When all we have is not enough: a search for the optimal method of quantifying inflation expectations

Figures & data

Table 1. Expectation quantification methods.

	Probabilistic methods	Regression methods
Author(s)	Theil (1952); Carlson and Parkin (1975)	Anderson (1952); Pesaran (1984)
Non-technical description	Percentages of responses to survey questions are expressed in terms of the probabilities of future inflation being in certain intervals. Sensitivity intervals are applied for (i) respondents declaring that prices will be stable (they do not necessarily mean that future inflation will be exactly equal to zero) and for (ii) respondents reporting that prices will increase at the same rate (they do not necessarily mean that future inflation will be exactly unchanged in comparison to the current situation). Solving a set of equations returns the expected inflation rate, its standard deviation and sensitivity intervals.	This relies on the estimation of the relationship between actual inflation as reported by official statistics and its survey respondents’ perceptions. The survey question on perceived inflation (Q5) is used to find coefficients of equations describing this relationship. It is assumed that the same relationship holds between respondents’ qualitative opinions concerning future price changes and expected inflation. Thus, inflation perception is a yardstick for the quantification of respondents’ expectations. Coefficients derived from Q5 answers are used–together question Q6 answers to reveal expected inflation.
Assumptions	The original Carlson and Parkin approach assumes the unbiasedness of inflation expectations and their normal distribution. Extensions remove the former and allow for the modifications of the latter. There is a need for an assumption about the scaling factor to which respondents refer to while answering the survey’s questions.	Unbiasedness of inflation perception. Additional assumptions differ across the models applied. No assumption about distribution is needed.
Versions/Extensions	Regarding the scaling factor: objectified (actual inflation rate serves as a scaling factor) and subjectified (a reference to perceived inflation). Regarding the distribution: normal, uniform, logistic, Student’s- t distributions, triangular distribution. The most commonly applied extension is that of Batchelor and Orr (1988). It adjusts the probabilistic method to polychotomous (five-questions) surveys which are currently a standard tool for examining expectations. The literature that presents methods that limits shortcomings and adjust the probabilistic procedure to modern economic conditions expands.	Models by Pesaran (1984), Smith and McAleer (1995) allowed for different model solutions regarding general price level change. Pesaran introduced asymmetrical relation of perceived inflation and expected price changes. Smith and McAleer presented a model with time varying parameters. Simmons and Weiserbs (1992) model was designed to work for the polychotomous survey.
Critique	Unrealistic assumptions and difficulties regarding the empirical application of the method; empirical distributions do not mimic theoretical distributions of variables; accuracy of quantified expectations.	The primary version of the procedure assumes constant model coefficients even if the inflation dynamic varies over time. Problems related to models’ estimations including standard econometric problems as the choice of estimator. Designed mostly for use in three-question surveys.

Source: Authors.

Table 2. Regression methods applied.

Anderson’ model (Anderson, 1952): $x_t=\alpha R_t-\beta F_t+{\epsilon }_t$$x_t$ - the dependent variable, $R_t, F_t$ - the percentage share reporting an increase (decrease) in variable $x_t,$ $\alpha , \beta$- parameters of the model, ${\epsilon }_t$ - error term

Balance statistic’s based model (Cunningham, 1997): $x_t={\alpha }_0+{\alpha }_1{B}_t+{\epsilon }_t$$x_t$ - the dependent variable, $B_t$ - the balance statistic, ${\alpha }_{0,}{\alpha }_1$ - parameters of the model, ${\epsilon }_t$ - error term

Pesaran’s model (Pesaran, 1984): $x_t=\frac{\alpha R_t-\beta F_t}{1-\gamma R_t}+{\epsilon }_t$$x_t$ - the dependent variable, $R_t, F_t$ - the percentage share reporting an increase (decrease) in variable $x_t,$ $\alpha , \beta , \gamma$ - parameters of the model, ${\epsilon }_t$ - error term

Smith and McAleer’s model (Smith & McAleer, 1995): $x_t=\frac{\alpha R_t-\beta F_t}{1-\gamma R_t-\delta F_t}+{\epsilon }_t$$x_t$ - the dependent variable, $R_t, F_t$ - the percentage share reporting an increase (decrease) in variable $x_t,$ $\alpha , \beta , \gamma , \delta$ - parameters of the model, ${\epsilon }_t$ - error term

Smith and McAleer’s time varying parameters model (Smith & McAleer, 1995): $x_t=\frac{\alpha R_t-\beta F_t+R_t{\sum }_{i=1}^l{\theta }_{1i}{\pi }_{t-i}+F_t{\sum }_{j=1}^q{\theta }_{2j}{\pi }_{t-j}}{1-\gamma R_t-\delta F_t}+{\epsilon }_t$$x_t$ - the dependent variable, $R_t, F_t$ - the percentage share reporting an increase (decrease) in variable $x_t,$ $\alpha , \beta , \gamma , \delta , {\theta }_{1i}, {\theta }_{2i}$ - parameters of the model, ${\epsilon }_t$ - error term

Cunningham’ s model (Cunningham, 1997): $\mathit{ln}\left(\frac{R_t }{1-R_t}\right)={{\alpha }_0+{\alpha }_{1}x}_t{+\epsilon }_{1t},$ $\mathit{ln}\left(\frac{F_t }{1-F_t}\right)={{\beta }_0+{\beta }_{1}x}_t{+\epsilon }_{2t}$$\mathit{ln}\left(·\right)$ - logistic transformation, $x_t$ - the dependent variable, $R_t, F_t$ - the percentage share reporting an increase (decrease) in variable $x_t,$ ${\alpha }_{0,}{\alpha }_1,{\beta }_{0,}{\beta }_1$ - parameters of the model, ${\epsilon }_{1t}{, \epsilon }_{2t}$ - error terms

Notes: we present a general specification of models applicable with Q5 (or BS) answers to estimate parameters and thereafter quantify expected inflation rate.

Source: Authors.

Table 3. Probabilistic methods applied.

No.	Distribution	Scaling factor	Referred to as
1	Normal	Subjectified version – inflation perception	CP_normal_sub
2		2M lagged inflation	CP_normal_obj_inf
3		Food prices inflation	CP_normal_obj_food
4		Electricity prices inflation	CP_normal_obj_electricity
5		Fuel prices inflation	CP_normal_obj_fules
6		Averaged evolution of food, energy, and fuel prices	CP_normal_obj_expectation_av
7		Weighted average evolution of food, energy, and fuel prices	CP_normal_obj_expectation_wav
8	Uniform	Subjectified version – inflation perception	CP_uniform_sub
9		2M lagged inflation	CP_uniform_obj_inf
10		Food prices inflation	CP_uniform_obj_food
11		Electricity prices inflation	CP_uniform_obj_electricity
12		Fuel prices inflation	CP_uniform_obj_fules
13		Averaged evolution of food, energy, and fuel prices	CP_uniform_obj_expectation_av
14		Weighted average evolution of food, energy, and fuel prices	CP_uniform_obj_expectation_wav
15	Logistic	Subjectified version – inflation perception	CP_logistic_sub
16		2M lagged inflation	CP_logistic_obj_inf
17		Food prices inflation	CP_logistic_obj_food
18		Electricity prices inflation	CP_logistic_obj_electricity
9		Fuel prices inflation	CP_logistic_obj_fules
20		Averaged evolution of food, energy, and fuel prices	CP_logistic_obj_expectation_av
21		Weighted average evolution of food, energy, and fuel prices	CP_logistic_obj_expectation_wav
22	Student’s-t	Subjectified version – inflation perception	CP_t-Student_sub
23		2M lagged inflation	CP_t-Student_obj_inf
24		Food prices inflation	CP_t-Student_obj_food
25		Electricity prices inflation	CP_t-Student_obj_electricity
26		Fuel prices inflation	CP_t-Student_obj_fules
27		Averaged evolution of food, energy, and fuel prices	CP_t-Student_obj_expectation_av
28		Weighted average evolution of food, energy, and fuel prices	CP_t-Student_obj_expectation_wav
29	Skewed Student’s-t	Subjectified version – inflation perception	CP_skewed_t-Student_sub
30		2M lagged inflation	CP_skewed_t-Student_obj_inf
31		Food prices inflation	CP_skewed_t-Student_obj_food
32		Electricity prices inflation	CP_skewed_t-Student_obj_electricity
33		Fuel prices inflation	CP_skewed_t-Student_obj_fules
34		Averaged evolution of food, energy, and fuel prices	CP_skewed_t-Student_obj_expectation_av
35		Weighted average evolution of food, energy, and fuel prices	CP_skewed_t-Student_obj_expectation_wav

Source: Authors.

Table 4. Evaluation criteria setup.

Evaluation criterion	Assumed assumptions	Way of selecting the optimal method
Criterion first: correlation only	Pearson’s correlation coefficient accounting for the relationship between expectations and balance statistics.	Correlation of expectations and balance statistics in the highest percentile possible.
Criterion second: correlation and directional co-movements	Correlation coefficient and directional co-movements coefficient (see Eq. 2(2) $$DC=\frac{\frac{1}{2}\sum _{i=1}^n\left|\mathit{sign}\right(\Delta x_i)+\mathit{sign}(\Delta B{S}_i)|-\frac{1}{2}\sum _{i=1}^n\left|\mathit{sign}\right(\Delta x_i)-\mathit{sign}(\Delta B{S}_i)|}{n},$$(2) ) of expectations and balance statistics.	Combining a correlation coefficient in the highest percentile possible with statistics detecting directional co-movement of expectations and BS in the highest percentile possible.
Criterion third: correlation and forecast accuracy	Correlation and forecast accuracy (Theil’s coefficient of inequality, see Eq. 3(3) $$U=\frac{\sqrt{\frac{1}{n}\sum _{i=1}^n{\left(y_i-x_i\right)}^2}}{\sqrt{\frac{1}{n}\sum _{i=1}^n\mathrm{ }{y}_i^2}+\sqrt{\frac{1}{n}\sum _{i=1}^n{x}_i^2}},$$(3) ) of expectations and balance statistics.	Combining a correlation coefficient in the highest percentile possible with Theil’s coefficient for forecast errors in the lowest percentile possible.

Source: Authors.

Table 5. The first evaluation criterion: a summary of results.

Quantification method	Eurozone		Non-Eurozone
	Full sample	Since 2010	Full sample	Since 2010
CP_uniform_sub			HU	HU
CP_uniform_obj_inf	PT	GR	UK	UK
CP_uniform_obj_food			RO
CP_uniform_obj_expectation_av	NL	NL, PT
CP_uniform_obj_expectation_wav		AT, DE
CP_logistic_sub			DK
CP_logistic_obj_inf	AT, BE, FI	FI	SE	SE
CP_logistic_obj_expectation_wav	FR		HR	HR
CP_t-Student_obj_inf	GR			RO
CP_skewed_t-Student_sub			CZ, PL	PL
CP_skewed_t-Student_obj_inf	DE, ES, IT	BE, ES, FR, IT		BG
CP_skewed_t-Student_obj_expectation_wav			BG	CZ

Notes: We use ISO two-letter country codes to denote economies covered. Bolded country codes indicate that the same procedure is optimal for full sample and 2010+ subsample. Correlations between quantified proxies of expectations and balance statistics for all economies are statistically significant and strong. DK is the only exception: moderate correlation for the full sample was detected and no statistically important correlation for shortened subsample. We linked this result with the low-quality survey data for DK (their publication was suspended). Results for DK are not reported for shortened sample. For numerical results please check supplementary materials.

Source: Authors.

Table 6. The second evaluation criterion: a summary of results.

Quantification method	Eurozone		Non-Eurozone
	Full sample	Since 2010	Full sample	Since 2010
CP_normal_sub	BE, GR
CP_uniform_sub			PL	HU, PL
CP_uniform_obj_inf	AT, PT	NL, PT	RO	CZ
CP_logistic_sub		FI	CZ, DK, SE
CP_logistic_obj_inf	NL			HR, SE, UK
CP_logistic_obj_expectation_wav	FR			BG
CP_t-Student_sub	DE		HU	RO
CP_t-Student_obj_expectation_wav	ES
CP_skewed_t-Student_sub	FI		HR
CP_skewed_t-Student_obj_inf		AT, BE, FR
CP_skewed_t-Student_obj_food	IT	GR, IT	UK
CP_skewed_t-Student_obj_expectation_wav		DE	BG

Notes: see details under Table 3.

Source: Authors.

Table 7. The third evaluation criterion: a summary of results.

Quantification method	Eurozone		Non-Eurozone
	Full sample	Since 2010	Full sample	Since 2010
CP_normal_sub	BE		HU
CP_normal_obj_inf	ES, GR, IT	ES
CP_normal_obj_food			RO
CP_normal_obj_expectation_wav	NL	NL
CP_uniform_sub			CZ
CP_uniform_obj_inf	DE, PT	PT, AT	UK	UK
CP_logistic_obj_inf	AT	GR, FI	DK, SE	CZ, RO
CP_logistic_obj_food
CP_logistic_obj_expectation_av	FI
CP_logistic_obj_expectation_wav			HR, PL	HR, PL
CP_t-Student_sub				HU
CP_t-Student_obj_inf		IT
CP_skewed_t-Student_obj_inf		FR, DE, BE		BG
CP_skewed_t-Student_obj_expectation_av	FR
CP_skewed_t-Student_obj_expectation_wav			BG	SE

Notes: see details under Table 3.

Source: Authors.

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