Forward interest rate premium and asymmetric adjustment: Evidence from 16 countries

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Int. Fin. Markets, Inst. and Money 19 (2009) 258–273

Forward interest rate premium and asymmetric adjustment: Evidence from 16 countries David G. McMillan ∗ School of Management, University of St Andrews, St Andrews, KY16 9SS, UK Received 17 March 2006; accepted 31 December 2007 Available online 9 January 2008

Abstract This paper examines the ability of the forward premium to provide an unbiased estimate of the future spot rate allowing for potential asymmetries. Extant evidence suggests that forward rates provide a biased predictor of future spot rates. Examining the forward premium for 16 countries, only for 2 countries does the linear expectations hypothesis holds. For the remaining countries, results generally support the view that the larger the forward premium the better a predictor for future spot rates it is, however, this result is not unique across all countries. Furthermore, although the asymmetric model improves data fit over the linear model, only in four cases does the model support an unbiased predictor interpretation. Further research is therefore required to understand the nature of this relationship, not least given the importance of correctly priced forward and long rates in terms of expected returns to future investments and the conduct of monetary policy. © 2008 Elsevier B.V. All rights reserved. JEL classification: C22; G12 Keywords: Asymmetric adjustment; Forward premium; Interest rates

1. Introduction A major research interest in the empirical analysis of interest rate dynamics is the relationship between forward and spot rates and whether forward rates are an unbiased predictor of the future spot rate. The standard expectation hypothesis approach argues that the forward rate indeed proxies for the expected future spot rate, subject to a constant liquidity or term premium. The general ∗

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consensus view from the literature is, however, that forward rates provided a biased predictor of the expected future spot rate, and whilst this could be consistent with market inefficiencies and systematic errors in the formation of expectations, typically it is believed this result arises from time-variation in the term premium (for a range of studies across different maturities and series, see for example, Shiller et al., 1983; Mankiw and Miron, 1986; Fama and Bliss, 1987; Hardouvelis, 1988; Mishkin, 1988; Simon, 1989; Cook and Hahn, 1990; Campbell and Shiller, 1991; Campbell, 1995; Roberds and Whiteman, 1999; Cooray, 2003). Understanding the nature of the relationship between forward and spot rates is an important issue. For instance, it has been shown that the term structure of interest rates is a useful predictor for the future movement of important economic variables, not least of all, short-term interest rates (Fama and Bliss, 1987; Hardouvelis, 1988), inflation (Mishkin, 1990; Fama, 1990), and economic activity (Harvey, 1988; Estrella and Hardouvelis, 1991; Hu, 1993). Moreover, correctly priced forward rates and long rates are important in terms of both the expected returns to future investments (such that if the forward rate was a biased predictor of the spot rate then this may impact the efficient allocation of resources and investments) and the conduct of monetary policy (whereby policy affects the long rates through altering expectations of future short rates). This paper attempts to reconsider the relationship between the forward and spot rates in the light of a concurrent line of empirical research that suggests that many financial assets are characterised by asymmetric or non-linear behaviour. More specifically, in the context of interest rates, either asymmetric or non-linear dynamics have been reported by, amongst others, Balke and Fomby (1997), Enders and Granger (1998), Tzavalis (1999), Andreou et al. (2000), van Dijk and Franses (2000), Enders and Siklos (2001), Coakley and Fuertes (2002) and McMillan (2004). In particular, this study considers the logistic smooth-transition model that has been supported in several of the papers cited above. This smooth-transition model, which captures asymmetric behaviour, can be viewed as exhibiting two extreme regimes that correspond to ‘large’ positive and negative values of the forward premium, and a middle transition regime where the forward premium is ‘small’ and of either sign. Asymmetric and potential non-linear behaviour within the context of interest rates can be motivated through the interactions within arbitrage relationships as well as the influence of monetary policy (notably when examining short rate dynamics). More specifically, where central banks are primarily interested in an inflation target they may respond asymmetrically in periods of rising and falling inflation (or inflationary expectations).1 Indeed, the results from several of the studies cited above generally support this view of asymmetric intervention through reporting evidence that adjustment of the short rate, which is effectively set by the central bank, is quicker when it is exceeded by the long rate, which is indicative of rising future inflation, than in the converse case. Furthermore, of course, the bond market is an arbitrage market and bonds of different maturity will be linked by an arbitrage relationship, such that we would expect any deviations from an equilibrium position to be arbitraged away. That is, should the one-period return on bonds of different maturity drift apart, for example, should the return on a short rate bond exceed that of a long rate bond, then arbitrageurs will buy the short bond and sell the long bond, so pushing up the price of the short bond and pushing down the price of long bond, until their respective one-period returns are equalised subject to any liquidity or other risk premium on the long bond. Furthermore, where arbitrageurs’ actions may be limited due to the presence of market frictions such as transactions costs, then non-linearities can arise. Indeed in the context of exchange rate

1

See, for example, Murchison and Siklos (1999) and Svensson (1999).

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dynamics more formal models of arbitrage have demonstrated the possibility of non-linearities arising from the presence of transactions costs (Baldwin, 1990; Dumas, 1992, 1994; Sercu et al., 1995). The remainder of the paper is as follows: Section 2 presents the theoretical arguments underlying the paper. Section 3 presents the LSTR model, which is of primary interest in that it captures the potential asymmetry in the behaviour of interest rates, but we also consider alternative empirical models, including one which allows for non-linear but symmetric adjustment. Section 4 introduces the data and presents our empirical results. Section 5 concludes. 2. Theoretical background Let R(t,m) be the continuously compounded yield to maturity of a m period pure discount (zero-coupon) bond at time t. In a standard ‘no arbitrage’ view this implies:   m 1  R(t, m) = Et {R(t + i − 1, 1)} + L(t, m) (1) m i=1

where Et denotes expectations at time t and L(t,m) represents the term premia. Eq. (1) can be thought of as the fundamental relationship that states that the long rate is a weighted average of current and expected future short rates. The forward rate obtained from this relationship is implied by the interest rate difference between the current yield on the long bond and the current yield on the shorter term bond (or yield spread) that would leave an investor indifferent between two investments strategies based on the different bonds. That is, the forward rate serves as a proxy for market expectations of the future short rate and is given by: f (t, 1) = Et (Rt+m ) + L(t, m)

(2)

where f(t,1) represents the forward rate for period m on a one-period bond. This equation shows that together, the expected future short rate and the expected term premium comprise the expected return to investing in a longer term bond, relative to investing in short-term bonds over the longer term maturity. Hence, forward rates are proxies for expected returns to future investments. Moreover, since forward rates reflect the behaviour of the yield spread, the relationship between short and forward rates are key to understanding the term structure of interest rates, and the effects of any changes in short rates on long-term interest rates are transmitted by forward rates. From Eq. (2) we can define the forward premium as the difference between the current forward rate and the current short rate, and which can be decomposed into the expected change in the short rate and the term premium: f (t, 1) − R(t, 1) = Et (Rt+m ) + L(t, m) = (Et (Rt+m ) − Rt ) + (f (t, 1) − Et (Rt+m )) (3) where f(t,1)−Et (Rt+m ) represents the term premium. As all the terms on the right-hand side of Eq. (3) can be interpreted as ‘quasi-difference’ terms, they are expected to be stationary. Under the pure form of the expectations hypothesis the term premium component will be zero, such that all movement in the forward premium will arise due to movement in expected future short rates. Assuming rational expectations, i.e. Rt+m = Et (Rt+m ) + νt , where νt is a white noise error term, we can determine the following regression: Rt+m − Rt = α1 + α2 (ft − Rt ) + υt

(4)

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where should the expectations hypothesis of the term structure hold, α1 will equal zero and α2 equal one. Allowance for a constant term premium would give rise to a non-zero constant term, whilst should the slope coefficient, α2 , not equal one, then this is suggestive of a time-varying term premium. As noted in Section 1, there is a substantial amount of empirical evidence that suggests that the conditions for the pure expectations hypothesis do not hold, and that α2 is not equal to one (in addition to the papers cited above, see also the review paper of Cook and Hahn, 1990). 3. Logistic smooth-transition model In order to capture the potential asymmetric dynamics within the relationship between forward and spot interest rates, we implement the logistic smooth-transition (LSTR) model.2 This model essentially distinguishes between two extreme regimes defined as whether the threshold variable is above or below a threshold value, and where the transition between these two regimes is smooth (alternatively it can be thought of as a continuum of regimes between the two extremes). Moreover, whilst an alternative modelling approach would be utilise a standard threshold regression, where the transition between regimes is abrupt, the smooth adjustment is preferred on theoretical grounds as the threshold model assumes that all market agents act simultaneously. Where agents are faced by different constraints and costs, such that their action are not synchronised then a smoothtransition is more appropriate. The LSTR model for the forward premium is given by: Rt+m − Rt = [α1 + α2 (ft − Rt )] + [β1 + β2 (ft − Rt )]F (zt ) + ut

(5)

where F(zt ) is the transition function and zt the transition variable. More specifically, the logistic transition function is given by: F (zt ) = {1 + exp[−γ(zt − c)]}−1 ;

γ>0

(6)

where zt , the transition variable, is given by the forward premium, ft –Rt , γ is the smoothing parameter which governs the speed of transition between regimes of behaviour and c is the threshold parameter, the point at which the process changes regime.3 This function thus allows a smooth-transition between the differing dynamics of positive and negative values of zt , or more precisely values of zt above or below c. That is, the function allows the parameters to change monotonically with zt . As γ → ∞ then F(zt ) becomes as Heaviside function where F(zt ) = 0 if zt < c and F(zt ) = 1 if zt > c, and hence the LSTR model nest a threshold model. As γ → 0 then F(zt ) = 0.5 and the LSTR model reduces to a linear model. As noted above the LSTR model can be viewed as having two extreme regimes and a transition regime, where c represents the threshold between the two regimes. In the lower regime, that is

2 For further discussion of the smooth-transition class of models see Ter¨ asvirta (1998). Furthermore, in a recent paper examining forward exchange rate dynamics Baillie and Kilic¸ (2006) utilise the LSTR model, given similarities in the modelling approach to forward rates, although different in context, some of our discussion follows theirs. 3 As noted by Ter¨ asvirta (1998) amongst others the in order to improve convergence of the LSTR model the transition function is scaled by the standard deviation of the transition parameter as such: F (zt ) = {1 + exp[−γ(zt − c)/σzt ]}−1 ; γ > 0. Furthermore, as noted by Baillie and Kilic¸ (2006) this allows us to interpret the transition variable as the risk-adjusted forward premium. More generally, lags of the transition variable could also be considered, however, we follow Baillie and Kilic¸ (2006) in only reporting results for zt . Nevertheless, a lag of zt was considered with results, which are available upon request, typically poorer in term of model fit.

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where F(zt ) = 0 and zt < c, our model is given by: Rt+m − Rt = α1 + α2 (ft − Rt ) + ut

(7)

whilst in the upper regime where F(zt ) = 1 and zt > c our model is given by: Rt+m − Rt = (α1 + β1 ) + (α2 + β2 )(ft − Rt ) + ut

(8)

finally, whilst, where F(zt ) = 0.5 and zt = c our model is given by: Rt+m − Rt = (α1 + 0.5β1 ) + (α2 + 0.5β2 )(ft − Rt ) + ut

(9)

Hence, we are able to capture differing dynamic behaviour according to the sign and size of the forward premium. Moreover, we are able to capture regimes where the forward premium may act as an unbiased estimator and regimes where deviations from the equilibrium condition occur. As noted above, several authors have suggested that the asymmetric adjustment may arise due to the actions of monetary policy authorities that can impart asymmetric dynamics, where, for example, the monetary authorities act quicker in adjusting short rate in periods of rising long rates, due to their inflationary implications, than in periods of falling long rates (see for example Murchison and Siklos, 1999; Svensson, 1999; McMillan, 2004; Martin and Milas, 2004). In terms of Eq. (3) this rationale for asymmetry in the forward premium would be transmitted through expectations about future short rates. However, asymmetry in the forward premium may also arise through the term, or liquidity, premium, where investors require different premiums, that is, have different preferences for liquidity or have different risk considerations, in periods of rising and falling rates.4 In contrast, an alternative view that imparts non-linear, but symmetric, adjustment is given by the QLSTR model (Jansen and Ter¨asvirta, 1996). The quadratic–logistic function given by: F (zt ) = 1 + exp[−γ(zt − c1 )(zt − c2 )]−1 ;

γ>0

(10)

where if γ → 0 the model becomes linear, while if γ → ∞ the function is equal to one for zt < c1 and zt > c2 and zero in-between. This model thus captures differing behaviour of large and small values, and additionally, does allow for some asymmetry through a different threshold value for positive and negative values of the forward premium. Such non-linearities within hypothesised equilibrium relationships are typically introduced through the presence of transaction costs and other market frictions, which can limit the arbitrage process, at least until the benefits from engaging in trade outweigh the costs of trade. Such non-linear dynamics have been extensively discussed in the context of exchange rates, for example, Baldwin (1990), Dumas (1992) and Sercu et al. (1995) all discuss models whereby the introduction of transaction costs impart non-linear dynamics. While Lyons (2001) has suggested that arbitrageurs only enter the market if the deviation (as measured by the Sharpe ratio) passes some threshold designed to signal profitable trading opportunities. Further, in the context of the cost-of-carry model between equity index spot and futures Dwyer et al. (1996) and Brooks and Garrett (2002) have proposed that, due to the presence of transaction costs, deviations from the equilibrium relationship will only be arbitraged away once such deviations have become sufficiently large so that the benefits from engaging in trade out way the costs. Furthermore, Tse (2001) argues that given different constraints faced by different arbitrageurs a smooth-transition 4 One plausible scenario would be in periods of falling rates, where reinvestment risk is high, investors may be willing to pay a higher premium to hold long-term assets, than in periods of rising rates.

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back to equilibrium is more appropriate when aggregating over the market than the Heaviside (threshold) switching approach. Finally, in the specific context of interest rates Anderson (1997) similarly argues that arbitrageurs’ actions may be limited due to the presence of market frictions such as transactions costs, such that the arbitrage activity must be delayed until the benefits from engaging in trade outweigh the costs, and that the heterogeneous nature of arbitrageurs will result in a smooth-transition between regimes of behaviour. Whilst the above models rely on transaction costs to generate non-linearity, Shleifer (2000) has argued that limits to arbitrage may be in place, particularly where arbitrageurs have to borrow assets to put on their trades (a practise more common in interest rates markets than equity markets), then the effectiveness of arbitrage will be affected by the willingness of investors to loan funds. More specifically, where, for example, an arbitrageur has an open book at the time an arbitrage opportunity arises (i.e. at the time disequilibrium arises), such that their book may show a (temporary) loss, precisely the time the arbitrageur now requires funds to engage in profitable trading, then these funds may be withheld by investors who only judges arbitrage traders by their current book position. Finally, we consider an alternate transition function in which we allow a one-time (in the case of the LSTR model) or a two-time (in the case of the QLSTR model) change between regimes over time. That is, we replace, as the transition variable, a time trend function instead of the forward premium. Whilst, there is, perhaps, no theoretical grounds for this transition function, we consider this to be an interesting empirical alternative and it may capture policy regime effects present within individual countries which are not well captured by the forward premium itself. 4. Data and empirical results Monthly data relating to the 3-month and 6-month interbank rates are collected for 16 countries. Table 1 presents the sample dates and some summary statistics. All the data was sourced from Datastream. The forward rates are calculated as ft = 2R6t − R3t where R6t represents the 6-month Table 1 Descriptive statistics Country

Australia Belgium Canada Denmark France Germany Ireland Italy Japan Norway NZ Singapore Spain Sweden UK US

Sample

86:9–03:4 89:10–03:4 90:5–03:4 88:6–03:4 89:1–03:4 86:1–03:4 84:1–03:4 90:5–03:4 86:7–03:4 93:3–03:4 86:4–03:4 86:4–03:4 90:5–03:4 92:12–03:4 86:1–03:4 86:1–03:4

Change in the spot rate

Forward premium

Mean

S.D.

Skew

Kurt

Mean

S.D.

Skew

Kurt

−0.178 −0.145 −0.196 −0.103 −0.122 −0.035 −0.136 −0.178 −0.072 −0.075 −0.182 −0.061 −0.237 −0.174 −0.125 −0.096

0.902 0.706 0.899 1.263 0.974 0.489 1.997 0.981 0.404 0.769 1.475 0.902 0.868 0.521 0.848 0.584

−0.212 0.382 0.843 −1.364 −0.466 0.793 −0.948 0.714 0.037 1.074 0.542 −0.365 0.315 −0.645 0.346 −0.248

3.963 11.453 5.115 13.615 12.005 4.834 21.764 8.812 4.684 6.597 11.388 4.621 13.779 3.526 8.257 3.092

0.143 −0.009 0.213 0.114 −0.047 0.056 −0.054 0.0004 0.002 0.025 0.020 0.262 −0.104 0.085 0.013 0.166

0.399 0.537 0.396 0.370 0.562 0.326 1.093 0.468 0.188 0.453 0.496 0.262 0.573 0.395 0.432 0.287

0.881 −4.011 0.402 0.227 −3.512 −1.351 −4.255 0.449 −0.792 −0.711 −2.740 0.825 −2.163 −0.188 −0.305 0.528

5.013 30.484 4.725 4.487 20.569 8.791 28.548 4.841 4.347 3.489 17.710 4.048 13.155 3.421 4.361 3.524

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Table 2 STR tests Country

Australia Belgium Canada Denmark France Germany Ireland Italy Japan Norway NZ Singapore Spain Sweden UK US

Transition function—forward premium

Transition function—time trend

F

F3

F2

F1

F

F3

F2

F1

0.034 0.810 0.044 0.001 0.003 0.015 0.008 0.050 0.600 0.481 0.002 0.756 0.503 0.002 0.356 0.026

0.082 0.983 0.480 0.002 0.001 0.052 0.312 0.402 0.344 0.367 0.015 0.300 0.158 0.152 0.201 0.751

0.112 0.738 0.265 0.078 0.063 0.024 0.278 0.0472 0.328 0.313 0.898 0.749 0.736 0.0004 0.478 0.023

0.013 0.355 0.012 0.139 0.087 0.208 0.002 0.075 0.922 0.426 0.003 0.936 0.633 0.858 0.295 0.44

0.028 0.625 0.420 0.0501 0.0009 0.037 0.723 0.553 0.021 0.0001 0.313 0.973 0.862 0.158 0.161 0.109

0.130 0.606 0.670 0.787 0.0004 0.209 0.0297 0.817 0.242 0.004 0.083 0.816 0.967 0.095 0.429 0.336

0.861 0.646 0.398 0.0731 0.354 0.046 0.777 0.429 0.007 0.001 0.579 0.954 0.400 0.179 0.351 0.905

0.007 0.282 0.160 0.031 0.007 0.012 0.717 0.238 0.251 0.251 0.614 0.675 0.716 0.588 0.064 0.176

Notes. Entries are p-values from the F-tests discussed in Section 4. F is the test of linearity vs. STR, while the remaining tests discriminate between LSTR and QLSTR behaviour.

interbank rate and R3t represents the 3-month rate.5 The summary statistics in Table 1 have the usual characteristics of financial data, that is, a mean which is smaller than the standard deviation and evidence of non-normality, in particular, excess kurtosis. Prior to proceeding to the estimation of the linear model and the asymmetric LSTR and nonlinear QLSTR models, we first report the usual test statistics of linearity against STR behaviour. This procedure is becoming increasingly well-known and is discussed in, for example, van Dijk et al. (2002). Briefly stated, given the presence of parameters which are not identified under the null of linearity the transition function is approximated by a third-order Taylor expansion, yielding the auxiliary regression: yt = δ0 xt + δ1 xt zt + δ2 xt z2t + δ3 xt z3t

(11)

where in general, yt and xt are the dependent and independent variables of interest and zt is the transition variable.6 The resultant test statistic is H0 : δ1 = δ2 = δ3 = 0 and can be F or ␹2 distributed. To discriminate between the alternate transition functions a sequential testing procedure exists as such: F3 : δ3 = 0;

F2 : δ2 = 0|δ3 = 0;

F1 : δ1 = 0|δ2 = δ3 = 0

where if test statistic F2 has the lowest p-value then the QLSTR model will be selected, otherwise the LSTR model will be selected. The results (p-values) from this procedure are presented in Table 2. Considering the test results with the forward premium as the transition variable, the results from this procedure suggest the following: the countries which appear 5

This definition of the forward rate was introduced by Shiller et al. (1983). That is, in the current context, the change in the short rate (yt ), and the forward premium (xt and zt , additionally alternative lags of zt can be considered). 6

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Table 3 Linear model Country

α1

α2

Wald

R2

Australia Belgium Canada Denmark France Germany Ireland Italy Japan Norway NZ Singapore Spain Sweden UK US

−0.292 (−4.66) −0.141 (−2.84) −0.383 (−5.01) −0.200 (−2.08) −0.077 (−1.27) −0.079 (−2.66) −0.077 (−0.74) −0.182 (−2.81) −0.074 (−3.10) −0.104 (−1.62) −0.210 (−2.21) −0.150 (−1.68) −0.129 (−2.54) −0.245 (−6.42) −0.138 (−2.60) −0.227 (−5.24)

0.782 (5.36) 0.587 (6.40) 0.852 (5.03) 0.828 (3.34) 1.011 (9.49) 0.755 (8.42) 1.082 (11.26) 1.188 (8.65) 1.149 (9.05) 0.724 (5.05) 1.179 (6.19) 0.336 (1.41) 1.049 (12.14) 0.789 (8.40) 0.845 (6.90) 0.781 (6.01)

32.78 (0.00) 27.99 (0.00) 39.42 (0.00) 6.25 (0.04) 1.62 (0.44) 17.71 (0.00) 1.35 (0.51) 9.74 (0.01) 10.94 (0.00) 6.93 (0.03) 5.59 (0.06) 34.37 (0.00) 7.54 (0.02) 55.48 (0.00) 8.63 (0.01) 52.70 (0.00)

0.20 0.20 0.14 0.24 0.34 0.25 0.35 0.32 0.29 0.17 0.15 0.01 0.49 0.36 0.18 0.14

Notes. See Eq. (4) for specification. Numbers in parentheses are t-statistics. Wald test is for the restrictions: α1 = 0 and α2 = 1.

to exhibit linear behaviour are Belgium, Japan, Norway, Singapore, Spain and the UK; the countries which appear to be best described by a LSTR model are Australia, Canada, Denmark, France, Ireland and New Zealand; while those best described by the QLSTR model are Germany, Italy, Sweden and the US. Turning to the results which allow for a deterministic trend function as the transition variable, then the following countries may be better characterised in this fashion on the basis of lowest p-value: Australia, France (LSTR); Japan, Norway (QLSTR). Table 3 reports the estimation results of linear model, given by Eq. (4). We can see for the vast majority of series that a Wald test (presented in column 3) of α1 = 0 and α2 = 1 is rejected for nearly all series, with only the rates for France and Ireland supporting the restrictions (of interest, of course, the tests reported in Table 2 suggested that both these series could be characterised as not linear, and conversely the estimated linear model is not supported for those series for which the tests in Table 2 do suggest linearity). This suggests that the forward premium is not an unbiased predictor of the future spot rate and thus provides implicit support for the existence of a risk premium. Further, support for the existence of a constant risk premium is given by a significant constant term for 12 of our series. While, evidence of a time-varying risk premium is also present for 12 series. Specifically, for 10 series the slope parameter is less than one, while for two countries it appears to be greater than one. Given these results we therefore proceed to consider whether allowing asymmetric dynamics, through our preferred LSTR model, within the forward premium is able to better describe the data for each country. Table 4 presents the estimates of the LSTR model. From this table we can see that, in addition to France and Ireland from the linear model, the coefficient restriction test results for Denmark, Japan, Spain and the UK now support the restriction that α1 + β1 = 0 and α2 + β2 = 1, and that the forward rate now provides an unbiased estimate of the future spot rate. This result suggests that as the forward premium becomes larger then the probability of it acting as an unbiased estimator of

266

Country

α1

α2

β1

β2

γ

c

Wald

t=1

R2

Australia Belgium Canada Denmark France Germany Ireland Italy Japan Norway NZ Singapore Spain Sweden UK US

−0.36 (−5.59) 0.27 (0.91) −0.39 (−5.28) 14.59 (3.94) 0.18 (0.53) −0.30 (−3.65) 15.15 (6.46) 1.71 (2.86) −0.28 (−2.75) −0.83(−2.45) 1.41 (2.06) 0.97 (0.41) −0.16 (−2.60) −1.76 (−4.25) 0.02 (0.14) −0.23 (−5.56)

1.19 (4.20) 0.81 (4.42) 0.85 (4.34) 3.10 (5.04) 1.17 (5.63) 0.38 (2.10) 3.48 (9.17) 3.47 (5.01) 0.59 (1.77) −0.28 (−0.60) 2.46 (5.62) 3.93 (1.65) 1.02 (9.55) −1.56 (−2.53) 1.21 (4.41) 1.04 (6.64)

1.22 (3.53) −0.44 (−1.41) 8.42 (2.32) −14.71 (−3.97) −4.11 (−0.41) 0.35 (3.18) −15.24 (−6.50) −1.91 (−3.17) 0.27 (2.18) 0.89 (2.24) −1.80 (−2.20) −1.18 (−0.50) 0.83 (1.96) 1.58 (3.67) −0.92 (−1.08) −1.78 (−2.85)

−1.60 (−3.25) −0.21 (−2.75) −6.18 (−2.44) −2.24 (−3.53) 5.77 (0.45) 0.20 (1.93) −2.59 (−6.28) −2.31 (−3.25) 0.20 (0.45) 0.70 (1.22) −1.28 (−1.78) −3.45 (−1.57) −1.07 (−1.74) 2.24 (3.64) 0.54 (0.59) 1.89 (2.28)

25.05 (1.25) 7.73 (1.14) 35.36 (0.10) 6.08 (1.22) 0.96 (0.98) 17.81 (0.56) 28.98 (1.42) 27.72 (1.90) 15.29 (0.56) 4.49 (0.78) 2.22 (0.97) 22.74 (1.78) 11.65 (0.40) 4.20 (1.11) 2.20 (1.05) 10.42 (0.57)

0.37 (20.26) 0.42 (1.04) 0.99 (4.06) −0.33 (−17.08) 0.60 (0.71) −0.004 (−1.32) −0.36 (3.17) −0.50 (−14.35) −0.04 (−1.24) −0.15 (−2.06) −0.27 (−2.46) 0.64 (−1.64) 0.31 (6.41) −0.27 (−5.71) 0.29 (2.03) 0.50 (5.47)

15.31 (0.00) 20.56 (0.00) 12.65 (0.01) 2.04 (0.36) 1.30 (0.52) 11.66 (0.00) 2.02 (0.36) 6.81 (0.03) 1.85 (0.40) 4.94 (0.08) 5.24 (0.08) 24.06 (0.00) 3.45 (0.17) 48.78 (0.00) 3.33 (0.19) 13.92 (0.00)

0.46 (0.50) 1.12 (0.29) 0.55 (0.46) − − 6.53 (0.01) − 0.87 (0.35) − 2.14 (0.14) 0.07 (0.78) 0.85 (0.36) − 4.14 (0.04) − 0.07 (0.79)

0.26 0.27 0.22 0.38 0.41 0.30 0.45 0.39 0.32 0.25 0.23 0.04 0.53 0.48 0.22 0.20

Notes. See Eqs. (5) and (6) for specification. Numbers in parentheses are t-statistics. Wald test is for the restrictions: α1 + β1 = 0 and α2 + β2 = 1. t = 1 is a test of α2 = 1 for Australia, Belgium, Canada, Singapore and the US, and a test of α2 + β2 = 1 for the remaining series.

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Table 4 LSTR model

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the future spot rate increases.7 This suggests that for these four countries the asymmetric LSTR model is able to explain the dynamic interactions between the forward interest rate and the change in the spot rate. Nevertheless, this still suggests that for the majority of our series the restriction is rejected. This could arise, as the estimated model is not adequately able to capture time-variation in the risk-premia. Thus, we further consider a test on whether part of the restriction is satisfied, that is we consider either α2 = 1 or α2 + β2 = 1 for the remaining series. For five series (Australia, Belgium, Canada, Singapore and the US) the restriction α2 = 1 is supported. For a further three series (Italy, Norway and New Zealand) the restriction α2 + β2 = 1 is supported. These latter two results suggest that the LSTR model is able to capture time-variation in the risk-premia inherent in the forward data, but a further source of unaccounted structure remains. Furthermore, the latter result also suggests that the larger the forward premium the better a predictor of the future spot rate. In contrast the former result states the opposite, namely the smaller the premium the better a predictor it is. However, it is worth noting that for these series the point of transition given by the parameter c is greater than for other series, suggesting that moderate positive values of the forward premium are reasonable predictors but that extreme positive values of the forward premium may indicate disequilibrium. Nevertheless, for two series (Germany and Sweden) both the Wald test concerning the overall model fit, or the test of the slope restrictions are not met. Finally, we can note that the R2 values from the LSTR regression are greater than those of the linear model.8 As a further exercise into the nature of these results Table 5 imposes the restrictions that support the forward premium being an unbiased predictor (i.e. α1 + β1 = 0 and α2 + β2 = 1) to examine how this affects the models performance. Notably, the R2 values are lower than those found in Table 4 where no restrictions are imposed, with the exceptions of France, Ireland, (for which the linear model is adequate), Denmark, Japan, Spain and UK (for which the LSTR model fits, although the last two values are marginally lower). In sum, the results presented in Tables 4 and 5 are generally supportive of the LSTR specification. As a final exercise we reconsider the models for those series that, on the basis of the tests reported in Table 2, maybe best characterised by an alternate model. More specifically, for Australia and France we consider a LSTR model of Eqs. (5) and (6) with a time trend in the transition function, for Germany, Italy, Sweden and the US we consider a QLSTR model (Eqs. (5) and (10)), while for Japan and Norway we also consider the QLSTR model but with a time trend in the transition function. The estimation results from this exercise are presented in Table 6.9 7 Examination of the coefficients would certainly support this view for Denmark and Japan. For Spain and the UK it appears that switching behaviour in the constant term is more important than switching in the slope parameters that are insignificant in the upper regime. 8 For the interested reader, plots of the transition functions are available upon request. In short, the following observations can be made. For the series that the LSTR model supports (Denmark, Japan, Spain and the UK) we observe data points appearing in each of the three noted regimes (that is, a regime associated with large negative values of the forward premium, large positive values of the forward premium and small values of the forward premium of either sign). For series where the restriction α2 = 1 is supported (Australia, Belgium, Canada, and the US) most observations lie in the lower regime, which can be viewed as the equilibrium regime, with departures from equilibrium for larger values of the forward premium. Where α2 + β2 = 1 (Italy, Norway and New Zealand) most data observations lie in upper regime, suggesting that the forward premium is a better predictor of the future spot rate when the premium is large. We can also note that the transition function for Sweden is similar to those of Italy, Norway and New Zealand suggesting that a large forward premium is a better predictor even if the unbiasedness conditions are not met. Examination of the transition function for Germany reveals it is similar to those for which the LSTR models does provide a good characterisation (especially Denmark and Spain), suggesting that over a larger sample better results may be obtained. 9 Again plots of the transition functions are available upon request.

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Table 5 Restricted LSTR Country

β1 = −α1

α2 = 1−β2

γ

c

R2

Australia Belgium Canada Denmark France Germany Ireland Italy Japan Norway NZ Singapore Spain Sweden UK US

−0.36 (−5.34) 0.61 (1.94) −0.39 (−5.06) 14.59 (3.94) −0.08 (−1.25) −0.30 (−3.55) 0.08 (0.47) 1.77 (2.90) −0.29 (−2.69) 0.85 (2.62) 1.47 (2.96) −0.37 (−1.81) −0.16 (−2.51) −1.96 (−2.86) −0.17 (−2.58) −0.24 (−5.33)

−0.21 (−0.70) 0.004 (0.03) 0.15 (0.70) 3.10 (5.04) −0.01 (−0.10) 0.63 (3.41) −0.20 (−1.74) −2.53 (−3.56) 0.42 (1.22) 1.29 (2.90) −1.49 (−3.91) 0.30 (0.76) −0.03 (−0.24) −0.64 (5.74) 0.15 (1.21) 0.10 (0.60)

25.30 (0.18) 10.03 (0.18) 32.77 (0.02) 6.08 (1.22) 5.33 (0.94) 19.39 (0.51) 4.35 (0.08) 28.46 (2.64) 12.26 (0.89) 6.32 (0.54) 14.31 (0.20) 2.29 (2.01) 8.07 (0.17) 1.72 (3.26) 1.80 (1.77) 9.32 (0.49)

0.36 (6.10) 0.42 (1.27) 0.91 (13.45) −0.33 (−17.08) 0.58 (1.15) −0.01 (−0.43) 0.24 (0.45) −0.49 (−1.03) −0.04 (−1.10) −0.18 (−2.86) −0.34 (−2.83) −0.04 (2.09) 0.28 (1.64) −0.25 (−2.43) 0.77 (3.47) 0.80 (28.86)

0.22 0.24 0.12 0.38 0.41 0.29 0.45 0.36 0.32 0.24 0.23 0.02 0.50 0.44 0.21 0.15

Notes. See Eqs. (5) and (6) for specification, but with imposed restrictions that β1 = −α1 and α2 = 1−β2 . Numbers in parentheses are t-statistics.

Examining the QLSTR models with the forward premium in the transition function (Germany, Italy, Sweden and the US) we can see that estimation results do not bring us closer to accepting the belief that the forward rate is an unbiased predictor of the future spot rate. In none of these four cases is the Wald test of α1 + β1 = 0 and α2 + β2 = 1 not rejected.10 With regard to the models which have a time trend in the transition function we can see again that the Wald test of α1 + β1 = 0 and α2 + β2 = 1 is rejected for Australia and Japan, whilst for France the restriction is accepted, as similarly occurred with the previous linear and LSTR models where the restriction was also accepted. The transition function plot for Norway reveals that only a few data points lie in the middle regime, thus, we reconsidered the results for Norway by estimating a LSTR model with time in the transition function, this time the restriction α1 + β1 = 0 and α2 + β2 = 1 is accepted.11 Finally, we repeat the non-linearity tests presented in Table 2 on the residuals from the fitted STR models using the method of Eitrheim and Ter¨asvirta (1996).12 The test assumes that any remaining non-linearity is again of the smooth-transition type. The alternative hypothesis can thus be defined as: Rt+m − Rt = [α1 + α2 (ft − Rt )] + [β1 + β2 (ft − Rt )]F (zt ) + [θ1 + θ2 (ft − Rt )]G(zt ) + ut

(12)

10 Inspection of the transition functions reveals that for each series the majority of the data points lie on one side of the ‘U’-shaped function. This suggest that in fact the QLSTR model is largely capturing the same dynamics as the LSTR model above but a few data points are driving the test in Table 2 towards the finding of QLSTR behaviour. 11 Of interest the transition function plots for Australia and Norway reveal that the switch in parameter values are associated with a change in the variability of the forward premium. Thus, while the results from this exercise do not take us away from our initial beliefs that a source of information in examining the forward premium lies in asymmetry within the sign of the forward premium, however, these results do suggest that perhaps taking into account variability in the forward premium is also important. 12 The results of further tests discussed by Eitrheim and Ter¨ asvirta are available upon request but suppressed here for space considerations.

Country

α1

α2

β1

β2

γ

C1

C2

Wald

R2

Australia France Germany Italy Japan Norway Norway Sweden US

−0.57 (−5.63) 0.31 (2.23) −0.09 (−5.48) 2.04 (1.41) 1.23 (4.68) 4.20 (4.42) −0.36 (−3.29) −3.55 (−3.10) 1.69 (1.47)

1.31 (6.30) −1.11 (−1.61) 1.37 (2.71) 3.93 (1.87) −1.80 (−1.67) 3.95 (4.17) 0.79 (2.96) −5.04 (−2.24) −1.82 (−1.19)

0.44 (3.38) −0.44 (−2.87) −0.19 (−0.36) −2.25 (−1.55) −1.37 (−5.16) −4.46 (−4.68 0.51 (3.62) 3.38 (2.98) −1.94 (−1.68)

−0.65 (−2.28) 2.43 (3.46) −1.10 (−2.94) −2.77 (−1.32) 2.88 (0.2.56 −3.01 (−3.05) 0.27 (0.84) 5.70 (2.49) 2.69 (1.75)

15.47 (0.73) 33.15 (1.80) 0.48 (1.41) 61.41 (0.40) 20.29 (4.46) 24.04 (3.53) 54.53 (0.66) 22.99 (2.55) 3.11 (0.52)

74.05 (5.52) 43.89 (43.18) −0.60 (−0.75) −1.50 (−0.38) 36.26 (7.27) 67.47 (0.00) 56.42 (28.57) −0.51 (−10.99) 0.52 (7.76)

– – 0.04 (0.12) −0.50 (−0.25) 41.60 (8.93) 67.47 (0.00) – −0.11 (−1.92) 1.37 (1.79)

12.15 (0.00) 1.37 (0.50) 7.44 (0.02) 8.45 (0.01) 35.62 (0.00) – 1.30 (0.52) 52.31 (0.00) 37.63 (0.00)

0.26 0.43 0.30 0.39 0.44 0.49 0.28 0.51 0.20

Notes. See Eqs. (5), (6) and (11) for specifications. Numbers in parentheses are t-statistics. Wald test is for the restrictions: α1 + β1 = 0 and α2 + β2 = 1.

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Table 6 Alternate STR models

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Table 7 STR model residual tests Country

Australia Belgium Canada Denmark France Germany Ireland Italy Japan Norway NZ Singapore Spain Sweden UK US

LSTR models

Alternate STR models

F

F3

F2

F1

F

F3

F2

F1

0.53 0.22 0.23 0.19 0.12 0.98 0.12 0.65 0.35 0.07 0.69 0.46 0.78 0.09 0.16 0.92

0.15 0.18 0.20 0.28 0.45 0.99 0.37 0.45 0.38 0.84 0.88 0.57 0.58 0.12 0.42 0.50

0.72 0.64 0.34 0.58 0.38 0.68 0.59 0.40 0.16 0.02 0.42 0.54 0.66 0.39 0.32 0.89

0.91 0.63 0.18 0.87 0.03 0.98 0.03 0.55 0.45 0.27 0.37 0.17 0.45 0.07 0.96 0.99

0.08

0.74

0.02

0.25

0.00 0.87

0.01 0.76

0.00 0.71

0.01 0.50

0.55 0.52 0.04

0.67 0.55 0.16

0.25 0.99 0.58

0.45 0.16 0.01

0.27

0.20

0.15

0.63

0.89

0.47

0.75

0.99

Notes. Entries are p-values from the F-tests discussed in Section 4. F is the test of a single transition function vs. an additional transition function, again the remaining tests discriminate between LSTR and QLSTR behaviour. For Norway the alternate model is a LSTR model with time trend in the transition function.

where G(zt ) is another transition function. To test the alternative an extension to the auxiliary model in (12) is run: yt = δ0 xt + F (zt ) + δ1 xt zt + δ2 xt z2t + δ3 xt z3t

(13)

where the test has a null hypothesis of no remaining non-linearity that is H0 : δ1 = δ2 = δ3 = 0. The p-values from the F-test are reported in Table 7 for both the LSTR and alternate STR models where appropriate. The results in Table 7 suggest that for the LSTR model there appears to be no remaining non-linearity for any series, albeit marginally for Norway and Sweden. For the alternate STR models remaining non-linearity is reported in those models that use time as the transition function (except Japan), whilst no remaining non-linearity is reported in the remaining models. These results thus support the LSTR model in attempting to capture the asymmetry. 5. Summary and conclusion There has been an increased interest in the dynamic behaviour of interest rate series, and in particular, whether they exhibit reversion to some long-run equilibrium and whether that reversion is asymmetric. Concurrently, a major and ongoing research interest in the empirical analysis of interest rate dynamics is the relationship between forward and spot rates and whether forward rates are an unbiased predictor of the future spot rate. The standard expectation hypothesis approach argues that the forward rate does indeed proxy for the expected future spot rate, subject to a constant liquidity or term premium. However, the general consensus view from the literature is that forward rates provided at best a biased predictor of the expected future spot rate, and whilst this could be consistent with market inefficiencies and systematic errors in the formation of expectations, typically it is believed this result arises from time-variation in the term premium.

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This paper attempts to reconsider the relationship between the forward and spot rates in the light of potential asymmetric behaviour within the forward premium. In particular, we consider the logistic smooth-transition model that can be viewed as exhibiting two extreme regimes that correspond to ‘large’ positive and negative values of the forward premium, and a middle transition regime where the forward premium is ‘small’. Examining the forward premium behaviour of 16 countries this paper reports evidence that, for two series the expectations hypothesis holds in a linear fashion, however, for the remainder of the series the forward premium is a biased predictor and implying evidence of risk-premia. For four series the LSTR model is adequately able to describe the data, and is suggestive that the unbiased nature of the forward rate as a predictor is more likely the larger is the forward premium. For a further eight series the models appears to capture some time-variation in the term premia, but does not fully account for the structure of the data, such that further confounding factors are as yet unaccounted. Although for five series the results suggest that when the forward premium is small or moderately positive, or negative, it acts as a better predictor, while for the remaining three series the forward premium appears to be a better predictor when it is large. Finally, for two series, the LSTR model is unable to describe the process, although the estimated models are similar in nature to those for which the LSTR model does provide a good description of the data. Further, examination of these series using alternate asymmetric or non-linear models does not typically improve our results, although there is some suggestion that taking into account variability in the forward premium may also be important in accounting for the failure of the forward premium to provide an unbiased predictor of future spot rates. Overall, these results appear to suggest that no unique dynamic process accounts for the failure of the forward premium to typically provide an unbiased predictor for the future spot rate. Although in general a larger forward premium is associated with improved prediction, this result is not ubiquitous. Thus, further research is therefore required to understand the nature of the relationship between forward and spot rate, not least because, correctly priced forward rates and long rates are important in terms of both the expected returns to future investments (such that if the forward rate was a biased predictor of the spot rate then this may impact the efficient allocation of resources and investments) and the conduct of monetary policy (whereby policy affects the long rates through altering expectations of future short rates). Acknowledgements The author gratefully acknowledges numerous helpful comments on an earlier version of the paper from the editor (Ike Mathur) and an anonymous referee. Any remaining errors are my own. References Anderson, H.M., 1997. Transaction costs and non-linear adjustment towards equilibrium in the US treasury bill market. Oxford Bulletin of Economics and Statistics 59, 465–484. Andreou, E., Osborn, D.R., Sensier, M., 2000. A comparison of the statistical properties of financial variables in the USA, UK and Germany over the business cycle. The Manchester School 68, 396–418. Baillie, R.T., Kilic¸, R., 2006. Do asymmetric and nonlinear adjustments explain the forward premium anomaly. Journal of International Money and Finance 25, 22–47. Baldwin, R., 1990. Re-interpreting the failure of foreign exchange market efficiency tests: small transaction costs, big hysterisis bands. NBER Working Paper No. 3319. Balke, N.S., Fomby, T., 1997. Threshold cointegration. International Economic Review 38, 627–643.

272

D.G. McMillan / Int. Fin. Markets, Inst. and Money 19 (2009) 258–273

Brooks, C., Garrett, I., 2002. Can we explain the dynamics of the UK FTSE 100 stock and stock index futures markets. Applied Financial Economics 12, 25–31. Campbell, J.Y., 1995. Some lessons from the yield curve. Journal of Economic Perspectives 9, 129–152. Campbell, J.Y., Shiller, R.J., 1991. Yield spreads and interest rate movements: a bird’s eye view. Review of Economic Studies 58, 495–514. Coakley, J., Fuertes, A.M., 2002. Asymmetric dynamics in UK real interest rates. Applied Financial Economics 12, 379–387. Cook, T., Hahn, T., 1990. Interest rate expectations and the slope of the money market yield curve. Economic Review, Federal Reserve Bank of Richmond, 3–26. Cooray, A., 2003. A test of the expectations hypothesis of the term structure of interest rates for Sri Lanka. Applied Economics 35, 1819–1827. Dumas, B., 1992. Dynamic equilibrium and the real exchange rate in a spatially separated world. Review of Financial Studies 5, 153–180. Dumas, B., 1994. Partial equilibrium versus general equilibrium models of the international capital market. In: Van Der Ploeg, F. (Ed.), The Handbook of International Macroeconomics. Blackwell, Oxford. Dwyer Jr., G.P., Locke, P., Yu, W., 1996. Index arbitrage and nonlinear dynamics between the S&P 500 futures and cash. Review of Financial Studies 9, 301–332. Eitrheim, O., Ter¨asvirta, T., 1996. Testing the adequacy of smooth transition autoregressive models. Journal of Econometrics 74, 59–75. Enders, W., Granger, C.W.J., 1998. Unit-roots tests and asymmetric adjustment with an example using the term structure of interest rates. Journal of Business and Economic Statistics 16, 304–311. Enders, W., Siklos, P.L., 2001. Cointegration and threshold adjustment. Journal of Business and Economic Statistics 19, 166–176. Estrella, A., Hardouvelis, G.A., 1991. The term structure as a predictor of real economic activity. Journal of Finance 46, 555–576. Fama, E.F., 1990. Term structure forecasts of interest rates, inflation and real returns. Journal of Monetary Economics 25, 59–76. Fama, E.F., Bliss, R.R., 1987. The information in long-maturity forward rates. American Economic Review 77, 680–692. Hardouvelis, G.A., 1988. The predictive power of the term structure during recent monetary regimes. Journal of Finance 43, 339–356. Harvey, C.R., 1988. The real term structure and consumption growth. Journal of Financial Economics 22, 305–333. Hu, Z., 1993. The yield curve and real activity. IMF Staff Papers 40, 781–806. Jansen, E.S., Ter¨asvirta, T., 1996. Testing parameter constancy and super exogeneity in econometric equations. Oxford Bulletin of Economics and Statistics 58, 735–763. Lyons, R.K., 2001. The Microstructure Approach to Exchange Rates. MIT Press, London. Mankiw, N.G., Miron, J.A., 1986. The changing behavior of the term structure of interest rates. Quarterly Journal of Economics 101, 211–228. Martin, C., Milas, C., 2004. Modelling monetary policy: inflation targetting in practise. Economica 71, 209–221. McMillan, D.G., 2004. Non-linear error-correction: evidence for UK interest rates. Manchester School 72, 626–640. Mishkin, F.S., 1988. The information in the term structure: some further results. Journal of Applied Econometrics 3, 307–314. Mishkin, F.S., 1990. What does the term structure tell us about future inflation? Journal of Monetary Economics 25, 77–95. Murchison, S., Siklos, P.L., 1999. Central bank reaction functions in inflation and non-inflation targeting economies. Working Paper. Wilfrid Laurier University. Roberds, W., Whiteman, C.H., 1999. Endogenous term premia and anomalies in the term structure of interest rates: explaining the predictability smile. Journal of Monetary Economics 44, 555–580. Sercu, P., Uppal, R., Van Hulle, C., 1995. The exchange rate in the presence of transactions costs: implications for tests of purchasing power parity. Journal of Finance 50, 1309–1319. Shiller, R., Campbell, J., Schoenholtz, K., 1983. Forward rates and future policy: interpreting the term structure of interest rates. Brookings Papers on Economic Activity 1, 173–217. Shleifer, A., 2000. Inefficient Markets. An Introduction to Behavioural Finance. Clarendon Lectures in Economics. Oxford University Press, Oxford. Simon, D.P., 1989. Expectations and risk in the treasury bill market: an instrumental variables approach. Journal of Financial and Quantitative Analysis 24, 357–365. Svensson, L.E.O., 1999. Inflation targeting as a monetary policy rule. Journal of Monetary Economics 43, 607–654.

D.G. McMillan / Int. Fin. Markets, Inst. and Money 19 (2009) 258–273

273

Ter¨asvirta, T., 1998. Modelling economic relationships with smooth transition regressions. In: Ullah, A., Giules, D. (Eds.), Handbook of Applied Economic Statistics. Marcel Dekker, New York. Tse, Y., 2001. Index arbitrage with heterogeneous investors: a smooth transition error-correction analysis. Journal of Banking and Finance 25, 1829–1855. Tzavalis, E., 1999. A common shift in real interest rates across countries. Applied Financial Economics 9, 365–369. van Dijk, D., Franses, P.H., 2000. Non-linear error-correction models for interest rates in the Netherlands. In: Barnett, W.A., Hendry, D.F., Hylleberg, S., Ter¨asvirta, T., Tjøstheim, D., W¨urtz, A. (Eds.), Nonlinear Econometric Modelling—Proceedings of the 6th (EC)2 Meeting. Cambridge University Press, Cambridge. van Dijk, D., Ter¨asvirta, T., Franses, P.H., 2002. Smooth transition autoregressive models—a survey of recent developments. Econometric Review 21, 1–47.