Are outcomes driving expectations or the other way around? An I(2) CVAR analysis of interest rate expectations in the dollar/pound market

Transcription

1 Are outcomes driving expectations or the other way around? An I(2) CVAR analysis of interest rate expectations in the dollar/pound market Abstract This paper uses consensus forecasts to address empirical puzzles in international macro using the Cointegrated VAR model. The data, consisting of three-month libor rates, their three-month ahead forecasts, prices and exchange rates for the US and UK, were all found to be near I(2) consistent with imperfect knowledge expectations. The I(2) analysis showed that over the medium run the nominal exchange rate has moved away from equilibrium values with interest rates following suit, whereas over the long run the nominal exchange rate was adjusting while the interest rate forecasts pushed the system away from steady state. Evidence of self-reinforcing feedback mechanisms in the system signals the importance of speculative bubbles for the determination of the exchange rate and the interest rates. 1 Introduction The use of expectations in economic theory is far from new. Many earlier economists, including A. C. Pigou, John Maynard Keynes, and John R. Hicks, assigned a central role to people s expectations for the determination of the business cycle. When financial actors rush to desert a currency that they expect to lose value, they contribute to its loss in value. Also, the price of a stock or bond depends partly on what prospective buyers and sellers believe it will be in the future. Keynes referred to this as waves of optimism and pessimism that helped determine the level of economic activity. John F. 1

2 Muth was the first to formulate the theory of rational expectations as a way to model expectations in economic theory. Standard economic models with "Rational expectations" (RE) assert that agents expectations should not differ systematically from actual outcomes. Thus, RE-based theory accepts that people make forecast errors, but assumes that these errors are unsystematic. Model consistent rational expectations are used to obtain internal consistency in a theoretical model assumed to be a "true" description of the economy. Recently, as a result of the failure of RE-based models to foresee the financial and economic crisis, the assumption of a "perfectly" known model has been questioned and alternative models based on imperfect knowledge have been proposed. Common to these models is that today s asset price depends on tomorrow s price forecast which is assumed to be formed under imperfect knowledge. For example, Hommes (2005) and Hommes et al. (2005a, 2005b) develop models for a financial market populated by fundamentalists and chartists where fundamentalists use expectations based on economic fundamentals and chartists are trend-followers using naive expectations. Positive feedback prevails when the latter dominate the market. Heemeijer et al. (2009), using experiments, found that prices converge to their fundamental values under negative feedback but fail to do so under positive feedback. Adam and Marcet (2011) proposed a separation of standard RE rationality into an internal and an external component. By internal rationality, Adam and Marcet mean that agents maximize utility relative to their subjective beliefs about the future, but they may not know the true process underlying asset prices. They showed that positive feedback can arise in a model where internal rationality is maintained but external rationality is relaxed due to imperfect market knowledge. Simulations in the stock market find support for both imperfect market knowledge (Adam, Marcet, and Nicolini, 2016) and heterogenous expectations among chartists and fundamentalists (Hommes and in t Veld, 2017). Frydman and Goldberg (2007, 2011) developed a theoretical framework where agents expectations are formed in the context of imperfect knowledge about the underlying causal mechanisms, which are subject to structural change. Supporting this, several recent studies have documented the importance of model ambiguity and structural change for forecasting ex- 2

3 change rates and explaining sovereign bond yields. 1 Frydman and Goldberg argue that expectations based on imperfect knowledge and radical uncertainty about future outcomes, including unpredictable structural changes, have strong consequences for an individual s optimal decision making. For example, when an individual is faced with imperfect knowledge about economic mechanisms it may be optimal, and therefore rational, to use many different models to predict future outcomes. When actual outcomes tend to deviate from expectations, it is also optimal to revise one s forecasting model. Also, it can be beneficial, and therefore optimal, to include psychological factors, such as the market s tendency for herding behavior, in one s expectations. Thus, under imperfect knowledge, expectations are likely to play an autonomous role in the model rather than endogenously adjust to the optimal path given by the theoretical model. In this sense, the long swings we see in asset price data may very well be the result of expectations that have driven prices persistently away from (and towards) historical benchmark levels. Juselius (2017b) derived a theory-consistent Cointegrated VAR (CVAR) scenario for an imperfect knowledge based monetary model of the nominal exchange rate and illustrated it with an application to the dollar/dmk rate in the post Bretton Woods - pre EMU period. Using a simple rule for how to treat unobservable expectations in a CVAR model exclusively based on observed variables, the paper found that forecast shocks in a world of imperfect knowledge have the potential of changing the properties of the data generating process in a more persistent I(2) direction. This was found without assuming systematic forecast errors. Estimated results in an I(2) CVAR, which tests and allows for persistent growth rates of the variables, showed remarkable support for almost every testable hypothesis of the imperfect knowledge monetary model. 2 But, despite a remarkably good empirical fit, the question remains whether the assumptions made on the unobserved expectations formation are consistent with empirical practice. The present paper addresses this question by using a survey of professional forecasts as a measure of the market s expectations. Numerous studies have analyzed data on traders exchange rate 1 See e.g. De Haan, Hessel, and van den End (2014), Beckmann and Schüssler (2016), and Kouwenberg et al. (2017). 2 Stillwagon (2015) also apply the I(2) CVAR to examine the term structure of interest rates and finds support for a similar notion of risk to that of Frydman and Goldberg (2007). 3

4 forecasts. 3 Most of these studies focus on testing for predictable forecast errors (i.e. whether REH holds) and whether a time-varying risk premium is present in UIP (i.e. whether it holds ex ante), as opposed to testing what determines expectations or risk premia. One exception is Goldbaum and Zwinkles (2014), who find that survey expectations are well described by switching expectations between chartists and fundamentalists. Another exception is Stillwagon (2017) who finds that the survey risk premium co-moves positively with deviations from purchasing power parity consistent with the theory of imperfect knowledge (Frydman and Goldberg, 2007). 4 The purpose of this paper is to study the role of interest rate expectations for the long swings in the dollar/pound currency market. By using observed consensus forecasts on three-month Libor rates, we are able to make inference on how market interest rates, prices and exchange rates have been affected by market expectations without having to make (untestable) assumptions about how these expectations have been formed. 5 To allow the data to speak as freely as possible about the empirical mechanisms (Hoover et al., 2008), we use the "general-to-specific" CVAR approach to address the following questions: 1. Are the persistency properties of the data more consistent with rational expectations or imperfect knowledge expectations? 2. Is there evidence of self-reinforcing feed-back mechanisms in the system consistent with speculative bubbles? 3. Are expectations primarily adjusting to interest rates or the exchange rate? 4. Are interest rate expectations causing the long persistent swings characterizing foreign currency markets? 3 See for example, Dominguez (1985), Frankel and Froot (1987), MacDonald and Torrance (1988), Frankel and Chinn (1993), Cavaglia, Verschoors, and Wolff (1993), Bachetta, Mertens, and Van Wincoop (2009), and Stillwagon (2014). 4 Stillwagon (2017) uses surveyed exchange rate forecasts to measure expected currency returns. 5 Other studies using interest rate forecast data in the literature include: Gourinchas and Tornell (2004) who focus on underreaction to interest rate shocks in the forward discount anomaly, and Dick, MacDonald, and Menkhoff (2015) who examine the correlation between interest rate forecasts and exchange rate forecasts. 4

5 To summarize our findings: we document that interest rates, goods prices, and the real exchange rate were all rejected as at most I(1) and, hence, have to be considered near I(2). This added persistence contradicts rational expectations models of the exchange rate, while being consistent with imperfect knowledge expectations. Even though near I(2) may seem logically implausible for a variable such as interest rates, our results merely state that these variables tend to undergo highly persistent changes for extended periods of time. In no way do the results mean that the shocks are infinitely lived. In the case of interest rates such persistent changes are consistent with extended cycles of tightening and accommodation by central banks, where the target interest rate moves persistently in one direction for potentially several years at a time and therefore affects other interest rates to do so. While near I(2) persistence may seem inconsistent with the vast number of studies which find these variables to be at most I(1), Juselius (2014) shows that this is likely to be an artifact of relying on univariate unit root tests, which have low power to detect a double unit root when the variance of the I(2) component is small relative to that of the I(1) component, i.e. a low signal to noise ratio typical of financial data. 6 Treating a near I(2) trend as at most I(1) would ignore some significant persistence and more elaborate interactions between the variables including highly relevant self-reinforcing feedback loops typical of financial bubbles. The latter is exactly what we find in our CVAR model: in the medium run the exchange rate has moved away from equilibrium with interest rates following suit. This evidence of self-reinforcing feedback mechanisms signals the importance of speculative bubbles in the determination of the exchange rate and interest rates, and is consistent with behavioral models of extrapolation/momentum trading. In the long-run, however, the exchange rate adjusts back to equilibrium with interest rate expectations pushing the system away from steady state. This autonomous, exogenous influence of interest rate expectations is consistent with models of imperfect knowledge expectations. 6 The following papers find evidence of I(2) trends using multivariate trace tests: Juselius, K. (1995), Fanelli and Bacciocchi (2005), Johansen et al. (2010), Juselius and Assenmacher (2017), Hetland and Hetland (2017), Salazar (2017), and Stillwagon (2017). 5

6 2 Basic regularities in the foreign currency data Rational expectations based models would in general be consistent with the real exchange rate q t = s p d,t + p f,t (1) holding as a stationary or near I(1) process and the uncovered interest rate parity corrected for a stationary risk premium, rp t, as a market clearing mechanism i d,t i f,t = s e t + rp t. (2) For a more detailed discussion, see Juselius (2017a) and Juselius and Assenmacher (2017). The risk premium is typically associated with exchange rate volatility or, more recently, its covariance with aggregate consumption. While a risk premium may be associated with short-term exchange rate volatility, it usually cannot account for the persistent swings in the interest rate spread. This is precisely what the uncertainty premium proposed by Frydman and Goldberg (2007, 2011) is supposed to do. In general, it is a measure of a so called gap effect measuring how far away the price has moved from its longrun benchmark value. In foreign currency markets, the uncertainty premium is often assumed to be related to the divergence from PPP, but other gap effects can also be relevant. To provide intuition for the gap effect, consider the case where a currency is overvalued relative to PPP. An investor may reasonably expect a further appreciation in the near-term for a number of reasons e.g. chartist behavior, expectations of interest rate increases, etc. But, they would also acknowledge the greater probability for a reversal, given the stylized fact that exchange rates tend to mean revert back to PPP over suffi ciently long horizons or once the deviation is suffi ciently large (Lothian and Taylor, 1996; Taylor and Peel, 2002). Due to this increased potential for loss (the increased downside risk or negative skewness to forecasted returns), the bull betting on a continued appreciation will require a higher expected return or premium to maintain their long position. A bear, meanwhile, will predict a reversal and respond in the converse fashion. Even though the bear know that his prediction may be wrong, their fear of a further large appreciation diminishes as the gap increases. Both effects would act to increase the aggregate market premium, of bulls minus bears, as the currency becomes more overvalued relative to 6

7 PPP (Frydman and Goldberg, 2007). Thus, in a world of imperfect knowledge the two parity conditions (1) and (2) have to be modified to adequately account for the increased persistence in the data. In the case of the PPP condition, Johansen et al. (2010) propose to describe the persistence to changes in the real exchange rate as a random walk with a time-varying drift term q t = ω t + ε q,t, (3) where ε q.t is stationary and the drift term, ω t, is assumed to follow an autoregressive process ω t = ρ t ω t 1 + ε ω,t, where ε ω,t is a white noise process. The parameter ρ t may vary over different periods but its average value ρ is likely to be close to 1.0 provided a relatively long sample period. Thus, while the differenced real exchange rate is assumed to be white noise in an REH-based monetary model, it is likely to be a near I(1) process in an imperfect knowledge based model. 7 Following Frydman and Goldberg (2007) the conventional uncovered interest rate parity is augmented with an uncertainty premium up t : i d,t i f,t = s e t + rp t + up t. Measuring the uncertainty premium with the PPP gap gives us the relationship of key interest: i d,t i f,t = s e t + rp t + f(s p d,t + p f,t ). (4) 3 A graphical analysis The upper panel of Figure 1 shows that US prices had increased more than the UK ones up to the collapse of Lehman Brothers in 2008:9, after which a reverse development took place. The real exchange rate in the middle panel shows how the dollar/pound rate has fluctuated around the US-UK relative price in long persistent swings, except for a period after the Lehman Brothers collapse. As can be seen from the upper panel, the dollar/pound nominal 7 Tabor (2014) demonstrates using simulations that forecasting models with timevarying coeffi cients can generate a pronounced persistence in the variables. 7

8 exchange rate fell rapidly and substantially more than the US-UK relative price when the crisis peaked. The lower panel shows that the interest rate differential exhibited similar long and persistent swings as the real exchange rate. While these swings may look like I(1), formal testing in Section 7 shows that this was rejected when tested against I(2). This is often the case when the drift component (as in (3)) is tiny but very persistent. Such a variable, while near I(2), would be visually indistinguishable from an I(1) variable as the drift term might be diffi cult to catch sight of because of massive shortterm volatility. According to standard theory, model-based rational expectations can deviate from outcomes but, since agents are assumed to know the correct model, expectations will adjust back towards "true" model values. In such a world, expectations are purely adjusting and expectational errors would have no exogenous effect on the system, whereas imperfect knowledge based models do not make such strong assumptions. The question we ask here is whether it is exogenous shocks to the exchange rate or the interest rates that have pushed the variables away from their long-run equilibrium value. Previous studies of similar hypotheses, but without using observed interest rate expectations, have found that it is the twice cumulated interest rate shocks that are the long-run exogenous drivers (Juselius, 2006; Johansen et al., 2010; Stillwagon, 2017). With the present data on consensus forecasts, we can investigate whether it is actual or expected interest rates that have played an important role in these persistent swings. Juselius (2017b) shows that in a world of imperfect knowledge, forecast shocks have the ability to change the data generating process and may play an exogenous role in the model. Figure 2 shows the forecast errors, i t+3 i e t+3 t (blue line), where ie t+3 t is the consensus forecast, compared to the forecast error of a simple random walk prediction where i e t+3 t = i t (red line). In the case of the US (upper panel), the consensus forecast seems to closely follow the random walk forecast except for the period from mid 2004 to 2007 when the consensus forecasts were more precise 8. In the case of the UK (lower panel), the consensus forecasts may have done slightly worse than the random walk. But, except for large forecast failures of the US rate at the beginning of 2008 and several failures in connection with the Lehman Brothers bankruptcy in both countries, the 8 This period, however, coincided with the Fed s rather mechanical tightening of the monetary cycle. 8

9 0.80 The dollar pound rate The relative prices The deviation from purchasing power parity The 3m interest rate spread Figure 1: The graphs of the relative US-UK price together with the nominal dollar/pound exchange rate (upper panel), the deviation from the long-run purchasing power parity level (middle panel) and the US-UK interest rate differential (lower panel). forecast errors seem moderately sized. The mean squared prediction error relative to the random walk variance was 0.63 for the US and 1.22 for the UK supporting the deductions from the visual images. 4 Model specification Juselius (2017a and 2017b) derives theory-consistent CVAR scenarios for a rational expectations based contra an imperfect knowledge based monetary model for the determination of the exchange rate. In the former model, the nominal exchange rate and the interest rates are I(1), whereas in the latter model they are near I(2). In both models, price levels are I(2) and the VAR can encompass both hypotheses. It is convenient to formulate our empirical VAR in acceleration rates, changes and levels: 9

10 D3US3m FEUS3m D3UK3m FEUK3m Figure 2: The graphs of the forecast errors for the US/UK 3 months treasury bill rates using consensus forecasts and using a random walk. 2 x t = Γ 1 2 x t 1 + Γ x t 1 + Πx t 1 + ΦD t + ε t. (5) where the vector of economic variables x t = [i e US,t, ie UK,t, i US,t, i UK,t, s, p US,t, p UK,t ] and i, i e t stands for a three-month interest rate with a superscript e denoting an expectation made at time t for three-months ahead, s stands for the log of the nominal US dollar per British pound exchange rate, and p t for the log of CPI. The vector of dummy variables D t = [D s,t, D p,t, D tr,t ] contains a step dummy, D s,08,t = 1 for t = 2008:9-2013:7, 0 otherwise, to control for the equilibrium mean shift occurring at 2008:9 after the Lehman Brothers collapse; nine impulse dummies, D p,xx = 1 for t = xx, 0 otherwise, effective for 2002:11, 2003:01, 2003:07, 2004:07, 2005:03, 2005:09, 2008:01, 2008:04, 2009:01 to control for extraordinary large changes in any of the variables; and three transitory dummies D p,xx effective for 2008:9, 2008:10, 2008:11 to control for the extraordinary wild fluctuations after the Lehman Brothers collapse. 9 Ignoring these outlying observations is likely to cause residual 9 Except for the Lehman Brothers dummies, all dummies enter exclusively at time t (i.e without lags) and hence only control for large unanticipated shocks (given the chosen 10

11 skewness and autocorrelation which may bias the estimates. Gonzalo (1994) documents through simulations that cointegration performs poorly in the presence of skewness and serial correlation, but is robust to many other forms of misspecification. The model also includes 11 seasonal dummies, S t. After controlling for the dummy variables mentioned above, the interest rate residuals were found to be symmetrical. Nonetheless, they were clearly fat-tailed and normality was rejected due to excess kurtosis rather than skewness. 10 The sample covers the period 2001:3 to 2013:7, and the beginning of the sample is limited by the availability of the interest rate forecast data. The hypothesis that x t is I(1), i.e. there are unit roots in the data, is formulated as a reduced rank restriction on Π = αβ where α and β are p r where p represents the number of variables in the information set and r is the rank of Π. The β vectors define long-run, stationary equilibrium relationships among the variables and the α vectors describe how the system adjusts (error-corrects) to the disequilibrium β x t. If the differenced process also exhibits strong persistence consistent with x t I(1) and, hence, x t I(2), then (a linear transformation of) Γ also has reduced rank. This is formulated as an additional reduced rank hypothesis, α Γβ = ξη, where ξ, η are (p r) s 1 and α, β are orthogonal complements of α, β respectively (Johansen 1992, 1995). The total number of stochastic trends, (p r), are divided into s 1 trends of order I(1) and s 2 of order I(2). Note that the I(1) reduced rank condition is associated with the levels of the variables, whereas the I(2) condition with the differenced variables. The intuition is that the differenced process also contains unit roots when data are I(2). Because the Γ matrix is no longer unrestricted as in the I(1) model, Johansen (1997) suggested a different parameterization more suitable for likelihood based inference: 11 information set) at time t. At time t + 1 they are no longer unanticipated and become part of the model dynamics. 10 While the former is less problematic for the estimation (see Gonzalo, 1994), the VAR model is therefore only a rough approximation of the true data generating process. 11 Doornik (2017) discusses computational issues in this model as well as in a few optional representations. 11

12 2 x t = α β β s β trend x t 1 D s,08,t 1 t 1 + d d p d 0 x t 1 D p,t 1 +ζ τ τ p τ 0 t = 1,..., T x t 1 D p 1 + ΦD t + ε t, (6) The parameterization includes a linear trend and a step dummy in the cointegration relations, thus allowing for the possibility of trend-stationary relations with an equilibrium mean shift at the time of the Lehman Brothers collapse. The relation in the hard brackets corresponds to the cointegrated relation, β x t 1 + d x t 1, with β = [β, β s, β trend ], d = [d, d p, d 0 ] and x t = [x t, D s,08, t]. It describes a situation where the deviations from a long-run static equilibrium, β x t, is a (near) I(1) process and, therefore, has to be combined with the differenced process, d x t, to become stationary. Such a relation can often be interpreted as a dynamic rather than static equilibrium relation typical of the I(1) model. The relation in soft brackets, ζ τ x t 1, where τ = [τ, τ p, τ 0 ] and τ = [β, β 1 ], describes medium-run relations among the differenced variables. The cointegration relations τ 1x t, which take the process from I(2) to I(1), consist of r relations, β x t, and s 1 relations, β 1x t. The difference between the two is that the former can be stationary either by polynomial cointegration ( β x t + d x t I(0)), or by differencing ( β x t I(0)), whereas the latter only by differencing ( β 1 x t I(0)). The economic interpretation of τ 1 x t is not always straightforward, but Juselius and Assenmacher (2017) and Juselius (2017b) give them an interpretation as medium-run relationships among changes in the process describing momentum trading along the trend in the foreign exchange market. The latter can be due to technical trading and/or because there are a majority of chartist among financial speculators (Hommes 2006). 12

13 Table 1: Determination of the rank indices Trace test statistics for the I(2) model p r r s 2 = 4 s 2 = 3 s 2 = 2 s 2 = 1 s 2 = [0.04] 95.4 [0.03] 59.0 [0.28] [0.79] 78.9 [0.03] 41.6 [0.41] 17.4 [0.88] [0.57] 70.1 [0.08] 35.0 [0.27] 12.8 [0.76] 1.8 [0.97] The seven largest roots of the characteristic polynomial Unrestricted VAR r = 3 s 1 = 4 s 2 = r = 4 s 1 = 1 s 2 = Rank determination The first step in the empirical analysis is to estimate (5), ensure that it is well-specified, and then determine the reduced rank indices. 12 The task is to determine the number of long-run relations, r, and then the number of s 2 trends among the total number of stochastic trends p r = s 1 + s 2. As a basis for these choices Table 1 reports the trace test by Rahbek and Nielsen (2007) and the value of the seven largest characteristic roots in the system. The latter give a rough indication of the total number of (near) unit roots (s 1 + 2s 2 ) in the VAR model. 13 Table 1 shows that the unrestricted VAR has two roots almost on the unit circle and three large near unit roots. Hence, the choice of {r, s 1, s 2 } should be consistent with five unit roots i.e. s 1 + 2s 2 = 5. The upper part of Table 1 reports the test statistics of the joint tests of r, s 1 and s 2. The tests of r = 0, 1, 2 were all strongly rejected, so we have omitted the first three rows from the table. The test procedure starts with the most restricted model (r = 3, s 1 = 0, s 2 = 4) in the upper left hand corner, continues to the end of the row (r = 3, s 1 = 4, s 2 = 0), and proceeds similarly row-wise from left to right until the first failure to reject at (r = 3, s 1 = 4, s 2 = 0). However, this case has a fairly low p-value (0.08), corresponds to only four unit roots, and hence would leave one large root 12 All estimates and tests in Section 5-10 have been obtained by using a Beta version of the software package CATS 3 (Doornik and Juselius, 2017). 13 An I(1) trend creates one unit root, while an I(2) trend creates a double unit root. 13

14 Table 2: Tests of no levels feedback and unit vector in alpha Test of a zero row in α r DGF 5%C.V. i e US,t i e UK,t i US,t i UK,t p US,t p UK,t s [0.77] [0.07] [0.10] = [0.59] [0.47] = Test of unit vector in α 8.39 [0.08] 8.36 [0.04] 7.32 [0.03] 5.83 [0.21] 5.46 [0.14] 3.67 [0.16] [0.13] 6.66 [0.08] 6.64 [0.04] 9.03 [0.06] 9.59 [0.09] [0.02] 7.46 [0.11] [0.01] [0.10] 0.99 [0.80] [0.03] [0.38] 0.15 [0.93] (0.89) in the model. The next case (r = 4, s 1 = 1, s 2 = 2) has a p-value of 0.28 and accounts for all five large roots in the model with the two largest unrestricted roots being 0.74 and 0.68, i.e. identical to the corresponding roots in the unrestricted model. It describes four polynomially cointegrating relations β x t + d x t and one relation, β,1 x t 1, which can only become stationary by differencing. This is our preferred model. 6 Are expectations pulling or pushing? To get a first idea of which variables, if any, have primarily pushed the system away from equilibrium and which variables, if any, have pulled the system back, we report the results of two simple tests in Table 2. The first test is similar to the weak exogeneity test in the I(1) model and is formulated as a zero row in α. If not rejected, the variable in question is not significantly affected by the disequilibrium β x t + d x t. Hence, it is sometimes called a test for "no levels feedback". But, as shown in Rahbek and Paruolo (1999), a zero row in α is not suffi cient for concluding that the variable is weakly exogenous. The latter requires an additional test of a corresponding zero row in the adjustment matrix ζ of the medium-run relations τ x t. For the preferred choice, r = 4, the US interest rate forecast is a candidate for no levels feed-back with a p-value of The individual tests of US and UK prices are also borderline not rejected but with quite small p-values. The 14

15 joint tests of every combination of the three variables were rejected. As a sensitivity check, we also report the results for r = 3, 5. The conclusions are robust to either choice, except that a zero row in α in the nominal exchange rate could not be rejected when r = 3. This shows that the nominal exchange rate is significantly adjusting exclusively to the fourth cointegration relation. A unit vector in α implies that the variable in question has been purely adjusting to the long-run relations, and signifies that the variable has the characteristic of an endogenous variable in this system. For the preferred choice of r = 4 the two interest rates and the nominal exchange rate cannot be rejected as being purely adjusting. This result is robust to the choice of r = 5 or r = 3. However, when testing them jointly, every possible combination of the three was rejected. Based on this, the US consensus forecasts seem primarily to be pushing, whereas the actual interest rates and the nominal exchange rate are primarily adjusting. As the subsequent results will show, both actual and expected interest rates exhibit significant feed-back effects from the medium-run relations τ x t. 7 Integration properties of the data To derive the time-series properties of nominal interest rates assuming that the market is demanding an uncertainty premium for holding any of the two currencies, Juselius (2017b) suggests the following data generating process: i j,t = ω j,t + ε j,t, and ε j,t Niid(0, σ 2 ε,j) j = 1, 2 (7) where ε j,t stands for an unanticipated interest rate shock and ω j,t is a drift term measuring a change in the domestic uncertainty premium assumed to follow an AR(1) process similarly as in (3): ω j,t = ρ t,j ω j,t 1 + ε ω j,t, and ε ω j,t (0, σ 2 ε ω,j) j = 1, 2 (8) where ρ t,j 1.0 in periods when the P P P gap is moderately sized and ρ t,j 1.0 when the gap is large. Since, according to Frydman and Goldberg (2007, 2011), the periods when ρ t,j 1.0 are likely to be short compared to the ones when ρ t,j 1.0, the average ρ j is assumed to be close to 1.0 so that ω j,t can be considered a near I(1) process. Integrating (7) over t gives the data-generating process for the interest 15

16 Table 3: Testing hypotheses of I(1) versus I(2) i e US i e UK i US i UK s p US p UK D s,08 t χ 2 (v) p val H 1 : (2) 0.00 H 2 : (2) 0.01 H 3 : (2) 0.02 H 4 : (4) 0.00 H 5 : (4) 0.01 H 6 : (4) 0.00 H 7 : (4) 0.00 H 8 : (4) 0.15 H 9 : (4) 0.00 H 10 : (4) 0.00 H 11 : (4) 0.00 rate: i j,t = i j,0 + t ε j,s + s=1 t ω j,s, j = 1, 2. (9) where t s=1 ω j,s = up t. Hence, the nominal interest rate is near I(2) under the imperfect knowledge assumption (8), but I(1) when corrected for the uncertainty premium assumed to capture persistent swings of shorter and longer durations typical of nominal interest rates. Starting from (9), Juselius (2017b) derives the time-series properties of prices and the nominal exchange rate and shows that the deviations from basic parities such as the PPP, the Fisher parities, and the terms spreads are all likely to be near I(2). Thus, the parities are assumed to be one degree more persistent under imperfect knowledge than under REH, where they would generally be stationary, or at most near I(1) (Juselius, 2017a). Table 3 report tests of whether the variables, as well as relevant transformations of them, are at most I(1). For example H 1 = [0, 0, 0, 0, 0, 1, 1, 0, 0] is a test of whether the relative price is a "unit vector in τ". As τ x t I(1), a rejection implies that p 1 p 2 does not cointegrate from I(2) to I(1) and, therefore, has to be considered I(2). The hypotheses that the relative price, the nominal and the real exchange rate, expected and actual interest rates, and expected and actual interest rate s=1 16

17 differentials are at most I(1) are all rejected. By contrast, the hypothesis H 8, that the differential between the US interest rate and its 3 months ahead expected value is at most I(1), cannot be rejected with a p-value of The latter is consistent with Assumption A in Juselius (2017b) which states that (x e t+1 t x t) I(1) when x t I(2), i.e. when a variable has a unit root in its growth rate, the expected change will also have a unit root. The corresponding UK differential in H 9 is, however, more persistent and rejected as at most I(1). This in itself indicates some misalignment between the underlying process of the variable and its expectation. Altogether, the results suggest a considerable degree of persistence in the nominal exchange rate, prices, interest rates, and co-movements among them. This is broadly consistent with imperfect knowledge economics and lends support to the conclusion in Johansen et al. (2010) that the change of the variables is a highly persistent near I(1) process due to the shocks to the drift term in (8). 8 The pulling force As discussed in Section 4, polynomial cointegration combines levels and differences of the variables into a dynamic long-run relationship, β x t 1 + d x t 1, where the components are individually I(1) but cointegrate to I(0). Juselius and Assenmacher (2017) argues that the coeffi cients α and d can often be interpreted as two levels of equilibrium correction: the α adjustment describing how 2 x t adjust to the dynamic equilibrium relations, β x t + d x t and the d adjustment describing how x t adjusts to the disequilibrium β x t. The signs of β, d, and α determine whether the variable x i,t is error increasing or error correcting in the medium and/or the long run. 14 If α ij β ij < 0 or/and α ij d ij < 0, then the acceleration rate, 2 x i,t, is equilibrium correcting to ( β j x t + d j x t ); if d ij β ij > 0 (given α ij 0), then x i,t, is equilibrium error correcting to β j x t ; if ζ ij β ij < 0 then 2 x i,t is equilibrium correcting to β j x t 1. In all other significant cases, the system is equilibrium error increasing. Note, however, that in a stable system like the present one where all characteristic roots are either inside or on the unit circle, any error-increasing behavior has to be counteracted by error-correcting behavior. 14 The interpretation of d as an adjustment coeffi cient is, however, conditional on α 0. 17

18 Since there are several ways of statistically identifying the coeffi cients β, d, and α, the error-increasing and -correcting behavior discussed above may differ among differently identified structures. But, if a variable is subject to self-reinforcing feed-back behavior, then this will show up as errorincreasing adjustment somewhere in the system. Thus, finding evidence of error-increasing behavior in the model is a sign of self-reinforcing behavior. As long as the long-run structure satisfies the conditions for formal, empirical and economic identification (Johansen and Juselius, 1994) it represents a specific perspective on the empirical problem. 15 In the present model with four cointegration relations there are potentially several cointegration structures which satisfy formal and empirical identification. However, not many, if any at all, may pass the condition for economic identification. In the present application, our primary goal has been to identify a relation that could be interpreted in the framework of an uncertainty adjusted UIP relation. Such a framework was developed in Juselius (2017b) which showed that the theory of imperfect knowledge is consistent with not just the uncertainty adjusted UIP relation, but also a number of additional "reduced-form" type of relations. 16 Our choice of identified structure contains the hypothetical uncertainty adjusted UIP relation (4) and three relations associating interest rates and their forecasts with price and exchange rate movements. While the former relation is of primary interest, the latter ones play an important role for the analysis of error-correcting and error-increasing behavior of the system variables. Table 4 reports the estimates of four dynamic long-run relations subject to eight over-identifying restrictions accepted with a p-value of The d coeffi cients are determined to be proportional to τ, thereby excluding any stationary terms in d x t. This orthogonality condition implies uniqueness, but at the price of fixing the coeffi cients. Hence, it is not possible to impose additional restrictions on d without violating the orthogonality condition. But, to improve readability, insignificant coeffi cients (t < 1.5) have instead 15 Of course, different choices of identifying restrictions lead to different long-run structures.this is similar to the simultaneous equation model for which the reduced form often can be consistent with not just one but several structural forms. For a comparison of identification in the cointegrated VAR and the Cowles Commission models see Juselius (2015). 16 The number of additional relations depends on the dimension of the VAR system. 17 The standard errors of β are derived in Johansen (1997) and those of d by the delta method in Doornik (2016). 18

19 been replaced by a *. To distinguish between equilibrium error-increasing (positive feed-back) and error-correcting behavior (negative feed-back) significant α, d and ζ coeffi cients that are describing error-increasing behavior are in bold face. The first β-relation shows that the difference between the forecasted and actual interest rate differential has been positively co-moving with the PPP gap. Thus, the consensus forecast differential is cointegrated with the expected excess return corrected for an uncertainty premium, proxied by the PPP gap, and a risk premium proxied by d 1 x t. Altogether, the results provide remarkable support for the imperfect knowledge based monetary model as given by (4). The second β-relation connects the US interest rate to its forecast and the relative US-UK price, while the third relation connects the UK interest rate to its forecast and the UK dollar price. The estimates show that the US consensus forecast is almost proportional to the actual interest rate, whereas the corresponding UK rates differ somewhat more. The fourth β-relation describes a homogeneous relationship between actual interest rates and their consensus forecasts as a function of the two CPI prices. As the two consensus forecasts are closely co-moving with the two interest rates, this may be evidence of a common (world) interest rate level. This differential between the forecasted and actual interest rates is co-moving with relative prices. The estimated α 1 coeffi cients show that it is the UK three-month interest rate that has adjusted to the first relation in an error-correcting manner, whereas the nominal exchange rate has done so in an error-increasing manner. This is consistent with the assumption in Section 2 that speculative behavior tends to drive the nominal exchange rate away from its long-run fundamental value. That prices do not significantly error-correct means that the ppp-gap, i.e. the uncertainty premium, will tend to increase as the nominal exchange rate moves in long persistent swings. Only the UK interest rate is significantly adjusting to this relation in an error-correcting manner. The estimate of α 2 shows that both US and UK prices have been equilibrium error increasing to the second β-relation whereas not the nominal exchange rate. In contrast, the estimate of α 3 shows that the nominal exchange rate is error-correcting to the third β relation but not prices, whereas the UK interest rate is error-increasing and its forecast is error-correcting. The estimate of α 4 shows that the nominal exchange rate is significantly adjusting to β 4, now in an error-increasing manner. Thus, when the consensus 19

20 Table 4: An identified structure of polynomially cointegrating relations i e US,t i e UK,t i US,t i UK,t p US,t p UK,t s t Ds t 10 3 Test of over-identifying restrictions χ 2 (8) = 6.23[0.62] β [ 11.8] d ( 2.6) ( 2.2) 0.06 ( 2.6) ( 3.0) α [ 5.8] β [ 177.5] d ( 6.0) (5.8) 0.12 ( 6.0) α (2.8) 0.04 (2.5) [ 6.7] ( 5.0) 2.31 [4.2] β [ 45.2] d (3.5) α ( 2.5) β [61.2] d ( 7.0) (6.3) 1.52 [ 6.4] 0.87 [ 66.2] 0.26 ( 7.0) 0.84 [ 60.4] ( 7.2) [11.8] 0.27 (2.6) [11.8] 79.2 [5.2] 0.08 (6.0) 4.54 ( 2.0) [6.7] 0.51 (6.0) 2.85 [1.7] [ 13.2] [ 13.3] 0.04 (8.6) 0.03 [17.0] (4.3) 0.04 (3.5) [ 5.7] [ 13.2] 0.02 (1.8) 56.2 [5.6] 0.03 [ 9.3] 0.17 (7.0) 0.02 [6.2] 1.14 (7.1) 0.76 [ 3.1] 0.01 (4.0) [12.8] 0.04 ( 11.3) (4.9) 0.04 (12.8) (1.9) 47.9 [6.2] 0.02 (3.4) 0.08 (14.6) α [ 3.0] (1.7) β Note: The standard errors of β are derived in Johansen (1997) and those of d by the delta method in Doornik and Juselius (2017). t-ratios are in the brackets. Bolded coeffi cients indicate error-increasing behavior 20

21 Table 5: The estimated adjustment coeffi cients to taudxt ζ 1 ( β 1 x t ) ζ 2 ( β 2 x t ) ζ 3 ( β 3 x t ) ζ 4 ( β 4 x t ) ζ 5 ( β 1 x t ) 2 i e US,t 0.16 [4.3] 2 i e UK,t 2.29 [ 2.0] 2 i US,t 7.12 [ 6.75] 2 i UK,t 6.88 [ 2.32] 2 p US,t [ 4.60] 2 p UK,t [1.72] [ 7.08] [ 2.68] [ 5.15] [1.83] 3.80 [5.64] 2.42 [1.8] 6.17 [7.75] 8.15 [3.64] [3.44] 9.08 [ 1.50] [6.03] [ 1.73] 0.19 [ 4.3] 0.37 [ 14.10] 0.31 [ 4.17] 2 s t Note: Bolded coeffi cients indicate error-increasing behavior [2.48] forecasts deviate from the level of actual interest rates, financial actors tend to behave as trend followers in the nominal exchange rate. It is notable, however, that the magnitude of the price coeffi cients is much smaller than those to the exchange rate, suggesting that price changes have been too small to compensate for the large changes in the nominal exchange rate, thus pushing the nominal exchange rate away from long-run equilibrium PPP values. Figure 1 illustrated the resulting real exchange rate persistence that Krugman (1986) discussed in the context of pricing-to-market. Altogether, neither the US nor the UK consensus forecast shows any significant α adjustment consistent with the test result in Table 2. Instead, it is the UK rate that has been significantly adjusting to the β-relations in an error-correcting manner to the first two and an error-increasing manner to the last two. A similar pattern characterizes the nominal exchange rate which is equilibrium error correcting to the second and third relation and error-increasing to the first and the last relation. This illustrates how equilibrium error increasing behavior in one part of the system is compensated by error-correcting behavior in another part, thereby maintaining stability of the system. Table 5 reports the estimated adjustment coeffi cients ζ of the mediumrun relations, τ x t. The first four columns correspond to the differenced β relations and the fifth column to the β 1 x t relation. These medium run relations are all describing relationships among changes in the process and 21

22 are, therefore, likely to capture how interest rates, exchange rates and prices respond to short-run price movements in the market. A mixture of errorincreasing and error-correcting behavior (the former indicated in bold face) may be interpreted as evidence of momentum trading typical of speculative markets. The error-increasing behavior is most dominant in the adjustment to β 2 x t (describing a change in the disequilibrium between the US interest rate and its forecast), whereas the adjustment to the remaining relations is primarily error-correcting. While the US interest rate forecast is neither significantly adjusting to any of the four relations β x t, nor to the polynomially cointegrated relations β x t + d x t, it is now significantly error-increasing to β 1 x t implying a minor violation of the condition for weak exogeneity. Interestingly, while the nominal exchange rate was found to strongly adjust to the levels relations, β x t + d x t, it has not been significantly affected by the changes in the long-run disequilibria, τ x t. By contrast, the US interest rate exhibits the opposite adjustment pattern: while it did not adjust significantly to the levels relations, it has been significantly adjusting to all five medium-run changes relations. To summarize: over the medium run the nominal exchange rate has been pushing the foreign exchange market, while interest rates have followed suit, whereas over the long run the nominal exchange rate has adjusted to the levels relations, β x t + δ x t, while interest rate forecasts have been pushing. 9 The long persistent swings and the estimated I(2) trends The moving average representation of (5) subject to the two reduced rank restrictions expresses the variables x t as a function of once and twice cumulated errors and deterministic terms as well as stationary components: x t = C 2 t j=1 i=1 j t (ε i + ΦD i + µ 0 ) + C 1 (ε j + ΦD j + µ 0 ) j=1 +C (L)(ε t + ΦD t + µ 0 ) + A + Bt. (10) As such it is a summary description of the exogenous forces that have generated the data x t. Most importantly, it shows how permanent exogenous 22

23 Table 6: The estimated common trends and their loadings i e US,t i e UK,t i US,t i UK,t p US,t p f,t s The estimated loadings to the I(2) trends β 2, β 2, The estimated I(2) trends α 2, α 2, shocks to the system have given rise to stochastic trends which push the variables into nonstationary trajectories. The MA parameters describing these pushing forces are complicated functions of the parameters in (6) (see Johansen, 1992, for a formal definition). However, for the purpose of this paper, it suffi ces to focus on C 2 = β 2 α 2 where α 2 is informative about the sources of exogenous shocks and β 2 about the weight with which the I(2) trends, α t j 2 j=1 i=1 ε i, load into the variables x t. Table 6 reports the estimates. The first I(2) trend, generated from the twice cumulated shocks to the spread between the US three-month interest rate and its forecast with a smaller weight to the actual rate, is primarily loading into the nominal exchange rate. The second I(2) trend, generated from the twice cumulated shocks to the UK consensus forecast relative to the US interest and its forecast, loads into the interest rates, prices and the nominal exchange rate. Thus, the US expectational shocks seem more exogenous than the UK ones, the effect of which are important only in relation to the rest of the system. Altogether, the results suggest that expectational shocks have a significant effect on the level of interest rates as well as prices in both countries. This provides some evidence for the hypothesis that it is shocks to the consensus forecasts that have driven prices away and towards long-run benchmark values in long persistent swings. To further check this hypothesis, we re-estimated model (5) but now as a partial model conditional on the two consensus forecasts. The rank tests showed no significant evidence of I(2), the closest case (r = 4, s 2 = 1) was rejected with a p-value of 0.025, whereas (r = 4, s 1 = 1) could not be rejected based on a p-value of When the partial model was re- 23

24 estimated, now assuming exogeneity of the two interest rates, the case (r = 4, s 2 = 1) could not be rejected with a p-value of While these results are only indicative (neither actual, nor forecasted interest rates were found to be truly exogenous) they seem to suggest that the consensus forecasts are the ones that contain the most relevant information regarding the long persistent movements in the data. 10 Conclusions A number of questions were raised in the introduction which we have tried to address using the rich structure of the CVAR model for I(2) data. First of all, the results gave fairly strong support to the hypothesis that it is the interest rate expectations, measured as consensus forecasts by professional forecasters, that have been pushing the interest rates and the exchange rate in the long run. The results also showed that the shocks to the US consensus forecasts have been more significant for the long persistent swings typical of the dollar/pound market than the UK one, pointing to the dominant role of the dollar in the foreign currency markets. While neither the consensus forecasts nor the US interest rate exhibited significant levels feedback effects, they showed significant adjustment to the changes in the disequilibria. Interestingly, the nominal exchange rate showed the opposite reaction pattern. Thus, over the medium run, changes in the nominal exchange rate have been pushing the foreign currency market (consistent with behavioral models of extrapolative expectations) while interest rates have followed suit. By contrast, the nominal exchange rate has been adjusting in the long-run while interest rate expectations have been pushing. This autonomous role for interest rate expectations is congruent with models emphasizing imperfect knowledge. All variables were found to be (near) I(2) exhibiting a pronounced persistence consistent with an imperfect knowledge based monetary model for the exchange rate. This conclusion was further confirmed by the strong evidence of self-reinforcing feed-back mechanisms in the dynamic adjustment structure. The latter was evidenced by a combination of error increasing and correcting behavior (positive and negative feedback) both in the medium run and the long run. Altogether the results suggest that positive feedback between market expectations and outcomes has played a crucial role in generating the long per- 24