Bayesian averaging of classical estimates in forecasting macroeconomic indicators with application...

ORI GIN AL PA PER

Bayesian averaging of classical estimates in forecastingmacroeconomic indicators with application of businesssurvey data

Piotr Białowolski • Tomasz Kuszewski •

Bartosz Witkowski

� Springer Science+Business Media New York 2013

Abstract In this paper, we develop a methodology for forecasting key macro-

economic indicators, based on business survey data. We estimate a large set of

models, using an autoregressive specification, with regressors selected from busi-

ness and household survey data. Our methodology is based on the Bayesian aver-

aging of classical estimates method. Additionally, we examine the impact of

deterministic and stochastic seasonality of the business survey time series on the

outcome of the forecasting process. We propose an intuitive procedure for incor-

porating both types of seasonality into the forecasting process. After estimating the

specified models, we check the accuracy of the forecasts.

Keywords Bayesian averaging of classical estimates � Business survey data �Seasonality � Automatic forecasting

JEL Classification C10 � C83 � E32 � E37

1 Introduction

Business and consumer surveys have a long history of describing changes in

economic activity and providing information on the expected future shape of

P. Białowolski (&)

Institute of Statistics and Demography, Warsaw School of Economics,

Warsaw, Poland

e-mail: piotr.bialowolski@sgh.waw.pl

T. Kuszewski � B. Witkowski

Institute of Econometrics, Warsaw School of Economics, Warsaw, Poland

e-mail: tkusze@sgh.waw.pl

B. Witkowski

e-mail: bwitko@sgh.waw.pl

123

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DOI 10.1007/s10663-013-9227-x

economic processes. In Poland, the history of tendency survey based research dates

from the mid-1980s, as traced in detail in Adamowicz (2008).

The main advantage of business and consumer survey data is their accessibility.

In particular, information gained from surveys can be included in models before

changes in economic indicators are reported in official statistics. Although there is

debate on the soundness and reliability of survey data, economic agents are believed

to have deep intuitive insight into various aspects of macroeconomic processes

(Drabarek 2006). An intuitive understanding of economic processes is only partly

determined by the level of knowledge and expertise one has acquired. Most

economic knowledge and acquisition of economic concepts is believed to be

obtained during an economic socialisation process (Tyszka 2004).

Business survey data are widely used in econometric analyses aimed at

forecasting key macroeconomic indicators. A comprehensive review of existing

approaches and models can be found in Bialowolski et al. (2007). However, the only

macroeconomic multi-equation prognostic model, for the Polish economy, that

uses business survey data is the CLIMA model developed by the team led by

M. Drozdowicz-Biec (Bialowolski et al. 2007, 2008). This model describes cyclical

changes in gross domestic product, unemployment and inflation and is constructed

as a tool for generating medium-term forecasts for these variables on a quarterly

basis. The specification of the model’s equations is based on the fundamentals

of economic theory, while business sentiment indicators are used to improve the

model fit.

The aim of this study is to show that it is possible to build a range of econometric

models based on regressors selected through the Bayesian averaging of classical

estimates (BACE) method. We use a set of business survey data from both Poland

and Poland’s main trading partner, Germany. We start with the assumption that

business survey indicators exhibit properties of a preemptive response to changes in

the values and relationships among key macroeconomic indicators, such as gross

domestic product (GDP) growth, inflation and unemployment. Additionally, we

analyse the impact of seasonality of the reference time series on the fit of

econometric models and on the accuracy of the forecasts generated on the basis of

these models. In the next step of the procedure, the developed econometric models

describing the Polish economy are used to generate short-term quarterly forecasts

of GDP dynamics, inflation dynamics, and the unemployment rate. Finally, the

accuracy of generated forecasts is examined by comparing them with other

economic forecasts.

The paper is arranged as follows. The following (Sect 2) briefly discusses

existing views on macroeconomic forecasting in times of economic crisis. Section 3

focuses on the data used to estimate the econometric models and on the statistical

properties of the time series used in the analysis. Section 4 describes the Bayesian

Averaging of Classical Estimates—the method in general and a modification to it

adopted in this paper. Section 5 examines the fit of obtained forecasting models and

analyses the impact of the seasonality of the time series and of the adopted approach

to modelling seasonality on the quality of the forecasts obtained. Finally, the

forecasts are compared with other predictions available on the issue date of our

forecast. Section 6 concludes.

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2 Macroeconomic forecasting before and after (?) the financial crisis

Macroeconomic forecasting has lost a considerable amount of public confidence in

recent years, due to a large number of inaccurate forecasts that cast doubt on the

ability of commonly used models to produce accurate forecasts. This is reflected in a

new and enhanced version of the well-known academic economics textbook by

Olivier Blanchard (2011). The author presents forecasts made by various American

economists just after the first signs of crisis in December 2007. The economists’

views with respect to short-term prospects for the U.S. economy were diverse. The

optimists among them argued in effect that nothing was wrong and that all

unfavourable economic developments could be neutralised through appropriate

decisions by the Federal Reserve Board. The pessimists, by contrast, suggested that

the decline in demand might prove so strong that it could not be overcome with

simple interest rate cuts and would eventually lead to recession in the U.S. Just nine

months later, it was clear that almost no one (with a few exceptions), at the

beginning of the crisis period, had been able to accurately predict further

developments in the U.S. economy. As Blanchard observes, ‘‘even the most

pessimistic opinions were too optimistic.’’ This view is supported by Coenen (2010)

who gave a presentation during a Workshop on Central Bank Forecasting.

According to Coenen, the models failed to recognise the size and persistence of the

shocks. Especially notable was a lack of proper recognition of developments in the

financial markets. Coenen suggested several remedies, including adoption of

weighting schemes in modelling approaches, to eliminate the worst performing

models from analysis. He also suggested that survey data should be included in

analysis, a view shared by Banbura et al. (2010) who show that survey data lead to

significant improvements in what is known as nowcasting of macroeconomic

variables. Banbura et al. (2010) take a set of variables from a survey conducted

using the methodology of the European Commission (similar to that used by the

Research Institute for Economic Development at the Warsaw School of Economics)

and show a significant improvement in data ‘‘nowcasting’’ up to the point where

official data are released.

Thus, there appears to be a need for further development of forecasting models in

times of poor economic performance. Based on standard statistical procedures, the

economic crisis in the eurozone has resulted in a shift from above-potential output

to below-potential output. What is more, this transition did not result in a significant

decline in inflation, which to a large extent was externally driven (Hucek et al.

2010). Forecasts made in uncertain economic environments associated with large

output gaps and continually high inflation are potentially more prone to bias, if

standard specifications of models are applied. As data from business and consumer

tendency surveys possess the element of common knowledge of economic agents

and therefore should be free of bias in the long run. It is also reasonable to assume

that such data tend to be relatively resistant to structural changes and quick to adapt

to changes in the economic environment. This is well stated in Benes et al. (2010):

‘‘Additional information in the high-frequency data may assume crucial importance

during a crisis or more generally during periods of high uncertainty, when the

regular relationships built into the model may cease to hold’’.

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Taking all the events of recent years into account, it might seem reasonable to

take the rigid approach of Roubini (2010) who state that the economic system

changes and that these changes are not incorporated into economic theory quickly

enough, while the new era requires changes in ways of thinking. Changes in the

economic environment should also have a significant impact on approaches to

economic modelling. As reported in the paper of Lee et al. (2010), developments in

the U.S. economy—where a significant decline in consumption, along with an

increase in the savings, was noted—are expected to have a significant influence on

global growth. The change in global growth patterns should induce changes in the

modelling strategies of other economies.

With these various arguments in mind, we undertake the task of modelling the

key macroeconomic variables of the Polish economy for the period 1996–2011.

Aware of the flaws of current mainstream modelling methods (such as DSGE or

SVAR), we decided to use tendency survey data, as we expect survey data will

provide additional, leading information, while initiating a novel approach to

forecasting with survey data. Our method is based on Bayesian averaging, an

approach to enhancing the process of variable selection and reducing the set of

regressors in the final model to those with the highest information content.

3 Data

To build our forecasting model, we use quarterly data covering the 1996–2011

period. The data on GDP, the consumer price index (CPI) and the unemployment

rate (UNE) are from statistics produced by Poland’s Central Statistical Office

(CSO). The unemployment rate is determined on the basis of a Labour Force Survey

also conducted at Poland’s CSO. The GDP indices come in two variants. ‘‘GDP’’

presents dynamics, calculated according to the ‘‘corresponding period of the

previous year = 100’’ principle, while ‘‘GDP_VOL’’ covers information on

quarterly seasonally unadjusted GDP values at constant 2005 prices, calculated

from the 2010 values, and using the link chain method retroactively for the

1996–2009 period and prospectively for 2011.

In forecasting models, lagged endogenous variables were used as explanatory

variables, in addition to the indicators from the business survey data. The dataset

applied in the modelling procedure comprised time series from the Research

Institute for Economic Development (RIED) at the Warsaw School of Economics

(WSE) on sentiment in the manufacturing industry, trade, construction and

households. In all fields (except for the manufacturing industry, where the survey

is conducted monthly), RIED conducts quarterly surveys, which are collected in the

first month of each quarter, i.e., in January, April, July and October.1 An additional

source of data were indices published by the Centre for European Economic

Research (ZEW), the Leibniz Institute for Economic Research at the University of

1 For the survey of the manufacturing sector, conducted on a monthly basis, data from January, April,

July and October were used.

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Munich (Ifo Institute), the Bureau for Investments and Economic Cycles (BIEC)

and the data on Purchasing Managers’ Index (PMI) for Polish industry.

For most of the time series from the business and consumer surveys (except for

zew_ies and PMI), values are expected to hover at around 100 because this value

represents an equal proportion of positive and negative responses to a given

question. However, in order to provide a common base for all variables, a constant

of 100 was subtracted from the Ifo, BIEC and RIED indices. From the PMI index,

we subtracted 50, because in this case this value denoted a neutral level.

With the set of time series prepared in this way, the order of integration and

seasonal properties were examined for each variable. ADF and KPSS tests were

used to determine order of integration. No time series with an order of integration

higher than 1 were identified in the database. It turned out that, in many cases, the

two tests did not provide consistent conclusions about a given series. In particular,

for the variables2 biec_wwk, biec_wrp, biec_wd, ind_q4f, ind_q5s, ind_q5f,

ind_q6s, ind_q6f, ind_q8s, hhs_q4, hhs_q5, hhs_q7, and hhs_q8, the conclusions

about the order of integration of the time series resulting from the two tests were

different. The general outcome of the analysis is that the time series for responses to

business survey questions targeted at the industrial sector are stationary I(0) time

series, while the time series for responses to business survey questions targeted at

households are integrated I(1) time series. However, the remaining time series are

also stationary. This explains why we decided against differentiating the values of

the series I(1) and instead studied the statistical properties of the residual series of

the estimated models.

We also faced the problem of seasonality in the time series of business survey

data. We treat seasonality as a consequence of the presence of different social and

climate factors. However, we do not limit our analysis to regular seasonality, but

allow it to change from one year to the next. For that reason, our investigation

covers both deterministic and stochastic seasonality. The task of testing for

deterministic seasonality in the time series used was reduced to that of estimating

the parameters of a trend model for a given series (linear or nonlinear), with binary

variables designed to detect seasonal deviations. Stochastic seasonality was assessed

using the Demetra? software package (Grudkowska 2011), but we decided to apply

a simple approach to testing the values of the regressor parameters, specifically, the

value of the time series component lagged four periods (Enders 1995).

In the time series for the variables3 zew_ies, pmi, ifo_bs, ifo_be, biec_wwk,

biec_wpi, biec_wrp, biec_wd, we did not find deterministic seasonality, but the

presence of stochastic seasonality was not rejected. In all time series of the RIED

data, deterministic seasonality was detected and the presence of stochastic

seasonality was not rejected. Because taking into account the occurrence of each

type of seasonality influences the quality of forecasts (Franses and Lof 2000), we

decided to consider seasonality in the final versions of the forecasting models.

2 The list of symbols used for the further analyses was presented in ‘‘Appendix’’.3 Full names can be found in ‘‘Appendix’’.

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4 Prognostic models

4.1 Modelling strategy

The main aim of our research was to estimate a model capable of forecasting the

quarterly figures of macroeconomic indicators—GDP (gross domestic product), CPI

(consumer price index) and UNE (unemployment rate), using business and

consumer survey data. We decided to avoid adopting an economic theory that

would suggest which indicators were of key importance for a given dependent

variable and should thus be selected as regressors in the model. In our previous work

(Bialowolski et al. 2010), we used a partly modified stepwise regression technique

for regressor selection. Depending on the selected stepwise algorithm, different

models were found as the best fitting ones. In this study, the selection of regressors

was based on the BACE, a method recently proposed by Sala-i-Martin et al. (2004)

and that has gained popularity among macroeconomists. In most previous research,

it has mainly been used to find the ‘‘best possible’’ growth factors. We believe it can

also be applied to the selection of a set of relevant indicators, as in the present

analysis.

Let M = {X1, X2,…, XK} be the set of K potential regressors in a single

regression equation in which Y is the dependent variable. Suppose that we are

interested in evaluating the influence of individual components of M on Y. In brief,

the idea of BACE is to first estimate all 2K - 1 possible models, with every possible

subset of M as independent variables. However, in the case of large values of K, the

estimation of all models might exceed current computing capabilities. In many

applications, therefore, instead of estimating all possible models (with each possible

subset of M), a large number of possible subsets of M is drawn and only models

involving those variables are estimated. The next step is to calculate the parameters

in the ‘‘final’’ regression (bk). These are derived as weighted averages of the

estimates of the parameters on Xk, obtained in each of the estimated models by using

a measure of ‘‘goodness of fit’’ of each model as the weight.4

This approach, based on BACE, has two main advantages. First, the number of a

priori assumptions regarding the set of independent variables is reduced as much as

possible: with a predefined set of possible independent variables Xk, the only prior

assumption made concerns the number of variables in the relevant model. Under the

BACE approach, it is possible to select the relevant regressors for the equation of Y

by assigning a probability of relevance to each of the estimated models and then

considering as relevant those elements of M included in the models whose estimated

probability of relevance is the highest. Second, the final bk’s are calculated using all

the obtained estimates—without limiting our focus to a given subset of M, which

would be based purely on a researcher’s views.

We here create three-equation models. To improve the quality of the forecasts,

we include as variables in the equations lagged values of GDP, CPI and UNE. We

also believe that there are interactions between the three macroindicators, which

compels us to include them not only as dependent, but also independent, variables.

4 The weight used in the procedure is given by Eq. 4.

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In order to avoid endogeneity problems, we construct a recurrent model in

accordance with the following procedure: let Y1, Y2 and Y3 be the macroindicators of

interest (GDP, CPI and UNE) ordered in a given way (one of the six possible ways).

The general form of our model is then:

Y1;t ¼ f1ðY1;t�1;M1; e1;tÞY2;t ¼ f2ðY2;t�1; Y1;t;M2; e2;tÞY3;t ¼ f3ðY3;t�1; Y1;t; Y2;t;M3; e3;tÞ;

ð1Þ

where: Yi,t—value of i-th macroindicator (GDP, CPI, UNE) from period t, i = 1, 2,

3, Yi;t—theoretical value of the i-th macroindicator (from the i-th equation of the

model), Mit—business survey indicators selected for explaining i-th macroindicator,

ei,t—error term in i-th equation, fi—linear function in i-th equation.

The model is estimated following a 2SLS-logic, which ensures that there are no

direct endogeneity problems with the Yi indicators, as these do not appear directly as

independent variables—only the theoretical values of these variables are used for

this purpose. Thus, we are able to use a regular equation-by-equation OLS

estimation of the model. Of the six possible ways of ordering the variables in a

three-equation model, we decided, based on results of our previous research

(Bialowolski 2010, 2011), to use the following order: Y1 = GDP, Y2 = UNE and

Y3 = CPI.

Our goal was to produce out-of-sample forecasts for three quarters ahead. All of

the Yi,t’s can be used to calculate Yi,t?10s, which can be treated as independent

variables in the subsequent models. As business survey indicators are not predicted

in our models and our forecasting horizon is three quarters ahead, these indicators

are lagged three periods. Thus the model (1) can be written:

GDP1;t ¼ f1ðGDP1;t�1;M1;t�3; e1;tÞUNE2;t ¼ f2ðUNE2;t�1;GDP1;t;M2;t�3; e2;tÞCPI3;t ¼ f3ðCPI3;t�1;GDP1;t;UNE2;t;M3;t�3; e3;tÞ:

ð2Þ

Relevant Mi,t-3 for i = 1, 2, 3 are selected through application of a modified

BACE algorithm.5 In the classical BACE approach, each independent variable is

potentially irrelevant and subject to elimination from the model, whereas in our

case, similarly to (Prochniak and Witkowski 2013), we retain certain variables

(lagged dependent variables and the theoretical values of the variables from

previous equations) in each model. The BACE algorithm that we use can be briefly

summarised as follows:

The number of variables in M (denoted as K) is 41, which yields a total number

of possible models of (241 – 1)3. As so large a number cannot be handled

computationally, we instead draw one million subsets of M with an average number

of 20 Xk’s per subset and an equal probability of entering each model for each

variable. This reflects our strategy of not having any economic preference for

particular Xk’s. In the next step, we estimate the model defined by (2), with a prior

5 A detailed description of Bayesian averaging algorithms can be found, e.g., in Moral-Benito (2011).

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assumption of the number of Xk’s in each equation of an ‘‘ideal’’ model (2). We

denote this number as �s. In our case, based our previous research, we assume that

�s = 10.

The next steps of the procedure are based on the following assumptions: let us

denote a model with the j-th drawn subset of M as Wj and the number of variables of

the subset of M in Wj as Kj. Further, assuming the independence of Xk’s and no

knowledge of which variables should be introduced into the model, using a negative

binomial, we can state that the prior probability that a given Xk is in the model is �sK,

whereas the prior probability of the relevance of Wj is given by

PðWjÞ ¼�s

K

� �Kj

1� �s

K

� �K�Kj

ð3Þ

and is equal for every Wj that shares the same Kj. Using Bayes’ rule and the Schwarz

approximation, the posterior probability of the relevance of Wj, that is, the proba-

bility ‘‘supported’’ by the data, denoted as P(Wj|D), is given by

PðWjjDÞ ¼PðWjÞn�Cj;i=2SSE

�n=2jP1;000;000

p¼1 PðWpÞn�Cp;i=2SSE�n=2p

; ð4Þ

where: n—number of observations in the sample (effectively the length of the time

series); cj,i, cp,i—the total number of independent variables in the ith equation of

models Wj and Wp, respectively, which equals Kj ? 1 for the GDP equation, Kj ? 2

for the UNE equation and Kj ? 3 for the CPI equation; SSEj, SSEp—sum of squared

residuals in models Wj and Wp, respectively.

The probabilities given by (4) can be treated as measures of ‘‘probability of

relevance’’ of a given model Wj. Such probability measures are computed for each

of the 1 million GDP equations, each of the 1 million UNE equations and each of

the 1 million CPI equations.

The final step of our adapted BACE procedure is to select the variables for the

three equations of interest. To do so, for each Xk in M, we compute its posterior(‘‘confirmed’’ by the dataset) probability of relevance for each of the three equations

separately, for a given �s:

kiðXkj�s;DÞ ¼X1;000;000

j¼1

PðWjjDÞ � Ik;j; ð5Þ

where Ik,p is an indicator function that takes a value of 1 if Xk 2 Kj and 0 otherwise.

Posterior probabilities for each of the Xk’s are used to confirm the relevance of a

given Xk in a given equation—its relevance is confirmed if its posterior probability

is higher than its prior probability. Thus, for each of the three equations, we select

those Xks for which the value of (5) exceeds �sK, which equals 0.24 in the present case.

It must also be emphasised that, although there is one common model, variables for

each equation are selected separately, and the final sets of business survey indicators

used as regressors for GDP, UNE and CPI need not be the same ones across the

different equations.

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Having selected the sets of regressors for each of the equations, we then consider

two further problems:

1. possible nonstationarity of the dependent variables, resulting in spurious

regressions and thus inaccurate forecasts;

2. possible violation of the OLS assumptions, in particular those concerning

properties of the error terms in each of the equations.

To evaluate the problem of nonstationarity of the dependent variables, we check

for cointegration of the estimated models. A lack of cointegration might raise

serious doubts about the quality of the obtained results. In order to eliminate the

influence of trend, we simply try to introduce it into the model. Finally, we check for

violation of the OLS assumptions, for autocorrelation of the error terms and for

collinearity among the selected Xk’s.

4.2 Estimation

The procedure described in Sect. 4.1 was run twice, separately for GDP and for

GDP_VOL. The initial M used to select particular Mi’s was the same for each

equation and for each of the two models (which differ in the measure of aggregate

of GDP used). A full list of variables in M is provided in ‘‘Appendix’’.

For each of the two types of model (2), we first selected business survey

indicators for the final model, using BACE. In Table 1, we present the posteriorprobabilities of inclusion (4) of particular variables in each of the equations. The

probabilities for the variables selected for a given equation are bolded. It should be

noted that the distribution of kiðXkj�s;DÞ for some of the equations is such that there

are a number of variables whose posterior probabilities of inclusion exceed their

prior probabilities. In such cases, we reduce the final set of independent variables in

the i-th equation to an initial set of the ten variables with the highest kiðXkj�s;DÞ.However, among the set of variables selected in this way, near multicollinearity is

present in some cases. To address this issue, we sequentially drop variables that are

strongly collinear (that is, variables whose variance inflation factor in a given

equation exceeds 10). The posterior probabilities for such variables are shaded in

the appropriate columns of Table 1.

As noted in the previous section, we ran selected validation tests and procedures.

Specifically, we ran the ADF tests for nonstationarity in the residuals, finding that it

is not present. As the dependent variables are mostly nonstationary, we concluded

that there is cointegration in the tested equations and thus that the modelled

variables remain in a balanced state in the long term. Similarly, using both Durbin

and LM tests for autocorrelation, we concluded that there is no autocorrelation up to

order 4.

With application of the BACE procedure for two model specifications (1—GDP,

UNE, CPI; 2—GDP_VOL, UNE, CPI), we established the sets of regressors in

individual equations. In the following step, we estimated the parameters of the

models with the previously identified regressors, additionally taking into account

seasonal components in three variants: deterministic (introducing a set of quarterly

dummy seasonal variables), stochastic (imposing a MA (4) process on the error

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Table 1 Posterior probabilities of inclusion of particular variables for equations in the GDP and

GDP_VOL models

Regressor Xk Model for GDP Model for GDP_VOL

GDP UNE CPI GDP_VOL UNE CPI

zew_ies 0.529973 0.473289 0.570007 0.204261 0.289795 0.172322

Pmi 0.143643 0.120608 0.144214 0.235723 0.343614 0.186498

ifo_bs 0.244707 0.046347 0.064009 0.087284 0.533400 0.197818

ifo_be 0.111710 0.147197 0.125835 0.362892 0.193616 0.094349

biec_wwk 0.092843 0.477494 0.036627 0.999998 0.999703 0.291972

biec_wpi 0.045370 0.754221 0.049681 0.377101 0.723594 0.168876

biec_wrp 0.186897 0.156539 0.284151 0.077910 0.150351 0.327932

biec_wd 0.324782 0.049047 0.084054 0.273071 0.136324 0.165020

ind_q1s 0.247212 0.973989 0.215227 0.877355 0.292578 0.274490

ind_q1f 0.179209 0.468517 0.25861 0.284972 0.307676 0.205343

ind_q2 s 0.142753 0.059925 0.236697 0.213152 0.260536 0.238870

ind_q2f 0.447050 0.085075 0.115459 0.115784 0.316939 0.110308

ind_q3s 0.097127 0.204787 0.260067 0.255725 0.208179 0.327371

ind_q3f 0.071420 0.179672 0.268869 0.435574 0.123372 0.235761

ind_q4s 0.200863 0.109203 0.137070 0.127792 0.104256 0.068615

ind_q4f 0.194507 0.133494 0.068642 0.270591 0.062994 0.172786

ind_q5s 0.141234 0.170285 0.044506 0.135926 0.262133 0.205454

ind_q5f 0.341719 0.054075 0.127928 0.169695 0.112532 0.232022

ind_q6s 0.181444 0.514534 0.258962 0.058501 0.084833 0.229098

ind_q6f 0.073308 0.095002 0.267408 0.193924 0.286455 0.068994

ind_q7s 0.308216 0.980561 0.237513 0.523176 0.376295 0.187030

ind_q7f 0.335049 0.077684 0.339904 0.374239 0.301672 0.235311

ind_q8s 0.181177 0.455188 0.087633 0.282741 0.383942 0.165216

ind_q8f 0.358626 0.603194 0.073181 0.202558 0.622445 0.145072

ind_r 0.077118 0.099542 0.116935 0.163334 0.296612 0.164803

hhs_q1 0.254419 0.281131 0.038463 0.091679 0.083131 0.086198

hhs_q2 0.086899 0.239642 0.228539 0.608905 0.205476 0.215549

hhs_q3 0.043738 0.418542 0.055954 0.193661 0.292155 0.127054

hhs_q4 0.069227 0.056893 0.146240 0.315878 0.144906 0.157511

hhs_q5 0.043636 0.22991 0.270003 0.224784 0.180944 0.149217

hhs_q6 0.945867 0.137937 0.263677 0.146575 0.240750 0.220438

hhs_q7 0.038287 0.131539 0.228455 0.130678 0.164926 0.073209

hhs_q8 0.121711 0.436050 0.106511 0.703570 0.187004 0.341998

hhs_q9 0.258586 0.172663 0.302928 0.070369 0.730140 0.427353

hhs_q10 0.178904 0.157873 0.235332 0.263119 0.075909 0.178285

hhs_q11 0.162894 0.270946 0.183867 0.302393 0.215743 0.204204

hhs_q12 0.156168 0.183020 0.301902 0.068248 0.244099 0.191933

Hhs 0.212643 0.028864 0.119975 0.292017 0.154258 0.099906

Trade 0.108158 0.600264 0.217376 0.099199 0.330489 0.235338

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term), and both simultaneously. Tables 2, 3 and 4 show the results of the estimation

of equations for each variable: GDP, UNE, and CPI.

In none of the three models in which we attempted to account for the seasonality

of changes in GDP were variables characterising this seasonality statistically

significant. This is probably due to the fact that the reference variable is expressed

as a year-on-year growth rate.

In all three models in which we attempted to account for seasonality in changes in

UNE, variables that characterise seasonality were found to be statistically significant.

For most regressors, including seasonality does not drastically influence the value of

the estimates of the regressors’ parameters. Thus we conclude that one can hypothesise

that changes in the value of the unemployment rate are seasonal in nature. It is also

worth noting that no variables among the regressors come from the household survey.

An analysis of the seasonality of changes in CPI reveals that only the model with

stochastic seasonality contains statistically significant variables describing the seasonal

process. The introduction of stochastic seasonality results in a change of the parameter

signs only for the ind_q3f and ind_q7f variables. Additionally, in the CPI models, we

note significant variation in the constant term under different seasonal specifications.

Table 2 Estimates of parameters in GDP equations

Regressor Models

No

seasonality

Deterministic

seasonality

Stochastic

seasonality

Both types of

seasonality

GDP(-1) 0.87099 0.85162 0.91933 0.89558

zew_ies -0.00554 -0.00595 -0.00296 -0.00397

biec_wd -0.01502 -0.01823 -0.00937 -0.01967

ind_q2f -0.03630 -0.03892 -0.04256 -0.04436

ind_q5f -0.00990 -0.01095 -0.01614 -0.01532

ind_q7s -0.00133 0.01353 -0.00202 0.01520

ind_q8f 0.02008 0.01584 0.02659 0.02053

hhs_q1 -0.01207 -0.01505 -0.01538 -0.01487

hhs_q6 -0.04353 -0.04997 -0.04533 -0.05078

hhs_q9 0.01487 0.02028 0.01595 0.02100

Const 3.21615 3.97296 3.06701 3.89963

Own estimates. All regressors lagged by 3 periods unless specified otherwise

Table 1 continued

Regressor Xk Model for GDP Model for GDP_VOL

GDP UNE CPI GDP_VOL UNE CPI

Agri 0.146806 0.999196 0.507167 0.279091 0.964015 0.667514

Constr 0.138002 0.469658 0.079768 0.913880 0.105246 0.193564

Own estimates. Columns show posterior probabilities for particular Xk’s in the respective models.

Probabilities for variables selected for particular equations are bolded. The probabilities for variables

selected for particular equations and those dropped due to multicollinearity are bolded and italicized

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Two variables, zew_ies and ind_q7, were present in all the estimated equations of

the model. The first describes changes in business sentiment in Germany, which is

found to have a very significant influence on the Polish economy. In recent years,

the Polish economy has been largely driven by the growth of exports, its main

export market being Germany. The second variable (ind_q7) indicates business

executives’ views about the financial conditions of their companies.

Table 3 Estimates of parameters in UNE equations

Regressor Models

No

seasonality

Deterministic

seasonality

Stochastic

seasonality

Both types of

seasonality

UNE(-1) 0.92739 0.95250 0.93913 0.96021

GDP -0.19378 -0.18612 -0.22500 -0.21489

zew_ies -0.00325 -0.00174 -0.00136 -0.00120

biec_wpi 0.04949 0.04951 0.04399 0.04184

ind_q1s 0.04152 -0.00321 0.02138 -0.00322

ind_q7s -0.06273 -0.02539 -0.04271 -0.02769

ind_q8f 0.02755 0.02526 0.02622 0.02589

Trade 0.02128 0.01772 0.03317 0.02889

Agri -0.04430 -0.04260 -0.04535 -0.04511

Constr -0.01193 -0.00916 -0.01178 -0.00311

Const 1.55463 1.08792 1.58558 1.01892

Own estimates. All regressors lagged by 3 periods unless specified otherwise

Table 4 Estimates of parameters in CPI equations

Regressor Models

No

seasonality

Deterministic

seasonality

Stochastic

seasonality

Both types of

seasonality

CPI(-1) 0.83816 0.85233 0.86764 0.84039

GDP 0.36635 0.36457 0.41948 0.28196

UNE 0.09501 0.05464 0.05638 0.05672

zew_ies -0.00284 -0.00330 -0.00422 -0.00165

ind_q3f -0.00396 0.00832 0.02927 0.01366

ind_q7f 0.01330 0.00230 -0.03800 -0.01461

hhs_q5 0.01982 0.01486 0.00583 0.01105

hhs_q6 0.00889 0.00943 0.03026 -0.00105

hhs_q9 0.01442 0.01119 0.01599 0.02219

hhs_q12 0.03610 0.03047 0.03885 0.03729

Agri -0.02242 -0.03466 -0.03189 -0.02299

Const -2.83184 -2.66463 -3.62635 -1.36974

Own estimates. All regressors lagged by 3 periods unless specified otherwise

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5 Forecasting

For forecasting purposes, we decided to discard the model that included the value of

gross domestic product at constant prices (GDP_VOL). The estimates for this model

were not satisfactory and forecasts generated on the basis of this model would very

likely be highly unstable. We also estimated a forecasting model called the final

model (Final), which maintains the previously identified sets of regressors while

incorporating information about the significance of seasonal components. In the

GDP equation, seasonality components were ignored; in the UNE equation, both

types of seasonality were modelled; while in the CPI equation, only stochastic

seasonality was taken into account.

Before conducting an out-of-sample forecast of GDP, UNE and CPI, we

compared the models in terms of how well they fit the data. We decided to check

their performance using RMSEs for individual quarters of 2009–2011 and

2010–2011, treating the theoretical values of the endogenous variables as ex post

forecasts. We used the period 2009–2011, in order to check the model

performance during a rapid transition between high growth rates and stagnation.

The period 2010–2011 represents a period of moderate growth for the Polish

economy.

The RMSE values are expressed in the same units as the endogenous variables,

which in our case are percentage points. The values given in Table 5 show that

taking seasonality into account slightly improves the data fit, making it less probable

that a generated forecast will exhibit a major error. In the final model, the error

decreases markedly only for the UNE variable. At the same time, it can be seen that

the mean errors for the 2010–2011 period are generally lower than those for the

2009–2011 period. This reflects the fact that significant change in economic

conditions, that had the strongest impact on the economy at the end of 2008 and in

2009, significantly reduced the forecasting ability of our model.

Table 5 Root mean square errors in 2009–2011 and 2010–2011

Regressand Models

No

seasonality

Deterministic

seasonality

Stochastic

seasonality

Both types of

seasonality

Final

Quarters 2009–2011

GDP 0.74 0.68 0.71 0.66 0.74

UNE 0.62 0.42 0.38 0.37 0.30

CPI 0.54 0.54 0.39 0.39 0.42

Quarters 2010–2011

GDP 0.54 0.52 0.57 0.56 0.54

UNE 0.59 0.42 0.25 0.30 0.20

CPI 0.53 0.58 0.37 0.39 0.42

Own estimates

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6 Concluding remarks

This paper is a follow-up to our previous research conducted in 2010 and 2011. In

this study, we constructed a prognostic model of three key macroeconomic

indicators: GDP growth, the unemployment rate and the consumer price index. We

used the Bayesian averaging of classical estimates method to build our model,

making necessary modifications to it, as needed, to serve the purposes of the study.

The model was modified to enable us to predict the values of the endogenous

variables without making any assumptions about the values of the regressors in the

forecast period. The modifications also ensured a specific sequence of relationships

between endogenous variables. Another change from our former approach is that

here we considered the possibility of seasonality in both the exogenous and

endogenous variables. The results of our study are promising. They show that it is

possible to build models that take into account both deterministic and stochastic

seasonality and that this procedure improves the goodness-of-fit of such models. It

also turns out that our forecasts are almost identical to those generated by the DSGE

model of the National Bank of Poland and similar to the 2012 projections of the

Gdansk Institute for Market Economics. It might be interesting to examine to what

extent applying the general Bayesian averaging technique to DSGE models might

yield even better results than those obtained with the atheoretical models considered

here, although such a process could be extremely time-consuming. Given that even

with relatively simple linear models, estimating hundreds of thousands of equations

takes considerable time, one might expect such computations to take far longer

when a more complex structure or estimation method is applied, as with DSGE

models. Nevertheless, this appears to be an interesting area for future research.

There is still a need for further development of this project. We would like to

examine the impact of the progressive aging of past data on the quality of the

estimated models and on the accuracy of their forecasts. We plan to test the

prognostic usefulness of business activity indicators other than those already

included in the model.

Acknowledgment The authors would like to thank the Research Institute of Economic Development at

the Warsaw School of Economics and especially prof. Elzbieta Adamowicz for support with the project.

An earlier version of this paper was published in Polish in Prace i Materiały IRG SGH.

Appendix: The list of symbols adopted for variables for the estimationprocedure

zew_ies ZEW indicator of economic sentiment

ifo_bs Ifo business situation indicator

ifo_be Ifo business expectations indicator

biec_wwk BIEC leading index

biec_wpi BIEC future inflation index

biec_wrp BIEC future unemployment rate index

biec_wd BIEC well-being index

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pmi Purchasing Managers’ index (PMI) for Polish industry

ind_xxx Balance of responses to question ‘xxx’ from a business sentiment

survey in industry RIED

hhs_xxx Balance of responses to question ‘xxx’ from a consumer sentiment

survey RIED

trade Business sentiment indicator RIED in trade

agri Business sentiment indicator RIED in agriculture

cons Business sentiment indicator RIED in construction

See Tables 6 and 7.

Table 6 Questions from the business sentiment survey in industry

Symbol Question (ind_xxs—current state, ind_xxf—projection)

ind_q1 Production

ind_q2 Total orders

ind_q3 Export orders

ind_q4 Stock of finished products

ind_q5 Prices of goods produced by enterprise

ind_q6 Employment

ind_q7 Financial standing

ind_q8 Poland’s macroeconomic performance

Business sentiment survey in industry, Research Institute for Economic Development, Warsaw School of

Economics

Table 7 Questions from the consumer sentiment survey

Symbol Question

hhs_q1 Assessment of household financial status, compared with the situation 12 months earlier

hhs_q2 Projected household financial status in the next 12 months

hhs_q3 Performance of the Polish economy in the last 12 months

hhs_q4 Projected performance of the Polish economy in the next 12 months

hhs_q5 Comparison of maintenance costs now and 12 months earlier

hhs_q6 Projection for the inflation rate in the next 12 months

hhs_q7 Projection for the unemployment rate in the next 12 months

hhs_q8 An advantage to make major purchases at the present time

hhs_q9 Projected spending on durable consumer goods over the next 12 months in relation to the level

reported in the last 12 months

hhs_q10 Assessment of savings and the climate for saving in the context of the country’s

macroeconomic performance

hhs_q11 Projected household’s saving in the next 12 months

hhs_q12 Financial position of the household

Survey of households, Research Institute for Economic Development, Warsaw School of Economics

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