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The Degree of Independence in Goods and Capital
Markets: An econometric degustation
Roger Hammersland
Thesis submitted for assessment with a view to obtaining
the degree ofDoctor o f the European University Institute
European University Institute
EUROPEAN UNIVERSITY INSTITUTE
Department of Economics
The Degree of Independence in Goods and Capital
Markets: An econometric degustation
Roger Hammersland
The Thesis Committee consists of:
Prof. Michael Artis, EUI, Co-Supervisor
"
Giorgio Calzolari, Università di Firenze
"
Soren Johansen, external EUI & University of Copenhagen, Supervisor
"
Grayham Mizon, University of Southampton
To my son Oystein, who suffered most from this ego trip.
Writing a thesis indebts one to many. To render justice to all would require a book in itself. I therefore forced to limit myself to few. I hope that my friends at the EUI in particular, will excuse me for the oversight.
My wannest thanks goes o f course to those nearest and dearest to me, my wife Monica and her wonderful Sardinian family for having helped and encouraged me in every respect the last couple o f years. Without them this thesis would never have seen it’s final stage and I would probably also have been fifteen kilos lighter. Mind you Rita!
Then o f course, I want to thank Soren Johansen for giving me the opportunity to w rite a thesis under his supervision. I want particularly to thank him for giving me support and help some years ago in an, to me, extremely difficult period o f life. Thanks also to my second supervisor, Mike Artis, for having taken the time to thoroughly read through all my papers, some o f them several times, and for offering suggestions, amendments and comments, all o f which have contributed significantly to this work. Then o f course I wish to extend a particular word o f thanks to Katarina Juselius, who through a countless number o f encouraging discussions, seminars and workshops gave me the inspiration and motivation I needed to be able to finish this thesis.
I also want to extend a particular word o f thanks to Anja Haensch and Matthias Rau for helping me to combat occasional problems with a lack o f self-confidence and for encouraging me to go on in times I felt most like giving up. Thanks also to my three “drinking” companions, Hans Van Der Veen, Erik Tangerstad and James Kay, to whom I owe an awful lot, last mentioned also for proofreading parts o f Chapter 2 and the introduction o f this thesis. Finally, I want to thank those not mentioned in the above who in some way o r other has helped and encouraged me during my stay in Italy. I want you all to know that I have not forgotten a single one o f you. Thanks.
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---Contents
Introduction i-v
Chapter 1: Who is in the driving seat in Europe, International 1-30 financial markets or the BUBA?
Chapter 2: “We are arrogant because we are good”, revisited: A 31-58 critical appraisal o f Central Banking versus fiscal policy
in accomplishing the community wide convergence o f the eighties and the nineties.
Chapter 3: Large T small N: A two-step approach to the 59-120 identification o f cointegrating relationships in time series
models with a small cross-sectional dimension.
Chapter 4: Bootstrapping or train-spotting: A note on small sample 121-137 properties o f the trace statistics related to specific VARS.
Chapter 5: The degree o f independence in European goods markets. 139-180
Appendix The Determination o f export volumes and export prices 181-196 within a theory o f monopolistic competition.
Introduction
The purposes o f this thesis are manifold. Nevertheless one may say that one o f its governing ideas has been to make use o f some o f the most advanced and recently developed state of the art time series techniques that exist to analyse actual data in as coherent, scientific and correct environment as possible. The overriding concern for this has primarily been to detach as far as it can be done, discussions o f economic matters from non-scientific and manipulative approaches where the focus as suggested by the title, has particularly been on revealing the degree o f independence in capital and goods markets. In this respect, as this thesis focuses on the analysis o f long-run relationships among the times series o f individual data sets, a matter o f particular concern has been to avoid resorting to dummies in the process o f m odel design and identification, beyond what has been necessary, o f course, to get residuals with sufficiently “nice” properties not to invalidate the statistical analyses. This choice has been made upon a strong and a priori personal belief that long-run common features among time series, like shared common trends, if they at all are to be considered as robust, should not be as heavily affected by outliers that they legitimate a whole battery o f dummies. Thus, to let them in should, in any case, not influence the outcome o f the analysis o f the long-run relationships as opposed to how they might affect a dynamic model specification. The development o f fully specified dynamic models has thus been deemed less urgent, though I admit that this might potentially have had the effect o f creating certain difficulties with regard to the possibility o f developing a congruent dynamic representation o f the information contained in data based on the identified long-term structures. Another important purpose o f this thesis has been to spark what is meant to be a creative discussion o f the different time series techniques involved, hopefully to enhance and elaborate their understanding as well as to pinpoint their implicit limitations and advantages. Finally, in some innocent way I have also tried to contribute to the econometric literature by suggesting new ways to deal with certain kinds o f problems and data. Below follows a brief summary o f the individual chapters and their aims.
The first chapter deals with the identification of international interest rate linkages between European and international capital markets. Besides identifying the long-term cointegrating relationships among the times series, using the method developed by S. Johansen (1988), the paper also seeks to develop a structural VAR model. The results are rather mixed as the outcome o f the cointegration analysis suggests that long-term European interest rates are
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driven in the long run by the corresponding international ones at the same time as the dynamic structure implies short-run effects on long-term European rates o f changes in short-term European interest rates. Even though the first result might suggest some kind o f impotence on the part o f Central Banks in the conduct o f monetary policy, the second could indeed be taken to indicate quite the contrary. That is that monetary policy is effective through affecting long term interest rates and expectations w ith regard to future interest rates in such a way that it neutralizes an eventual effect that short rates might have on long-term interest rates in the long run. However as paper number two, to which we now turn, strongly suggests, this potential explanation might be an illusion as the dynamic structure given to the VAR model o f this paper turns out to be highly arbitrary. Furthermore, the choice made in this paper with regard to the num ber o f cointegrating vectors is not a trivial one as the analysis indicates the potential existence o f an additional cointegrating vector. The elaboration o f this possibility and the discussion o f its implications are placed in Chapter 3 o f this thesis.
The second paper, Chapter 2, aims at critically discussing the widely adopted perception o f central banks having been the crux o f the two periods o f convergence observed during the eighties and nineties. Based on the results o f Chapter one, a paper by Juselius and McDonald (2000) and an independent analysis herein, the paper concludes that there is evidence that the dynamic short-run effect o f short rates on long rates identified in the first paper o f this thesis, is a spurious one. If so, this could seriously jeopardize the conclusions made w ith regard to the ECB’s ability to run an independent monetary policy; as long-term interest rates would be totally determined by what is going on in international markets an independent monetary policy can have no bearing whatsoever on long-term European interest rates. W hether this is the case rests on the assumption that monetary policy works mainly through the way that policy rates affect long-term interest rates and in so far as it is correct, this would imply that Central Bank policy cannot have been the only factor influencing the convergence process during the last tw o decades. Other factors must have had important roles as well and the paper points to fiscal policy and the effect o f improved mobility in capital and goods markets as alternative and supplementary explanations. The second part o f the paper then tries to bring this discussion one step further as a potential loss o f control on part o f central banks with regard to the long end o f the capital market, could have clear policy implications, particularly with regard to solving the issue o f unemployment in Europe. In this respect the paper ends up suggesting the use o f regionally directed policies geared towards stimulating investment and boosting demand.
The aim o f Chapter three has been to develop a new technique to deal with cointegration when data in addition to varying along a time series dimension, vary along a cross sectional dimension that is not too large. The suggested strategy is a simple one and implies undertaking the analysis in two-steps w ith the possibility o f adding a third step to improve upon the estimates. The first step involves making an ordinary section-wise cointegration analysis. The second step then treats the cointegrating relations o f the first step as known and looks for long-run relationships across sectors conditional on these by again using Johansen’s Maximum Likelihood procedure in a straight-forward way, explicit account o f course taken o f the fact that the distribution o f the trace statistics now will deviate from the ordinary asymptotic one. A motivation for the idea o f treating some “known” cointegrating vectors as fixed when in fact these have been estimated in a preliminary step, is given in Chapter 4. The recommended third step, after having identified the long-term structure, is to estimate all parameters simultaneously; this is to take into account the potential non-diagonality o f the covariance matrix and thus to improve upon the estimates. In the paper the suggested procedure is used to identify cointegrating relationships between and within sectors related to two applied studies, respectively, the international interest rate study o f the first two chapters o f this thesis and a study o f Norwegian exports. In both studies the sector dimension is equal to two, which o f course makes them particularly suitable for illustrative purposes, and the suggested procedure turns out to be able to identify cointegrating relationships across sectors as well as between for both o f them. W ith regard to the first o f these studies, the procedure is in addition able to pinpoint the existence o f a third cointegrating relation, a possibility that to some degree had already been addressed in the first paper o f this thesis. Even though the identification o f this third long-run relationship does not have a bearing on the conclusion made with regard to the ability o f Central Banks to control the long end o f the yield curve on the basis o f two cointegrating vectors, it nevertheless turns out to have substantial implications with respect to the status o f short-term interest rates and how they might be affected by long-tem interest rates through their potential capacity o f informing policy rules on the part o f Central Banks.
Chapter four I have chosen to call: “Bootstrapping or train-spotting: A note on small sample properties o f the trace statistics related to specific VARS”, and as the name suggests, not only aims at discussing the small sample properties o f the trace test statistics related to the studies o f Chapter 1 and 3 herein, but also to critically discuss the value added o f undertaking Monte
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Carlo, and Bootstrapping in particular, on non robust coefficients and test statistics when the central premise that one knows the DGP probably is far from satisfied and one in fact is forced to use a substitute that might potentially deviate from it significantly. As the paper is partly based on the preceding chapter, the text also aims at discussing the idea o f treating some o f the cointegrating vectors as fixed when in fact these have been estimated in a first step.
The final chapter, Chapter 5, is a study o f German and Norwegian exports and aims particularly at addressing the issue o f I(2)-ness. A central goal has therefore been to unveil potential signs o f higher order non-stationarity and in the case this is found to be evident, to identify potentially multi-cointegrating relationships. In this, there has been no intention o f forcing I(2)-ness upon the data and as stated in the introduction o f the chapter the approach has been more to cling to a null o f 1(1) than to continue along the dimension o f an artificially made supposition. In this respect it may also be added that the 1(2) analysis is deemed less urgent as the aim o f both studies is to reveal generic properties o f the underlying data generating processes. However, when this is said, it must also be stressed that an 1(2) analysis may be an interesting exercise to carry out even in the case one might not feel confident about its premises; i f nothing else, to compare with and eventually to support the outcome o f an 1(1) analysis. This m ore pragmatic view is the preferred one w hen interpreting the results o f the 1(2) analysis in Section 4 o f this chapter. The paper is also given an economic motivation: to test the claims o f foreign trade entrepreneurs that their businesses are extremely vulnerable to vagaries o f foreign demand and prices as well as to shocks to supply, like hikes in wages and prices o f intermediate products. To be able to discuss this issue in its full generality the theoretical framework has been an encompassing on, meaning that most special cases constitute restrictions on a parameterised version o f a general model. M y results indicate that there is monopolistic power in the process determining export prices, not only in a big country like Germany but also in a small open one like Norway. Furthermore, exports and export prices seem to be heavily affected by shocks to world quantities like world demand and world prices. This at least is in accordance with the claims o f foreign entrepreneurs. Their businesses are heavily influenced by the vicissitudes in international trade.
References
Johansen, S. (1988), Statistical analysis o f cointegration vectors. Journal of
Economic Dynamics and Control 12, 231-254.
Juselius, K. and MacDonald R. (2000), International Parity Relationships
between Germany and the United States: A joint Modelling Approach,
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Chapter 1
“W ho’s in the driving seat in Europe,
International financial markets or th e
BUBA? ” *
Roger Hammersland
European University Institute
Florence
A b stra c t
The purpose of this chapter is to reexamine empirically the rela tionship between long-term interest rates in well integrated financial markets. The analysis focuses on long-term interest rates in the US and Germany and has been carried out within the framework of a five dimensional VAR for the simultaneous determination of short- and long-term interest rates in the US and Germany and the rate of de preciation. The results strongly support the existence of a long-run relationship between the long-term German and the long-term US in terest rate and imply a full pass-through of changes in the long-term US rate into the corresponding German rate. The analysis also sub stantiates that the direction of causality goes from the long-term US to the long-term German interest rate. With regard to the possibility of controlling the long end of the market on the part of the Bundes bank, the paper apparently takes on a rather pessimistic view, as there is nothing to indicate a long-run relationship between domestic short- and long-term German interest rates. However, the strong influence that short-term German interest rates exhibit on German long-term interest rates in the very short run according to the structural model of this paper, might be taken to indicate that the opposite is the case, *1 am grateful for comments by Soren Johansen, Grayham Mizon and participants at the first year student forum at the EUI, Florence. I want also to thank Birger Vikoren with whom I wrote the forerunner of this paper*
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as effects originating from expectations with regard to future short term interest rates might totally neutralize an unequivocally positive short-run portfolio effect in the long rim. If this is the case, there is nothing strange in the fact th at one is unable to identify a long run relationship between domestic short- and long-term German in terest rates. On the contrary it is exactly w hat to be expected if the monetary transmission mechanism works appropriately.
C o n ten ts
1 Introduction
2
2 Integration, C oin tegration and W eak E xogen eity
7
3 A conditional error correction m odel for th e long- and sh ort
term German in terest rate.
18
4 Sum m ary and conclu sions
21
A Tables and graphs
24
1
In tr o d u c tio n
Recently, there has been som e focus o n w h at im p a c t increased capital mo bility could have on th e determ in atio n o f long-term interest ra te s (e.g. Borio and M cCauley (1996) and O EC D (1996)). T hese studies have been initiated by th e striking co-movements o f long-term in terest ra te s in US and E urope so far in th e 1990s (Figure 1). A recent s tu d y of th is relationship su b stan tiates this high degree of correlation and also suggests th a t long-term E uropean in terest ra te s seem m ainly to b e determ ined by US long-term interest ra te s in the long run, the causality going only one way, from th e US to th e E uropean economy (H am m ersland an d Vikoren (1997)). However, the model developed in this p ap er does n o t really seem to explain th e events of 1997, when th e two interest rates start to diverge. Also, problem s w ith interpreting th e m odel’s long-run relationship suggest extending th e inform ation set to improve on th e model. However, before s ta rtin g th e analysis, I will look a t two types of explanations, one m acroeconom ic and one microeconomic, which have been suggested as reasons for th e stro n g co-movements in long-term interest rates
* * * *
Figure 1: Long-term interest ra te s in Germany (R10DME) and th e US (R10USE).
in recent years. It is im p o rtan t to realize th a t these explanations are all based on tim e series being statio n ary and th a t a high degree of correlation may be spurious as a consequence of non stationarity. W hen analyzing th e actual d a ta in th e next sections to come, it is therefore extremely im portant to use a m ethodology th a t is capable of identifying th e fundam ental factors behind the correlation p a tte rn s observed between th e tim e series. T his is th e m ain reason w hy when analyzing th e data, I pursue a reduced ra n k VAR analysis in th is paper.
A typical macroeconomic explanation for th e correlation between nom inal long-term interest rates across countries assumes th a t these rates are roughly equal to th e sum of real long-term interest ra te s and inflation ex pectations. D isregarding for a m om ent th e problem commented on above w ith regard to spurious correlation w hen dealing w ith non-stationaxy data, correlation between nom inal in terest rates m ust therefore entail th a t there is a correlation between real interest rates a n d /o r a correlation betw een in flation expectations. T h e jo in t hypothesis of uncovered interest ra te parity (UIP) and ex a n te Purchasing Power P arity (P P P ) leads to real interest ra te parity (R IP). A lthough it is a widely held view th a t R IP does not hold in th e short run, K ing (1992) argues th a t R IP is more likely to hold in th e long run. In this case, real long-term interest rates will be highly correlated between
countries. There m ight also b e a correlation betw een inflation expectations in different countries due to eith er significant changes in com m odity prices or to synchronized changes in th e assessm ent of th e business cycles in various countries.
A microeconomic explanation looks a t th e tra d in g strategies of large in stitu tio n a l investors. For instance, th e increase in bond rates in th e US and E u ro p e during 1994 has been explained by th e observation th a t th e fall in bond prices in the US prom pted highly leveraged investors to sell US as well as E u ro p ean bonds. T his explanation is su p p o rted by Borio an d M cCauley (1996) w ho examine th e rise in long-term interest ra te s in 1994 and conclude th a t m ark ets’ own dynam ics seem to provide a stronger explanation th a n m arket participants’ apprehensions ab o u t economic fundam entals.
So far, I have focused on th e relationship betw een foreign long-term in terest ra te s across countries. However, th e expectations th eo ry of th e term stru c tu re entails t h a t th ere should also b e a relationship betw een sh o rt-term and long-term in terest rates in each country. According to th is theory, th e long-term interest ra te is equal to a weighted average of th e current an d ex pected future short-term in terest ra te (see Schiller (1979)). T hus, th e im pact on th e long-term interest ra te from a change in th e current sh o rt-term interest ra te depends on how expected fu tu re sh o rt-term interest ra te s are affected. A rise in th e current sh o rt-term interest ra te th a t is regarded as p erm anent will lead to a full pass-through from sh o rt-term to long-term interest rates. On th e other hand, if an increase in th e current sh o rt-term interest r a te leads to a significant reduction in inflation expectations, long-term interest rates m ay even decline.
T h e discussion above shows t h a t b o th dom estic short-term in terest rates and foreign long-term in terest ra te s could have a n im pact on dom estic long term in terest rates. G o o d h art (1995) recognizes this and argues th a t in creased capital m obility has led to a g reater tension betw een international pressure (e.g. foreign long-term in terest rates) and dom estic factors (e.g. th e expected tim e-p ath of future sh o rt rates) in th e determ ination o f long te rm in terest rates. However, uncovered interest p arity and relative purchas ing pow er parity, used to explain th e correlation betwreen long-term interest rates, also suggests effects from differences in inflation rates or th e expected ra te o f depreciation, and a unified tre a tm e n t of all these possibilities m ay be given w ith in th e fram ew ork o f a loanable funds equilibrium approach where in te re st rates are determ ined by th e dem and an d supply of funds (B randson (1977)).
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Figure 2: In terest spreads between long-term interest rates in th e US and Germ any (SlODMUS) and between domestic long-and short-term G erm an interest rates (SD103).
In Figure 2 I plot th e spreads between long-term interest rates in Ger many and th e US and between domestic long- and short-term G erm an in terest rates, respectively. G raphical inspection indicates a possible long-run relationship between Germ an and US long-term interest rates, although ex tended periods are observed in which th e long-run relationship does not seem to hold. However, a similar relationship between G erm an short and long-term interest rates does not seem to exist.
Below, I shed further light on these issues by undertaking an empirical analysis of nom inal short- and long-term interest rates in Germany, (iGL and iGS), an d the US, (iUL and i u s ). T h e empirical proxies for long-term interest rates have been effective interest ra te s on Government bonds with ten years to m atu rity while short-term interest rates are represented by the corresponding three m onths money m arket interest rates1. T he inform ation set also consists
l The concept effective interest rates refers to the fact that one has taken into account the compound interest rate effect. In the general case with a deposit with a term to maturity less than one year this might be given the following representation:
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o f th e actual rate o f depreciation (D v), w here th e exchange ra te is th e log of G erm an marks per US dollar. The ratio n ale for including th is variable was alluded to in the above and comes from th e a rb itrag e condition of uncovered interest ra te parity, saying th a t in a stea d y s ta te th e re tu rn o f investing one u n it o f domestic currency a t hom e o r ab ro ad should b e equal. T h u s, the dom estic interest ra te , i D, should be eq u al to th e foreign in terest ra te , iF, plus th e expected percentage increase in th e value of th e foreign currency relative to th e domestic currency, th a t is th e expected depreciation o f the bilateral domestic exchange ra te , over th e horizon we are looking a t * 2. The analysis has been und ertak en using m o n th ly d a ta for th e perio d 1990 (1) to 1997 (12) and has been carried o u t w ithin th e fram ew ork of a five dim ensional VAR m odel for th e sim ultaneous determ in atio n o f th e four interest ra te s and th e ra te o f depreciation. To b e able to te s t th e F ish e r hypothesis and to build a m odel of inflation, inform ation sets including inflation rates and indicators of dom estic activity have b een tried o u t prior to th e em pirical analysis of this p ap er. However, th ese a tte m p ts have so far n o t succeeded and belong to th e field to be further explored. C om pared to a stu d y undertaken o n a d a ta set comprising only th e four interest ra te s, it tu rn s o u t th a t th e widening of th e inform ation set to also include th e b ilateral exchange ra te revises results and m akes it possible to identify an in terp retab le long-run relationship. The m odel’s forecast ability is also strongly im proved com pared to a m odel of interest rates only.
T h e rest of this ch ap ter is organized as follows. Section 2 exam ines coin teg ratio n and exogeneity. Section 3 th e n presents th e outcom e o f a stru c tu ra l reinterpretation of th e reduced form analysis. Section 4 contains concluding rem arks.
where: i is the effective interest rate, r the nominal coupon interest rate and n the number o f periods per year. The implicit assumption in the above example is that the principal amount and the accrued interest rate are re-invested at the same nominal rate o f interest rate throughout the period. In the case of bonds with fixed coupon dividends the formulas become slightly more elaborate and the interested reader is referred to The Norwegian Society of Financial Analysts (2001).
2 In the chapter I have used the monthly change in the logarithm of the bilateral ex change rate, being aware of the fact that it would have been more correct from a theoretical perspective to use the change over three months. However, one may argue that investors operating in the markets are using the monthly change as an indicator because it is a more updated proxy for what it after all seeks to capture, namely the expected rate of depreciation.
2
In teg ra tio n , C o in te g r a tio n a n d W eak E x o
g e n e ity
This section presents statistics for testin g stationarity of th e individual tim e series in the information set. Jo h an sen ’s maximum likelihood procedure is applied to te st for cointegration an d the direction of causality among th e short-and long-term interest rates in Germany and th e US.
Prior to modelling, it is useful to determ ine th e orders o f integration of th e variables in th e inform ation set. Below, I therefore first present th e results of using ordinary univariate augm ented Dickey-Fuller (A D F) tests for un it roots in individual tim e series (Dickey a n d Fuller (1981)). However, I also present th e results w hen using th e Johansen m ethod to te st for statio n arity in a multivariate framework. These tw o approaches for testin g statio n arity differ in two im portant respects. First, w hen using th e Johansen approach th e null- hypothesis is th a t the individual tim e series is stationary, while Dickey-Fuller tests have non-stationarity as their null-hypothesis. Second, th e m ultivariate test statistics are conditional on th e number of cointegrating vectors in the information set.
Table 2.1 lists augm ented Dickey-Fuller te st statistics for th e long- and short-term interest rates in G erm any and th e US. T he la st column also gives the tests for th e ra te of depreciation. T he absolute value o f th e deviation from unity of the estim ated largest root appears in parentheses below each Dickey- Fuller statistic: this deviation should b e approxim ately zero if th e series has a unit root. U nit root tests are given for th e variables in levels and for th eir first differences. T his perm its testing w hether a given series is 1(0), 1(1) or I(2)3, albeit in a pairwise fashion for adjacent orders of integration. According to the un it root tests all variables except for th e ra te o f depreciation appear to be integrated of order one4. T h e ra te of depreciation on the o ther hand seems to be a stationary variable.
Table 2.2 below reports values o f a m ultivariate statistic for testing the times series properties of a given variable. Specifically, these LR -test
statis-3 For identification of the cointegration indices using the two-step procedure of Johansen (1995), the reader is referred to the international interest rate analysis in Chapter 3.
4 The diagnostics of the fourth order autoregressive model of the German long-term interest rate reveal problems with heteroscedasticity and autocorrelation as well as non normality. Strictly speaking therefore, the results of the Dickey Fuller test for this variable is not valid. However, with regard to the other variables all diagnostics are fine.
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Table 1:
A D F(4) S ta tistics for T esting for a unit R oot.
E stim ates of ^ — 1 in parenthesis1)’2^V ariable Ho i GL i UL i GS i US D v m -1.1465 -1.5053 -0.9477 -1.9276 -5.072** (0.0217) (0.036) (0.0074) (0.0231) (0.859) 1(2) -3.5098** 3) -4.5702** -3.1376** -2.7768** -7.3708 (0.5419) (0.747) (0.469) (0.3662) (2.5954)
1For any variable x and a null hypothesis of 1(1), the A D F statistics are testing a null hypothesis of a unit root in x against an alternative of a stationary root. For a null hypothesis of 1(2), the statistics are testing a null hypothesis of an unit root in A x against the alternative of a stationary root in Ax.
2For a given variable and the null hypotheses of 1(1) and 1(2), two values are reported. The 4 ’th-order augmented Dickey-Fuller (1981) statistics, denoted ADF(4) and (in pa rentheses) the absolute value o f the estimated coefficient on the lagged variable, where that coefficient should be equal to zero under the null. A constant-term is included in all regressions. The effective sample is 1990(1)-1997(12).
3Here and elsewhere in the chapter, asterisks * and ** denote rejection of the null hypo* theses at the 5% and 1% significance level, respectively. The critical values for the ADF statistics are -2.892 at a level o f 5% and -3.499 at a level of 1 %(MacKinnon (1991))
tics te st th e hypothesis th a t one o f th e cointegrating vectors contains all zeros except for th e coefficients corresponding to th e variable under con sideration and a non-restricted con stan t term , where th e te st as alluded to above, is conditional on th e num ber o f cointegrating vectors. For instance, the null hypothesis of a statio n ary long-term G erm an interest ra te implies th a t one of th e cointegrating vectors is ( 1 0 0 0 0 ¡3 where I have implicitly assum ed th a t long-term G erm an interest ra te s and th e constant are respectively th e first and last variable of th e variable vector. In Table 2, the statistics quoted are conditional on there being two cointegrating vectors and refer to th e sam e VAR model th a t is used later to identify th e long-run relationships. Empirically, all th e statio n arity tests, except for th e depreci ation rate, reject w ith p-values less th a n one per cent. These rejections o f stationarity a re consistent w ith th e inability to reject th e null hypothesis o f a unit root in all th e interest rates w hen using th e Dickey Fuller te st statistic. Thus, all four interest rates are tre a te d below as if th ey are 1(1). T h e rate of depreciation, however, seems to b e statio n ary and will b e treated likewise.
T h e m ethodology developed by S. Johansen (Johansen (1988), (1992) an d Johansen and Juselius (1990)) is used to identify th e long-run relationships and to test w hether some variables m ay be considered as exogenous w ith regard to estim ation of th e param eters of th e long-run relationships. T he results of th e analysis are given in Table 3. However, th e order of th e VAR is not known a priori, hence some testin g of lag order m ay be beneficial in order to ensure reasonable power in th e Johansen procedure. Beginning w ith a fifth-order VAR in iGLi i UL, i GS, i u s and D v th a t includes a restricted constant term , we show in A ppendix A, Table 8, th a t it is statistically ac ceptable to simplify to a second-order VAR. Further reduction to a first-order VAR is rejected. T he empirical cointegration analysis is therefore made on a 5 dimensional VAR of order two.
Table 3 show's th e results of Johansen’s maximum likelihood procedure. Looking first a t Table 4 which gives th e diagnostics of th e individual equa tions as w'ell as for the system, it is w'orth noting th a t all diagnostics are fine except for a five per cent rejection of norm ality for th e residuals in th e equation of th e US long-term interest rate and a m arginal rejection of th e corresponding vector test statistic. Table 3 supports th e existence of three cointegrating vectors at a significance level of five per cent, but only two using a test level of one per cent. However, we know' th a t th e ra te of depreciation is stationary', so if v re accept th a t th ere are only tw’o cointegrating vectors, we
Table 2:
M ultivariate te st sta tistics for testin g for station arity
Two cointegrating vectors and constant in
C l-space1^ 2^Variables
iGL i UL iGS iUS D v
X 2(3) 17.556** 10.387* 18.814** 12.095** 4.1717 [0.0005] [0.0155] [0.0003] [0.0071] [0.2435]
the Johansen framework. Specifically, these statistics test the restriction that one of the cointegrating vectors contains all zeros except for a unity corresponding to the coefficient of the variable we are testing whether is stationary and a non-restricted constant coefficient. In Table 2, the statistics quoted are conditional on there being two CI-vectors and refer to the same VAR model that later is used to identify the long-run relationships. The figures in brackets under each test statistics are the tests’ significance probabilities and * and ** denote rejection at 5% and 1% critical levels, respectively.
Table 3: Johansens cointegration tests
System: i GL, iULt i GS, i u s , D v.
D eterm inistic part: R estricted constant1)
VAR order: 2. Sample period: 1990 (1)-1997 (12).
Eigenvalues of II: 0.4508 0.2551 0.2064 0.1219 0.0520 Max Eigenvalue Tests2) TYace Eigenvalue Tests Null A lt. Statistics 95% Null Alt. Statistics 95%
r= 0 r < l 57.53** 34.4 r= 0 r< 5 125.6** 76.1 r < l r < 2 28.27* 28.1 r < l r< 5 68.08** 53.1 r< 2 r < 3 22.2 22.0 r< 2 r< 5 39.8* 34.9 r< 3 r < 4 12.48 15.7 r< 3 r< 5 17.61 20.0 r< 4 r < 5 5.13 9.2 r< 4 r< 5 5.13 9.2
^The constant is restricted to lie in the space spanned by the columns of Ck
2)The 5 per cent critical values shown in brackets are taken from Osterwald Lenum (1992). An asterisk indicates that a test is significant to a level of five per cent, while two asterisks indicate that the test is significant to a level of one per cent.
■naniiii i
i i i b i U U U
only have to identify the second5. The u n restricted estim ated cointegrating linear combinations and th e loading m atrix in case of only tw o cointegrating vectors are given in Table 5 below. T he following table, Table 6, q u o tes tests of different hypotheses w ith regard to th e cointegration space and th e space spanned by the a ’s. As already noted w hen te ste d for statio n arity th e te st of th e restrictions identifying th e ra te of depreciation as th e first cointegrating vector is fine. Also, the te sts do not reject a homogenous linear com bination of th e long-term interest ra te s and th e sh o rt-term G erm an ra te to be the second cointegrating vector. However, th e spread between th e two long-term interests rates is not rejected either. A nticipating th e outcom e of th e te sts for G ranger non-causality and exogeneity, th is suggests th a t th e G erm an short ra te is superfluous and th a t th ere is a full pass through o f changes in the US ra te into the Germ an ra te in the long run. T h is agrees w ith th e former graphical inspection of th e spreads m ade in th e introduction. A sim ple test of weak exogeneity, proposed by Johansen (1992a, 1992b) (see also Urbain (1992)), is simply to test zero restrictions on a subset of th e w eights in the loading m atrix. T he results o f these te sts give su p p o rt to tre a tin g th e long term US interest ra te as exogenous w ith respect to estim ation of th e long-run param eters of the two restricted cointegrating vectors. W ith reg ard to the sh o rt-term US interest ra te th e statu s is m ore u n certain as th e individual test conditional on th e two identified cointegrating relationships an d n o e rro r cor rection in th e equation of long-term US interest rates, is significant to a level of five p e r cent (p-value equal to 0.0246). However, th e sam e te st w hen not conditioning on long-term US interest rates as exogenous h as a p-value th a t is only marginally below five p e r cent which is also th e case w ith reg ard to th e te s t of considering b o th US rates as jo in tly exogenous. T his im plies th a t we probably are no t making to o big a m istake by restricting th e tw o cointe gratin g vectors and th e feedback coefficients to en ter only th e eq u atio n s of th e long- and short-term G erm an in terest ra te to g eth er w ith th e eq u atio n of th e ra te of depreciation. If so, th e two US in terest rates can b e considered as being exogenous w ith regard to estim ation of th e long-run p aram eters and inference w ith regard to th ese would b e possible to conduct from a three dim ensional model where we condition on US in terest rates w ithout a signif icant loss of information. However to take th e additional step of justifying 5 The analysis of this chapter is based on the existence of only two cointegrating long- run relationships. For an elaboration of the alternative of three cointegating vectors the reader is referred to Chapter 3.
on this basis th e simpler modelling strategy implied by a three dim ensional conditional system analysis when building a dynam ic stru ctu ral m odel ne cessitates fu rth er investigation as to whether th e two US interest ra te s m ight also be considered as weakly exogenous with regard to estim ation o f th e dy namic sh ort-run param eters. A te s t of strict exogeneity w ith regard to th e two US interest rates related to th e structural m odel developed in th e next section, does however not reject6. T his is indicative o f b o th US in terest rates also being w’eakly exogenous w ith regard to th e dynam ic coefficients and to gether w ith th eir statu s of being exogenous w ith regard to estim ation of th e long-run param eters legitimates th e sort of conditional analysis pursued in the next section to come. T h a t is a three dimensional stru ctu ral dynam ic analysis of th e system consisting of German short- an d long-term interest rates an d th e actual rate of depreciation conditional on th e tw'o US interest rates7.
The two identified cointegrating relationships together w ith th e restricted loading m atrix, are given in Table 7 below. T h e te st of th e restrictions is also quoted and does not reject to a level of five per cent. The long-run rela tionship implies th a t a 100 basis points change in th e long-term US interest rate leads to th e same change in th e German long-term interest ra te in th e long run. T hus there is a full pass-through of changes in US long-term in terest rates into th e corresponding German rates. T he recursively estim ated eigenvalues of Figure 3 in th e appendix show signs o f instability. However, taking th e scale on th e vertical axes into consideration, th is instability seems mainly to be a graphical illusion.
6 The test of strict exogeneity has been undertaken by plugging the residuals of the structural model of Section 3 into the autoregressive marginal processes of order one of the two American interest rates and restricting their coefficients to zero. The joint test of restricting all residual coefficients to zero is x 2(6) and gave a test statistic equal to 6.25826 [0.3949], where the number in parenthesis is the respective test’s significance probability.
7The outcome of an unconditional analysis does not significantly change the outcome of our analysis as the restrictions implied by both US interest rates being univariate autoregressive processes of order one constitute valid restrictions on the full djmamic structure. However, there is some indication of a simultaneous dynamic effect of changes in long-term US interest rates on changes in the corresponding short term US interest rates. For a discussion of this possibility the reader is referred to Chapter 2.
n e & A U i i i i i i
Table 4:
Individual equation and system diagnostics o f th e
u n restricted V A R 1)
E quation/T ests A R 1-6 F[6,79] A RCH 6 F[6,73] N o n n ality x 2 (2)
A iGL 0.6045[0.7260] 0.5329(0.7815] 4.510(0.1049]
A 1.2169(0.3065] 0.8787(0.5149] 9.128(0.0104]*
A iGS 1.3716(0.2364] 0.6138(0.7185] 1.933(0.3804]
A i u s 2.0817(0.0646] 1.9673(0.0814] 0.759(0.6842]
A D v 0.6376(0.6998] 0.2909(0.9394] 1.631(0.4424]
System tests: A R 1-5(150,257] V N orm ality x 2(10) VX2 F[300,646] Statistics: 1.2128(0.0886] 14.707(0.1431] 1.1967(0.0324]* 1The Values shown in brackets are the individual te st’s significance probability. * and ** denote as usual rejection of the corresponding null at levels of 5 and 1 per cent, respectively. VNonnality and V X 2 denote th e Vector tests of normality and hetero-scedasticity. For an explanation of the various test statistics the reader is referred to Chapter 14 of the PcFiml manual (Doornik and Hendry (1999)).
Table 5: T he unrestricted cointegrating linear com binations and th e loading m atrix
$ ( i cL i uL i G* %vs d v i y
fill i GL + 021*UL + h \ i GS + 04l*US + 051&v + 061
012^GL + 022^UL + 0Z2^GS + 0 tf}US + 0 b 2 ^ v + 062
i GL - 0.28iUL - 0.2\ i GS - 0.2Zius + 0.64Dv - 0.025
- 0 .7 9zgl + iUL - 0.06zg s - 0 . m i u s + 0.014DV - 0.016
E quation Loading m atrix 1
A i GL a n «12 - 0.025 [0.0136) 0.143 [0.0527]
A i UL «21 «22 - 0.010(0.0170] 0.101(0.0656]
A i GS «31 «32 = - 0.016 [0.0149] 0.201 [0.0575]
A i u s Q41 «42 - 0.017 [0.0143] 0.022 [0.0552]
A D v «51 «52 - 1.340 [0.1777] 1.468 [0.6836]
^ h e values shown in brackets to the right of the estimated loading coefficients are the respective coefficients’ standard error.
!KH
Table 6: Test of Hypotheses related to the parameterisation of Table 5
Hypotheses LR -test, R an k = 2 1): @ 11 — @21 = - @ 51 ~ @ 41 ~ ^61 “ ® ’ @ 51 = 1 X2 (4) = 4.18 [0.383] 2): @ 11 ~ @21 = “ @ 31 = @ 4 1 = @ 61 ~ 0 , 0 a = 1 X2 (7) = 8.46 [0.294] @ 42 ~ @52 ~ = @e2 = 0) @ 12 = 1 = ~ i@ 22 + @32) 3): @ 11 ~ @21 ~= @ 31 “ @ 41 ~ @ 61 “ @51 = 1 X2 (8) = 13.65 [0.091] n c * II C O = @ 52 “ @ 52 — @12 = 1 = ~ ~ @ 2 2 4): @ 11 “ @21 -“ @ 31 ~ @ 41 = @ 61 = 0)^51 = X X2 (1 0 )== 13.78 [0.183] @ 32 ~ @ 42 ~ = @ 52 — @ 6 2 ~ 0) @ 12 = 1 = ~ @ 2 2 a 2i = a2 2 == 0 5): @ 11 = @21 ~= @ 31 = @ 41 ~ @ 61 = 0)^ 51 = 1 X2( i o ) == 18.75 [0.044]* @ 32 = @42 =~ @ 52 ~ @ 6 2 ~ @12= 1 = ~ @ 2 2 Q41 = a 42 == 0 6): @ 11 = @21 ~= @ 31 = @ 4 1 = @ 61 ~ ®i@ 51 = 1 X2 (1 2 )== 21.06 [0.05]* @ 32 = @42 ~= @ 52 ~ @ 6 2 = 0 , @ 12 = 1 = ~ @ 2 2 «21 “ «22 =- 0) «41 = «42 = 0
LThe value shown in brackets after each individual LR-test is the test’s significance pro bability, One star, *, behind a test statistic means as before that the test is significant to a level below five per cent.
U U d M ifT
h t i w l i teTable 7: The restricted cointegrating linear combinations and the restricted loading m atrix _______________________________________________________
R estricted cointegrating linear com binations1
D vt
Equation: R estricted estim ated loading m atrix
A iGL « n Oi2 = -0.0112 [0.0078] -0.1067(0.0324] A i UL «21 «22 = 0.0000 0.0000 A iGS «31 S32 = -0.0094 [0.0096] -0.1 0 2 1 [0.0409] A i u s «41 «42 = 0.0000 0.0000 A D v «51 S52 = -0.8302 [0.1097] 0.0000 X2 (13) = 21.08 [0.0714]
:The values in brackets to the right o f the estimated loading coefficients are the res pective coefficients’ standard error
. . t i n u m n f i h h h t i h h Éi i u i m n i
3
A co n d itio n a l error c o rrectio n m o d e l for
th e lon g- an d sh o r t-te r m G erm an in te r e st
rate.
B ased on the results of th e vector autoregressive analysis above, I sta rte d by specifying a three dimensional conditional stru c tu ra l error correction model incorporating only one lag of differences and th e two error correction mech anism s given by th e ra te of depreciation a n d th e long-term in terest rate sp read lagged one period8. T h e stru ctu re identified was inform ed by theory a n d th e desire to explain th e correlation p a tte rn of th e reduced form resid uals as the result of a solved d a ta generating structure. From prelim inary d a ta analysis we know th a t th e regression m odel is balanced, i.e. th a t the m odel includes only variables w ith consistent tem p o ral properties. T h e error correction specification makes it easy to distinguish between short- an d long- ru n effects. The short-run effects are represented by th e differenced variables, w hile th e long-run effects are associated w ith th e level variables. In order to find a parsimonious simplification, we th e n imposed restrictions on the sh o rt-term coefficients of th e model. T h e restrictions- like th e identification- schem e were informed by theory and th e desire to explain th e correlation p a tte rn of th e reduced form.
T h e structural m odel below shows th e regression result w hen using Full Inform ation M aximum Likelihood (FIM L) on m onthly d a ta for th e period J a n u a ry 1990 to Decem ber 1997. Looking a t th e diagnostics q u o ted below th e identified stru ctu ral model, th e L R te st for over-identifying restrictions im plies th a t th e stru c tu re im posed co n stitu tes a valid reduction o f a just- identified structure. Also, we cannot reject a jo in t te st o f im posing linear hom ogeneity in th e equation for long-term G erm an interest rates, together w ith a linear restriction identifying th e first difference of th e long-term in te re st spread as an explanatory variable in th e equation determ ining the b ilateral exchange rate. T h e negative im pact from th e first difference of the sp read on the first difference of the depreciation rate is consistent w ith an overshooting effect in case o f changes to long-term interest rates. T h a t is, to generate increased depreciation expectations in th e wake of long-term in te re st hikes th e depreciation ra te will have to decrease. From th e identified 8N ote that one lag of a difference includes the second lag of the level, m atching the order of the VAR in Section 2.
a IMMM iM ItM M EM M M m illHM M H lM rM IIM M U M lIim *titH
stru ctu re we note th a t th e two variables iGS and \UL b o th seem to explain th e G erm an long-term interest ra te in th e short rim, b u t only th e la tte r in th e long run. T h e coefficient on A if x shows the im pact (after one m onth) on th e G erm an long-term interest ra te of a change in the US long-term rate. T h e estim ate of this coefficient is 0.40, implying th a t a 100 basis point change in US long-term interest rates leads to a 40 basis points change in th e G erm an long-term interest rate after one month. Moreover, we note th a t this effect is considerably weaker th an th e short-run impact from a 100 basis points change in th e short-term G erm an interest ra te which changes th e G erm an long-term interest ra te by as m uch as 60 basis points. As th e long-run effect of a change in short rates on long-term interest ra te s is neutral this sug gests th a t th e strong short-run effect is neutralized in th e long ru n thro u g h affecting expectations of future short-term interest rates. The long ru n re lationship is derived by setting all the differenced variables in th e reduced form of th e stru ctu re equal to zero and implies as commented on before, th a t there is a complete pass-through into G erm an long-term interest rates of a change in th e US long-term interest rate. Thus, a 100 basis points change in th e long-term US interest ra te leads in th e long ru n to an equal change in th e G erm an long-term rate. Hence, US long-term interest rates have a considerably stronger im pact on G erm an long-term interest rates in th e long run th a n in th e short run.
The Identified S tru ctu ral model
A i f £ 0.156 A i? }.+ 0.398 A iY L+ 0.602 A i? s (0.0803) (0.066) (0.066) 0.001756 H- £\t A r? s O2 0.234 A i S + 0.0151 A D v c- 0.114 ( i GL (0.0854) (0.0094) (0.0316) 1 0.00201395 f-i + ?2i A D vt = 0.173 A D i’w - 3.819 A U GL - ¡“ 1, , - 0.834 D v , , (0.0856) ( 1.107) 1 J i ' 1 (0.107) + 2.517 A if75- 3.742 A iu£ + eZt (1.142) 4 ( 1.18) 1 1 (t3 = 0.022361
Some Diagnostics of T h e S tru ctu ral model T = 9 6 (1990(1)-1997(12)) LR: x 2 (17) = 14.4772 [0.6331] LR : x 2 (2) = 0.4298 [0.807] V AR 1-6 F (54,218) = 0.8356[0.7809] V N orm * 2(6) = 12.12 [0.0594] X 2 (36) = 21.346 [0.9749] F (36,80) = 0.59294 [0.9583] X 2(36) = 20.947 [0.9786] F (36,80) = 0.58186 [0.9634]
In Section 2 we found th a t we could e stim ate th e long-run param eters conditionally on b o th US in terest rates, w ith o u t having to pay atten tio n to th eir m arginal distributions. This suggests th a t th e direction of causality goes from the US to th e G erm an economy. To further su b sta n tia te this result, however, we have to te s t against lagged effects of G erm an long- and sh o rt-term interest ra te s as well as of th e ra te o f depreciation on b o th US interest rates. However, a te s t for G ranger non-causality (G ranger (1969)) does n o t reject th e null o f no lagged effects on US interest rates of these variables9. Thus, th e re is evidence of a one-way causality between US and G erm an interest rates, th e direction of causality going from th e US economy to th e G erm an economy.
T h e system diagnostics for serial correlation, non-norm ality and param eter constancy are all fine. However, b o th te sts for vector heteroscedasticity reject t o a level of one p er c e n t10. Also, by form ulating a stru ctu ral m odel we were u n ab le to get rid of th e residual correlations across equations in th e un re stricted reduced form of th e system , th e co rrelatio n between the residuals of th e tw o German in terest ra te s in fact increasing in stead of decreasing. These facts b o th indicate som e s o rt o f misspecification; th e two obvious candidates are w rongly imposed s tru c tu ra l restrictions a n d a to o small inform ation set.
9T he test of Granger non-causality is made on an error correction model for US long- and short-term interest rates where we together w ith th e lagged error correction terms and lagged changes in US long- and short-term interest rates, only have regressed on lagged changes o f German long- and short-term interest rates in addition to lagged changes in the rate o f depreciation. W hen incorporating only tw o lags o f differences, the joint reduction of all lagged effects from these m odel endogenous variables and error correction terms in the m arginal models o f th e two US interest rates gives a test statistic with a significance probability of 0.09.
10The vector x 2 and vector X* + X j tests are respectively F (108,402)—1.63 [0.0004] * * and F(324, 206)=1.58[0.0002]**.
--- --- .. ... - ... ... ■ . ■ . . u y y M t o M M M . » .
However, as mentioned in th e introduction, I have so far not been able to find an adequate understanding of th e stru ctu re underlying alternative inform a tion sets a n d will therefore leave th e ground open for further research. W ith regard to w hether th e stru c tu ra l representation m ight represent wrongly im posed identifying as well as over identifying restrictions th e reader is again referred to C h ap ter 2. T h e forecast statistics together with Figures 6 to 7, which show static (one step ahead) and dynam ic ex post forecasts for th e ra te of depreciation and th e G erm an long- and sh o rt-term interest ra te for 1997, indicate th a t our model seems to make fairly good ex p o st forecasts w ithin th e sam ple period even though th e error b an d s are wide. Figure 5 shows dynam ic forecasts of th e differenced series. All these forecasts have been undertaken by a model estim ated on d a ta only for th e period 1990 (1) to 1996 (12) and thus are ex p ost forecasts in the sense th a t th ey are m ad e for th e period after th e estim ation period. Figure 4 shows some graphical test statistics for param eter stability. These graphs do not indicate a serious problem w ith unstable param eters during th e sample period and are in line w ith the form al tests given under th e stru ctu ral m odel above.
4
S u m m ary a n d co n clu sio n s
The purpose of this chapter has been to reexamine empirically th e relation ship between long-term interest rates in well integrated financial m arkets. T he analysis has been carried o u t w ithin th e framework of a five dimensional VAR for th e simultaneous determ ination of short- an d long-term interest rates in th e US and Germany, an d th e ra te of depreciation. A n im p o rtan t m otivation for using this framework has been to carefully examine cointegra tion and exogeneity. Interestingly, our results indicate th a t both US interest rates are exogenous w ith regard to estim ation of th e long-run coefficients in a three dimensional regression model for Germ an long- and short-term in terest rates and th e ra te of depreciation. Also, G erm an long-term interest rates do n o t seem to G ranger cause US interest rates. Thus, th e direction of causality seems to be unidirectional, namely from th e US to th e European economy. T his could have im p o rtan t macroeconomic consequences in G er many since much of th e debt to households and firms is linked to long-term rates. Moreover, we find th a t short-term Germ an and long-term US in ter ests rates b o th have a significant impact on long-term German rates in th e short run. However, domestic interest rates do not seem to enter the
long-i U U i l i l J
ru n relationship. In addition to implying th a t th e re is a full pass-through of long-term US interest rates into th e corresponding G erm an ra te in th e long run, th is suggests th a t m onetary policy could b e effective th ro u g h affecting expectations with regard to future short-term interest rates in a way th a t neutralizes th e short-run effect th a t sh o rt-term domestic interest ra te s have on long-term German interest rates in th e long ru n . The forecastability of th e m odel improves significantly com pared to a model w here one excludes th e bilateral exchange rate in th e inform ation set, and does give decent fore casts even for 1997. This suggests th a t th e increase in th e long-term interest spread in 1997 is p a rtly d ue to increased depreciation expectations as a con sequence of different grow th p attern s in th e US a n d G erm any and probable overvaluation of th e dollar in th is period.
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IMftJUÜUUtMMMMJIMJ m i M i ( n n ' i > t (1i f ‘M" ‘‘ “ 'i < i ) l i i i
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¿1 j_ x iL m m u L J ü ü ü it il--- — T t iiin w i^
A
T ab les an d grap h s
Table 8: F and related S tatistics for the Sequential R eduction from th e fifth- order VAR to the First-order VAR.
Null H ypothesis1 M aintained H ypothesis2
System k SC V A R (5) VAR (4) V AR (3) V A R (2) V A R (5) i 130 -5 7 .0 5 0.884 V A R (4) l 105 -5 4 .8 9 [0.63] (25,246) 0.974 1.074 V A R (3) 80 -5 5 .7 2 [0.53] [0.37] 1 (50,304) (25,265) 1.048 1.139 1.204 V A R (2) 55 -5 6 .5 4 [0.38] [0.25] [0.23] i (75,320) (50,327) (25,283) 1.388* 1.570** 1.814** 2.417** V A R ( l) 20 -5 7 .0 5 [0.017] [0.004] [0.001] [0.000] (100,33) (75,34) (50,35) (25,30) Notes:
1The first three columns report the vector autoregression w ith its order, and for th at model: the number of unrestricted parameters k and the Schwartz criterion SC. 2The three entries within a given block of numbers in the last four columns are: the approximate F-statistic for testing the null hypothesis (indicated by the model to the left of the entry) against the maintained hypothesis (indicated by the model above the entry), the tail probability associated with that value of the F -statistic(in square brac kets), and the degrees of freedom for the F-statistic (in parentheses). See Dooraik and • Hendry (1994) for details on the algebra underhung these calculations. * and ** denote
Jb MW «MUM MiMu
