*3.1. Hypotheses*

Three hypotheses have been proposed referred to the informational content of the considered surveys and the relationship between uncertainty and economic growth:


With the aim of testing the proposed hypotheses we firstly describe the available information, respectively provided by the barometer of the Spanish Center of Sociological Research and the regional Industrial Trend Survey. Besides supplying synthetic indicators, both sources allow the estimation of probabilities and uncertainty levels through entropy-based measures. More specifically in this paper we used Shannon's and quadratic Indexes, thus allowing a comparison of the uncertainty levels estimated by both expressions.

Furthermore, the estimation of econometric models allows a more detailed analysis about the causal relationship and the impact of uncertainty on economic growth. Thus, vector autoregresive (VAR) models were estimated, and their results are described in Section 4.

## *3.2. Data Description: Confidence Barometers and Industrial Trend Surveys*

CIS is an independent entity assigned to the Ministry of the Presidency, and gathers the necessary data for research in very different fields, carrying out a wide variety of surveys, whose data is in the public domain. The CIS databank includes confidence barometers, polls carried out since 1994 on a monthly basis (except in August), with the aim of measuring Spanish public opinion. As described in the CIS website [20] these polls involve interviews with around 2500 randomly-chosen people from all over the country, including a block of variable questions which focuses on the assessment of both the economic situation in Spain and the personal economic situation, as described in Table 2.


**Table 2.** Spanish Center for Sociological Research (CIS) confidence barometer.

Microdata provided by the monthly polls can be downloaded from the CIS website www.cis.es and allow the calculation of probabilities based on relative frequencies assigned to the alternative options.

Regarding the Spanish industrial trend surveys, the Ministry of Industry, Trade and Tourism, and also some regional statistical offices develop qualitative surveys with the aim of catching the opinion of industrial managers about the current situation and future prospects. More specifically, the questionnaire is directed to the managemen<sup>t</sup> industrial personnel and compiles qualitative

information referred to the present levels of the portfolio orders and the production, sale prices and employment expected for the next months.

Three alternative answers (high, normal or low) are provided for those questions reflecting the present level, while the options to increase, to stay or to diminish can be selected if the questions refer to the expected tendency. The individual answers given to the different questions are aggregated in order to obtain series by classes and categories and the balance between the extreme options provides an indicator with values oscillating between +100 and −100 (totally' optimistic and pessimistic situations). The results for each variable can also be summarized through the industrial climate indicator (ICI) computed as an arithmetic mean of the balances of the portfolio orders, the production expectations and, with the opposite sign, the level of finished product stocks. This composite indicator is widely used to provide a global vision of the industrial confidence in relation to the conjunctural evolution. In fact, as the leading indicator signals summarized in the ICI are assumed to happen before the economy turning points, this index can be used as a leading indicator of economic activity allowing the obtention of economic turning point forecasts as shown in [16].

Since the estimation of uncertainty requires detailed information about individuals perceptions we focus on the regional industrial trend survey referred to Asturias, whose databank is fully available from [21] allowing the estimation of the corresponding probabilities.

## *3.3. Shannon's and Quadratic Entropy Measures*

Although qualitative surveys have been extensively used to obtain synthetic indicators, few attempts have been made in order to quantify the uncertainty level perceived by the respondents. In this paper we aim at filling this gap, and also analyzing to which extent the level of uncertainty perceived by the experts is related with the economic situation.

Entropy measures provide a suitable framework for our goal, as entropy is a function of the probability distribution and not a function of the actual values taken by the random variable. Since microdata of qualitative surveys allow the estimation of the probabilities assigned to each possible outcome, entropy measures can also be estimated. Thus, given the set of *n* distinct mutually exclusive options for a specific question, the individual responses allow the estimation of frequency probabilities *pi*, ∀*i* = 1, ... *n* such that *pi* ≥ 0, ∑*i pi* = 1. Shannon [22] defines the information content of a single outcome as *h*(*pi*) = log - 1 *pi* . According to this definition, observing a rare event provides much more information than observing another, more probable outcome.

In this context, Shannon's entropy is defined as the expected amount of information and can be computed as *H* = − ∑*i pi* log(*pi*). This expression plays a central role since it fulfills a number of interesting properties which, as shown in [22] substantiate it as a reasonable measure of information, choice or uncertainty:


Following a similar approach, Pérez [23] proposes the individual quadratic entropy, which can be computed for a single outcome as *<sup>h</sup>*<sup>2</sup>(*pi*) = 2(1 − *pi*). According to this proposal, the quadratic entropy is quantified as twice the distance of the probability of an event from the true outcome, and similarly to Shannon's measure, the information provided by a rare event is higher than the information corresponding to a more likely one.

Given a set of probabilities *pi*, ∀*i* = 1, ... , *n* such that *pi* ≥ 0, ∑*i pi* = 1, the quadratic entropy is defined in [23] as the expected value of the individual quadratic entropies, given by the expression *H*<sup>2</sup> = 2 ∑*i pi*(<sup>1</sup> − *pi*). This is a suitable measure of uncertainty since it fulfils the requirements proposed by Shannon. More specifically:


The quadratic measure has been successfully used in different economic applications, including the evaluation of forecasts [24,25]. Taking into account its suitable behavior, in this paper we propose the joint use of Shannon's and quadratic entropy to approach the level of uncertainty.
