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DYNAMIC ECONOMETRIC MODELS 
Vol. 9 – Nicolaus Copernicus University – Toruń – 2009 

Mariola Piłatowska  
Nicolaus Copernicus University in Toruń 

Combined Forecasts Using the Akaike Weights 

A b s t r a c t. The focus in the paper is on the information criteria approach and especially the 
Akaike information criterion which is used to obtain the Akaike weights. This approach enables 
to receive not one best model, but several plausible models for which the ranking can be built 
using the Akaike weights. This set of candidate models is the basis of calculating individual fore-
casts, and then for combining forecasts using the Akaike weights. The procedure of obtaining the 
combined forecasts using the AIC weights is proposed. The performance of combining forecasts 
with the AIC weights and equal weights with regard to individual forecasts obtained from models 
selected by the AIC criterion and the a posteriori selection method is compared in simulation 
experiment. The conditions when the Akaike weights are worth to use in combining forecasts 
were indicated. The use of the information criteria approach to obtain combined forecasts as an 
alternative to formal hypothesis testing was recommended.  

K e y w o r d s: combining forecasts, weighting schemes, information criteria. 

1. Introduction  
 The development of time series analysis and computing power of computers  
made that many different forecasts can be obtained when forecasting the same 
economic variable with different methods. Many selection criteria based on the 
performance of ex post forecasts are used to choose the best forecast (Arm-
strong, 2001). Combining forecasts can be treated as an alternative approach to 
the selection of the best individual forecast. Since the seminal paper of Bates 
and Granger (1969) has been known that combining forecasts can produce 
a forecast superior to any element in the combined set1. Hence, instead of seek-
ing the best forecasting model the combined forecasts based on competing 
models are received.  
 Moreover, the reason for combining forecasts (or model averaging) is that 
the data generating model (true model) is unknown. Therefore each model 
                                                 

1 The paper of Bates and Granger (1969) caused the development of research on combining 
forecasts (for overview see Timmermann, 2006). 



Mariola Piłatowska  6 

should be treated as an approximation of unknown data generating model. 
These models may be incomplete (or incorrectly specified) in different ways; 
forecast based on them might be biased. Even if forecasts are unbiased, there 
will be covariances between forecasts which should be taken into account. 
Then, combining forecasts produced by misspecified models may improve the 
forecast in comparison to any individual forecast obtained from the underlying 
models. As a consequence, the problem of selecting the individual forecasts 
over the set of available forecasts and the weighting schemes is occurred. Espe-
cially, the selection of weighting scheme is important.  
 Most frequently the following weighting schemes can be distinguished: 
equal weights (Stock, Watson, 2004, 2006; Marcellino, 2004), Akaike weights 
(Atkinson, 1980; Swanson, Zeng, 2001; Kapetanios et al., 2008), optimized and 
constrained weights (Jagammathan, Ma, 2003), Bayesian weights (Min, Zellner, 
1993; Diebold, Pauly, 1980; Wright, 2003).  
 In the paper the focus is on the information criteria approach, especially the 
Akaike information criterion which is used to produce the Akaike weights. This 
approach enables to obtain not only one, but several plausible models for which 
the ranking can be built using the Akaike weights. The individual forecasts, 
calculated from the considered models, are aggregated with the Akaike weights.  
The paper propagates the application of the Akaike weights, previously un-
known in Polish literature, and evidence ratios in selecting a model over the 
underlying set of models and in producing the combined forecasts.  
 The purpose of the paper is to propose the procedure of combining forecasts 
using the Akaike weights, and also to compare the combined forecasts (with the 
Akaike weights and equal weights) with individual forecasts obtained from the 
best model selected according to: (1) the Akaike information criterion and (2) 
traditional hypothesis testing. The analysis will be conducted in the simulation 
study in which autoregressive models and causal models are taken as approx-
imating models provided that the data generating model is unknown.  
 The structure of the paper is as follows. In section 2 the construction of the 
Akaike weights will be presented. In section 3 the procedure of combining fore-
casts using the Akaike weights will be depicted. Next, the results of simulation 
experiment will be showed and at the very end – some conclusions.  

2. The Akaike Weights  
 The Akaike information criterion is applied to select the best model from 
among the candidate models considered. The Akaike’s (1973) seminal paper 
proposed the use of the Kullback-Leibler information or distance as a funda-
mental basis for model selection. The Kullback-Leibler (K-L) information be-
tween models f (true model or probability distribution) and g (approximating 
model in terms of a probability distribution) is defined for continuous functions 
as the integral:  



Combined Forecasts Using the Akaike Weights 7

∫= ,
)|(

)(ln)(),( dx
xg

xfxfgfI
θ

 (1) 

where ,0),( ≥gfI  0),( =gfI  only if ,gf =  ).,(),( fgIgfI ≠    
),( gfI  denotes the information lost when the model g is used to approximate 

the model f. The purpose is to seek an approximating model that loses as little 
information as possible. This is equivalent to minimizing ),( gfI  over g. 
Akaike (1973) found a rigorous way to estimate K-L information based on the 
empirical log-likelihood function at its maximum point. This result took the 
form of an information criterion:  

,2)ˆ(ln2AIC KL +−= θ  (2) 

where )ˆ(θL  is the maximum likelihood for the candidate model ,i  which is 
corrected by K  the number of estimated parameters.  
Akaike has showed that choosing the model with the lowest expected informa-
tion loss (i.e. the model which minimizes the expected Kullback-Leibler discre-
pancy) is asymptotically equivalent to choosing the model iM  ),...,2,1( Ri =  
that has the lowest AIC value.  
 To obtain the Akaike weights a simple transformation of the raw AIC values 
should be performed. For each model the difference in AIC with respect to the 
AIC of the best candidate model is computed: 

.AICAIC min−=Δ ii  (3) 

These iΔ  are easy to interpret and allow a quick comparison and ranking of 
candidate models. The best model over the candidate models has 0min =Δ≡Δ i . 
The larger iΔ  is, the less plausible is that the fitted model iM  is the K-L best 
model, given the data. For nested model some rough rules of thumb are availa-
ble in selecting the model (Burnham, Anderson, 2002), i.e. models with 2<Δ i  
have substantial support, models with 74 <Δ< i  – considerable less support. 
Models with 10>Δ i  have either essentially no support and might be omitted 
from further consideration, because they fail to explain some substantial ex-
plainable variation in the data. In empirical data analysis the models with 

4<Δ i  are accepted.  

 From the differences iΔ  we can obtain the relative plausibility of model 

iM  over the set of candidate models by estimating the relative likelihood 
)|( xML i  of model iM  given the data x  (Akaike, 1983): 

),5.0exp()|( ii xML Δ−∝   (4) 

where ∝  stands for „is proportional to”.  



Mariola Piłatowska  8 

Finally, the relative model likelihoods are normalized (divided by the sum of 
the likelihoods of all models) and the Akaike weights iw  are obtained:   

,
)5.0exp(

)5.0exp(

1
∑
=

Δ−

Δ−
= R

r
r

i
iw      .1

1
=∑

=

R

i
iw  (5) 

Weight iw  can be interpreted as the probability that iM  is the best model  
(in the AIC sense, i.e. the model minimizing the K-L information) given the 
data and the set of candidate models.  
 Additionally weights iw  can be useful in evaluating the relative strength of 
evidence for the best model (with biggest weight) over the other in the set of R  
models. Thus, the evidence ratios or the ratio of Akaike weights ji ww /   

(in particular the ratio ,/1 jww  where 1w  is the weight for the best model, and 

jw  – weights for models in the set) are calculated. It is worth pointing out that 
this approach does not assume that any of the candidate models is necessarily 
true, but rather the ranking of models in the sense of K-L information is consi-
dered2.  
 The AIC weights (5) can be generalized into the form (Burhnham, Ander-
son, 2002, 2004): 

,
)5.0exp(

)5.0exp(

1
r

R

r
r

ii
i

q

qw
∑
=

Δ−

Δ−
=  (6) 

where iq  is a prior probability of our prior information (or lack thereof) about 
which of the R  models is the K-L best model for the data. By the AIC weights 
we mean the expression (6) with the equal prior probabilities, i.e. ./1 Rqi = The 
inclusion of prior probabilities in (6) makes that the AIC weights can be treated 
as an approximation of the Bayesian posterior model probability (Burnham, 
Anderson, 2002, 2004). However, it is not a true Bayesian approach. The full 
Bayesian approach to model selection requires both the prior iq  on the model 
and a prior probability distribution on the parameter θ  in model iM  for each 
model. Then the derivation of posterior results requires integration (usually 
achievable only by Markov chain Monte Carlo methods). In that context the 
AIC weight seem to be useful because they are much easier to compute and 
additionally the researcher is not required to determine prior densities for the 
parameters.  

                                                 
2 It is the main difference in the comparison with the Bayesian model averaging which as-

sumes that the true generation model is in the set of candidate models and measures the degree of 
belief that a certain model is the true data-generating model.  



Combined Forecasts Using the Akaike Weights 9

3. The Procedure of Obtaining The Combined Forecasts Using 
The Akaike Weights 

 When calculating combined forecasts using the Akaike weights some condi-
tions should be satisfied. Namely, all models in the set of candidate models 
should be fitted to exactly the same set of data because the inference based on 
information criteria is conditional on the data in hand. Moreover, all models in 
the set should represent the same response variable. A common type of mistake 
is to compare models of ty  with models of transformed variable, e.g. tyln  or 

.tyΔ  

 The steps in the procedure of obtaining forecasts aggregated with the 
Akaike weights are following.  
Step 1. Establishing the initial set of R  models describing a given variable and 

their specification. The guidelines on specifying causal models should be de-
rived from an economic theory explaining the phenomenon in interest. In the 
case of large number of variables it is not recommended to run all possible 
regressions because the set of candidate models should be plausible with re-
spect to the economic theory, and not be automatically selected. The true 
generation model does not have to be included in the set of models.  

Step 2. Fitting the models )...,,2,1( Ri =  to the data, calculating the AIC values 
and differences .iΔ  Models should satisfy statistical and economic require-
ments.  

Step 3. Creating the reduced set of models )...,,2,1( *Ri =  for which ,4<Δ i  
i.e. models plausible in the sense of K-L information.  

Step 4. Calculating the Akaike weights (eq. (4)) and combined forecasts accord-
ing to formula: 

∑
=

++ =
*

1
,,, ,ˆ

R

i
htihihtt fwy     ,1

*

1
, =∑

=

R

i
hiw  (7) 

where htty +,ˆ   – combined forecast, hiw ,  – the weight assigned to the fore-

cast httif +,,  based on the ith individual model.  

When combining forecasts the problem is to estimate the weights ,, hiw  so as to 
minimize a penalty function depending on the forecast errors. Very often, the 
penalty function is simply the mean square forecast error (MSFE).  

4. Simulation Experiment Results  
 The purpose of simulation experiment is to compare combining forecasts 
(using the Akaike weights and equal weights) with individual forecasts obtained 
from the best model selected according to: (1) the Akaike information criterion 



Mariola Piłatowska  10

and (2) traditional hypothesis testing. In experiment the autoregressive models 
and causal models are taken as approximating models provided that the data 
generating model is unknown.  
 Simulation scenario is following. 
The data-generating model of tY  has the form:  

,,33,22,110 ttttt XXXY εββββ ++++=    ),,0(~ εσε Nt  ,3,2,1=εσ  

with parameters: 2,2.1,5.1,10 3210 ==== ββββ , for samples: 50,100=n  
(number of replications 1000=m ). Processes tt XX ,2,1 ,  and tX ,3  have follow-
ing structure:  
 ,6.08.012 11,1,1 −− +++= tttt XX ζζ  ),1,0(~ Ntζ  

 ,8.07.014 11,2,2 −− +++= tttt XX ηη  ),1,0(~ Ntη    

 ,4.02.18 2,31,3,3 tttt XXX ξ+−+= −−  ).1,0(~ Ntξ  

As approximating models are taken: the autoregressive models: 
,...110 qtqtt YYY −− +++= γγγ    ,4,3,2,1=q  

and causal models3: 

tttttttt vXXYYYYY +++++++= −−−−− 1,12,11443322110 γγααααα   (M1) 

ttttttt vXXYYYY ++++++= −−−− 1,12,113322110 γγαααα   (M2) 

 tttttt vXXYYY +++++= −−− 1,12,1122110 γγααα   (M3) 

 .1,12,11110 ttttt vXXYY ++++= −− γγαα  (M4) 

 ,,1122110 ttttt vXYYY ++++= −− γααα  (M5) 

 .,11110 tttt vXYY +++= − γαα  (M6) 

It is assumed that the true generation model is unknown, therefore specifying 
the causal models the variables tX ,2  and tX ,3  were omitted.  

The analysis was carried out separately for autoregressive models and causal 
models. In each replication only the models with 4<Δ i  were taken. For those 

models the Akaike weights and equal weight )/1( *R were received, and having 
calculated the individual4 forecasts, the h-period ahead combined forecasts were 
obtained. To compare forecasts the mean square forecast error (MSFE) was 
calculated for combined forecasts and individual forecasts obtained from the 

                                                 
3 Having conducted initial simulation, these models are accepted as plausible.   
4 Individual forecasts were dynamic, and as values of explanatory variables X1t in forecast pe-

riod the generated values are taken.  



Combined Forecasts Using the Akaike Weights 11

best model (selected by: the AIC and a posteriori selection method applied to 
the causal model5 M1). The results present Table 1 (for )100=n  and Table 2 
(for ).50=n  

 Results presented in Table 1 and 2 show that the differences between MSFE 
obtained for combined forecasts (using AIC weights and equal weights) and 
individual forecasts (from the best model selected by the minimum of AIC and 
by a posteriori selection method) are small. However, certain regularities indi-
cating the usefulness of combined forecasts can be observed.  
 In the case of small size of disturbance (σε = 1) the combined forecasts 
(with the Akaike weights, wAIC) obtained from causal models give smaller mean 
square forecast error (MSFE) than forecasts combined with equal weights (wEQ.) 
at the whole forecast horizon (Table 1). This slight dominance of combined 
forecast with the AIC weights is hold for σε = 2, 3 at longer horizons (h > 5), 
and for shorter horizons – the forecasts combined with equal weights (wEQ.) 
have lower MSFE.   
 In general the combined forecasts (with AIC weights and equal weights) 
outperform the individual forecasts obtained from the best model (selected by 
a posteriori method, mSEL.), because the MSFE for combined forecasts are visi-
bly lower than the MSFE for individual forecasts; this occurs for all sizes of 
disturbance σε  (Table 1). Such performance indicates the dominance of com-
bined forecasts. However this dominance is not complete, because the lowest 
MSFE are obtained for individual forecasts calculated from the best model se-
lected by the minimum of AIC (for σε =1). For bigger size of disturbance, i.e. σε 

=2 and 3, the lower MSFEs at the horizon h ≤ 5 give the forecasts combined 
using equal weights, and at the horizon h ≥ 6 – forecasts combined using the 
AIC weights. These lower MSFEs for forecasts from models selected by the 
minimum of AIC (for σε =1) refer to the cases when the set of candidate models 
is small (in the considered experiment it were models M3 and M4), and addi-
tionally one model in the set has the dominating AIC weight (wAIC > 0.7). For 
the bigger size of disturbance, i.e. σε = 2 and 3, the set of competing models, 
being used in combining forecasts, consisted most frequently of models M3, 
M4, M5, M6, and none had the dominating AIC weight. Then, the combined 
forecasts (using the AIC weights or equal weights) outperformed the individual 
forecasts from model selected by the minimum of AIC, i.e. they gave the lower 
MSFEs.  

                                                 
5 Variables elimination in a posteriori selection method was realized at the 5% significance 

level.  



Table 1. Mean square forecast errors (MSFE) for sample n = 100 

h 
Causal models Autoregressive models 

wAIC wEQ. minAIC mSEL. wAIC wEQ. minAIC 
 σε = 1 
1 2.950 2.973 2.935 3.050 3.102 3.117 3.096 
2 2.290 2.301 2.287 2.357 2.530 2.547 2.523 
3 2.471 2.500 2.456 2.604 2.568 2.595 2.548 
4 3.280 3.320 3.254 3.436 2.569 2.602 2.537 
5 4.064 4.097 4.043 4.192 3.706 3.748 3.653 
6 3.949 3.985 3.926 4.092 3.413 3.452 3.367 
7 3.807 3.842 3.785 3.944 3.192 3.225 3.156 
8 3.589 3.622 3.568 3.717 3.038 3.073 3.001 
9 3.503 3.528 3.488 3.607 2.931 2.960 2.906 
10 3.903 3.915 3.900 3.961 3.266 3.280 3.265 
 σε = 2 
1 1.958 1.944 2.008 2.035 2.220 2.217 2.229 
2 2.200 2.175 2.280 2.207 3.111 3.105 3.121 
3 3.236 3.221 3.284 3.266 2.890 2.885 2.899 
4 4.795 4.794 4.823 4.901 3.200 3.198 3.204 
5 6.334 6.342 6.350 6.497 4.696 4.698 4.697 
6 6.333 6.341 6.347 6.484 4.836 4.838 4.835 
7 5.971 5.978 5.984 6.107 4.596 4.598 4.595 
8 5.675 5.681 5.689 5.792 4.359 4.361 4.358 
9 5.397 5.403 5.410 5.508 4.172 4.173 4.171 
10 5.224 5.229 5.237 5.324 4.087 4.088 4.086 
 σε = 3 
1 3.172 3.177 3.168 3.222 4.430 4.415 4.459 
2 3.456 3.455 3.459 3.477 5.061 5.048 5.087 
3 5.283 5.251 5.321 5.211 6.934 6.921 6.961 
4 7.320 7.320 7.331 7.361 9.907 9.897 9.933 
5 8.008 8.014 8.010 8.066 11.199 11.189 11.225 
6 7.621 7.624 7.625 7.670 10.966 10.957 10.992 
7 7.171 7.172 7.178 7.215 10.380 10.371 10.405 
8 6.794 6.796 6.802 6.836 9.846 9.837 9.870 
9 6.514 6.518 6.520 6.563 9.339 9.331 9.362 
10 6.260 6.262 6.267 6.305 8.993 8.985 9.015 

Note: In columns wAIC, weq. are the MSFEs for forecasts combined using the Akaike weights and equal 
weights, and in columns minAIC., mSEL – the MSFEs for individual forecasts obtained from model selected by 
the minimum of AIC and model received after applying a posteriori selection method to model M1 at the 5% 
significance level.  



Combined Forecasts Using the Akaike Weights 13

Table 2. Mean square forecast errors (MSFE) for sample n = 50 

h 
Causal models Autoregressive models 

wAIC wEQ. minAIC mSEL. wAIC wEQ. minAIC 
 σε = 1 
1 1.019 1.018 1.018 1.023 1.069 1.069 1.076 
2 1.502 1.587 1.409 1.888 2.712 2.705 2.721 
3 1.564 1.576 1.570 2.002 2.804 2.796 2.815 
4 2.440 2.393 2.493 2.494 2.489 2.482 2.498 
5 3.599 3.486 3.722 3.251 2.288 2.283 2.296 
6 3.685 3.609 3.761 3.274 2.136 2.132 2.144 
7 3.455 3.391 3.518 3.144 2.219 2.212 2.229 
8 3.279 3.212 3.352 3.093 2.443 2.435 2.456 
9 3.164 3.110 3.224 2.988 2.342 2.334 2.355 
10 3.669 3.642 3.699 3.412 2.569 2.565 2.576 
 σε = 2 
1 3.522 3.508 3.539 4.230 5.419 5.399 5.414 
2 3.854 3.803 3.918 4.644 5.739 5.711 5.744 
3 4.199 4.100 4.318 4.900 5.669 5.641 5.677 
4 4.832 4.671 5.018 5.261 5.483 5.458 5.489 
5 5.379 5.222 5.555 5.668 5.679 5.660 5.685 
6 5.493 5.324 5.683 5.616 5.356 5.340 5.362 
7 6.133 5.955 6.329 5.949 5.273 5.262 5.280 
8 6.171 6.016 6.346 5.890 5.139 5.130 5.146 
9 6.311 6.195 6.456 6.048 5.362 5.357 5.369 
10 6.286 6.189 6.411 6.034 5.380 5.376 5.387 
 σε = 3 
1 5.088 5.096 5.085 5.335 7.026 6.997 7.064 
2 6.378 6.371 6.389 6.498 8.233 8.203 8.276 
3 6.968 6.952 6.987 6.993 8.696 8.683 8.720 
4 6.578 6.562 6.597 6.599 8.389 8.386 8.399 
5 6.193 6.175 6.213 6.205 7.944 7.945 7.948 
6 5.803 5.788 5.823 5.821 7.391 7.393 7.396 
7 5.599 5.579 5.627 5.593 6.938 6.939 6.942 
8 5.504 5.484 5.532 5.492 6.593 6.594 6.597 
9 5.325 5.308 5.351 5.334 6.456 6.458 6.459 
10 5.181 5.166 5.206 5.228 6.601 6.603 6.603 

Note: See Table 1.  

 In the case of autoregressive models the forecasts from models selected by 
the AIC gave the lowest MSFE for small size of disturbance σε =1 for the same 
reasons as in the case of causal models, i.e. the set of candidate models con-
tained the small number of models (here AR(2), AR(3) and AR(4)) and one 



Mariola Piłatowska  14

model had a dominating AIC weight (0.5 < wAIC < 0.6). Hence, in the sense of 
the MSFE, the individual forecasts outperformed the combined forecasts. For 
the bigger size of disturbance, i.e. σε = 2 (and bigger uncertainty) and at the 
shorter horizon (h ≤ 5) the dominance of forecasts combined using equal 
weights is observed, and at the longer horizons (h ≥ 6) the dominance of fore-
casts combined using the AIC weights occurs. For the disturbance σε = 3 the 
forecasts combined using equal weights slightly outperform the forecast com-
bined using the AIC weights and individual forecast (minAIC) – see Table 1. 
Generally, the combined forecasts gave the lower MSFE than the individual 
forecasts. This refers to the cases when the set of models consisted of many 
autoregressive models of different order and none of them had the dominating 
AIC weight. Then, the lower MSFEs are received for combined forecasts.  
 The results tabulated in Table 2, for sample n = 50 indicate that for the 
causal models the MSFEs are lower for forecasts combined using equal weights 
than those using the AIC weights for all size of disturbance (except σε = 1 and  
h = 1, 2). The dominance of combined forecasts (wAIC, wEQ.) or individual fore-
casts (minAIC, mSEL.) depends on the forecast horizon and size of disturbance σε . 
For horizon h ≤ 4 and disturbance σε = 1 (also σε = 2 and h ≤ 7; σε = 3) the 
MSFEs for combined forecasts are lower than for forecasts from the best model 
selected by a posteriori method (mSEL.), but for longer horizon h ≥ 5 (for σε = 1 ) 
and h ≥ 7 (for σε = 2) the MSFEs are lower for forecasts from mSEL.. Forecasts 
from models selected by the minimum of AIC have the higher MSFEs than 
combined forecasts and in general also higher than forecasts from mSEL..  
 In the case of autoregressive models the MSFEs for combined forecasts are 
always lower than for individual forecasts (minAIC) – see Table 2. Simultaneous-
ly the combined forecasts using equal weights (wEQ.) outperform those using the 
AIC weights (wAIC).  

5. Summary  
 From comparison of forecasts combined using the AIC weights and equal 
weights results that in the case when the set of candidate models contains the 
model with dominating AIC weight (wAIC > 0.7) the combination of forecasts 
using the AIC weights is not effective. Then the MSFEs are higher than those 
for forecasts combined using equal weights. However, in such case the AIC 
weights can be useful in building ranking of models, and additionally in calcu-
lating the evidence ratios informing about the relative strength of evidence for 
the best model (with biggest weight) in the sense of AIC over the other models 
in the set of candidate models. The benefits from applying the AIC weights 
occur when the number of candidate models in the set is big and none has the 
dominating weight wAIC.  
 The results of experiment indicate that the combined forecasts outperform 
the individual forecasts in the case of autoregressive models. For causal models 



Combined Forecasts Using the Akaike Weights 15

this dominance of combined forecasts is hold at the shorter horizon for distur-
bance σε = 1, 2 and at the whole horizon for bigger size of disturbance σε = 3.  
 Summing up, the information criteria approach, particularly the use of AIC 
weights to build the ranking of models and to calculate the combined forecasts, 
can be treated as alternative to the traditional hypothesis testing approach di-
rected to select the best model and calculate individual forecasts.  

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Timmermann, A. (2006), Forecast Combinations, [in:] Elliott G., Granger C. W. J., Timmermann 
A. (ed.), Handbook of Economic Forecasting, ch. 4, Elsevier. 

Prognozy kombinowane z wykorzystaniem wag Akaike’a 

Z a r y s  t r e ś c i. W artykule uwaga jest skupiona na podejściu wykorzystującym kryteria in-
formacyjne, a w szczególności kryterium Akaike’a, które jest wykorzystywane do wyznaczenia 
wag Akaike’a. Podejście to umożliwia otrzymanie nie jednego, a kilku wiarygodnych modeli, dla 
których można stworzyć ranking stosując wagi Akaike’a. Modele te stanowią podstawę oblicze-



Mariola Piłatowska  16

nia prognoz indywidualnych, a te z kolei służą do wyznaczenia ostatecznej prognozy kombino-
wanej, przy formułowaniu której wykorzystuje się wagi Akaike’a.   

S ł o w a  k l u c z o w e: prognozy kombinowane, systemy wag, kryteria informacyjne. 


