Macroeconomic Forecasting Using
Diffusion Indexes
James H. Stock
Kennedy School of Government, Harvard University, and National Bureau of Economic Research,
Cambridge, MA 02138
Mark W. Wa tson
Woodrow Wilson School, Princeton Un iversity, Princeton, NJ 08544, and National Bureau of
Economic Research
This article studies forecasting a macroecono mic time series variable using a large number of predictors.
The predictors are summarized using a small number of indexes constructed by principal component
analysis. An approximate dynamic factor model serves as the statistical framework for the estimation of
the indexes and construction of the forecasts. The method is used to construct 6-, 12-, and 24-month-
ahead forecasts for eight monthly U.S. macroeconomic time series using 215 predictors in simulated real
time from 1970 through 1998. During this sample period these new forecasts outperformed univariate
autoregressions, small vector autoregressions, and leading indicator models.
KEY WORDS: Factor model; Forecasting; Principal components.
1. INTRODUCTION
Recent advances in information technology make it possi-
ble to access in real time, at a reasonable cost, thousands of
economic time series for major developed economies. This
raises the prospect of a new frontier in macroeconomic fore-
casting, in which a very large n umber of time series are used
to forecast a few key economic quantities, such as aggregate
production or in ation. Time series models currently used for
macroeconomic forecasting, however, incorporate only a few
series: vector autoregressions, for example, typically contain
fewer than 10 varia bles. Although variable selection proce-
dures can be used to choose a small subset of predictors from
a large set of potentially useful variables, the performance of
these methods ultimately rests on the few variables that are
chosen. For example, real economic activity is often used to
predict in ation (the so-called Philips curve), but is the unem-
ployment rate, the rate of capacity utilization, or the Gross
Domestic Product gap the best measure of rea l activity for
this purpose? An alternative to selecting a few predictors is to
pool the information in all the candidate predictors, averaging
away idiosyncratic variation in the individual series. In this
paper, we use an approximate factor model for this purpose.
The p remise i s that for forecasting purposes, the information
in the l arge number of predictors can b e replaced by a handful
of estimated factors.
This idea has a long tradition in ma croeconomics. For
example, the notion of a common business cycle underlies the
classic work of Burns and Mitchell (1947) and the indexes
of leading and coincident indicators originally developed at
the National Bureau of Economic Research (NBER). This
notion was formally modeled by Sarg ent and Sims (1977)
in their dynamic generalization of the classic factor analy-
sis model. Versions of their model have been used by several
researchers to study dynamic covariation among sets of vari-
ables (Geweke 1977; Singleton 1980; Engle and Watson 1981;
Stock and Watson 1989, 1991; Quah and Sargent 1 993; Forni
and Reichlin 1996, 199 8). Modern dynamic general equilib-
rium macroeconomic models often postulate th at a small set
of driving variables is responsible for variation in macro time
series, and these variables can be viewed as a set of common
factors. Although the previous empirical research focused on
estimating indexes of covariation, this paper uses the estimated
factors for prediction.
The approximate dynamic factor model, which relates the
variable to be forecast,
y
t
C
1
, to a set of predictors collected in
the vector
X
t
, is presented in Section 2. Forecasting is carried
out in a two-step process: rst the factors are estimated (by
principal compon ents) using
X
t
, then these estimated factors
are used to forecast
y
t
C
1
. Focusing o n the forecasts implied
by the factors rather than on the factors themselves permits
sidestepping the dif cult problem of identi cation (or rotation)
inherent in factor models. One interpretation of the estimated
factors is in terms of d iffusion indexes developed by NBER
business cycle analysts to measure common movement in a
set of macroeconomic variables, and accordingly we call the
estimated factors diffusion indexes.
The performance of the diffusion index (D I) forecasts is
examined i n Sections 3 and 4. The experiment reported in
these sections simulates real-time forecasting during the 1970–
1998 period of eight U.S. macroeconomic variables, four mea-
sures each of real economic activity and of price in ation.
The D I forecasts are constructed at horizons of 6, 12, and 24
months using as many as 215 predictor seri es. These forecasts
are compared t o several conventional benchmarks: univari-
ate autogressions, small vector autoregressions, leading indi-
cator models, and, for in ation, unemployment-based Phillips
curve models. Generally speaking, the diffusion index fore-
casts based on a sma ll number of factors (in most cases, one
or two) are found to perform wel l, with relative performance
improving as the horizon increases. The improvement over the
benchmark forecasts can be dramatic, in several cases produc-
© 2002 American Statistical Association
Journal of Business & Economic Statistics
April 2002, Vol. 20, No. 2
147
148 Journal of Business & E conomic Statistics, April 2002
ing simulated out-of-sample mean square forecast errors that
are one-third less than those of the benchmark models.
2. ECONOMETRIC FRAMEWORK
2.1 An Approximate Dynamic Factor Model
We begin with a discussion of the s tatistical model that
motivates the D I forecasts. Let
y
t
C
1
denote the scalar series
to be forecast and let,
X
t
be an
N
-dimensional multiple time
series of predictor variables, observed for
t
D
1
1 : : : 1 T
, where
y
t
and
X
t
are both taken to have mean 0. (The different time
subscripts used for
y
and
X
emphasize the forecasting rela-
tionship.) We suppose that
4X
t
1 y
t
C
1
5
admit a dynamic factor
model representation with
N
r
common dynamic factors
f
t
,
y
t
C
1
D
‚4L5f
t
C
ƒ4L5y
t
C
…
t
C
1
1
(2.1)
X
it
D
‹
i
4L5f
t
C
e
it
1
(2.2)
for
i
D
1
1 : : : 1 N
, where
e
t
D
4e
1
t
1 : : : 1 e
N t
5
0
is the
N
1
idiosyncratic disturbance and
‹
i
4L5
and
‚4L5
are lag polyno-
mials in nonnegative powers of
L
. It is assumed that
E4…
t
C
1
—
f
t
1 y
t
1 X
t
1 f
t
ƒ
1
1 y
t
ƒ
1
1 X
t
ƒ
1
1 : : : 5
D
0. Thus, if
8f
t
91 ‚4L5
, and
ƒ4L5
were kn own, th e minimum mean square error forecast
of
y
T
C
1
would be
‚4L5f
T
C
ƒ4L5y
T
.
We make two important modi cations to (2.1) and (2.2).
First, the lag polynomials
‹
i
4L5
,
‚4L5
, and
ƒ4L5
are modeled
as having nite orders of at most
q
, so
‹
i
4L5
D
P
q
j
D
0
‹
ij
L
j
and
‚4L5
D
P
q
j
D
0
‚
j
L
j
. The nite lag assumption permits rewriting
(2.1) and (2.2) as
y
t
C
1
D
‚
0
F
t
C
ƒ4L5y
t
C
…
t
C
1
1
(2.3)
X
t
D
åF
t
C
e
t
1
(2.4)
where
F
t
D
4f
0
t
1 : : : 1 f
0
t
ƒ
q
5
0
is
r
1, where
r µ 4q
C
1
5
N
r
, the
i
th
row of
å
in (2.4) is
4‹
i
0
1 : : : 1 ‹
iq
5
, and
‚
D
4‚
0
1 : : : 1 ‚
q
5
0
. The
main advantage of this static representation of the dynamic
factor model is tha t the factors can be estimated using prin-
cipal comp onents. This comes at a cost, because the assump-
tion is inconsistent with in nite distributed lags of t he factors.
Whether this cost is large is ultimately an empirical question,
addressed here by studying whether (2.3) and (2.4) can be
used to produce accurate forecasts.
Second, our emp irical ap plication focuses on
h
-step-ahead
forecasts. At least two approaches to multistep forecasting are
possible. One is to develop a vector time series model for
F
t
, to estimate this u sing the estimated factors, and to roll
the
4y
t
1 F
t
5
model forward, but this entails estimating a large
number of parameters that could erode fo recast performance.
Another approach is to recognize that the ensuing multistep
forecasts woul d be linear
F
t
and
y
t
(and lags) an d to use an
h
-step-ahead projection to construct the forecasts directly. We
adopt the latter approach, and the resulting multistep ahead
version of (2.3) is
y
h
t
C
h
D
h
C
‚
h
4L5F
t
C
ƒ
h
4L5y
t
C
…
h
t
C
h
1
(2.5)
where
y
h
t
C
h
is the
h
-step-ahead variable to be forecast, the con-
stant term is introduced explicitly, and the subscripts re ect
the dependence of the projecti on on the horizon.
2.2 Estimation and Forecasting
Because
8F
t
91
h
1 ‚
h
4L5
, and
ƒ
h
4L5
are unknown, forecasts
of
y
T
C
h
based on (2.4) and (2.5) are constructed using a two-
step procedure. First , the sample data
8X
t
9
T
t
D
1
are used t o esti-
mate a time series of factors (the diffusion indexe s),
8
b
F
t
9
T
t
D
1
.
Second, the estimators
O
h
1
O
‚
h
4L5
and
O
ƒ
h
4L5
are obtained
by regressing
y
t
C
1
onto a constant,
b
F
t
and
y
t
(and lags). The
forecast of
y
h
T
C
h
is then formed as
O
h
C
O
‚
h
4L5
b
F
T
C O
ƒ
h
4L5y
T
.
Stock and Watson (1998) developed theoretical results for
this two-step procedure applied to (2.3) and (2.4). The factors
are estimated by principal components because these estima-
tors are rea dily calculated even for very large
N
and because
principal components can be generalized to handle data irreg -
ularities as di scussed later. Under a set of moment conditions
for
4…1 e1 F5
and an asymptotic rank condition on
å
, the feasi-
ble forecast is asymptotically rst-order ef cient in the sense
that its mean square forecast error (MSE) approaches the
MSE of the optimal infeasible forecast as
N 1 T
! ˆ
, where
N
D
O4T
5
for any
>
1. This result suggests that feasible
forecasts are likely to be nearly optimal when
N
and
T
are
large, regardless of the ratio of
N
to
T
. The assumptions by
Stock and Watson (1998) are similar to assumptions made in
the literature on approximate factor models (Chamberlain and
Rothschild 1983; Connor and Koraj czyk 19 86, 1988, 1993),
generalized to allow for serial correlation. A related d ynamic
generalization and estimation (but not forecasting) results were
discussed by Forni, Hallin, Lippi, and Reichlin (2000). Stock
and Watson (1998) also showed that the principal components
remain consistent when there is some time variation in
å
and
small amounts of data contamination, as long as the number
of predictors is very large,
N T
.
2.3 Data Irregularities and Computational Issues
In our dataset, some series contain missing observations or
are available over a diminished time span. Although our data
are all monthly, further complications would arise in applica-
tions in which mixed sampling frequencies are used, such as
monthly and quarterly. In these cases standard principal com-
ponents analysis does not apply. However, the expectation-
maximization (EM) algorithm can be used to estimate the fac-
tors by solving a suitable minimization problem iteratively.
Details are given in Appendix A.
Although the components of
X
t
typically will be distinct
time series,
X
t
could contain multiple lags of one or more
series. Because the estimated factors
F
t
could include lags of
the dynamic factors
f
t
, estimation of
F
t
might be enhanced
by augmenting a vector o f distinct time series with its lags.
This is referred to later as stacking
X
t
with its lags, in which
case the principal co mponents of the stacked da ta vector are
computed.
3. THE DATA AND FORECASTING
EXPERIMENTAL DESIGN
3.1 Forecasting Models and Data
The forecasting experiment simulates real-time forecast-
ing for eight major monthly macroeconomic variables for the
Stock and Watson: Macroeconomic Forecasting Using Diffusion Indexes 149
United States. The complete dataset spans 1959:1 to 1 998:12.
Four of these eight variables are the measures of real economic
activity used to construct the Index of Coincident Economic
Indicators maintained by the Conference Board (formerly by
the U.S. Department o f Commerce): total industrial production
(ip); real personal income less transfers (gmyxpq); real man-
ufacturing and trade sales (msmtq); and number of employ-
ees on nonagricultural payrolls (lpnag). (Additional details are
given in Appendix B, which lists series by the mnemonics
given here in parenthesis.) The remaining four series are price
indexes: the consumer price index (punew); the personal con-
sumption expenditure implicit price de ator (gmdc); the con-
sumer price index (CP I) less food and energy (puxx); and the
producer p rice index for nished goods (pwfsa). These series
and the predictor series were taken from the May 1999 release
of the DR I/McGraw–Hill Basic Economics database (formerly
Citibase). In general these series represent the fully revised
historical series available as of May 1999, and in this regard
the forecasting results will differ from results that would be
calculated using real-time data.
For each series, several forecasting models are compared at
the 6-, 12 -, and 24-month forecasting horizons: D I forecasts
based on estimated factors, a benchmark univariate autoregres-
sion, and benchmark multivariate models. For both the real
and the price series, one of the benchmark multivariate models
is a trivariat e vector autoregression, and a second is based on
leading economi c indicators. As a further comparison, in a-
tion forecasts are also computed using an unemployment-
based Phillips curve.
Our focus is on multistep-ahead prediction, and most of the
forecasting regressions are projections of an
h
-step-ahead vari-
able
y
h
t
C
h
onto
t
-dated predictors, sometimes including lagged
transformed values
y
t
of the variable of interest. The real vari-
ables are modeled as being I(1) in logarithms. Because all
four real variables are treated identically, consider indu strial
production, for which
y
h
t
C
h
D
4
1200
=h5
ln
4
IP
t
C
h
=
IP
t
5
and
y
t
D
1200 ln
4
IP
t
=
IP
t
ƒ
1
50
(3.1)
The price indexes are mo deled as being I(2) in logarithms.
The I(2) speci cation is consistent with standard Phillips curve
equations and is a good description of the series over much of
the sample period. However, I(1) speci cations also provide
adequate descriptions of the data, particularly in the early part
of the sample. Stock and Wat son (1 999) found little difference
in I(1) and I(2) factor model forecasts for these prices over the
sample period studied here, so for the sake of brevity we limit
our analysis to the I(2) speci cation. Accordingly, for the CPI
(and similarly for the other price series),
y
h
t
C
h
D
4
1200
=h5
ln
4
CP I
t
C
h
=
CP I
t
5
ƒ
1200 ln
4
CP I
t
=
CP I
t
ƒ
1
5
and
y
t
D
1200
ã
ln
4
CP I
t
=
CP I
t
ƒ
1
50
(3.2)
Diffusion Index Forecasts.
Following (2.5), the most gen-
eral DI forecasting function is
O
y
h
T
C
h
—
T
D O
h
C
m
X
j
D
1
O
‚
0
hj
b
F
T
ƒ
j
C
1
C
p
X
j
D
1
O
ƒ
hj
y
T
ƒ
j
C
1
1
(3.3)
where
b
F
t
is the vector of
k
estimated factors. Results for three
variants of (3.3) are reported. The rst, denoted in the tables
by D I-AR, Lag, includes lags of the factors and lags of
y
t
, with
k
and lag orders
m
and
p
estimated by Bayesian information
criterion (B IC), with 1
µ k µ
4, 1
µ m µ
3, and 0
µ p µ
6.
Thus the smallest c andidate model th at B IC can choose here
includes only a single contemporaneous factor and excludes
y
t
. The sec ond, denoted DI-AR, includes contemporaneous
b
F
t
,
that is,
m
D
1, and
k
and
p
are chosen by B IC with 1
µ
k µ
12 and 0
µ p µ
6. The third, denoted D I, includes only
contemporaneous
b
F
t
, so
p
D
0
1 m
D
1, and
k
is chosen by B IC,
1
µ k µ
12.
The full dataset used to estimate the factors contains 215
monthly time series for the United States from 1959:1 to
1998:12. The series were selected judgmentally to represent 14
main categories of macroeconomic time series: real output and
income; employment and hours; real retail, manufacturing, and
trade sales; consumption; housing starts an d sales; real inven-
tories and inventory-sales ratios; orders and un lled orders;
stock prices; exchange rates; interest rates; money and credit
quantity aggregates; price indexes; average hourly earnings;
and miscellaneous. The list of series is given in Appendix B
and is similar to lists we have used elsewhere (Stock and
Watson 1996, 1999). These series were taken from a some-
what longer list, from which we eliminated series with gross
problems, such as rede nitions. However, no further pruning
was performed.
The theory o utlined in Section 2 assumes that
X
t
is I(0),
so these 215 series were subjected to three preliminary steps:
possible transformation by taking logarithms, possible rst dif-
ferencing, and screening for outliers. The decision to take log-
arithms or to rst difference the series was made judgmentally
after preliminary data analysis, including inspection of the data
and unit root tests. In general, logarithms were taken for all
nonnegative series that were not already in rates or p ercentage
units. Most series were rst differenced. A code summarizing
these transformations is given for each series in Appendix B.
After these transformations, all series were further standard-
ized t o have sample mean zero and unit sample variance.
Finally, the transformed data were screened automatically for
outliers (generally taken to be coding errors or exceptional
events such as labor stri kes), and observations exceeding 10
times the interquartile range from the median were replaced
by missing val ues.
Using this transformed and screened dataset, three sets of
empirical factors were constructed. The rst was computed
using principal components from the subset of 149 variables
available for the full sample period (the balanced panel). The
second set of factors was computed using the nonbalanced
panel of all 215 series using the methods of Appendix A. The
third set of factors was computed by stacking the 149 variables
in the balanced panel with their rst lags, so the augmented
data vector has dimension 298. Em pirical factors were then
estimated by the principal components of the stacked data, as
discussed in Section 2 .
Autoregressive Forecast.
The autoregressive forecast is a
univariate forecast based on (3.3), where the terms involving
b
F
are excluded. The lag order
p
was selected recursively by
B IC with 0
µ p µ
6, where
p
D
0 indicates that
y
t
and its lags
are excluded.
150 Journal of Business & E conomic Statistics, April 2002
Vector Autoregressive Forecast.
The rst multivariate
benchmark model is a vector autoregression (VAR) with
p
lags each of three variables. One version of the VAR used
p
D
4 lags, and another version sel ected
p
recursively by B IC.
The xed-lag VARs performed somewhat better than the B IC
selected lag lengths (which often set
p
D
1), and we report
results for the xed lag speci cations in the results to follow.
The variables in the VAR are a measure of the monthly growth
in real activity, the change in monthly in ation, and the change
in the 9 0-day U.S. treasury bill rate. When used to forecast the
real series, the relevant real activity vari able was used and the
in ation measure was CP I in ation. For forecasting in ation,
the relevant price series was used and the real activity measure
was in dustrial production. Multistep forecasts were computed
by iterating the VAR forward. Th is contrasts to the autoregres-
sive forecasts, which were computed by
h
-step-ahead projec-
tion rather than iteration.
Multivariate Leading Indicator Forecasts.
The leading
indicator forecasts have the form
O
y
h
T
C
h
—
T
D
O
„
h
0
C
m
X
j
D
1
O
„
0
hi
W
T
ƒ
j
C
1
C
p
X
j
D
1
O
ƒ
hj
y
T
ƒ
j
C
1
1
(3.4)
where
W
t
is a vector of leading indicators that have been fea-
tured in the literature or in real-time forecasting applications
and
O
„
h
0
and so forth are ordinary least squares coef cient
estimates.
For the real variables,
W
t
consists of 11 leading indicators
that we used for real-time monthly forecasting in experimen-
tal leading and recession indicators (Stock and Watson 1989).
(The list use d here consists of the leading indicators used to
produce the XR I and the XR I-2, which are released monthly
and documented at the web site
http://www.nber.org.
) Five o f
these leading indicators are also used in the factor estima-
tion step in the diffusion index forecasts. These are average
weekly hours of production workers in manufacturing (lphrm),
the capacity utilization rate in manufacturing (ipxmca), hous-
ing starts (building permits) (hsbr), t he index of help-wanted
advertising in newspapers (lhel), and the interest rat e on
10-year U.S. treasury bonds (fygt10). The remaining six lead-
ing indicators are the interest rate spread between 3-month
U.S. treasury bills and 3-month commercial paper; the spread
between 10-year and 1-year U.S. treasury bonds; the num-
ber of people working part-time in nonagricultural industries
because o f slack work; real manufacturers’ un lled orders in
durable goods industries; a trade-weighted index o f nominal
exchange rates b etween the United States and the U.K., West
Germany, France, Italy, and Japan; and the National Associ-
ation of Purchasing Managers’ index of vendor performance
(the percent of companies reporting slower deliveries).
For the in ation forecasts, eight l eading indicators are used.
These variables were chosen because of their good individ-
ual performance in previous in ation forecasting exercises. In
particular these variables performed well in at least one of the
historical episodes considered by Staiger, Stock, and Watson
(1997) (also see Stock and Watson 1999). Seven of these vari-
ables are also used in the factor-estimation step in the diffu-
sion index forecasts: the total unemployment rate (lhur), re al
manufacturing and trade sales (msmtq), housing starts (hsbr),
new orders in durable goods industries (mdoq), the nominal
M1 money supply (fm1), the federal funds overnight interest
rate (fyff), and the interes t rate spread between 1-year U.S.
treasury bonds and the federal funds rate (sfygt1). The remain-
ing variable is the trade-weighted exchange rate listed in the
previous paragraph.
In all cases, the leading indicators were transformed so that
W
t
is I(0 ). This entailed taking logarithms of variables not
already in rates and differencing all variables except the inter-
est rate spreads, housing starts, the index of vendor perfor-
mance, and the help wanted index.
For each variable to be fo recast,
p
and
m
in (3.4) were
determined by recursive B IC with 1
µ m µ
4 and 0
µ p µ
6,
so 28 possible models were compared in each time period.
Phillips Curve Forecasts.
The unemployment-based
Phillips curve is considered by many to have been a reliable
method for forecasting in ation over this period (Gordon
1982; Congressional Budget Of ce 1996; Fuhrer 1995; Gor-
don 1997; Staiger et al. 1997; Tootel 1994). The Phillips
curve in ation forecasts considered here have the form (3.4),
where
W
t
consists of the unemployment rate (LHUR) and
m
ƒ
1 of its lags, the relative price of food and energy (current
and one lagged val ue only), and Gordon’s (1982) variable that
controls for the imposition and removal of the Nixon wage
and price controls. The wage and price control vari able is
introduced for forecasts made in 1971
2
7
C
h
, before which it
produces singular regressions. The lag lengths
m
and
p
were
chosen by recursive B IC, where 1
µ m µ
6 and 0
µ p µ
6.
3.2 Simulated Real-Time Experimental Design
Estimation and forecasting was conducted to simulate real-
time forecasting. This entailed fully recursive parameter esti-
mation, factor estimation, model selection, and so forth. The
rst simulated out of sample forecast was made in 1970:1.
To construct this forecast, the data were screened for outliers
and standardized, the parameters and factors were estimated,
and the models were selected, using only data available from
1959:1 through 1970:1. (The rst date for the regressions was
1960:1, and earlier observations were used for initial co ndi-
tions as needed.) Thus regressions (3.3) and (3.4) were ru n for
t
D
1960:1
1 : : : 1
1970
2
1
ƒ
h
, then the values of the regressors
at
t
D
1970
2
1 were used to forecast
y
h
1970
2
1
C
h
. All parameters,
factors, and so forth were then reestimated, in formation cri-
teria were reco mputed, and models were selected using data
from 1959:1 through 1970:2, and forecasts from these m odels
were th en computed for
y
h
1970
2
2
C
h
. The nal simulated out of
sample forecast was made in 1998
2
12
ƒ
h
for
y
h
1998
2
12
.
4. EMPIRICAL RESULTS
4.1 Forecasting Results
The results for the real variables are reported in detail in
Table 1 for 12-month-ahead forecasts, and summaries for 6-
and 24-month-ahead forecasts are reported in Table 2. Two
sets of statistics are reported. The rst is the MSE of the can-
didate forecasting model, computed relative to the MSE of the
univariate autoregressive forecast (so the autoregressive fore-
cast has a relative MSE of 1.00). For example, the simulated
Stock and Watson: Macroeconomic Forecasting Using Diffusion Indexes 151
Ta ble 1. Simulated Out-of-Sample Forecasting Results: Rea l Variables, 12-Month Horizon
Industrial production Personal income Mfg & trade sales Nonag. employment
Forecast
method Rel. MSE
O
Rel. MSE
O
Rel. MSE
O
Rel. MSE
O
Benchmark models
AR 1000 1000 1000 1000
LI 086 (027) 057 (013) 097 (021) .52 (.15) 082(025) 063 (017) 089 (023) .56 (.14)
VAR 097 (007) 075 (068) 098 (005) .68 (.34) 098(004) 073 (058) 1005 (009) .22 (.41)
Full dataset (N
D
215)
DI-AR, Lag 057 (027) 076 (013) 077 (014) .76 (.13) 048(025) 099 (015) 091 (013) .63 (.18)
DI-AR 063 (025) 071 (012) 086 (016) .61 (.12) 057(024) 084 (018) 099 (031) .51 (.20)
DI 052 (026) 088 (017) 086 (016) .61 (.12) 056(023) 094 (020) 092 (026) .55 (.20)
Balanced panel (N
D
149)
DI-AR, Lag 067 (025) 070 (013) 082 (015) .70 (.13) 056(023) 091 (016) 088 (014) .68 (.18)
DI-AR 067 (025) 070 (012) 092 (014) .57 (.12) 061(023) 080 (017) 088 (022) .58 (.17)
DI 059 (025) 081 (017) 092 (014) .57 (.12) 057(023) 091 (018) 084 (021) .62 (.16)
Stacked balance panel
DI-AR 065 (025) 070 (012) 093 (015) .56 (.12) 061(022) 089 (019) 1002 (030) .49 (.14)
DI 062 (025) 081 (018) 093 (015) .56 (.12) 066(021) 085 (020) 095 (024) .53 (.14)
Full dataset; m
D
1, p
D
BIC, k ’ xed
DI-AR, k
D
1 1006 (011) 027 (034) 1003 (008) .34 (.41) 098(006) 063 (046) 1001 (009) .49 (.24)
DI-AR, k
D
2 063 (025) 076 (014) 078 (014) .77 (.14) 053(024) 093 (015) 077 (013) .82 (.15)
DI-AR, k
D
3 056 (026) 084 (014) 077 (015) .77 (.13) 052(023) 099 (016) 084 (014) .75 (.20)
DI-AR, k
D
4 054 (026) 085 (014) 076 (015) .78 (.14) 051(023) 1001 (016) 083 (015) .73 (.19)
Full dataset; m
D
1, p
D
0, k ’ xed
DI, k
D
1 1003 (007) 030 (049) 1001 (009) .46 (.34) 098(005) 067 (049) 1001 (009) .48 (.24)
DI, k
D
2 055 (025) 089 (015) 078 (014) .76 (.13) 057(024) 095 (017) 078 (013) .83 (.16)
DI, k
D
3 051 (025) 1000 (016) 077 (015) .77 (.13) 060(021) 1002 (019) 084 (014) .76 (.19)
DI, k
D
4 049 (025) 1000 (016) 076 (015) .78 (.14) 059(022) 1003 (020) 082 (015) .75 (.18)
RMSE, AR Model .049 .027 .045 .017
out of sample MSE of the leading indicator (L I) forecast of
industrial production is 86% that of the autoregressive fore-
cast at the 12-month horizon. Autocorrelation consistent stan-
dard errors for these relative MSEs, calculated following West
(1996), are reported in parentheses. The second set of statistics
is the coef cient on the candidate forecast from the forecast
combining regression,
y
h
t
C
h
D
O
y
h
t
C
h
—
t
C
4
1
ƒ
5
O
y
h1
AR
t
C
h
—
t
C
u
h
t
C
h
1
(4.1)
where
O
y
h
t
C
h
—
t
is the candidate
h
-step-ahead forecast and
O
y
h1
AR
t
C
h
—
t
is the benchmark
h
-step-ahead autoregressive forecast. Het-
eroscedastic autocorrelation robust (HAC) st andard errors for
are reported in parentheses. For example,
is estimate d
to be .57 when the candidate forecast is the leading indica-
tor forecast at the 12-month horizon, with a standard error of
.13, so the hypothesis that the weight on the leading indica-
tor forecast is 0 (
D
0) is rejected at the 5% level, but so i s
the hypothesis that the leading indicator forecast receives unit
weight.
We now turn to the results for the real variables. First con-
sider the D I forecasts with factors estimated using the full
dataset (the unbalanced panel). These forecasts with B IC fac-
tor selection generally improve substantially over the bench-
mark univariate and multivariate forecasts. The D I-AR, Lag
model, wh ich allows recursive B IC selection across own lags
and lags of the factors, outperforms all three benchmark
models in 10 of the 12 variable-horizon combinations, the
exceptions being 6- and 12-month-ah ead forecasts of employ-
ment. In most cases the performance of the simpl er D I fore-
casts, which exclude lags of
b
F
t
and
y
t
, is comparable to or
even better than that of the D I-AR, Lag forecasts. This is
rather surprising, b ecause it implies that essentially all the
predictable dynamics of these series are accounted for by
the estimated factors. In some cases, the improvement over
the benchmark forecasts are quite substantial; for ex ample,
for industrial production at t he 12-month horizon the D I-AR,
Lag forecast has a forecast error variance 57% that of the
autoregressive model and two-thirds that of the leading indi-
cator model. The relative improvements are more modest at
the 6-month horizon. At the 24-month horizon, the multivari-
ate benchmark forecasts break down and perform worse than
the univariate forecast; however, the D I-AR, Lag, D I-AR, and
D I forecasts continue to outperform the autoregressive bench-
mark very substantially.
The performance of comparable mo dels is usually better
when the empirical factors from the full dataset are used, rel-
ative t o those from the balanced panel subset. Performance is
not improved by using empirical factors from augmenting the
balanced panel with its rst l ag; for these real series, doing
so does comparably to, or somewhat worse than, using the
empirical factors from the unstacked balanced panel.
Inspection of the nal panels of Tables 1 and 2 reveals
a striking nding: simply using D I or DI-AR forecasts with