Non-Linear Dynamics in Output, Real Exchange Rates and Real Money Balances: Norway, 1830-2003

Akram, Q. Farooq; Eitrheim, Øyvind; Sarno, Lucio Working Paper Non-Linear Dynamics in Output, Real Exchange Rates and Real Money Balances: Norway, 1830-2003 Working Paper, No. 2005/2 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Akram, Q. Farooq; Eitrheim, Øyvind; Sarno, Lucio (2005) : Non-Linear Dynamics in Output, Real Exchange Rates and Real Money Balances: Norway, 1830-2003, Working Paper, No. 2005/2, ISBN 82-7553-295-7, Norges Bank, Oslo, https://hdl.handle.net/11250/2498443 This Version is available at: https://hdl.handle.net/10419/209849 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/deed.no ANO 2005/2 Oslo June 1, 2005 Working Paper Research Department Non-linear dynamics in output, real exchange rates and real money balances: Norway, 1830-2003 by Q. Farooq Akram, Øyvind Eitrheim and Lucio Sarno ISSN 0801-2504 (printed) 1502-8143 (online) ISBN 82-7553-294-9 (printed), 82-7553-295-7 (online) Working papers from Norges Bank can be ordered by e-mail: [email protected] or from Norges Bank, Subscription service, P.O.Box. 1179 Sentrum N-0107Oslo, Norway. Tel. +47 22 31 63 83, Fax. +47 22 41 31 05 Working papers from 1999 onwards are available as pdf-files on the bank’s web site: www.norges-bank.no, under “Publications”. Norges Bank’s working papers present research projects and reports (not usually in their final form) and are intended inter alia to enable the author to benefit from the comments of colleagues and other interested parties. Views and conclusions expressed in working papers are the responsibility of the authors alone. Working papers fra Norges Bank kan bestilles over e-post: [email protected] eller ved henvendelse til: Norges Bank, Abonnementsservice Postboks 1179 Sentrum 0107 Oslo Telefon 22 31 63 83, Telefaks 22 41 31 05 Fra 1999 og senere er publikasjonene tilgjengelige som pdf-filer på www.norges-bank.no, under “Publikasjoner”. Working papers inneholder forskningsarbeider og utredninger som vanligvis ikke har fått sin endelige form. Hensikten er blant annet at forfatteren kan motta kommentarer fra kolleger og andre interesserte. Synspunkter og konklusjoner i arbeidene står for forfatternes regning. Non-linear dynamics in output, real exchange rates and real money balances: Norway, 1830–2003∗ Q. Farooq Akram Norges Bank Øyvind Eitrheim Norges Bank Lucio Sarno Warwick Business School, CEPR and Norges Bank Abstract We characterise the behaviour of Norwegian output, the real exchange rate and real money balances over a period of almost two centuries. The empirical analysis is based on a new annual data set that has recently been compiled and covers the period 1830–2003. We apply multivariate linear and smooth transition regression models proposed by Ter¨asvirta (1998) to capture broad trends, and take into account non-linear features of the time series. We particularly investigate and characterise the form of the relationship between output and monetary policy variables. It appears that allowance for statedependent behaviour and response to shocks increases the explanatory powers of the models and helps bring forward new aspects of the dynamic behaviour of output, the real exchange rate and real money balances. Keywords: Business cycles, real exchange rates, money demand, non-linear modelling, smooth transition regressions. JEL classifications: C51, E32, E41, F31. ∗A version of this paper has been accepted for publication in C. Milas, P. Rothman and D. van Dijk (eds.), Nonlinear Time Series Analysis of Business Cycles, which is forthcoming in the series Contributions to Economic Analysis of Elsevier Science, Amsterdam. We would especially like to thank two anonymous referees, the editors C. Milas and D. van Dijk, Hilde C. Bjørnland and participants at the Annual Meeting of Norwegian Economists 2005 for comments and useful suggestions. The views expressed are those of the authors, and should not be interpreted as reflecting those of Norges Bank (Central Bank of Norway). 1 1 Introduction For a long time, linear empirical models of business cycles were the standard tool of trade in analyses of fluctuations in output and its interaction with monetary and fiscal policy and foreign shocks. Accordingly, symmetric behaviour of output in recessions and expansions and symmetric response to positive and negative impulses, irrespective of their sizes, were imposed by the choice of model. Economic theory, however, has long recognised the presence of real and nominal rigidities in labour and product markets, uncertainty, coordination failure, credit rationing and other constraints facing economic agents that may lead to non-linear demand and supply curves, unemployment hysteresis, multiple equilibria in growth rates, and size and sign dependent response of output to various shocks; see e.g. Akerlof (1973), Stiglitz and Weiss (1981), Diamond (1982), and Murphy et al. (1989). The literature on international trade also notes that large entry costs in a market may lead to size and sign dependent response of imports and exports to real exchange rate shocks; see e.g. Baldwin and Krugman (1989). Real exchange rates themselves are known to undergo different adjustments in the face of small and large shocks; cf. Heckscher (1916) and Sercu et al. (1995). Moreover, it is well known that the response of output to a shock may even depend on the persistence of the shock as perceived by economic agents. Such persistence dependent response may be ascribed to e.g. the asymmetric response of consumption to transitory versus permanent shocks to income and wealth, or to the effects of transitory versus permanent shocks to pricing, manning, investment and production decisions of firms; see e.g. Taylor (2001), Dixit (1992) and the references therein. In addition, the literature on time inconsistency of policies has drawn attention to state-dependent commitment of policy makers to announced policies. In particular, the literature on currency crises focuses on trade-offs faced by policy makers when deciding whether or not to honour their commitment to an announced exchange rate target, making their commitment dependent on the state of the business cycle and the size and signs of, e.g., terms of trade shocks; see inter alia Dumas and Svensson (1994) and Ozkan and Sutherland (1998). This suggests that not only may the response of output to policy shocks be state-dependent, but even the response of monetary and fiscal authorities can depend on the state of the economy. A number of studies ascribe such state-dependent policy response to preferences of policy makers; see Bec et al. (2002) and the references therein. Development of non-linear time series and econometric models have vastly increased the scope of empirical analyses. Non-linear autoregressive models have made it possible to investigate and characterise the presence of non-linear behaviour of economic variables and have turned out to be useful forecasting devices. Notable applications of such models to represent the behaviour of key macroeconomic variables include Neft¸ci (1984), Hamilton (1989), Ter¨asvirta and Anderson (1992), Dumas (1992), Granger and Ter¨asvirta (1993), Rothman (1998), and Skalin and Ter¨asvirta (1999). So far, development and applications of non-linear univariate models seem to dominate the literature on non-linear models relative to multivariate non-linear 2 models. Yet, multivariate models are required to examine the response of indicators of business cycles to various shocks and to test non-linear behaviour implied by different kinds of frictions highlighted by economic theories. Important recent contributions to the development of multivariate non-linear models include Engle and Hamilton (1990), Krolzig (1997) and Ter¨asvirta (1998). These models have been successfully employed in analyses of business cycles, employment, money demand and exchange rates by, inter alios, Clements and Krolzig (1998), Burgess (1992), Ter¨asvirta and Eliasson (2001), Meese and Rose (1991), Michael et al. (1997), and Taylor et al. (2001). This paper applies both linear and non-linear multivariate models to characterise and explain Norwegian output, the real exchange rate and real money balances. By developing models of the real exchange rate and real money balances, we are able to investigate their interaction with output and with each other. Moreover, they enable us to investigate whether and to what extent the response of monetary policy to domestic and foreign shocks varies with the state of the economy. Once one leaves the realm of linear models, however, one is faced with a choice between various types of non-linear models. Such models generally differ in the extent and form of implicit and explicit restrictions on model formulation and estimation algorithms. We have, however, limited our choice to the smooth transition regression (STR) class of models; see Granger and Ter¨asvirta (1993) and Ter¨asvirta (1998). These models are quite flexible and enable one to represent many forms of non-linear behaviour. In particular, they allow for both smooth and abrupt transitions between different states. We employ a new data set that covers a period of more than 170 years, from 1830 to 2003. The time series for the Norwegian economy has recently been extended so far back to the 19th century. The paper is organised as follows. Section 2 outlines the modelling and evaluation framework of STR models. Section 3 provides a brief description of the time series that we use in the empirical analysis. Section 4 develops linear and non-linear multivariate models of output (real gross domestic product, GDP), the real exchange rate against pound sterling and narrow money balances (M0) in real terms over the full sample. The UK represents the foreign sector in our study as it has been among Norway’s major trading partners over the whole sample period and because of its technological lead, which may potentially account for a stochastic trend in the Norwegian time series. More specifically, in Section 4 we first develop linear equilibrium correction models of GDP, the real exchange rate and real narrow money balances and undertake a comprehensive investigation of potentially neglected non-linear effects of various foreign and domestic variables in these models. In light of these results, we develop non-linear equilibrium correction models that improve on the corresponding linear models in terms of explanatory power and bring forward new aspects of the behaviour of GDP, the real exchange rate and real money. Section 5 is devoted to the specification and evaluation of the non-linear models. Section 6 concludes and finally, an appendix provides a detailed account of 3 deterministic variables used in this study. 2 STR models A smooth transition regression (STR) model of variable xcan be formulated as follows: xt=z0 t(ϕ0+ϕ1×F(γ, c;st)) + εt,(1) where ztis a vector of explanatory variables, which may include lags of xt.ϕ0and ϕ1are vectors of the associated coefficients and F(γ, c;st) is a transition function (hereafter denoted as F), which is characterised by two parameters γand c, and a variable stthat governs the transition function.1 In STR models, Fis assumed to increase monotonically with the level of st and it is bounded. It can either be specified as a logistic function (LSTR) or as an exponential function (ESTR). These specifications of Fdefine LSTR and ESTR models, respectively. The logistic function is specified as: F(γ, c;st) = (1 + exp{−γ(st−c)})−1, γ > 0.(2) The STR model allows the coefficients to change with the value of F. Accordingly, the process determining xtchanges with the state variable st. Specifically, the LSTR specification allows the process for xtto vary between z0 tϕ0+tand z0 t(ϕ0+ϕ1) + t as (st−c)→ −∞ and (st−c)→ ∞, respectively. The parameter γdetermines the speed of transition between these two extreme regimes, for a given deviation stfrom a presumably constant threshold value c. In general, LSTR models allow one to take into account effects of both the size and sign of ston the xt-process. The exponential smooth transition function (ESTR) is specified as: F(γ, c;st) = 1 −exp{−γ(st−c)2}.(3) In this case Frises symmetrically when stdeviates from c. Moreover, small deviations have smaller effects on xtthan large deviations due to the quadratic term in the transition function. The parameter γdetermines the speed of transition between regimes when stdeviates from c. The exponential specification of Fallows the process determining xtto shift between z0 tϕ0+εtand z0 t(ϕ0+ϕ1) + εtdepending on the size of the deviation (st−c). In general, ESTR models are well suited to capture size-dependent effects of ston the xt-process. STR models are quite general and allow for both smooth and abrupt transitions between two regimes z0 tϕ0+εtand z0 t(ϕ0+ϕ1)+εtfor a process xt. Sufficiently large values of γmay lead to an abrupt transition from one regime to another upon a typical deviation st−c, while small values lead to smooth transitions. In the former case, STR models resemble threshold models where even small deviations between st 1A smooth transition autoregressive (STAR) model is obtained when ztonly contains an intercept and lags of xt, and st=xt−l, where the integer l > 0. 4 and cmake Fshift from one extreme value to another. Furthermore, a linear model is nested in a STR model. Specifically, if γ→0, Fconverges towards a constant, and hence Fbecomes independent of st. The generality of STR models make them suitable for allowing state dependent responses of a variable to changes in other variables, e.g. for allowing signand size-dependencies in the adjustment towards equilibrium, or for representing asymmetric responses of a variable to various shocks. 2.1 Testing for non-linearity and its form In this sub-section, we outline the STR modelling strategy which is described in detail in Ter¨asvirta (1998). This modelling strategy consists of three stages. In the first stage, we test linear dynamic model specifications against non-linear STR alternatives. If the null hypothesis of linearity is rejected, we conclude that a non-linear modelling approach is warranted and the next two stages consist of specification and evaluation of non-linear STR models until a set of model design criteria is met. In the first stage, residuals from a linear model of x, say (4), are subjected to tests for neglected state-dependent (non-linear) effects of a set of variables z: xt=z0 tϕ0+ut.(4) The potentially neglected non-linear effects of STR form are approximated by cross products of ztand a state variable sraised to the power of 1–3. The relevance of these terms is thereafter tested in an auxiliary regression model, such as: but=z0 tβ0+ (ztst)0β1+ (zts2 t)0β2+ (zts3 t)0β3+vt,(5) where butis a residual from model (4) and vtis an error term. The test of a linear model against a ST(A)R model characterised by a state variable sis equivalent to conducting a joint test of: H0:β1=β2=β3= 0. Empirically, scan be determined by conducting this test for several variables in e.g. the vector z. If linearity is rejected for more than one variable, the variable causing the strongest rejection of the null hypothesis, i.e. the variable corresponding to the lowest p-value of the joint test, is likely to be an appropriate state variable s. If the linear model is rejected in this test, one needs to test the appropriateness of a logistic specification of Fagainst an exponential specification. For this purpose, the following sequence of tests within the auxiliary regression has been suggested: H04 :β3= 0, H03 :β2= 0 |β3= 0, H02 :β1= 0 |β2=β3= 0. An LSTR model is chosen if H04 or H02 is rejected, but an ESTR model is chosen if H03 is rejected for the chosen st; see Ter¨asvirta (1998). If all hypotheses are rejected, an LSTR (ESTR) specification of Fis chosen if H04 or H02 is rejected more (less) strongly than H03. When testing H03 and H02,β2and β1are tested by prior imposition of β3= 0 and β2=β3= 0, respectively. 5 2.2 Evaluation of STR models After deriving a certain specification of an STR model for st=s∗ t, say (6), it remains to be seen whether it adequately characterises the non-linearity of STR form: xt=z0 tβ0+z0 tθ×F(γ, c;s∗ t) + t.(6) For this purpose we formulate the following auxiliary regression: bt=z0 tθ0+z0 t×F(bγ, bc;s∗ t)θ1+z0 tbϕ1×(∂F(.)/∂bγ)θ2+z0 tbϕ1×(∂F(.)/∂bc)θ3 +(ztst)0e β1+ (zts2 t)0e β2+ (zts3 t)0e β3+wt,(7) where btis the residual from the non-linear model (6), wtis an error term and “b”indicates the estimated value of a parameter or an error term. The null hypothesis of no remaining non-linearity dependent on sis tested by conducting the joint test of H0:e β1=e β2=e β3= 0. If this null hypothesis is rejected for a transition variable sincluding s∗, the form of the remaining nonlinearity can be determined by undertaking the test sequence specified above with e βireplacing βiwhere i= 1, 2, 3. The model is then respecified accordingly to obtain a satisfactory characterisation of the remaining non-linearity. The adequacy of the respecified model is examined by testing for remaining non-linearity within a new auxiliary regression analogous to (7), where the estimated first derivatives of the terms defining the additional non-linearity with respect to their parameters are added to the auxiliary regression. Evaluation of a non-linear model also includes tests for parameter non-constancy, residual autocorrelation, heteroscedasticity of different form and alternative tests of model misspecification. These tests may be conducted in the following way. Tests for parameter constancy with respect to, e.g., the initial parameters defining the linear model, can be performed by testing the null hypothesis of non-linearity with st=t, which denotes a deterministic trend. If this null hypothesis is rejected, one can characterise the non-constancy of being either of the LSTR or the ESTR form; see Lin and Ter¨asvirta (1994) for an elaboration. A test for residual autocorrelation of order pcan be conducted by replacing the regressors in the second row of the auxiliary regression (7) with lagged residuals up to order pand testing their significance; see Eitrheim and Ter¨asvirta (1996). The test for heteroscedasticity is based on the regressors and their cross products and can be undertaken by replacing the regressors in the second row of (7) with the squares of the regressors in the first row and testing their significance; cf. White (1980). A test for autoregressive conditional heteroscedasticiy (ARCH) up to order pcan be performed by regressing squares of the residuals (b2 t) on a constant and their lagged values up to order pand testing for their significance. Finally, model specification can be examined by conducting a RESET (Regression Specification Error Test) by replacing the regressors in the second row of (7) with the square and/or the third power of the fitted value of x, i.e. bx2and bx3, from the non-linear model and testing whether they become significant in the model. A 6 (∆rext<0) and a real depreciation in the opposite case. It also follows that the real exchange rate remains constant in the face of equal growth rates at home and abroad. These implications are consistent with the Balassa-Samuelson hypothesis; see Balassa (1964) and Samuelson (1964). Similar results were obtained by Edison and Klovland (1987) who examined the behaviour of the real exchange rate between Norway and the UK over the period 1874–1971. Notably, these results differ from studies that support PPP on long spans of data; see Sarno and Taylor (2002) and the references therein. The t-value for the cointegration test is about −5.40. Thus, the null hypothesis of no cointegration between the real exchange rate and the GDP ratio can be clearly rejected at the 5% level, even when we compare against MacKinnon’s critical values. Finally, Panel III first presents the unrestricted estimate of narrow money demand. The estimated income, price and nominal interest rate effects are consistent with standard models of money demand. Numerically, the long-run income and price elasticities are close to one, as implied by e.g. the quantity theory of money. Accordingly, (m0−cpi−y) can be interpreted as the inverse of the velocity of money, which is often assumed to rise with nominal interest rates. Hence, the negative interest rate effect could be proxying the inverse of the velocity of money. Statistically, however, there is only weak evidence in support of this constituting a valid long-run relationship. The t-value is just −2.35 while the Dickey-Fuller and MacKinnon critical values are as above, –2.90 and –3.40. Nevertheless, preliminary analysis indicates that if one controls for relatively large shocks to the money balances over the sample period, it is possible to find statistical support for the suggested long-run relationship for narrow money. This becomes evident in Table 4, which presents a linear dynamic model of the real money demand. Also, if the relationship investigated is characterised by non-linear dynamics, the cointegration test may have low power. 4.1 Linear dynamic models Table 4 presents a vector equilibrium correction model (VECM) of Norwegian GDP, real exchange rate and real money. The three equations are treated as a system of simultaneous equations and estimated by the method of full information maximum likelihood (FIML) over a common sample period 1834–2000. These equations were developed by following a “general to specific”model specification strategy, cf. Hendry (1995). The general versions of the equations initially allowed for three lags of each of the explanatory variables, except for the equilibrium correction terms (and dummy variables). Thereafter, statistically insignificant variables were sequentially left out for the sake of parsimony. [Table 4 about here.] The VECM characterises the short-run behaviour of these variables and their adjustment towards their long-run relationships. In this model we have allowed for 13 short-run effects of variables that have long-run effects, but also of those that are only short-run determinants. We assume that the domestic real interest rate, RRt, public consumption at home, cot, public consumption and narrow money in the UK, couktand mukt, are valid conditioning variables for inference purposes, i.e., they are weakly exogenous variables with respect to parameters of interest. A test of this assumption requires that we develop models of these variables which are beyond the scope of this study. We control for relatively large shocks that remain unexplained by our information set by using impulse dummies. It appears that these impulse dummies can be mainly associated with relatively extreme movements in GDP, real exchange rate and money balances during and between the two World Wars and other well known economic and financial crises.6 We note that the left-hand-side variables respond such that they partly correct past deviations from their long-run relationships, buy,burex and burm, respectively. The t-values associated with the deviation terms are –2.94, –4.71 and –5.23, respectively. This implies that the null hypotheses of no response to lagged deviations can be rejected at the standard 5% level of significance. Broadly, the VECM suggests many interactions between the modelled variables and strong influence of foreign shocks on the domestic economy. More specifically, money growth and public consumption affect output and the real exchange rate in the short-run. It appears that real interest rates and the real exchange rate have also short-run effects on output. Domestic output follow the foreign output in the longrun and a deterministic trend representing evolution of physical capital and labour force over time. These variables also influence the course of output in the short-run through the equilibrium reversion process. Furthermore, foreign output together with domestic output appear as important short-run and long-run factors in the VECM. They determine the real exchange rate in the long-run and have substantial effects on it in the short-run (owing to the equilibrium reversion process). Moreover, domestic output has strong influence on money balances in both the long-run and the short-run. In more details, the equation of ∆ytshows that terms of trade shocks (as represented by changes in the real exchange rate) and monetary and fiscal policies are among the main determinants of GDP in the short-run. As expected, an increase in real interest rates and in the growth of real money tend to have negative and positive short-run effects on GDP, respectively. Their effects are of almost equal magnitude on GDP growth. Higher growth in public consumption also tends to boost the activity level. This effect as well as the long-run effects of public consumption appear explicitly in the model only after 1950. Depreciation of the real exchange rate have a positive effect on GDP in the short-run, though it is statistically insignificant at the 5% level. We also note evidence of persistence in the growth rate given that the lagged growth rate appears with a positive coefficient in the equation. The equation for the real exchange rate, ∆rextsuggests that difference between 6The use of impulse dummies helps us avoid the influence of probably unique events on the parameter estimates. Moreover, they contribute to bringing about symmetric/normal distributions of residuals. 14 growth rates of domestic and foreign public expenditures and between growth rates of domestic and foreign money affect the real exchange rate in the short-run. Specifically, relatively higher growth in domestic public consumption and money relative to abroad lead to a real appreciation of the exchange rate. The latter two growth ratios are assumed to work through their effects on relative prices between home and abroad. Government expenditures are commonly believed to be biased towards purchases of non-tradables and thus tend to raise their prices relative to those of tradables. Therefore, an increase in government expenditures may increase the overall price level. If growth in public expenditure at home is higher than abroad, the overall domestic price level is likely to rise faster than the foreign price level, which in turn leads to a real appreciation of the exchange rate, ceteris paribus. Similarly, domestic money growth that is relatively higher than money growth abroad is likely to raise the domestic price level faster than the foreign price level and thereby lead to a real appreciation of the exchange rate. There is also some evidence that differences in money growth were particularly important to exchange rate movements during WWI and the interwar period, represented by the step dummy W2W. There is also an indication of some persistence in changes in the real exchange rate, e.g., a depreciation tends to be followed by depreciation in the subsequent year. However, the equilibrium correction mechanism largely counteracts such persistence and ensures that movements in the real exchange are determined only by diverging growth paths between home and abroad. Some impulse dummies are required to control for the relatively large exchange rate fluctuations around the end of WWI and during the 1920s. The remaining dummies may be associated with large changes in the nominal exchange rate and domestic prices since the late 1960s.7 The model for real money, ∆rmt, suggests that apart from reversion towards it long-run level, which is determined by GDP and the nominal interest rate, GDP growth tends to have a substantial effect on real money growth. The model also indicates a fairly small degree of persistence in real money growth. It appears that we are able to obtain fairly stable parameters over time once we use the dummy variables. For example, allowance for separate income effects on real money growth before and after WWI does not suggest a change in the income effects; note the coefficient estimates of ∆y×preW1tand ∆y×postW1t. The impulse dummies can be ascribed to episodes of excess money growth and relatively high inflation in 1916 and 1918, 16% deflation in 1926, 78% increase in money growth in 1941 and about 20% in 1947, which coincided with zero inflation. However, system diagnostic tests suggest that the VECM could be misspecified. 7Specifically, they may be associated with the devaluation of pound sterling in November 1967, the appreciation and the subsequent revaluation of the krone in 1973, the relatively high wage and price growth in Norway during the mid 1970s and the subsequent devaluations in 1977 and 1978. Moreover, the Norwegian government imposed wage and price control in the period 1978–1980, which may explain the real exchange rate depreciation in 1979–1980. Finally, the real exchange rate depreciation indicated by the impulse dummy for 1997 may be ascribed to the relatively strong appreciation of pound sterling against European currencies, about 14% against the krone, in this period. 15 We note that null hypothesis of normality can be rejected at the 5% level. In addition, the null hypothesis of no heteroscedasticity can be rejected even at the 1% level. The null hypothesis of no autocorrelation for the vector of the three residuals cannot be rejected at standard levels of significance. A comparison of these system tests with tests based on singleequation models suggests that the apparent non-normality of equation errors and the absence of homoscedasticity can mainly be ascribed to the GDP equation, cf. Table 8. All three tests mentioned as well as other tests for model misspecification, i.e., ARCH and RESETs, suggest no misspecification of the equations for the real exchange rate and money growth, respectively, see Tables 9–10 in the next section. 5 Non-linear conditional models Specification and estimation of non-linear multivariate models while conditioning on a number of variables can be undertaken more conveniently within the context of single-equation models rather than in a system. However, valid inference on key parameters such as those measuring the degree of equilibrium reversion in each period and those characterising the long-run relationships presupposes that variables in the system can be considered as weakly exogenous with respect to the parameters of interest. In the following, we test whether our key variables (GDP, real exchange rate and real money) can be considered as weakly exogenous with respect to the long-run parameters and the associated adjustment coefficients. The outcome of these tests may also lend some support to our estimation of the long-run parameters within the static single-equation models. In addition, we examine possible simultaneity bias in the coefficient estimates, owing to endogenous right-hand-side variables, when their equations are estimated individually by OLS rather than as a system by the FIML method. [Table 5 about here.] Table 5 presents the outcome of the weak exogeneity tests. It appears that the real exchange rate and real money can be considered weakly exogenous with respect to the long-run parameters and the adjustment coefficient in the GDP equation and vice versa. Furthermore, the real exchange rate and real money seem to be weakly exogenous with respect to the long-run parameters and the adjustment coefficients in each others’ equation. A joint test of weak exogeneity of all the three variables with respect to the parameters of interest does not reject the null hypothesis of weak exogeneity; the p-value is 28%. Hence, inference on these parameters may be valid within single-equation models of these variables. In order to investigate possible simultaneity bias in the parameter estimates when moving from system to single-equation modelling, we estimated each of the equations in Table 4 by OLS and compared the coefficient estimates with their FIML 16 estimates in Table 4. The OLS estimates of the linear ECMs were generally comparable to their corresponding FIML estimates, indicating negligible bias, especially in the ECMs of GDP and the real money (not reported). The OLS estimates of the (linear) real exchange rate model, however, differed somewhat from their FIML estimates. In particular, the estimated effects of differences in money growth (∆(m0−m0uk)t) and of the lagged real exchange rate (∆rext−1) became weaker when estimated by OLS, see Table 6. [Table 6 about here.] 5.1 STR models of output, the real exchange rate and real money This section develops non-linear single-equation equilibrium correction models (ECMs) of GDP, the real exchange rate and real money. We begin their development by formal tests of the adequacy of linear ECMs that are obtained by OLS estimation of each of the three equations in Table 4. To ease comparison with properties of the non-linear versions of the linear ECMs, the outcomes of a number of standard misspecification tests for all of the linear models are reported in Tables 8–10. [Table 7 about here.] Table 7 presents tests for non-linear effects of STR form for different state variables in each of the three linear ECMs. The tests are based on the residuals from these models. We have limited the set of state variables mainly to the regressors in each of the three models and the time trend, t. In the latter case, the linearity test can be considered a test for smooth variation in the parameters of the linear ECM. In the case of the ECM for GDP, Panel I shows that the null hypothesis of linearity can be rejected at the 1% level for st= ∆RRt; see the row for H0.8The remaining test sequence shows the rejection of H3and H2at the 5% and 1% levels, respectively; see Section 2.1 for an explanation of the tests. We therefore assume that a logistic function of ∆RRtis required to characterise the non-linear effects of the explanatory variables. We also note that linearity is nearly rejected, i.e., at the 10% level, for st∈[t, ∆co, ∆rmt], but we consider this evidence to be too weak against linearity to pursue non-linear modelling in these directions. The evidence against linearity for the monetary policy and fiscal policy variables is largely consistent with a number of previous studies. A large number of studies 8We have also tested the null hypothesis of linearity with contemporaneous and lagged levels of the real interest rates as transition variables, that is with s=RRtand RRt−1. However, the null hypotheses was not rejected at the standard levels of significance as the p-values turned out to be 12% and 15%, respectively. 17 point out that contractionary monetary policy has a more pronounced effect on output than expansionary monetary policy; see inter alia, Cover (1992), Karras (1996) and Parker and Rothman (2004). Furthermore, effects of large monetary shocks may have a larger impact on output than small shocks if there are threshold effects in e.g. consumption and investment and if aggregate supply curves are highly convex and upward sloping, cf. the literature on multiple equilibria. The effects of monetary policy may also depend on the stage of the business cycle. For example Sensier et al. (2002) argue that monetary policy has stronger effects in expansions than in recessions. They also find evidence of non-linear effects of changes in the nominal interest rates on GDP growth in the UK. Moreover, the non-linear effects are of logistic form with annual changes in the nominal interest rate as the transition variable with a threshold value of 2.89 percentage points. Asymmetric effects seem to be observed less often in connection with expansionary and contractionary fiscal policies than in connection with monetary policy. Nevertheless, a number of studies have reported evidence of smaller effects of fiscal expansions relative to those of contractions, and that fiscal policy has a stronger impact in recessions than in booms; see e.g. Kandil (2001) and the references therein. In the case of the ECM of the real exchange rate, linearity is rejected at the 5% level for both st=burex,t−1and st= ∆rext−1. In addition, there seems to be strong evidence of smooth variation in parameters over time as linearity is rejected also for st=t. In this case, a permanent shift seems to occur in the parameters over time since a logistic function of tis favoured against an exponential function. For the other two transition variables, however, exponential functions turn out to be the preferred functions for characterising non-linear effects. We note that H4is rejected in the case of st=t, while H3is rejected for both st= ∆rext−1and st=burex,t−1, all at the 1% level. Previously, Michael et al. (1997), Sarno (2000), Taylor et al. (2001) have developed STR models to characterise the behaviour of real exchange rates for a number of countries. These models suggest that the speed at which a real exchange rate moves towards its equilibrium level increases with the size of the deviation from its equilibrium, which is assumed to be a constant, as implied by the PPP hypothesis. More specifically, these studies reject the linearity of the real exchange rate process against the STR form of non-linearity for the lagged real exchange rate as the transition variable. The transition function is commonly specified as an exponential function of the lagged real exchange rate, though evidence of a logistic function is also found; see Michael et al. (1997). Deviations from the equilibrium exchange rate define non-linear effects in our case as well. In contrast to the above studies, however, the equilibrium level of the real exchange rate is not constant but depends on the growth difference between home and abroad. In addition, our evidence of non-linearity and its form is based on a multivariate model where the equilibrium correction term is embedded in a model which controls for short-run effects of a number of presumably exogenous variables. In contrast, the evidence in previous studies is mainly based on autoregressive models of real exchange rates. The last panel of Table 7 presents tests of the linearity of the ECM of real money. 18 It appears that linearity can be rejected at the 5% level for three of the explanatory variables: st∈[∆rmt−1,burm,t−1,∆yt]. However, it is more strongly rejected in the case of st= ∆rmt−1than in the other cases, that is, at the 1% level rather that at the 5% level. The sequence of tests conducted to determine the form of the non-linearity suggests logistic functions of both st= ∆rmt−1and st=burm,t−1. In the case of st= ∆yt, however, none of the tests aimed at determining the form of non-linearity is rejected at the 5% level which undermines the evidence against linearity with ∆ytas the state variable. In addition to the right-hand-side variables appearing explicitly in the linear ECM for real money, we have also tested for possible non-linear effects with both the level and changes in real interest rates as transition variables. However, the hypothesis of linear effects was not rejected in either case, see the last two columns of Table 7. The rejection of linearity for st=burm,t−1is consistent with a number of studies of money demand. The suggested logistic form of the transition function is at variance with some of the well-known studies, though. Previously, the STR form of equilibrium correction models of money have been developed for e.g. the US, the UK, Italy and Germany; see Sarno et al. (2003), Ter¨asvirta and Eliasson (2001), Sarno (1999) and L¨utkepohl et al. (1999). These studies specify the transition function as an exponential function of the lagged value of equilibrium correction terms. However, evidence for the UK and Germany also supports logistic transition functions of income growth and inflation, respectively. 5.2 The STR models We specify the transition functions in light of the results in Table 7 and initially allow for non-linear effects of all explanatory variables in a model, except for the dummy variables. These general models are estimated by NLS and sequentially reduced to more parsimonious versions. In cases where several state variables were suggested, we make an effort to condition non-linear effects on each of these state variables, individually and jointly. Upon convergence of parameter estimates, we compare the performance of the different models of a variable in terms of explanatory power, interpretability and the extent to which they are able to represent the non-linear effects suggested by Table 7. Tables 8–10 present the preferred models. To ease comparison with the linear models, statistically insignificant variables appearing in the linear models have not been left out to achieve more parsimonious models. The tables also report comprehensive evaluations of the models. Specifically, they lay out outcomes of a number of tests aimed at detecting possible violations of the standard assumptions about residuals and functional form misspecification. Moreover, these tables report the outcome of the corresponding tests for the linear models that were estimated by OLS, cf. Table 4. In Table 11, we examine to what extent the proposed models capture the state-dependent effects suggested by Table 7 through testing hypotheses of no remaining non-linear effects. 19 5.3 LSTR model of output Table 8 presents the model of GDP with a logistic transition function of changes in real interest rates ∆RRt. It appears that increases in real interest rates above 3.9 percentage points tend to substantially push up the speed of adjustment towards the long-run equilibrium for GDP, cf. Sensier et al. (2002). Specifically, the speed of adjustment increases up to –0.473 (= –0.069 –0.404) per annum compared with the typical speed of –0.069 when changes in real interest rates are relatively smaller. Moreover, the adjustment speed is more than four times higher than that implied by the linear model (–0.106), see Table 4.9 [Table 8 about here.] Furthermore, the partial effect of the change in real interest rates becomes about ten times higher than suggested by the linear model of GDP, see Table 4. The LSTR model, however, indicates that only particularly large interest rate increases, i.e., above 3.9 percentage points, tend to have contractionary effects on GDP, see Figure 4. In particular, the estimated logistic transition function implies that cuts in real interest rates do not raise GDP growth. [Figure 4 about here.] Figure 2 shows that relatively large increases in the real interest rate occurred numerous times until about the early 1970s. Values of the transition function were mostly close to 1 during these occasions, owing to the step-form of the transition function, see Figure 4. Thus, the non-linear effects were quite active until the early 1970s. A closer examination of changes in the real interest rates suggests that large positive increases in the real interest rates mostly occurred during periods of large deflations until the late 1920s and due to sharp increases in nominal interest rates 9It should be noted that in the case of LSTR models in Tables 8–10, estimates of the transition parameter γs are relatively large even when scaled by the sample standard deviations of the corresponding transition variables. Moreover, they are imprecisely estimated. In general, numerically large values of γof a logistic transition function F(γ, c;st) make it change rapidly at even small deviations between stand c, and its shape becomes consistent with a broad range of values of γ. The high standard deviations of γare assumed to reflect this feature. In such cases, many observations in the neighbourhood of care required to obtain precise estimates of γ, cf. Ter¨asvirta (1994). Given that threshold values coften represent non-typical values of st, imprecise estimates of γare commonly encountered in the literature. This occurs particularly when st=t, as observations in the neighbourhood of care few by the nature of t. In the case of ESTR models, however, relatively large standard errors, may indicate relatively poor fit of the model, relative to a linear version of the model. Moreover, particularly large values of the transition parameters γs in an estimated ESTR model may suggest that the model can be considered linear in practice. The transition function converges to a single value in such cases, and acts as an impulse dummy. This is, however, not the case for the real exchange rate model (M2) in Table 9. 20 in the period afterwards, see Figure 5. [Figure 5 about here.] The addition of the state-dependent effects leaves the coefficient estimates of the remaining variables largely unaltered. Numerically, the coefficient estimate of ∆rext increases, while those of the impulse dummies d1862tand d22tfall. The explanatory power of the LSTR model is 9% higher than that of the linear models, as measured by the ratio of the standard deviations. The diagnostics show that the standard assumption about the error term and the presumed adequacy of the functional form are not rejected at the 5% level. There is an indication of ARCH effects in the residual as the p-value of the test statistics is 4.6%. In contrast, the outcome of the corresponding tests of the linear model suggests that most of the tested assumptions are rejected at the 5% level, while the null hypotheses of no autocorrelation and the extended RESET (with both cubic and square terms) can be nearly rejected at the 10% level. Finally, Table 11 shows that the null hypothesis of no remaining non-linearity of STR form with st= ∆RRtis not rejected. Also, the null hypothesis of time variation in parameters is not rejected, which indicates absence of time variation in the model’s parameters. 5.4 STR model of the real exchange rate As noted above, comparison of the OLS estimates with the FIML estimates for the model of ∆rextindicated some numerical differences. To separate the effect of nonlinearisation on parameter estimates from that of potential simultaneity bias, we use the linear model with OLS estimates in Table 6 as the reference model. [Table 9 about here.] Panel I of Table 9 presents the NLS estimates of the LSTR model with s=t, that is with time variation in a subset of parameters. The logistic function implies a permanent shift in the coefficient of ∆(co−couk)t−1, the deviation between domestic and foreign growth rates of public consumption, quite early in the sample period: around 1845. Accordingly, the coefficient estimate falls from −0.484 to −0.044 (= −0.484+0.440), becoming virtually equal to that in the linear model. The remaining coefficient estimates remain comparable to those in the linear model. By allowing for such time variation, the explanatory power of the model increases by 3% relative to the linear model. Panel II reports an extended model of the real exchange rate.10 This model 10Initially, we allowed its parameters to change over time and with lagged deviations of the real exchange rate from its long-run relationship (burex,t−1). Owing to non-convergence of the parameter estimates, we had to condition on the logistic specification of time in Panel I and then 21 supports the shift in the coefficients of ∆(co−couk)t−1around 1845. In this model the intercept term also becomes significant over time. However, the intercept term varies significantly with lagged deviations of the real exchange rate from its longrun relationship (burex,t−1). Due to the exponential transition function of burex,t−1the intercept rises (at most) to 0.019 for particularly large positive or negative values of burex,t−1. Accordingly, the negative intercept of –0.0183, which induces a negative drift in the real exchange rate of 1.83% per annum, is virtually cancelled out whenever burex,t−1is large (in absolute terms). [Figure 6 about here.] [Figure 7 about here.] On the other hand, the influence of burex,t−1on changes in the real exchange rate becomes larger whenever the negative intercept term is cancelled out. Thus the real exchange rate appreciation (or depreciation) can be larger than usual, i.e. when burex,t−1is not particularly large and the negative intercept term is active. Figure 7 suggests that estimated values of the exponential transition function of burex,t−1were often high and close to one when there was a downward trend in the real exchange rate, i.e. a tendency to appreciate, see Figure 1. One also gets the impression that relatively low values of the transition function often coincided with periods of relatively stable real exchange rate. Thus it seems that the downward trend in the real exchange rate does not vanish when the negative intercept term is outweighed. On the contrary, the downward trend over several periods becomes more pronounced on such occasions owing to the Balassa-Samuelson effect working through the burex,t−1term. The explanatory power of the extended model is 5% higher than that of the linear model and 2% higher than that of the non-linear model in Panel I. The diagnostics shows that the model satisfies the standard residual assumptions and that its functional form is adequate. We note that these tests are not rejected in the case of the linear model either. The tests for no remaining non-linearity in Table 11 indicate that time variation in the parameters have been adequately characterised though not fully satisfactorily. The results in the table also suggests that non-linearity with st=burex,t−1is still a feature of the model. Moreover, there is also weak evidence of non-linearity with st= ∆rext−1in the extended model (M2), although not in the simpler model (M1). The p-values are 4% and 7%, respectively. 5.5 LSTR model of real money The non-linear model of real money growth has been developed with its lagged value (∆rmt−1) as the transition variable. The linearity hypothesis was also rejected with allow coefficients to vary with burex,t−1. Except for the intercept term, none of the other coefficients seemed to vary significantly with burex,t−1, hence they were excluded from the model for the sake of parsimony. 22 where denotes element-wise multiplication of each element i, j in the matrices Φ2, Ψ2with the corresponding element i, j in the matrix Fwith nonlinear transition functions Fij(γij, cij, st,ij ) as its elements i, j or 0. If we index by F†() a given choice of values for the transition function elements F† ij, we can write the corresponding coefficient matrices of the companion form representation of the model as Φ†= [Φ1+ Φ2F†(Γ, C, S)] Ψ†= [Ψ1+ Ψ2F†(Γ, C, S)] The corresponding companion form is given by Zt= Φ†Zt−1+ Ψ†xt+Ut(13) The eigenvalues (characteristic roots) of the system matrix Φ†are useful to summarise the characteristics of the dynamic behaviour of the system, and by varying F†() we can explore the dynamic properties of the system when the transition function takes on different values. Admittedly, this is a rather crude approximation to the dynamic properties of the system, but it gives a rough indication about the dynamics of particular regime combinations stemming from the matrix of transition functions F†(). To calculate the roots of the system matrix Φ†, we have used the Gauss function EIG, which calculates the eigenvalues of a general matrix. References Akerlof, G. A. (1973), The Demand for Money: A General-Equilibrium InventoryTheoretic Approach, Review of Economic Studies, 40:115–1130. Akram, Q. F. (2004), Oil wealth and real exchange rates: The FEER for Norway, Working Paper 2004/16, Norges Bank. Balassa, B. (1964), The purchasing power doctrine: A reappraisal, Journal of Political Economy, 72:584–596. Baldwin, R. and P. A. Krugman (1989), Persistent trade effects of large exchange rate shocks, Quarterly Journal of Economics, 104:635–654. Bec, F., M. Salem and F. Collard (2002), Asymmetries in monetary policy reaction functions: Evidence for U.S. French and German central banks, Studies in Nonlinear Dynamics and Econometrics, 6:1–20. Burgess, S. M. (1992), Nonlinear dynamics in a structural model of employment, Journal of Applied Econometrics, 7:101–118. Clements, M. and H. M. Krolzig (1998), A comparison of the forecast performance of Markov-switching and threshold autoregressive models of US GNP, Econometrics Journal, 1:C47–C75. 29 Cover, J. P. (1992), Asymmetric effects of positive and negative money-supply shocks, Quarterly Journal of Economics, 107, 4:1261–1282. Diamond, P. (1982), Aggregate demand management in search equilibrium, Journal of Political Economy, 90:881–894. Dixit, A. (1992), Investment and hysteresis, Journal of Economic Perspectives, 6:107–132. Doornik, J. A. and D. F. Hendry (2001), Modelling Dynamic Systems using PcGive 10, Volume II (Timberlake Consultants LTD., London). Dumas, B. (1992), Dynamic equilibrium and the real exchange rate in spatially separated world, Review of Financial Studies, 5:153–180. Dumas, B. and L. E. O. Svensson (1994), How long do unilateral target zones last?, Journal of International Economics, 36:467–481. Edison, H. and J. T. Klovland (1987), A quantitative reassessment of the purchasing power parity hypothesis: Evidence from Norway and the United Kingdom, Journal of Applied Econometrics, 2:309–333. Eitrheim, Ø. and T. Ter¨asvirta (1996), Testing the adequacy of smooth transition autoregressive models, Journal of Econometrics, 74, 1:59–75. Engle, C. and J. D. Hamilton (1990), Long swings in the dollar: Are they in the data and do markets know it?, American Economic Review, 80:689–713. Engle, R. F. and C. W. J. Granger (1987), Co-integration and error correction: Representation, estimation and testing, Econometrica, 55:251–276. Feinstein, C. H. (1991), A new look at the cost of living 1870-1914, in J. ForemanPeck, ed., New perspectives on the late Victorian economy: Essays in quantitative economic history 1860-1914, pages 151–179 (Cambridge University Press, Cambridge). Feinstein, C. H. (1998), Pessimism perpetuated: Real wages and the standard of living in Britain during and after the industrial revolution, Journal of Economic History, 58:625–658. Granger, C. W. J. and T. Ter¨asvirta (1993), Modelling nonlinear economic relationships (Oxford University Press, Oxford). Grytten, O. H. (2004a), A consumer price index for Norway 1516–2003, in Ø. Eitrheim, J. T. Klovland and J. F. Qvigstad, eds., Historical Monetary Statistics for Norway 1819–2003, Occasional Papers No. 35, Ch. 3 (Norges Bank, Oslo). Grytten, O. H. (2004b), The gross domestic product for Norway 1830–2003, in Ø. Eitrheim, J. T. Klovland and J. F. Qvigstad, eds., Historical Monetary Statistics for Norway 1819–2003, Occasional Papers No. 35, Ch. 6 (Norges Bank, Oslo). 30 Hamilton, J. D. (1989), A new approach to the economic analysis of nonstationary time series and the business cycles, Econometrica, 49:1057–1072. Heckscher, E. F. (1916), V¨axelkursens grundval vid pappersmyntfont, Ekonomisk Tidskrift, 18:309–312. Hendry, D. F. (1995), Dynamic Econometrics (Oxford University Press, Oxford). Johansen, S. (1995), Likelihood-based Inference in Cointegrated Vector Autoregressive Models (Oxford University Press, Oxford). Kandil, M. (2001), Asymmetry in the effects of US government spending shocks: Evidence and implications, Quarterly Review of Economics and Finance, 41:137– 165. Karras, G. (1996), Are the output effects of monetary policy asymmetric? Evidence from a sample of European countries, Oxford Bulletin of Economics and Statistics, 58:267–278. Klovland, J. T. (2004a), Bond markets and bond yields in Norway 1820–2003, in Ø. Eitrheim, J. T. Klovland and J. F. Qvigstad, eds., Historical Monetary Statistics for Norway 1819–2003, Occasional Papers No. 35, Ch. 4 (Norges Bank, Oslo). Klovland, J. T. (2004b), Historical exchange rate data 1819–2003, in Ø. Eitrheim, J. T. Klovland and J. F. Qvigstad, eds., Historical Monetary Statistics for Norway 1819–2003, Occasional Papers No. 35, Ch. 7 (Norges Bank, Oslo). Klovland, J. T. (2004c), Monetary aggregates in Norway 1819–2003, in Ø. Eitrheim, J. T. Klovland and J. F. Qvigstad, eds., Historical Monetary Statistics for Norway 1819–2003, Occasional Papers No. 35, Ch. 5 (Norges Bank, Oslo). Krolzig, H. M. (1997), Markov-Switching Vector Autoregression (Springer-Verlag, Berlin). Lin, C.-F. J. and T. Ter¨asvirta (1994), Testing the constancy of regression parameters against continuous structural change, Journal of Econometrics, 62:211–228. L¨utkepohl, H., T. Ter¨asvirta and J. Wolters (1999), Investigating stability and linearity of a German M1 money demand function, Journal of Applied Econometrics, 14:511–525. L¨utkepohl, H. (1991), Introduction to Multiple Time Series Analysis (SpringerVerlag). MacKinnon, J. G. (1991), Critical values for cointegration tests, in R. F. Engle and C. W. J. Granger, eds., Long-Run Economic Relationships (Oxford University Press, Oxford). Maddison, A. (1991), Dynamic forces in capitalist development: A long-run comparative view (Oxford University Press, Oxford). 31 Meese, R. and A. K. Rose (1991), An empirical assessment of non-linearities in models of exchange rate determination, Review of Economic Studies, 58, 3:603– 619. Michael, P., A. R. Nobay and D. A. Peel (1997), Transaction costs and nonlinear adjustment in real exchange rates: An empirical investigation, Journal of Political Economy, 105:862–879. Mitchell, B. R. (1975), European Historical Statistics 1750-1970 (Columbia University Press, New York). Mitchell, B. R. (1998), International Historical Statistics: Europe 1750-1993 (4th ed.) (Macmillan, London). Murphy, K., A. Shleifer and R. Vishny (1989), Industrialization and the big push, Journal of Political Economy, 51:1003–1026. Neft¸ci, S. (1984), Are economic time series asymmetric over the business cycle?, Journal of Political Economy, 92:307–328. Officer, L. H. (2003), The Annual Real and Nominal GDP for the United Kingdom, 1086 - 2000., Economic History Services, June:http://www.eh.net/hmit/ukgdp/. Ozkan, F. G. and A. Sutherland (1998), A currency crisis model with an optimising policymaker, Journal of International Economics, 44:339–364. Parker, R. E. and P. Rothman (2004), An examination of the asymmetric effects of money supply shocks in the pre-World War I and Interwar periods, Economic Inquiry, 42:88–100. Rothman, P. (1998), Forecasting asymmetric unemployment rates, Review of Economics and Statistics, 80:164–168. Samuelson, P. (1964), Theoretical notes on trade problems, Review of Economics and Statistics, 46:145–154. Sarno, L. (1999), Adjustment costs and non-linear dynamics in the demand for money: Italy, 1861–1991, International Journal of Finance and Economics, 4:155– 177. Sarno, L. (2000), Real exchange rate behaviour in the Middle East: A reexamination, Economic Letters, 66:127–136. Sarno, L. and M. P. Taylor (2002), Purchasing power parity and the real exchange rate, IMF Staff Papers, 49, 1:65–105. Sarno, L., M. P. Taylor and D. A. Peel (2003), Non-linear equilibrium correction in US real money balances, Journal of Money Credit and Banking, 35:787–799. 32 Sensier, M., D. R. Osborn and N. ¨ Ocal (2002), Asymmetric interest rate effects for the UK real economy, Oxford Bulletin of Economics and Statistics, 64:315–339. Sercu, P., R. Uppal and C. van Hulle (1995), The exchange rate in the presence of transaction costs: Implications for tests of purchasing power parity, Journal of Finance, 50:1309–1319. Skalin, J. and T. Ter¨asvirta (1999), Another look at Swedish business cycle, Journal of Applied Econometrics, 14:359–378. Stiglitz, J. E. and A. Weiss (1981), Credit rationing in markets with imperfect information, American Economic Review, 70:393–421. Taylor, J. B. (2001), Low inflation, pass-through, and the pricing power of firms, European Economic Review, 44:1389–1408. Taylor, M. P., D. A. Peel and L. Sarno (2001), Nonlinear mean reversion in real exchange rates: Towards a solution to the purchasing power parity puzzles, International Economic Review, 42:1015–1042. Ter¨asvirta, T. (1994), Specification, estimation, and evaluation of smooth transition autoregressive models, Journal of the American Statistical Association, 89:207– 218. Ter¨asvirta, T. (1996), Power Properties of Linearity Tests for Time Series, Studies in Nonlinear Dynamics & Econometrics, 1:3–10. Ter¨asvirta, T. (1998), Modelling economic relationships with smooth transition regressions, in A. Ullah and D. E. Giles, eds., Handbook of Applied Economic Statistics, chapter 15, pages 507–552 (Marcel Dekker, Inc. New York). Ter¨asvirta, T. and H. M. Anderson (1992), Characterizing nonlinearities in business cycles using smooth transition autoregressive models, Journal of Applied Econometrics, 7:119–136. Ter¨asvirta, T. and A. Eliasson (2001), Non-linear equilibrium correction and the UK demand for broad money, 1878–1993, Journal of Applied Econometrics, 16:277– 288. White, H. (1980), A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica, 48:817–838. 33 9 10 11 12 13 14 15 1850 1875 1900 1925 1950 1975 2000 (a) Real GDP, log(Y) 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 1850 1875 1900 1925 1950 1975 2000 (b) Real exchange rate, log(REX) 0 1 2 3 4 5 1850 1875 1900 1925 1950 1975 2000 (c) Real narrow money, log(M0/CP I) 2 3 4 5 6 7 1850 1875 1900 1925 1950 1975 2000 (d) Consumer price index, log(CPI) -.3 -.2 -.1 .0 .1 .2 .3 1850 1875 1900 1925 1950 1975 2000 (e) Real interest rate, RR .02 .04 .06 .08 .10 .12 .14 1850 1875 1900 1925 1950 1975 2000 (f) Annual bond yield, R Figure 1: Historical data 1831–2003 (levels). Here and elsewhere in this paper, the shaded areas designate the Crimean war, WW I & II and the Korean war. 34 -.15 -.10 -.05 .00 .05 .10 .15 .20 1850 1875 1900 1925 1950 1975 2000 (a) ∆ log(Y) -.3 -.2 -.1 .0 .1 .2 .3 1850 1875 1900 1925 1950 1975 2000 (b) ∆ log(REX) -.4 -.2 .0 .2 .4 .6 .8 1850 1875 1900 1925 1950 1975 2000 (c) ∆ log(M0/CPI) -.2 -.1 .0 .1 .2 .3 1850 1875 1900 1925 1950 1975 2000 (d) ∆RR Figure 2: Growth rates in real GDP, the real exchange rate, narrow real money balances and changes in real interest rates 1831–2003. 35 -.2 -.1 .0 .1 .2 .3 5.6 6.0 6.4 6.8 7.2 7.6 1850 1875 1900 1925 1950 1975 2000 (a) Relative GDP, log(Y/Y UK) -1.5 -1.0 -0.5 0.0 0.5 1.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 1850 1875 1900 1925 1950 1975 2000 (b) Relative narrow money, log(M0/M0UK) -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 0 1 2 3 4 5 6 1850 1875 1900 1925 1950 1975 2000 (c) Relative public consumption, log(CO/COUK) Figure 3: Growth rates (left scale) and ratios (right scale) of Norwegian GDP, narrow money and public consumption relative to the UK. 36 0.0 0.2 0.4 0.6 0.8 1.0 -.2 -.1 .0 .1 .2 .3 (a) Transition function Fy×s 0.0 0.2 0.4 0.6 0.8 1.0 1850 1875 1900 1925 1950 1975 2000 (b) Transition function over time Figure 4: Logistic transition Fyfunction for the model of GDP, 1831–2000; st= ∆RRt. -.2 -.1 .0 .1 .2 .3 1850 1875 1900 1925 1950 1975 2000 Figure 5: Changes in real interest rates in periods with inflation (solid line) and deflation (dotted line). 37 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (a) Transition function Frex ×s 0.0 0.2 0.4 0.6 0.8 1.0 1850 1875 1900 1925 1950 1975 2000 (b) Transition function over time Figure 6: Logistic transition function for the real exchange rate model M1, 1831– 2000; st=t. 0.0 0.2 0.4 0.6 0.8 1.0 -.5 -.4 -.3 -.2 -.1 .0 .1 .2 .3 .4 (a) Transition function Frex ×s 0.0 0.2 0.4 0.6 0.8 1.0 1850 1875 1900 1925 1950 1975 2000 (b) Transition function over time Figure 7: Exponential transition function Frex for the real exchange rate model M2, 1831–2000; st=burex,t−1. 38 Table 7: Testing for non-linearity and for its form I. ECM of ∆yt s=t∆yt−1 buy,t−1∆cot∆rex t∆RRt∆rmt H0: F18,133 1.60[.07] 1.28[.21] 1.41[.13] 1.62[.06] 1.02[.44] 2.18[.01]∗∗ 1.55[.08] H04: F6,133 0.25[.96] H03: F6,139 2.72[.02]∗ H02: F6,145 3.53[.00]∗∗ II. ECM of ∆rext s=t∆rex t−1 burex ,t−1∆(m0−m0uk)t∆(co−couk)t−1 H0: F15,139 2.56[.00]∗∗ 1.88[.03]∗1.73[.05]∗1.04[.42] 1.67[.06] H04: F5,139 5.67[.00]∗∗ 1.56[.18] 1.03[.40] H03: F5,144 0.57[.72] 3.48[.01]∗∗ 3.47[.01]∗∗ H02: F5,149 1.16[.33] 0.48[.79] 0.63[.68] III. ECM of ∆rmt s=t∆rmt−1∆rmt−2 burm,t−1∆ytRRt∆RRt H0: F15,141 0.96[.51] 2.46[.00]∗∗ 1.38[.17] 1.96[.02]∗1.77[.04]∗0.44[0.96] 0.88[.59] H04: F5,141 3.05[.01]∗∗ 3.29[.01]∗∗ 1.55[.18] H03: F5,146 1.65[.15] 0.81[.54] 1.97[.09] H02: F5,151 2.34[.05]∗1.60[.16] 1.67[.15] Note: The F-tests associated with H0test the null hypotheses of linear effects from a variable against the alternative hypotheses of non-linear effects of STR form. The other F-tests are aimed at determining the form of non-linearity. 45 Table 8: Non-linear ECM of Norwegian GDP M1: LSTR ECM of ∆ytwith s= ∆RRt ∆byt= 0.019 (0.003) + 0.225 (0.053) ∆yt−1−0.069 (0.038) buy,t−1+ 0.238 (0.067) ∆co ×SD50t + 0.015 (0.049) ∆RRt+ 0.072 (0.033) ∆rext+ 0.073 (0.020) ∆rmt+ 0.048 (0.022) d1862 −0.125 (0.021) d17 + 0.173 (0.029) d19 −0.119 (0.026) d21 + 0.032 (0.025) d22 −0.114 (0.021) d31 −0.126 (0.023) d40 −0.083 (0.021) d44 + 0.105 (0.022) d45 + 0.064 (0.022) d46 + 0.072 (0.022) d47 +  0.033 (0.010) −0.464 (0.106) ∆RRt−0.404 (0.095) buy,t−1   ×  1 + exp(−231.196 (17655) ×1 0.0612 ×(∆RRt−0.039) (0.127)   −1 bσ∆y,M1= 0.0208; bσy= 0.023; bσ∆y,M1/bσ∆y= 0.91 Diagnostics LSTR ECM (M1) Linear model Normality χ2(2) = 2.804 [.246] χ2(2) = 9.20 [.010]∗∗ AR1−3F3,136 = 0.694 [.557] F3,145 = 2.132 [.099] HetXi2F32,134 = 1.131 [.307] F24,145 = 4.070 [.000]∗∗ ARCH1−3F3,158 = 2.735 [.046]∗F3,162 = 2.897 [.037]∗ RESET (sq.) F1,144 = 0.028 [.868] F1,150 = 4.003 [.047]∗ RESET (sq. and cub.) F2,143 = 0.405 [.668] F2,149 = 2.266 [.107] Note: M1 is our preferred LSTR model with st= ∆RRt. The transition parameter has been scaled by the empirical std. deviation of ∆RRt. The panel of diagnostics lays out observed test-statistics and the associate p-values in square bracket for a number of standard tests for model misspecification. Specifically, we test the following null hypotheses: the null hypothesis of normally distributed errors, tested by Jarque-Bera chi-square test; No residual autocorrelation up to order 3; No residual heteroscedasticity, which has been tested by including the regressors and their squares; No ARCH effects up to order 3; And finally, the null hypothesis of correct model specification, through two RESETs. The outcome of the first RESET refers to the case when the significance of the square of the fitted value is tested in the model while the second one refers to the case when the joint significance of the second and third power of the fitted value is tested. Sample 1832–2000; Method: NLS. 46 Table 9: Non-linear ECM of the real exchange rate M1: LSTR ECM of ∆rextwith s=t ∆drext= 0.095 (0.059) ∆rext−1−0.099 (0.028) burex,t−1−0.484 (0.132) ∆(co −couk)t−1 −0.083 (0.030) ∆(m0−m0uk)t−0.172 (0.060) ∆(m0−m0uk)W2Wt+IDs +  −0.004 (0.003) + 0.440 (0.133) ∆(co −couk)t−1   × 1 + exp(−11604 (NC) ×(t/T −0.074) (large)   −1 bσrex,M1= 0.0356,bσrex,M1/bσrex = 0.97 M2: STR ECM of ∆rextwith s1=tand s2=burex,t−1 ∆drext= 0.129 (0.057) ∆rext−1−0.117 (0.026) burex,t−1−0.533 (0.127) ∆(co −couk)t−1 −0.060 (0.029) ∆(m0−m0uk)t−0.192 (0.060) ∆(m0−m0uk)W2Wt −0.255 (0.036) d18 + 0.139 (0.037) d20 + 0.124 (0.036) d23 −0.324 (0.072) d29 −0.117 (0.034) d68 −0.148 (0.035) d73 −0.126 (0.036) d76 + 0.107 (0.036) d79 + 0.102 (0.036) d80 + 0.128 (0.035) d97 +  −0.0183 (0.005) + 0.491 (0.127) ∆(co −couk)t−1   × 1 + exp(−11603.97 ×(t/T −0.074 )  −1 + 0.019 (0.005) × 1−exp(−79.245 (50.892) ×(burex,t−1−0.00)2 (0.00)   bσrex,M2= 0.0349,bσrex =0.0367; bσrex,M2/bσrex = 0.95 Diagnostics STR ECM (M2) Linear ECM Normality χ2(2) = 4.153 [.125] χ2(2) = 3.369 [.190] AR1−3F3,141 = 2.072 [.107] F3,148 = 1.834 [.143] HetXi2F29,137 = 1.226 [.218] F20,148 = 1.230 [.238] ARCH1−3F3,162 = 0.213 [.888] F3,162 = 0.704 [.551] RESET (sq.) F1,146 = 2.324 [.130] F1,153 = 0.476 [.491] RESET (sq. and cub.) F2,145 = 1.313 [.272] F2,152 = 0.356 [.701] Note: M1 is our preferred LSTR model with s=t, which has been scaled by the total number of observations T (= 169). The dummies have been suppressed to save space. Their effects are represented by the term ”IDs” and are almost identical to those presented in M2. NC means Not Computed. M2 is our preferred model with both LSTR and ESTR type of effects triggered by the two transition variables tand burex,t−1, respectively. M2 has been estimated by conditioning on the estimate of γand cfrom M1. The transition parameter in the ESTR term has been scaled by the empirical std. deviation of burex,t−1. We have also computed the ratios between the estimated standard deviation of the residuals from M1 and M2 relative to that from the linear ECM of rex in Table 4. The tests are those proposed by Eitrheim and Ter¨asvirta (1996). The square brackets contain p-values. The sample period is 1832–2000; Method: NLS. 47 Table 10: Non-linear ECM of money demand M1: LSTR ECM of ∆rmtwith s= ∆rmt−1 ∆crmt= 0.172 (0.100) ∆rmt−1−0.189 (0.061) ∆rmt−2−0.086 (0.017) burm,t−1 + 0.442 (0.206) ∆y×preW1t+ 0.421 (0.157) ∆y×postW1t+ 0.129 (0.031) W1t+ 0.293 (0.041) W2t + 0.180 (0.068) d16 −0.257 (0.068) d18 + 0.240 (0.060) d26 + 0.383 (0.068) d41 + 0.237 (0.067) d47 +  0.036 (0.018) −0.371 (0.154) ∆rmt−1+ 0.060 (0.029) burm,t−1   ×   1 + exp(−216.322 (2980.639) ×1 0.0949 (∆rmt−1−0.058 (0.014))   −1 bσrm,M1 = 0.0588, bσrm = 0.0596; bσrm,M1 /bσrm = 0.99 Diagnostics LSTR ECM (M1) Linear model Normal χ2(2) = 4.84 [.089] χ2(2) = 8.282 [.016]∗ AR1−3F3,145 = 1.26 [.292] F3,150 = 1.384 [.250] HetXi2F27,140 = 1.21 [.231] F17,150 = 1.159 [.305] ARCH1−3F3,161 = 0.75 [.522] F3,161 = 0.231 [.875] RESET (sq.) F1,150 = 0.21 [.645] F1,155 = 0.224 [.637] RESET (sq. and cub.) F2,149 = 0.73 [.482] F2,154 = 0.323 [.724] Note: M1 is our preferred LSTR model with st= ∆rmt−1. The transition parameter has been scaled by the empirical std. deviation of ∆rmt−1. The sample period is 1834–2000; Method: NLS. 48 Table 11: Testing for no remaining non-linearity STR ECM of: ∆yt∆rext∆rmt s= ∆RRt∆rext−1∆rmt−1 M1: F18,127: 1.10[.36] F15,135: 1.63[.07] F15, 136: 2.15[.01]∗∗ M2: F15,132: 1.80[.04]∗ s=burex,t−1burm,t−1 M1: F15,135: 2.14[.01]∗∗ F15,136: 1.73[.05]∗ M2: F15,132: 2.08[.02]∗ s=t t t M1: F21,124: 1.39[.14] F18,132: 1.70[.05]∗F15,136: 1.13[.33] M2: F18,129: 1.49[.11] Note: The first row indicates the non-linear ECM of a given variable while the rows headed by spresent the transition variable (s) defining the non-linear model, see Tables 8–10. The Fdf1,df2-tests test whether there is any remaining non-linearity of STR type for a given s in the non-linear ECMs. We also test whether there is any remaining non-linearity when s = t, the time trend, cf. Table 7. 49 Table 12: Dynamic properties of the linear and non-linear systems Properties of the complete system Roots Modulus Period Linear VECM (Table 4) 0.82 +/- 0.44i 0.93 12.8 0.83 0.83 0.06 +/- 0.38i 0.38 4.5 0.28 0.28 System of the STR models (Tables 8–10) Fy= 0, Frex = 0, Frm = 0 0.84 +/- 0.39i 0.93 14.5 0.86 0.86 0.10 +/- 0.45i 0.46 4.7 0.24 0.24 0.15 0.15 Fy= 1, Frex = 1, Frm = 1 0.86 0.86 0.74 0.74 0.41 +/- 0.34i 0.53 9.1 -0.10 +/- 0.45i 0.46 3.5 0.15 0.15 Fy= 1, Frex = 1, Frm = 0 0.85 0.85 0.65 0.65 0.42 +/- 0.38i 0.57 8.5 0.12 +/- 0.44i 0.46 4.8 0.15 0.15 Fy= 0, Frex = 1, Frm = 1 0.86 +/- 0.36i 0.94 15.8 0.87 0.87 -0.10 +/- 0.45i 0.45 0.24 0.24 3.5 0.15 0.15 Note: We have used the Gauss function EIG to calculate the roots. Fy= 0: Weak response to ∆RRt.Fy= 1: Strong response to ∆RRt.Frex = 0: Weak response to ∆(co−couk)t−1, Frex = 1: Strong response to ∆(co −couk)t−1.Frm = 0: Strong equilibrium reversion. Frm = 1: Weak equilibrium reversion. 50 Q. Farooq Akram, Øyvind Eitrheim and Lucio Sarno: Non-linear dynamics in output, real exchange rates and real money balances: Norway, 1830-2003 Working Paper 2005/2 KEYWORDS: Business cycles Real exchange rates Money demand Non-linear modelling Smooth transition regressions - 32799