Testing for Near I(2) Trends When the Signal-to-Noise Ratio Is Small

Abstract Researchers seldom find evidence of I(2) in exchange rates, prices, and other macroeconomics time series when they test the order of integration using univariate Dickey-Fuller tests. In contrast, when using the multivariate ML trace test they frequently find double unit roots in the data. The paper demonstrates by simulations that this often happens when the signal-tonoise ratio is small.


Introduction
The global …nancial and economic crisis that began in 2007 has led to an increased interest in the mechanisms that cause asset prices to undergo persistent swings away and towards long-run equilibrium values. One important implication of the recent theory of Imperfect Knowledge Economics (IKE) by Goldberg (2007, 2011) is that fully rational behavior under imperfect knowledge will show such tendencies. While these wide swings are typical of most prices subject to speculation, such as nominal exchange rates and stock prices, they also characterize many variables in the real economy, for example unemployment rates, suggesting a close two-way interrelationship between the …nancial market and the real economy.
The purpose of this paper is to discuss di¤erent econometric characterizations of these persistent swings, in particular focussing on nominal and real exchange rates in periods of currency ‡oat. Figure 1, upper panel, illustrates how the German mark/US dollar nominal exchange rate has shown a tendency to move in long persistent swings around its I owe Andreas Noack Jensen a deep thanks for having carried out all simulations in Section 3 and for many useful comments. long-run purchasing power parity (PPP) value as given by the prices di¤erential between the two countries. The graphs illustrate that the longest swing took place from approximately 1976 to 1988, followed by shorter swings. Figure 1, lower panel, shows the real exchange rate (measured as p p s) together with the long bond rate di¤erential. It appears that the long swings in the real exchange rates (inherited from the nominal exchange rate) move almost in parallel with the swings in the bond rate di¤erential. Given the assumption that purchasing power parity holds as a stationary condition one would a priori expect relative prices and nominal exchange rates to be I(1) and the real exchange rate to be I(0). However, econometric tests often …nd many real and nominal exchange rates to be I(1) or even more persistent. For example, Engel and Hamilton (1990) found that the random walk model is strongly rejected in favor of a segmented-trends model for nominal exchange rates. 1 Shocks to a segmented-trends process display a high degree of persistence because the segmented trends have a long-lasting impact on both the level and the …rst di¤erence of the variable. This is in contrast to shocks to a random walk series which are persistent only in the level. Also an I(2) process has a tendency to generate longer-lasting swings similar to a segmented-trends process, because the shocks have a persistent impact both on the levels and the …rst di¤erences (Johansen, 1997, 2006a, Paruolo and Rahbek, 1999. In line with this, nominal and real exchange rates have frequently been well approximated by an I(2) process (Juselius, 1995, 2006, Johansen et al. 2011) but only when using multivariate unit root tests. Based on univariate Dickey-Fuller tests they are mostly found to be at most I(1). To explore why this is the case, we have simulated data designed to replicate typical features of relative prices and nominal exchange rates relying on results in Frydman and Goldberg (2007). Given an assumption that economic actors make forecasts under imperfect information they show that real exchange rates are likely to have a small signal-to-noise ratio. When testing the order of integration of the simulated series the results show that the univariate DF test tends to reject the second (near) unit root in almost all cases, whereas the multivariate test almost always …nds it. The former result can be explained by the low power of the univariate DF tests to detect a second unit root when the shocks to the drift term of the di¤erenced process are small compared to the shocks to the di¤erenced process itself, i.e. when the signal-to-noise ratio is small.
We apply the above results to a data set consisting of the German Mark/US dollar exchange rate, German and US prices and interest rates . Based on the multivariate tests we could not reject the I(2) type characterization of the nominal and real exchange rate at high signi…cance levels against the null of an I(1) characterization. Based on the univariate Dickey-Fuller tests the double unit root was rejected.
2 Testing for a double unit root when the signal-to noise ratio is small We …rst introduce the baseline autoregressive model of order k and discuss the augmented D-F model, the Engle-Hamilton segmented trends model and the Imperfect Knowledge model. We then simulate time series with double (near) unit roots and a small signal-to-noise ratio using parameter values that closely replicate the characteristic features of actual prices and exchange rates. To start with, just one single case is used to illustrate in detail why it is often di¢ cult to discover a second unit root when the signal-to-noise ratio is small. We also illustrate the di¤erence between a random walk, a near I(2) and an I(2) process and then proceed to test the series using univariate versus multivariate test procedures.

Univariate models
The pronounced persistence of many real exchange rates is often associated with a unit root in the data and mostly tested based on univariate Dickey-Fuller type models such as: q t = 1 q t 1 + 11 q t 1 + ::: where is the …rst di¤erence operator, q t stands for the log of the real exchange rate in a period of currency ‡oat, and " qt is an error term. The null of a unit root in q t is formulated as the composite null hypothesis ( 1 = 0 and = 0) where the second condition re ‡ects the fact that one would not consider deterministic linear trends in real exchange rates to be economically plausible. When the null cannot be rejected we conclude that there is at least one unit root in the real exchange rates. To test whether there is a double unit root, (1) is respeci…ed as: 2 q t = 2 q t 1 + 21 2 q t 1 + ::: where 2 = 0 is the null. In this case, the real exchange rate would exhibit a very pronounced persistence as its change is a unit root process and the real exchange rates would be likely to drift o¤ from it its long-run value for extended periods of time. If, instead, 2 deviates from zero with a very small amount, then q t would be mean-reverting, but the meanreversion would be very slow. In this case; moderately sized realizations of q t could easily exhibit a similar persistence as an I(2) process and, for practical purposes, would be di¢ cult to distinguish from an I(2) process.
Another possibility is to model the long swings in real exchange rate as a combination of unit roots and piecewise linear trends as in Engle and Hamilton (1990) implying the following change in the speci…cation of (1): q t = t + 11 q t 1 + ::: where t = i i = 1; 2; 3; :::. In this case, the real exchange rate would be described by piecewise linear trends with shifting slope parameters i : If the shifts in the secular trends take place at ever smaller intervals, t could instead be modelled as a random walk where " t would describe the change in slope parameter from time t to t + 1 and the piece-wise linear trend speci…cation (3) would converge to a double unit root process.
The Imperfect Knowledge Economics (IKE) model described in Goldberg (2007, 2011) resembles the secular trends model but di¤ers with respect to the speci…cation of t : where t is a drift term measuring the change in the real exchange rate due to a change in individuals'forecasting strategies and in the underlying fundamentals. When q t is in the neighborhood of the long-run benchmark value individuals change their forecasts conservatively and we would expect t ' 1; whereas when q t is far away from the benchmark, we would expect t < 1. Since the far-from equilibrium period is likely to be much shorter than the "close-to-neighborhood" period, the average value, ; of t for t = 1; :::; T is likely to be close to unity. This is in particular so when T is su¢ ciently large to cover several long swings in the real exchange rate. 2 Another important feature of the IKE model to be subsequently discussed is that the signal-to-noise ratio 2 " = 2 "q can be very small implying that the second near unit root associated with (5) may not be easily detectable.

Illustrating di¤erent degrees of persistence
Because the drift term (5) is unobservable in actual time-series, it is useful to simulate the persistency properties of processes described by (3) -(5) under di¤erent assumptions of the drift term. For this purpose, we have generated time-series according to: and where is {0.0, 0.95, 1.0} and lagged di¤erences have been set to zero without loss of generality. The length of the simulated sample is set to 500 corresponding to roughly 40 years of monthly observations. Figure 2, upper panel, illustrates swings that have been generated by a random walk ( = 0) and a near-I(2) process ( = 0:95). The lower panel compares the same near-I(2) process ( = 0:95) with an I(2) process ( = 1:00). The range of variation of the near-I(2) process is 50 compared to 220 for the I(2) process, which explains the di¤erence in appearance of the identical near I(2) variables in the two panels. To The same near I(2) variable (thick line) together with an I(2) variable (thin line). Signal-to-noise ratio = 0.15 Figure 2: The graph of a near-I(2) variable together with a random walk (upper panel) and the same near-I(2) variable together with an I(2) variable isolate the e¤ect of the persistency parameter , all three series have been generated from an identical realization of the random shocks " qt and " t and the signal-to-noise ratio is 0.15 for the near I(2) and the I(2) series. While both series in the upper panel exhibit persistent swings, they are much more pronounced for the near-I(2) series compared to the random walk. The two series in the lower panel exhibit persistent swings, but the swings of the I(2) series are less bounded, signifying the absence of signi…cant mean reversion in the changes of an I(2) process. This becomes even more apparent at the end of the sample illustrating that we often need a long sample period to distinguish a near I(2) from an I(2) process. A small, but persistent, drift term can be almost hidden for the eye when variance of the …rst di¤erences is large and may not be easily detectable in a time graph. A (su¢ ciently long) MA of the original series will smoothen out the highly volatile short-term movements and, therefore, can provide a …rst rough indication of a persistent drift term in the data. To illustrate this, Figure 3 upper panel, shows the graph of t together with a 12 period MA of q t for the simulated process (6) with = 0:95: While not identical, the moving average component seems to provide a fairly good description of the near I(2) drift term t . But, as successive moving average values share 2k-2 identical observations, a moving average component is inherently a time dependent process. It is, therefore, likely to exhibit swings also when there are no swings in the data. For example, when q t is a random walk process, i.e. when To illustrate the di¤erence between a small and a large signal-tonoise ratio, the lower panel shows a di¤erenced near-I(2) series that was generated with a large ratio of " = "q . The persistence of the drift term of this series is identical to that of the near-I(2) series in the upper panel, i.e. both are generated with = 0:95: Hence the series in the upper panel and the lowest panel di¤er only in terms of the signal to noise ratio, i.e. by the relative magnitude of the shocks " t and " qt . When the shocks to t and q t are of similar magnitude the di¤erence is striking: no moving average is needed to see the persistent drift in the data.
Finally, Figure 4 illustrate how di¤erent realizations of " t and " qt can produce series that look widely di¤erent. Both panels show a near I(2) ( = 0:95) and an I(2) ( = 1:00) series with a signal-to-noise ratio of 0.15, so the di¤erence between the panels is only due to di¤erent stochastic realizations of the process. The two series in the upper panel exhibit long swings around the zero line whereas in the lower panel they show a much more pronounced tendency to move away from the zero line. As expected, the I(2) series tend to drift away from the zero line more persistently than the near I(2) series. Also, the divergence of the near I(2) and the I(2) series tend to be much stronger at the end of the sample. As will be demonstrated in Section 3 based on a simulation study of 5000 replications, a long sample is often needed to be able to statistically discriminate between the two. The "true" underlying process that have generated an observed economic variable is of course much more complex than all the models discussed in Section 2.1. The graphs of the simulated series suggest that a near I(2) process (with a small signal-to-noise ratio) can reproduce the long swings behavior we often see in actual real exchange rate data and that similar behavior also can be generated by an I(2) process for moderately sized samples. Whether these models are su¢ ciently close approximations to allow us to make inference on the broad features of the underlying data generating process will be studied by simulation in Section 3.
Another issue to discuss is how the asymptotic I(2) inference is affected when data are near I(2) rather than exactly I(2). It is useful to distinguish between the case when a near unit root is treated as (1) stationary or (2) nonstationary. In the …rst case Johansen (2006b) showed by simulation that some inference (on steady-state values) became very fragile when a near unit root was treated as stationary. For example, up to 5000 observations were needed for the empirical distribution to converge to Students t when the near unit root was 0.998.
In the second case, Elliot (1998) showed both analytically and by simulations that the asymptotic distribution is no longer mixed Gaussian and that standard asymptotic inference can be misleading. However, Corollary 1 in Johansen (1997) can be used to show that inference on and in the I(2) model is e¢ cient and unbiased also in the near I(2) case 3 . Since all results discussed in the subsequent sections have been obtained by cointegration analysis in the I(2) model, the corollary result allows us to attach a fair degree of con…dence to our empirical …ndings. Nonetheless, robustness is an important issue which needs to be further studied.

Univariate Dickey-Fuller Tests
The near I(2) and the I(2) series were all simulated for a fairly small signal-to-noise ratio (0.15) and the drift term was not easily detectable as it was well hidden in the very volatile …rst di¤erences. Both processes contain two large characteristic roots, one associated with a high error variance, 2 "q ; the other with a small error variance, 2 " . In this case, the power of univariate unit root tests to detect the second (near) unit root is likely to be low because the estimated residual is a function of " qt and " t . With a small signal-to-noise ratio, " qt will completely dominate " t and the small but persistent drift that is associated with the second  large root becomes hard to detect. To illustrate, we test the null of a unit root with the augmented Dickey-Fuller test based on (1) for the three series depicted in Figure 2. Table 1 presents the results of testing the null of a unit root in the levels ( 1 = 0) and …rst di¤erences ( 2 = 0) of the simulated series using Dickey-Fuller regressions. 4 The results in the upper part of the table are for the simulated I(2), near I(2) with = 0:95, both with a signal-tonoise ratio of 0:15; and for the I(1) series. The lower part of the table, compare the results for similarly simulated I(2) and near I(2) series but with a signal-to-noise ratio of 1:0. The results show that the null of a second unit root is strongly rejected for both the I(2) and near-I(2) cases when " = "q is small, whereas it cannot be rejected when " = "q is large.
The results indicate that the univariate Dickey-Fuller test has great di¢ culty detecting the large root associated with the persistent drift term, t when the signal-to-noise ratio is small. The null of a double unit root was strongly rejected both when q t I(2) and when it was a near I(2) series with = 0:95. When q t was a random walk it was also rejected but now correctly so. In all cases the (correct) null of (at least) one unit root could not be rejected. Section 3 reports a more comprehensive simulation study.

Multivariate I(2) trace tests
In a univariate model, it is straightforward to determine whether a variable has two large roots (with a modulus that is large but less than one) whereas in a multivariate model we can only determine the number of common stochastic trends in the system and whether they are of …rst order or second order. The classi…cation of variables according to their order of integration is, therefore, more involved in the latter case which 4 The regressions for levels and …rst di¤erences were q t = 1 q t 1 + 11 q t 1 + 0 + " t and 2 q t = 2 q t 1 + " t , respectively. might suggest the use of univariate tests. 5 But, as demonstrated in Section 3, univariate Dickey-Fuller type of tests have low power to detect a second unit root, in particular when the signal to noise ratio is small. Since the multivariate model accounts for all information in the data a multivariate model is more powerful in this respect. Besides,the order of integration of individual series can also be determined within this model as shown in Section 4.3.
To give the intuition for the main issues we …rst use one set of simulations to analyze a three-dimensional VAR model in detail and then improve the generality of our …ndings based on a much larger simulation study. The VAR model has two lags and is conveniently expressed in acceleration rates, changes and levels: where ; are p p matrices, 0 ; 1 are p 1 vectors and " t is N ID(0; 2 " ): The matrices and are variation free but 0 and 1 are restricted to exclude the possibility of quadratic trends.
The hypothesis that x t I(2) is formulated as two reduced rank hypothesis: = 0 where ; are p r; with r the cointegration rank, and 0 ?
where ? ; ? are p p r orthogonal complements to ; ; and ; are p r p r s 2 : The number of common stochastic trends is p r = s 1 + s 2 of which s 1 are integrated of order one and s 2 of order two. If s 2 = 0; (10) is a full rank matrix and x t I(1): Thus testing whether x t I(2) amounts to testing s 2 > 0: Paruolo and Rahbek (1999) suggested the following parameterization of (8): ; is a p s 2 matrix of polynomially cointegrating parameters, such that ( 0 x t 1 + 0 x t 1 ) I(0); and is a p p s 2 matrix of medium run adjustment coe¢ cients.
The three variables have been simulated to re ‡ect typical time series properties of the real exchange rate, q t ; the interest rate di¤erential between the two countries, b t , and the long-term drift term, t (measured for example by the change in nominal exchange rate, s t ): They are simulated in accordance with the IKE models (3) and (5): where "q = 1:0; " b = 0:2; q t ; " = 0:15; and = 0:95: Thus, the variables q t and b t share the same realization of t . For an application to a problem with similar characteristics, see Johansen et al. (2010). As in the univariate case, we have generated 500 observations. Provided the near unit root = 0:95 is approximated with a unit root in an empirical CVAR application, then the results would be consistent with r = 1; s 1 = 1; and s 2 = 1. In this case, q t and b t would share one common stochastic near I(2) trend and would be CI(2; 1). Since the two I(1) trends in (11) cannot cancel by any linear combination 0 x t ; (11) corresponds to the 0 ?1 x t relation (s 1 = 1) which can only become stationary by di¤erencing. The polynomially cointegrated relation (r = 1) corresponds to t 1 q t 2 b t where 1 + 2 = 1: Table 2 reports the multivariate rank test results where the …rst row shows the trace test for s 2 = 3; 2; 1; 0; given r = 0 and the second row for s 2 = 2; 1; 0; given r = 1: The hypothesis fr = 1; s 2 = 1g cannot be rejected based on a p-value of 0.23. It implies three unit roots in the characteristic polynomial. The lower part of Table 2 reports the modulus of the four largest roots in the characteristic polynomial. The unrestricted VAR contains two roots almost exactly equal to one and a third root, 0.96, which is very close to the simulated value of 0.95. If we approximate the latter with a unit root then the highest unrestricted root is 0.07 for the choice of fr = 1; s 1 = 1; s 2 = 1g : But if we ignore the possibility of I(2) then the model would contain an unrestricted root of 0.96 for the choice of fr = 1; s 1 = 2; s 2 = 0g : Such a large root is likely to jeopardize standard inference on stationarity at least for some hypotheses (Johansen, 2006b).   To conclude the main di¤erence between testing the order of integration based on a univariate versus a multivariate model is that the former is generally unable to detect a large root in t when the signal to noise ratio is small, whereas the multivariate trace test is able to do so. Also, approximating a root of 0.95 with a unit root allows us to structure the data in terms of polynomial cointegration and common trends of different order, thereby exploiting the di¤erent persistency pro…les of the data. Evidence of the second large root can also be found by checking the characteristic roots of the multivariate model whereas such evidence is usually not present in the roots of a univariate model, in particular when the signal-to-noise ratio is small.

A simulation study
We consider now the IKE model (6) - (7) x 1;t = x 1;t 1 + t + " 1;t (12) x 2;t = x 2;t 1 + t + 0:2" 2;t (13) t = 0:95 t 1 + 0:15" u;t : where x 1;t represents the real exchange rate and x 2;t the real bond rate di¤erential. They share the same drift term but di¤ers in terms of the short-term volatility with x 1;t ‡uctuating much more than x 2;t re ‡ecting typical behavior of real exchange rates versus real interest rate di¤erentials. The drift term t is highly persistent and the signal-to-noise ratio between " 1;t and " u;t is 1.0/0.15. The model (12)-(14) represents the baseline model. Since t is generally not directly observable we have simulated three versions of the three-dimensional process x 0 t = [x 1;t ; x 2;t ; x 3;t ] which di¤ers with respect to the choice of x 3;t : 3;t = t + 0:2" 4;t Case 3 : x 3;t = t + " 5;t where " i;t N ID(0; 2 i ); i = 1; :::; 5. Table 4 reports the simulated results of testing the hypotheses (r; s 2 ) for r = 0; 1; 2; 3 and s 2 = p r; p r 1; :::; 0 based on the multivariate rank test in a CV AR model: The latter is estimated for the three cases S 1 = (x 1;t ; x 2;t ; x (1) 3;t ); S 2 = (x 1;t ; x 2;t ; x (2) 3;t ); S 3 = (x 1;t ; x 2;t ; x (3) 3;t ) using two lags. S 1 represents the case when the drift term, t ; is known, S 2 when it can be measured with a small error, S 3 when it is very imprecisely measured.
The results show generally that the multivariate I(2) test: rarely rejects cointegration, except when t is imprecisely measured (S 3 ); seldom accepts two or three cointegration relations and no I(2) trends, Table 4: Simulated frequences of testing the cases (r = i and s 2 = 3 i j), i = 0; 1; 2; 3 and j = 3 i; :::; 0 rarely rejects the preferred hypothesis H(1; 1) unless T is large(500) and t is imprecisely measured (S 3 ), frequently fails to reject the hypothesis H(1; 0) when T is large and t is imprecisely measured (S 3 ); i.e. = 0:95 is found to be signi…cantly di¤erent from a unit root, and seldom accepts that the process can be stationary.
In general, the larger the sample the smaller the standard errors and the higher the power of the test to discriminate between a root of  0.95 and 1.00. Thus, with a su¢ ciently large sample size even a tiny deviation from unity will be found signi…cant even though the drift term t is highly persistent. Also, the more imprecisely t is measured, the more di¢ cult it is to detect the large near I(2) root in the model. Table 5 reports simulated univariate augmented Dickey-Fuller tests of the hypothesis x 1;t I(2); I(1); I(0) based on two respective three lags for T = 50; 100; 250 and 500. Each case has been replicated 5.000 times. The simulations of x 1;t are identical to the ones being analyzed in Table 3 above. The upper part of the table reports the results for = 1:0 and the lower part for = 0:95: The results show that the ADF test rejects I(2) in essentially all cases whether = 1 or 0:95, fails to reject I(1) in the absolute majority of cases Thus, the results suggest that the univariate Dickey-Fuller test will essentially never detect a double unit root when the signal-to-noise ratio is small.

Empirical illustration: the dollar Dmk rate
This section illustrates the test procedures based on actual real exchange data for USA and Germany for a sample from May 1975 to December 1998 comprising the post Bretton Woods period of currency ‡oat. Section 4.1 …rst reports the univariate Dickey-Fuller test to determine the order of integration of the real and nominal exchange rate, US and German prices and long-term interest rates. Section 4.2 then reports the order of integration and cointegration based on the multivariate trace test  in a …ve-dimensional VAR model of x 0 t = [p 1;t ; p 2;t ; s 12;t ; b 1;t ; b 2;t ] where p stands for prices, s for the nominal dollar-Dmk rate, b for long-term bond rates and the subscript 1 stands for USA and 2 for Germany. Finally Section 4.3 reports tests of the order of integration of the individual series within the multivariate model. Table 6 report the Dickey-Fuller univariate tests of I(1) based on (1), and of I(2) based on (2) for the …ve variables as well as for the following transformations: real exchange, p 1;t p 2;t s 12;t ; the interest rate differential, b 1;t b 2;t and relative prices, p 1;t p 2;t . The I(1) test failed to reject the null of one unit root in all cases except for the relative price for which^ 1 = 0:005 was found to be signi…cant due to a very small standard error. The I(2) version of the Dickey-Fuller test failed to detect a second (near) unit root in almost all cases except for US long-term bond rate and US prices (borderline). For both the nominal and real exchange rate a double unit root was strongly rejected with almost identical test statistics signifying the fact that the two variables have moved almost in parallel. That a second large root was detected in the US bond rate is interesting and suggests that it has exhibited similar long swings as the real exchange rates while not exhibiting the same short-term volatility. That the test borderline failed to reject a double unit root in the US CPI is also interesting as consumer prices are not likely to be subject to speculation in any signi…cant manner and, hence, should not be a¤ected by the long swings drift term t : Therefore, a double root in US CPI prices suggests that CPI shocks, while not necessary large, have been persistent over this period and that the signal-to-noise ratio is rather high. This suggests that the power of Dickey-Fuller test to detect a double unit root is reasonable provided the signal-to-noise ratio is not too small.

Multivariate VAR based unit root tests
We estimate the VAR model (8) for x 0 t = [p 1;t ; p 2;t ; s 12;t ; b 1;t ; b 2;t ] augmented with a few dummy variables, primarily to account for the German reuni…cation as explained in Johansen et al. (2010) and in Juselius (2012). Table 7 reports the I(2) trace tests as well as the characteristic roots of the model. Since all versions for r = 0 were strongly rejected, they are not reported in the table. The estimated characteristic roots in Table  7 suggest a total number of …ve large roots in the unrestricted VAR, four of which are almost exactly on the unit circle, while the …fth, while large (0.88), is not equally close to one. Juselius (2012) shows that the case (r = 2; s 1 = 1; s 2 = 2) is theoretically consistent with an IKEbased model. The …rst acceptable choice is {r = 1; s 1 = 3; s 2 = 1g: Both the preferred and the …rst acceptable case are consistent with …ve unit roots in the characteristic polynomial. Jensen (2013) derives a Likelihood ratio test for the choice between two non-nested models with equal number of unit roots, LR fH(r 1; s 1 + 2) j H(r; s 1 )g : The test of H(1; 3) against H(2; 1) gives a test statistic of 13:6 > Q(:95) = 13:3: Since H(2; 1) is consistent with our theoretical prior we continue with the case fr = 2; s 1 = 1; s 2 = 2g : The given this choice is 0:51. If, instead, we had chosen s 2 = 0 (and treated the variables in the model as I(1)), our model would contain two large roots with a modulus of 0:96 and any inference on stationarity would have been jeopardized.
To summarize: the case fr = 2; s 1 = 1; s 2 = 2g is supported by the data, is able to account for all …ve large roots in the unrestricted VAR, and is consistent with the economic prior.

Testing the order of integration of individual variables
As discussed above, the multivariate trace tests are informative about the order of integration of the vector process but are generally uninformative about the individual variables. It is, however, straightforward to formulate and test hypotheses of the order of integration of these variables given the choice of r; s 1 and s 2 . We …rst de…ne = ; ?;1 : The hypothesis that a variable is I(1) in the I(2) CVAR model can be formulated as a known vector b 1 in = (b 1 ; b 1? ') where b 1? ' de…nes the other vectors to be restricted to lie in the orthogonal space of b 1 : For example b 1 = [0; 0; 1; 0; 0; 0; 0] is a test of the hypothesis that the nominal exchange rate, s 12;t ; is a unit vector in : If not rejected, it can be considered I(1); otherwise I(2). See Johansen et al. (2006) for further details.
As prices generally are subject to linear deterministic (as well as stochastic) trends we need to test the hypothesis that the price as well as the trend-adjusted price is I(1). As discussed in Johansen et al. (2010) this can be formulated as a test on = (H 1 ' 1 ; H 2 ' 2 ) : For example, H 0 1 = 1 0 0 0 0 0 0 0 0 0 0 0 0 1 would be a test of trend-adjusted US price being I(1). As the VAR model was speci…ed to allow for a change in the slope of the linear trend at the time of the reuni…cation we formulate the test to allow also for this possibility. Table 8 reports the Likelihood Ratio test results. Except for the German bond rate, all hypotheses were strongly rejected, indicating that the di¤erenced processes are persistent enough to reject the I(1) hypothesis in favor of an I(2)-type characterization. 6

Conclusions
Macro-…nancial data (nominal exchange rates, stock prices, etc.) are typically characterized by very volatile short-run changes around smooth persistent trends. In this case, the drift term in the changes of the process is a very persistent process with a small variance compared to the large variance of the short-run changes, i.e. the process is (near) I(2) with a small signal-to-noise ratio. A priori we expect near-I(2) trends to be prevalent in asset prices strongly a¤ected by …nancial speculation, such as exchange rates, stock prices, and commodity prices. However, univariate Dickey-Fuller tests seldom …nd evidence of a double unit root in such data whereas multivariate trace tests frequently do. Our paper Table 8: Testing hypotheses of I(1) versus I(2) p 1;t p 2;t s 12;t b 1;t b 2;t t 91 t 2 (v) p val Is the price di¤erential I (1) demonstrates by simulations that this is often the case when the signalto-noise-ratio is small.