PPP Strikes Out: The e ect of common factor shocks on the real exchange rate. Nelson Mark, University of Notre Dame and NBER

Transcription

1 PPP Strikes Out: The e ect of common factor shocks on the real exchange rate Nelson Mark, University of Notre Dame and NBER and Donggyu Sul, University of Auckland Tufts University November 17, 2008

2 Background and motivation Quest for realistic (i.e., conform to our priors) half-life measurement q = ln SP! P qt = qt 1 + et ln (2) h = ln ()

3 Univariate analyses of PPP over post 1973 era cannot reject the unit root. Implies half-life of in nity Example: Real $/DM rate 1973Q2 to 2007Q3. Augmented Dickey-Fuller t-statistic P-value Test critical values: Is this because sample is too small? If so, get more from the cross-section, do panel data work.

4 Panel Data Analysis. AR(1) example with xed e ects i qit = i + qit 1 + eit Econometric Issues ^ is biased down. Kendall (Nickell) bias from constant ( xed e ect). Errors are not independent. They are cross-sectionally correlated. eit = ift + vit Ft is a common factor. i is a factor loading. vit is the idiosyncratic shock. Panel data studies (e.g., Papell, Murray and Papell, Choi, Mark and Sul) account for these issues. They estimate the half-life to be between 3 and 5 years. Less than in nity, but still not good enough.

5 Imbs, Mumtaz, Ravn, Rey (IMRR) use sectoral price index data. They say one should not constrain to be the same across sectors (goods). Doing so creates an upward aggregation bias in the half-life estimate. Prices from 19 sectors and 11 countries. The log relative price of good i between the US and country c! qict = ln S ctpict P0ct Sct is the nominal US dollar price of a unit of country c s money and Pict is the country c currency price of good i in country c:

6 IMRR speci cation. Take cross-sectional average of the panel as common factor Run this regression qt = 1 CI I X i=1 C X c=1 qict; qict = X p icj qict j j=1 {z } qit 1 X H ich q t h + ict h=0 {z } ift {z } Error components model + ic + ic good-and-country xed e ect, ict idiosyncratic shock. Require lags of qt because common factor is persistent. (1)

7 PPP convergence applies to the real exchange rate between the U.S. and country c, X I Qct =!icqict; where i=1 X I i=1!ic = 1; Link deviations from LOOP to PPP with Peseran s mean group common correlated elements (MG-CCE) estimator of the AR coe cients for Qct. MG-CCE for k th autoregressive lag ^ k = 1 IC I X i=1 C X c=1 ^ ick IMRR punchline: Half-life estimate is less than 1 year!ppp has struck back. Claim: Di erent goods have di erent speeds of adjustment. Aggregation bias caused long half life estimates due to contamination of heterogeneous speeds of adjustment. Note: Crucini and Shintani, nd similar results with more disaggregated price data.

8 What s wrong with that? qit = qit {z 1 } Transitory + i + ift + {z vit } eit {z } Permanent Factor structure treated as nuisance parameters one estimates so as to get good estimates of : Forgot that common factor also drives exchange rate dynamics. Critique leveled not just at IMRR, but to (almost?) all panel data studies on PPP.

9 PPP strikes out We use IMRR s data which we obtained from their website. Ask these questions: How does the common factor behave? How important is the factor in deviation from LOOP dynamics? Revisit tests for PPP

10 Recast dynamics of relative prices as two-component model qict = eict + icft (2) with eict orthogonal to Ft; and each component is a p-th order autoregression, eict = ic + Ft = p X j=1 p X j=1 icj eict j + ict (3) j Ft j + vt (4) is observationally equivalent to IMRR s speci cation (2) with Ft = qt: qict = ic + X p j=1 icj qict j + icft X p j=1 ic icj Ft j + ict; (5)

11 Let us examine measurements of Ft The rst principle component of deviations from LOOP ^Ft: Plus three others (of little concern here)

12 ^Ft is nearly identical to the cross-sectional average used by IMRR. They look like unit root processes.

13 Quantify the persistence Fit AR(12) to IMRR measure, qt = a + X 12 k=1 Do a Jackknife correction for bias and nd X 12 k=1 k q t k + t: (6) ^ k = 1:028 Preliminary evidence says Ft is unit-root nonstationary.

14 How big is the unit root in the real exchange rate? Ft is orthogonal to eit. Therefore, short-run variance of the relative price of the i th good qi is Var (qict) = Var (eict) + Var (icft) : Table 1 uses the cross-sectional average qt to measure Ft and regression estimates of the factor loading coe cients ic:

15 Table 1: Decomposition of Var (qi) i Total Common Common share Fruits Comm Tobacco Sound Hotel Pub.Trans Dairy Fuel Rents Meat Clothing Footwear Vehicles Bread Leisure Book Alcohol Dom.Appl Furniture Average

16 Statistical tests for international price convergence We should probably re-examine the evidence for PPP in panel data. But we ve just identi ed a pitfall in panel unit root tests: Suppose dynamics are driven (in part) by a unit-root factor structure. Then a single unit root process drives dynamics of all the prices. The cross-sectional dimension creates no advantages for unit root tests.

17 A log-t test of law-of-one price convergence Log-t test suggested by Phillips and Sul (2007b) which they employed in growth context. Gives a test of relative convergence. Concept: Crosssectional dispersion of USD price of a good is constant or shrinking over time. ict = SctPict Idea: Regress cross-sectional variance of prices on a trend. Is slope zero or negative? DGP that underlies test is a two-component model ln ict = ictft + eict (7) There is relative convergence to the law-of-one price if ict! i as t! 1

18 To implement test, give time-varying factor loadings a parametric form ict = i + ict (8) ict iid 0; (t) 2 i ln (t) 2 ic! : (9) i 0 =)Var( ict )! 0 =) ict! i =)Asymptotic convergence i < 0 =)Var( ict )! 1 as t! 1 =)Asymptotic divergence i is convergence rate to the asymptotic law-of-one price. Can say ln (ict) and ln (irt) have a common (stochastic or deterministic) trend. If trend ift is stochastic, then ln (ict) and ln (irt) are cointegrated with CI vector (1; 1) :

19 Collect pieces for log(t) test ln (Hi1=Hit) {z } Normalization 2 (ln (ln (t))) = ai + 2i ln (t) + uit; ln ict = ictfit + eict; ict C ikt = i + ict ; iid (t) 2 i ict 0; ln (t) 2 ic ict 1 P Ck=1 (relative to cross-sectional mean) hict =! Hit = 1 C X C c=1 (hict 1) 2 ( Cross-sectional variance ) H0 : ic 0 =) ict! i for all c 2 [1; 11]; H A : ic < 0 =) ict 9 i for some c hict: price of good i in country c normalized by (controlling for) the common factor Fit; Hit has dimension of cross-sectional variance of ict

20 Behavior of the cross-sectional variances Hit: Price data are indices not levels. If base year is last observation, ict will all appear to converge. Set base to rst year. Prices will diverge from this point. See Figure 2: A view of the Hit

21 Plots of Hit (cross-sectional variance of good i 0 s price).

22 Attenuate e ect of initial conditions. Discard the rst half of the sample. So from the original 175 timeseries observations, we work with the 87 observations from October 1988 to December From this subsample, we discard an additional fraction r of the sample.

23 Table 2: t ratios for slope coe cients in ln (Hi1=Hit) 2 (ln (ln (t))) = ai + 2i ln (t) + uit First Observation Clothing -287 Vehicles -120 Fruits Comm Dom.Appl Footwear Meat Sound Furniture Bread Tobacco Rent ln (Hi1=Hit) 2 (ln (ln (t))) = ai + Pub.Trans Leisure 2.75 Hotel 2.49 Book 5.25 Alcohol 8.86 Dairy 14.7 Fuel 15.1 (i) Strongest evidence against: clothing t-statistic is 287. (ii) Nontraded goods prices (public transportation and hotels) show evidence of law-of-one price convergence. (iii) Relative convergence found for only 1/3 of the goods.

24 Figure 3: hict for the U.S. dollar price of clothing for each of 11 countries

25 Figure 4: hict for the U.S. dollar price of Fuel for each of 11 countries

26 Split up the cross-sectional variances (the Hit) among the convergent and nonconvergent goods

27 Figure 5: Hit: Cross-sectional variance of prices for convergent group of goods

28 Figure 6: Hit :Cross-sectional variance of prices of divergent group of goods

29 Test for convergence to PPP. Apply the convergence test to the overall price levels across countries measured in U.S. dollars. Looking for evidence that the cross-sectional dispersion in national price levels is decreasing over time. Regress ln H 1 Ht! 2 ln (ln t) = a + 2 ln (t) + ut ^ = 1:42 t = 81:4 No evidence of asymptotic convergence to PPP.

30 Conclusion Panel-data studies on PPP have left out the dynamics of a common factor, which is persistent and maybe unit-root nonstationary. Once this is taken into account, the evidence for PPP (and nite half-lives) goes away. We seem to be back where we started.

31 Convergence concepts Absolute convergence: ln (ict) ln (irt)! 0. Relative convergence: ln( ict) ln(irt)! 1: What s the di erence? Let eict be white noise and ignore it. Let ict = 1 and irt = 1 1= p t : Suppose common factor is deterministic time trend, Ft = t: Relative convergence holds b/c ln p ict = ln p irt 1: Absolute convergence does not hold b/c ln p ict Ft= p t = p t! 1: = 1= 1 1= p t! ln p irt =