991 0.128436 0.125674 0.010621 0.002074

1992 0.119938 0.129845 0.017606 0.002656

1993 0.102943 0.138187 0.021735 0.002554

1994 0.101439 0.142358 0.025180 0.003143

1995 0.096886 0.138524 0.028624 0.003731

1996 0.092332 0.138524 0.032068 0.004909

1997 0.080786 0.138861 0.035512 0.005628

1998 0.076232 0.139029 0.038956 0.005497

1999 0.071679 0.143200 0.033095 0.006086

2000 0.119939 0.102258 0.023114 0.011242

2001 0.115616 0.102258 0.031815 0.011134

2002 0.113505 0.102258 0.023984 0.011084

2003 0.113505 0.102258 0.023984 0.011242

2004 0.113505 0.102258 0.023114 0.011242

2005 0.119939 0.102258 0.023114 0.011242

2006 0.119939 0.102258 0.023114 0.011242

2007 0.115616 0.102258 0.031815 0.011134

[Page 4]{.underline}

Page 4 presents several key regression diagnostics. The primary
regression diagnostic used by the Postal Service to evaluate econometric
demand equations, particularly to compare alternate specifications, is
mean-squared error.

Mean Square Error is calculated as the sum of squared residuals divided
by degrees of freedom. Unadjusted Mean Square Error is calculated based
on residuals without AR or ARCH corrections (see below). The Full Sample
Unadjusted Mean Square Error is calculated as the sum of the unadjusted
residuals (see below) squared divided by degrees of freedom. Unadjusted
Mean Square Errors calculated over recent years, on the other hand, are
calculated as the sum of the unadjusted residuals squared (over the
relevant time period) divided by the number of observations within that
time period (e.g., 20 for the MSE over the last 5 years).

REGRESSION DIAGNOSTICS

Sum of Sq Resids 0.004318

Mean Square Error 0.000111

Unadjusted Mean Square Error

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

Full Sample 0.000126

Last 5 Yrs 0.000099

Last 4 Yrs 0.000116

Last 3 Yrs 0.000120

Last 2 Yrs 0.000107

Last 1 Yr 0.000022

Durbin-Watson 1.839727

R-Square 0.994060

Adj. R-Square 0.989796

Degrees of Freedom 39

The Shapiro-Francia test statistic is a simple correlation between the
final residuals from the equation and a standard normal distribution.
The null hypothesis of the Shapiro-Francia test is that the residuals
are normally distributed; hence, a P-Value greater than 0.05 means that
one cannot reject the null hypothesis so that the residuals are not not
normally distributed.

SHAPIRO-FRANCIA TEST FOR NORMALITY OF RESIDUALS

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

Test Statistic P-Value

\-\-\-\-\-\-\-\-\-\-\-\-\-- \-\-\-\-\-\--

0.971 0.118

The Dickey-Fuller test results should be largely self-explanatory here.
A Test Statistic greater (in absolute value) than the Critical Value
indicates that one can reject the null hypothesis and state that the
variable does [not]{.underline} contain a unit root, which indicates
that the variable in question is stationary.

AUGMENTED DICKEY FULLER TEST FOR STATIONARITY

\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\-\--

Test Statistic Critical Value

Dependent Variable -11.474 -3.417

Residuals -7.476 -1.927

Null hypothesis: Variable contains a unit root

For dependent variable, model is AR(4) with constant and time trend

For residuals, model is AR(4) with no constant, no trend

Critical values are calculated for a 5% significance level from \'Lag
Order

by Yin-Wong Cheung and Kon S. Lai, Journal of Business & Economic
Statisti

Mechanical net trends are the trends which would be necessary to have
achieved a perfect forecast of the four quarters identified in the top
row, starting from the base identified in the left-most column.

Mechanical Net Trends

Forecast Period

4 Yrs Ago 3 Yrs Ago 2 Yrs Ago 1 Yr Ago Last 4 Qtrs

Base 5 Yrs Ago 0.992066 0.992726 1.001660 0.999531 0.999368

Base 4 Yrs Ago 0.993387 1.006491 1.002032 1.001202

Base 3 Yrs Ago 1.019768 1.006383 1.003821

Base 2 Yrs Ago 0.993173 0.995941

Base 1 Yr Ago 0.998716

The final figure presented on Page 4 is the Shiller k^2^ value (or
values) used in estimation. The smaller the k^2^ value, the less
restrictive the Shiller restriction.

SHILLER K-SQUARED VALUES

PX1_3WS 0.401563

[Page 5]{.underline}

Page 5 presents any autocorrelation (AR) and heteroskedasticity (ARCH)
corrections that are made and evaluates the autocorrelation structure of
the final regression residuals. AR and ARCH corrections were described
above.

The first table on Page 5, if any AR-corrections are made, presents the
AR-coefficients, along with standard errors, and t-ratios. These
coefficients are estimated simultaneous with the coefficients presented
earlier.

AR-Coefficients

Coefficient Std. Error T-Ratio

Rho-1 0.000000 0.000000 0.000000

Rho-2 0.000000 0.000000 0.000000

Rho-4 -0.437416 0.141968 -3.081084

If any ARCH corrections are made, the next table presents ARCH
coefficients, standard errors, and t-ratios. The First-Class workshared
letters equation does not include any ARCH corrections. The table below
is taken from the Certified Mail equation.

ARCH-Coefficients

Coefficient Std. Error T-Ratio

Alpha-1 0.207291 0.081259 2.550995

Finally, the last table on page 5 presents full and partial
autocorrelations of the final regression residuals, along with standard
errors and t-ratios for the partial autocorrelations. The table shown
here is from the First-Class workshared letters equation.

AUTOCORRELATION STRUCTURE OF FINAL RESIDUALS

Auto- Partial Auto- Standard T-Stat on

Lag Correlation Correlation Error Partial

1 0.049953 0.062104 0.125988 0.492934

2 -0.263385 -0.247068 0.127000 -1.945415

3 -0.226659 -0.185816 0.128037 -1.451270

4 0.187550 0.170211 0.129099 1.318445

5 0.188409 0.090820 0.130189 0.697604

6 0.102919 0.141143 0.131306 1.074916

7 -0.351871 -0.245670 0.132453 -1.854765

8 -0.243391 -0.151263 0.133631 -1.131946

9 0.033922 -0.002413 0.134840 -0.017894

[Page 6]{.underline}

Page 6 presents a Recursive Residuals analysis. Recursive residuals are
calculated by estimating the demand equation ending the sample period as
early as possible and then calculating one-quarter-ahead forecast
errors. This process is continued for all possible quarters. The results
are then normalized, so that the results shown here can be interpreted
as something akin to t-statistics.

Recursive residuals greater than two in absolute value may indicate
periods where something has changed in the demand for the particular
mail category.

Recursive Residuals

(normalized: (Ln(Actual) - Ln(Forecast)) / SE

Quarter 1 Quarter 2 Quarter 3 Quarter 4

1996 1.122

1997 -0.131 -0.064 -0.766 0.357

1998 0.236 -0.891 -1.475 0.365

1999 0.665 0.144 -0.293 -0.316

2000 0.637 -0.725 0.713 0.328

2001 0.508 -0.121 -0.071 0.420

2002 0.976 -0.094 -0.699 -1.285

2003 1.095 -1.629 0.052 1.100

2004 -0.664 0.208 -0.169 0.636

2005 1.695 0.989 0.141 -0.176

2006 -0.625 0.998 1.205 -0.803

2007 -0.168 0.372 -0.807 0.848

The second table on page 6 shows the cumulative sum of the Recursive
residuals (CUSUM). This is the sum of the data from the above table from
the first period estimated (1996Q4 in this case) through the current
period. The significance level for these CUSUM statistics is much
greater than for the individual Recursive residuals and increases over
time. The 95% significance critical value for the last period of the
regression period is shown at the top of the table.

Cumulative Sum of One-Quarter-Ahead Residuals

95% confidence range: 2007:4 19.078

Quarter 1 Quarter 2 Quarter 3 Quarter 4

1996 1.122

1997 0.991 0.927 0.161 0.517

1998 0.753 -0.138 -1.613 -1.248

1999 -0.583 -0.438 -0.731 -1.047

2000 -0.410 -1.136 -0.422 -0.094

2001 0.414 0.292 0.221 0.641

2002 1.617 1.522 0.824 -0.461

2003 0.634 -0.995 -0.943 0.157

2004 -0.508 -0.299 -0.468 0.168

2005 1.863 2.852 2.993 2.817

2006 2.192 3.190 4.394 3.591

2007 3.423 3.795 2.988 3.836

[Page 7 (if needed)]{.underline}

Regression residuals are shown next. If the equation includes any AR or
ARCH corrections, unadjusted residuals are shown on Page 7. Residuals
take the following functional form:

Residual = Ln(Volume) -- Ln(Fitted Volume)

REGRESSION RESIDUALS (w/o AR- and ARCH-Corrections)

Quarter 1 Quarter 2 Quarter 3 Quarter 4

1991 -0.023693 0.012004 -0.033602 -0.022988

1992 0.039590 -0.012315 0.039155 0.019362

1993 -0.016545 -0.009641 -0.019940 0.009176

1994 -0.019424 -0.008933 0.011180 -0.005067

1995 0.021060 0.015922 0.017659 -0.008373

1996 -0.015674 -0.011182 -0.007388 -0.001023

1997 -0.000173 0.014896 0.010876 0.002995

1998 -0.006402 -0.016490 -0.022649 0.000540

1999 0.005671 0.000935 -0.009906 -0.010680

2000 -0.006102 0.007065 0.017320 -0.003699

2001 -0.000573 -0.000968 0.004614 0.006275

2002 0.008974 0.006785 -0.000610 -0.003531

2003 0.002526 -0.026978 -0.005496 0.011554

2004 -0.017929 -0.013153 -0.015684 0.000731

2005 0.024148 0.009659 -0.000213 -0.002324

2006 -0.008862 0.014766 0.014783 -0.016378

2007 -0.001470 0.004840 -0.013327 0.009459

[Page 8]{.underline}

Final regression residuals, incorporating all adjustments including AR-
and ARCH-corrections are shown at the top of the final page of
econometric output. As with the unadjusted residuals, these residuals
take the following functional form:

Residual = Ln(Volume) -- Ln(Fitted Volume)

REGRESSION RESIDUALS (Final)

Quarter 1 Quarter 2 Quarter 3 Quarter 4

1991

1992 0.029227 -0.007065 0.024457 0.009307

1993 0.000772 -0.015028 -0.002813 0.017645

1994 -0.026662 -0.013150 0.002458 -0.001053

1995 0.012563 0.012014 0.022549 -0.010589

1996 -0.006462 -0.004217 0.000336 -0.004686

1997 -0.007029 0.010005 0.007645 0.002547

1998 -0.006478 -0.009974 -0.017891 0.001850

1999 0.002871 -0.006278 -0.019812 -0.010444

2000 -0.003621 0.007474 0.012987 -0.008371

2001 -0.003242 0.002123 0.012190 0.004657

2002 0.008723 0.006362 0.001408 -0.000786

2003 0.006451 -0.024010 -0.005763 0.010010

2004 -0.016824 -0.024953 -0.018088 0.005785

2005 0.016306 0.003906 -0.007073 -0.002005

2006 0.001701 0.018991 0.014690 -0.017394

2007 -0.005346 0.011299 -0.006861 0.002296

The final table in the Econometric Output decomposes the contribution of
the various econometric inputs on changes in mail volume over the last
five years.

CONTRIBUTIONS TO CHANGE IN

Workshared First-Class Letters

VOLUME OVER LAST FIVE YEARS

Volume 5 Years Ago 47658.076

Effect of

Percent Change Variable on

Variable In Variable Elasticity Volume

Own-Price 1.65% -0.246 -0.40%

EMPLOY -0.45% 0.424 -0.19%

BROADBAND(-4)\*D02Q4 23.67% -0.441 -8.95%

D3R_NCR_L 5.75% -0.055 -0.31%

INSTR_FIT 88.79% 0.134 8.86%

Adult Population 6.31%

Interventions Starting in:

1996Q4 0.81%

Seasonality -0.06%

Other Factors -0.32%

Mechanical Net Trend 0.999368

Base Volume 49978.441

Total Change in Volume 4.87%

IV. [Equation-Specific Changes]{.underline}

This section lists all of the specific changes made to the explanatory
variables included in each of the Postal Service's econometric demand
equations for market-dominant mail.

[First-Class Single-Piece Letters]{.underline}

> ♦ Removed dummy variable equal to one starting in 1993Q1 (MSADJ)
>
> ♦ Replaced dummy variable for classification reform (MC95) with
> non-linear Intervention variable starting in 1996Q4
>
> ♦ Replaced dummy variable equal to one in 2004Q1 and 2005Q1 with a set
> of quarterly dummies equal to zero through 2003
>
> ♦ Eliminated stochastic constraint on average First-Class worksharing
> discount (D1_3WS); elasticity is freely estimated in the most recent
> equation
>
> ♦ Added variable R2006PHOP, which is equal to -1 in 2006Q1 and +1 in
> 2006Q2.

[First-Class Workshared Letters]{.underline}

> ♦ Replaced Retail Sales (STR) with Employment (EMPLOY)
>
> ♦ Removed dummy variable equal to one starting in 1993Q1 (MSADJ)
>
> ♦ Replaced dummy variable for classification reform (MC95) with
> non-linear Intervention variable starting in 1996Q4
>
> ♦ Removed number of Broadband subscribers not interacted with dummy
> variable starting in 2002Q4

[First-Class Single-Piece Cards]{.underline}

> ♦ Replaced dummy variable for classification reform (MC95) with
> non-linear Intervention variable starting in 1996Q4
>
> ♦ Added variable interacting ISP Consumption (CS_ISP) with a time
> trend starting in 2002Q4 (T02Q4)
>
> ♦ Added dummy variable equal to one since 2006Q1 (D2006ON)
>
> ♦ Added variable R2006PHOP, which is equal to -1 in 2006Q1 and +1 in
> 2006Q2.

[First-Class Workshared Cards]{.underline}

> ♦ Changed starting period for regression from 1997Q1 to 1998Q1

[Standard Regular Mail]{.underline}

> ♦ Added 2002Q1 dummy variable lagged one quarter -- D2002Q1(-1)
>
> ♦ Added dummy variable equal to one since 2006Q1 (D2006ON)
>
> ♦ Added dummy variable equal to one with the implementation of R2006-1
> rates (D_R06)

[Standard ECR Mail]{.underline}

> ♦ Removed price of newspaper advertising (WP_NWS)
>
> ♦ Removed Internet Advertising Expenditures (I_ADV_S)
>
> ♦ Added dummy variable for October of off-year Federal elections since
> 2000 (EL_OFF_OCT00)
>
> ♦ Added dummy variable equal to one in third quarter of Federal
> election years (D_EL3)
>
> ♦ Replaced dummy variable for R97-1 (D_R97) with non-linear
> Intervention variable starting in 1999Q2
>
> ♦ Added dummy variable equal to one with the implementation of R2006-1
> rates (D_R06)

[Standard Nonprofit Mail]{.underline}

> ♦ Replaced Retail Sales (STR) with Employment (EMPLOY)
>
> ♦ Removed the number of Broadband subscribers
>
> ♦ Removed dummy variable for a 1994 rule change restricting nonprofit
> eligibility (RULE94)
>
> ♦ Removed dummy variable for October of off-year Federal elections
> since 2000 (EL_OFF_OCT00)
>
> ♦ Added non-linear Intervention variable starting in 1997Q1
> (implementation of MC96-1)
>
> ♦ Added dummy variable equal to one with the implementation of R2006-1
> rates (D_R06)

[Standard Nonprofit ECR Mail]{.underline}

> ♦ Removed Retail Sales (STR)
>
> ♦ Removed consumption expenditures on Internet Service Providers
> (CS_ISP)
>
> ♦ Combined dummy variables for September and October of off-year
> Federal elections (EL_OFF_SEP, EL_OFF_OCT) into a single variable
> (EL_OFF_FALL)
>
> ♦ Removed dummy variable for October of off-year Federal elections
> since 2000 (EL_OFF_OCT00)
>
> ♦ Added non-linear Intervention variable starting in 1997Q1
> (implementation of MC96-1)
>
> ♦ Added dummy variable equal to one with the implementation of R2006-1
> rates (D_R06)

[Retail Parcel Post (Non-Destination-Entry Parcel Post)]{.underline}

> ♦ Added Parcel Select Return Service (PSRS) to the volume used as the
> dependent variable in this regression
>
> ♦ Added a linear time trend
>
> ♦ Removed dummy variable equal to one in 1997Q4 (D97Q4)
>
> ♦ Added dummy variable equal to one starting in 2004Q1 (D2004)
>
> ♦ Added variable R2006PHOP, which is equal to -1 in 2006Q1 and +1 in
> 2006Q2.
>
> ♦ Added dummy variable equal to one with the implementation of R2006-1
> rates (D_R06)

[Bound Printed Matter]{.underline}

> ♦ Replaced Media and Library mail price index with Media Mail price
> index and replaced fixed restriction on cross-price elasticity with
> stochastic restriction
>
> ♦ Replaced dummy variable for cancellation of Sears catalog (SEARS)
> with non-linear Intervention variable starting in 1993Q2
>
> ♦ Replaced dummy variable equal to one starting in 1998Q1 (D98Q1) with
> non-linear Intervention variable starting in 1998Q1

[Media and Library Rate Mail]{.underline}

> ♦ Replaced mail-order retail sales (STR_MO) with mail-order retail
> sales lagged two quarters
>
> ♦ Replaced dummy variable equal to one starting in 1998Q1 (D98Q1) with
> non-linear Intervention variable starting in 1998Q1
>
> ♦ Replaced dummy variable equal to one starting in 2001Q1 (DVOL29)
> with non-linear Intervention variable starting in 2001Q1

[Periodical Regular Rate Mail]{.underline}

> ♦ Replaced employment lagged three quarters with retail sales lagged
> two

[Periodical Within-County Mail]{.underline}

> ♦ Replaced Employment with real GDP (GDPR)
>
> ♦ Replaced Time Trend starting in 1993Q1 (T93) with a Time Trend
> starting in 1997Q1 (T97)
>
> ♦ Removed dummy variable equal to one starting in 1987Q1 (D87)
>
> ♦ Removed dummy variable equal to one starting in 1993 (PANELCHG)

[Periodical Nonprofit and Classroom Mail]{.underline}

> No changes

[Postal Penalty Mail]{.underline}

> ♦ Added dummy variable equal to one since 2006Q1 (D2006ON)

[Free-for-the-Blind Mail]{.underline}

> ♦ Changed starting period for regression from 1971Q1 to 1996Q1
>
> ♦ Removed consumption expenditures on Internet Service Providers
> (CS_ISP)
>
> ♦ Replaced linear time trend (TREND) with logistic time trend --
> \@LOG(TREND-100)
>
> ♦ Removed dummy variable equal to one in 2000 and 2001 (D2000_01)
>
> ♦ Removed dummy variable equal to one in 2003Q1 - 3
>
> ♦ Added dummy variable equal to one in 2001Q1, zero elsewhere
> (D2001Q1)
>
> ♦ Added dummy variable equal to one in 2001Q2, zero elsewhere
> (D2001Q2)

[Registered Mail]{.underline}

> No changes

[Insured Mail]{.underline}

> ♦ Replaced Parcel Post volume with the sum of Priority Mail, Retail
> Parcel Post, and Media Mail volumes
>
> ♦ Replaced two trend variables (LTMC96_3, T03Q4) and two dummy
> variables (D97Q4, MC96_3) with a non-linear Intervention variable
> starting in 1997Q4

[Certified Mail]{.underline}

> ♦ Replaced dummy variable for the introduction of Delivery
> Confirmation (D_DC) with a non-linear Intervention variable starting
> in 1999Q2
>
> ♦ Replaced dummy variables for the first three quarters of 2002with a
> non-linear Intervention variable starting in 2002Q1

[COD Mail]{.underline}

> ♦ Removed Parcel Post volume
>
> ♦ Replaced dummy variables for 2002Q1 (D2002Q1) and thereafter
> (D2002Q2ON) with a non-linear Intervention variable starting in 2002Q1
>
> ♦ Added variable R2006PHOP, which is equal to -1 in 2006Q1 and +1 in
> 2006Q2.

[Money Orders]{.underline}

> ♦ Replaced Employment with Retail Sales lagged three quarters
>
> ♦ Removed Time Trend starting in 2002Q4 (T02Q4)

[Return Receipts]{.underline}

> ♦ Removed dummy variables for 2002Q1 (D2002Q1) and thereafter
> (D2002Q2ON)

[Delivery and Signature Confirmation]{.underline}

> ♦ Added Parcel Select (destination-entry Parcel Post) volume
>
> ♦ Added dummy variable equal to one in 2002Q1, zero elsewhere
>
> ♦ Replaced dummy variable for implementation of R2001-1 (D_R01) with a
> non-linear Intervention variable starting in 2002Q4
>
> ♦ Added dummy variable equal to one with the implementation of R2006-1
> rates (D_R06)

[Stamped Cards]{.underline}

> ♦ Changed starting period for regression from 2000Q1 to 2001Q2
>
> ♦ Removed dummy variable equal to one starting in 2000Q2 (D_SC)
>
> ♦ Removed dummy variable equal to one in 2001Q4
>
> ♦ Removed dummy variable equal to one in 2003Q4
>
> ♦ Replaced dummy variable equal to one in 2002Q4 (D2002Q4) with a
> dummy variable equal to one in 2002Q4 and 2003Q1 (D2002Q4_03Q1)
>
> ♦ Added dummy variables equal to one in 2004Q3, 2005Q2, 2006Q3,
> 2006Q4, 2007Q1, and 2007Q3

[^1]: ARCH models generally assume zero correlation in the errors.

[^2]: January 1^st^ is a Postal holiday. Hence, the time period from
    December 25^th^ through December 31^st^ will be identical to the
    time period from December 25^th^ through January 1^st^ for the
    purposes of constructing these variables.

[^3]: Combined seasonals were given names which should be obvious in the
    econometric output. For example, when DEC11_17, DEC18_24, and
    DEC25_JAN1 were combined, the combined variable was called DEC11_31.