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Macroeconomic Forecasting Using Diffusion

Indexes
a b
James H Stock & Mark W Watson
a
Kennedy School of Government, Harvard University, and National Bureau of
Economic Research, Cambridge, MA 02138
b
Woodrow Wilson School, Princeton University, Princeton, NJ 08544, and
National Bureau of Economic Research
Published online: 01 Jan 2012.

To cite this article: James H Stock & Mark W Watson (2002) Macroeconomic Forecasting Using Diffusion Indexes,
Journal of Business & Economic Statistics, 20:2, 147-162, DOI: 10.1198/073500102317351921

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 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. Watson
Woodrow Wilson School, Princeton University, Princeton, NJ 08544, and National Bureau of
Economic Research

This article studies forecasting a macroeconomic 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
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autoregressions, small vector autoregressions, and leading indicator models.

KEY WORDS: Factor model; Forecasting; Principal components.

1. INTRODUCTION of driving variables is responsible for variation in macro time

series, and these variables can be viewed as a set of common
Recent advances in information technology make it possi-
factors. Although the previous empirical research focused on
ble to access in real time, at a reasonable cost, thousands of
estimating indexes of covariation, this paper uses the estimated
economic time series for major developed economies. This
factors for prediction.
raises the prospect of a new frontier in macroeconomic fore-
The approximate dynamic factor model, which relates the
casting, in which a very large number of time series are used
to forecast a few key economic quantities, such as aggregate variable to be forecast, ytC1 , to a set of predictors collected in
production or in ation. Time series models currently used for the vector Xt , is presented in Section 2. Forecasting is carried
macroeconomic forecasting, however, incorporate only a few out in a two-step process: Ž rst the factors are estimated (by
series: vector autoregressions, for example, typically contain principal components) using Xt , then these estimated factors
fewer than 10 variables. Although variable selection proce- are used to forecast ytC1 . Focusing on the forecasts implied
dures can be used to choose a small subset of predictors from by the factors rather than on the factors themselves permits
a large set of potentially useful variables, the performance of sidestepping the difŽ cult problem of identiŽ cation (or rotation)
these methods ultimately rests on the few variables that are inherent in factor models. One interpretation of the estimated
chosen. For example, real economic activity is often used to factors is in terms of diffusion indexes developed by NBER
predict in ation (the so-called Philips curve), but is the unem- business cycle analysts to measure common movement in a
ployment rate, the rate of capacity utilization, or the Gross set of macroeconomic variables, and accordingly we call the
Domestic Product gap the best measure of real activity for estimated factors diffusion indexes.
this purpose? An alternative to selecting a few predictors is to The performance of the diffusion index (D I) forecasts is
pool the information in all the candidate predictors, averaging examined in Sections 3 and 4. The experiment reported in
away idiosyncratic variation in the individual series. In this these sections simulates real-time forecasting during the 1970–
paper, we use an approximate factor model for this purpose. 1998 period of eight U.S. macroeconomic variables, four mea-
The premise is that for forecasting purposes, the information sures each of real economic activity and of price in ation.
in the large number of predictors can be replaced by a handful The D I forecasts are constructed at horizons of 6, 12, and 24
of estimated factors. months using as many as 215 predictor series. These forecasts
This idea has a long tradition in macroeconomics. For are compared to several conventional benchmarks: univari-
example, the notion of a common business cycle underlies the ate autogressions, small vector autoregressions, leading indi-
classic work of Burns and Mitchell (1947) and the indexes cator models, and, for in ation, unemployment-based Phillips
of leading and coincident indicators originally developed at curve models. Generally speaking, the diffusion index fore-
the National Bureau of Economic Research (NBER). This casts based on a small number of factors (in most cases, one
notion was formally modeled by Sargent and Sims (1977) or two) are found to perform well, with relative performance
in their dynamic generalization of the classic factor analy- improving as the horizon increases. The improvement over the
sis model. Versions of their model have been used by several benchmark forecasts can be dramatic, in several cases produc-
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 1993; Forni © 2002 American Statistical Association
and Reichlin 1996, 1998). Modern dynamic general equilib- Journal of Business & Economic Statistics
rium macroeconomic models often postulate that a small set April 2002, Vol. 20, No. 2

147
 148 Journal of Business & Economic Statistics, April 2002

ing simulated out-of-sample mean square forecast errors that 2.2 Estimation and Forecasting
are one-third less than those of the benchmark models.
Because 8Ft 91  h 1 ‚h 4L5, and ƒh 4L5 are unknown, forecasts
of yT Ch based on (2.4) and (2.5) are constructed using a two-
2. ECONOMETRIC FRAMEWORK step procedure. First, the sample data 8X t 9TtD1 are used to esti-
2.1 An Approximate Dynamic Factor Model mate a time series of factors (the diffusion indexes), 8Fbt 9TtD1 .
Second, the estimators O h 1 ‚O h 4L5 and ƒOh 4L5 are obtained
We begin with a discussion of the statistical model that by regressing ytC1 onto a constant, Fbt and yt (and lags). The
motivates the D I forecasts. Let ytC1 denote the scalar series forecast of yTh Ch is then formed as O h C ‚O h 4L5FbT C ƒOh 4L5yT .
to be forecast and let, Xt be an N -dimensional multiple time Stock and Watson (1998) developed theoretical results for
series of predictor variables, observed for t D 11 : : : 1 T , where this two-step procedure applied to (2.3) and (2.4). The factors
yt and Xt are both taken to have mean 0. (The different time are estimated by principal components because these estima-
subscripts used for y and X emphasize the forecasting rela- tors are readily calculated even for very large N and because
tionship.) We suppose that 4Xt 1 ytC1 5 admit a dynamic factor principal components can be generalized to handle data irreg-
model representation with rN common dynamic factors ft , ularities as discussed later. Under a set of moment conditions
for 4…1 e1 F 5 and an asymptotic rank condition on å, the feasi-
ytC1 D ‚4L5ft C ƒ4L5yt C …tC1 1 (2.1)
ble forecast is asymptotically Ž rst-order efŽ cient in the sense
Xit D ‹i 4L5ft C eit 1 (2.2) that its mean square forecast error (MSE) approaches the
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MSE of the optimal infeasible forecast as N 1 T ! ˆ, where

for i D 11 : : : 1 N , where et D 4e1t 1 : : : 1 eN t 50 is the N € 1 N D O4T  5 for any  > 1. This result suggests that feasible
idiosyncratic disturbance and ‹i 4L5 and ‚4L5 are lag polyno- forecasts are likely to be nearly optimal when N and T are
mials in nonnegative powers of L. It is assumed that E4…tC1 — large, regardless of the ratio of N to T . The assumptions by
ft 1 yt 1 Xt 1 ftƒ1 1 ytƒ1 1 Xtƒ1 1 : : : 5 D 0. Thus, if 8ft 91 ‚4L5, and Stock and Watson (1998) are similar to assumptions made in
ƒ4L5 were known, the minimum mean square error forecast the literature on approximate factor models (Chamberlain and
of yT C1 would be ‚4L5fT C ƒ4L5yT . Rothschild 1983; Connor and Korajczyk 1986, 1988, 1993),
We make two important modiŽ cations to (2.1) and (2.2). generalized to allow for serial correlation. A related dynamic
First, the lag polynomials ‹i 4L5, ‚4L5, and ƒ4L5 Pqare modeled generalization and estimation (but not forecasting) results were
as having Ž nite orders of at most q, so ‹ 4L5 D jD0 ‹ ij L and
j
discussed by Forni, Hallin, Lippi, and Reichlin (2000). Stock
Pq i
‚4L5 D jD0 ‚j L . The Ž nite lag assumption permits rewriting
j
and Watson (1998) also showed that the principal components
(2.1) and (2.2) as remain consistent when there is some time variation in å and
small amounts of data contamination, as long as the number
ytC1 D ‚0 Ft C ƒ4L5yt C …tC1 1 (2.3)
of predictors is very large, N T.
Xt D åFt C et 1 (2.4)
2.3 Data Irregularities and Computational Issues
where Ft D 4ft0 1 : : : 1 ftƒq
0
50 is r € 1, where r µ 4q C 15rN, the ith
row of å in (2.4) is 4‹i0 1 : : : 1 ‹iq 5, and ‚ D 4‚0 1 : : : 1 ‚q 50 . The In our dataset, some series contain missing observations or
main advantage of this static representation of the dynamic are available over a diminished time span. Although our data
factor model is that the factors can be estimated using prin- are all monthly, further complications would arise in applica-
cipal components. This comes at a cost, because the assump- tions in which mixed sampling frequencies are used, such as
tion is inconsistent with inŽ nite distributed lags of the factors. monthly and quarterly. In these cases standard principal com-
Whether this cost is large is ultimately an empirical question, ponents analysis does not apply. However, the expectation-
addressed here by studying whether (2.3) and (2.4) can be maximization (EM) algorithm can be used to estimate the fac-
used to produce accurate forecasts. tors by solving a suitable minimization problem iteratively.
Second, our empirical application focuses on h-step-ahead Details are given in Appendix A.
forecasts. At least two approaches to multistep forecasting are Although the components of Xt typically will be distinct
possible. One is to develop a vector time series model for time series, X t could contain multiple lags of one or more
Ft , to estimate this using the estimated factors, and to roll series. Because the estimated factors Ft could include lags of
the 4yt 1 Ft 5 model forward, but this entails estimating a large the dynamic factors ft , estimation of Ft might be enhanced
number of parameters that could erode forecast performance. by augmenting a vector of distinct time series with its lags.
Another approach is to recognize that the ensuing multistep This is referred to later as stacking Xt with its lags, in which
forecasts would be linear Ft and yt (and lags) and to use an case the principal components of the stacked data vector are
h-step-ahead projection to construct the forecasts directly. We computed.
adopt the latter approach, and the resulting multistep ahead
version of (2.3) is
3. THE DATA AND FORECASTING
h
ytCh D  h C ‚h 4L5Ft C ƒh 4L5yt C …tCh
h
1 (2.5) EXPERIMENTAL DESIGN
3.1 Forecasting Models and Data
where ytCh
h
is the h-step-ahead variable to be forecast, the con-
stant term is introduced explicitly, and the subscripts re ect The forecasting experiment simulates real-time forecast-
the dependence of the projection on the horizon. 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 1998:12. where Fbt is the vector of k estimated factors. Results for three
Four of these eight variables are the measures of real economic variants of (3.3) are reported. The Ž rst, denoted in the tables
activity used to construct the Index of Coincident Economic by D I-AR, Lag, includes lags of the factors and lags of yt , with
Indicators maintained by the Conference Board (formerly by k and lag orders m and p estimated by Bayesian information
the U.S. Department of Commerce): total industrial production criterion (B IC), with 1 µ k µ 4, 1 µ m µ 3, and 0 µ p µ 6.
(ip); real personal income less transfers (gmyxpq); real man- Thus the smallest candidate model that B IC can choose here
ufacturing and trade sales (msmtq); and number of employ- includes only a single contemporaneous factor and excludes
ees on nonagricultural payrolls (lpnag). (Additional details are yt . The second, denoted D I-AR, includes contemporaneous Fbt ,
given in Appendix B, which lists series by the mnemonics that is, m D 1, and k and p are chosen by B IC with 1 µ
given here in parenthesis.) The remaining four series are price k µ 12 and 0 µ p µ 6. The third, denoted D I, includes only
indexes: the consumer price index (punew); the personal con- contemporaneous Fbt , so p D 01 m D 1, and k is chosen by B IC,
sumption expenditure implicit price de ator (gmdc); the con- 1 µ k µ 12.
sumer price index (CP I) less food and energy (puxx); and the The full dataset used to estimate the factors contains 215
producer price index for Ž nished goods (pwfsa). These series monthly time series for the United States from 1959:1 to
and the predictor series were taken from the May 1999 release 1998:12. The series were selected judgmentally to represent 14
of the DR I/McGraw–Hill Basic Economics database (formerly main categories of macroeconomic time series: real output and
Citibase). In general these series represent the fully revised income; employment and hours; real retail, manufacturing, and
historical series available as of May 1999, and in this regard trade sales; consumption; housing starts and sales; real inven-
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the forecasting results will differ from results that would be tories and inventory-sales ratios; orders and unŽ lled orders;
calculated using real-time data. stock prices; exchange rates; interest rates; money and credit
For each series, several forecasting models are compared at quantity aggregates; price indexes; average hourly earnings;
the 6-, 12-, and 24-month forecasting horizons: D I forecasts and miscellaneous. The list of series is given in Appendix B
based on estimated factors, a benchmark univariate autoregres- and is similar to lists we have used elsewhere (Stock and
sion, and benchmark multivariate models. For both the real Watson 1996, 1999). These series were taken from a some-
and the price series, one of the benchmark multivariate models what longer list, from which we eliminated series with gross
is a trivariate vector autoregression, and a second is based on problems, such as redeŽ nitions. However, no further pruning
leading economic indicators. As a further comparison, in a- was performed.
tion forecasts are also computed using an unemployment- The theory outlined in Section 2 assumes that Xt is I(0),
based Phillips curve. so these 215 series were subjected to three preliminary steps:
Our focus is on multistep-ahead prediction, and most of the possible transformation by taking logarithms, possible Ž rst dif-
forecasting regressions are projections of an h-step-ahead vari- ferencing, and screening for outliers. The decision to take log-
able ytCh
h
onto t-dated predictors, sometimes including lagged arithms or to Ž rst difference the series was made judgmentally
transformed values yt of the variable of interest. The real vari- after preliminary data analysis, including inspection of the data
ables are modeled as being I(1) in logarithms. Because all and unit root tests. In general, logarithms were taken for all
four real variables are treated identically, consider industrial nonnegative series that were not already in rates or percentage
production, for which units. Most series were Ž rst differenced. A code summarizing
these transformations is given for each series in Appendix B.
h
ytCh D 41200=h5 ln4 IPtCh = IPt 5 After these transformations, all series were further standard-
ized to have sample mean zero and unit sample variance.
and yt D 1200 ln4 IPt = IPtƒ1 50 (3.1) Finally, the transformed data were screened automatically for
The price indexes are modeled as being I(2) in logarithms. outliers (generally taken to be coding errors or exceptional
The I(2) speciŽ cation is consistent with standard Phillips curve events such as labor strikes), and observations exceeding 10
equations and is a good description of the series over much of times the interquartile range from the median were replaced
the sample period. However, I(1) speciŽ cations also provide by missing values.
adequate descriptions of the data, particularly in the early part Using this transformed and screened dataset, three sets of
of the sample. Stock and Watson (1999) found little difference empirical factors were constructed. The Ž rst was computed
in I(1) and I(2) factor model forecasts for these prices over the using principal components from the subset of 149 variables
sample period studied here, so for the sake of brevity we limit available for the full sample period (the balanced panel). The
our analysis to the I(2) speciŽ cation. Accordingly, for the CP I second set of factors was computed using the nonbalanced
(and similarly for the other price series), 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 Ž rst lags, so the augmented
h
ytCh D 41200=h5 ln4CP ItCh =CP It 5 ƒ 1200 ln4CP It =CP Itƒ1 5
data vector has dimension 298. Empirical factors were then
and yt D 1200ã ln4CP It =CP Itƒ1 50 (3.2) estimated by the principal components of the stacked data, as
discussed in Section 2.
Diffusion Index Forecasts. Following (2.5), the most gen- Autoregressive Forecast. The autoregressive forecast is a
eral D I forecasting function is univariate forecast based on (3.3), where the terms involving
X
m X
p Fb are excluded. The lag order p was selected recursively by
yOTh Ch—T D O h C ‚O 0hj F
bT ƒjC1 C ƒOhj yT ƒjC1 1 (3.3) B IC with 0 µ p µ 6, where p D 0 indicates that yt and its lags
jD1 jD1 are excluded.
 150 Journal of Business & Economic Statistics, April 2002

Vector Autoregressive Forecast. The Ž rst multivariate new orders in durable goods industries (mdoq), the nominal
benchmark model is a vector autoregression (VAR) with p M1 money supply (fm1), the federal funds overnight interest
lags each of three variables. One version of the VAR used rate (fyff), and the interest rate spread between 1-year U.S.
p D 4 lags, and another version selected p recursively by B IC. treasury bonds and the federal funds rate (sfygt1). The remain-
The Ž xed-lag VARs performed somewhat better than the B IC ing variable is the trade-weighted exchange rate listed in the
selected lag lengths (which often set p D 1), and we report previous paragraph.
results for the Ž xed lag speciŽ cations in the results to follow. In all cases, the leading indicators were transformed so that
The variables in the VAR are a measure of the monthly growth Wt is I(0). This entailed taking logarithms of variables not
in real activity, the change in monthly in ation, and the change already in rates and differencing all variables except the inter-
in the 90-day U.S. treasury bill rate. When used to forecast the est rate spreads, housing starts, the index of vendor perfor-
real series, the relevant real activity variable was used and the mance, and the help wanted index.
in ation measure was CP I in ation. For forecasting in ation, For each variable to be forecast, p and m in (3.4) were
the relevant price series was used and the real activity measure determined by recursive B IC with 1 µ m µ 4 and 0 µ p µ 6,
was industrial production. Multistep forecasts were computed so 28 possible models were compared in each time period.
by iterating the VAR forward. This contrasts to the autoregres- Phillips Curve Forecasts. The unemployment-based
sive forecasts, which were computed by h-step-ahead projec- Phillips curve is considered by many to have been a reliable
tion rather than iteration. method for forecasting in ation over this period (Gordon
1982; Congressional Budget OfŽ ce 1996; Fuhrer 1995; Gor-
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Multivariate Leading Indicator Forecasts. The leading

indicator forecasts have the form don 1997; Staiger et al. 1997; Tootel 1994). The Phillips
p
curve in ation forecasts considered here have the form (3.4),
m
X X where Wt consists of the unemployment rate (LHUR) and
yOTh Ch—T D „O h0 C „O 0hi WT ƒjC1 C ƒOhj yT ƒjC1 1 (3.4)
jD1 jD1 m ƒ 1 of its lags, the relative price of food and energy (current
and one lagged value only), and Gordon’s (1982) variable that
where Wt is a vector of leading indicators that have been fea- controls for the imposition and removal of the Nixon wage
tured in the literature or in real-time forecasting applications and price controls. The wage and price control variable is
and „O h0 and so forth are ordinary least squares coefŽ cient introduced for forecasts made in 19712 7 C h, before which it
estimates. produces singular regressions. The lag lengths m and p were
For the real variables, Wt consists of 11 leading indicators chosen by recursive B IC, where 1 µ m µ 6 and 0 µ p µ 6.
that we used for real-time monthly forecasting in experimen-
tal leading and recession indicators (Stock and Watson 1989). 3.2 Simulated Real-Time Experimental Design
(The list used here consists of the leading indicators used to
produce the XR I and the XR I-2, which are released monthly Estimation and forecasting was conducted to simulate real-
and documented at the web site http://www.nber.org.) Five of time forecasting. This entailed fully recursive parameter esti-
these leading indicators are also used in the factor estima- mation, factor estimation, model selection, and so forth. The
tion step in the diffusion index forecasts. These are average Ž rst simulated out of sample forecast was made in 1970:1.
weekly hours of production workers in manufacturing (lphrm), To construct this forecast, the data were screened for outliers
the capacity utilization rate in manufacturing (ipxmca), hous- and standardized, the parameters and factors were estimated,
ing starts (building permits) (hsbr), the index of help-wanted and the models were selected, using only data available from
advertising in newspapers (lhel), and the interest rate on 1959:1 through 1970:1. (The Ž rst date for the regressions was
10-year U.S. treasury bonds (fygt10). The remaining six lead- 1960:1, and earlier observations were used for initial condi-
ing indicators are the interest rate spread between 3-month tions as needed.) Thus regressions (3.3) and (3.4) were run for
U.S. treasury bills and 3-month commercial paper; the spread t D 1960:11 : : : 1 197021 ƒ h, then the values of the regressors
between 10-year and 1-year U.S. treasury bonds; the num- at t D 197021 were used to forecast y197021
h
Ch . All parameters,

ber of people working part-time in nonagricultural industries factors, and so forth were then reestimated, information cri-
because of slack work; real manufacturers’ unŽ lled orders in teria were recomputed, and models were selected using data
durable goods industries; a trade-weighted index of nominal from 1959:1 through 1970:2, and forecasts from these models
exchange rates between the United States and the U.K., West were then computed for y197022
h
Ch . The Ž nal simulated out of

Germany, France, Italy, and Japan; and the National Associ- sample forecast was made in 1998212 ƒ h for y1998212h
.
ation of Purchasing Managers’ index of vendor performance
(the percent of companies reporting slower deliveries). 4. EMPIRICAL RESULTS
For the in ation forecasts, eight leading indicators are used.
4.1 Forecasting Results
These variables were chosen because of their good individ-
ual performance in previous in ation forecasting exercises. In The results for the real variables are reported in detail in
particular these variables performed well in at least one of the Table 1 for 12-month-ahead forecasts, and summaries for 6-
historical episodes considered by Staiger, Stock, and Watson and 24-month-ahead forecasts are reported in Table 2. Two
(1997) (also see Stock and Watson 1999). Seven of these vari- sets of statistics are reported. The Ž rst is the MSE of the can-
ables are also used in the factor-estimation step in the diffu- didate forecasting model, computed relative to the MSE of the
sion index forecasts: the total unemployment rate (lhur), real univariate autoregressive forecast (so the autoregressive fore-
manufacturing and trade sales (msmtq), housing starts (hsbr), cast has a relative MSE of 1.00). For example, the simulated
 Stock and Watson: Macroeconomic Forecasting Using Diffusion Indexes 151

Table 1. Simulated Out-of-Sample Forecasting Results: Real 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)
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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 models in 10 of the 12 variable-horizon combinations, the
industrial production is 86% that of the autoregressive fore- exceptions being 6- and 12-month-ahead forecasts of employ-
cast at the 12-month horizon. Autocorrelation consistent stan- ment. In most cases the performance of the simpler D I fore-
dard errors for these relative MSEs, calculated following West casts, which exclude lags of F bt and yt , is comparable to or
(1996), are reported in parentheses. The second set of statistics even better than that of the D I-AR, Lag forecasts. This is
is the coefŽ cient on the candidate forecast from the forecast rather surprising, because it implies that essentially all the
combining regression, predictable dynamics of these series are accounted for by
the estimated factors. In some cases, the improvement over
h1 AR
h
ytCh D  yOtCh—t
h C 41 ƒ 5yOtCh—t C uhtCh 1 (4.1) the benchmark forecasts are quite substantial; for example,
for industrial production at the 12-month horizon the D I-AR,
h1 AR
where yOtCh—t
h
is the candidate h-step-ahead forecast and yOtCh—t Lag forecast has a forecast error variance 57% that of the
is the benchmark h-step-ahead autoregressive forecast. Het- autoregressive model and two-thirds that of the leading indi-
eroscedastic autocorrelation robust (HAC) standard errors for cator model. The relative improvements are more modest at
 are reported in parentheses. For example,  is estimated the 6-month horizon. At the 24-month horizon, the multivari-
to be .57 when the candidate forecast is the leading indica- ate benchmark forecasts break down and perform worse than
tor forecast at the 12-month horizon, with a standard error of the univariate forecast; however, the D I-AR, Lag, D I-AR, and
.13, so the hypothesis that the weight on the leading indica- D I forecasts continue to outperform the autoregressive bench-
tor forecast is 0 ( D 0) is rejected at the 5% level, but so is mark very substantially.
the hypothesis that the leading indicator forecast receives unit The performance of comparable models is usually better
weight. when the empirical factors from the full dataset are used, rel-
We now turn to the results for the real variables. First con- ative to those from the balanced panel subset. Performance is
sider the D I forecasts with factors estimated using the full not improved by using empirical factors from augmenting the
dataset (the unbalanced panel). These forecasts with B IC fac- balanced panel with its Ž rst lag; for these real series, doing
tor selection generally improve substantially over the bench- so does comparably to, or somewhat worse than, using the
mark univariate and multivariate forecasts. The D I-AR, Lag empirical factors from the unstacked balanced panel.
model, which allows recursive B IC selection across own lags Inspection of the Ž nal panels of Tables 1 and 2 reveals
and lags of the factors, outperforms all three benchmark a striking Ž nding: simply using D I or D I-AR forecasts with
 152 Journal of Business & Economic Statistics, April 2002

Table 2. Simulated Out-of-Sample Forecasting Results: Real Variables, 6- and 24-Month Horizons

Industrial production Personal income Mfg & trade sales Nonag. employment
Forecast
method Rel. MSE O Rel. MSE O Rel. MSE O Rel. MSE O

A. Horizon D 6 months
Benchmark models
AR 1000 1000 1000 1000
LI 070 (025) .68 (.13) 083 (015) 064 (011) 077 (019) .68 (.14) 075 (019) 067 (012)
VAR 1001 (005) .43 (.39) 099 (003) 063 (043) 099 (004) .64 (.45) 1006 (007) 012 (034)
Full dataset (N D 215)
DI-AR, Lag 069 (025) .69 (.14) 077 (012) 086 (015) 063 (018) .89 (.17) 094 (016) 056 (018)
DI-AR 077 (030) .62 (.16) 081 (016) 066 (013) 070 (020) .76 (.17) 1002 (032) 049 (019)
DI 074 (025) .68 (.17) 081 (016) 065 (013) 067 (020) .79 (.18) 096 (028) 052 (019)
Balanced panel (N D 149)
DI-AR, Lag 073 (025) .68 (.16) 079 (013) 078 (013) 066 (017) .87 (.17) 093 (017) 058 (021)
DI-AR 078 (028) .62 (.16) 081 (015) 066 (011) 076 (019) .70 (.17) 097 (028) 052 (019)
DI 073 (024) .69 (.15) 081 (015) 066 (011) 068 (019) .81 (.17) 095 (026) 053 (018)
Full dataset; m D 1, p D BIC, k ’ xed
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DI-AR, k D 1 097 (015) .58 (.33) 091 (007) 080 (023) 099 (011) .52 (.29) 094 (012) 060 (019)
DI-AR, k D 2 067 (022) .77 (.15) 076 (011) 090 (014) 064 (018) .86 (.16) 084 (013) 071 (016)
DI-AR, k D 3 064 (023) .81 (.15) 075 (012) 089 (014) 064 (018) .88 (.17) 088 (014) 066 (017)
DI-AR, k D 4 064 (023) .80 (.15) 074 (013) 087 (014) 063 (018) .87 (.15) 091 (016) 060 (018)
RMSE, AR Model .030 .016 .028 .008
B. Horizon D 24 months
Benchmark models
AR 1000 1000 1000 1000
LI 1009 (028) .45 (.14) 1029 (031) 030 (020) 1008 (021) .45 (.14) 1007 (031) 047 (015)
VAR 1001 (010) .44 (.48) 098 (006) 063 (034) 1003 (006) .13 (.85) 1006 (013) 035 (031)
Full dataset (N D 215)
DI-AR, Lag 057 (024) .88 (.13) 070 (020) 094 (023) 066 (018) .95 (.18) 082 (015) 088 (026)
DI-AR 059 (025) .88 (.15) 076 (022) 080 (026) 070 (020) .89 (.19) 074 (019) 097 (024)
DI 055 (026) .91 (.14) 076 (022) 080 (025) 070 (020) .89 (.19) 074 (019) 097 (024)
Balanced panel (N D 149)
DI-AR, Lag 057 (025) .87 (.14) 076 (019) 086 (023) 064 (020) .94 (.18) 074 (017) 1006 (025)
DI-AR 058 (025) .87 (.14) 083 (020) 074 (024) 067 (019) .93 (.18) 076 (018) 094 (025)
DI 058 (025) .87 (.14) 083 (020) 074 (024) 067 (020) .94 (.19) 075 (018) 094 (024)
Full dataset; m D 1, p D BIC, k ’ xed
DI-AR, k D 1 1012 (019) .10 (.46) 1007 (009) 081(1000) 097 (004) .90 (.62) 1003 (007) 033 (046)
DI-AR, k D 2 076 (019) .68 (.11) 088 (013) 068 (017) 065 (020) .87 (.14) 072 (016) 099 (017)
DI-AR, k D 3 058 (024) .89 (.13) 072 (019) 090 (018) 070 (017) .89 (.14) 079 (016) 095 (024)
DI-AR, k D 4 056 (024) .90 (.14) 070 (020) 093 (023) 067 (018) .95 (.18) 078 (016) 096 (024)
RMSE, AR Model .075 .046 .070 .031

two factors captures most of the forecasting improvement. In on the D I-AR lag. Thus most of the forecasting gains seem to
most cases, incorporating B IC factor and lag order selection come from using a single factor.
provides little or no improvement over just using two fac- As with the real variables, forecasts based on the stacked
tors, with no lags of the factors and no lagged dependent data perform less well than those based on the unstacked data.
variables. Although the full dataset forecasts are typically better than
The results for the price series are given in Tables 3 and 4. the balanced panel subset forecasts for the 6- and 12-month
There are three notable differences in these results, relative to horizons, at the 24-month horizon the balanced panel forecasts
those for the real variables. First, the D I-AR, Lag forecasts slightly outperform the full dataset forecasts.
outperform all the benchmark forecasts less often, in only 6 Additional analysis of factor-based forecasts of CP I and
of the 12 variable-horizon combinations. Second, including consumption de ator in ation, and additional comparisons of
lagged in ation dramatically improves the forecasts, and with- these forecasts to other Phillips-curve forecasts and to fore-
out this the D I forecasts are actually worse than the autore- casts based on other leading indicators, were presented by
gressive forecasts. Third, other factor forecasts generally out- Stock and Watson (1999). Three Ž ndings from that study are
perform the D I-AR, Lag forecasts. Notably, the full data set worth noting here. First, the D I-AR and D I-AR, Lag forecasts
D I-AR forecast with k D 1 (and no lagged factors) outperforms are found to perform well relative to a large number of addi-
all the benchmarks in 11 of 12 cases and typically improves tional multivariate benchmarks. Second, the forecasts reported
 Stock and Watson: Macroeconomic Forecasting Using Diffusion Indexes 153

Table 3. Simulated Out-of-Sample Forecasting Results: Price In‘ ation, 12-Month Horizon

CPI Consumption de‘ ator CPI exc. food & energy Producer price index
Forecast
method Rel. MSE O Rel. MSE O Rel. MSE O Rel. MSE O

Benchmark models
AR 1000 1000 1000 1000
LI 079 (015) 076 (015) 095 (012) 058 (017) 1000 (016) 050 (021) 082 (015) 075 (019)
Phillips Curve 082 (013) 095 (020) 092 (010) 072 (023) 079 (018) 080 (022) 087 (014) 096 (030)
VAR 091 (009) 074 (020) 1002 (006) 045 (020) 099 (005) 056 (021) 1029 (014) 025 (012)
Full dataset (N D 215)
DI-AR, Lag 072 (014) 091 (014) 090 (009) 065 (013) 084 (015) 076 (020) 083 (013) 078 (021)
DI-AR 071 (016) 083 (013) 090 (010) 062 (013) 085 (015) 074 (020) 082 (014) 075 (020)
DI 1030 (016) 034 (008) 1040 (016) 025 (008) 1055 (031) 024 (006) 2040 (088) 013 (007)
Balanced panel (N D 149)
DI-AR, Lag 070 (014) 094 (012) 090 (008) 067 (015) 084 (015) 077 (021) 086 (011) 077 (021)
DI-AR 069 (015) 088 (013) 087 (010) 066 (012) 085 (015) 073 (020) 085 (014) 071 (019)
DI 1030 (016) 032 (008) 1034 (013) 026 (009) 1057 (033) 020 (007) 2044 (087) 014 (006)
Stacked balance panel
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DI-AR 073 (015) 082 (012) 087 (009) 065 (012) 085 (015) 077 (021) 081 (014) 075 (020)
DI 1054 (031) 028 (008) 1051 (018) 025 (008) 1055 (032) 023 (006) 3006(1089) 011 (006)
Full dataset; m D 1, p D BIC, k ’ xed
DI-AR, k D 1 064 (015) 1014 (014) 077 (012) 096 (016) 071 (017) 1025 (023) 076 (016) 095 (024)
DI-AR, k D 2 067 (014) 1007 (013) 083 (009) 083 (014) 072 (017) 097 (019) 077 (015) 093 (023)
DI-AR, k D 3 076 (013) 091 (015) 094 (007) 061 (014) 086 (014) 073 (020) 086 (011) 078 (021)
DI-AR, k D 4 074 (014) 089 (015) 091 (009) 064 (014) 087 (015) 072 (021) 082 (013) 079 (021)
Full dataset; m D 1, p D 0, k ’ xed
DI, k D 1 1060 (034) 025 (007) 1056 (020) 022 (009) 1055 (031) 023 (006) 2076(1061) 012 (007)
DI, k D 2 1056 (031) 026 (007) 1058 (020) 021 (008) 1062 (039) 022 (007) 2072(1056) 013 (007)
DI, k D 3 1057 (032) 024 (008) 1060 (020) 017 (008) 1069 (043) 018 (007) 2068(1049) 013 (007)
DI, k D 4 1056 (025) 025 (007) 1056 (019) 021 (008) 1067 (040) 019 (007) 2055 (099) 016 (006)
RMSE, AR Model .021 .015 .019 .033

here can be further improved on using a single-factor fore- Nevertheless, the Ž nding that good forecasts can be made with
cast, where the factor is computed from a set of variables that only one or two factors suggests brie y characterizing the Ž rst
measure only real economic activity. Forecasts based on this few factors.
real economic activity factor have MSEs approximately 10% Figure 1 therefore displays the R2 of the regressions of the
less than the best forecasts reported in Table 3. Finally, sim- 215 individual time series against each of the Ž rst six empiri-
ilar rankings of methods are obtained using I(1) forecasting cal factors from the balanced panel subset, estimated over the
models, rather than the I(2) models used here, that is, when full sample period. These R2 are plotted as bar charts with
Ž rst rather than second differences of log prices are used for one chart for each factor. (The series are grouped by category
the forecasting equation and factor estimation. and ordered numerically using the ordering in the Appendix.)
In interpreting these results, it should be stressed that the Broadly speaking, the Ž rst factor loads primarily on output
multivariate leading indicator models are sophisticated fore- and employment; the second factor on interest rate spreads,
casting tools that provide a stiff benchmark against which unemployment rates, and capacity utilization rates; the third,
to judge the diffusion index forecasts. In our judgment, the on interest rates; the fourth, on stock returns; the Ž fth, on in a-
performance of the leading indicator models reported here tion; and the sixth, on housing starts. Taken together, these six
overstates their true potential out of sample performance, factors account for 39% of the variance of the 215 monthly
because the lists of leading indicators used to construct the time series in the full dataset, as measured by the trace-R2 ;
forecasts were chosen by model selection methods based the Ž rst 12 factors together account for 53% of the variance
on their forecasting performance over the past two decades, of these series. (The contributions to the trace-R2 by the Ž rst
as discussed in Section 3. In this light, we consider the six factors are, respectively, .137, .085, .048, .040, .034, and
performance of the various diffusion index models to be par- .041, for a total of .385.)
ticularly encouraging.
5. DISCUSSION AND CONCLUSIONS
4.2 Empirical Factors
We Ž nd two features of the empirical results surprising and
Because the factors are identiŽ ed only up to a k € k matrix, intriguing. First, only six factors account for much of the
detailed discussion of the individual factors is unwarranted. variance of our 215 time series. One interpretation of this
 154 Journal of Business & Economic Statistics, April 2002

Table 4. Simulated Out-of-Sample Forecasting Results: Price In‘ ation, 6- and 24-Month Horizons

CPI Consumption de‘ ator CPI exc. food & energy Producer price index
Forecast
method Rel. MSE O Rel. MSE O Rel. MSE O Rel. MSE O

A. Horizon D 6 months
Benchmark models
AR 1000 1000 1000 1000
LI 082 (012) 078 (016) 1004 (009) 042 (016) 1010 (016) 032 (027) 1000 (009) 051 (019)
Phillips Curve 090 (011) 080 (027) 099 (006) 054 (023) 090 (011) 068 (019) 1002 (004) 034 (037)
VAR 1004 (008) 041 (016) 1015 (007) 008 (020) 1000 (005) 050 (021) 1034 (016) 019 (012)
Full dataset (N D 215)
DI-AR, Lag 073 (014) 1005 (018) 091 (008) 071 (017) 083 (013) 089 (025) 087 (011) 087 (026)
DI-AR 074 (014) 1001 (019) 089 (008) 079 (018) 083 (013) 089 (025) 087 (010) 087 (026)
DI 1057 (025) 021 (008) 1068 (026) 010 (008) 1074 (043) 013 (007) 2042 (074) 005 (007)
Balanced panel (N D 149)
DI-AR, Lag 079 (013) 1000 (022) 097 (007) 059 (018) 085 (013) 085 (025) 091 (009) 078 (027)
DI-AR 078 (013) 094 (021) 096 (008) 060 (018) 085 (013) 085 (025) 091 (009) 082 (029)
DI 1059 (026) 019 (008) 1064 (021) 009 (008) 1073 (043) 013 (007) 2042 (070) 007 (007)
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Full dataset; m D 1, p D BIC, k ’ xed

DI-AR, k D 1 071 (014) 1015 (019) 085 (009) 091 (020) 085 (011) 1013 (029) 085 (012) 090 (026)
DI-AR, k D 2 072 (014) 1003 (018) 088 (008) 078 (017) 080 (013) 1000 (024) 086 (012) 086 (026)
DI-AR, k D 3 076 (013) 097 (018) 093 (008) 066 (017) 086 (012) 082 (025) 091 (010) 076 (026)
DI-AR, k D 4 076 (013) 096 (019) 093 (008) 065 (017) 088 (012) 079 (025) 090 (010) 075 (025)
RMSE, AR Model .010 .007 .009 .017
B. Horizon D 24 months
Benchmark models
AR 1000 1000 1000 1000
LI 070 (021) 076 (012) 070 (020) 078 (011) 099 (029) 051 (025) 065 (022) 084 (019)
Phillips Curve 084 (012) 077 (008) 081 (015) 080 (009) 072 (021) 093 (019) 077 (019) 1000 (006)
VAR 092 (008) 080 (022) 098 (006) 057 (018) 1000 (006) 049 (034) 1018 (012) 029 (010)
Full dataset (N D 215)
DI-AR, Lag 074 (023) 074 (018) 075 (016) 079 (013) 092 (026) 058 (028) 082 (014) 068 (012)
DI-AR 075 (025) 067 (016) 071 (021) 073 (012) 096 (033) 053 (027) 077 (017) 068 (013)
DI 1018 (022) 040 (012) 1021 (018) 038 (012) 1040 (022) 030 (007) 2009 (072) 019 (009)
Balanced panel (N D 149)
DI-AR, Lag 059 (022) 095 (012) 067 (018) 084 (010) 084 (022) 069 (024) 076 (014) 078 (013)
DI-AR 070 (024) 072 (013) 070 (020) 075 (012) 087 (029) 061 (025) 086 (015) 062 (011)
DI 1007 (020) 046 (012) 1008 (018) 045 (012) 1043 (022) 027 (007) 2010 (070) 019 (008)
Full dataset; m D 1, p D BIC, k ’ xed
DI-AR, k D 1 063 (020) 1004 (018) 068 (017) 097 (015) 060 (025) 1012 (020) 073 (017) 093 (022)
DI-AR, k D 2 061 (021) 1007 (017) 072 (016) 092 (013) 064 (024) 096 (017) 068 (019) 097 (020)
DI-AR, k D 3 080 (017) 082 (023) 080 (012) 083 (013) 094 (025) 056 (029) 081 (011) 080 (014)
DI-AR, k D 4 076 (020) 081 (021) 074 (015) 083 (014) 092 (026) 059 (029) 078 (014) 078 (014)
RMSE, AR Model .052 .038 .046 .077

result is that there are only a few important sources of macro- estimating the factors with mixed frequency data is outlined
economic variability. Second, just a few factors are needed to in Appendix A. Third, we considered only U.S. data, and it
forecast real activity, and the most accurate forecasts of in a- would be useful to study the relative forecasting performance
tion use lags of in ation together with a single factor. This of these methods for other countries. Fourth, the estimated
suggests that a very small state vector may be necessary for factors that we used here were based on simple estimators
forecasting macroeconomic time series. and it would be useful to study other estimators designed to
These results raise several issues for future empirical and exploit the heteroscedasticity and serial correlation in the data
theoretical research. We mention Ž ve here. First, classical dif- to improve efŽ ciency. Finally, our results are based on 215
fusion indexes are computed using nonlinear transformations time series chosen judgementally from the large number of
of the data, but our indexes are linear functions of the data. available macroeconomic time series. Would there be addi-
This raises the possibility that further forecasting gains can tional improvements if we were to use 500 series or much
be realized using a nonlinear version of the dynamic factor loss by restricting ourselves to only 100 series? Alternatively,
model. Second, the results reported here rely on monthly data, the problem of systematically selecting many series from
but data from other sampling frequencies (weekly, quarterly) very many series is a difŽ cult problem that requires further
may improve the forecasts. A computational algorithm for research.
 Stock and Watson: Macroeconomic Forecasting Using Diffusion Indexes 155
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Figure 1. R2 Between Factors and Individual Time Series, Grouped by Category (see Appendix B). Categories: real output and income
(Out); employment and hours (Emp); real retail, manufacturing, and trade sales (RTS); consumption (PCE); housing starts and sales (HSS); real
inventories and inventory-sales ratios (Inv); orders and un’ lled orders (Ord); stock prices (SPr); exchange rates (EXR); interest rates (Int); money
and credit quantity aggregates (Mon); price indexes (Pri); average hourly earnings (AHE); miscellaneous (Oth).
 156 Journal of Business & Economic Statistics, April 2002

ACKNOWLEDGMENTS To carry out the calculations, note that

The material in this article originally appeared in our paper
titled “Diffusion Indexes.” We thank Michael Boldin, Frank Q4X ! 1 Fb1 b
å1 F 1 å5
Diebold, Gregory Chow, Andrew Harvey, Lucrezia Reichlin, XX ¢
D 8EFb1 b 2 —
å Xit X
! C bit 4‹0i Ft 591
4‹0i Ft 52 ƒ 2X (A.4)
Ken Wallis, Charles Whiteman, and several referees for help-
i t
ful discussions and comments and Lewis Chan and Alexei
Onatski for skilled research assistance. This research was sup-
ported in part by National Science Foundation grants SBR- where X bit D EFb1 bå 4Xit — X ! 5. The Ž rst term on the right side of
9409629 and SBR-9730489. (A.4) does not depend on F or å,P and
Psobfor purposes of min-
2
imization it can be replaced by i t X it . This implies that
the values of F and å that minimize (A.4) can be calculated
as the minimizers of V b4F 1 å5 D Pi Pt 4X bit ƒ ‹0i Ft 52 . At the
APPENDIX A: EM ESTIMATION WITH AN
UNBALANCED PANEL AND DATA IRREGULARITIES jth step, this reduces to the usual principal component eigen-
value calculation where the missing data are replaced by their
In practice, when N is large one encounters various expectation conditional on the observed data and using the
data irregularities, including occasionally missing observa- parameter values from the previous iteration. If the full dataset
tions, unbalanced panels, and mixed frequency (for example, contains a subset that constitutes a balanced panel, then start-
Downloaded by [Linköping University Library] at 10:51 12 August 2013

monthly and quarterly) data. In this case, a modiŽ cation of ing values for Fb in the EM iteration can be obtained using
standard principal component estimation is necessary. To moti- estimates from the balanced panel subset.
vate the modiŽ cation, consider the least squares estimators of We now provide some additional details on the calcu-
å and Ft from (2.4) from a balanced panel. The objective lation of X bit for some important special cases. Let X i D
function is 4Xi 1 : : : 1 XiT 50 , and let X !i be the vector of observations on the
ith variable. Suppose that X !i D Ai X i for some known matrix
X
N X
T
V 4F 1 å5 D 4Xit ƒ ‹0i Ft 52 1 (A.1) Ai , as can be done in the cases of missing values and tempo-
ral aggregation, for example. Then E4X i — X ! 5 D E4X i — Xi 5 D
!
iD1 tD1

F ‹i C Ai 4Ai Ai 5 4X i ƒ Ai F‹i 5, where 4Ai Ai 5 is the general-

0 0 ƒ ! 0 ƒ

where ‹i is the ith row of å. (A.1) can be minimized by the ized inverse of Ai A0i . The particulars of these calculations are
usual eigenvalue calculations and Fbt are the principal compo- now presented for some important special cases. In the Ž rst
nents of Xt . four special cases discussed, this level of generality is unnec-
When the panel is unbalanced, least squares estimators of essary and the formula for X bit follows quite simply from the
Ft can be calculated from the objective function nature of the data irregularity.
A. Missing Observations. Suppose some observations on
X
N X
T
Xit are missing. Then, during iteration j, the elements of
V ! 4F 1 å5 D Iit 4Xit ƒ ‹0i Ft 52 1 (A.2)
iD1 tD1 the estimated balanced panel are constructed as X bit D X it if
b b b
Xit observed and Xit D ‹i Ft otherwise. The estimate of F
0

where Iit D 1 if Xit is available and 0 otherwise. Minimization is then updated by computing the eigenvectors correspond-
P bb bD
of (A.2) requires iterative methods. This appendix summarizes ing to the largest r eigenvalues of N ƒ1 i X i X i , where X i
an iterative method based on the EM algorithm that has proved 4Xbi1 1 X
bi2 1 : : : 1 X
biT 5 . The estimate of å is updated by the ordi-
0

to be easy and effective. nary least squares regression of X b onto this updated estimate
To motivate this EM algorithm, notice that V 4F 1 å5 is pro- of F .
portional to the log-likelihood under the assumption that Xit B. Mixed Monthly and Quarterly Data— I(0) Stock Vari-
are iid N 4‹0i Ft 1 15, in which case the least squares estimators ables. A series that is observed quarterly and is a stock vari-
are the Gaussian maximum likelihood estimators. Because V ! able would be the point-in-time level of a variable at the end
is just a missing data version of V and because minimization of the quarter, say, the level of inventories at the end of the
of V is computationally simple, a simple EM algorithm can quarter. If this series is I(0), then it is handled as in case A;
be constructed to minimize V ! . that is, it is treated as a monthly series with missing observa-
The jth iteration of the algorithm is deŽ ned as follows. Let tions in the Ž rst and second months of the quarter.
b and Fb denote estimates of å and F constructed from the
å C. Mixed Monthly and Quarterly Data— I(0) Flow Vari-
4j ƒ 15st iteration, and let ables. A quarterly  ow variable is the average (or sum)
of unobserved monthly values. If this series is I(0), it can
Q4X ! 1 Fb1 b
å1 F 1 å5 D EFb1 b — !
å 6V 4F 1 å5 X 71 (A.3) be treated as follows. The unobserved monthly series, Xit ,
q q
is measured only as the time aggregate Xit , where Xit D
q
where X ! denotes the full set of observed data and 41=354Xi1 tƒ2 C Xi1 tƒ1 C Xit 5 for t D 31 61 91 121 : : : 1 and Xit
EFb1 b — !
å 6V 4F 1 å5 X 7 is the expected value of the complete data is missing for all other values of t. In this case estima-
log-likelihood V 4F 1 å5, evaluated using the conditional den- tion proceeds as in case A but with X bit D ‹O 0i Fbt C eOit , where
sity of X — X ! evaluated at Fb and b eOit D Xi’ ƒ ‹O 0i 4Fb’ ƒ2 C Fb’ ƒ1 C Fb’ 5=3, where ’ D 3 when t D
q
å. The estimates of F and
b1 b
å at iteration j solve MinF 1 å Q4X ! 1 F å1 F 1 å5. 11 21 31 ’ D 6, when t D 41 51 6, and so forth.
 Stock and Watson: Macroeconomic Forecasting Using Diffusion Indexes 157

D. Mixed Monthly and Quarterly Data— I(1) stock vari- APPENDIX B: DATA DESCRIPTION
ables. Suppose that underlying monthly data are I(1) and let
Xit1 denote the quarterly Ž rst difference stock variable, assumed The time series used to construct the diffusion index fore-
to be measured in the third month of every quarter, and let casts discussed in Section 5 are presented here. The format is
Xit denote the monthly Ž rst difference of the variable. Then
q q as follows: series number, series mnemonic, data span used,
Xit D 4Xi1 tƒ2 C Xi1 tƒ1 C Xit 5 for t D 31 61 91 121 : : : 1 and X it is
transformation code, and brief series description. The transfor-
missing for all other values of t. In this case estimation pro-
bit D ‹O 0i Fbt C 41=35eOit , where eOit D mation codes are 1 D no transformation, 2 D Ž rst difference,
ceeds as in case A but with X
q O b’ ƒ2 C Fb’ ƒ1 C Fb’ 5, where ’ D 3 when t D 11 21 31 ’ D 4 D logarithm, 5 D Ž rst difference of logarithms, 6 D second
Xi’ ƒ ‹i 4F
0
difference of logarithms. An asterisk after the date denotes a
6, when t D 41 51 6, and so forth.
E. Mixed Monthly and Quarterly Data— I(1) Flow Vari- series that was included in the unbalanced panel but not the
ables. Construction of X bit is more difŽ cult here than in the balanced panel, either because of missing data or because of
earlier cases. Here the general regression formula given above gross outliers that were treated as missing data. The series
can be implemented after specifying X !i and Ai . Let the quar- either were taken directly from the DR I-McGraw–Hill Basic
q
terly Ž rst differences be denoted by Xit , which is assumed to Economics database, in which case the original mnemonics
be observed at the end of every quarter. The vector of obser- are used, or were produced by author calculations based on
q q q
vations is then X !i D 4Xi3 1 Xi6 1 : : : 1 Xi’ 50 , where ’ denotes the data from that database, in which case the author calcula-
tions and original DR I-McGraw series mnemonics are sum-
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month of the last quarterly observation. If the underlying quar-

terly data are averages of monthly series, and if the monthly marized in the data description Ž eld. The following abbrevia-
q
Ž rst differences are denoted by Xit , then Xit D 41=354Xi1 t C tions appear in the data deŽ nitions: SA D seasonally adjusted,
2Xi1 tƒ1 C 3Xitƒ2 C 2Xitƒ3 C Xitƒ4 5 for t D 31 61 91 121 : : : 1 and NSA D not seasonally adjusted, SAAR D seasonally adjusted
this implicitly deŽ nes the rows of Ai . Then the estimate of X i at an annual rate, FRB D Federal Reserve Board, AC D Author
is given by X bi D F‹i C A0i 4Ai A0i 5ƒ1 4X !i ƒ Ai F‹i 5. calculations.

Real output and income (Out)

1. ip 1959:01–1998:12 5 industrial production: total index (1992 D 100, sa)
2. ipp 1959:01–1998:12 5 industrial production: products, total (1992 D 100, sa)
3. ipf 1959:01–1998:12 5 industrial production: Ž nal products (1992 D 100, sa)
4. ipc 1959:01–1998:12 5 industrial production: consumer goods (1992 D 100, sa)
5. ipcd 1959:01–1998:12 5 industrial production: durable consumer goods (1992 D 100, sa)
6. ipcn 1959:01–1998:12 5 industrial production: nondurable consumer goods (1992 D 100, sa)
7. ipe 1959:01–1998:12 5 industrial production: business equipment (1992 D 100, sa)
8. ipi 1959:01–1998:12 5 industrial production: intermediate products (1992 D 100, sa)
9. ipm 1959:01–1998:12 5 industrial production: materials (1992 D 100, sa)
10. ipmd 1959:01–1998:12ü 5 industrial production: durable goods materials (1992 D 100, sa)
11. ipmnd 1959:01–1998:12 5 industrial production: nondurable goods materials (1992 D 100, sa)
12. ipmfg 1959:01–1998:12 5 industrial production: manufacturing (1992 D 100, sa)
13. ipd 1959:01–1998:12 5 industrial production: durable manufacturing (1992 D 100, sa)
14. ipn 1959:01–1998:12 5 industrial production: nondurable manufacturing (1992 D 100, sa)
15. ipmin 1959:01–1998:12 5 industrial production: mining (1992 D 100, sa)
16. iput 1959:01–1998:12 5 industrial production: utilities (1992– D 100, sa)
17. ipx 1967:01–1998:12ü 1 capacity util rate: total industry (% of capacity, sa)(frb)
18. ipxmca 1959:01–1998:12 1 capacity util rate: manufacturing, total (% of capacity, sa)(frb)
19. ipxdca 1967:01–1998:12ü 1 capacity util rate: durable mfg (% of capacity, sa)(frb)
20. ipxnca 1967:01–1998:12ü 1 capacity util rate: nondurable mfg (% of capacity, sa)(frb)
21. ipxmin 1967:01–1998:12ü 1 capacity util rate: mining (% of capacity, sa)(frb)
22. ipxut 1967:01–1998:12ü 1 capacity util rate: utilities (% of capacity, sa)(frb)
23. pmi 1959:01–1998:12 1 purchasing managers’ index (sa)
24. pmp 1959:01–1998:12 1 NAPM production index (percent)
25. gmpyq 1959:01–1998:12ü 5 personal income (chained) (series #52) (bil 92$, saar)
26. gmyxpq 1959:01–1998:12 5 personal income less transfer payments (chained) (#51) (bil 92$, saar)
Employment and hours (EMP)
27. lhel 1959:01–1998:12 5 index of help-wanted advertising in newspapers (1967 D 100; sa)
28. lhelx 1959:01–1998:12 4 employment: ratio; help-wanted ads:no. unemployed clf
29. lhem 1959:01–1998:12 5 civilian labor force: employed, total (thous., sa)
 158 Journal of Business & Economic Statistics, April 2002

30. lhnag 1959:01–1998:12 5 civilian labor force: employed, nonagric. industries (thous., sa)
31. lhur 1959:01–1998:12 1 unemployment rate: all workers, 16 years & over (%, sa)
32. lhu680 1959:01–1998:12 1 unemploy. by duration: average (mean) duration in weeks (sa)
33. lhu5 1959:01–1998:12 1 unemploy. by duration: persons unempl. less than 5 wks (thous., sa)
34. lhu14 1959:01–1998:12 1 unemploy. by duration: persons unempl. 5 to 14 wks (thous., sa)
35. lhu15 1959:01–1998:12 1 unemploy. by duration: persons unempl. 15 wks C (thous., sa)
36. lhu26 1959:01–1998:12 1 unemploy. by duration: persons unempl. 15 to 26 wks (thous., sa)
37. lpnag 1959:01–1998:12 5 employees on nonag. payrolls: total (thous., sa)
38. lp 1959:01–1998:12 5 employees on nonag. payrolls: total, private (thous., sa)
39. lpgd 1959:01–1998:12 5 employees on nonag. payrolls: goods-producing (thous., sa)
40. lpmi 1959:01–1998:12ü 5 employees on nonag. payrolls: mining (thous., sa)
41. lpcc 1959:01–1998:12 5 employees on nonag. payrolls: contract construction (thous., sa)
42. lpem 1959:01–1998:12 5 employees on nonag. payrolls: manufacturing (thous., sa)
43. lped 1959:01–1998:12 5 employees on nonag. payrolls: durable goods (thous., sa)
44. lpen 1959:01–1998:12 5 employees on nonag. payrolls: nondurable goods (thous., sa)
45. lpsp 1959:01–1998:12 5 employees on nonag. payrolls: service-producing (thous., sa)
46. lptu 1959:01–1998:12ü 5 employees on nonag. payrolls: trans. & public utilities (thous., sa)
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47. lpt 1959:01–1998:12 5 employees on nonag. payrolls: wholesale & retail trade (thous., sa)
48. lpfr 1959:01–1998:12 5 employees on nonag. payrolls: Ž nance, insur. & real estate (thous., sa)
49. lps 1959:01–1998:12 5 employees on nonag. payrolls: services (thous., sa)
50. lpgov 1959:01–1998:12 5 employees on nonag. payrolls: government (thous., sa)
51. lw 1964:01–1998:12ü 2 avg. weekly hrs. of prod. wkrs.: total private (sa)
52. lphrm 1959:01–1998:12 1 avg. weekly hrs. of production wkrs.: manufacturing (sa)
53. lpmosa 1959:01–1998:12 1 avg. weekly hrs. of prod. wkrs.: mfg., overtime hrs. (sa)
54. pmemp 1959:01–1998:12 1 NAPM employment index (percent)

Real retail, manufacturing and trade sales (RTS)

55. msmtq 1959:01–1998:12 5 manufacturing & trade: total (mil of chained 1992 dollars)(sa)
56. msmq 1959:01–1998:12 5 manufacturing & trade: manufacturing; total (mil of chained 1992 dollars)(sa)
57. msdq 1959:01–1998:12 5 manufacturing & trade: mfg; durable goods (mil of chained 1992 dollars)(sa)
58. msnq 1959:01–1998:12 5 manufact. & trade: mfg; nondurable goods (mil of chained 1992 dollars)(sa)
59. wtq 1959:01–1998:12 5 merchant wholesalers: total (mil of chained 1992 dollars)(sa)
60. wtdq 1959:01–1998:12 5 merchant wholesalers: durable goods total (mil of chained 1992 dollars)(sa)
61. wtnq 1959:01–1998:12 5 merchant wholesalers: nondurable goods (mil of chained 1992 dollars)(sa)
62. rtq 1959:01–1998:12 5 retail trade: total (mil of chained 1992 dollars)(sa)
63. rtnq 1959:01–1998:12 5 retail trade: nondurable goods (mil of 1992 dollars)(sa)
Consumption (PCE)
64. gmcq 1959:01–1998:12 5 personal consumption expend (chained)-total (bil 92$, saar)
65. gmcdq 1959:01–1998:12 5 personal consumption expend (chained)-total durables (bil 92$, saar)
66. gmcnq 1959:01–1998:12 5 personal consumption expend (chained)-nondurables (bil 92$, saar)
67. gmcsq 1959:01–1998:12 5 personal consumption expend (chained)-services (bil 92$, saar)
68. gmcanq 1959:01–1998:12 5 personal cons expend (chained)-new cars (bil 92$, saar)
Housing starts and sales (HSS)
69. hsfr 1959:01–1998:12 4 housing starts: nonfarm (1947–58); total farm & nonfarm (1959-) (thous., sa)
70. hsne 1959:01–1998:12 4 housing starts: northeast (thous.u.) s.a.
71. hsmw 1959:01–1998:12 4 housing starts: midwest (thous.u.) s.a.
72. hssou 1959:01–1998:12 4 housing starts: south (thous.u.) s.a.
73. hswst 1959:01–1998:12 4 housing starts: west (thous.u.) s.a.
74. hsbr 1959:01–1998:12 4 housing authorized: total new priv housing units (thous., saar)
75. hsbne 1960:01–1998:12ü 4 houses authorized by build. permits: northeast (thous.u.) s.a.
76. hsbmw 1960:01–1998:12ü 4 houses authorized by build. permits: midwest (thous.u.) s.a.
77. hsbsou 1960:01–1998:12ü 4 houses authorized by build. permits: south (thous.u.) s.a.
78. hsbwst 1960:01–1998:12ü 4 houses authorized by build. permits: west (thous.u.) s.a.
79. hns 1963:01–1998:12ü 4 new 1-family houses sold during month (thous, saar)
80. hnsne 1973:01–1998:12ü 4 one-family houses sold: northeast (thous.u., s.a.)
81. hnsmw 1973:01–1998:12ü 4 one-family houses sold: midwest (thous.u., s.a.)
82. hnssou 1973:01–1998:12ü 4 one-family houses sold: south (thous.u., s.a.)
 Stock and Watson: Macroeconomic Forecasting Using Diffusion Indexes 159

83. hnswst 1973:01–1998:12ü 4 one-family houses sold: west (thous.u., s.a.)

84. hnr 1963:01–1998:12ü 4 new 1-family houses, month’s supply @ current sales rate (ratio)
85. hniv 1963:01–1998:12ü 4 new 1-family houses for sale at end of month (thous, sa)
86. hmob 1959:01–1998:12 4 mobile homes: manufacturers’ shipments (thous. of units, saar)
87. contc 1964:01–1998:12ü 4 construct. put in place: total priv & public 1987$ (mil$, saar)
88. conpc 1964:01–1998:12ü 4 construct. put in place: total private 1987$ (mil$, saar)
89. conqc 1964:01–1998:12ü 4 construct. put in place: public construction 87$ (mil$, saar)
90. condo9 1959:01–1998:10ü 4 construct. contracts: comm’l & indus.bldgs (mil.sq.ft. oor sp.; sa)
Real inventories and inventory-sales ratios ( Inv)
91. ivmtq 1959:01–1998:12 5 manufacturing & trade inventories: total (mil of chained 1992)(sa)
92. ivmfgq 1959:01–1998:12 5 inventories, business, mfg (mil of chained 1992 dollars, sa)
93. ivmfdq 1959:01–1998:12 5 inventories, business durables (mil of chained 1992 dollars, sa)
94. ivmfnq 1959:01–1998:12 5 inventories, business, nondurables (mil of chained 1992 dollars, sa)
95. ivwrq 1959:01–1998:12 5 manufacturing & trade inv: merchant wholesalers (mil of chained 1992 dollars)(s
96. ivrrq 1959:01–1998:12 5 manufacturing & trade inv: retail trade (mil of chained 1992 dollars)(sa)
97. ivsrq 1959:01–1998:12 2 ratio for mfg & trade: inventory/sales (chained 1992 dollars, sa)
98. ivsrmq 1959:01–1998:12 2 ratio for mfg & trade: mfg; inventory/sales (87$)(s.a.)
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99. ivsrwq 1959:01–1998:12 2 ratio for mfg & trade: wholesaler; inventory/sales (87$)(s.a.)
100. ivsrrq 1959:01–1998:12 2 ratio for mfg & trade: retail trade; inventory/sales (87$)(s.a.)
101. pmnv 1959:01–1998:12 1 napm inventories index (percent)
Orders and unŽ lled orders (Ord)
102. pmno 1959:01–1998:12 1 napm new orders index (percent)
103. pmdel 1959:01–1998:12 1 napm vendor deliveries index (percent)
104. mocmq 1959:01–1998:12 5 new orders (net)-consumer goods & materials, 1992 dollars (bci)
105. mdoq 1959:01–1998:12 5 new orders, durable goods industries, 1992 dollars (bci)
106. msondq 1959:01–1998:12 5 new orders, nondefense capital goods, in 1992 dollars (bci)
107. mo 1959:01–1998:12 5 mfg new orders: all manufacturing industries, total (mil$, sa)
108. mowu 1959:01–1998:12 5 mfg new orders: mfg industries with unŽ lled orders (mil$, sa)
109. mdo 1959:01–1998:12 5 mfg new orders: durable goods industries, total (mil$, sa)
110. mduwu 1959:01–1998:12 5 mfg new orders: durable goods indust with unŽ lled orders (mil$, sa)
111. mno 1959:01–1998:12 5 mfg new orders: nondurable goods industries, total (mil$, sa)
112. mnou 1959:01–1998:12 5 mfg new orders: nondurable gds ind. with unŽ lled orders (mil$, sa)
113. mu 1959:01–1998:12 5 mfg unŽ lled orders: all manufacturing industries, total (mil$, sa)
114. mdu 1959:01–1998:12 5 mfg unŽ lled orders: durable goods industries, total (mil$, sa)
115. mnu 1959:01–1998:12 5 mfg unŽ lled orders: nondurable goods industries, total (mil$, sa)
116. mpcon 1959:01–1998:12 5 contracts & orders for plant & equipment (bil$, sa)
117. mpconq 1959:01–1998:12 5 contracts & orders for plant & equipment in 1992 dollars (bci)
Stock prices (SPr)
118. fsncom 1959:01–1998:12 5 NYSE common stock price index: composite 412=31=65 D 505
119. fsnin 1966:01–1998:12ü 5 NYSE common stock price index: industrial 412=31=65 D 505
120. fsntr 1966:01–1998:12ü 5 NYSE common stock price index: transportation 412=31=65 D 505
121. fsnut 1966:01–1998:12ü 5 NYSE common stock price index: utility 412=31=65 D 505
122. fsnŽ 1966:01–1998:12ü 5 NYSE common stock price index: Ž nance 412=31=65 D 505
123. fspcom 1959:01–1998:12 5 S&P’s common stock price index: composite (1941–43 D 10)
124. fspin 1959:01–1998:12 5 S&P’s common stock price index: industrials (1941–43 D 10)
125. fspcap 1959:01–1998:12 5 S&P’s common stock price index: capital goods (1941–43 D 10)
126. fsptr 1970:01–1998:12ü 5 S&P’s common stock price index: transportation (1970 D 10)
127. fsput 1959:01–1998:12 5 S&P’s common stock price index: utilities (1941–43 D 10)
128. fspŽ 1970:01–1998:12ü 5 S&P’s common stock price index: Ž nancial 41970 D 105
129. fsdxp 1959:01–1998:12 1 S&P’s composite common stock: dividend yield (% per annum)
130. fspxe 1959:01–1998:12 1 S&P’s composite common stock: price-earnings ratio (%, nsa)
131. fsnvv3 1974:01–1998:07ü 5 NYSE mkt composition: reptd share vol by size, 5000 C shrs,%
Exchange rates (EXR)
132. exrus 1959:01–1998:12 5 United States effective exchange rate (merm) (index no.)
133. exrger 1959:01–1998:12 5 foreign exchange rate: Germany (deutsche mark per U.S.$)
 160 Journal of Business & Economic Statistics, April 2002

134. exrsw 1959:01–1998:12 5 foreign exchange rate: Switzerland (swiss franc per U.S.$)
135. exrjan 1959:01–1998:12 5 foreign exchange rate: Japan (yen per U.S.$)
136. exruk 1959:01–1998:12ü 5 foreign exchange rate: United Kingdom (cents per pound)
137. exrcan 1959:01–1998:12 5 foreign exchange rate: Canada (canadian $ per U.S.$)
Interest rates ( Int)
138. fyff 1959:01–1998:12ü 2 interest rate: federal funds (effective) (% per annum, nsa)
139. fycp90 1959:01–1998:12ü 2 interest rate: 90 day commercial paper, (ac) (% per ann, nsa)
140. fygm3 1959:01–1998:12ü 2 interest rate: U.S. treasury bills, sec mkt, 3-mo. (% per ann, nsa)
141. fygm6 1959:01–1998:12ü 2 interest rate: U.S. treasury bills, sec mkt, 6-mo. (% per ann, nsa)
142. fygt1 1959:01–1998:12ü 2 interest rate: U.S. treasury const maturities, 1-yr. (% per ann, nsa)
143. fygt5 1959:01–1998:12 2 interest rate: U.S. treasury const maturities, 5-yr. (% per ann, nsa)
144. fygt10 1959:01–1998:12 2 interest rate: U.S. treasury const maturities, 10-yr. (% per ann, nsa)
145. fyaaac 1959:01–1998:12 2 bond yield: moody’s aaa corporate (% per annum)
146. fybaac 1959:01–1998:12 2 bond yield: moody’s baa corporate (% per annum)
147. fwaŽ t 1973:01–1994:04ü 1 weighted avg foreign interest rate (%, sa)
148. fyfha 1959:01–1998:12 2 secondary market yields on fha mortgages (% per annum)
149. sfycp 1959:01–1998:12 1 spread fycp - fyff
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150. sfygm3 1959:01–1998:12 1 spread fygm3 - fyff

151. sfygm6 1959:01–1998:12 1 spread fygm6 - fyff
152. sfygt1 1959:01–1998:12 1 spread fygt1 - fyff
153. sfygt5 1959:01–1998:12 1 spread fygt5 - fyff
154. sfygt10 1959:01–1998:12 1 spread fygt10 - fyff
155. sfyaaac 1959:01–1998:12 1 spread fyaaac - fyff
156. sfybaac 1959:01–1998:12 1 spread fybaac - fyff
157. sfyfha 1959:01–1998:12 1 spread fyfha - fyff
Money and credit quantity aggregates (Mon)
158. fm1 1959:01–1998:12 6 money stock: m1 (curr, trav.cks, dem dep, other ck’able dep) (bil$, sa)
159. fm2 1959:01–1998:12 6 money stock: m2 (m1 C o’nite rps, euro$, g/p&b/d mmmfs&sav&sm time dep) (bil$,
160. fm3 1959:01–1998:12 6 money stock: m3 (m2 C lg time dep, term rp’s&inst only mmmfs) (bil$, sa)
161. fml 1959:01–1998:09ü 6 money stock: l (m3 C other liquid assets) (bil$, sa)
162. fm2dq 1959:01–1998:12 5 money supply-m2 in 1992 dollars (bci)
163. fmfba 1959:01–1998:12 6 monetary base, adj for reserve requirement changes (mil$, sa)
164. fmrra 1959:01–1998:12 6 depository inst reserves: total, adj for reserve req chgs (mil$, sa)
165. fmrnbc 1959:01–1998:12 6 depository inst reserves: nonborrow C ext cr, adj res req cgs (mil$, sa)
166. fcls 1973:01–1998:12ü 5 loans & sec @ all coml banks: total (bils, sa)
167. fcsgv 1973:01–1998:12ü 5 loans & sec @ all coml banks: U.S. govt securities (bil$, sa)
168. fclre 1973:01–1998:12ü 5 loans & sec @ all coml banks: real estate loans (bil$, sa)
169. fclin 1973:01–1998:12ü 5 loans & sec @ all coml banks: loans to individuals (bil$, sa)
170. fclnbf 1973:01–1994:01ü 5 loans & sec @ all coml banks: loans to nonbank Ž n inst (bil$, sa)
171. fclnq 1959:01–1998:12ü 5 commercial & industrial loans outstanding in 1992 dollars (bci)
172. fclbmc 1959:01–1998:12ü 1 wkly rp lg com’l banks: net change com’l & indus loans (bil$, saar)
173. cci30m 1959:01–1995:09ü 1 consumer instal. loans: delinquency rate, 30 days & over, (%, sa)
174. ccint 1975:01–1995:09ü 1 net change in consumer instal cr: total (mil$, sa)
175. ccinv 1975:01–1995:09ü 1 net change in consumer instal cr: automobile (mil$, sa)
176. ccinrv 1980:01–1995:09ü 1 net change in consumer instal cr: revolving (mil$, sa)
Price indexes (Pri)
177. pmcp 1959:01–1998:12 1 napm commodity prices index (percent)
178. pwfsa 1959:01–1998:12 6 producer price index: Ž nished goods (82 D 100, sa)
179. pwfcsa 1959:01–1998:12 6 producer price index: Ž nished consumer goods (82 D 100, sa)
180. pwimsa 1959:01–1998:12ü 6 producer price index: intermed mat. supplies & components (82 D 100, sa)
181. pwcmsa 1959:01–1998:12ü 6 producer price index: crude materials (82 D 100, sa)
182. pwfxsa 1967:01–1998:12ü 6 producer price index: Ž nished goods, excl. foods (82 D 100, sa)
183. pw160a 1974:01–1998:12ü 6 producer price index: crude materials less energy (82 D 100, sa)
184. pw150a 1974:01–1998:12ü 6 producer price index: crude nonfood mat less energy (82 D 100, sa)
185. psm99q 1959:01–1998:12 6 index of sensitive materials prices 41990 D 1005 (bci-99a)
186. punew 1959:01–1998:12 6 cpi-u: all items (82–84 D 100, sa)
 Stock and Watson: Macroeconomic Forecasting Using Diffusion Indexes 161

187. pu81 1967:01–1998:12ü 6 cpi-u: food & beverages (82–84 D 100, sa)
188. puh 1967:01–1998:12ü 6 cpi-u: housing (82–84 D 100, sa)
189. pu83 1959:01–1998:12 6 cpi-u: apparel & upkeep (82–84 D 100, sa)
190. pu84 1959:01–1998:12 6 cpi-u: transportation (82–84 D 100, sa)
191. pu85 1959:01–1998:12 6 cpi-u: medical care (82–84 D 100, sa)
192. puc 1959:01–1998:12 6 cpi-u: commodities (82–84 D 100, sa)
193. pucd 1959:01–1998:12 6 cpi-u: durables (82–84 D 100, sa)
194. pus 1959:01–1998:12 6 cpi-u: services (82–84 D 100, sa)
195. puxf 1959:01–1998:12 6 cpi-u: all items less food (82–84 D 100, sa)
196. puxhs 1959:01–1998:12 6 cpi-u: all items less shelter (82–84 D 100, sa)
197. puxm 1959:01–1998:12 6 cpi-u: all items less medical care (82–84 D 100, sa)
198. pcgold 1975:01–1998:12ü 6 commodities price: gold, london noon Ž x, avg of daily rate, $ per oz
199. gmdc 1959:01–1998:12 6 pce, impl pr de : pce 41987 D 1005
200. gmdcd 1959:01–1998:12 6 pce, impl pr de : pce; durables 41987 D 1005
201. gmdcn 1959:01–1998:12 6 pce, impl pr de : pce; nondurables 41987 D 1005
202. gmdcs 1959:01–1998:12 6 pce, impl pr de : pce; services 41987 D 1005
Average hourly earnings (AHE)
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203. leh 1964:01–1998:12ü 6 avg hr earnings of prod wkrs: total private nonagric ($, sa)
204. lehcc 1959:01–1998:12 6 avg hr earnings of constr wkrs: construction ($, sa)
205. lehm 1959:01–1998:12 6 avg hr earnings of prod wkrs: manufacturing ($, sa)
206. lehtu 1964:01–1998:12ü 6 avg hr earnings of nonsupv wkrs: trans & public util ($, sa)
207. lehtt 1964:01–1998:12ü 6 avg hr earnings of prod wkrs: wholesale & retail trade (sa)
208. lehfr 1964:01–1998:12ü 6 avg hr earnings of nonsupv wkrs: Ž nance, insur, real est ($, sa)
209. lehs 1964:01–1998:12ü 6 avg hr earnings of nonsupv wkrs: services ($, sa)

Miscellaneous (Oth)
210. fste 1986:01–1998:12ü 5 U.S. mdse exports: total exports (f.a.s. value) (mil.$, s.a.)
211. fstm 1986:01–1998:12ü 5 U.S. mdse imports: general imports (c.i.f. value) (mil.$, s.a.)
212. ftmd 1986:01–1998:12ü 5 U.S. mdse imports: general imports (customs value) (mil.$, s.a.)
213. fstb 1986:01–1998:12ü 2 U.S. mdse trade balance: exports less imports (fas/cif) (mil.$, s.a.)
214. ftb 1986:01–1998:12ü 2 U.S. mdse trade balance: exp. (fas) less imp. (custom) (mil.$, s.a.)
215. hhsntn 1959:01–1998:12 1 u. of mich. index of consumer expectations (bcd-83)

[Received May 2000. Revised March 2001.] (1998), “Lets Get Real: A Dynamic Factor Analytical Approach to
Disaggregated Business Cycle,” Review of Economic Studies, 65, 453–474.
Fuhrer, J. C. (1995), “The Phillips Curve is Alive and Well,” New England
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