The Measurement-Error Model for Consumption

Monthly Interpolation and Adjustment of Consumption

For expositional purposes, we assume that the accurately measured low-frequency observa- tions are available at quarterly frequency (instead of annual frequency as in the main text). Correspondingly, we define the time subscript t = 3(j−1) +m, where month m = 1,2,3 and quarter j = 1, . . .. We use uppercase C to denote the level of consumption and low- ercase c to denote percentage deviations from some log-linearization point. Growth rates are approximated as log differences and we use a superscriptoto distinguish observed from “true” values.

The measurement-error model presented in the main text can be justified by assuming that the statistical agency uses a high-frequency proxy series to determine monthly con- sumption growth rates. We use Z3(j−1)+m to denote the monthly value of the proxy series and Z₍q_j) the quarterly aggregate. Suppose the proxy variable provides a noisy measure of monthly consumption. More specifically, we consider a multiplicative error model of the form

Z_3(j−1)+m =C3(j−1)+mexp(3(j−1)+m). (3.59)

The interpolation is executed in two steps. In the first step we construct a series ˜

Co

3(j−1)+m, and in the second step we rescale the series to ensure that the reported monthly consumption data add up to the reported quarterly consumption data within the period. In Step 1, we start from the level of consumption in quarter j−1,C₍q_j−1), and define

˜ C₃₍o_j−1)+1 = C₍q,o_j−1) Z3(j−1)+1 Z₍q_j−1) ! (3.60) ˜ C₃₍o_j−1)+2 = C₍q,o_j−1) Z3(j−1)+1 Z₍q_j−1) ! _Z 3(j−1)+2 Z3(j−1)+1 =C₍q,o_j−1) Z3(j−1)+2 Z₍q_j−1) ! ˜ C₃₍o_j−1)+3 = C₍q,o_j−1) Z3(j−1)+1 Z₍q_j−1) ! _Z 3(j−1)+2 Z3(j−1)+1 Z3(j−1)+3 Z3(j−1)+2 =C₍q,o_j−1) Z3(j−1)+3 Z₍q_j−1) ! .

Thus, the growth rates of the proxy series are used to generate monthly consumption data for quarter q. Summing over the quarter yields

˜ C₍q,o_j) = 3 X m=1 ˜ C₃₍o_j−1)+m=C₍q,o_j−1) " Z3(j−1)+1 Z₍q_j−1) + Z3(j−1)+2 Z₍q_j−1) + Z3(j−1)+3 Z₍q_j−1) # =C₍q,o_j−1) Z q (j) Z₍q_j−1)(3.61).

In Step 2, we adjust the monthly estimates ˜C₃₍o_j−1)+m by the factor C₍q,o_j)/C˜₍q,o_j), which leads to C₃₍o_j−1)+1 = C˜₃₍o_j−1)+1 Cq,o (j) ˜ C₍q,o_j) =C₍q,o_j)Z3(j−1)+1 Z₍q_j) (3.62) C₃₍o_j−1)+2 = C˜₃₍o_j−1)+2 Cq,o (j) ˜ C₍q,o_j) =C₍q,o_j)Z3(j−1)+2 Z₍q_j) C₃₍o_j−1)+3 = C˜₃₍o_j−1)+3 Cq,o (j) ˜ C₍q,o_j) =C₍q,o_j)Z3(j−1)+3 Z₍q_j)

and guarantees that

C₍q,o_j) = 3 X

m=1

C₃₍o_j−1)+m.

We now define the growth rates go_c,t = logC_to−logC_to−1 and gc,t = logCt−logCt−1. By taking logarithmic transformations of (3.59) and (3.62) and combining the resulting equations, we can deduce that the growth rates for the second and third month of quarter

q are given by

g_c,o_3(j−1)+2 = gc,3(j−1)+2+3(j−1)+2−3(j−1)+1 (3.63)

g_c,o_3(j−1)+3 = g_c,3(j−1)+3+3(j−1)+3−3(j−1)+2.

The derivation of the growth rate between the third month of quarter j−1 and the first month of quarter j is a bit more cumbersome. Using (3.62), we can write the growth rate as

go_c,3(j−1)+1 = logC₍q,o_j)+ logZ3(j−1)+1−logZ₍q_j) (3.64)

−logC₍q,o_j−1)−logZ_3(j−2)+3+ logZ q (j−1).

To simplify (3.64) further, we are using a log-linear approximation. Suppose we log- linearize an equation of the form

X₍q_j) =X_3(j−1)+1+X3(j−1)+2+X3(j−1)+3

aroundX∗q andX∗=X∗q/3, using lowercase variables to denote percentage deviations from

the log-linearization point. Then,

xq_(j)≈ 1

3(x3(j−1)+1+x3(j−1)+2+x3(j−1)+3).

Using (3.59) and the definition of quarterly variables as sums of monthly variables, we can apply the log-linearization as follows:

logC₍q,o_j) −logZ₍q_j)= log(C_∗q/Z_∗q) +q_(j)−1

3 3(j−1)+1+3(j−1)+2+3(j−1)+3

. (3.65)

Substituting (3.65) into (3.64) yields

g_c,o_3(j−1)+1 = gc,3(j−1)+1+3(j−1)+1−3(j−2)+3+q_(j)−q_(j−1) (3.66) −1 3 3(j−1)+1+3(j−1)+2+3(j−1)+3 +1 3 3(j−2)+1+3(j−2)+2+3(j−2)+3 .

Chapter 4

Improving GDP Measurement: A

Measurement-Error Perspective

4.1

Introduction

Aggregate real output is surely the most fundamental and important concept in macroe- conomic theory. Surprisingly, however, significant uncertainty still surrounds its measure- ment. In the U.S., in particular, two often-divergent GDP estimates exist, a widely-used expenditure-side version,GDPE, and a much less widely-used income-side version,GDPI.1 Nalewaik (2010) and Fixler and Nalewaik (2009) make clear that, at the very least, GDPI deserves serious attention and may even have properties in certain respects superior to those of GDPE.2 That is, if forced to choose between GDPE and GDPI, a surprisingly strong case exists forGDPI. But of course one isnot forced to choose betweenGDPE andGDPI, and a GDP estimate based onboth GDPE andGDPI may be superior to either one alone. In this paper we propose and implement a framework for obtaining such a blended estimate. Our work is related to, and complements, Aruoba, Diebold, Nalewaik, Schorfheide, and Song (2012). There we took a forecast-error perspective, whereas here we take a

1_{Indeed we will focus on the U.S. because it is a key egregious example of unreconciledGDP}

EandGDPI estimates.

2

For additional informative background on GDPE,GDPI, the statistical discrepancy, and the national accounts more generally, see of Economic Analysis (2006), McCulla and Smith (2007), Landefeld, Seskin, and Fraumeni (2008), and Rassier (2012).

measurement-error perspective.3 In particular, we work with a dynamic factor model in the tradition of Geweke (1977) and Sargent and Sims (1977), as used and extended by Watson and Engle (1983), Edwards and Howrey (1991), Harding and Scutella (1996), Jacobs and van Norden (2011), Kishor and Koenig (2011), and Fleischman and Roberts (2011), among others.4 That is, we view “true GDP” as a latent variable on which we have several indicators, the two most obvious being GDPE and GDPI, and we then extract true GDP using optimal filtering techniques.

The measurement-error approach is time honored, intrinsically compelling, and very dif- ferent from the forecast-combination perspective of Aruoba, Diebold, Nalewaik, Schorfheide, and Song (2012), for several reasons.5 First, it enables extraction of latent trueGDP using a model with parametersestimated with exact likelihood or Bayesian methods, whereas the forecast-combination approach forces one to use calibrated parameters. Second, it deliv- ers not only point extractions of latent true GDP but also interval extractions, enabling us to assess the associated uncertainty. Third, the state-space framework in which the measurement-error models are embedded facilitates exploration of the relationship between

GDP measurement errors and the economic environment, such as stage of the business cycle, which is of special interest. Fourth, the state-space framework facilitates real-time analysis and forecasting, despite the fact that preliminary GDPI data are not available as quickly as those for GDPE.

We proceed as follows. In section 4.2 we consider several measurement-error models and assess their identification status, which turns out to be challenging and interesting in 3_{Hence the pair of papers roughly parallels the well-known literature on “forecast error” and “measure-}

ment error” properties of of data revisions; see Mankiw, Runkle, and Shapiro (1984), Mankiw and Shapiro (1986), Faust, Rogers, and Wright (2005), and Aruoba (2008).

4

See also Smith, Weale, and Satchell (1998), who take a different but related approach, and the indepen- dent work of Greenaway-McGrevy (2011), who takes a closely-related approach but unfortunately estimates a model that we will show to be unidentified.

5

the most realistic and hence compelling case. In section 4.3 we discuss the data, estimation framework and estimation results. In section 4.4 we explore the properties of our newGDP

series. We conclude in section 4.5.