The tale of two expectations

Abstract

This paper aims to shed some light on the way lay consumers forecast by comparing survey expectations on two different—but linked—fundamentals. Specifically, we study the central tendencies and cross-sectional dispersions of predictions on individual-level and aggregate income dynamics. The proposed joint analysis highlights several interesting outcomes. Agents’ predictions on micro and macroeconomic evolutions do not drift apart despite (possibly composite) shocks have permanent effects on expectations. When shocks create a gap between the two expectations, in fact, individuals revise only forecasts about GDP dynamics. Otherwise stated, predictions on personal stances are much stickier. With respect to these latter, then, expectations on aggregate dynamics overreact to shocks and, amazingly from an objective standpoint, they are systematically bleaker. As per second moments evidence shows, in sharp contrast with the typical assumption maintained in the macroeconomic literature, that disagreement among agents is persistently high. Moreover, the comparison makes astonishingly clear that when predicting the same macro fundamental the consensus is even lower. Finally, our setting allows testing whether cross sectional disagreement and time series volatility in expectations are equal.

Appendices

Appendix 1: Robustness check. VECM recursive estimations

We re-write here the long-run relationships shown in Sect. 4 for two generic variables, x _i,t and x _t :

$$ x_{it} - x_{it - 1} = \updelta_{i} \left( {x_{it - 1} + \upbeta x_{t - 1} + \uptheta } \right) $$

(7.1)

$$ x_{t} - x_{t - 1} \; = \;\updelta \left( {x_{it - 1} + \upbeta x_{t - 1} + \uptheta } \right) $$

(7.2)

Figure 6 collects the recursive coefficients of Eqs. (7.1) and (7.2) referring to the central tendencies, i.e., whereas $x_{it} =\,_{t - h} y_{it}^{e}$ and $x_{t} =\,_{t - h} y_{t}^{e}$. Alike, Fig. 7 reports the recursive coefficients of Eqs. (7.1) and (7.2) referring to the two signal to noise ratios, i.e., whereas $x_{it} = \;_{t - h} y_{it}^{e} /\upsigma_{it}^{e}$ and $x_{t} = \;_{t - h} y_{t}^{e} /\upsigma_{t}^{e}$.

Fig. 6

Recursive VECM coefficients. Central tendencies. In the figure are reported the recursive coefficients of the VECM (cf. Sect. 4, Eqs. 4.6 and 4.7): $_{t - h} y_{it}^{e} -\,_{t - h} y_{it - 1}^{e} = \updelta_{\text{i}} \left( {_{t - h} y_{it - 1}^{e} + \upbeta_{t - h} y_{t - 1}^{e} + \uptheta } \right)$; $_{t - h} y_{t}^{e} -\,_{t - h} y_{t - 1}^{e} = \updelta \left( {_{t - h} y_{it - 1}^{e} + \upbeta_{t - h} y_{it - 1}^{e} = \uptheta } \right)$. $_{t - h} y_{it}^{e}$ is the central tendency of the responses to the query “How do you expect the economic situation in your household to change over the next 12 months?” $_{t - h} y_{t}^{e}$ is the central tendency of the responses to the query “How do you expect the general economic situation in the country to develop over the next 12 months?”. Data for Italy. The recursions start estimating the VECM from January 1995 to January 2005, we then add a month each new estimation (last sample: Jan. 1995–Jul. 2013)

Fig. 7

Recursive VECM coefficients. Signal-to-noise ratios. Note. In the figure are reported the recursive coefficients of the VECM: $_{t - h} SNR_{it}^{e} -\,_{t - h} SNR_{it - 1}^{e} = \updelta_{i} \left( {_{t - h} SNR_{it - 1}^{e} + \upbeta_{t - h} SNR_{t - 1}^{e} + \uptheta } \right)$; $_{t - h} SNR_{t}^{e} -\,_{t - h} SNR_{t - 1}^{e} = \updelta \left( {_{t - h} SNR_{it - 1}^{e} + \upbeta_{t - h} SNR_{t - 1}^{e} + \uptheta } \right)$. $_{t - h} SNR_{it}^{e} \equiv ({{_{t - h} y_{it}^{e} } \mathord{\left/ {\vphantom {{_{t - h} y_{it}^{e} } {\upsigma_{it}^{e} }}} \right. \kern-0pt} {\upsigma_{it}^{e} }})$ is the signal-to-noise ratio referring to the query “How do you expect the economic situation in your household to change over the next 12 months?” $_{t - h} SNR_{t - 1}^{e} \equiv ({{_{t - h} y_{t}^{e} } \mathord{\left/ {\vphantom {{_{t - h} y_{t}^{e} } {\upsigma_{t}^{e} }}} \right. \kern-0pt} {\upsigma_{t}^{e} }})$ is the signal-to-noise ratio referring to the query “How do you expect the general economic situation in the country to develop over the next 12 months?”. Data for Italy. The recursions start estimating the VECM from January 1995 to January 2005 and then add a month each new estimation (last sample: Jan. 1995–Jul. 2013). See also Sect. 2

Figures 6 and 7 point to a remarkable robustness of the findings of the main text.

Appendix 2: Robustness check. The Carlson–Parkin method

In this Appendix we calculate the first two moments of the distribution of the interviewers’ replies with a different approach with respect to that used in the main text. Specifically, we take advantage of the Carlson–Parkin method (CP, Carlson and Parkin 1975) in the five option replies version of Batchelor (1986), Batchelor and Orr (1988). Essentially, the method interprets the share of respondents as maximum likelihood estimates of areas under the density function of aggregate expectations, that is, as probabilities. More formally, the first two moments are:

$$ y_{t}^{e,CP} = - y_{t}^{r} \left( {\frac{{z_{t}^{3} + z_{t}^{4} }}{{z_{t}^{1} + z_{t}^{2} + z_{t}^{3} + z_{t}^{4} }}} \right) $$

(8.1)

$$ \upsigma_{t}^{e,CP} = y_{t}^{r} \left( {\frac{2}{{z_{t}^{1} + z_{t}^{2} + z_{t}^{3} + z_{t}^{4} }}} \right) $$

(8.2)

where $y_{t}^{r}$ is the agent’s reference GDP growth rate, $z_{t}^{1} = {\rm N}^{ - 1} \left[ {1 - {\text{LB}}_{\text{t}} } \right],\;\;z_{t}^{2} = {\rm N}^{ - 1}\left[ {1 - {\text{LB}}_{\text{t}} - {\text{B}}_{\text{t}} } \right],\;\;z_{t}^{3} = {\rm N}^{ - 1}\left[ {1 - {\text{LB}}_{\text{t}} - {\text{B}}_{\text{t}} - {\text{E}}_{\text{t}} } \right],\;z_{t}^{4} = {\rm N}^{ - 1}\left[ {{\text{LW}}_{\text{t}} } \right]$, ${\text{N}}^{ - 1} \left[ {} \right]$ is the inverse of the cumulative normal distribution.¹²

As done in the main text, we add the suffix “i” to refer to individual-level conditions. Thus, mutatis mutandis and applying the formulae (8.1) and (8.2) to the data referring to expectations on personal stances, we obtain the first, $y_{it}^{e,CP}$, and the second moment, $\upsigma_{it}^{e,CP}$, of expectations on personal stances. Analogously to what said about the parameters α and α_i in the Anderson-Theil measures (Sect. 2), $y_{t}^{r}$ and $y_{it}^{r}$ can be chosen to ensure that the CP-statistics have the same average value as the (perceived) underlying income process in the sample period. As before, it is a critical choice.¹³ To increase the comparability of the outcomes, we set $y_{t}^{r} = y_{it}^{r} = 1$ as done for the parameters α and α_i in the balance statistics. This said, it is worth noticing that (i) our aim is not the quantification of survey expectations, that (ii) we just match survey data and that (iii) this issue disappears when working with CP-signal-to-noise ratios.

The following tables report the same empirical exercises presented in the main text. To ease comparisons we maintain the same table headings of the main text, just adding “B” to the corresponding number of the table.

Alike Table 1 of the main text, Table 8 shows that the persistence of the two central tendencies is practically unchanged irrespective of the method used to compute them. Table 9 informs that the unit-root tests for $y_{t}^{e,CP}$ cannot reject the null of I(1) only at the 10 % level.¹⁴ We therefore go on with Johansen’s cointegration tests. Since two series with different orders of integration cannot be cointegrated, the cointegration test may offer further evidence on whether both central tendencies are I(1) processes.

Table 8 Survey expectations: persistence

Table 9 Survey expectations: efficient unit root tests

Table 10 reassures that the two statistics are cointegrated.

Table 10 Central tendency: cointegration test

Working with the Eqs. (7.1) and (7.2) presented in Appendix 1, we estimate another VECM whereas x _it and x _t are now the two CP-central tendencies. We collect the results in the following Table 11.

Table 11 Central tendency: VECM

Once again, the results of the main text are substantially validated.

Finally, using the two above mentioned CP statistics, define the signal to noise ratio (CP-SNR) as central tendency on cross sectional disagreement, i.e., $_{t - h} SNR_{t}^{e,CP} \equiv y_{t}^{e,CP} /\upsigma_{t}^{e,CP}$ and $_{t - h} SNR_{it}^{e,CP} \equiv y_{it}^{e,CP} /\upsigma_{it}^{e,CP}$. Similarly to what done for the two CP-central tendencies, Tables 12, 13 collects, respectively, the results of efficient unit root tests, Johansen’s cointegration tests. Table 14 the results of the bivariate VECM sets out in Eqs. (7.1) and (7.2) (cf. Appendix 1) relative to the two CP-SNRs.

Table 12 Signal-to-noise ratios: efficient unit root tests

Table 13 Signal-to-noise ratios: cointegration test

Table 14 Signal-to-noise ratios: VECM

All in all, the evidence obtained using the Carlson–Parkin method is substantially reassuring about the high degree of robustness of the outcomes reported in the main text.

Appendix 3: Robustness check. Evidence from Euro Area

In this section we exploit Euro Area (EA) data to replicate for the same sample period (Jan. 1995–Jul. 2013) exactly the same exercises done for Italy in the main text. This validation generalizes our arguments and it increases the robustness of our evidence. To underline that we are redoing the very same empirical exercises already performed with Italian data, we define the variables in the same way as before but for the fact that EA wide variables are in bold. Thus, e.g., $_{t - h} y_{it}^{e}$ stands for the balance dealing with personal situations in Italy, whereas $_{t - h} {\bf{y}}_{it}^{e}$ stands for the corresponding balance at the EA level. In the EA each interviewed faces—obviously in the language of the country (s)he lives—virtually the same queries that deal with the change over the next 12 months of (i) the “economic situation in your household” (Personal) and of (ii) the “economic situation in your country” (General). Data at national level are aggregate as follows. The percentage for each alternative answer to each question for the EA is calculated as the weighted average of the corresponding percentages in each EA member. The weights are the shares of each of the member states in the EA reference series, and are smoothed by calculating a 2 year moving average. The weights are usually updated every year in January (European Commission 2014).

Figure 8 highlights that since 2002 the two balances show no zero crossing and a tendency to stay below zero with very few exceptions. We interpret it as survey data correctly reflecting the declining macroeconomic developments in EA over the last decade and as suggesting that negative shocks have had greater impact than positive ones on both hard data and households’ expectations. More importantly, $_{t - h} \varvec{y}_{it}^{e}$ is almost always higher than $_{t - h} \varvec{y}_{t}^{e}$, suggesting that, as expected, better-than-average effects and the like (cf. Sect. 3) are an immanent trait of human being that, accordingly, are present everywhere. Even at continental level.

Fig. 8

Survey expectations. Central tendency. “Personal” ($_{t - h} {\bf{y}}_{it}^{e}$) is the central tendency (cf. Sect. 2) of the responses to the query “How do you expect the economic situation in your household to change over the next 12 months?” “General” ($_{t - h} {\bf{y}}_{t}^{e}$) is the central tendency of the responses to the query “How do you expect the general economic situation in the country to develop over the next 12 months?” They vary from −1 (all replies are: it will get a lot worse) to +1 (all replies are: it will get a lot better). Shaded area indicates periods of below-full-sample-average GDP annual growth rate. Data for the Euro Area (Jan. 95–Jul. 2013). These data are obtained aggregating national data using the weighted averages of the corresponding percentages in each euro-area member. The weights are the shares of each of the Member States in the EA reference series (European Commission 2014)

Figure 9 very clearly displays that the level of disagreement among European lay forecasters is persistent. Indeed, $\varvec{\upsigma}_{it}^{e}$ never goes below 0.55 and it has a sample mean of 0.63 (recall that it ranges between 0 and 1 = total heterogeneity). As already argued in the main text, this is less unsurprising than the fact that $\varvec{\upsigma}_{t}^{e}$, in a sample covering two-hundred and thirty months, has a minimum value of 0.8 and an average value of 0.87. That is to say, forecast heterogeneity across European households remains systematically very close to its maximum value. As stressed in Sect. 3, this finding is even more striking because it is detected in a period characterized by poor economic performances—in this environment more homogeneous predictions should show up because the cost of erring is higher and the cost of information is lower.

Fig. 9

Survey expectations. Cross sectional dispersion. Note. “Personal” ($\varvec{\upsigma}_{it}^{e}$) is the second moment (index of qualitative variation) of the responses to the query “How do you expect the economic situation in your household to change over the next 12 months?” “General” ($\varvec{\upsigma}_{t}^{e}$) is the index of qualitative variation of the responses to the query “How do you expect the general economic situation in the country to develop over the next 12 months?” (Cf. Sect. 2). $0 \le\varvec{\upsigma}_{it}^{e} ,\;\varvec{\upsigma}_{t}^{e} \le 1$. $\varvec{\upsigma}_{it}^{e} ,\;\varvec{\upsigma}_{t}^{e} = 0$ means totally homogeneous expectations. Data for the Euro Area (Jan. 95–Jul. 2013). These data are obtained aggregating national data using the weighted averages of the corresponding percentages in each euro-area member. The weights are the shares of each of the Member States in the EA reference series (European Commission 2014)

Following the same reasoning of the main text, we now turn our attention on some preliminary statistics aimed to see whether there is the need to perform cointegration analyses.

Table 15 informs that in the EA the sample persistence of the balances is even larger than in Italy. This leads to unit root tests. We collect the outcomes of these latter in Table 16.

Table 15 Survey expectations: persistence

Table 16 Survey expectations: efficient unit root tests

Table 16 robustly suggests that it is not possible to reject the null of unit root at the usual confidence levels, calling for cointegration tests. We report the relative statistics in Table 17.

Table 17 Central tendency: cointegration test

Table 17 confirms the previous findings about the presence of unit roots, and supports the view that the central tendencies of the two expectations are cointegrated at the Euro Area level too. We therefore go on with VECM analyses (cf. Table 18).

Table 18 Central tendency: VECM

The visual inspection of Table 18 once again sustains what already pointed out in the main text regarding Italian lay forecasters. Only some quantitative results change. For instance, the residual correlation (0.66) and the cointegrating vector [1, −0.57] are virtually the same in Italy and in the EA. By contrast, the speed of adjustment, δ, is now much lower (0.33 vs 0.61). As it should be clear, to our aim it is much more important that δ_i is (i) very close to zero and (ii) much lower than δ. At this point we can go on examining the SNR for the Euro Area as a whole.

These two tables clarify that the two SNRs are first-order integrated processes (Table 19) and that they are linked by an error correction model (Table 20). The outcomes stemming from the logically consequent VEC model are collected in the following Table 21.

Table 19 Signal-to-noise ratios: efficient unit root tests

Table 20 Signal-to-noise ratios: cointegration test

Table 21 Signal-to-noise ratios: VECM

Table 21 highlights that δ_i is only marginally significant and, in any case, it is still very close to zero and much smaller than δ, indicating that European households modify $_{t - h} \varvec{y}_{t}^{e}$ in order to reduce the divergence with $_{t - h} \varvec{y}_{it}^{e}$. The unrestricted cointegrating vector is [1, −0.85], but the last row of Table 21 demonstrates that the cointegrating vector is statistically [1, −1]. This offers corroborating evidence on the circumstance that, just like Italians, European lay consumers’ expectations are such that their cross sectional disagreement and time series volatility is statistically equal.

In order to save space we do not report other exercises referring to EA data.¹⁵ We just note, to conclude, that what we have said about Italy in the main text finds here and in the unreported findings strong confirmation at the European level.