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Initial Beliefs Uncertainty

  • Jaqueson K. Galimberti ORCID logo EMAIL logo
Published/Copyright: October 23, 2023
The B.E. Journal of Macroeconomics
From the journal The B.E. Journal of Macroeconomics Volume 24 Issue 1

Abstract

This paper evaluates how initial beliefs uncertainty can affect data weighting and the estimation of models with adaptive learning. One key finding is that misspecification of initial beliefs uncertainty, particularly with the common approach of artificially inflating initials uncertainty to accelerate convergence of estimates, generates time-varying profiles of weights given to past observations in what should otherwise follow a fixed profile of decaying weights. The effect of this misspecification, denoted as diffuse initials, is shown to distort the estimation and interpretation of learning in finite samples. Simulations of a forward-looking Phillips curve model indicate that (i) diffuse initials lead to downward biased estimates of expectations relevance in the determination of actual inflation, and (ii) these biases spill over to estimates of inflation responsiveness to output gaps. An empirical application with U.S. data shows the relevance of these effects for the determination of expectational stability over decadal subsamples of data. The use of diffuse initials is also found to lead to downward biased estimates of learning gains, both estimated from an aggregate representative model and estimated to match individual expectations from survey expectations data.

Keywords: expectations; adaptive learning; bounded rationality; macroeconomics
JEL Classification: E70; D83; D84; E37; C32; C63
Corresponding author: Jaqueson K. Galimberti, Asian Development Bank, 6 ADB Avenue, 1550 Mandaluyong, Metro Manila, Phillipines; Centre for Applied Macroeconomic Analysis, Australian National University, Canberra, Australia; and KOF Swiss Economic Institute, ETH Zurich, Zürich, Switzerland, E-mail: , https://sites.google.com/site/jkgeconoeng/

I thank the Associate Editor Eva Carceles-Poveda and an anonymous reviewer for their helpful and insightful assessments. Paper presented at the 27th International Conference Computing in Economics and Finance, held online in 2021. I thank the conference organizers and participants for the comments that helped shape the current draft. Earlier versions of this paper circulated under the title ‘[Initial beliefs uncertainty and] Information weighting in the estimation of models with adaptive learning.’ Any remaining errors are my own. The views expressed in this paper are those of the author and do not necessarily represent the views of his corresponding institutional affiliations.

Appendix A: Proofs and Derivations

A.1 Correspondence Between Penalized WLS and RLS

To see how the RLS of (2) and (3) can be derived from the penalized WLS formulation of (5) and (6), first notice that iterating (3) recursively from R 0 we have that

R t = ∑ i = 1 t ω t , i x i x i ′ + ω t , 0 R 0 ,

which is the inverse of the first term in (5), leading to

(22) ϕ ̂ t = R t − 1 ∑ i = 1 t ω t , i x i y i + ω t , 0 R 0 ϕ 0 .

For the second term notice that

∑ i = 1 t ω t , i x i y i = ∑ i = 1 t − 1 ω t , i x i y i + γ t x t y t , = 1 − γ t ∑ i = 1 t − 1 ω t − 1 , i x i y i + γ t x t y t ,

and

ω t , 0 R 0 ϕ 0 = 1 − γ t ω t − 1,0 R 0 ϕ 0 ,

where we use

ω t , i = 1 − γ t ω t − 1 , i ,

which follows from (6). Hence, (22) is equivalent to

(23) ϕ ̂ t = R t − 1 γ t x t y t + 1 − γ t ∑ i = 1 t − 1 ω t − 1 , i x i y i + ω t − 1,0 R 0 ϕ 0 .

Lagging (22) one period we find that

R t − 1 ϕ ̂ t − 1 = ∑ i = 1 t − 1 ω t − 1 , i x i y i + ω t − 1,0 R 0 ϕ 0 ,

which can be substituted into (23) to yield

(24) ϕ ̂ t = R t − 1 γ t x t y t + 1 − γ t R t − 1 ϕ ̂ t − 1 .

From (3) notice that

1 − γ t R t − 1 = R t − γ t x t x t ′ ,

which substituted into (24) and after rearranging leads to

ϕ ̂ t = R t − 1 γ t x t y t + R t − γ t x t x t ′ ϕ ̂ t − 1 , = γ t R t − 1 x t y t + ϕ ̂ t − 1 − γ t R t − 1 x t x t ′ ϕ ̂ t − 1 , = ϕ ̂ t − 1 + γ t R t − 1 x t y t − x t ′ ϕ ̂ t − 1 ,

establishing the correspondence between the penalized WLS solution of (5) and the RLS of (2).

A.2 Absolute and Relative Weights

Letting W t n stand for the sum of weights starting from weight n up to weight t, from the definition of the absolute weights, (6), this sum of weights can be expanded according to

(25) W t 0 = ∑ i = 0 t ω t , i , = ∏ j = 1 t 1 − γ j + ∑ i = 1 t − 1 γ i ∏ j = i + 1 t 1 − γ j + γ t .

Expanding the first term of (25) we have that

(26) ω t , 0 = 1 − γ 1 1 − γ 2 … 1 − γ t − 1 1 − γ t , = 1 − γ 2 … 1 − γ t − 1 1 − γ t − γ 1 ∏ j = 2 t 1 − γ j , = 1 − γ t − ∑ i = 1 t − 1 γ i ∏ j = i + 1 t 1 − γ j .

Returning to (25) we then have

W t 0 = 1 − γ t − ∑ i = 1 t − 1 γ i ∏ j = i + 1 t 1 − γ j + ∑ i = 1 t − 1 γ i ∏ j = i + 1 t 1 − γ j + γ t , = 1 .

A.3 Equivalent Time-Varying Gains Under Diffuse Initials

The sequence of gains, γ ̃ t , that generates equivalent weightings as a constant-gain under diffuse initials needs to solve

(27) ϖ t , l = ϖ t , l dcg , = κ 1 − γ ̄ t 1 + κ − 1 1 − γ ̄ t   for l = t , γ ̄ 1 − γ ̄ l 1 + κ − 1 1 − γ ̄ t   for 0 ≤ l < t ,

for all t and l. From Equation (7), starting with l = 0 we simply have that

(28) γ ̃ t = γ ̄ / 1 + κ − 1 1 − γ ̄ t .

It only remains to validate if Equation (28) also solves Equation (27) for l > 0. Substituting Equation (28) into Equation (7) for 0 < l < t,

ϖ t , l = γ ̄ 1 + κ − 1 1 − γ ̄ t − l ∏ j = 0 l − 1 1 − γ ̄ 1 + κ − 1 1 − γ ̄ t − j , = γ ̄ 1 − γ ̄ l 1 + κ − 1 1 − γ ̄ t − l ∏ j = 0 l − 1 1 + κ − 1 1 − γ ̄ t − j − 1 1 + κ − 1 1 − γ ̄ t − j , = γ ̄ 1 − γ ̄ l 1 + κ − 1 1 − γ ̄ t − l 1 + κ − 1 1 − γ ̄ t − l 1 + κ − 1 1 − γ ̄ t , = γ ̄ 1 − γ ̄ l 1 + κ − 1 1 − γ ̄ t ,

which solves Equation (27) for 0 < l < t. Similarly, for the initial, l = t,

ϖ t , l = ∏ j = 1 t 1 − γ ̄ 1 + κ − 1 1 − γ ̄ j , = 1 − γ ̄ t ∏ j = 1 t 1 + κ − 1 1 − γ ̄ j − 1 1 + κ − 1 1 − γ ̄ j , = 1 − γ ̄ t κ 1 + κ − 1 1 − γ ̄ t , = κ 1 − γ ̄ t 1 + κ − 1 1 − γ ̄ t .

Finally, notice that under diffuse initials, κ = 0, the weight given to the learning initials is null, i.e. ϖ t , t dcg = 0 . This is equivalent to using a γ 1 = 1, which is again satisfied by Equation (28).

Appendix B: Supplementary Results

Figure 12: Estimates of simulated Phillips curve model with constant-gain learning – using diffuse WLS pre-sample initials. Notes: Same as Figures 4 and 5 except that using diffuse WLS pre-sample estimates for R 0.
Figure 12:

Estimates of simulated Phillips curve model with constant-gain learning – using diffuse WLS pre-sample initials. Notes: Same as Figures 4 and 5 except that using diffuse WLS pre-sample estimates for R 0.

Figure 13: Estimates of simulated Phillips curve model with constant-gain learning – using OLS pre-sample initials. Notes: Same as Figures 4 and 5 except that using OLS pre-sample estimates for R 0.
Figure 13:

Estimates of simulated Phillips curve model with constant-gain learning – using OLS pre-sample initials. Notes: Same as Figures 4 and 5 except that using OLS pre-sample estimates for R 0.

Figure 14: Estimates of simulated Phillips curve model with constant-gain learning – using OLS plus diffuse WLS pre-sample initials. Notes: Same as Figures 4 and 5 except that using OLS plus diffuse WLS pre-sample estimates for R 0.
Figure 14:

Estimates of simulated Phillips curve model with constant-gain learning – using OLS plus diffuse WLS pre-sample initials. Notes: Same as Figures 4 and 5 except that using OLS plus diffuse WLS pre-sample estimates for R 0.

Figure 15: Estimates of simulated Phillips curve model with constant-gain learning – using OLS plus WLS pre-sample initials. Notes: Same as Figures 4 and 5 except that using OLS plus WLS pre-sample estimates for R 0.
Figure 15:

Estimates of simulated Phillips curve model with constant-gain learning – using OLS plus WLS pre-sample initials. Notes: Same as Figures 4 and 5 except that using OLS plus WLS pre-sample estimates for R 0.

Figure 16: Estimates of simulated Phillips curve model with constant-gain learning – with jointly estimated learning gains. Notes: Same as Figures 4 and 5 except that jointly estimating the learning gains as in Figure 8.
Figure 16:

Estimates of simulated Phillips curve model with constant-gain learning – with jointly estimated learning gains. Notes: Same as Figures 4 and 5 except that jointly estimating the learning gains as in Figure 8.

Figure 17: Estimates of simulated Phillips curve model with constant-gain learning – alternative initials uncertainty re-scaling. Notes: Same as Figures 4 and 5 with additional re-scaling of initials uncertainty, R 0.
Figure 17:

Estimates of simulated Phillips curve model with constant-gain learning – alternative initials uncertainty re-scaling. Notes: Same as Figures 4 and 5 with additional re-scaling of initials uncertainty, R 0.

Figure 18: Estimates of individual learning gains from survey forecasts – using OLS pre-sample initials. Notes: See notes to Figure 11.
Figure 18:

Estimates of individual learning gains from survey forecasts – using OLS pre-sample initials. Notes: See notes to Figure 11.

Table 2:

Joint estimates of simulated Philips curve model with constant-versus decreasing-gain.

Sample Constant-gain Decreasing-gain
Size Correct Diffuse Δ Correct Diffuse Δ
(a) Median β estimates
T = 50 0.617 0.183 −0.434 0.360 0.152 −0.208
T = 100 0.689 0.333 −0.356 0.528 0.283 −0.245
T = 250 0.787 0.572 −0.215 0.717 0.497 −0.220
T = 1000 0.866 0.804 −0.062 0.867 0.764 −0.103
(b) Median λ estimates
T = 50 0.339 0.496 0.157 0.445 0.510 0.065
T = 100 0.298 0.434 0.136 0.373 0.460 0.087
T = 250 0.254 0.341 0.087 0.293 0.375 0.082
T = 1000 0.218 0.246 0.028 0.224 0.268 0.044

  1. Notes: The estimates under constant-gain are from the same exercise as reported in 5. The estimates under decreasing-gain only differ in that a sequence of decreasing gains, given by γ t = 1/t + 1, is used instead of a constant-gain.

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Received: 2023-05-02
Accepted: 2023-10-06
Published Online: 2023-10-23

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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https://doi.org/10.1515/bejm-2023-0069
Keywords for this article
expectations; adaptive learning; bounded rationality; macroeconomics

Articles in the same Issue

  1. Frontmatter
  2. Advances
  3. Optimal Taxation of Informal Firms: Misreporting Costs and a Tax Reform in Brazil
  4. Initial Beliefs Uncertainty
  5. Credit Resource Misallocation and Macroeconomic Fluctuations in China: From the Perspective of Heterogeneous Financial Frictions
  6. The Bitcoin Premium: A Persistent Puzzle
  7. Contributions
  8. Intermediate Goods–Skill Complementarity
  9. Perfect Competition and Fixed Costs: The Role of the Ownership Structure
  10. Optimal Monetary Policy with Government-Provided Unemployment Benefits
  11. Employment Protection in Dual Labor Markets: Any Amplification of Macroeconomic Shocks?
  12. A Tide that Lifts Some Boats: Assessing the Macroeconomic Effects of EU Enlargement
  13. Current Account Balances’ Divergence in the Euro Area: An Appraisal of the Underlying Forces
  14. Merging Structural and Reduced-Form Models for Forecasting
  15. Trust in Government in a Changing World: Shocks, Tax Evasion, and Economic Growth
  16. The Fiscal Multiplier of Public Investment: The Role of Corporate Balance Sheet
  17. Does Uncertainty Matter for the Fiscal Consolidation and Investment Nexus?
  18. Government Spending Between Active and Passive Monetary Policy: An Invariance Result
  19. A DSGE Model with Government-owned Banks
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