## wpiea2024252-print-pdf — selected sections (4.1, 4.2, 8.1–8.2, 9, 10, Part III)

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### 4.1 Intuition
- Key observations from monthly survey data (2000–2024)
  - Expected exchange rate changes are not zero and "can be quite substantial."
  - Expected exchange rate changes exhibit high persistence over time.
  - Notable negative relationship between exchange rate levels and expected future changes; strong for most countries, weaker for a few, including Great Britain.
  - Data: monthly data of 12-month expected exchange rate changes vis-à-vis the US dollar for the period 2000–2024 from Das et al. (2022).
  - Empirical focus: nine inflation-targeting countries with freely floating exchange rates (selected because their interest rate differentials are exogenous to expected exchange rate changes).
- Conceptual framework: dual-component decomposition
  - s_t decomposed into:
    - μ_t: a slowly evolving stochastic trend (random walk driven by fundamentals).
    - s^c_t: a stationary cyclical component that reverts to its mean.
  - Implications:
    - Long-term unpredictability from μ_t (unit root behavior).
    - Medium-term predictability from s^c_t: expected exchange rate changes depend solely on s^c_t—the larger s^c_t, the higher expected depreciation.
    - Expected exchange rate changes are zero only if s^c_t = 0.
    - If μ_t did not evolve slowly, persistence of expected changes and the level–change link would weaken.
- Predicted predictability pattern
  - Inverted U-shaped predictability:
    - Least predictable short term (noise dominates).
    - Most predictable medium term (stationary component dominates).
    - Least predictable long term (stochastic trend dominates).
- Illustrative numerical scenarios (20% initial appreciation)
  - Scenario 1 (cyclical predominant, 90%):
    - 90% of 20% = 18 percentage points from s^c_t; 2 percentage points from μ_t.
    - Expected future depreciation ≈ 18%.
    - Interpretation: strong mean reversion and strong predictability.
  - Scenario 2 (stochastic-trend predominant, 90%):
    - 10% of 20% = 2 percentage points from s^c_t; 18 percentage points from μ_t.
    - Expected future depreciation ≈ 2%.
    - Interpretation: weak mean reversion and reduced predictability.
- Broader implication
  - Dual-component framework reconciles unit root behavior with observed persistent expected exchange rate changes when μ_t evolves slowly and s^c_t is significant.

### 4.2 The Math
- Model setup and key equations
  - s_t = μ_t + s^c_t  (equation 4.2.1).
  - μ_t = μ_{t−1} + η_t, with η_t white noise, mean zero, variance σ^2_η  (equation 4.2.2).
  - E_t[s_{t+h}] = E_t[μ_{t+h}] + E_t[s^c_{t+h}]  (equation 4.2.3).
  - E_t[s^c_{t+h}] = ρ(h) s^c_t  (equation 4.2.4).
  - E_t[μ_{t+h}] = μ_t  (equation 4.2.5).
  - Implication: E_t[s_{t+h} − s_t] = −(1 − ρ(h)) s^c_t  (equation 4.2.6).
  - Conclusion: expected change determined by s^c_t; zero expected change only if s^c_t = 0.
- HP-filter settings referenced for trend/cycle extraction (Monthly data)
  - λ = 14400, λ = 1e6, λ = 1e10 (Figure 4.4).
- Why the stochastic trend must be moving slowly
  - If μ_t moved quickly, expected changes would not be highly persistent and the level–change link would break.
  - Variance composition matters: larger proportion of variance in cyclical component and small σ^2_η produce large, persistent expected changes.
- Simulation evidence (Figures 4.5 and 4.6)
  - AR(1) stationary coefficient = 0.97.
  - Figure 4.5 parameters: AR(1) std dev = 0.01; stochastic trend std dev = 0.001.
    - Small sample n = 300: regression y = 0.01 + 0.96 * x, R^2 = 0.929 (top panel).
    - Large sample n = 1 million: regression y = −0.02 + 0.28 * x, R^2 = 0.293 (bottom panel).
  - Figure 4.6 contrast when stochastic trend variance increases:
    - Small σ_η = 0.001: regression y = 0.01 + 0.96 * x, R^2 = 0.929.
    - Large σ_η = 0.01: regression y = 0 + 0.09 * x, R^2 = 0.061.
  - Conclusion: smaller stochastic-trend variance relative to cyclical variance → strong small-sample level–change link; increasing stochastic-trend variance weakens link.
- Summary of explanatory power
  - Model accounts for:
    - (a) Exchange rates have a unit root.
    - (b) Expected changes are highly persistent.
    - (c) Strong link between levels and expected changes.
  - Mechanism: slowly moving μ_t + mean-reverting s^c_t → long-run random walk with medium-term predictability.

### 8.1 Potential Sources of Apparent Bias: Interest Rate Differential Projections
- Theory and hypothesis
  - Bias could arise if convergence of interest rate differentials is slower than investors expect, causing predicted exchange rate changes to differ from realized changes.
  - Slower-than-expected convergence would imply slower exchange rate adjustment.
- Empirical finding (2000–2024)
  - Evidence shows interest rate differentials converged faster than expected; therefore bias in forecasted exchange rate changes cannot be attributed to slow-moving interest rate adjustments for 2000–2024.
- Regression coefficients and standard errors (Table 8.1) — coefficients are regression of x-year change in interest rate differential on x-year lagged expected x-year change (Monthly data)
  - EUR: Coefficients 1 year 1.66, 2 years 1.65, 3 years 1.54, 4 years 1.43; Standard errors 1 year 0.12, 2 years 0.10, 3 years 0.08, 4 years 0.07
  - JPN: Coefficients 1 year 1.04, 2 years 1.19, 3 years 1.08, 4 years 0.98; Standard errors 1 year 0.12, 2 years 0.13, 3 years 0.11, 4 years 0.09
  - GBR: Coefficients 1 year 1.63, 2 years 1.77, 3 years 1.21, 4 years 1.20; Standard errors 1 year 0.11, 2 years 0.09, 3 years 0.10, 4 years 0.08
  - CAN: Coefficients 1 year 1.36, 2 years 1.37, 3 years 1.49, 4 years 1.22; Standard errors 1 year 0.12, 2 years 0.11, 3 years 0.11, 4 years 0.10
  - SWE: Coefficients 1 year 2.10, 2 years 2.06, 3 years 1.68, 4 years 1.51; Standard errors 1 year 0.13, 2 years 0.13, 3 years 0.11, 4 years 0.09
  - CHE: Coefficients 1 year 1.13, 2 years 1.63, 3 years 1.43, 4 years 1.11; Standard errors 1 year 0.12, 2 years 0.12, 3 years 0.11, 4 years 0.10
  - NOR: Coefficients 1 year 1.73, 2 years 1.49, 3 years 1.24, 4 years 1.17; Standard errors 1 year 0.12, 2 years 0.11, 3 years 0.11, 4 years 0.08
- R-squared of regressions (Table 8.2) — R2 of regression of x-year change in interest rate differential on x-year lagged expected x-year change, 2004-24 (Monthly data)
  - EUR: 1 year 0.43, 2 years 0.54, 3 years 0.59, 4 years 0.63
  - JPN: 1 year 0.23, 2 years 0.27, 3 years 0.29, 4 years 0.37
  - GBR: 1 year 0.46, 2 years 0.64, 3 years 0.40, 4 years 0.56
  - CAN: 1 year 0.33, 2 years 0.37, 3 years 0.46, 4 years 0.38
  - SWE: 1 year 0.50, 2 years 0.54, 3 years 0.54, 4 years 0.57
  - CHE: 1 year 0.26, 2 years 0.47, 3 years 0.44, 4 years 0.39
  - NOR: 1 year 0.44, 2 years 0.43, 3 years 0.39, 4 years 0.56

### 8.2 Potential Sources of Apparent Bias: Timing of Exchange Rate Projections
- Timing concern
  - Forecasts produced earlier than publication month imply expected exchange rates used in regressions should be characterized by E_{t−1} s_{t+12} rather than E_t s_{t+12}, which can bias regression coefficients.
- Analytical framework and bias formula
  - Regress x_{t+h} − x_t on (E_{t−1} x_{t+h} − x_t), where E_{t−1} x_{t+h} = ρ(h+1) x_{t−1}.
  - For a stationary ARMA(p,q) with autocorrelation ρ(h), the regression coefficient is:
    - β(h) = [ ρ(h+1) ρ(h+1) − ρ(h) − ρ(h+1) ρ(1) + 1 ] / [ ρ(h+1)^2 + 1 − 2 ρ(h+1) ρ(1) ]  (equation 8.2.1).
- Illustrative AR(2) example
  - AR(2) with roots 0.98 and 0.65:
    - Regressing s_{t+12} − s_t on ρ(12) s_t − s_t yields expected coefficient = 1.
    - Regressing s_{t+12} − s_t on ρ(13) s_{t−1} − s_t yields β(12) = 0.56.
    - Bias (β(h) < 1) diminishes as h increases; β(h) → 1 as h increases for this AR(2).

### 9 Out-of-Sample Forecasts of Multi-Year Exchange Rate Changes
- Methodology (MSE definitions)
  - MSE Model = (1/N) Σ_{i=1}^N (∆s_{t+h,i} − ∆ŝ^{Model}_{t+h,i})^2  (9.1.1).
  - MSE RW = (1/N) Σ_{i=1}^N (∆s_{t+h,i})^2  (9.1.2).
  - MSE Ratio = MSE Model / MSE RW  (9.1.3).
  - ∆s_{t+h,i} = s_{t+h,i} − s_{t,i}; random walk predicts zero change.
- Model specification
  - Et[s_{t+h} − s_t] = b_h [Et s_{t+12} − s_t]  (9.2.1).
  - Coefficients b_h set a priori (Table 9.1):
    - 12 months: 1.00
    - 24 months: 1.50
    - 36 months: 2.00
    - 48 months: 2.25
    - 60 months: 2.50
- Results and discussion — MSE ratios (Table 9.2), ratio of mean squared error to that of random walk, 2005-24 (Monthly data. Data start in January 2005)
  - JPN: 1 year 1.28, 2 years 1.13, 3 years 0.98, 4 years 0.71, 5 years 0.51
  - EUR: 1 year 0.88, 2 years 0.70, 3 years 0.67, 4 years 0.61, 5 years 0.52
  - AUS: 1 year 0.97, 2 years 0.91, 3 years 0.86, 4 years 0.73, 5 years 0.69
  - GBR: 1 year 0.91, 2 years 0.76, 3 years 0.79, 4 years 0.83, 5 years 0.74
  - SWE: 1 year 1.01, 2 years 0.92, 3 years 0.91, 4 years 0.82, 5 years 0.78
  - CHE: 1 year 1.06, 2 years 1.16, 3 years 1.21, 4 years 0.93, 5 years 0.79
  - CAN: 1 year 0.89, 2 years 0.95, 3 years 0.93, 4 years 0.87, 5 years 0.83
  - NZL: 1 year 1.07, 2 years 1.09, 3 years 1.17, 4 years 0.92, 5 years 0.87
  - NOR: 1 year 1.01, 2 years 1.06, 3 years 1.16, 4 years 1.05, 5 years 0.95
- Key empirical insights
  - At longer horizons, MSE ratio is often below one across multiple currencies, indicating superior predictive power relative to the random walk.
  - Example: JPN at five-year horizon, MSE Ratio = 0.51 (model forecast errors are 51% of random walk errors; 49% improvement).
  - Predictive improvement generally grows with horizon and is consistent across several major currencies.
- Sensitivity analysis (Table 9.3) — MSE ratio for five-year forecasts for various b60 values, 2005-24 (Monthly data)
  - b60 columns: 1, 1.5, 2, 2.5, 3, 3.5, 4
  - EUR: 0.72, 0.62, 0.56, 0.52, 0.52, 0.55, 0.61
  - JPN: 0.72, 0.62, 0.55, 0.51, 0.50, 0.52, 0.56
  - GBR: 0.86, 0.81, 0.77, 0.74, 0.72, 0.71, 0.72
  - CAN: 0.91, 0.87, 0.84, 0.83, 0.82, 0.81, 0.82
  - AUS: 0.80, 0.73, 0.70, 0.69, 0.70, 0.74, 0.81
  - NZL: 0.83, 0.80, 0.82, 0.87, 0.97, 1.10, 1.28
  - CHE: 0.80, 0.76, 0.75, 0.79, 0.87, 0.98, 1.13
  - SWE: 0.80, 0.75, 0.75, 0.78, 0.85, 0.96, 1.10
  - NOR: 0.93, 0.92, 0.92, 0.95, 0.99, 1.04, 1.12
- Robustness result
  - For b60 between 1 and 3, the model outperforms the random walk across most currencies; model captures direction and approximate magnitude compared with random walk’s zero-change forecast.

### 10 Conclusion (selected findings)
- Proposed hybrid model: stochastic trend + stationary component reconciles random walk behavior and mean reversion.
- Key features captured
  - Exchange rates can have a unit root while maintaining medium-term forecastability.
  - Stochastic trend governs long-term movements; stationary component generates medium-term predictability.
  - Model predicts inverted U-shaped predictability, with peak accuracy at intermediate horizons.
  - Multi-year exchange rate changes are increasing multiples of one-year changes.
- Empirical confirmation
  - Tests using nine inflation-targeting, freely floating currencies confirm model predictions and show outperformance of the random walk benchmark, especially over multi-year horizons.
- Suggested future research directions
  - Apply framework to other variables with persistent trends and cycles (example: unemployment rate).
  - Extend to real-time trend/cycle distinction, regime changes, structural breaks, and multivariate interactions.

### Part III — Technical appendices (A, B, C)
- Appendix A: The Exchange Rate Level and Expected Exchange Rate Changes
  - Regression setup: s_t = a + b_k s_{t−k} + ξ_t  (A.0.1).
  - b_k = cov(s_t, s_{t−k}) / var(s_t)  (A.0.2).
  - Autocovariance recurrence: γ(k) = (α+β) γ(k−1) − αβ γ(k−2)  (A.0.5).
  - General solution: γ(k) = A α^k + B β^k  (A.0.6).
  - AR(2) form: s_t = (α+β) s_{t−1} − αβ s_{t−2} + ε′_t  (A.0.7).
  - Using A and B, b_k = γ(k) / γ(0) and explicit form:
    - b_k = [ α^{1+k} (1−β^2) − β^{1+k} (1−α^2) ] / [ (α−β)(1 + αβ) ]  (A.0.19).
  - Properties: 0 < b_k < 1; b_{k+1} < b_k; lim_{k→∞} b_k = 0.
  - Regression implication (A.0.20): E_{t−k} s_t − s_{t−k} = a + [ expression in α,β,k ] − 1 all times s_{t−k}; implies expected depreciation larger when exchange rate is more appreciated and coefficient goes to −1 as k increases.
- Appendix B: Regressing the Change in the Stationary Part on the Total Change
  - Decomposition: S_t = X_t + Z_t with X_t stationary ARMA(p,q) and Z_t a random-walk trend (B.0.1–B.0.3).
  - Change decomposition: ΔS_t = ΔX_t + ΔZ_t  (B.0.4).
  - Regression coefficient b = cov(ΔX_t, ΔS_t) / var(ΔS_t)  (B.0.5).
  - Because ΔX_t and ΔZ_t uncorrelated, cov(ΔX_t, ΔS_t) = var(ΔX_t)  (B.0.10).
  - var(ΔS_t) = γ σ^2_ε + σ^2_η, where var(ΔX_t) = γ σ^2_ε and var(ΔZ_t) = σ^2_η  (B.0.13–B.0.17).
  - Hence b = γ σ^2_ε / (γ σ^2_ε + σ^2_η)  (B.0.18–B.0.19).
  - Interpretation: b ∈ (0,1); b near 1 → changes driven by stationary component; b near 0 → changes driven by stochastic trend.
  - Example: AR(1) x_t = 0.98 x_{t−1} + ε_t → var(Δx_t) = 1.0101 σ^2_ε (γ = 1.0101) (footnote example).
- Appendix C: Regressing Z(t+h) − Z(t) on (ρ(h) − 1) Z(t)
  - For stochastic ARMA(p,q) Z(t) with autocorrelation ρ(h):
    - Regress Y = Z(t+h) − Z(t) on X = (ρ(h) − 1) Z(t).
    - Regression coefficient β = 1 (C.1.9).
    - Var(Y) = 2(1 − ρ(h)) σ^2  (C.2.2–C.2.5).
    - R^2 = (1 − ρ(h)) / 2, and as h increases for stationary ARMA, ρ(h) → 0 → lim_{h→∞} R^2 = 0.5 (C.2.11).
  - Conclusions: coefficient = 1; R^2 approaches 0.5 as h increases.

*Source: wpiea2024252-print-pdf — sections 4.1, 4.2, 8.1–8.2, 9, 10, Part III*

### 4.1  Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  10

### 4.1 Intuition

### Key observations from monthly survey data (2000–2024)
- Expected exchange rate changes are not zero and "can be quite substantial."
- Expected exchange rate changes exhibit high persistence over time.
- There is a notable negative relationship between exchange rate levels and expected future changes; this link is strong for most countries, though weaker for a few, including Great Britain.
- Data used: monthly data of 12-month expected exchange rate changes vis-à-vis the US dollar for the period 2000–2024 from a dataset of monthly survey data provided by Das et al. (2022).
- Empirical focus: nine inflation-targeting countries with freely floating exchange rates (selected because their interest rate differentials are exogenous to expected exchange rate changes).

### Conceptual framework: dual-component decomposition
- Exchange rates are decomposed into two parts:
  - A slowly evolving stochastic trend (μt): represents the long-run equilibrium exchange rate and evolves as a random walk driven by fundamentals (differential inflation rates, productivity growth).
  - A stationary cyclical component (sct): captures temporary deviations from equilibrium and reverts to its mean over time.
- Implications of decomposition:
  - Long-term unpredictability arises from the stochastic trend (unit root behavior).
  - Medium-term predictability arises from the stationary cyclical component: expected exchange rate changes depend solely on the cyclical component—the larger the cyclical component, the higher the expected exchange rate depreciation.
  - Without a stationary component, expected exchange rate changes would be zero.
  - If the stochastic trend did not evolve slowly, the relationship between exchange rate levels and expected changes would break down, and the persistence of expected exchange rate changes would diminish.

### Predicted predictability pattern
- The interplay between the stochastic trend and the stationary cyclical component generates an inverted U-shaped predictability pattern:
  - Least predictable in the short term (noise dominates).
  - Most predictable in the medium term (stationary component dominates).
  - Least predictable in the long term (stochastic trend dominates).

### Illustrative numerical scenarios (20% initial appreciation)
- Scenario 1: Predominance of the cyclical component (90%)
  - 90% of the 20% appreciation = 18 percentage points originates from sct.
  - Remaining 2 percentage points reflect a shift in μt.
  - Since sct is stationary and reverts to its mean, expected future depreciation ≈ 18%.
  - Interpretation: strong mean reversion and strong predictability.
- Scenario 2: Predominance of the stochastic trend (90%)
  - 10% of the 20% appreciation = 2 percentage points ascribed to sct.
  - Remaining 18 percentage points driven by μt.
  - Expected future depreciation ≈ 2%.
  - Interpretation: weak mean reversion and reduced predictability.

### Broader implications
- The dual-component framework reconciles the unit root (random walk) hypothesis with observed persistent expected exchange rate changes.
- When the stochastic trend evolves slowly and the cyclical component is significant, exchange rates can display both a unit root and medium-term predictability.

*Source: wpiea2024252-print-pdf — section 4.1 Intuition (2000–2024 monthly data).*

### 4.2  The Math

### 4.2  The Math

### Why Exchange Rates Should Have a Stationary Component
- Model setup:
  - Exchange rate at time t: s_t = μ_t + s^c_t (equation 4.2.1), where μ_t is a stochastic trend and s^c_t is a stationary component.
  - Stochastic trend evolves as a random walk: μ_t = μ_{t−1} + η_t (equation 4.2.2), with η_t white noise, zero mean, variance σ^2_η.
  - Expectations decompose: E_t[s_{t+h}] = E_t[μ_{t+h}] + E_t[s^c_{t+h}] (equation 4.2.3).
  - Expectation of cyclical component uses the autocorrelation function: E_t[s^c_{t+h}] = ρ(h) s^c_t (equation 4.2.4).
  - Expectation of stochastic trend equals current value: E_t[μ_{t+h}] = μ_t (equation 4.2.5).
- Implication for expected changes:
  - E_t[s_{t+h} − s_t] = −(1 − ρ(h)) s^c_t (equation 4.2.6).
  - The best forecast for future exchange rate changes is the expected change in the cyclical component; expected change is zero only if s^c_t = 0 (pure random walk).
- Figures and numerical settings referenced:
  - HP-filter lambda values used in examples: λ = 14400, λ = 1e6, λ = 1e10 (Figure 4.4).
  - Note: Trend and cyclical exchange rates calculated using an HP-filter with different values of lambda (Monthly data).

### Why the Stochastic Trend Must Be Moving Slowly
- Two reasons slow-moving stochastic trend is required:
  - If trend moved quickly, expected exchange rate changes would not be highly persistent.
  - Rapidly moving trend would weaken the link between exchange rate levels and expected changes.
- Role of variance composition:
  - If exchange rate changes are mainly driven by cyclical component, stochastic trend evolves slowly, producing large and persistent expected changes.
  - If stochastic trend drives most changes, cyclical component is small and expected changes are smaller and less persistent.
- Simulation evidence (Figure 4.5 and Figure 4.6):
  - AR(1) stationary component coefficient = 0.97.
  - Simulation parameters in Figure 4.5: AR(1) coefficient = 0.97, standard deviation AR(1) process = 0.01, standard deviation stochastic trend = 0.001.
    - Small sample: n = 300 (equivalent to 25 years of monthly data): regression y = 0.01 + 0.96 * x, R^2 = 0.929 (top panel).
    - Large sample: n = 1 million: regression y = −0.02 + 0.28 * x, R^2 = 0.293 (bottom panel).
  - Figure 4.6 shows contrast when stochastic trend variance is larger:
    - Small standard deviation stochastic trend (0.001): regression y = 0.01 + 0.96 * x, R^2 = 0.929.
    - Large standard deviation stochastic trend (0.01): regression y = 0 + 0.09 * x, R^2 = 0.061.
  - Conclusion: With a much smaller variance of the stochastic trend relative to the cyclical component, the link between exchange rate level and cyclical component is strong in small samples; increasing stochastic-trend variance weakens that link.

### Summary of How the Dual-Component Model Explains Stylized Facts
- The model accounts for three observations simultaneously:
  - (a) Exchange rates have a unit root.
  - (b) Expected exchange rate changes are highly persistent.
  - (c) There is a strong link between exchange rate levels and expected exchange rate changes.
- Mechanism: A slowly moving stochastic trend combined with a stationary (mean-reverting) cyclical component produces long-run random-walk behavior while generating medium-term predictability via the cyclical component.

*Source: wpiea2024252-print-pdf — Section 4.2 "The Math"*

### 8.1  Potential Sources of Apparent Bias: Interest Rate Differential Pro-

### 8.1 Potential Sources of Apparent Bias: Interest Rate Differential Projections

### Theory and Hypothesis
- One possible source of bias in exchange rate forecast regressions is the behavior of interest rate differentials: if convergence of interest rate differentials is slower than investors anticipate, expected exchange rate changes may diverge from actual changes.
- Slower-than-expected convergence would imply slower adjustment in exchange rate levels, potentially explaining why observed twelve-month exchange rate change is often smaller than predicted.

### Empirical Finding (2000–2024)
- Contrary to the slower-than-expected convergence hypothesis, for the period 2000 to 2024 the evidence shows interest rate differentials converged faster than expected; thus the bias in forecasted exchange rate changes cannot be attributed to slow-moving interest rate adjustments.

### Regression coefficients and standard errors (Table 8.1)
- Coefficients and Standard errors of regression of x-year change in interest rate differential on x-year lagged expected x-year change (Monthly data)
  - EUR: Coefficients 1 year 1.66, 2 years 1.65, 3 years 1.54, 4 years 1.43; Standard errors 1 year 0.12, 2 years 0.10, 3 years 0.08, 4 years 0.07
  - JPN: Coefficients 1 year 1.04, 2 years 1.19, 3 years 1.08, 4 years 0.98; Standard errors 1 year 0.12, 2 years 0.13, 3 years 0.11, 4 years 0.09
  - GBR: Coefficients 1 year 1.63, 2 years 1.77, 3 years 1.21, 4 years 1.20; Standard errors 1 year 0.11, 2 years 0.09, 3 years 0.10, 4 years 0.08
  - CAN: Coefficients 1 year 1.36, 2 years 1.37, 3 years 1.49, 4 years 1.22; Standard errors 1 year 0.12, 2 years 0.11, 3 years 0.11, 4 years 0.10
  - SWE: Coefficients 1 year 2.10, 2 years 2.06, 3 years 1.68, 4 years 1.51; Standard errors 1 year 0.13, 2 years 0.13, 3 years 0.11, 4 years 0.09
  - CHE: Coefficients 1 year 1.13, 2 years 1.63, 3 years 1.43, 4 years 1.11; Standard errors 1 year 0.12, 2 years 0.12, 3 years 0.11, 4 years 0.10
  - NOR: Coefficients 1 year 1.73, 2 years 1.49, 3 years 1.24, 4 years 1.17; Standard errors 1 year 0.12, 2 years 0.11, 3 years 0.11, 4 years 0.08

### R-squared of regressions (Table 8.2)
- R2 of regression of x-year change in interest rate differential on x-year lagged expected x-year change, 2004-24 (Monthly data)
  - EUR: 1 year 0.43, 2 years 0.54, 3 years 0.59, 4 years 0.63
  - JPN: 1 year 0.23, 2 years 0.27, 3 years 0.29, 4 years 0.37
  - GBR: 1 year 0.46, 2 years 0.64, 3 years 0.40, 4 years 0.56
  - CAN: 1 year 0.33, 2 years 0.37, 3 years 0.46, 4 years 0.38
  - SWE: 1 year 0.50, 2 years 0.54, 3 years 0.54, 4 years 0.57
  - CHE: 1 year 0.26, 2 years 0.47, 3 years 0.44, 4 years 0.39
  - NOR: 1 year 0.44, 2 years 0.43, 3 years 0.39, 4 years 0.56

---

### 8.2 Potential Sources of Apparent Bias: Timing of Exchange Rate Projections

### Timing Concern
- Forecasts published, for example, in mid-March for end-March of the following year are typically produced earlier in the month. This implies expected exchange rates used in regressions should be characterized by Et−1 st+12 rather than Et st+12, potentially biasing regression coefficients.

### Analytical framework and bias formula
- To account for timing discrepancy, regress xt+h − xt on (Et−1 xt+h − xt), where Et−1 xt+h = ρ(h+1) xt−1, instead of on (Et xt+h − xt).
- For a stationary ARMA(p,q) process with autocorrelation function ρ(h), the regression coefficient for st+h − st on ρ(h+1)st−1 − st is:
  - β(h) = [ ρ(h+1) ρ(h+1) − ρ(h) − ρ(h+1) ρ(1) + 1 ] / [ ρ(h+1)^2 + 1 − 2 ρ(h+1) ρ(1) ]  (equation 8.2.1)

### Illustrative AR(2) example
- Consider an AR(2) process with roots 0.98 and 0.65:
  - Regressing st+12 − st on ρ(12) st − st yields an expected coefficient of 1.
  - Regressing st+12 − st on ρ(13) st−1 − st yields an expected coefficient of β(12) = 0.56.
- The bias (β(h) < 1) diminishes as h increases; β(h) converges to 1 as h increases for this AR(2) process (Figure 8.1).

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### 9 Out-of-Sample Forecasts of Multi-Year Exchange Rate Changes

### 9.1 Methodology (MSE definitions)
- MSE Model = (1/N) Σ_{i=1}^N (∆s_{t+h,i} − ∆ŝ^{Model}_{t+h,i})^2  (9.1.1)
- MSE RW = (1/N) Σ_{i=1}^N (∆s_{t+h,i})^2  (9.1.2)
- MSE Ratio = MSE Model / MSE RW  (9.1.3)
  - ∆s_{t+h,i} = s_{t+h,i} − s_{t,i}
  - For the random walk, predicted change is zero, so MSE RW is average squared actual change.

### 9.2 Model specification
- Model assumption: Et[s_{t+h} − s_t] = b_h [Et s_{t+12} − s_t]  (9.2.1)
- Coefficients b_h are set a priori (Table 9.1):
  - Horizon (months) 12: Coefficient (b_h) 1.00
  - 24: 1.50
  - 36: 2.00
  - 48: 2.25
  - 60: 2.50

### 9.3 Results and discussion — MSE ratios (Table 9.2)
- Ratio of mean squared error to that of random walk, 2005-24 (Monthly data. Data start in January 2005)
  - JPN: 1 year 1.28, 2 years 1.13, 3 years 0.98, 4 years 0.71, 5 years 0.51
  - EUR: 1 year 0.88, 2 years 0.70, 3 years 0.67, 4 years 0.61, 5 years 0.52
  - AUS: 1 year 0.97, 2 years 0.91, 3 years 0.86, 4 years 0.73, 5 years 0.69
  - GBR: 1 year 0.91, 2 years 0.76, 3 years 0.79, 4 years 0.83, 5 years 0.74
  - SWE: 1 year 1.01, 2 years 0.92, 3 years 0.91, 4 years 0.82, 5 years 0.78
  - CHE: 1 year 1.06, 2 years 1.16, 3 years 1.21, 4 years 0.93, 5 years 0.79
  - CAN: 1 year 0.89, 2 years 0.95, 3 years 0.93, 4 years 0.87, 5 years 0.83
  - NZL: 1 year 1.07, 2 years 1.09, 3 years 1.17, 4 years 0.92, 5 years 0.87
  - NOR: 1 year 1.01, 2 years 1.06, 3 years 1.16, 4 years 1.05, 5 years 0.95

- Key empirical insights:
  - At longer horizons, the MSE ratio is consistently below one across multiple currencies, indicating superior predictive power relative to the random walk.
  - Example: For JPN at the five-year horizon, MSE Ratio = 0.51 → model forecast errors are 51% of the random walk model’s errors, a 49% improvement.
  - Predictive improvement grows with forecast horizon and is consistent across multiple major currencies.

### Sensitivity analysis (Table 9.3)
- Ratio of mean squared error to that of random walk for five years exchange rate change forecasts for various levels of b60, 2005-24 (Monthly data)
  - Coefficient b60 columns: 1, 1.5, 2, 2.5, 3, 3.5, 4
  - EUR: 0.72, 0.62, 0.56, 0.52, 0.52, 0.55, 0.61
  - JPN: 0.72, 0.62, 0.55, 0.51, 0.50, 0.52, 0.56
  - GBR: 0.86, 0.81, 0.77, 0.74, 0.72, 0.71, 0.72
  - CAN: 0.91, 0.87, 0.84, 0.83, 0.82, 0.81, 0.82
  - AUS: 0.80, 0.73, 0.70, 0.69, 0.70, 0.74, 0.81
  - NZL: 0.83, 0.80, 0.82, 0.87, 0.97, 1.10, 1.28
  - CHE: 0.80, 0.76, 0.75, 0.79, 0.87, 0.98, 1.13
  - SWE: 0.80, 0.75, 0.75, 0.78, 0.85, 0.96, 1.10
  - NOR: 0.93, 0.92, 0.92, 0.95, 0.99, 1.04, 1.12

- Robustness result:
  - For b60 between 1 and 3, the model outperforms the random walk across most currencies; results are not very sensitive to the precise coefficient because the model captures direction and approximate magnitude of expected multi-year changes versus the random walk’s zero-change forecast.

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### 10 Conclusion (selected findings)
- The paper proposes a hybrid model combining a stochastic trend and a stationary component to reconcile random walk behavior and mean reversion in exchange rates.
- Key features captured:
  - Exchange rates can have a unit root while maintaining medium-term forecastability.
  - The stochastic trend governs long-term movements; the stationary component generates medium-term predictability.
  - The model predicts an inverted U-shaped pattern of predictability, with peak forecast accuracy at intermediate horizons.
  - Multi-year exchange rate changes are increasing multiples of one-year changes.
- Empirical tests using nine inflation-targeting, freely floating currencies confirm the model’s predictions and show outperformance of the random walk benchmark, especially over multi-year horizons.
- Suggested future research directions:
  - Apply the hybrid framework to other economic variables with persistent trends and cycles (example: unemployment rate).
  - Extend to real-time trend/cycle distinction, regime changes, structural breaks, and multivariate interactions.

*Source: wpiea2024252-print-pdf - 8.1  Potential Sources of Apparent Bias: Interest Rate Differential Pro-*

### Part III

### Part III

### A  The Exchange Rate Level and Expected Exchange Rate Changes
- Regression setup: s_t = a + b_k s_{t−k} + ξ_t (A.0.1)
- Coefficient expression: b_k = cov(s_t, s_{t−k}) / var(s_t) (A.0.2)
- Autocovariance definition: γ(k) = cov(s_t, s_{t−k}) (A.0.3)
- Recurrence from model (Equation (5.1.4)):
  - cov(s_t, s_{t−k}) = (α+β) cov(s_{t−1}, s_{t−k}) − αβ cov(s_{t−2}, s_{t−k}) + cov(ε′_t, s_{t−k}) (A.0.4)
  - with ε′_t = [ε_t / (1−β)]
- Difference equation for autocovariance: γ(k) = (α+β) γ(k−1) − αβ γ(k−2) (A.0.5)
- General solution: γ(k) = A α^k + B β^k (A.0.6)
- AR(2) representation: s_t = (α+β) s_{t−1} − αβ s_{t−2} + ε′_t (A.0.7)
- Moment equations:
  - γ(0) = (α+β) γ(1) − αβ γ(2) + σ^2_ε (A.0.8)
  - γ(1) = (α+β) γ(0) − αβ γ(1) (A.0.9)
  - γ(2) = (α+β) γ(1) − αβ γ(0) (A.0.10)
- Solutions:
  - γ(0) = [ (1 + αβ) / (1 − αβ) ] σ^2_{ε′} / [ (1−α^2)(1−β^2) ] = [ (1 + αβ) / (1 − αβ) ] σ^2_e / (1 + αβ)^2 − (α+β)^2  (A.0.11)
  - γ(1) = [ (α+β) / (1 + αβ) ] [ (1 + αβ) / (1 − αβ) ] σ^2_{ε′} / (1−α^2)(1−β^2) (A.0.12)
- Relations to A and B:
  - γ(0) = A + B (A.0.14)
  - γ(1) = A α + B β (A.0.15)
  - A = α σ^2_{ε′} (1−α^2)^{-1} (α−β)^{-1} (1−αβ)^{-1} (A.0.16)
  - B = − β σ^2_{ε′} (α−β)^{-1} (1−αβ)^{-1} (1−β^2)^{-1} (A.0.17)
- Using A and B:
  - b_k = γ(k) / γ(0) (A.0.18)
  - b_k = α^{1+k} (1−β^2) − β^{1+k} (1−α^2)  all over  (α−β)(1 + αβ)  (A.0.19)  [explicitly: b_k = α^{1+k} (1−β^2) − β^{1+k} (1−α^2) / (α−β)(1 + αβ)]
- Properties of b_k:
  - 0 < b_k < 1
  - b_{k+1} < b_k
  - lim_{k→∞} b_k = 0
- Regression implication:
  - E_{t−k} s_t − s_{t−k} = a + [ α^{1+k} (1−β^2) − β^{1+k} (1−α^2) ] / [ (α−β)(1 + αβ) ] − 1  all times s_{t−k}  (A.0.20)
- Two important implications from Equation (A.0.20):
  - Expected exchange rate changes depend on exchange rate levels: the more appreciated the exchange rate level, the larger the expected future depreciation.
  - The longer the lag k, the higher the coefficient; as lim_{k→∞} b_k = 0, the coefficient of a regression of exchange rate changes on past exchange rate levels should go to −1 as k increases.
- Demonstrations and inequalities shown:
  - 1 + αβ > α + β → 1 + αβ − α − β > 0 → (1−α)(1−β) > 0
  - For α > β: proofs that b_k > 0, b_k < 1, and b_k < b_{k−1} using algebraic bounds on the expression for b_k.

### B  Regressing the Change in the Stationary Part on the Total Change
- Decomposition: S_t = X_t + Z_t (B.0.1)
  - X_t is a stationary ARMA(p,q): X_t = φ_1 X_{t−1} + φ_2 X_{t−2} + ··· + φ_p X_{t−p} + ε_t + θ_1 ε_{t−1} + ··· + θ_q ε_{t−q} (B.0.2)
  - ε_t is white noise with mean zero and variance σ^2_ε
  - Z_t is a stochastic trend random walk: Z_t = Z_{t−1} + η_t (B.0.3)
  - η_t is white noise with mean zero and variance σ^2_η
- Change decomposition: ΔS_t = ΔX_t + ΔZ_t (B.0.4)
- Regress ΔX_t on ΔS_t; regression coefficient b = cov(ΔX_t, ΔS_t) / var(ΔS_t) (B.0.5)
- Covariance:
  - cov(ΔX_t, ΔS_t) = cov(ΔX_t, ΔX_t + ΔZ_t) = var(ΔX_t) + cov(ΔX_t, ΔZ_t) (B.0.7–B.0.8)
  - Since ΔX_t and ΔZ_t are uncorrelated, cov(ΔX_t, ΔZ_t) = 0 (B.0.9)
  - Therefore cov(ΔX_t, ΔS_t) = var(ΔX_t) (B.0.10)
- Variances:
  - var(ΔX_t) = γ σ^2_ε, where γ depends on φ_i and θ_j (B.0.13)
    - Example: for AR(1) x_t = 0.98 x_{t−1} + ε_t, var(Δx_t) = 1.0101 σ^2_ε, so γ = 1.0101 (note example) (footnote)
  - var(ΔS_t) = var(ΔX_t) + var(ΔZ_t) since uncorrelated (B.0.15)
  - var(ΔZ_t) = var(η_t) = σ^2_η (B.0.16)
  - Thus var(ΔS_t) = γ σ^2_ε + σ^2_η (B.0.17)
- Regression coefficient:
  - b = γ σ^2_ε / (γ σ^2_ε + σ^2_η) (B.0.18–B.0.19)
- Interpretation:
  - b represents the proportion of the change in the exchange rate attributable to the change in its stationary component.
  - b ranges from 0 to 1: values closer to 1 indicate changes predominantly driven by the stationary component; values closer to 0 indicate changes mainly due to the stochastic trend.

### C  Regressing Z(t+h) − Z(t) on (ρ(h) − 1) Z(t)
- Setup: stochastic ARMA(p,q) process Z(t) with autocorrelation function ρ(h).
- Regression target: regress Y = Z(t+h) − Z(t) on X = (ρ(h) − 1) Z(t).
- Regression coefficient β = Cov(X,Y) / Var(X) (C.1.1)
- Covariance computation:
  - Cov(X,Y) = (ρ(h) − 1) Cov(Z(t), Z(t+h)) − (ρ(h) − 1) Var(Z(t)) (C.1.2–C.1.4)
  - = (ρ(h) − 1)^2 σ^2 (C.1.5)
- Variance of X:
  - Var(X) = (ρ(h) − 1)^2 Var(Z(t)) = (ρ(h) − 1)^2 σ^2 (C.1.6–C.1.8)
- Regression coefficient result:
  - β = 1 (C.1.9)
- R^2 analysis:
  - R^2 = [Cov(X,Y)]^2 / [Var(X) · Var(Y)] (C.2.1)
  - Var(Y) = Var(Z(t+h) − Z(t)) = 2(1 − ρ(h)) σ^2 (C.2.2–C.2.5)
  - Substitution yields R^2 = (1 − ρ(h))^2 / [2(1 − ρ(h))] = (1 − ρ(h)) / 2 (intermediate algebra shown) (C.2.6–C.2.10)
  - As h increases for stationary ARMA processes, ρ(h) → 0, so:
    - lim_{h→∞} R^2 = 0.5 (C.2.11)
- Conclusions:
  - The regression coefficient of Z(t+h) − Z(t) on (ρ(h) − 1) Z(t) is 1.
  - The R^2 of this regression approaches 0.5 as h increases.

*Source: wpiea2024252-print-pdf - Part III*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024252-print-pdf.pdf_