Forecasting contemporaneous aggregates with stochastic aggregation weights

Forecasting contemporaneous aggregates with stochastic aggregation weights

Ralf Brliggemann

a,*,

Helmut Llitkepohl

b

• Department of Economics, University of Konstanz, Box 129, D-78457 Konstanz, Germany

b Department of Economics, Freie Universitiit Berlin and DIW Berlin, Mol1renstr. 58, D-10 117 Berlin, Germany

ABSTRACT

Keywords:

Aggregation Autoregressive process Mean squared error

Many contemporaneously aggregated variables have stochastic aggregation weights.

We compare different forecasts for such variables, including univariate forecasts of the aggregate, a multivariate forecast of the aggregate that uses information from the disaggregated components, a forecast which aggregates a multivariate forecast of the disaggregate components and the aggregation weights, and a forecast which aggregates univariate forecasts of individual disaggregate components and the aggregation weights.

In empirical illustrations based on aggregate GOP and money stock series, we find forecast mean squared error reductions when information in the stochastic aggregation weights is used.

1. Introduction

Many economic variables which are contemporaneous aggregates of a number of disaggregate variables have time-varying aggregation weights. For example, the Euro- pean Union (EU) growth rate is an average of the growth rates of the individual member states, weighted by the rel- ative shares of overall output. The EU unemployment rate is the weighted average of the individual member states' unemployment rates, with the weights being the relative shares of the respective labor forces. As another example, consider the North American output, which is the sum of the outputs of the northern American countries, weighted by the exchange rates. In these examples, the aggregation weights are actually best thought of as stochastic.

Despite the stochastic nature of the weights of many aggregates. most previous studies on forecasting contem- poraneously aggregated variables have focused on aggre- gation with fixed, time-invariant weights. Examples are Ansley. Spivey, and Wrobleski (1977), Tiao and Guttman

*

Corresponding author. Tel.: +497531882643; fax: +497531883324.

[-mail addresses: ralf.brueggemann@uni-konstanz.de (R. Brliggemann). hluetkepohl@diw.de (H. Llitkepohl).

(1980), Wei and Abraham (1981), Kohn (1982). and Liitke- pohl (1984a,b. 1986. 1987). See also the survey by Li.itke- pohl (2010). These studies suggest that. theoretically, taking disaggregate information into account is help- ful for reducing the forecast mean squared error (MSE).

However, specification and estimation uncertainty may reduce or even reverse the gains. particularly when higher- dimensional multivariate models are fitted to disaggre- gate data. Therefore. some studies have also compared aggregates of univariate forecasts of the disaggregate com- ponents and found that such forecasts may outperform aggregated multivariate forecasts. Parameter reduction methods such as subset vector autoregressions. as in Hubrich (2005). or factor models, as in Hendry and Hubrich (2011), have also been considered in this context. The re- sults are not uniform across studies and depend to some extent on the data generation process (DGP). Overall, there is evidence that taking disaggregate information into ac- count can improve the forecast efficiency for contempora- neous aggregates with fixed weights. as long as methods are used which limit the estimation and specification uncertainty. Empirical studies confirming this conclusion include those by Marcellino, Stock, and Watson (2003). Es- pasa. Senra, and Albacete (2002) and Carson. Cenesizoglu.

and Parker (2011).

Ersch. in: International Journal of Forecasting ; 29 (2013), 1. - S. 60-68 http://dx.doi.org/10.1016/j.ijforecast.2012.05.007

Konstanzer Online-Publikations-System (KOPS) URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-221868

The fact that many aggregates have time-varying weights was recognized by Uitkepohl (2011), who devel- oped a general framework for the comparison of predic- tors for such aggregates based on aggregate and disaggre- gate information. However, in that framework, the process generating the time-varying, possibly stochastic weights is not considered explicitly. Hence, any information in that process is' ignored or only indirectly taken into account for forecasting purposes. In practice, such an approach has its advantages if the aggregation weights are either unavail- able or unobservable. On the other hand, there are also many cases where past aggregation weights are available. In this study, we focus on that case and investigate whether it is worthwhile to take the information in the weights into account explicitly when forecasting.

There are a number of different predictors which can be used in this situation. For instance, one may model the disaggregate series and the aggregation weights separately, forecast them, and then aggregate the forecasts using the predicted weights, or one may construct a joint model for the disaggregate series and the aggregation weights and aggregate the forecasts from such a model.

Obviously, this forecasting strategy can rapidly lead to very high-dimensional models, even if only a few disaggregate components are involved, and in practice we often have to deal with very large panels of dis aggregate components, as the aforementioned examples suggest. Hence, one may instead consider forecasting all components and aggregation weights with univariate models and then aggregating thes.e forecasts. A range of other possibilities may be useful, and some of them will be discussed in Section 2.

The main objective of this study is'to check whether there is the potential for it to be beneficial for forecasting to take the information in stochastically varying aggregation weights into account. Therefore, we will focus on a small number of plausible predictors and compare their forecasting efficiencies on a limited set of example series.

We find that taking the information in the aggregation weights into account explicitly may indeed help to improve the forecasts in a MSE sense. It is not the purpose of this study to suggest a universally optimal predictor, but rather to point out a source of information which it may be worthwhile to consider. We are fully aware that, in practice, the most suitable predictors may be different for each specific forecasting problem.

The structure of the study is as follows. In Section 2, some possible predictors for contemporaneous aggregates with stochastic weights are presented and discussed. In Section 3, a small set of real life examples is investigated, and it is demonstrated that taking the information in the stochastic aggregation weights into account explicitly helps to improve the forecast efficiency. Finally, Section 4 concludes.

The following notation is used throughout. E denotes the expectation operator and Er denotes the corresponding conditional operator, which conditions on information up to and including period r. The natural logarithm is denoted by log, and 6. is the differencing operator. We use the following abbreviations: AR for autoregressive, VAR for vector autoregressive, DCP for data generation process,

MSE and RMSE for mean squared error and root mean squared error, respectively, CDP for real gross domestic product, PPP for purchasing power parity, US for United States of America, and NAFTA for North American Free Trade Agreement.

2. Possible predictors

Suppose that Yt

=

(Ylt, ... , YKt)' is the vector of dis- aggregate component series and the aggregate of interest is at = w;Yt, where Wt = (WH, "" WKt)' is a vector of stochastic (time-varying) weights. Furthermore, suppose that Yt and Wt are generated by stochastic processes, or possibly by a joint stochastic process. In the empirical sec- tion, it will be assumed that all DCPs are AR or VAR pro- cesses which can at least be approximated well by finite order versions. However, for discussing the predictors to be used later, such an assumption is not required. The fol- lowing l1-step predictors at origin r will be considered:

Univariate forecast. Direct forecast of the univariate process at:

ar+"lr = E(a r

+"

lar , at-1, ... ) = Er (ar+,,).

This predictor serves as a benchmark. It does not use any disaggregate information. If such information is useful, then forecasts based on it should improve on this predictor.

Multivariate linear forecast. Linear forecast taking into ac- count disaggregate information:

a~+"lr = E(ar+hlar, ar-1, ... ,Yr, Yr-J, .•. ),

that is, a multivariate model is fitted to (at, y;)' and used for forecasting. The first component of the vector forecast is a~+"lt' As with Liitkepohl (2011), the forecast may be based on selected components ofYt only, rather than the full dis- aggregate vector.

Aggregation of multivariate forecasts. Forecast based on multivariate predictions of disaggregate components and weights:

a~';:::lt = E(wt+hlwr, Wr-1,·, .)'E(Yr+"IYr,Yr-1, .. ,).

Aggregation of univariate forecasts. Forecast based on univariate predictions for disaggregate components and weights:

[

Er(Y1. r+h)]

a~~~hlr = [Et(W1.t+h),···, Er(WK,r+")] : ' Er(YK,r+h) where Er(Wk,r+") = E(Wk.r-l-hlwk,r, Wk,r-J" . . ), etc.

The last predictor is included because it may not be possible to construct multivariate forecasts of Yt and Wt if there are many disaggregate components. Of course, other predictors are conceivable. For example, one could use a multivariate forecast of Yt 'and still predict the components of Wt with univariate models, or vice versa.

Also, it is possible that the disaggregate components and aggregation weights are related. In that case, it may be plausible to model and forecast the joint process (y;, w;)', then compute the aggregate forecast on that basis. The obvious disadvantage here is that it quickly leads to a very

62

high-dimensional prediction problem. As was mentioned earlier, the objective of this study is not to find a universally optimal predictor for the case of aggregates with stochastic weights, as we believe that the most suitable predictor will depend on the problem at hand. The small selection of predictors described in the foregoing is enough to enable us to make our main points. Hence, we limit our attention to them. We will, however, consider the possibility of structural shifts in the aggregation weights and use robustified versions of the forecasts which account for structural breaks. Details of this will be provided in the next section.

3. Empirical examples

Two examples based on real economic data are considered. In the first, forecasts of real GDP growth in the NAFTA are studied, while the second is based on European money stock variables. In both examples, only three component series are aggregated. With such a small number of disaggregate components, multivariate methods based on VARs are still feasible, and may in fact have an advantage over univariate methods. This is the reason why we have chosen these examples, even though there are many examples in practice where one has many more components.

3.1. NAFfA real GDP growth

Quarterly real GDP data for the three NAFTA countries US, Canada and Mexico, measured at price levels and PPPs of 2005, are considered. Details of the data sources are given in Appendix A.1. The aggregate NAFTA real GDP growth rate is computed by aggregating growth rates using real GDP shares as weights. More precisely,

(i)

NAFfA "qC-l ⁽ⁱ⁾

!:llog qc = ~ NAFfA !:llog qc ,

i qC-l

(3.1)

where q~i) denotes output in country i, with i

=

US, Canada, Mexico. Notice that the weights are based on the output share in the previous period, as with Beyer, Doornik, and Hendry (2001), so that the weights are actually known for one-step-ahead forecasts.¹ Data are available for the period 1970Q1-201OQ4, although NAFTA only started in 1994. [n what follows, only data from 1985Q1 are used, in order to alleviate potential problems related to structural breaks.

The three disaggregate series and the aggregate series are plotted in Fig. 1, and the aggregation weights are depicted in Fig. 2. Apparently the weights vary substantially and are quite persistent. [n fact, augmented Dickey-Fuller tests (results not shown) suggest that the Canadian series has a unit root. This persistence may also be due to level shifts

1 In practice, the final GOP figures are typically not available at the time when the forecasts are performed because revisions continue to be made for a number of years. We ignore this problem here and use the data from the sources given in Appendix A.1. Otherwise. issues of now casting would become relevant for variables such as GOP. In turn. the issues of interest in the present study may also be important for nowcasting (see, e.g .. Castle

& Hendry. 2010).

rather than unit roots, and the graphs in Fig. 2 suggest that there may be level shifts in the weights during our sample period. Therefore we also use forecasts which are robustified for possible structural breaks in the weights.

Details will be given later.

We conduct a recursive pseudo out-of-sample forecast- ing experiment for growth rates of GDP and log-levels based on the forecasts of the growth rates. Log-levels are included because they may be of independent interest, and the relative rankings of forecasts of log-levels and growth rates more than one period ahead may differ (Clements

& Hendry, 1993). Estimation and model selection are re- peated for every sample considered. We use data from 1985Q1 onwards, and the actual starts of the estimation periods are adjusted according to the presample values needed. We fit AR and VAR processes only, and choose the lag order by the model selection criteria AIC and SC, with the first one being more generous and the second one more restrictive if they differ. The maximum lag length is four in all cases. Potential breaks and outliers in the time series or weights are ignored in our baseline forecast comparison.

Sensitivity checks accounting for such data features will be reported at the end of this section. The estimation and eval- uation period varies because we wanted to check the sen- sitivity of the results to changes in the forecast period. To check the effect of the recent recession on the outcome, one set of results is reported using data only until2007Q4 and another one with data until 201OQ4. [n both cases there are evaluation periods of different lengths, to investigate the sensitivity of the results with respect to variations in the forecast period. As an additional sensitivity check, we also consider forecasts evaluated on rolling periods of five years. The forecasting horizons are h

=

^{1 and h = 4.}

RMSEs relative to the univariate AR benchmark fore- casts for different evaluation periods are presented in Fig. 3. They are based on the models selected by the A[e.

The RMSEs for growth rates and log-levels are identical for forecast horizon h = 1, while they can be quite dif- ferent and may not even have the same relative rankings for h > 1, as was pointed out by Clements and Hendry (1993). The panels in the left-hand column of Fig. 3 are based on forecast evaluation periods with an end point of 2007Q4, the middle column shows the corresponding RMSEs obtained for an evaluation period up to 201 OQ4, and the right-hand column contains the relative RMSEs based on 5-year rolling window evaluation periods. The horizon- tal axis of each panel shows the starting points of the evalu- ation periods, ranging from 1995Q1 to 2005Q1. For exam- ple, in the left-hand panels, where the evaluation period ends in 2007, the RMSEs for 1995Q1 and h

=

1 are based on 49 forecasts, whereas the RMSEs for 2005Q1 are only based on 9 forecasts.² Analogous comments apply to the middle column of Fig. 3. On the other hand, the RMSEs in the right-hand column of Fig. 3 are each based on a con- stant number of20 forecasts.

2 We have chosen to use the same number of forecasts at all forecast horizons. and thus for 11 = 1 and the initial split into estimation and forecasting samples. we use 49 forecasts for 199sQl-2007Q1. while for h = 4 we also use 49 forecasts for 199sQ4-2007Q4 (see e.g. Hubrich.

2005. for a similar setup).

OYAGG OYCAN

2,--- - - - ,

3,---,

-1

-2

-4 -6

-8 -2 -h-,.-,.-,--,-,-,-"-,.-,.-,--,-,-,-,,-,.-,.-,--,-,-,-,,c-!

86 88 90 92 94 96 98 00 02 04 06 08 10 86 88 90 92 94 96 98 00 02 04 06 08 10 Fig_ 1_ GOP growth rates of NAFrA and NAFrA countries.

0.85 , - - - -- - - -- - - ,

0.84

0.83

0.82

0.09

0.08 ____ ._"

0.07

0.06

1 -

wus ----WCPN - -WMEX

1

86 88 90 92 94 96 98 00 02 04 06 08 10 Fig. 2. Weights for NAFrA GOP growth rates.

The results in Fig. 3 show that the predictors which utilize disaggregate information are typically better than the benchmark if the evaluation period ends before 2008.

In fact, in the left and right columns, the predictors which

& t f th . t' . I t muir d u,,;

use lorecas so e aggl ega Ion welg 1 s, a_r+_h1r an a_r+_h1r,

are often superior to those which do not forecast the weights. Of course, for h = 1 the weights are known at the time of the forecast, because the lagged shares are used (see Eq. (3.1 )). Therefore, it is important to note that efficiency gains are also obtained for forecast horizons of 11 = 4. When only data until 2007Q4 are used (left-hand panels of Fig. 3), that is, when the recent crisis period is excluded, a~~hlr provides the smallest RMSEs for most of the evaluation periods. This holds for both growth rates

and log-levels. The superior performance of the predictor which aggregates univariate forecasts of the disaggregate components and the weights may reflect the small sample size used for some of the forecasts. For example, when a long evaluation period starting in 1995Q1 is used, the associated estimation and specification period from 1985Ql-1994Q4 is rather small and leaves only a sample size of T = 40 when forecasts for 1995 are determined. Fitting three-dimensional VARs with such a small sample period may well lead to large estimation uncertainty and a reduced forecast precision relative to a predictor which is based exclusively on univariate forecasts.

It must be noted, however, that the situation is slightly different for evaluation periods up to 201OQ4 (middle

64

Forecast evaluatioll period until 2007Q4

Forecast evaluation period until 201OQ4

5-year rolling evaluatiOll window

~~---, ~r_---' ~r_---~

gro(~ili ;·~te ~ ':~~:;:\:i.:.~:~:::~:::~::.~~.--:.- .-: .... ~y . ...

^'": '.;::~:.:.<;l::::<;l:::.'l.::.~:::\iI""'.-0"""~':"'YI'::'

" I---~.= .. ~ ... '"'" .. '"'" .. ""'::-:""'--'----"""""'--

ci 'n-..

~ 1995 1997 1999 2001 2003 2005 d ^{19950 1997 1999 2001 2003 2005} d '::'."" •• :---:-::'.':":.,:----:-:'.=-= •• - - - ,2::-:00:-:"'---,2:c:O"='O':---d:2005

~~---,

Nr_---,

~r_---~

h = 4,

growth rate ~

;a;-: ::e:: ::e: ...

o-... o::: :O::::O::::ij:::ij:::

m ci

d ^{1995 1997 1999 2001 2003 2005} d ^{1995 1997 1999 2001 2003 2006} d ~,.:-:-•• :---::,.':":.,:----:-::,.':": •• ---2::'OO':":'---:2:-:0':":O':--~200·5

~r_---, ~r_---,

'":

-.

h=4 log-lev~l

"

m ci

..

ci _{1995 1997 1999 2001 2003 2005} ~L---₁₉₉₅ __ ₁₉₉₇ ---~ ~L---_{1999 2001 200l 2005 1995} _ _ ₁₉₉₇ - - - -_ _ ₁₉₉₉ ---~ _{2001 200l}

2005

Fig. 3. RMSEs relative to univariate rorecasts ror NAFrA CDP; horizontal axis: start' orrorecast evaluation period (total sample period: 1985Ql-2010Q4.

lag order selection based on the AIC).

column of Fig. 3). In that case, the forecasts based on disaggregate information lead to only small gains, or are even inferior to the benchmark. For the 5-year rolling window evaluation period (right-hand column of Fig. 3), the situation is consistent with the previous results, in that a~~hlr is typically optimal if the evaluation period ends prior to 2008, but, at least for one-step forecasts, the optimality is lost when the evaluation period extends into the crisis years. One may conjecture that the crisis in 2008-2010 has caused structural change which is reflected in the change in the relative rankings of the forecasts.

Hence, one may want to account for structural breaks, and we will consider robustified forecasts shortly.

Before that, it may be worth commenting on the results obtained by using SC for lag order selection. They are a bit different from those based on the AIC, but convey a similar general message, and therefore are not presented here.

They also show that RMSE gains from using disaggregate information are possible. The SC forecasts are often quite close to the benchmark, however. In fact, they are inferior to the benchmark for a number of evaluation periods

I ^inuit d uni . . h b d th

w lere _{a r}+h1r an ar +h1r ale superIor w en ase on e Ale.

Given the possible level shifts in the aggregation weights visible in Fig. 2 and the forecasting results for the crisis years 2008-2010, we have investigated whether our results are driven by structural change. More precisely, we have robustified the forecasts of the aggregation weights to structural change by using five different devices for the

4-step forecasts.³ Ifthe forecasting model of the weights is a VAR(1), the robustified forecasts are obtained as;

1.

w

r +4lr = _{w r}+llr (no change forecast)

2.

W

r +4lr

=

w r +llr

+

~Wr+llr (trend break correction) 3. Wr+4lr = Wr+4lr

+ LL oAiu

r (intercept correction 1) 4. Wr+4lr = Wr+4lr

+

A²

u

r (intercept correction 2) 5. w;+4Ir =

w

r+4lr

+ u

r (intercept correction 3), where

w

r+4lr denotes the uncorrected forecast, A is the VAR( 1) coefficient matrix and

u

r is the period T residual of the weight forecasting model. The first forecast wr+4lr

is simply a no change forecast, which has performed well in many related contexts and can be justified as a mean correction in the case of a level shift (Clements & Hendry, 1998, Section 8.1). The second forecast

W

r+4lr can be viewed as a trend break correction, and the last three forecasts are different types of intercept corrections for AR models, as proposed by Clements and Hendry (1996).

More precisely, the forecasts wr+41 r, Wr +4lr and w;+4Ir are obtained by adding

u

r at each step ahead, by only adjusting the l-step-ahead forecast, and by adjusting the l1-step- ahead forecast with the full amount of Ur, respectively.

Notice, however, that in the present case wr+4lr is actually just a 3-step-ahead forecast, because lagged weights are used in Eq. (3.1), as was mentioned earlier. Corrections for univariate predictors of the weights are obtained in a

3 We thank a rereree ror suggesting them.

h=4. growth rote. uncorrected

~r_---' -6\

,8'--' ,-0 .... '

,0,"·'0. \

'"

o ... ,-99-5- - -,-9 ... 97---, .... 99~9---2 ... 00-'---20 ... 0-3---'2005

h=4. growth rote. trend breok correction

~~---~

-EJ'\

h=4. growth rote. no chonge forecost

~ , . - - - - -

... ---...--- ...

-EJ'\

ci

1..'-99-5---'-9"'9 7- - - - -' .... 99~9---2 ... 00-'---20-0-3---..I2005 h=4. growth rote. intercept correction ,

~~----

...

---...---~

-ff.. ,-G--' '.-0"

0> ' - -_ _ _ _ _ ".... _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ J. 0> ' - -_ _ _ _ _ ".... _ _ _ _ _ _ _ _ _ ~ _ _ _ _ _ _ _ _ _ _ _ ...

o ^'995 '991 '999 200' 2003 2005 0 '995 1991 1999 2001 2003 2005

h=4. growth rote. intercept correction 2

~r_---...---~ ^h=4. growth rote. intercept correction 3

~r_----... - - - . . . - - - , I;].'-Q

,8'-_: -",

-6\ I·E-roO' -I!l\ ,-G--'

\-8" '. -[ ] , '

ci

1..'-99-5---'-9""9""' ----'-9 ... 99---2 .... 00-'---2 .... 00-3---2^.J005

ci

l'-99-5---'-9""9-7 ----'-9-99---2 ....

00~'----..:2::;00=3===2~005

Fig. 4. RMSEs of various forecasts which are robust to structural change, relative to univariate forecasts of NAFTA GDP; horizontal axis: start of forecast evaluation period. end of forecast evaluation period: 2007Q4 (lag order selection based on the Ale).

similar way. If higher order AR or VAR models are used, the intercept corrections can easily be obtained from the corresponding companion forms of the models.

Fig. 4 shows the resulting relative RMSEs of the 4-

t ^~ t mult d un; ~ . th b d

s ep .orecas s, a_r+_4lr an a_{r +4lr}' .or glOw rates ase on the Ale. These RMSEs are based on evaluation periods starting at the time indicated on the horizontal axis and ending in 2007Q4. Hence, the crisis years are excluded, in order to check the sensitivity of our previous results to possible shifts in the aggregation weights before 2008.

We also show the corresponding RMSEs of the unadjusted forecasts from Fig. 3 for comparison purposes in the upper left panel of Fig. 4. Apparently the six graphs are basically indistinguishable. The underlying RMSEs are very close numerically. Hence, there are basically no differences between the original and robustified forecasts. The same result is obtained for log-levels, forecasts based on SC and the other forecast evaluation periods used in Fig. 3, and therefore we do not show them here. In other words, structural shifts in the aggregation weights may not be a major problem for the present example. Alternatively, our simple robustification methods may not capture the shifts well. This does not necessarily imply that the choice of forecasting method for the aggregation weights does

not matter. Obviously, whether univariate or multivariate methods are used can make a substantial difference. It is only the robustification to structural change in the weights that does not make much difference.

The overall conclusion from this empirical exercise is that using forecasts of the aggregation weights may improve the forecasts of the aggregate, and hence, may be worth considering. In what follows, a further empirical example is presented to show that the result is not specific to the data set at hand, but is more generally an issue to be considered.

3.2. European M3 growth

The second example is based on the quarterly real money stock M3 series from three European countries:

Germany, France and Italy. Details on the data sources are given in Appendix A.2. The weights are computed based on real M3 shares, and the aggregate real M3 growth rate is computed by aggregating growth rates using these weights. Data are available for the period 1981Q2-2010Q3; that is, our sample period starts well before the introduction of the euro. The series and the weights are plotted in Figs. 5 and 6. Again, the aggregation

66

DYAGG DYGER

o

-2 -4

1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010

DYFRA DYIT

o

-2

-4 -4

1985 1990 1995 2000 2005 2010 1985 1990 1995 2000 2005 2010

Fig. 5. Aggregate and individual real M3 growth rates for Germany. France alld Italy.

0.50

Tr=========:===::::::;-- - - -- 1 -

^{WGER n n _ WFRA - -WIT}

1 ,

0.45

0.40

0.35

0.30

0.25

82 84 86 88 90 92 94 96 98 00 02 04 06 08 10 Fig. 6. Weights for aggregate real M3 growth rates.

weights vary substantially and show a considerable degree of persistence. In fact, in this case, unit roots are not rejected in either of the series, and a visual inspection again suggests possible structural shifts.

Forecasting is done as in the previous example; that is, recursive pseudo out-of-sample forecasting for growth rates and log-levels of real M3 is carried out. Estimation and model selection are repeated for every sample considered. Different initial estimation and evaluation periods are used. We use data from 1981Q2 onward, and again the beginning of the estimation periods depends on the number of presample values needed. We compare results for samples until both 2007Q4 and 2010Q3, and rolling S-year evaluation windows. The forecasting horizons are again h = 1 and 4. Relative RMSEs are reported, with the univariate forecasts as the benchmark.

Initially, we fit only AR and VAR models, without accounting for potential breaks or outliers, then discuss the results for forecasts which have been robustified for

structural change. The model orders are chosen by the AIC and SC using a maximum order of four..

Some relative RMSEs are plotted in Fig. 7 in a fashion similar to that of Fig. 3. Clearly, the predictors based on forecasts of the aggregation weights tend to have smaller RMSEs than the other two predictors. Note that the one-step-ahead forecasts a^l11l1_r+1lr ^/t and a"_r^lli _+1lr again use known weights, whereas the 4-step-ahead forecasts use predicted weights. As in the previous example, a~'~"lr is the best predictor overall. However, in contrast to the GDP example, a~~"lr is now also superior for the final years 2008-2010. Only during the second half of the 1990s is there a period where it cannot beat the benchmark when a S-year rolling evaluation window (right-hand column of Fig. 7) is used. For this case, it is seen that even a change in the relative rankings of the growth rate and log-levels forecasts can occur. In fact, there is a considerable degree of variability in the RMSEs if a S-year rolling evaluation period is used. It may be worth noting, however, that there

Forecast evaluation period until 2007Q4

Forecast evaluation period until 2010Q3

5-year rolling evaluation window

-~---~--~ -~---~--~ ~r-~---~

~ .;a::@:;~~;;::,;;::· .. ,}-a~:.&,.\\::.!.:",:.'A/ _{~ ::o:~;~;::} :;:~::~::~:~~:~ ; ~;~ . . ::~~:';;'·

~ '''0';1:::( '

~ -'(L- .

d 1992 199-4 1996 1998 2000 2002 2004 2006 d 1992 19904 1996 1998 2000 2002 2004 2006 1994 1996 1998 2000 2002 2004 2006

_ r---,

_r---~--__, ~r---~

~ ~"8!;\i::!il::~:'l::~::~:{3::[]::~·······

.~

...

d 1992 .99-4 1996 1998 2000 2002 2004 2006 d 1992 1994 1996 1998 2000 2002 2004 2006 d \992 1994 1996 1998 2000 2002 200.. 2006 -.r---~--__,

_r---,

~: :~::':: :~::~: ~::~ :

:'!.:

:':~·El."tp" ... .

^~ ... _---_._---_._----_ ... -.-..

"' ... A ..

P .. /x ••

A ··"·\ . ... ,, .. l·, .g::e·:.

n· -8 -J:h 0--f} _ -(3--O· -Et -G-,

0'1 O'-f)

o ~ ...... . 1> •• ··A·. . •

'A... ··"~.fi"

d '992 1994 1996 1998 2000 2002 2004 2006 0 1992 \99.11 1996 1998 2000 2002 2004 2006 0 \992 199.. 1996 1998 1000 2002 2004 2006

Fig. 7. RMSEs relative to univariate forecasts for European real M3; horizontal axis: start of forecast evaluation period (total sample period: 1981Q2-2010Q3, lag order selection based on the Ale).

are never substantial losses due to using the information in the aggregation weights.

For this example, the results obtained with SC are similar to those for the AIC, and are not shown. Moreover, we have checked the robustified forecasts mentioned in the previous subsection. For the present example, the differences to the results in Fig. 7 are very minor, and are therefore not presented. Thus. this example again shows that exploiting the information in the stochastically varying aggregation weights can increase the forecast precision, as measured by the RMSE.

4. Conclusions

In this study we have considered forecasting contem- poraneous aggregates with stochastic aggregation weights. We have pointed out that such aggregates are quite com- mon in practice, and that taking the information in the weights into account may lead to better forecasts in a MSE sense. We have compared four predictors for such vari- ables: (1) a standard direct univariate AR forecast which is based only on the past ofthe aggregate series, (2) a mul- tivariate linear VAR forecast of the aggregate which takes into account information from the disaggregate compo- nents, (3) a forecast which aggregates a multivariate fore- cast of the disaggregate components and the aggregation weights, and (4) a forecast which is based on aggregat- ing univariate AR forecasts for the individual disaggregate components and the aggregation weights. In two empirical examples, we have shown that the last two forecasts may

lead to lower forecast MSEs than the first two forecasts. In other words, using the information in the stochastic aggre- gation weights explicitly may indeed improve the forecast efficiency in practice,

There are a number of related problems which we have not addressed in this study but which may be of interest for future work. First. we have investigated the potential for forecast efficiency gains by using the information in the aggregation weights for only a very small set of empirical examples. A larger scale investigation may shed light on the general potential for gains in forecast precision, and perhaps on the aggregates for which they can be expected. Second, we have considered a rather limited number of possible predictors in our comparison. While they are sufficient to demonstrate that there is scope for improvements in efficiency by using information in -the stochastic aggregation weights, there are a number of other predictors that seem to be natural competitors and may further improve the exploitation of the information in the aggregation weights. For example, one could either consider modelling the joint OGP of the disaggregate components and the aggregation weights or combine univariate forecasts of the weights with multivariate forecasts of the disaggregate components, or vice versa.

Another related strand of research could consider the precise stochastic structure of the aggregate for given OGPs of the weights and disaggregate components. In general, this is not likely to be an easy problem, because the aggregate is a product of two multivariate processes.

A very limited set of properties under rather special conditions for such processes are provided in Appendix B of

68

Ltitkepohl (2011). More general results may well be helpful in assessing the potential for forecast improvements in the aggregation weights. These issues are left for future research.

Aclmowledgments

We thank an anonymous referee for detailed comments and for pointing out a number of additional references.

We are also grateful for helpful discussions with seminar participants at the University of Mtinster and at CREATES, Aarhus University.

Appendix A. Data sources A.1. NAFfA COP data

The real GDP series, denoted as q~i), are taken from Thomson Datastream and correspond to the seasonally adjusted gross domestic product, measured at constant 2005 PPPs in millions of US Dollars, as reported by the OECD. The Datastream mnemonics for the US, Canada and Mexico are USOCFGVOD, CNOCFGVOD, MXOCFGVOD, respectively. Growth rates and weights are computed as described in Section 3.1.

A.2. European M3 data

Cermany. Seasonally adjusted monthly values of nomi- nal money supply M3 (in billions of EUR), as reported by the Deutsche Bundesbank, are taken from Thomson Datastream (Mnemonic: BDM3 ... B). Quarterly values cor- respond to observations of the last month in the re- spective quarters. The real M3 is obtained by using the GDP deflator with base year 2005 (Datastream Mnemonic:

BDONA001E). German unification effects are accounted for by regressing the growth rate of real M3 on a constant.

four lags and a unification dummy that takes a value of 1 in 1990Q3 and 0 elsewhere. The estimated effect on the growth rates is 0.143, and thus the pre-unification figures are multiplied by 1.143.

France. Seasonally non-adjusted monthly values of nomi- nal money supply M3 (in millions of EUR), as reported by the Banque de France, are taken from Thomson Datastream (Mnemonic: FRM3 ... A). The data have been seasonally ad- justed by the X12-ARIMA method and converted into bil- lions of EUR. Quarterly values correspond to observations of the last month in the respective quarters. The real M3 is obtained by using the GDP deflator, with base year 2005 (Datastream Mnemonic: FRONA001E).

/ta/y. Seasonally non-adjusted monthly values of nomi- nal money supply M3 (in millions of EUR), as reported by the Banca d'ltalia, are taken from Thomson Datastream (Mnemonic: ITM3 ... A). The data have been seasonally ad- justed by the X12-ARIMA method and converted into bil- lions of EUR. Quarterly values correspond to observations of the last month in the respective quarters. The real M3 is

obtained by using the GDP deflator, with base year 2005, which is obtained by rebasing a price deflator that cor- responds to the base year 2000 (Datastream Mnemonic:

ITESGDDFE).

References

Ansley. C. F .. Spivey. w. A., & Wrobleski, W.J. (1977). On the structure of moving average processes. journal of Econometrics, 6, 121-134.

Beyer, A .. Doornik, J. A., & Hendry, D. F. (2001). Reconstructing historical euro-zone data. Economicjoumal, 111,308-327.

Carson, R. T., Cenesizoglu, T .. & Parker, R. (2011). Forecasting (aggregate) demand for us commercial air travel. International journal of Forecasting, 27,923-941.

Castle,J. L .. & Hendry, D. F. (2010). Nowcasting from disaggregates in the face of location shifts. journal of Forecasting, 29, 200-214.

Clements, M. p .. & Hendry, D. F. (1993). On the limitations of comparing mean square forecast errors. journal of Forecasting, 12,617-637.

Clements, M. P., & Hendry, D. F. (1996). Intercept corrections and structural change. journal of Applied Econometrics, 11.475-494.

Clements, M. P., & Hendry, D. F. (1998). Forecasting economic time series.

Cambridge: Cambridge University Press.

Espasa, A .. Senra, E .. & Albacete, R. (2002). Forecasting inflation in the European monetary union: a disaggregated approach by country and by sectors. Europeanjournal of Finance, 8, 402-421.

Hendry, D. F .. & Hubrich, K. (2011). Combining disaggregate forecasts or combining disaggregate information to forecast an aggregate. journal of Business and Economic Statistics, 29. 216-227.

Hubrich, 1<. (2005). Forecastingeuro area inflation: does aggregating fore- casts by HICP component improve forecast accuracy? International journal of Forecasting, 21, 119-136.

I<ohn, R. (1982). When is an aggregate of a time series efficiently forecast by its past? journal of Econometrics, 18,337-349.

Liitkepohl, H. (1984a). Linear transformations of vector ARM A processes.

journal of Econometrics, 26. 283-293.

Liitkepohl, H. (1984b). Forecasting contemporaneously aggregated vector ARMA processes. journal of Business and Economic Statistics, 2, 201-214.

Liitkepohl, H. (1986). Comparison of predictors for temporally and contemporaneously aggregated time series. International journal of Forecasting, 2, 461-475.

Liitkepohl, H. (1987). Forecasting aggregated vector ARMA processes. Berlin:

Springer-Verlag.

Uitkepohl, H. (2010). Forecasting aggregated time series variables - a survey. journal of Business Cycle Measurement and Analysis, 2010/2, 37-62.

Liitkepohl, H. (2011). Forecasting nonlinear aggregates and aggregates with time-varying weights. jahrbacller fiir Nationaliikonomie und Statistik, 231, 107-133.

Marcellino, M .. Stock, J. H .. & Watson, M. W. (2003). Macroeconomic forecasting in the Euro area: country specific versus area-wide information. European Economic Review, 47, 1-18.

Tiao, G. C .. & Guttman, I. (1980). Forecasting con temporal aggregates of multiple time series.journal of Econometrics, 12,219-230.

Wei, W. w. S., & Abraham, B. (1981). Forecasting comtemporal time series aggregates. Communications in Statistics: Theory and Methods, AIO, 1335-1344.

Ralf Briiggemann is Professor of Econometrics and Statistics at the University of I<onstanz. His research is related to time series econometrics and its applications. He has published in refereed journals including the journal of Econometrics, journal of Applied Econometrics, Oxford Bulletin of Economics and Statistics and Empirical Economics.

Helmut Liitkepohl is Bundesbank Professor of Methods of Empirical Economics at the Free University Berlin and Dean of the Graduate Center of the DIW Berlin. He is Associate Editor of Econometric Theory, Macroeconomic Qynamics and Empirical Economics. He has published extensively in learned journals and books and he is author, coauthor and editor of a number of books in econometrics and time series analysis.