Purchasing Power Parity FX 3

• Relative PPP is weaker than Absolute PPP: Rt can be different from 1. • Relative PPP: Testing Key: On average, what we expect to happen, 𝑒  On average: 𝑒 or E[𝑒  𝑒 , , , , ] = E[ 𝑒 ,,  𝐼 , should happen, 𝑒 , , . 𝐼 ]  E[ 𝐼 ,, ,, 𝐼 ] A linear regression is a good framework to test theories. Recall, 𝑒 =α+β 𝐼 , , where 𝜀 : regression error. That is, E[𝜀 Then, E[𝑒 , , ] = α + β E[ 𝐼  E[𝑒 , , 𝐼 ] = α + β E[ 𝑒 𝐼 +𝜀 , ] = 0. ] + E[𝜀 ,, ] = α + β E[ 𝑒 ,, ] ]  For Relative PPP to hold, on average, we need α=0 & β=1. • Relative PPP: General Evidence Under Relative PPP: 𝑒 , ,  𝐼 𝐼 1. Visual Evidence Plot 𝐼 𝐼 against 𝑒 , (JPY/USD), using monthly data 1975 - 2022. Test: Is there a 45° line? No 45° line  Visual evidence rejects PPP. 23 • Relative PPP: General Evidence Under Relative PPP: 𝑒 , ,  𝐼 𝐼 1. Visual Evidence Plot (IUSD – IGBP) against 𝑒 , (USD/GBP), using monthly data 1975 - 2022. Test: Is there a 45° line? • Relative PPP: General Evidence Under Relative PPP: 𝑒 , ,  𝐼 𝐼 1. Visual Evidence Plot (IUSD – IGBP) against 𝑒 , (USD/GBP), using monthly data 1975 - 2022. Test: Is there a 45° line? No 45° line  Visual evidence rejects PPP. 24 • Relative PPP: General Evidence 1. Visual Evidence Test: Is Rt Constant? (Under Absolute PPP 1) mean(Rt) Some evidence for mean reversion, though slow, for Rt (average = 0.77). • Relative PPP: General Evidence (continuation) In the long run, Rt moves around some mean number (long-run PPP parity?). But, the deviations from long-run parity are very persistent. Economists report the number of years that a PPP deviation is expected to decay by 50%, the half-life. The half-life is in the range of 3 to 5 years for developed currencies. Very slow! • Descriptive Stats (1975:Jan – 2022:Dec) Mean SD Min Median Max IJPY IUSD 0.00125 0.00485 -0.01095 0.00102 0.02558 0.00303 0.00322 -0.01786 0.00266 0.01420 Long-run, on average. IJPY – IUSD ef,t,T (JPY/USD) -0.00179 0.00502 -0.01981 -0.00184 0.02104 -0.00139 0.02622 -0.08065 0.00022 0.08066 Big difference in volatility. 25 2. Statistical Evidence Formal test: Regression 𝑒 , , = α+β 𝐼 The null hypothesis is: 𝐼 +𝜀 , (𝜀 : error term, E[𝜀 ] = 0). H0 (Relative PPP true): α=0 and β=1 H1 (Relative PPP not true): α≠0 and/or β≠1 • Tests: t-test (individual tests on α and β) & F-test (joint test) (1) Individual test: t-test t-test = tθ = [θ – θ0]/S.E.(θ) where θ represents α or β  ( θ0 = α or β evaluated under H0). Statistical distribution: tθ ~ tv (v = N – K = degrees of freedom) K = # parameters in model, & N = # of observations. Rule: If |t-test| > |tv,α/2|, reject H0 at the α level. When v = N – K > 30, t30+,.025 ≈ 1.96  2-sided C.I. α = .05 (5 %) 2. Statistical Evidence (2) Joint Test: F-test F= [RSS(H0) – RSS(H1)]/J RSS(H1)/(N K) Statistical distribution: F ~ FJ,N-K J = # of restrictions in H0 (under PPP, J=2: α=0 & β=1) K = # parameters in model (under PPP model, K=2: α & β) N = # of observations RSS = Residuals Sum of Squared, εt = 𝑒 = 𝑒 , , – [α β 𝐼 , 𝐼 , ]. 2 𝑠t 𝐼 , 𝐼, ] RSS(H0) = ∑ 2 RSS(H1) = ∑ (εt) Rule: If F > FJ,N-K,α, reject at the α level. Usually, α = .05 (5 %) When N > 300, FJ=2,300+,α=.05 ≈ 3. 26 Example: Using monthly Japanese and U.S. data (1975:Jan - 2022:Dec), we fit the following regression (Observations = 576): 𝑒 , , (JPY/USD) = (𝑆 – 𝑆 )/𝑆 =α+β 𝐼 𝐼 +𝜀 . R2 = 0.005621 Standard Error (σ) = .02617 F-stat (slopes=0 –i.e., β=0) = 3.244 (p-value = 0.07219) Observations (N) = 576 Intercept (α ) Coefficient -0.00209 Stand Err 0.001157 t-Stat -1.804 P-value 0.0717 (IJAP – IUS) (β ) -0.39148 0.217343 -1.801 0.0722 We will test the H0 (Relative PPP true): α=0 & β=1 Two tests: (1) t-tests (individual tests) (2) F-test (joint test) Example: Using monthly Japanese and U.S. data (1975:Jan - 2022:Dec), we fit the following regression (Observations = 576): =α+β 𝐼 𝐼 +𝜀 . 𝑒 , , (JPY/USD) = (𝑆 – 𝑆 )/𝑆 R2 = 0.005621 Standard Error (σ) = .02617 F-stat (slopes=0 –i.e., β=0) = 3.244 (p-value = 0.07219) F-test (H0: α=0 & β=1): 19.185 (p-value: < 0.00001)  reject H0 at 5% level (F2,550,.05= 3.012) Coefficient Stand Err t-Stat P-value Intercept (α ) -0.00209 0.001157 -1.804 0.0717 (IJAP – IUS) (β ) -0.39148 0.217343 -1.801 0.0722 Test H0, using t-tests (t574,.05 = 1.96 – Note: when N-K > 30, t.05 = 1.96): tα=0: (-0.00209 – 0)/0.001157 = -1.804 (p-value = .07)  cannot reject H0. tβ=1: (-0.39148 – 1)/0.217343 = -6.402 (p-value: < .00001)  reject H0. ¶ 27 • PPP Evidence: ⋄ Relative PPP tends to be rejected in the short-run. In the long-run, there is debate about its validity: Currencies with high inflation rate differentials tend to depreciate. ⋄ Some evidence for a mean reverting Rt (average Rt = 1.10). But deviations can last for years! • PPP: Rt and St Mussa (1986): Rt is more variable under a free float. Rt variability is highly correlated with St variability. Check Second Moments: Volatility (changes in Rt) = 2.706% & Volatility (changes in St) = 2.622 (correlation = .983). Almost the same! 28 Implications: Price levels play a minor role in explaining the movements of Rt (prices are sticky). Possible explanations: (a) Contracts: Prices cannot be continuously adjusted due to contracts. (b) Mark-up adjustments: Manufacturers and retailers moderate increases in their prices in order to keep market share. Changes in St are only partially transmitted or passthrough to import/export prices. Average ERPT (exchange rate pass-through) is around 50% over one quarter and 64% over the long run for OECD countries (for the U.S., 25% in the short-run and 40% over the long run). (c) Repricing costs (menu costs) Expensive to adjust continuously prices –a restaurant, re-printing the menu. (d) Aggregation Q: Is price rigidity a result of aggregation –i.e., the use of price index? Empirical work using micro level data –say, same good (exact UPC!) in Canadian and U.S. grocery stores– show that on average product-level Rt moves with St. But, evidence is not as solid. • PPP: Puzzle The fact that no single model of exchange rate determination can accommodate both the high persistent of PPP deviations and the high correlation between Rt and St has been called the “PPP puzzle.” 29 • PPP: Summary of Empirical Evidence ⋄ Rt and 𝑆 are highly correlated, Pd tends to be sticky. ⋄ In the short run, PPP is a poor model to explain short-term St movements. ⋄ PPP deviations are very persistent. They take years to disappear. ⋄ In the long run, there is some evidence of mean reversion, though slow, for Rt. That is, 𝐒𝐭𝐏𝐏𝐏 has long-run information: Currencies that consistently have high inflation rate differentials tend to depreciate. • The long-run interpretation is the one that economists like and use: 𝐒𝐭𝐏𝐏𝐏 is seen as a benchmark. • Calculating 𝐒𝐭𝐏𝐏𝐏 (Long-Run FX Rate) We want to calculate 𝐒𝐭𝐏𝐏𝐏 = Pd,t / Pf,t over time. (1) Divide 𝐒𝐭𝐏𝐏𝐏 by S (t = 0 is our starting point). (2) After some algebra, 𝐒𝐭𝐏𝐏𝐏 = S By assuming S * [Pd,t / Pd,0] * [Pf,0/Pf,t] = S0, we plot 𝐒𝐭𝐏𝐏𝐏 over time. Note: S = S0 assumes that at t=0, the economy was in equilibrium. This may not be true: Be careful when selecting a base year. 30 Let’s look at the MXN/USD case. - In the short-run, 𝐒𝐭𝐏𝐏𝐏 misses the target, 𝑆 . - But, in the long-run, 𝐒𝐭𝐏𝐏𝐏 gets trend right, reflecting a consistent higher inflation in Mexico. Another example, the JPY/USD case. As predicted by PPP, since IUS has been consistently higher than IJAP, in the long-run, the USD depreciates against the JPY. 31 Another example, the USD/GBP case. As predicted by PPP, IUS was consistently lower than IUK until the mid-90s, the USD appreciated against the GBP. Since then, it has been moving around a constant value. • PPP Summary of Applications: ⋄ Equilibrium (“long-run”) exchange rates. ⋄ Explanation of St movements. ⋄ Indicator of competitiveness or under/over-valuation. ⋄ International GDP comparisons: Instead of using 𝑆 , 𝐒𝐭𝐏𝐏𝐏 is used to translate local currencies to USD. For example, Chinese per capita GDP (World Bank figures, in 2017): Nominal GDP per capita: CNY 59,670.52; 𝑺𝒕 = 0.14792 USD/CNY; - Nominal GDP_cap (USD)= CNY 59,670.52 * 0.1479 USD/CNY=USD 8,827 𝐒𝐭𝐏𝐏𝐏 = 0.2817 USD/CNY ⇒ “U.S. is 90% more expensive” - PPP GDP_cap (USD)= CNY 59,670.52 * 0.2817 USD/CNY = USD 16,807. 32 Country Luxembourg USA Japan Italy Czech Republic Costa Rica Brazil China Lebanon Algeria India Ethiopia Mozambique GDP per capita (in USD) - 2017 Nominal PPP 104,103 103,745 59,532 59,532 38,428 43,279 31,953 39,427 20,368 36,504 11,631 17,044 9,821 15,484 8,827 16,807 8,524 14,676 4,123 15,275 1,937 7,056 767 1,899 416 1,247 Note: PPP GDP/Nominal GDP = USD 16,807/ USD 8,827 = 1.9040 ⇒ “U.S. is 90% more expensive.” ¶ 33