## Appendix I: Data

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---

### Key findings (from the paper’s introduction)
- DE fits the data better: the diagnosticity parameter 휃 that governs overreaction is robustly estimated to be greater than zero; improvements in the log marginal likelihood confirm superior empirical performance of the DE specification.
- DE generates greater volatility and stronger exchange-rate responses:
  - Monetary policy shocks yield larger real effects: a policy rate increase induces a stronger exchange rate appreciation, reducing exports and contributing to a sharper output contraction.
  - Higher government spending’s expansionary impact is partly offset by a real appreciation that crowds out exports and dampens overall output response.
- DE shifts the importance of shocks (Canadian data):
  - Under RE, demand shocks (for example, government spending) are key drivers of GDP.
  - Under DE, supply shocks (for example, productivity) become the key drivers of GDP.
  - Cost-push shocks gain prominence for inflation; risk-premium shocks matter more for the exchange rate.
- DE worsens the policy trade-off:
  - Inflation–output volatility frontiers shift outward—DE induces greater endogenous volatility and strengthens the role of supply shocks. Stabilizing inflation implies larger output swings relative to RE.
- DE improves forecasting performance:
  - DE better captures turning points (for example, the output trough during the Global Financial Crisis) and provides earlier signals of economic distress, even though both frameworks struggle with extreme shock magnitudes.

### Diagnostic Expectations (DE) — primer and formalization
- Conceptual core:
  - DE implements the representativeness heuristic: agents overweight outcomes that appear more representative based on recent data, producing systematic overreaction to news.
  - The DE operator 피휃 replaces the RE operator 피 in expectations. With 휃=0, DE reduces to RE.
- Formal representation (as given in the source):
  - 피휃[red hair | Irish] = 피[red hair | Irish] + 휃(피[red hair | Irish] − 피[red hair | World])
  - In time-series macro context:
    - 피푡휃[xt+1] = 피푡[xt+1] + 휃(피푡[xt+1] − 피푡−1[xt+1])  (equation (1))
    - Interpreted as: 피푡휃[xt+1] = 피푡[xt+1] + 휃 ξt+1x  (equation (2)), where ξt+1x is the surprise at t regarding expectation of xt+1.
- DE introduces surprise terms into linearized equilibrium conditions and creates three main channels of effect in the Euler equation for consumption: surprises for current consumption (ξtC), surprises for expected future consumption (ξt+1C), and surprises related to inflation expectations (ξt+1π and ξtπ).

### Model structure and agents (overview)
- Model: a small open-economy (SOE) New Keynesian DSGE model with DE integrated; builds on Justiniano and Preston (2010) and Galí and Monacelli (2005).
- Key features:
  - Two traded goods, incomplete asset markets, habit formation in consumption, price stickiness, inflation indexation, imperfect pass-through, deviations from the law of one price (LOOP).
  - Domestic economy is small relative to rest of world; foreign variables denoted with an asterisk (e.g., X*).
  - Steady-state notation: X̅; log deviations X̂t = log(Xt/X̅).
- Households:
  - Lifetime utility under DE: U(Ct, Nt) + 피푡휃[∑_{s=1}^{∞} β^{t+s} U(C_{t+s}, N_{t+s})]  (equation (3)).
  - Period utility: U(Ct, Nt) = (Ct − h Ct−1)^{1−σ}/(1−σ) − N_t^{1+φ}/(1+φ)  (equation (4)), with β ∈ (0,1), h>0, σ>0, φ>0.
  - Consumption composite: CES of domestic CH,t and imported goods CF,t purchased at prices PH,t and PF,t:
    - CH,t = (1−α)(PH,t/Pt)^{−η} Ct  and CF,t = α (PF,t/Pt)^{−η} Ct  (equation (5)),
      with η>0 and α ∈ (0,1).
  - CPI: Pt = [ (1−α) PH,t^{1−η} + α PF,t^{1−η} ]^{1/(1−η)}; CPI inflation Πt = Pt/Pt−1.
- Household budget constraint (nominal):
  - Pt Ct + Dt Rt + et Bt Rt* φt(At) = Dt−1 + et Bt−1 + Wt Nt + F_H,t + F_F,t − Tt, ∀ t>0  (equation (6).
  - φt(.) captures a time-varying risk premium/financial friction on foreign borrowing: φt = exp{χ[−At] − ε_{rp,t}}  (equation (7)), where −At = −et Bt Pt Y̅^{-1} denotes real outstanding foreign debt as fraction of steady-state GDP Y̅, ε_{rp,t} is a risk premium shock, and in steady state φ̅ = 1.

### Linearized implications: Euler equation, UIP, and monetary transmission
- Euler equation for consumption (linearized, under DE):
  - Ĉt = h/(1+h) (Ĉt−1 − 휃 ξtC) + h/(1+h) {피t[Ĉt+1] + 휃 ξt+1C} − 1/σ (1+h) {R̂t − 피t[Π̂t+1] − 휃(ξt+1π + ξtπ)}  (equation (8)).
- DE channels in the Euler equation:
  - Surprise on current consumption, ξtC, negatively affects current consumption Ĉt.
  - Surprise on future consumption, ξt+1C, amplifies the reaction of Ĉt to expected future consumption.
  - Inflation expectation surprises (ξt+1π, ξtπ) cause overreaction to expected future and current inflation, distorting the real interest rate term R̂t − 피t[Π̂t+1] − 휃(ξt+1π + ξtπ), and thereby affecting intertemporal consumption choices.
- UIP condition (linearized, under DE):
  - êt = R̂t* − R̂t + φ̂t + 피t[êt+1] + 휃[ξt+1Δe + ξtΔe]  (equation (9)),
    where Δe denotes the change in the linearized exchange rate.
  - Implication: nominal exchange rate is driven by interest rate differentials (R̂t* − R̂t), the risk premium φ̂t, expected future exchange rate 피t[êt+1], and additional surprise components in exchange rate expectations captured by 휃[ξt+1Δe + ξtΔe], amplifying exchange rate volatility.
- Monetary policy transmission:
  - DE strengthens exchange-rate amplification: policy-rate changes influence exchange rate expectations and surprise terms, which feed back into consumption and output via the distorted real interest rate and the UIP channel, thereby increasing both nominal and real exchange-rate responses and associated real effects (trade and output).

### Exchange rates, LOOP, and real exchange rate decomposition
- LOOP departures: model assumes LOOP does not hold; price gap Γ_{F,t} exists between foreign CPI P_t* and domestic-currency price of imports PF,t, defined Γ_{F,t} = e_t P_t* / P_{F,t}.
- CPI-based real exchange rate: qt = e_t P_t* / P_t.
- Terms of trade (TOT): St = P_{F,t} / P_{H,t}.
- Linearized real exchange rate decomposition:
  - q̂t = Γ̂_{F,t} + (1 − α) Ŝt.
  - Interpretation: real exchange rate fluctuations are driven by deviations from LOOP (the price gap Γ̂_{F,t}) and movements in TOT Ŝt.

### 2.3 Firms — final domestic goods, intermediate goods, retail import firms
- Final domestic goods firms:
  - Three firm types in the model: final domestic goods firms, intermediate goods firms, and retail import firms.
  - Final domestic goods firms operate in perfect competition, aggregating intermediate goods with a Dixit‑Stiglitz aggregator.
  - Demand for each variety i:
    - Y_H,t(i) = (P_H,t(i) / P_H,t)^(-ε_H) Y_H,t  (equation (10)), with ε_H > 1.
- Intermediate goods firms:
  - Production: Y_H,t(i) = ε_a,t N_t(i)  (equation (11)), where ε_a,t is TFP.
  - Price adjustment costs (Rotemberg):
    - AC_H,t(i) = ψ_H / 2 ( (P_H,t(i) / (Π_H,t−1^δ_H P_H,t−1(i))) − 1 )^2 P_H,t Y_H,t  (equation (12)), with ψ_H ≥ 0, δ_H ≥ 0.
  - Firms maximize expected discounted profits; real marginal cost:
    - MC_t = W_t / (P_H,t ε_a,t); subsidy ι_F calibrated such that MC̅ = 1.
  - Symmetric linearized NKPC:
    - (1 + β δ_H) Π̂_H,t = δ_H Π̂_H,t−1 + β π_t [Π̂_H,t+1] + (ε_H − 1)/ψ_H MĈ_t + β θ [ ξ_{t+1}^{Π_H} − δ_H ξ_t^{Π_H} ].  (14)
  - Marginal cost in log-deviations:
    - MĈ_t = α Ŝ_t + φ Ŷ_H,t + σ/(1−h) (Ĉ_t − h Ĉ_{t−1}) − (1+φ) ε̂_{a,t}.
  - Role of DE: adds β θ [ ξ_{t+1}^{Π_H} − δ_H ξ_t^{Π_H} ], generating additional volatility and overreaction in domestic inflation dynamics.
- Retail import firms:
  - Continuum of retailers import differentiated foreign goods; LOOP does not hold.
  - Imported varieties face time-varying elasticity ε_F,t > 1 to introduce cost-push shocks.
  - Price adjustment costs AC_F,t(i) with ψ_F ≥ 0 and indexation δ_F ≥ 0.
  - Symmetric linearized NKPC for import inflation:
    - (1 + β δ_F) Π̂_F,t = δ_F Π̂_F,t−1 + β π_t [Π̂_F,t+1] + (ε_F − 1) α / ψ_F Γ̂_F,t + β θ [ ξ_{t+1}^{Π_F} − δ_F ξ_t^{Π_F} ] + ε̂_{cp,t}.  (16)
  - DE similarly introduces β θ [ ξ_{t+1}^{Π_F} − δ_F ξ_t^{Π_F} ], amplifying imported goods inflation dynamics.

### Appendix III: Unit shocks and Rational Expectations counterfactual
- Overview:
  - Appendix III Figures 1-5 consider unit shocks to complement the one standard deviation analysis.
  - Counterfactual: all parameters set to their RE estimates except θ = 1.96 to illustrate effect of the overreaction parameter independently.
  - Even with RE parameters, setting θ = 1.96 leads to heightened volatility and more pronounced exchange-rate effects.
- Imported goods inflation and demand shock (DE vs RE):
  - Under DE, imported goods inflation declines less because the associated NKPC is flatter, damping responsiveness to the price gap.
  - Sharper initial real appreciation under DE dampens output response to the demand shock.
  - A government spending impulse raises output less under DE than under RE, driven by DE’s exchange rate transmission.
- Risk premium shock:
  - Under RE:
    - A one-standard-deviation increase in the country risk premium triggers nominal and real depreciation, increases the price gap, raises imported goods inflation and CPI inflation, stimulates exports, and prompts the central bank to tighten after a few quarters.
  - Under DE:
    - Depreciations are considerably larger due to depreciation surprise terms in UIP, stimulating activity more via exports.
    - Imported-goods NKPC is flatter under DE, limiting pass-through of depreciation to imported goods inflation.
    - Domestic inflation remains lower than under RE despite larger output increases; monetary authority raises policy rate initially more than under RE.
    - Output response features an initial spike under DE from a stronger exchange rate channel.
- Cost-push shock:
  - Under both RE and DE:
    - Cost-push shock increases imported goods inflation, raises domestic inflation, central bank tightens, inducing nominal and real appreciations, reducing the price gap, depressing exports and consumption, and contracting domestic output.
  - Under DE:
    - Imported goods inflation and domestic inflation impacts are amplified by inflation surprise terms.
    - Larger initial CPI inflation leads to stronger monetary tightening and sharper consumption contraction.
    - Tighter policy generates more pronounced initial real appreciation and a more severe decline in economic activity on impact.
  - Technical note (verbatim):
    - More specifically, note that the marginal cost for domestic goods is given by  푀퐶̂푡 = 훼푆̂푡 + 휑 푌̂퐻,푡 + 휎/(1−ℎ) (퐶̂푡 − h퐶̂푡−1) − (1+휑) 휀̂푎,푡. Therefore, under DE, a more pronounced decline in TOT (푆̂푡)—a relative price adjustment which is not shown in the figure—helps dampen the domestic inflationary pressures.
- Productivity shock:
  - Temporary increase in TFP raises domestic output and lowers domestic inflation under both RE and DE.
  - Central bank cuts policy rate, stimulating consumption and inducing nominal and real depreciation; depreciation raises the price gap of imported goods, pushing imported goods inflation upward and partially offsetting domestic disinflation.
  - Under DE:
    - Stronger real exchange rate response via depreciation surprise terms amplifies exports and generates an additional spike in output.
    - Households overreact by increasing consumption more strongly than under RE, reinforcing output expansion and dampening initial domestic disinflation.
- Variance decompositions (VD):
  - Table 4 (and Appendix I Table AI.2) and Figure 7 show DE changes:
    - Under RE, government spending shock explains most GDP variability; under DE, TFP becomes more prominent.
    - Countercyclicality of net exports is amplified under DE due to greater real exchange rate volatility.
    - Cost-push shocks gain prominence for headline inflation and the real interest rate under DE.
    - Monetary policy shocks are more influential for domestic inflation volatility under DE.
    - Risk premium shocks matter more for the real exchange rate under DE.
  - The prominence of supply shocks under DE aligns with commodities’ importance for the Canadian economy in the parsimonious setup.
- Output–inflation volatility trade-offs:
  - DE always yields higher output and inflation volatility than RE for corresponding models.
  - DE results in a worse inflation-output volatility trade-off; Figure 8 traces frontiers for RE, DE (θ=0.59 and θ=1.96).
  - All three frontiers show a non-linear inverse relationship between output and inflation volatility; increased diagnosticity (higher θ) moves frontier away from origin.

### 4.4 Forecasting Properties—An Initial Assessment
- Forecasting exercise design:
  - Variables: output (gap), CPI and domestic core inflation, output and real exchange rate (RER) growth.
  - Horizons: h = 1, h = 2, h = 4 (one-, two-, and four-quarters).
  - Periods: tranquil (2016 Q1) and turbulent (2008 Q3).
  - Metric: RMSE = √(1/H ∑(yt+h − ŷt+h)^2) for h = 1 and h = 4.
- Key forecasting findings:
  - DE (θ = 1.96):
    - Outperforms RE for output gap, core inflation, real GDP and RER growth in the near term (h = 1) during 2016 Q1.
    - Better forecasts the real exchange rate at all horizons.
    - Better captures dip in output growth in 2016 Q2 and the trough during the GFC (2008 Q3) for six-month-ahead forecasts.
    - Signals ensuing RER depreciation in turbulent episodes but underestimates magnitude of relative price shift.
  - DE (θ = 0.59):
    - Mixed performance; sometimes similar to RE, sometimes intermediate between RE and DE (θ = 1.96).
  - General caution: stylized framework; extreme global dislocation episodes remain challenging.
- Forecast evaluation: RMSE (Table 5)
  - Tranquil period: 2016 Q1 (RMSE)
    - h = 1
      - RE: Output (gap) 0.954; Domestic (core) inflation 0.399; Real GDP growth 0.644; CPI inflation 0.224; Real exchange rate growth 5.995
      - DE (θ=1.96): Output (gap) 0.802; Domestic (core) inflation 0.394; Real GDP growth 0.493; CPI inflation 0.330; Real exchange rate growth 4.763
      - DE (θ=0.59): Output (gap) 0.958; Domestic (core) inflation 0.401; Real GDP growth 0.649; CPI inflation 0.353; Real exchange rate growth 5.529
    - h = 2
      - RE: Output (gap) 0.487; Domestic (core) inflation 0.088; Real GDP growth 1.240; CPI inflation 3.446; Real exchange rate growth 14.627
      - DE (θ=1.96): Output (gap) 0.130; Domestic (core) inflation 0.089; Real GDP growth 1.857; CPI inflation 3.417; Real exchange rate growth 12.344
      - DE (θ=0.59): Output (gap) 0.267; Domestic (core) inflation 0.072; Real GDP growth 1.460; CPI inflation 3.472; Real exchange rate growth 13.728
    - h = 4
      - RE: Output (gap) 0.333; Domestic (core) inflation 0.011; Real GDP growth 0.821; CPI inflation 0.370; Real exchange rate growth 1.105
      - DE (θ=1.96): Output (gap) 0.464; Domestic (core) inflation 0.010; Real GDP growth 0.985; CPI inflation 0.328; Real exchange rate growth 0.591
      - DE (θ=0.59): Output (gap) 0.382; Domestic (core) inflation 0.012; Real GDP growth 0.886; CPI inflation 0.337; Real exchange rate growth 0.938
  - Turbulent period: 2008 Q3 (RMSE)
    - h = 1
      - RE: Output (gap) 0.509; Domestic (core) inflation 0.065; Real GDP growth 0.444; CPI inflation 0.275; Real exchange rate growth 0.842
      - DE (θ=1.96): Output (gap) 0.560; Domestic (core) inflation 0.084; Real GDP growth 0.242; CPI inflation 0.324; Real exchange rate growth 0.060
      - DE (θ=0.59): Output (gap) 0.560; Domestic (core) inflation 0.074; Real GDP growth 0.398; CPI inflation 0.322; Real exchange rate growth 0.616
    - h = 2
      - RE: Output (gap) 1.705; Domestic (core) inflation 0.296; Real GDP growth 2.192; CPI inflation 1.030; Real exchange rate growth 3.007
      - DE (θ=1.96): Output (gap) 1.544; Domestic (core) inflation 0.291; Real GDP growth 1.414; CPI inflation 1.017; Real exchange rate growth 6.462
      - DE (θ=0.59): Output (gap) 1.712; Domestic (core) inflation 0.286; Real GDP growth 1.978; CPI inflation 1.008; Real exchange rate growth 4.124
    - h = 4
      - RE: Output (gap) 0.272; Domestic (core) inflation 0.191; Real GDP growth 1.240; CPI inflation 6.994; Real exchange rate growth 3.253
      - DE (θ=1.96): Output (gap) 0.284; Domestic (core) inflation 0.695; Real GDP growth 1.211; CPI inflation 9.366; Real exchange rate growth 3.765
      - DE (θ=0.59): Output (gap) 0.273; Domestic (core) inflation 0.325; Real GDP growth 1.176; CPI inflation 7.856; Real exchange rate growth 3.460
- Interpretation and model properties relevant to forecasting:
  - DE amplifies endogenous volatility via agents’ systematic overreaction; this can improve short-term responsiveness to shocks and help capture troughs during turbulent episodes.
  - DE strengthens the exchange rate channel: a rise in the policy rate under DE results in a larger real appreciation and thus tighter monetary conditions, producing a larger economic contraction.
  - DE worsens the inflation–output volatility trade-off faced by policymakers.
- Robustness, caveats, and extensions:
  - Exercise is a first pass using a streamlined model and parsimonious DE.
  - Suggested extensions:
    - Relax parsimonious DE assumption and consider distant memory models.
    - Incorporate investment, capital accumulation, sticky wages, richer supply side.
    - Extend to emerging market economies and select financial frictions.
    - Explore more general and distant-memory variants of DE.
  - Additional robustness checks: 2017 and the COVID episode considered; 2017 broadly consistent with 2016, while for COVID inflation forecasts performed well but other macro outcomes were mixed.

### Appendix II: Microeconomic Foundations, the Steady State and the Linearized Model
- Households:
  - Intratemporal labor–consumption condition:
    - W_t / P_t = (C_t − h C_{t−1})^{−σ} N_t^{ϕ}. (A1)
  - Euler equation (micro foundations):
    - (C_t − h C_{t−1})^{−σ} / Π_t R_t = β E_t[θ ((C_{t+1} − h C_t)^{−σ} / Π_{t+1})]. (A2)
  - UIP with country risk premia:
    - R_t / Π_t = β E_t[θ ((C_{t+1} − h C_t)^{−σ} / Π_{t+1})] = R_t^* φ_t((e_t / e_{t−1})) β E_t[θ ((C_{t+1} − h C_t)^{−σ} / Π_{t+1}) ((e_{t+1}/e_t)(e_t/e_{t−1}))]. (A3)
- Firms (micro foundations):
  - Final goods aggregator and demand (A4–A6).
  - Non-linear NKPC for domestic inflation (A7); marginal cost expressions (A8–A9).
  - Retail import firms: price adjustment costs (A10), demand (A11), real profits (A13), and non-linear NKPC for imported goods inflation (A14).
- Steady state:
  - Simplifying assumptions: A̅ = 0; S̅ = Π̅_F = Π̅_H = Π̅ = Y̅_H = 1; subsidies ensure MC̅ = Γ̅_F = 1.
  - Implications: φ̅ = Q̅ = P̅ / P̅_H = P̅ / P̅_F = C̅ = 1; R̅ = R̅^* = 1/β; C̅_H = 1 − α; C̅_F = α; Y̅^* = C̅_H^* = α.
- Linearized model — structural behavioral equations:
  - Expectation surprise: ξ_{t+j} x = E_t[x̂_{t+j}] − E_{t−1}[x̂_{t+j}] for j = 0,1.
  - Key linearized equations preserved exactly (Euler, UIP, φ̂_t, NKPCs, MĈ_t, price gap, Taylor rule, government spending shock, market clearing, budget constraint).
  - Definitions: real interest rate r̂_t = R̂_t − E_t[Π̂_{H,t+1}], ∆Ŝ_t = Ŝ_t − Ŝ_{t−1}, ∆q̂_t = q̂_t − q̂_{t−1}, ∆q̂_t = ∆ê_t + Π̂_t^* − Π̂_t, ∆Ŝ_t = Π̂_{F,t} − Π̂_{H,t}, Π̂_t = α ∆Ŝ_t + Π̂_{H,t}.
- Foreign sector VAR(2): equations for Ŷ_t^*, R̂_t^*, Π̂_t^* with 2 lags and innovations μ_{·,t}^*.
- Shocks (autoregressive specifications):
  - ε̂_{a,t} = ρ_a ε̂_{a,t−1} + μ_{a,t}.
  - ε̂_{g,t} = ρ_g ε̂_{g,t−1} + μ_{g,t}.
  - ε̂_{rp,t} = ρ_{rp} ε̂_{rp,t−1} + μ_{rp,t}.
  - ε̂_{cp,t} = ρ_{cp} ε̂_{cp,t−1} + μ_{cp,t}.
  - ε̂_{m,t} = ρ_m ε̂_{m,t−1} + μ_{m,t}.
- Appendix III — supplementary IRF analysis (figures summary):
  - Figure AII.1: Impulse Responses to a Monetary Policy Shock (μ_m). “modified parameters v2” corresponds to θ = 1.96.
  - Figure AIII.2: Impulse Responses to a Government Spending Shock (μ_g). θ = 1.96 in modified parameters v2.
  - Figure AIII.3: Impulse Responses to a Risk Premium Shock (μ_rp). θ = 1.96 in modified parameters v2.
  - Figure AIII.4: Impulse Responses to an Imports Cost-Push Shock (μ_cp). θ = 1.96 in modified parameters v2.
  - Figure AIII.5: Impulse Responses to a Productivity Shock (μ_a). θ = 1.96 in modified parameters v2.

*Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025246-source-pdf.pdf*

### Appendix I: Data .......................................................................................................

### Appendix I: Data

### Key findings (from the paper’s introduction)
- DE fits the data better: the diagnosticity parameter 휃 that governs overreaction is robustly estimated to be greater than zero; improvements in the log marginal likelihood confirm superior empirical performance of the DE specification.
- DE generates greater volatility and stronger exchange-rate responses:
  - Monetary policy shocks yield larger real effects: a policy rate increase induces a stronger exchange rate appreciation, reducing exports and contributing to a sharper output contraction.
  - Higher government spending’s expansionary impact is partly offset by a real appreciation that crowds out exports and dampens overall output response.
- DE shifts the importance of shocks (Canadian data):
  - Under RE, demand shocks (for example, government spending) are key drivers of GDP.
  - Under DE, supply shocks (for example, productivity) become the key drivers of GDP.
  - Cost-push shocks gain prominence for inflation; risk-premium shocks matter more for the exchange rate.
- DE worsens the policy trade-off:
  - Inflation–output volatility frontiers shift outward—DE induces greater endogenous volatility and strengthens the role of supply shocks. Stabilizing inflation implies larger output swings relative to RE.
- DE improves forecasting performance:
  - DE better captures turning points (for example, the output trough during the Global Financial Crisis) and provides earlier signals of economic distress, even though both frameworks struggle with extreme shock magnitudes.

### Diagnostic Expectations (DE) — primer and formalization
- Conceptual core:
  - DE implements the representativeness heuristic: agents overweight outcomes that appear more representative based on recent data, producing systematic overreaction to news.
  - The DE operator 피휃 replaces the RE operator 피 in expectations. With 휃=0, DE reduces to RE.
- Formal representation (as given in the source):
  - 피휃[red hair | Irish] = 피[red hair | Irish] + 휃(피[red hair | Irish] − 피[red hair | World])
  - In time-series macro context:
    - 피푡휃[xt+1] = 피푡[xt+1] + 휃(피푡[xt+1] − 피푡−1[xt+1])  (equation (1))
    - Interpreted as: 피푡휃[xt+1] = 피푡[xt+1] + 휃 ξt+1x  (equation (2)), where ξt+1x is the surprise at t regarding expectation of xt+1.
- DE introduces surprise terms into linearized equilibrium conditions and creates three main channels of effect in the Euler equation for consumption: surprises for current consumption (ξtC), surprises for expected future consumption (ξt+1C), and surprises related to inflation expectations (ξt+1π and ξtπ).

### Model structure and agents (overview)
- Model: a small open-economy (SOE) New Keynesian DSGE model with DE integrated; builds on Justiniano and Preston (2010) and Galí and Monacelli (2005).
- Key features:
  - Two traded goods, incomplete asset markets, habit formation in consumption, price stickiness, inflation indexation, imperfect pass-through, deviations from the law of one price (LOOP).
  - Domestic economy is small relative to rest of world; foreign variables denoted with an asterisk (e.g., X*).
  - Steady-state notation: X̅; log deviations X̂t = log(Xt/X̅).
- Households:
  - Lifetime utility under DE: U(Ct, Nt) + 피푡휃[∑_{s=1}^{∞} β^{t+s} U(C_{t+s}, N_{t+s})]  (equation (3)).
  - Period utility: U(Ct, Nt) = (Ct − h Ct−1)^{1−σ}/(1−σ) − N_t^{1+φ}/(1+φ)  (equation (4)), with β ∈ (0,1), h>0, σ>0, φ>0.
  - Consumption composite: CES of domestic CH,t and imported goods CF,t purchased at prices PH,t and PF,t:
    - CH,t = (1−α)(PH,t/Pt)^{−η} Ct  and CF,t = α (PF,t/Pt)^{−η} Ct  (equation (5)),
      with η>0 and α ∈ (0,1).
  - CPI: Pt = [ (1−α) PH,t^{1−η} + α PF,t^{1−η} ]^{1/(1−η)}; CPI inflation Πt = Pt/Pt−1.
- Household budget constraint (nominal):
  - Pt Ct + Dt Rt + et Bt Rt* φt(At) = Dt−1 + et Bt−1 + Wt Nt + F_H,t + F_F,t − Tt, ∀ t>0  (equation (6)).
  - φt(.) captures a time-varying risk premium/financial friction on foreign borrowing: φt = exp{χ[−At] − ε_{rp,t}}  (equation (7)), where −At = −et Bt Pt Y̅^{-1} denotes real outstanding foreign debt as fraction of steady-state GDP Y̅, ε_{rp,t} is a risk premium shock, and in steady state φ̅ = 1.

### Linearized implications: Euler equation, UIP, and monetary transmission
- Euler equation for consumption (linearized, under DE):
  - Ĉt = h/(1+h) (Ĉt−1 − 휃 ξtC) + h/(1+h) {피t[Ĉt+1] + 휃 ξt+1C} − 1/σ (1+h) {R̂t − 피t[Π̂t+1] − 휃(ξt+1π + ξtπ)}  (equation (8)).
  - DE channels in the Euler equation:
    - Surprise on current consumption, ξtC, negatively affects current consumption Ĉt.
    - Surprise on future consumption, ξt+1C, amplifies the reaction of Ĉt to expected future consumption.
    - Inflation expectation surprises (ξt+1π, ξtπ) cause overreaction to expected future and current inflation, distorting the real interest rate term R̂t − 피t[Π̂t+1] − 휃(ξt+1π + ξtπ), and thereby affecting intertemporal consumption choices.
- UIP condition (linearized, under DE):
  - êt = R̂t* − R̂t + φ̂t + 피t[êt+1] + 휃[ξt+1Δe + ξtΔe]  (equation (9)),
    where Δe denotes the change in the linearized exchange rate.
  - Implication: nominal exchange rate is driven by interest rate differentials (R̂t* − R̂t), the risk premium φ̂t, expected future exchange rate 피t[êt+1], and additional surprise components in exchange rate expectations captured by 휃[ξt+1Δe + ξtΔe], amplifying exchange rate volatility.
- Monetary policy transmission:
  - DE strengthens exchange-rate amplification: policy-rate changes influence exchange rate expectations and surprise terms, which feed back into consumption and output via the distorted real interest rate and the UIP channel, thereby increasing both nominal and real exchange-rate responses and associated real effects (trade and output).

### Exchange rates, LOOP, and real exchange rate decomposition
- LOOP departures: model assumes LOOP does not hold; price gap Γ_{F,t} exists between foreign CPI P_t* and domestic-currency price of imports PF,t, defined Γ_{F,t} = e_t P_t* / P_{F,t}.
- CPI-based real exchange rate: qt = e_t P_t* / P_t.
- Terms of trade (TOT): St = P_{F,t} / P_{H,t}.
- Linearized real exchange rate decomposition:
  - q̂t = Γ̂_{F,t} + (1 − α) Ŝt.
  - Interpretation: real exchange rate fluctuations are driven by deviations from LOOP (the price gap Γ̂_{F,t}) and movements in TOT Ŝt.

*Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025246-source-pdf.pdf*

### 2.3 Firms

### 2.3 Firms

### Final domestic goods firms
- Three firm types in the model: final domestic goods firms, intermediate goods firms, and retail import firms.
- Final domestic goods firms:
  - Operate in perfect competition.
  - Aggregate a continuum of intermediate goods indexed by i ∈ [0,1] into composite good Y_H,t using a Dixit‑Stiglitz aggregator.
  - Intermediate goods Y_H,t(i) are supplied by monopolistically competitive intermediate firms charging prices P_H,t(i).
  - Demand for each variety i (from cost minimization subject to the aggregator) is:
    - Y_H,t(i) = (P_H,t(i) / P_H,t)^(-ε_H) Y_H,t,  (10)
    - where ε_H > 1 denotes the elasticity of substitution across domestic varieties of intermediate goods.

### Intermediate goods firms
- Each intermediate producer i supplies a differentiated variety Y_H,t(i) using linear technology:
  - Y_H,t(i) = ε_a,t N_t(i),  (11)
  - where ε_a,t is total factor productivity (TFP), common across firms and subject to exogenous productivity shocks.
- Production is demand-determined; each firm faces the downward-sloping demand curve in equation (10).
- Nominal rigidities: price adjustment costs à la Rotemberg (1982) with quadratic cost:
  - AC_H,t(i) = ψ_H / 2 ( (P_H,t(i) / (Π_H,t−1^δ_H P_H,t−1(i))) − 1 )^2 P_H,t Y_H,t,  (12)
  - where ψ_H ≥ 0 captures degree of price stickiness and δ_H ≥ 0 reflects indexation to past domestic inflation.
- Firms set P_H,t(i) to maximize expected discounted profits:
  - F_H,t(i)_P_H,t + π_t θ [ ∑_{s=1}^∞ Λ_{t,t+s} F_H,t+s(i)_P_H,t+s ],  (13)
  - Real profits defined as:
    - F_H,t(i)_P_H,t = P_H,t(i)/P_H,t (P_H,t(i)/P_H,t)^{−ε_H} Y_H,t − (1−ι_H) MC_t (P_H,t(i)/P_H,t)^{−ε_H} Y_H,t − AC_H,t(i)/P_H,t,  (14)
  - Real marginal cost: MC_t = W_t / (P_H,t ε_a,t), which is not firm-specific.
  - Subsidy ι_F calibrated such that MC̅ = 1 in steady state.
  - Stochastic discount factor: Λ_{t,t+s} = β^s (C_{t+s} − h C_{t+s−1})^{−σ} (C_t − h C_{t−1})^{−σ}.
- Under symmetry P_H,t(i) = P_H,t for all i, the linearized New Keynesian Phillips Curve (NKPC) is:
  - (1 + β δ_H) Π̂_H,t = δ_H Π̂_H,t−1 + β π_t [Π̂_H,t+1] + (ε_H − 1)/ψ_H MĈ_t + β θ [ ξ_{t+1}^{Π_H} − δ_H ξ_t^{Π_H} ].  (14)
  - Π̂_H,t is log-deviation of domestic inflation; ξ_{t+1}^{Π_H} and ξ_t^{Π_H} are inflation expectation surprises.
  - Marginal cost in log-deviations:
    - MĈ_t = α Ŝ_t + φ Ŷ_H,t + σ/(1−h) (Ĉ_t − h Ĉ_{t−1}) − (1+φ) ε̂_{a,t}.
- Role of Diagnostic Expectations (DE):
  - Compared with standard NKPC under Rational Expectations (RE), DE adds the term β θ [ ξ_{t+1}^{Π_H} − δ_H ξ_t^{Π_H} ].
  - This term captures surprises in inflation expectations, implying DE can generate additional volatility and overreaction in domestic inflation dynamics (e.g., ξ_{t+1}^{Π_H} > 0 adds to current inflationary pressures beyond the standard forward-looking RE term β π_t [Π̂_H,t+1]).

### Retail import firms
- Continuum of retail firms i ∈ [0,1] import differentiated foreign goods, operate under monopolistic competition.
- Each retailer purchases quantity C_F,t(i) of imported goods at price e_t P_t^* = Γ_F,t P_F,t (LOOP does not hold).
- Retailers differentiate and resell imported goods at retail price P_F,t(i), facing a demand curve analogous to equation (10) but with time-varying elasticity:
  - Elasticity of substitution between imported varieties ε_F,t > 1 is time-variant to introduce cost-push shocks (as in Justiniano and Preston (2010)).
- Price adjustment costs AC_F,t(i) à la Rotemberg (1982) with ψ_F ≥ 0 and indexation δ_F ≥ 0; Appendix II contains first-order conditions.
- Under symmetry P_F,t(i) = P_F,t for all i, the linearized NKPC for import inflation is:
  - (1 + β δ_F) Π̂_F,t = δ_F Π̂_F,t−1 + β π_t [Π̂_F,t+1] + (ε_F − 1) α / ψ_F Γ̂_F,t + β θ [ ξ_{t+1}^{Π_F} − δ_F ξ_t^{Π_F} ] + ε̂_{cp,t}.  (16)
  - DE similarly introduces the inflation expectation surprise term β θ [ ξ_{t+1}^{Π_F} − δ_F ξ_t^{Π_F} ], which can amplify volatility and induce overreactions in imported goods inflation dynamics.
  - Note: ε̂_{cp,t} denotes a normalized price-markup shock process as defined in the model.

*Source: wpiea2025246-source-pdf - 2.3 Firms*

### Appendix III Figures 1-5 consider unit shocks to complement the analysis shown here that illustrates one standard deviat

### Appendix III: Unit shocks and Rational Expectations counterfactual

### Overview
- Appendix III Figures 1-5 consider unit shocks to complement the analysis that illustrates one standard deviation shocks.
- The appendix figures of the IRFs also present a counterfactual scenario in which all parameters take their Rational Expectations (RE) estimates, except for θ = 1.96, thereby illustrating how the overreaction parameter influences model dynamics independently.
- Even in the case of RE parameters, setting θ = 1.96 leads to heightened volatility and more pronounced effects on both real and nominal exchange rates in response to different shocks. As a result, the exchange rate channel becomes a central mechanism for either amplifying or dampening shock transmission.

### Imported goods inflation and demand shock (DE vs RE)
- Under Diagnostic Expectations (DE), imported goods inflation declines less because the associated NKPC is flatter, making it less responsive to changes in the price gap.
- This dampening effect dominates the influence of the larger negative price gaps and the expectation-surprise terms that would otherwise push imported goods inflation further down.
- The sharper initial real appreciation under DE dampens the output response to the demand shock.
- A government spending impulse raises output less under DE than under RE, primarily driven by DE’s effects on the exchange rate transmission (relative to a closed-economy framework).

### Risk premium shock
- Under RE:
  - A one-standard-deviation increase in the country risk premium triggers a nominal and real depreciation.
  - This causes a positive LOOP deviation—increase in the price gap (marginal cost of imported goods).
  - Imported goods inflation rises, pushing up CPI inflation.
  - Real depreciation raises foreign demand for domestic goods, stimulating exports and adding domestic inflationary pressures.
  - The central bank tightens monetary policy—raising real interest rates after a few quarters—while the economy gradually converges to steady state.
- Under DE:
  - The nominal and real depreciation are considerably larger due to depreciation surprise terms in the UIP condition.
  - The initial sharp real depreciation stimulates economic activity more than under RE through stronger export effects.
  - Imported-goods NKPC is significantly flatter under DE, limiting pass-through of the larger depreciation to imported goods inflation.
  - Despite larger output increases and further real appreciation, domestic inflation remains lower than under RE because of smaller impacts on imported goods inflation.
  - With smaller increases in domestic and imported inflation, CPI inflation is lower under DE, yet the larger output increase and depreciation prompt the monetary authority to raise the policy rate initially more than under RE.
- The output response broadly aligns with Adrian and others (2021); a key distinction is the initial spike in output under DE driven by a stronger exchange rate channel due to depreciation surprise terms.

### Cost-push shock
- Under both RE and DE:
  - A cost-push shock increases imported goods inflation and necessitates a relative price adjustment that also raises domestic inflation (Figure 5).
  - CPI inflation rises, prompting the central bank to raise the nominal interest rate.
  - Tighter monetary stance induces nominal and real appreciations, generating a negative deviation in the price gap (marginal cost of imported goods), mitigating inflationary impact.
  - Real appreciation depresses exports; higher interest rates contract consumption, leading to a decline in domestic output.
- Under DE, differences are:
  - Impact on imported goods inflation is amplified by inflation surprise terms in the imported-goods NKPC.
  - Stronger increase in domestic inflation explained by surprise terms in the domestic NKPC.
  - Larger initial increase in CPI inflation leads to stronger monetary tightening and a sharper contraction in consumption.
  - Tighter policy generates a more pronounced initial real appreciation.
  - Decline in economic activity is more severe on impact due to stronger exchange rate channel under DE, higher interest rate response, and consumption surprise terms in the Euler equation.
- Technical note (verbatim):
  - More specifically, note that the marginal cost for domestic goods is given by  푀퐶̂푡 = 훼푆̂푡 + 휑 푌̂퐻,푡 + 휎/(1−ℎ) (퐶̂푡 − h퐶̂푡−1) − (1+휑) 휀̂푎,푡. Therefore, under DE, a more pronounced decline in TOT (푆̂푡)—a relative price adjustment which is not shown in the figure—helps dampen the domestic inflationary pressures.

### Productivity shock
- A temporary increase in total factor productivity:
  - Raises domestic output and lowers domestic inflation under both RE and DE (Figure 6).
  - Higher productivity reduces marginal costs, pushing down domestic and CPI inflation.
  - The central bank cuts the policy rate, stimulating consumption and inducing nominal and real depreciation.
  - Real depreciation raises the price gap of imported goods, pushing imported goods inflation upward and partially offsetting domestic disinflation.
  - Depreciation strengthens exports, which together with higher consumption further boosts domestic output.
- Under DE:
  - Stronger real exchange rate response—driven by depreciation surprise terms in the UIP condition—amplifies the export response and generates an additional spike in output.
  - Households overreact amid brighter prospects by increasing consumption more strongly than under RE, further reinforcing output expansion and dampening the initial decline in domestic inflation.

### Variance decompositions (VD)
- Table 4 presents VD results for RE and DE models (Appendix I Table AI.2 provides further details). Figure 7 visualizes how DE changes the VDs for a subset of variables: output, inflation, domestic inflation, real interest rate, real exchange rate; combining contributions of the three foreign shocks.
- Key changes under DE:
  - Structural shocks driving key macroeconomic variables shift in relative importance.
  - Under RE, government spending shock explains most variability in GDP; under DE, TFP becomes more prominent.
  - Countercyclicality of net exports is amplified under DE owing to greater real exchange rate volatility, helping stabilize output.
  - Cost-push shocks become more prominent for headline inflation and the real interest rate under DE.
  - Monetary policy shocks are more influential for domestic inflation volatility in the DE model.
  - Risk premium shocks gain traction under DE for the real exchange rate.
  - The greater role of cost-push and risk premium shocks reflects the larger role of exchange rate fluctuations in the DE model.
- The prominence of supply shocks (TFP, cost push) under DE is consistent with the importance of commodities for the Canadian economy (energy, metals, agricultural products), which are captured by exogenous shocks in the parsimonious setup; this may help explain why DE improves model fit for Canada.

### Output–inflation volatility trade-offs
- Standard inflation-output volatility frontiers are constructed to illustrate policy implications of DE.
- Relative to RE, volatility of output and inflation is always higher for corresponding DE models.
- The DE model results in a worse inflation-output volatility trade-off compared with RE, in part due to greater prominence of supply shocks.
- Figure 8 traces inflation-output volatility policy frontiers for three models: RE, DE (θ=0.59 and θ=1.96).
  - All three frontiers illustrate a non-linear inverse relationship between output and inflation volatility.
  - DE models present a worse policy trade-off since their frontiers are further away from the origin.
  - An increasing degree of diagnosticity results in a less favorable trade-off.

*Source: wpiea2025246-source-pdf - Appendix III Figures 1-5 consider unit shocks to complement the analysis shown here that illustrates one standard deviat*

### 4.4 Forecasting Properties—An Initial Assessment

### 4.4 Forecasting Properties—An Initial Assessment

### Forecasting exercise design
- Evaluated five macroeconomic variables: the output (gap), CPI and (domestic) core inflation, and output and real exchange rate (RER) growth.
- Horizons considered: one-, two-, and four-quarters (h = 1, h = 2, h = 4).
- Periods analyzed: a relatively tranquil period (2016 Q1) and a turbulent period (2008 Q3, the Global Financial Crisis).
- Metric of forecast accuracy: root mean squared error (RMSE), computed as RMSE = √(1/H ∑(yt+h − ŷt+h)^2) for h = 1 and h = 4 for one-quarter- and one-year-ahead calculations.

### Key forecasting findings
- DE model with θ = 1.96:
  - Outperforms the RE model for the output gap, core inflation, and real GDP and RER growth in the near term (h = 1) during 2016 Q1.
  - Better forecasts the real exchange rate at all horizons; improvement attributed to surprise terms DE introduces into the UIP conditions which capture a broader array of RER dynamics.
  - Better captures the dip in output growth in the second quarter of 2016, likely due to near-term overreaction that DE imbues in the model at shorter horizons.
  - During the GFC (2008 Q3), displays superior performance for output and inflation variables for six-month-ahead forecasts; best at capturing the trough in output growth.
  - Signals ensuing RER depreciation in turbulent episodes but falls short relative to the data in capturing the magnitude of the relative price shift.
- DE model with θ = 0.59:
  - Mixed performance relative to RE and DE (θ = 1.96); in some cases similar to RE, in others intermediate between RE and DE (θ = 1.96).
- General caution: accurately replicating dynamics in episodes of global economic dislocation is challenging for any model; DE shows promise but results are from a stylized framework.

### Forecast evaluation: RMSE (Table 5)
- Tranquil period: 2016 Q1 (RMSE)
  - h = 1
    - RE: Output (gap) 0.954; Domestic (core) inflation 0.399; Real GDP growth 0.644; CPI inflation 0.224; Real exchange rate growth 5.995
    - DE (θ=1.96): Output (gap) 0.802; Domestic (core) inflation 0.394; Real GDP growth 0.493; CPI inflation 0.330; Real exchange rate growth 4.763
    - DE (θ=0.59): Output (gap) 0.958; Domestic (core) inflation 0.401; Real GDP growth 0.649; CPI inflation 0.353; Real exchange rate growth 5.529
  - h = 2
    - RE: Output (gap) 0.487; Domestic (core) inflation 0.088; Real GDP growth 1.240; CPI inflation 3.446; Real exchange rate growth 14.627
    - DE (θ=1.96): Output (gap) 0.130; Domestic (core) inflation 0.089; Real GDP growth 1.857; CPI inflation 3.417; Real exchange rate growth 12.344
    - DE (θ=0.59): Output (gap) 0.267; Domestic (core) inflation 0.072; Real GDP growth 1.460; CPI inflation 3.472; Real exchange rate growth 13.728
  - h = 4
    - RE: Output (gap) 0.333; Domestic (core) inflation 0.011; Real GDP growth 0.821; CPI inflation 0.370; Real exchange rate growth 1.105
    - DE (θ=1.96): Output (gap) 0.464; Domestic (core) inflation 0.010; Real GDP growth 0.985; CPI inflation 0.328; Real exchange rate growth 0.591
    - DE (θ=0.59): Output (gap) 0.382; Domestic (core) inflation 0.012; Real GDP growth 0.886; CPI inflation 0.337; Real exchange rate growth 0.938
- Turbulent period: 2008 Q3 (RMSE)
  - h = 1
    - RE: Output (gap) 0.509; Domestic (core) inflation 0.065; Real GDP growth 0.444; CPI inflation 0.275; Real exchange rate growth 0.842
    - DE (θ=1.96): Output (gap) 0.560; Domestic (core) inflation 0.084; Real GDP growth 0.242; CPI inflation 0.324; Real exchange rate growth 0.060
    - DE (θ=0.59): Output (gap) 0.560; Domestic (core) inflation 0.074; Real GDP growth 0.398; CPI inflation 0.322; Real exchange rate growth 0.616
  - h = 2
    - RE: Output (gap) 1.705; Domestic (core) inflation 0.296; Real GDP growth 2.192; CPI inflation 1.030; Real exchange rate growth 3.007
    - DE (θ=1.96): Output (gap) 1.544; Domestic (core) inflation 0.291; Real GDP growth 1.414; CPI inflation 1.017; Real exchange rate growth 6.462
    - DE (θ=0.59): Output (gap) 1.712; Domestic (core) inflation 0.286; Real GDP growth 1.978; CPI inflation 1.008; Real exchange rate growth 4.124
  - h = 4
    - RE: Output (gap) 0.272; Domestic (core) inflation 0.191; Real GDP growth 1.240; CPI inflation 6.994; Real exchange rate growth 3.253
    - DE (θ=1.96): Output (gap) 0.284; Domestic (core) inflation 0.695; Real GDP growth 1.211; CPI inflation 9.366; Real exchange rate growth 3.765
    - DE (θ=0.59): Output (gap) 0.273; Domestic (core) inflation 0.325; Real GDP growth 1.176; CPI inflation 7.856; Real exchange rate growth 3.460

### Interpretation and model properties relevant to forecasting
- DE amplifies endogenous volatility via agents’ systematic overreaction; this can improve short-term responsiveness to shocks and help capture troughs during turbulent episodes.
- DE strengthens the exchange rate channel:
  - A rise in the policy rate under DE results in a larger real appreciation and thus tighter monetary conditions, producing a larger economic contraction.
  - The stronger real appreciation can dampen fiscal policy’s output response by overwhelming aggregate demand stimulus through a larger contraction of the trade balance.
- DE worsens the inflation–output volatility trade-off faced by policymakers, as illustrated by inflation–output volatility frontiers computed in the analysis.

### Robustness, caveats, and extensions
- Exercise presented as a first pass: streamlined model and parsimonious DE specification.
- Results are stylized and caution is warranted before generalizing.
- Suggested extensions and avenues for future research:
  - Relax parsimonious DE assumption and consider distant memory models similar to those used in Bianchi and others (2024).
  - Incorporate investment, capital accumulation, sticky wages, and a richer supply side (Christiano and others, 2005; Smets and Wouters, 2007; Adolfson and others, 2007).
  - Extend the model to emerging market economies and incorporate selected financial frictions.
  - Explore more general and distant-memory variants of DE to potentially further improve model fit and forecast accuracy.
- Additional robustness checks noted in the source:
  - 2017 and the COVID episode were also considered as alternative tranquil and turbulent periods; results for 2017 were broadly consistent with 2016, whereas for COVID the inflation forecasts performed well but other macroeconomic outcomes were more mixed.

*Source: wpiea2025246-source-pdf - 4.4 Forecasting Properties—An Initial Assessment*

### Appendix II: Microeconomic Foundations, the Steady State

### Appendix II: Microeconomic Foundations, the Steady State and the Linearized Model

### A. Microeconomic Foundations — Households
- Representative household chooses sequences for C_t, N_t, D_t, and B_t to maximize expected discounted utility (equations (3) and (4)) subject to the period budget constraint (equation (6)) and a standard transversality condition on bond holdings.
- First-order conditions and their equilibrium interpretations:
  - Intratemporal labor–consumption condition:
    - W_t / P_t = (C_t − h C_{t−1})^{−σ} N_t^{ϕ}. (A1)
    - Interpreted as the optimal intra-temporal trade-off implicitly defining labor supply.
  - Euler equation:
    - (C_t − h C_{t−1})^{−σ} / Π_t R_t = β E_t[θ ((C_{t+1} − h C_t)^{−σ} / Π_{t+1})]. (A2)
    - Characterizes optimal intertemporal consumption smoothing via domestic bond accumulation/decumulation.
  - Uncovered interest parity (UIP) with country risk premia:
    - R_t / Π_t = β E_t[θ ((C_{t+1} − h C_t)^{−σ} / Π_{t+1})] = R_t^* φ_t((e_t / e_{t−1})) β E_t[θ ((C_{t+1} − h C_t)^{−σ} / Π_{t+1}) ((e_{t+1}/e_t)(e_t/e_{t−1}))]. (A3)
    - Equates expected returns on domestic and foreign bonds adjusted for expected exchange rate depreciation and country-specific risk premia.

### A. Microeconomic Foundations — Firms
- Final goods firms:
  - Aggregate continuum of intermediate goods i ∈ [0,1] using Dixit–Stiglitz aggregator:
    - Y_{H,t} ≡ ( ∫_0^1 Y_{H,t}(i)^{(ε_H−1)/ε_H} di )^{ε_H/(ε_H−1)}. (A4)
  - Demand for variety i:
    - Y_{H,t}(i) = (P_{H,t}(i) / P_{H,t})^{−ε_H} Y_{H,t}. (A5)
  - Domestic price index:
    - P_{H,t} = ( ∫_0^1 P_{H,t}(i)^{1−ε_H} di )^{1/(1−ε_H)}. (A6)
- Intermediate domestic firms (monopolistic competition, nominal price rigidities):
  - Non-linear New Keynesian Phillips Curve (NKPC) for domestic inflation:
    - ψ_H ((Π_{H,t}/Π_{H,t−1})^{δ_H−1}) (Π_{H,t}/Π_{H,t−1})^{δ_H} = (1−ε_H) + ε_H MC_t + β ψ_H Π_{H,t} (Λ_t Y_{H,t})^{−1} E_t[Λ_{t+1} ( (Π_{H,t+1}/Π_{H,t})^{δ_H−1} ) (Π_{H,t+1}/Π_{H,t})^{δ_H} Y_{H,t+1} / Π_{H,t}]. (A7)
  - Marginal cost expression:
    - MC_t = W_t / P_{H,t} × ε_{a,t}. (A8)
  - Symmetric-equilibrium marginal-cost expression:
    - MC_t = [ (1−α) + α S_t^{1−η} ]^{1/(1−η)} Y_{H,t}^{ϕ} (ε_{a,t})^{1+ϕ} (C_t − h C_{t−1})^{−σ}. (A9)
- Retail import firms (monopolistic competition, Rotemberg price adjustment costs):
  - Price adjustment cost:
    - AC_{F,t}(i) = ψ_F / 2 × ( (P_{F,t}(i) / Π_{F,t−1}^{δ_F} P_{F,t−1}(i)) − 1 )^2 × P_{F,t} / C_{F,t}. (A10)
  - Demand for imported variety i:
    - C_{F,t}(i) = (P_{F,t}(i) / P_{F,t})^{−ε_{F,t}} C_{F,t}, with ε_{F,t} > 1 (time-varying). (A11)
  - Real profits (retailer i):
    - F_{F,t}(i)/P_{F,t} = (P_{F,t}(i)/P_{F,t})^{1−ε_{F,t}} C_{F,t,−} − (1−ι_F) Γ_{F,t} (P_{F,t}(i)/P_{F,t})^{−ε_{F,t}} C_{F,t,−} − AC_{F,t}(i)/P_{F,t}. (A13) (ι_F denotes subsidy ensuring Γ̅_F = 1)
  - NKPC for imported goods inflation (non-linear):
    - ψ_F ((Π_{F,t}/Π_{F,t−1})^{δ_F−1}) (Π_{F,t}/Π_{F,t−1})^{δ_F} = (1−ε_{F,t}) + ε_{F,t} Γ_{F,t} + β ψ_F Π_{F,t} (Λ_t C_{F,t})^{−1} E_t[Λ_{t+1} ( (Π_{F,t+1}/Π_{F,t})^{δ_F−1} ) (Π_{F,t+1}/Π_{F,t})^{δ_F} C_{F,t+1} / Π_{F,t}] . (A14)

### B. Steady State
- Simplifying steady-state assumptions:
  - Net foreign assets: A̅ = 0.
  - Normalizations: S̅ = Π̅_F = Π̅_H = Π̅ = Y̅_H = 1.
  - Subsidies ensure MC̅ = Γ̅_F = 1.
- Implications for steady-state variables (by construction and model equilibrium):
  - φ̅ = Q̅ = P̅ / P̅_H = P̅ / P̅_F = C̅ = 1.
  - Nominal interest rates:
    - R̅ = R̅^* = 1/β.
  - Consumption allocations:
    - C̅_H = 1 − α and C̅_F = α.
  - Foreign economy:
    - Y̅^* = C̅_H^* = α.

### C. Linearized Model — Structural Behavioral Equations (domestic economy)
- Expectation operator for linearized variable x̂:
  - ξ_{t+j} x = E_t[x̂_{t+j}] − E_{t−1}[x̂_{t+j}] for j = 0,1.
- Key linearized equations (variables denoted with hats):
  - Euler equation for consumption:
    - Ĉ_t = h/(1+h) (Ĉ_{t−1} − θ ξ_t^C) + h/(1+h) { E_t[Ĉ_{t+1}] + θ ξ_{t+1}^C } − (1/σ)(1+h) { R̂_t − E_t[Π̂_{t+1}] − θ (ξ_{t+1}^π + ξ_t^π) }.
  - Uncovered interest parity (UIP):
    - ê_t = R̂_t^* − R̂_t + φ̂_t + E_t[ê_{t+1}] + θ (ξ_{t+1}^{∆e} + ξ_t^{∆e}).
  - Country-risk premium:
    - φ̂_t = −χ Â_t − ε̂_{rp,t}.
  - NKPC for domestic inflation (linearized):
    - (1 + β δ_H) Π̂_{H,t} = δ_H Π̂_{H,t−1} + β E_t[Π̂_{H,t+1}] + (ε_H − 1)/ψ_H MĈ_t + β θ (ξ_{t+1}^{Π_H} − δ_H ξ_t^{Π_H}).
  - Real marginal cost (linearized):
    - MĈ_t = α Ŝ_t + ϕ Ŷ_{H,t} + σ/(1−h) (Ĉ_t − h Ĉ_{t−1}) − (1+ϕ) ε̂_{a,t}.
  - NKPC for imports inflation (linearized):
    - (1 + β δ_F) Π̂_{F,t} = δ_F Π̂_{F,t−1} + β E_t[Π̂_{F,t+1}] + (ε_F − 1) α / ψ_F Γ̂_{F,t} + β θ (ξ_{t+1}^{Π_F} − δ_F ξ_t^{Π_F}) + ε̂_{cp,t}.
  - Price gap:
    - Γ̂_{F,t} = q̂_t − (1−α) Ŝ_t.
  - Taylor rule (monetary policy):
    - R̂_t = ρ R̂_{t−1} + ψ_π Π̂_{H,t} + ψ_y Ŷ_t + ψ_{Δe} Δê_t + ε̂_{M,t}.
  - Government spending shock:
    - Ĝ_{H,t} = ε̂_{g,t}.
  - Market clearing for domestic goods:
    - Ŷ_{H,t} = (1−α) Ĉ_t + Ĝ_{H,t} + η α (2−α) Ŝ_t + α η Γ̂_{F,t} + α Ŷ_t^*.
  - Domestic economy budget constraint:
    - β Â_t − Â_{t−1} = NX̂_t = Ŷ_{H,t} − Ĉ_t − Ĝ_{H,t} − α (Ŝ_t + Γ̂_{F,t}).
- Additional definitions and dynamics:
  - Real interest rate:
    - r̂_t = R̂_t − E_t[Π̂_{H,t+1}].
  - Change in terms of trade (TOT):
    - ∆Ŝ_t = Ŝ_t − Ŝ_{t−1}.
  - Change in real exchange rate (rer):
    - ∆q̂_t = q̂_t − q̂_{t−1}.
  - Relation between change of rer, nominal depreciation, and foreign and domestic inflation:
    - ∆q̂_t = ∆ê_t + Π̂_t^* − Π̂_t.
  - Relation between TOT change and imports and domestic inflation:
    - ∆Ŝ_t = Π̂_{F,t} − Π̂_{H,t}.
  - Relation between CPI inflation, TOT change, and domestic inflation:
    - Π̂_t = α ∆Ŝ_t + Π̂_{H,t}.

### D. Foreign Sector VAR(2)
- Foreign output:
  - Ŷ_t^* = ρ_{yy,1} Ŷ_{t−1}^* + ρ_{yr,1} R̂_{t−1}^* + ρ_{yπ,1} Π̂_{t−1}^* + ρ_{yy,2} Ŷ_{t−2}^* + ρ_{yr,2} R̂_{t−2}^* + ρ_{yπ,2} Π̂_{t−2}^* + μ_{y,t}^*.
- Foreign interest rate:
  - R̂_t^* = ρ_{ry,1} Ŷ_{t−1}^* + ρ_{rr,1} R̂_{t−1}^* + ρ_{rπ,1} Π̂_{t−1}^* + ρ_{ry,2} Ŷ_{t−2}^* + ρ_{rr,2} R̂_{t−2}^* + ρ_{rπ,2} Π̂_{t−2}^* + μ_{r,t}^*.
- Foreign inflation:
  - Π̂_t^* = ρ_{πy,1} Ŷ_{t−1}^* + ρ_{πr,1} R̂_{t−1}^* + ρ_{ππ,1} Π̂_{t−1}^* + ρ_{πy,2} Ŷ_{t−2}^* + ρ_{πr,2} R̂_{t−2}^* + ρ_{ππ,2} Π̂_{t−2}^* + μ_{π,t}^*.

### E. Shocks (autoregressive specifications)
- Technology:
  - ε̂_{a,t} = ρ_a ε̂_{a,t−1} + μ_{a,t}.
- Government spending:
  - ε̂_{g,t} = ρ_g ε̂_{g,t−1} + μ_{g,t}.
- Risk premium:
  - ε̂_{rp,t} = ρ_{rp} ε̂_{rp,t−1} + μ_{rp,t}.
- Cost-push:
  - ε̂_{cp,t} = ρ_{cp} ε̂_{cp,t−1} + μ_{cp,t}.
- Monetary policy:
  - ε̂_{m,t} = ρ_m ε̂_{m,t−1} + μ_{m,t}.

### F. Appendix III — Supplementary Impulse Response Function (IRF) Analysis (figures summary)
- Figures reported (authors' calculations; percent deviations from steady state; shock is a unit increase):
  - Figure AII.1: Impulse Responses to a Monetary Policy Shock (μ_m). Note: “modified parameters v2” simulations correspond to the case in which all parameters are set to their Rational Expectations estimates, except for the diagnosticity parameter (θ = 1.96).
  - Figure AIII.2: Impulse Responses to a Government Spending Shock (μ_g). Same note on “modified parameters v2” and θ = 1.96.
  - Figure AIII.3: Impulse Responses to a Risk Premium Shock (μ_rp). Same note on θ = 1.96.
  - Figure AIII.4: Impulse Responses to an Imports Cost-Push Shock (μ_cp). Same note on θ = 1.96.
  - Figure AIII.5: Impulse Responses to a Productivity Shock (μ_a). Same note on θ = 1.96.

*Source: Appendix II and Appendix III of "The Importance of Diagnostic Expectations in Open Economies", IMF Working Paper No. WP/2025/246.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025246-source-pdf.pdf_