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E XCHANGE R ATE M ODELS ANDS PURIOUS R EGRESSIONSJón Steinsson and Emi NakamuraUC BerkeleyApril 2019Nakamura-SteinssonFX and Spurious Regressions1 / 50

“One of the most remarkable facts about G3 exchange rates is thatthey are so seemingly immune to systematic empirical explanation.”– Kenneth RogoffNakamura-SteinssonFX and Spurious Regressions2 / 50

DEM/USD Exchange -Steinsson1967197219771982FX and Spurious Regressions1987199219973 / 50

1970’ S M ONETARY M ODEL OF E XCHANGE R ATESPurchasing Power Parityet pt pt Nakamura-SteinssonFX and Spurious Regressions4 / 50

1970’ S M ONETARY M ODEL OF E XCHANGE R ATESPurchasing Power Parityet pt pt Money demandmt pt φy yt φi itmt pt φy yt φi it Nakamura-SteinssonFX and Spurious Regressions4 / 50

1970’ S M ONETARY M ODEL OF E XCHANGE R ATESPurchasing Power Parityet pt pt Money demandmt pt φy yt φi itmt pt φy yt φi it Combining money demandpt pt (mt mt ) φy (yt yt ) φi (it it )Nakamura-SteinssonFX and Spurious Regressions4 / 50

1970’ S M ONETARY M ODEL OF E XCHANGE R ATESPurchasing Power Parityet pt pt Money demandmt pt φy yt φi itmt pt φy yt φi it Combining money demandpt pt (mt mt ) φy (yt yt ) φi (it it )Exchange rate and “fundamentals”:et (mt mt ) φy (yt yt ) φi (it it )Nakamura-SteinssonFX and Spurious Regressions4 / 50

F RENKEL (1976)Sample: German Mark, February 1920 - November 1923.Hyperinflation: Ignore a bunch of terms.et (mt mt ) φy (yt yt ) φi (it it )et mt φi (it it )Nakamura-SteinssonFX and Spurious Regressions5 / 50

log S a' b[ log M b2 log n uwhere as before, 1 z*. The estimates are reerrors below the coefficients:logS -5.135 0.975 log M 0.591 log g(0.731) (0.050)(0.073)R2 0.994; s.e. 0.241; D.W. 1.91.As is evident these results are fully consistenSource: Frenkel (1976).Scand. J. of Economics 1976Nakamura-SteinssonFX and Spurious Regressions6 / 50

DATE2002 20052008 20-2102 2105-. .-2108--LOG MONEY2111 2202- 2205 2208 2211 Z23022305 LOG EXCHANGERATE-23082311 -Fig. 1.Source: Frenkel (1976).Scand. J. of Economics 1976Nakamura-SteinssonFX and Spurious Regressions7 / 50

F RANKEL (1979)Sample: DEM/USD, July 1974 - February 1978.et φ0 φm (mt mt ) φy (yt yt ) φi (it it ) φπ (πte πte ) tNakamura-SteinssonFX and Spurious Regressions8 / 50

SEPTEMBER 1979THE AMERICAN ECONOMIC REVIEW616\\\E \\lx1-197411975FIGURElitted\ 4a f\ NXlV I\ g actual197611. PLOT OF (log OF) MARK/DOLLAROLS REGRESSION FROM TABLE 1\19771RATE,Source: Frankel (1979).Nakamura-Steinssonand Spurious Regressions9 / 50 1As a final indication of the support Tablethe (negative)coefficient on theFX nominal

VOL. 69 NO. 4FRANKEL: FLOATING EXCHANGE RATESTABLEI-TESTOF REALINTERESTDIFFERENTIAL615HYPOTHESIS(Sample: July 1974-February 1978)TechniqueOLSCORCINSTFAIRConstantm - .25).96(.14).97(.21)y - y-.72(.22)-.33(.20)-.54(.18)-.52(.22)r- 65(2.70)7.72(4.47)27.42(2.26)29.40(3.33)R2D. W.80.76.91Number ofObservations44.9843421.00.4641Note: Standard errors are shown in parentheses.Definitions: Dependent Variable (log of) Mark/Dollar Rate.CORC Iterated Cochrane-Orcutt.INST Instrumental variables for expected inflation differential are Consumer Price Index (CPI) inflationdifferential (average for past year), industrial Wholesale Price Index (WPI) inflation differential (average for pastyear), and long-term commercial bond rate differential.FAIR Instrumental variables are industrial WPI inflation differential and lagged values of the following: exchangerate, relative industrial production, short-term interest differential, and expected inflation differential. The method ofincluding among the instruments lagged values of all endogenous and included exogenous variables, in order to insureconsistency while correcting for first-orderserial correlation, is attributed to Ray Fair.m -m* log of GermanM,/U.S. M,y - y* log of German production/U.S. productionr-r* Short-term German-U.S. interest differential(r - r*), Short-term German-U.S. interest differential lagged7r - 7r* Expected German-U.S. inflation differential, proxied by long-term government bond differential.Source: Frankel (1979).of the long-term interest differential is that itzero. This result is all the more striking when10 / 50Nakamura-SteinssonFX and Spurious Regressions

M EESE AND ROGOFF (1983)Do the monetary models of exchange rates fit out of sample?Nakamura-SteinssonFX and Spurious Regressions11 / 50

M EESE AND ROGOFF (1983)Do the monetary models of exchange rates fit out of sample?Generalized monetary model:et φ0 φm (mt mt ) φy (yt yt ) φi (it it ) φπ (πte πte ) φTB TBt φTB TBt tNakamura-SteinssonFX and Spurious Regressions11 / 50

M EESE AND ROGOFF (1983)Auto-regressive modelet φ0 JXφj et j tj 1Nakamura-SteinssonFX and Spurious Regressions12 / 50

M EESE AND ROGOFF (1983)Auto-regressive modelet φ0 JXφj et j tj 1Vector auto-regressive modele t φ0 JXj 1Nakamura-Steinssonφj et j JXΦj Xt j tj 1FX and Spurious Regressions12 / 50

M EESE AND ROGOFF (1983)Auto-regressive modelet φ0 JXφj et j tj 1Vector auto-regressive modele t φ0 JXj 1φj et j JXΦj Xt j tj 1Random Walk modelEt et j etNakamura-SteinssonFX and Spurious Regressions12 / 50

M EESE AND ROGOFF (1983)Sample period: March 1973 - June 1981Forecasts based on rolling regression starting November 1976Forecast horizons: 1, 6 and 12 monthsNakamura-SteinssonFX and Spurious Regressions13 / 50

M EESE AND ROGOFF (1983)Sample period: March 1973 - June 1981Forecasts based on rolling regression starting November 1976Forecast horizons: 1, 6 and 12 monthsMeasure of out-of-sample accuracy: RMSE k 1 NX Nakamura-Steinsson[F (t s k ) A(t s k )]2 /Nk 1/2 s 0FX and Spurious Regressions13 / 50

M EESE AND ROGOFF (1983)In structural models:Use actual realized future values of explanatory variables(as opposed to also forecasting explanatory variables)Nakamura-SteinssonFX and Spurious Regressions14 / 50

M EESE AND ROGOFF (1983)In structural models:Use actual realized future values of explanatory variables(as opposed to also forecasting explanatory variables)Two possible stories:Hard to predict exchange rate because it is hard to predictvariables that it depends onNakamura-SteinssonFX and Spurious Regressions14 / 50

M EESE AND ROGOFF (1983)In structural models:Use actual realized future values of explanatory variables(as opposed to also forecasting explanatory variables)Two possible stories:Hard to predict exchange rate because it is hard to predictvariables that it depends onHard to find any systematic relantionship between exchangerates and other variablesNakamura-SteinssonFX and Spurious Regressions14 / 50

.oII or,r E(3c C "G0 0 00 0 00 0NNN 0NN 0 0 00.080gJr.r m. h g 8( /Mark 1-month number should be 3.17 not 3.72, see Table 3)Source: Meese and Rogoff (1983).Nakamura-SteinssonFX and Spurious Regressions15 / 50

M EESE AND ROGOFF (1983)Nothing beats random walk out of sampleNakamura-SteinssonFX and Spurious Regressions16 / 50

M EESE AND ROGOFF (1983)Nothing beats random walk out of sampleStronger than just lack of predictability(since they use realized future values of explanatory variables)Nothing even explains exchange rates!!!Nakamura-SteinssonFX and Spurious Regressions16 / 50

M EESE AND ROGOFF (1983)Rogoff (2001) recounts:For a long time, no one did believe us.The editor ofthe American Economic Reivew (Robert Clower) sent ourmanuscript back in return mail with a scathing letter sayingthat the results are obviously garbage and if we wish to remain in the economics profession, we had better developa more positive attitude. . One then young and now preeminent MIT macroeconomist, when told the findings, forcefully commented (with a French accent) “You just cannot possibly have done it right.”Nakamura-SteinssonFX and Spurious Regressions17 / 50

M EESE AND ROGOFF (1983)Rogoff (2001) recounts:For a long time, no one did believe us.The editor ofthe American Economic Reivew (Robert Clower) sent ourmanuscript back in return mail with a scathing letter sayingthat the results are obviously garbage and if we wish to remain in the economics profession, we had better developa more positive attitude. . One then young and now preeminent MIT macroeconomist, when told the findings, forcefully commented (with a French accent) “You just cannot possibly have done it right.”As of April 2019: 4776 Google scholar citationsNakamura-SteinssonFX and Spurious Regressions17 / 50

T WO L ESSONS1. Economics lesson:Exchange rate dominated by unpredicatable shocks(unpredictable capital flows?)Exchange rate very forward looking variableNakamura-SteinssonFX and Spurious Regressions18 / 50

T WO L ESSONS1. Economics lesson:Exchange rate dominated by unpredicatable shocks(unpredictable capital flows?)Exchange rate very forward looking variable2. Econometric lesson:Beware regressing very persistent variable onanother very persistent variableNakamura-SteinssonFX and Spurious Regressions18 / 50

S IMPLIFIED E NGEL AND W EST (2005)Uncovered interest rate parity:it it Et et 1 etReturns should be equalized across countriesIf interest rate is higher abroad, exchange rate shouldfall enough on average to equalize returns(et is domestic currency price of foreign currency)Nakamura-SteinssonFX and Spurious Regressions19 / 50

S IMPLIFIED E NGEL AND W EST (2005)Rearranging and solving forward:it it Et et 1 etet (it it ) Et et 1et (it it ) X Et (it j it j ) lim Et et jj 1Nakamura-SteinssonFX and Spurious Regressionsj 20 / 50

S IMPLIFIED E NGEL AND W EST (2005)What determines the change in the exchange rate:et 1 et (it it ) X Et 1 (it j it j ) lim Et 1 et jj 1j where Et 1 xt j Et 1 xt j Et xt j (time t 1 news about xt j )Nakamura-SteinssonFX and Spurious Regressions21 / 50

S IMPLIFIED E NGEL AND W EST (2005)What determines the change in the exchange rate:et 1 et (it it ) X Et 1 (it j it j ) lim Et 1 et jj 1j where Et 1 xt j Et 1 xt j Et xt j (time t 1 news about xt j )Two components:Current interest rate differencialNews about all future interest rate differentialsNakamura-SteinssonFX and Spurious Regressions21 / 50

S IMPLIFIED E NGEL AND W EST (2005)What determines the change in the exchange rate:et 1 et (it it ) X Et 1 (it j it j ) lim Et 1 et jj 1j where Et 1 xt j Et 1 xt j Et xt j (time t 1 news about xt j )Two components:Current interest rate differencialNews about all future interest rate differentialsNot so implausible that the variance of the latter is hugecompared to the formerNakamura-SteinssonFX and Spurious Regressions21 / 50

S IMPLIFIED E NGEL AND W EST (2005)et 1 et (it it ) X Et 1 (it j it j ) lim Et 1 et jj 1j But (it it ) not only thing observedNakamura-SteinssonFX and Spurious Regressions22 / 50

S IMPLIFIED E NGEL AND W EST (2005)et 1 et (it it ) X Et 1 (it j it j ) lim Et 1 et jj 1j But (it it ) not only thing observedMovements in longer-term bonds allow one to back outestimates of X Et 1 (it j it j )j 1at least up to j 40 quarters (and assuming EHTS)Nakamura-SteinssonFX and Spurious Regressions22 / 50

S IMPLIFIED E NGEL AND W EST (2005)et 1 et (it it ) X Et 1 (it j it j ) lim Et 1 et jj 1j But (it it ) not only thing observedMovements in longer-term bonds allow one to back outestimates of X Et 1 (it j it j )j 1at least up to j 40 quarters (and assuming EHTS)limj Et 1 et j still a potential problemBut in real terms PPP should hold in the very long run(Clarida-Luo 14; Engel 15)Nakamura-SteinssonFX and Spurious Regressions22 / 50

I N - SAMPLE VERSUS O UT- OF - SAMPLEWhy was Frankel’s in-sample inference so much strongerthan Meese-Rogoff’s out-of-sample inference?Nakamura-SteinssonFX and Spurious Regressions23 / 50

I N - SAMPLE VERSUS O UT- OF - SAMPLEWhy was Frankel’s in-sample inference so much strongerthan Meese-Rogoff’s out-of-sample inference?Suggests that something is wrong with in-sample inference(This is a general concern)Nakamura-SteinssonFX and Spurious Regressions23 / 50

S PURIOUS R EGRESSIONSMonetary model of exchange rate:et φ0 φf ft tBoth et and ft have a unit-root.Granger and Newbold (1974):Usual methods massively understate standard errorsNakamura-SteinssonFX and Spurious Regressions24 / 50

115C. W.J. Granger, P. Newbold, Regressions in econometrics4. Some simulation resultsAs a preliminary, we looked at the regressionY, PO &X*9where Y, and X, were, in fact, generated as independent random walks each oflength 50. Table 1 shows values ofthe customary statistic for testing the significance of fil, for 100 simulations.Table 1Regressing two independent random walks.s:Frequency :o-113l-2102-3113-4134-5185-6a6-787-85s:Frequency g the traditional t test at the 5 % level, the null hypothesis of no relationSource: Grangerand Newbold(1974).ship betweenthe two serieswould be-rejected (wrongly) on approximately threequarters of all occasions. If fll/S.E.(i ) were distributed as N(0, l), then theNakamura-SteinssonSpuriousRegressionsexpected value of S would FXbeand1/2/nN 0.8.In fact, the observed average value25 / 50

(iv) changes in A.R.I.M.A. (0, 1, I), i.e., first order moving average.Table 2Regressions of a series on m independent‘explanatory’ series.Series either all random walks or all A.R.I.M.A. (0, 1, 1) series, or changes in these. Y, 100,Y, Y, l a,, Y,’ Y, kb,; X,., 100, XL, X ., l aj. Xj.,‘ XJ,t kbJ,r;a,,r,a,,b,,bJ,rsets of independent N(0, 1) white noises. k 0 gives random walks, k 1 gives A.R.I.M.A.(0, 1, 1) series. Ho no relationship, is true. Series length 50, number of simulations 100,R’ corrected RZ.Per cent timesHo rejected’Levelsm lm 2m 3m 4m 57678939596Changes m 1m 2m 3m 4m 5842106Levels6481829090m lm 2m 3m 4m 5Changes m 1m 2m 3m 4m 58127913AverageDurbin-WatsonRandom .M.A. (0, 1, R20.260.340.460.550.59Per centRf 0.1582534370.0040.001- 10.0050.0250.02700000‘Test at 5% level, using an overall test on RZ.Source: Granger and Newbold (1974).All error terms were distributed as N(0, 1) and the A.R.I.M.A. (0, 1, 1) serieswhite noise. TheNakamura-SteinssonFXaandSpuriousRegressionswas derived as the sum ofrandomwalkand independent26 / 50

S PURIOUS R EGRESSIONSTwo common responses:Use HAC standard errors (e.g., Newey-West, 1987)Series are persistent but don’t have a unit root.Nakamura-SteinssonFX and Spurious Regressions27 / 50

S PURIOUS R EGRESSIONSTwo common responses:Use HAC standard errors (e.g., Newey-West, 1987)Series are persistent but don’t have a unit root.Granger, Hyung, and Jeon (2001)Xt α βYt utXt θx Xt 1 x,tYt θy Yt 1 y ,tNakamura-SteinssonFX and Spurious Regressions27 / 50

900Table 1. Regressing between two independent AR series ³ ˆ ³x ˆ ³y †, percentage of jtj 1:96MethodNOBS³ˆ0³ ˆ 0:25³ ˆ 0:5³ ˆ 0:75³ ˆ 0:9³ ˆ 1:0OLS1005002 00010 00011005002 00010 079.686.492.5100.0BARTNotes: 1. The number of iteration 5 1000.2. % of rejection, i.e., absolute value of t-value 1:96.3. 1 means asymptotic case.4. To avoid the problem of xing X0 and Y0 , 100 pre-samples are generated and letX¡100 ˆ Y¡100 ˆ 0.5. The number of rejections (BART) depends on the number of lags (l) used to calculate v .1 4l ˆ integer ‰4 T 100† Š is set.FromEquation1 the andordinarysquares estimate of Source:Granger,Hyung,Jeon least(2001).is de ned by:Nakamura-SteinssonTFX and Spurious Regressions(c) The use of consistent estimatolem, and is thus clearly helplarger.28 / 50

S PURIOUS R EGRESSIONSBig problem even if series are stationary if they arepretty persistent and sample is smallNewey-West standard errors have very badsmall sample propertiesNakamura-SteinssonFX and Spurious Regressions29 / 50

S PURIOUS R EGRESSIONSBig problem even if series are stationary if they arepretty persistent and sample is smallNewey-West standard errors have very badsmall sample propertiesAccurate standard errors require more sophisticated methodsLazarus-Lewis-Stock 18 suggest improvementsEven this not so good. No really satisfactory methods existNakamura-SteinssonFX and Spurious Regressions29 / 50

S PURIOUS R EGRESSIONSWhy does this problem occur, intuitively?Nakamura-SteinssonFX and Spurious Regressions30 / 50

S PURIOUS R EGRESSIONSWhy does this problem occur, intuitively?Observations are not independent!!Observations that are close in time are very correlatedNakamura-SteinssonFX and Spurious Regressions30 / 50

S PURIOUS R EGRESSIONSWhy does this problem occur, intuitively?Observations are not independent!!Observations that are close in time are very correlatedIntuitively, the key question is:How many independent observations do I have?(With unit root, all observations are correlated)Nakamura-SteinssonFX and Spurious Regressions30 / 50

S PURIOUS R EGRESSIONSWhy does this problem occur, intuitively?Observations are not independent!!Observations that are close in time are very correlatedIntuitively, the key question is:How many independent observations do I have?(With unit root, all observations are correlated)Is higher frequency data useful?Nakamura-SteinssonFX and Spurious Regressions30 / 50

S PURIOUS R EGRESSIONSWhy does this problem occur, intuitively?Observations are not independent!!Observations that are close in time are very correlatedIntuitively, the key question is:How many independent observations do I have?(With unit root, all observations are correlated)Is higher frequency data useful?It does increase the number of data pointsBut the correlation between data points goes upIntuitively: No new information about low frequency stuffNakamura-SteinssonFX and Spurious Regressions30 / 50

S PURIOUS R EGRESSIONSWhether a sample is “small” or “large” is not so simple a questionDepends on how correlated observations areYou can have hundreds of thousands of observations but a “smallsample” problem if correlation between observations is very highCross-sections correlation can also be a problem(hence importance of “clustering” in constructing standard errors)Nakamura-SteinssonFX and Spurious Regressions31 / 50

R ISING E MPIRICAL S TANDARDS1. “Revolution of identification”More serious attention to credible identification of causal effects2. Accurate standard errorsClusteringAccounting for persistenceNakamura-SteinssonFX and Spurious Regressions32 / 50

H AS M EESE -ROGOFF 83 S TOOD THE T EST OF T IME ?Nakamura-SteinssonFX and Spurious Regressions33 / 50

H AS M EESE -ROGOFF 83 S TOOD THE T EST OF T IME ?Mostly yes!Rossi 13 provides comprehensive surveyMark 95 long-run predictability results most serious challengeSee also more recent work on Taylor rule fundamentals(Molodtsova-Papell JIE 09)Nakamura-SteinssonFX and Spurious Regressions33 / 50

M ARK (1995)Simple monetary model:et ft cft (mt mt ) λ(yt yt )Even if monetary model doesn’t work in the short run,it may work in the long runEstimates partial adjustment model:et k et αk βk (ft et ) νt k ,tNakamura-SteinssonFX and Spurious Regressions34 / 50

M ARK (1995)et k et αk βk (ft et ) νt k ,tSample period: 1973:2 - 1991:4Pseudo-out-of-sample period: 1981:4 - 1991:4Currencies: Canada, Germany, Switzerland, JapanHorizons: k 1, 4, 8, 12, 16 (quarters)Nakamura-SteinssonFX and Spurious Regressions35 / 50

M ARK (1995): E CONOMETRIC I SSUESet k et αk βk (ft et ) νt k ,tMultiperiod forecasts induce correlation in error termsNakamura-SteinssonFX and Spurious Regressions36 / 50

M ARK (1995): E CONOMETRIC I SSUESet k et αk βk (ft et ) νt k ,tMultiperiod forecasts induce correlation in error termsStambaugh 86/99 biasft et predetermined but not strictly exogenousPast values of et k et correlated with ft etCauses finite sample bias in βkNakamura-SteinssonFX and Spurious Regressions36 / 50

M ARK (1995): E CONOMETRIC I SSUESet k et αk βk (ft et ) νt k ,tMultiperiod forecasts induce correlation in error termsStambaugh 86/99 biasft et predetermined but not strictly exogenousPast values of et k et correlated with ft etCauses finite sample bias in βkStandard errors produced using bootstrap that assumesft et follows AR(p)But et and ft may not be cointegratedSmall sample bias in estimating AR(p)Nakamura-SteinssonFX and Spurious Regressions36 / 50

M ARK (1995): ROBUSTNESSWhy not use UK pound?Nakamura-SteinssonFX and Spurious Regressions37 / 50

M ARK (1995): ROBUSTNESSWhy not use UK pound?Mark calibrates λ 1:ft (mt mt ) (yt yt )also no interest rate term. Why not estimate?Nakamura-SteinssonFX and Spurious Regressions37 / 50

M ARK (1995): ROBUSTNESSWhy not use UK pound?Mark calibrates λ 1:ft (mt mt ) (yt yt )also no interest rate term. Why not estimate?GNP for US, GDP for all other countries. Why?M3 for Canada, M1 for all other countries. Why?Nakamura-SteinssonFX and Spurious Regressions37 / 50

TABLE 2-REGRESSIONESTIMATES AND BOOTSTRAP (A)AMSL-pMSL-nCanadian dollar:1 0.040 0.0274 0.155 0.1098 0.349 0.26412 0.438 0.32016 0.450 .2160.3430.456Deutsche mark:1 0.035 0.0124 0.205 0.1148 0.554 0.38012 0.966 0.73316 1.324 .1430.0480.025Swiss franc:1 0.0744 0.2858 0.56812 0.83716 0.6730.042 2.6810.145 3.2480.276 4.7700.452 8.0130.655 543.5164.8894.919(xiii)(xiv)Notes: The table presents OLS estimates of the regression et k - et ak k(ft - et) 1t k,k,where ft(mi - m*)- (yt - Y*). The (Gaussian) parametric and nonparametric bootstrap distributions are generated underthe null hypothesis that the regressor follows an AR(4) for the Canadian dollar, the Swiss franc, and the yen, and anAR(5) for the deutsche mark. Exchange rates are dollars per unit of foreign currency. Adj-p and Adj-n arebias-adjusted values obtained by subtracting median values generated by the parametric and nonparametricbootstrap distributions, respectively, from the estimates. MSL-p and MSL-n are, respectively, the parametric andnonparametric bootstrap marginal significance levels for a one-tail test. A is the truncation lag determined byAndrews's (1991) univariate AR(1) rule used for constructing the t ratios with the data.Source: Mark (1995). Note: Big β, big R 2 , large tk (20) for DM, CHF, JPYNakamura-SteinssonFX and Spurious Regressions38 / 50

0Q .00 0ite00culCas000c00000000000000000-0 00.0co0 19741976197819801982FIGURE 1. 2CHANGES IN THE LOGEXCHANGE RATESource: Mark (1995)Nakamura-SteinssonFX and Spurious Regressions39 / 50

0l andr theizonsn areed toindiactualre inillusthatened.isplayandbias-0 Fitted ValueActua Changes00200 07(D0 I1974197619781980198219841986198819901992FIGURE 5. SIXTEEN-QUARTERCHANGTES IN THEEXCIIANGELoG DOLLAR/DEUTSCHE-MARKRATESource: Mark (1995)Nakamura-SteinssonFX and Spurious Regressions40 / 50

TABLE 4-OUT-OF-SAMPLE(i)(ii)k IN/RWSource:(iii)(iv)OUT/INOUT/RW(v)(vi)FORECAST EVALUATION(vii)(viii)(ix)(x)(xi)MSL-p MSL-n 2DM(20) MSL-p MSL-n 2DM(A) A(xii)(xiii)MSL-p MSL-nCanadian dollar:1 0.96040.8898 0.67512 0.65416 0.6360.061- 1.270- 1.036- 70.5560.54210.0368-0.925-0.89017- 1.661 18- 1.857 55Deutsche mark:1 0.98840.9278 0.83312 0.67016 0.025-0.932- 1360.011Swiss franc:1 0.97240.8868 0.78012 0.62516 10.0800.2780.2720.2360.1370.0580.064 30.162 120.560 170.938 131.996 s: The table presents ratios of root-mean-squared errors for the regression's out-of-sample forecasts (OUT),the driftless random walk (RW), and the in-sample regression residual during the forecast period (IN). The firstforecast is made on 1981:4. D9M(20)and D9M(A)are the Diebold-Mariano statistics constructed using the method ofNewey and West (1987) with the truncation lag of the Bartlett window set to 20 and set by Andrews's (1991) AR(1)rule, respectively. In instances where the estimated spectral density at frequency zero of the squared errordifferential is nonpositive (see footnote 8), the Bartlett-window truncation lag is decreased by 1. MSL-p and MSL-nare marginal significance levels, generated by the parametric and nonparametric bootstrap distributions, respectively, for one-tail tests.aBartlett-window truncation lag 18.Mark(1995). Note: OUT/RW much smaller than 1.bBartlett-window truncation lag 17.Nakamura-SteinssonFX and Spurious Regressions41 / 50

T RUE O UT-O F -S AMPLE T ESTJon wrote a class paper on this for Jim Stock’s Time Seriesclass in 2003True out-of-sample period: 1992:1-2000:4Nakamura-SteinssonFX and Spurious Regressions42 / 50

T RUE O UT-O F -S AMPLE T ESTJon wrote a class paper on this for Jim Stock’s Time Seriesclass in 2003True out-of-sample period: 1992:1-2000:4Used slightly different data:M2 as opposed to M3 for CanadaGDP as opposed to GNP for USResults sensitive to this (not comforting)Main results do not survive in 1990sNakamura-SteinssonFX and Spurious Regressions42 / 50

Table 1 -- Replication, Extentions and Out of Sample PerformanceMark's publishedresultsbetaR2(ii)(iii)Mark's data -- MyreplicationbetaR2(iv)(v)Current Data -Mark's sample periodbetaR2(vi)(vii)Mark's RMSEresultsOUT/RW(viii)RMSE ratiosfor 1990'sOUT/RW(ix)Canadian dollar vs. U.S. 9330.9321.0371.208Deutsche mark vs. U.S. 9921.5111.957Swiss franc vs. U.S. 7860.7630.980Japanese Yen vs. U.S. 1771.1391.185Source: Steinsson (2003)Nakamura-SteinssonFX and Spurious Regressions43 / 50

F OLLOW U P ON M ARK (1995)Killian 99 makes same point as I did. Also critiques bootstrap.Faust-Rogers-Wright 03: Doesn’t work with other vintages ofdataNakamura-SteinssonFX and Spurious Regressions44 / 50

Fig. 6. Out-of-sample relative RMSE using different data vintages (Mark’s sample period).Source: Faust-Rogers-Wright (2003)this model posits that the nominal exchange rate is determined by home–foreigndifferentials in the monetary fundamentals used above as well as short-terminterest rates, expected inflation rates, and cumulated current account balances.Nakamura-SteinssonFX and Spurious Regressions45 / 50

Fig. 7. Bootstrap P-values for out-of-sample relative RMSE using different data vintages (Mark’ssample period).Source: Faust-Rogers-Wright (2003)Nakamura-SteinssonFX and Spurious Regressions46 / 50

F OLLOW U P ON M ARK (1995)Killian 99 ma

supply on the current rate of change of the exchange rate. The estimates of the distributed lags for the equation of the rates of change reveal that the cur- rent rate of change of the exchange rate depends only on the current rate of DATE 2002 - 2005- 2008 - 20 - 2102 - _- ._. LOG MONEY 2105- - 2108- - 2111 -

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