Juan Paez-Farrell

IN THEORY AND IN PRACTICE: EVIDENCE FROM THE UK AND THE US

Keywords: E52 -Monetary Policy (Targets, Instruments, and Effects

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

set of variables, with the Taylor rule being the most prominent example. Simple rules have the advantage of being clearly understandable and transparent, so that the objectives of monetary policy be well understood. Moreover, uncertainty regarding the future path of nominal interest rates would be reduced given the predictability of the rule.

Whilst no central bank has publicly stated that it has followed an instrument rule, the Bank of England follows a targeting rule to implement its policy 5 and the Fed's monetary policy objectives consist of achieving high employment, stable prices, economic growth and balance in the international accounts. 6 This would seem to imply that estimating monetary policy rules for these two economies would be a pointless exercise, 7 given that their central banks do not explicitly follow them.

However, Taylor-type rules do have some influence on policy 8 and it could be argued that central banks implicitly follow them. This would not be surprising given the two-way influence between monetary policy and recent theoretical developments 9 . An additional reason for investigating the relevance of a Taylor-type rule for the UK and the US also lies in the fact that, as mentioned above, an interest rate rule may provide a parsimonious description of actual policy even when it is not 5 Nikolov (2002). 6 This lack of specificity implies that it follows neither an instrument rule nor a target rule. 7 Carare and Tchaidze (2005) discuss additional dangers in the use of Taylor rules, especially when used for policy recommendations. 8 Nikolov (2002) states that Taylor rules are used at the Bank of England as an indicator of the stance of current policy; Yellen suggested that following a Taylor-type rule would represent good policy, which is remarkable when the alternative policy being considered was one of strict (over the medium term) inflation targeting, as argued by Broaddus (Federal Reserve Board 1995, p. 39-44). 9 Indeed, given that much research on monetary policy is conducted within central banks, the fact that some of the proposed rules analysed in this paper were published near the end of the sample period does not preclude their relevance. adhered to by the monetary authorities. This is evidently relevant when one considers the large amount of research (especially for the US) that has found evidence of Taylor-type rules. This paper aims to combine the two lines of research discussed above by considering to what extent some of the proposed rules emanating from theoretical models reflect actual central bank behaviour. To the extent that theoretical contributions have been able to model central bank objectives and constraints successfully, these will be shown in the paper.

II. PROPOSED RULES FOR MONETARY POLICY

The volume edited by [START_REF] Taylor | Monetary Policy Rules[END_REF] provided one of the first thorough analyses on the macroeconomic consequences of alternative monetary policy rules in the presence of nominal rigidities. One of the benefits of the Taylor rule lies in the fact that under reasonable parameter values it will generally ensure a unique and determinate rational expectations equilibrium. 10 Furthermore, from an empirical point of view its simplicity would help in being understood by the public. However, some authors have proposed similarly simple rules that nevertheless may possess superior features to the Taylor rule. In particular, the standard Taylor rule takes the form:

t t t x R 2 1 0 µ µ µ + + =
(1)

10 See Woodford (2003, p. 252-261). where denotes the inflation rate value and x denotes the output gap.

That is, the nominal interest rate is adjusted each quarter to respond to deviations in the target values of inflation and output. However, most empirical studies on Taylor rule variants have found that the lagged interest rate enters (1) and that it is strongly significant, modifying (1) to:

( )[ ] t t t t t v R x R + + + + = 1 3 2 1 0 3 1 µ µ µ µ µ (2)
A theoretical rationale for the inclusion of the lagged interest rate in (2) can be found in Woodford (2003, p.280), in that the maintained high interest rates in response to rises in inflation or the output gap, for given 1 µ , 2 µ , have larger stabilising effects upon current output gap and inflation. A crucial feature that any monetary policy rule must possess is that it will ensure determinacy -both real and nominal -in the economy. The Taylor principle, 1 1 > µ , is generally a necessary requirement for the uniqueness and stability of the equilibrium. 11 Nevertheless, as Carlstrom andFuerst (2000, 2001) demonstrate, the timing on which the monetary policy rule is based can also be crucial in order to prevent disastrous effects on the real economy. To avoid this Carlstrom and Fuerst propose either current or backward-looking Taylor rules. However, this conclusion runs counter to the professed approach of explicit inflation targeting central banks, which 11 The general condition can be found in Woodford (2003, p. 255). for the present paper is most relevant for the UK, as their approach is forwardlooking. Whereas backward-looking rules have been proposed on the grounds of avoiding indeterminacy, the rationale in support of forward-looking rules is based primarily on the fact that it takes into account the lags in the monetary transmission mechanism [START_REF] Batini | Forward-looking rules for monetary policy[END_REF].

These two different Taylor-type rules, which differ on the timing of the explanatory variables to which the monetary policy instrument reacts can be succinctly represented as:

(

) [ ]

t t m t k t j t t v R x E R + + + + = + + 1 3 2 1 0 3 1 µ µ µ µ µ (3)
Where j represents the possible information lag to which the central bank is subject.12 k (m) is a positive integer when the central bank reacts to expectations of future inflation (output gap), and a negative integer under a backward-looking Taylor-type rule.

In contrast to the proposed rules where the only difference concerns timing, Walsh (2003) has argued that a speed limit policy dominates inflation targeting as long as the model is predominantly forward-looking, since under a discretionary policy the central bank is able to achieve higher social welfare when it reacts to changes in the output gap. Whereas the Taylor rule can be derived from quadratic preferences, a F o r P e e r R e v i e w speed limit rule implicitly embodies a more general form. Consequently, the rule proposed by [START_REF] Walsh | Speed limit policies: the output gap and optimal monetary policy[END_REF] would take the form:

( )[ ] t t t t t v R x R + + + + = 1 3 2 1 0 3 1 µ µ µ µ µ (4)
Lastly, another Taylor-type rule that has been prominent in the literature arises when one allows for a more flexible form of the central bank's objective function or if the economy's structure is non-linear. If the central bank's loss function is asymmetric, so that negative and positive deviations in the inflation rate and the output gap are assigned different weights, the optimal Taylor rule would be non-linear. 13 Then the Taylor rule (2) would be modified to:

( ) [ ] t t t t t t t t t t v R x x x E R + + + + + + + = 1 3 6 2 5 2 4 2 1 0 3 1 µ µ µ µ µ µ µ µ (5)
Hence, the Taylor-type rules that this paper will focus on are forward and backwardlooking rules, speed limit and non-linear rules, as well as the basic Taylor rule (2) which will be considered as the benchmark.

III. PREVIOUS ESTIMATES OF SIMPLE MONETARY POLICY RULES

Results from empirical studies that have estimated monetary policy rules have generally adopted one of two approaches. Most have been descriptive [START_REF] Taylor | Discretion versus rules in practice[END_REF] and have attempted to determine whether simple rules provide a useful description 13 An insightful analysis of such preferences in a monetary policy context can be found in [START_REF] Nobay | Optimal discretionary monetary policy in a model of asymmetric central bank preferences[END_REF]. of actual monetary policy behaviour. On the other hand, some authors have adopted a more normative approach, by characterising the monetary policy rules during periods of successful stabilisation and therefore attributing the superior outcome to the monetary policy rule of the central bank [START_REF] Clarida | Monetary policy rules and macroeconomic stability: evidence and some theory[END_REF].

Although the Fed does not have an explicit inflation target, it is committed to achieving low and stable prices as well as promoting employment growth. This implies that a monetary policy rule as a function of inflation and some measure of real activity may provide a realistic description of actual behaviour. In this sense [START_REF] Taylor | Discretion versus rules in practice[END_REF] article, whilst using calibrated values in the policy rule, provided the first analysis of such a function. Furthermore, most estimated monetary policy rules for the Fed have found that a Taylor-type rule has provided a reasonable description of actual monetary policy. [START_REF] Clarida | Monetary policy rules and macroeconomic stability: evidence and some theory[END_REF] found that monetary policy during the Volcker-Greenspan years (1979:3 to 1996:4) was forward looking, in that the monetary policy instrument responded to forecasts of its target variables. More importantly, it satisfied the Taylor principle, 14 with 3 µ , the coefficient on the lagged interest rate around 0.80. Similar results are obtained by [START_REF] Nelson | UK monetary policy 1972-1997: a guide using Taylor rules[END_REF] for the UK during the period 1992-97, whilst [START_REF] Adam | Monetary frameworks and institutional constraints: UK monetary policy reaction functions, 1985-2003[END_REF] found that monetary policy was forward looking in terms of inflation, but responded to the current output gap.

The robustness of the Taylor principle has also been found in other studies. [START_REF] Ball | The Fed and the new economy[END_REF] analysed US monetary policy in terms of inflation and 14 The coefficient on the output gap (incorrectly defined as hp detrended output) was also high, at 0.93. (1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000). They found that the rule had been relatively stable throughout the period once one allowed for changes in the NAIRU, with a coefficient on inflation of 1.29 for the "old economy" period (1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995) and 1.54 during the "new economy" years (1996)(1997)(1998)(1999)(2000).

An additional issue concerns the inclusion of the lagged interest rate when estimating monetary policy rules. Although its inclusion is generally interpreted as reflecting partial adjustment in the behaviour of interest rates [START_REF] Clarida | Monetary policy rules and macroeconomic stability: evidence and some theory[END_REF], [START_REF] Rudebusch | Term structure evidence on interest rate smoothing and monetary policy inertia[END_REF] has argued that the significant lagged interest rate is the result of serially correlated residuals. However, in attempting to disentangle the two effects, [START_REF] English | Interpreting the significance of the lagged interest rate in estimated monetary policy rules[END_REF] found that the former argument is the dominant cause for the significance of the lagged interest rate.

Meanwhile, [START_REF] Surico | Asymmetric reaction functions for the Euro area[END_REF] found evidence in support of an asymmetric Taylor rule for the Fed for the period 1997-2002, whilst for the UK [START_REF] Martin | Modelling monetary policy: inflation targeting in practice[END_REF] results indicate that the Bank of England for the period 1992-2000 responded more to upward deviations of inflation away from the inflation target than when below the target, despite the official objective being symmetric.

More recently, [START_REF] Taylor | Interest rate setting and inflation targeting: evidence of a nonlinear Taylor rule for the United Kingdom[END_REF] using threshold models find that the Bank of England followed a forward looking Taylor rule when inflation was about half a percentage point above the inflation target. For inflation rates below this level the policy rule was best described as a random walk with a small positive coefficient on the output gap. Regarding evidence on the speed limit policy, Peel et al. (2004) did find empirical support for a speed limit policy for the US, although to this author's knowledge no research has been carried out regarding its relevance for the UK.

Nevertheless, despite the vast amount of research on estimating monetary policy rules for the US -and to a lesser extent, the UK -there is no clear consensus on the interest rate rule that best characterises each economy during a common sample period, such as the Greenspan era in the US and the period of inflation targeting in the UK.

IV. RESULTS FROM SOME PROPOSED SIMPLE INTEREST RATE

RULES

This section aims to compare the four monetary policy rules proposed by theoretical concerns using a common sample to determine if the behaviour of either the Fed or the Bank of England can be well characterised by one of these rules.

Rules where the interest rate responds directly to the exchange rate have not been included in this paper. [START_REF] Taylor | The role of the exchange rate in monetary policy rules[END_REF] is sceptical of including the exchange rate in the policy rule, as it is likely to worsen the outcome of stabilisation policies. Further support for this point of view emerges from [START_REF] Allsopp | United Kingdom inflation targeting and the exchange rate[END_REF], who argue that under an inflation targeting regime with a flexible exchange rate, the latter matters only to the extent that it affects the inflation rate, rendering a direct reaction to it unnecessary. The results for the US are shown in Table 1 for the four rules considered in this paper. The data are quarterly and cover the period 1987:3 to 2004:4, which span most of the Greenspan years. 15 Several measures of inflation were considered, including the GDP deflator, but the best fit was provided by the annual rate of change in personal consumption expenditures less food and energy, which is generally interpreted as a measure of core inflation in the US. The output gap is defined as current GDP as a proportion of the Congressional Budget Office-derived potential output, whilst the interest rate is measured as the average federal funds rate.

All rules are estimated by GMM 16 using four lags of each explanatory variable as instruments, with the validity of the overidentifying restrictions confirmed the Jstatistic (p-values in parentheses). The null of residual normality (Jarque-Bera) is only rejected for the non-linear Taylor rule.

In estimating the forward-looking monetary policy rule, various timing horizons were considered, and only the one with the best fit is reported, which is the Taylortype rule where interest rates react to the forecast of inflation (the output gap) eight (four) quarters ahead.

[TABLE I]

15 All data were taken from the Federal Reserve Bank of St. Louis. 16 The backward-looking rule is estimated by least squares, with Newey-West standard errors in parentheses. The models yield similar coefficients for the autoregressive parameter, over 0.80, which is consistent with the results obtained by [START_REF] Clarida | Monetary policy rules and macroeconomic stability: evidence and some theory[END_REF]. Whilst the Taylor principle is satisfied for all models -excluding the non-linear Taylor rule, which in any case performs poorly -although it ranges from 1.27 in the case of the pure Taylor rule (first column) to almost 3.9 under an inflation-forecast targeting rule. Similar results are obtained for the output gap, which is also significant in the first three models. Comparing the models on the basis of the Schwarz Bayesian information criterion (SBC) indicates that the standard Taylor rule provides the best description of the Fed's behaviour, which could lead one to conclude that despite the desirable features in the other monetary policy rules presented, these are not applicable to the US economy. 17 However, despite the superior performance of the basic Taylor rule, the Q statistic is indicative of strong serial residual correlation, so that all of the equations may be mis-specified. This issue will be pursued further below.

For the UK, the sample in which the monetary policy rules were estimated is 1992:4 to 2004:4, which can be regarded as a single monetary policy regime, 18 as the Bank of England had the official objective of an inflation target. The output gap is 17 Obviously, more complicated rules are likely provide a more accurate description of actual interest rate behaviour, but this paper focuses on only a selection of simple rules with theoretical foundations. 18 [START_REF] Adam | Monetary frameworks and institutional constraints: UK monetary policy reaction functions, 1985-2003[END_REF] reported a change in regime pre-and post-independence of the Bank of England; however, Lord George, in an interview for the Financial Times on 2 May 2007, has stated that it was the introduction of inflation targeting and the general economic consensus in the UK that mattered most. Again, several inflation measures were considered, with the measure that yielded the best fit being the annual rate of RPI inflation. 20 Interestingly, the regressions with RPIY inflation -which removes the effects of indirect taxes -yield poorer results, despite the fact that [START_REF] Cutler | Core inflation in the UK[END_REF] found that it was the best predictor of future inflation, in other words, it could be defined as core inflation.

Table 2 reports the estimates of the various models. It is interesting to note that the UK's monetary policy rule that best describes the data corresponds to the one that is closest to research emanating from the Bank of England [START_REF] Batini | Forward-looking rules for monetary policy[END_REF][START_REF] Batini | Forward-looking rules for monetary policy[END_REF][START_REF] Nikolov | Monetary policy at the Bank of England[END_REF], that is, a forward looking rule. For the sample considered the best fit is obtained when the forecast horizon is four quarters for both inflation and the output gap. As with the Fed, the Taylor principle is satisfied, 21 although the coefficient on the output gap is higher than that on inflation, albeit less precisely estimated. Moreover, whilst for the US the results on inflation were generally consistent across models, at least with regards to the coefficient on inflation, for the UK results are highly sensitive to the timing of the Taylor rule. This is especially the case with the backward-looking Taylor rule, indicating that the results are highly sensitive to the specified horizon. 19 Although this is common practice, as mentioned earlier one should be aware that this is not the equivalent measure from a theoretical output gap measure. 20 In the estimations, inflation is then RPI minus the official inflation target. Modifying this to the gap using first RPIX and then CPI inflation, the Bank's official target, has not effect on the results. 21 However, it is not significantly above unity. Although the results from Table 2 are consistent with the accepted wisdom regarding the behaviour of the Bank of England, the Q-statistic again indicates substantial residual autocorrelation. Given that this paper has argued that the monetary policy rules being estimated are only rough approximations to actual central bank behaviour, as no central bank explicitly (and from the results, implicitly) follows a mechanical interest rate rule, monetary policy may also be responding to other events. However, a potential reason for the high Q statistic concerns the behaviour of interest rates and the modelling framework.

Therefore, it could be argued that the dynamics of interest rates are not captured by the models presented above, but that an error correction form provides a superior representation of the data 22 . Early empirical support for this formulation can be found in [START_REF] Judd | Taylor's rule and the Fed: 1970-97[END_REF], in the form of a modified basic Taylor rule: 23

t t t t t t v R R x R + + + + + = 1 4 1 3 1 2 1 1 0 µ µ µ µ µ (6)
To determine whether the residual autocorrelation is due to the fact of the poorer dynamics in the previous models, they are now presented in Table 3 for the speed- 22 In effect, equation ( 6) can be seen as a re-parameterisation of an autoregressive distributed lag (ARDL) model embodying an error correction model. Hence it can be interpreted as a Taylor rule but with richer dynamics. 23 Their estimation was conducted by least squares on the grounds that given the lags in the monetary transmission mechanism, reverse causation from interest rates to inflation and output was unlikely. limit and non-linear Taylor rules. In addition, difference rules have also been found to be optimal under a variety of contexts, with the underlying rationale being an extension to the inclusion of the lagged interest rate in the Taylor rule. 24 The rationale is that the commitment to maintaining higher interest rates in the future in response to a positive inflationary or output gap shock in the present induces greater stabilisation in these two variables.

In addition, [START_REF] Orphanides | Robust monetary policy rules with unknown natural rates[END_REF] propose a difference rule on the grounds that it is more robust in the face of problems measuring the natural rate. The rule they put forward takes them form:

t t t ue R + + = 2 1 0 µ µ µ (7)
where ue denotes the unemployment rate. 25 [START_REF] Giannoni | How forward-looking is optimal monetary policy?[END_REF] proposed an alternative rule, which can be interpreted as a speed limit rule that is optimal from a timeless perspective within the context of the model they analyse, and is represented by equation ( 8): 24 An important methodological issue concerns the stationarity properties of the interest rate. For the US, both the ADF and Phillips-Perron test reject stationarity at the 1% level, but not at the 5% level. For the UK, although the ADF test rejects stationarity at the 1% level, the Phillips-Perron gives the same results as for the US. Hence the results are not conclusive, but are indicative of substantial persistence in the series. Moreover, there are strong theoretical reasons to believe that the interest rate is a stationary series, especially in the case of an inflation targeting regime. 25 This is measured as the civilian unemployment rate in the present paper. Lastly, [START_REF] Walsh | Parameter misspecification and robust monetary policy rules[END_REF] argued that a difference rule provide a robust simple rule when faced with possible parameter misspecification, especially regarding the degree of inflation inertia. 26 This leads to the simple interest rule:

1 4 1 3 2 1 0 + + + + = t t t t t R R x R µ µ µ µ µ (8)
t t t x R + + = 2 1 0 µ µ µ (9)
Table 3 presents the monetary policy rules in differences for the US. Of the different measures of inflation considered, the best fit was provided by the quarterly percentage change in the GDP deflator (at an annualised rate). Although the dynamic Taylor rule provides a reasonable description of movements in the Federal funds rate, the best model is given by the speed limit policy, as initially put forward by [START_REF] Walsh | Speed limit policies: the output gap and optimal monetary policy[END_REF], but with allowance for richer dynamics. 27 Moreover, the long-run response to the inflation rate of 2 is consistent with previous findings in the literature, whilst there is also a strong response to changes in the output gap, of 0.29.

The monetary policy rules considered by [START_REF] Walsh | Parameter misspecification and robust monetary policy rules[END_REF], and [START_REF] Giannoni | How forward-looking is optimal monetary policy?[END_REF] also include a speed-limit component, which is even stronger, although less precisely estimated. In both cases the coefficient on inflation is strongly significant. Nevertheless, the Giannoni-Woodford rationale for the particular rule they derive does not provide an appropriate characterisation of interest rate behaviour, as the coefficient on the lagged interest rate should be positive. Lastly, while in the non-linear specification the inflation rate provides little explanatory power, the Orphanides-Williams formulation yields a negative coefficient on the inflation rate. Their generally poor empirical performance is reflected in the high SBC values they yield.

Overall, in analysing the Fed's behaviour within the scope of simple interest rate rules, an error correction formulation in which interest rates follow a speed limit policy does seem to provide a reasonable characterisation of the data and this formulation has a theoretical basis to support it.

[TABLE 3]

The same models estimated for the UK are presented in Table 4, but none of the models can describe the data well. The Orphanides-Williams, and the Giannoni-Woodford specifications suffer from the same problems as in the US estimation. The coefficient on inflation is strongly significant under the specification considered by [START_REF] Walsh | Parameter misspecification and robust monetary policy rules[END_REF], although the change in the output gap is insignificant at the 10% level, whilst residual normality is also rejected. Once again, the non-linear formulation is unsuccessful at characterising the data, with the coefficients on inflation being insignificant. 28 However, in contrast to the US results, the interaction coefficient between output and inflation is positive and significant, as expected.

[TABLE 4]

Overall, within the model in differences, UK monetary policy is best described by a simple Taylor rule, although this is nevertheless a poor representation. There are two reasons why this result arises. First, UK monetary policy does seem to be more forward looking than in the US, so that rule (3) provides a better characterisation of interest rates. Secondly, and most importantly, there is evidence that inflation is a less persistent series under an inflation targeting regime; 29 this can account for the support of the ARDL model for the US but its rejection in the case of the UK.

V CONCLUSION

In recent decades there has been a large amount of interaction between central bank and academic researchers on monetary policy issues, as pointed out by [START_REF] Mccallum | Recent developments in monetary policy analysis: the roles of theory and evidence[END_REF]. Among the most prominent topics have been not only the objectives of monetary policy, but also the variables that the monetary policy instrument, typically a short term interest rate, reacts to.

It is generally agreed that a necessary criterion for any policy rule is that it should ensure determinacy and that this result be robust under a variety of models.

Nevertheless, within rules that do yield a unique rational expectations equilibrium, 28 A potential explanation is that the nonlinearity that best describes the data may be the one used by [START_REF] Taylor | Interest rate setting and inflation targeting: evidence of a nonlinear Taylor rule for the United Kingdom[END_REF] -a threshold model -and not equation ( 5). 29 See [START_REF] Benati | Investigating inflation persistence across monetary regimes[END_REF]. different rules may possess additional benefits, such as being robust to parameter uncertainty, being more efficient (in the sense of [START_REF] Ball | Policy rules for open economies[END_REF], or better capturing the preferences of policymakers.

Although no central bank has indicated that it follows a mechanical policy instrument rule, economic research normally assumes or derives particular rules with the aim of analysing their consequences under different modelling formulations.

Given that much of this research has been conducted within central banks, one could argue that some of the proposed monetary policy rules have emerged as a result of carrying out monetary policy, whilst at the same time policymakers are influenced by academic developments. In essence, although central banks may not adhere to a rule, given the interactions mentioned above it could be argued that a rule may provide a close description of actual monetary policy.

Constraining the analysis to simple interest rate rules, this paper has considered whether some prominent simple monetary policy rules that have been proposed in the academic literature have been reflected in practice at either the Federal Reserve or the Bank of England.

The introduction of inflation targeting seems to have led to a substantial decrease in inflation persistence, and this has been reflected in the rule that best characterises interest rate behaviour in the UK, that of a forward looking Taylor rule.

Given that the US does not have explicit numerical objectives -as well as having real objectives -the simple monetary policy rule that provides a reasonable description of interest rate behaviour is a speed limit rule. This follows [START_REF] Walsh | Speed limit policies: the output gap and optimal monetary policy[END_REF], except for the fact that it embodies richer dynamics in the form of an error correction model, or alternatively, in an ARDL representation.

Thus both central banks' estimated interest rate rules do have underlying theoretical support, and the particular rules that provide the best description of monetary policy in each country are consistent with their official policy objectives.

Lastly, while a considerable amount of research has been devoted to estimating monetary policy rules under varying assumptions, there has been little effort to assess which of these rules provides the best description of actual behaviour using the same sample period. In focusing on a selection of simple rules, the results of this paper can be regarded as a first step in that endeavour. Notes: B-P-L denotes the Box-Pierce-Ljung Q statistic for residual autocorrelation to the 4th order, which is distributed as chi-squared (4) with critical value of 9.49 SBC is the Schwarz Bayesian Criterion and J is a test of overidentifying restrictions. k is -1 for the backwardlooking model and 8 (4) for inflation (output gap) when forward looking. The backwardlooking rule is estimated by least squares (Newey-West standard errors in parentheses); all other rules are estimated by GMM with the Newey-West criterion being used to choose the lag truncation parameter. The different models are described in the text.

  Hodrick-Prescott filtered level of GDP 19 (with a smoothing parameter of 1600), whilst the interest measure is the official Bank of England rate.

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Table 1 .

 1 Estimated simple interest rate rules for theUS (1987US ( :3-2004:4) :4)

							Taylor Rule Backward Taylor	Forward	Non-linear	Speed
							Rule	Taylor Rule	Taylor Rule	limit
	µ	0				1.80 (0.75)	1.32 (1.07)	-4.19 (1.52)	-38.97 (125)	-4.84 (2.38)
	t R		1	0.86 (0.03)	0.82 (0.06)	0.88 (0.03)	0.96 (0.07)	0.92 (0.03)
			t				1.27 (0.35)	36.0 (104)	3.82 (0.78)
			t+	k		1.52 (0.40)	3.89 (0.61)
	x	t				1.36 (0.17)	-9.87 (17.9)
	t x +	k			1.28 (0.20)	2.09 (0.46)
			2 t					-5.68 (17.4)
	x	2 t					0.58 (1.70)
			t x	t		6.93 (12.1)
			x	t			11.57 (5.0)
	JB				6.4	4.48	1.63	13.0	0.15
	B-P-	15.0	59.1	50.1	51.6	18.5
	L
	SBC	-1.62	-1.50	-1.24	-0.91	-0.79
	J						8.7	5.7	5.15	6.92
							(0.36)	(0.68)	(0.53)	(0.55)

Table 2 .

 2 Estimated simple interest rate rules for theUK (1992UK ( :4-2004:4) :4) As in Table1, except that in the forward-looking model the forecast horizon for both inflation and the output gap are four quarters.

							Taylor	Forward	Backward Taylor	Non-linear	Speed
							Rule	Taylor Rule	Rule	Taylor Rule	Limit
	µ	0				5.16 (0.37)	5.01 (0.26)	5.19 (0.51)	14.88 (58.99)	4.58 (0.37)
	t R		1	0.86 (0.04)	0.79 (0.04)	0.85 (0.06)	0.99 (0.04)	0.76 (0.07)
			t				-0.43 (0.50)	7.38 (34.9)	0.71 (0.40)
			t+	k		1.10 (0.29)	-0.17 (0.67)
	x	t				3.93 (1.17)	33.45 (152.1)
	t x +	k			1.67 (0.72)	2.10 (1.06)
			2 t					-26.1 (135)
	x	2 t					-25.9 (133.6)
			t x	t		81.5 (391)
			x	t			0.70 (1.70)
	JB				1.12	4.76	1.1	1.1	2.64
	B-P-L	24.0	19.9	21.5	18.59	23.2
	SBC	-1.59	-1.96	-1.62	-1.31	-1.40
	J						6.5	6.12	4.27	5.68
							(0.99)	(0.99)	(0.99)	(0.99)
	Notes:

Table 3 .

 3 Estimated simple differenced interest rate rules for theUS (1987US ( :3-2004:4) :4)

							Taylor	Speed Limit Nonlinear Orphanides-	Walsh	Giannoni
							Rule		Williams	(2005)	Woodford
	µ	0				0.17 (0.10)	-0.04 (0.10)	0.21 (0.24)	0.03 (0.13)	-0.58 (0.19)	-0.28 (0.16)
			t						-0.10 (0.05)	0.28 (0.07)	0.17 (0.05)
			x	t				0.55 (0.18)	0.68 (0.15)
			t		1		0.10 (0.04)	0.10 (0.04)	0.11 (0.20)
	x	t	1		0.07 (0.03)		0.11 (0.05)
	t R		1		-0.08 (0.02)	-0.05 (0.02)	-0.09 (0.02)	-0.022 (0.03)
			t R	1	0.67 (0.11)	0.60 (0.07)	0.67 (0.12)	0.48 (0.10)
			x	t	1		0.29 (0.09)
			2 t		1				-0.002 (0.04)
	x	2 t	1				-0.01 (0.01)
			x	2 t	1
			1 t t x	1		-0.02 (0.02)
			ue	t			-1.95 (0.18)
	JB				3.95	0.70	5.03	3.03	2.90	3.17
	B-P-L	0.27	0.16	0.26	10.5	15.6	0.72
	SBC	-1.95	-2.07	-1.78	-1.59	-1.12	-1.45
	J								6.76	8.67	7.17
	Notes: As in

Table 1 . Page 27 of 28 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

 1

	Submitted Manuscript

Table 4 .

 4 Estimated simple differenced interest rate rules for theUK (1992UK ( :4-2004:4) :4)

							Taylor	Speed Limit Nonlinear Orphanides-	Walsh	Giannoni
							Rule		Williams	(2005)	Woodford
	µ	0				0.74 (0.32)	0.85 (0.49)	0.58 (0.25)	-0.54 (0.08)	-0.08 (0.05)	0.58 (0.41)
			t						-0.15 (0.08)	0.15 (0.03)	-0.02 (0.07)
			x	t				0.59 (0.37)	0.65 (0.35)
			t		1		-0.08 (0.06)	-0.04 (0.06)	-0.03 (0.09)
	x	t	1		0.16 (0.12)		0.14 (0.16)
	t R		1		-0.14 (0.06)	-0.16 (0.09)	-0.11 (0.06)	-0.12 (0.08)
			t R	1	0.44 (0.18)	0.56 (0.12)	0.38 (0.21)	0.53 (0.12)
			x	t	1		0.15 (0.11)
			2 t		1				-0.02 (0.11)
	x	2 t	1				-0.03 (0.10)
			x	2 t	1
			1 t t x	1		0.22 (0.10)
			ue	t
	JB						-3.88
									(0.59)
	B-P-L	7.4	31.1	0.48	0.16	155.3	57.5
	SBC	6.0	3.40	6.9	3.22	11.3	2.02
	J						-1.80	-1.73	-1.73	-0.99	-1.15	-1.51
									4.50	7.3	5.29
	Notes: As in

Table 1 . Page 28 of 28 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

 1

This would enable one to consider an operational monetary policy rule, as in[START_REF] Mccallum | Performance of operational monetary policy rules in an estimated semi-Classical structural model[END_REF].

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

However, as Walsh also points out, this rule performs substantially worse than the optimal rule under more general misspecification.

Previous empirical support for a speed limit rule was found byPeel et al. (2004).

ACKNOWLEDGEMENTS

I would like to thank an anonymous referee for helpful comments.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript


Alberiko Luis

Christian Gil-Alana

Fischer

Luis A Gil-Alana

Christian Fischer

INTERNATIONAL TRAVELLING AND TRADE: FURTHER EVIDENCE FOR THE CASE OF SPANISH WINE BASED ON FRACTIONAL VAR SPECIFICATIONS

Keywords: International trade, Multivariate models, Fractional VAR, Tourism JEL Classification: C22, F14, Q13, L83 International trade, Multivariate models, Fractional VAR, Tourism JEL Classification: C22, F14, Q13, L83 International trade, Multivariate models, Fractional VAR, Tourism

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

International trade of good and services has been shown to be influenced by many factors of either push or pull character. While no mainstream microeconomic model has yet been developed to establish theoretically the link between trade and tourism (see however, [START_REF] Fischer | The influence of immigration and international tourism on the import demand for consumer goods -a theoretical model[END_REF], for a first approach), a host of empirical work has emerged which shows that there may in fact be a connection between the flow of goods and people. Many studies attempt to test the hypothesis that movements of people can contribute to generate exports, but they may also stimulate imports. Tourism is thought to be able to promote cross-border exports by initiating entrepreneurial activities as a result of learning about new business opportunities while travelling. At the same time, demand for new products to be consumed back home may be created as a consequence of learning about them during foreign travel.

In previous work, for instance, [START_REF] Easton | Is Tourism Just Another Commodity? Links between Commodity Trade and Tourism[END_REF] analysed whether Canadian total exports are complementary or substitutive to tourist arrivals, using pooled data regressions. He finds "some evidence of substitution of Canadian exports for tourist excursions to Canada" (p. 542) by showing that when the relative price of exports goes up, the number of tourists visiting Canada increases. [START_REF] Kulendran | Is there a relationship between international trade and international travel?[END_REF] analysed the direction of causality between different travel and (aggregate) trade categories for Australia and its four main trading partners. Their results show that travel Granger causes international trade in some cases and vice versa in others. [START_REF] Shan | Causality Between Trade and Tourism: Empirical Evidence from China[END_REF] replicate this latter approach and also find two-way Granger causality using aggregate data for China. [START_REF] Aradhyula | Does Tourism Promote Cross-Border Trade?[END_REF] used a simultaneous bivariate qualitative choice model to show that cross-border business trips have a significant and positive effect on US agribusinesses' propensity to trade. [START_REF] Fischer | The influence of immigration and international tourism on the demand for imported food products[END_REF] explored the connection between aggregate imports and imports of individual products 3 and bilateral tourist flows, using an error correction model. His results show that tradetourism elasticities are consistently higher for individual products. [START_REF] Fischer | The Nature of the Relationship Between International Tourism and International Trade: the Case of German Imports of Spanish Wine[END_REF] quantified for the first time the length of the effect of tourism on international trade, using the case of German imports of Spanish wine. Depending on the wine type, the effect was estimated to last between 3 and 11 months, and on average 5.5 months.

What emerges from all these studies is that the existence, the direction, the strength (magnitude), and the length of the effect which tourism may have on international trade seems to depend on the analysed countries and products, and on the estimation technique used. Even if a range of results is now available, further empirical evidence is still useful in order to obtain a more complete and robust understanding of the actual nature of the relationship between international tourism and trade.

The aim of this study is therefore to expand existing knowledge on the traveltrade relationship for food products by generating further empirical evidence, based on recently developed fractional vector autoregressive (VAR) regression models. The analysis is more empirical than theoretical in the sense that it is attempted to test econometrically the hypothesis of a potentially existing relationship between travelling and exports. The used approach is not grounded in microeconomic supply or demand theory since no output, income or price data are taken into account in the econometric specification of the model. However, VAR models are generally accepted as theory-free methods for estimating economic relationships, thus being a legitimate alternative to the identification restrictions in structural models [START_REF] Sims | Macroeconomics and Reality[END_REF].

We focus exclusively on the relationship between the exports of Spanish wines to Germany and the number of German tourists travelling to Spain on a monthly basis.

We do this for several reasons. First, wine has become a truly globalised industry with [START_REF] Anderson | Introduction, in: The World's Wine Markets -Globalization at[END_REF]. Second, in industrialized nations, wine is a commonly available commodity offered in a large variety mostly differentiated by production origin. Given that objective wine quality is hard to assess for non-expert consumers, the origin of a wine is often used as a short-cut quality indicator in cases where the country of origin is associated with a preferred holiday destination [START_REF] Felzenstein | Is the Country of Origin the Fifth Element in the Marketing Mix of Wine? A Critical Review of the Literature[END_REF].

Last, wine imports have been shown to display a significant connection with tourism activities among a range of previously investigated products [START_REF] Fischer | The influence of immigration and international tourism on the demand for imported food products[END_REF].

Nevertheless, it must be stated that in other industries (e.g., automotives), the traveltrade link may be much weaker or even inexistent. This analysis differs from earlier work [START_REF] Fischer | The Nature of the Relationship Between International Tourism and International Trade: the Case of German Imports of Spanish Wine[END_REF] in various aspects. First, we extend the period of investigation both backwards and forwards implying that the results are expected to be more reliable. Moreover, we use export instead of import data, and more importantly, we use a different econometric modelling approach, which is multivariate rather than univariate as in [START_REF] Fischer | The Nature of the Relationship Between International Tourism and International Trade: the Case of German Imports of Spanish Wine[END_REF]. Our method allows us to compute impulse response functions across variables and extend previous methodologies (e.g., VAR) in the use of fractional orders of integration for the series under investigation. As far as we know, this is the first empirical application using fractional vector autoregressive models, and, given the plethora of work on univariate fractionally integrated models in macro series, a work on multivariate (fractional) systems seems clearly overdue.

The structure of this article is as follows. First, for the two series under consideration, we produce univariate results based on fractional integration models. This approach is more flexible than others usually employed in the literature since it allows us to consider the cases of stationarity I(0) and nonstationarity I(1) as particular cases of our approach. Then, an innovative bivariate fractionally integrated model is used, i.e., we estimate jointly the orders of integration of the two series, and then analyse the cross impulse response functions. In other words, the effect of a shock in one variable over the other across time is computed. Here, we implicitly assume that the direction is uni-directional in the sense that we believe that travelling has an influence on exports and not the reverse case. 1 The main innovation is that we allow fractional values for the orders of integration while standard methods suppose that they are either 0 or 1. In the fifth section, the obtained results are compared to similar ones from other studies, before some conclusions are drawn. The appendices contain the technical details of the paper. Appendix A presents the functional form of the test statistic for testing fractional integration in a multivariate context. Appendix B refers to the fractional VAR model, specified for the bivariate case, and the subsequent sub-sections in that appendix describe the restriction required to the identification of the system for the two cases of white noise and autocorrelated disturbances.

Econometric methodology

This section is based on econometric grounds and we describe the techniques employed in the empirical work in Section 4. A crucial point when modelling univariate (or multivariate) time series is to correctly determine the order(s) of integration. In other words, in order to make statistical inference, the series are required to be stationary I(0).

If they are not, a standard approach is to take first differences based on the assumption that the series are then nonstationary I(1). However, these two approaches (I(0) and I(1)) may be too restrictive in the sense that many series may present a behaviour that is far from these two cases. In particular, the series may present a degree of dependence 1 Fischer and Gil-Alana (2007) showed that German travellers to Spain cause (in the Granger causality tests) exports of Spanish wines to Germany. We should expect a similar pattern for imports. 6 across time that is higher than the one described by the I(0) models (e.g., the exponential degree associated to the AR specifications) but smaller than the one obtained through the I(1) case. In such cases fractional differencing may be a viable approach. Given their theoretical appeal, fractional integration methods are increasingly used among applied economists, see, for instance, [START_REF] Barkoulas | Fractional monetary dynamics[END_REF], [START_REF] Tolvi | Long memory and outliers in stock market returns[END_REF], [START_REF] Gil-Alana | Fractional integration in total factor productivity: evidence from US data[END_REF], [START_REF] Gschwandtner | Modelling profit series: nonstationarity and long memory[END_REF] or [START_REF] Assaf | Fractional integration in the equity markets of MENA region[END_REF].

For the purpose of the present paper we define an I(0) process as a covariance stationary process with spectral density function that is positive and finite. In this context, we say that a given raw time series {x t , t = 0, ±1, ..

} is I(d) if: ... , 2 , 1 , ) 1 ( = = t u x L t t d , (1)
, 0 , 0 = t x t
where u t is I(0) and where L means the lag operator (Lx t = x t-1 ).2 Note that the polynomial above can be expressed in terms of its Binomial expansion, such that for all real d,

+ = = = 0 2 . ... 2 ) 1 ( 1 ) 1 ( ) 1 ( j j j d L d d L d L j d L
The literature has usually stressed the cases of d = 0 and 1, however, d can be any real number. If d = 0 in (1), x t = u t , and a 'weakly autocorrelated' (e.g., AR) x t is allowed for. However, if d > 0, x t is said to be a long memory process, also called 'strongly autocorrelated', so-named because of the strong association between observations widely separated in time, and, as d increases beyond 0.5 and through 1, x t can be viewed as becoming "more nonstationary", in the sense, for example, that the variance of partial sums increases in magnitude. 3 These processes were introduced by [START_REF] Granger | Long memory relationships and the aggregation of dynamic models[END_REF][START_REF] Granger | Some properties of time series data and their use in econometric model specification[END_REF], [START_REF] Granger | An introduction to long memory time series and fractionally differencing[END_REF] and [START_REF] Hosking | Fractional differencing[END_REF], (though earlier work by [START_REF] Adenstedt | On large-sample estimation for the mean of a stationary random sequence[END_REF][START_REF] Taqqu | Weak convergence to fractional Brownian motion and to the Rosenblatt process[END_REF] shows an awareness of its representation), and were theoretically justified in terms of aggregation of ARMA processes with randomly varying coefficients by [START_REF] Robinson | Statistical inference for a random coefficient autoregressive model[END_REF], [START_REF] Granger | Long memory relationships and the aggregation of dynamic models[END_REF]. If d belongs to the interval (0, 0.5) x t is covariance stationary, but both the autocorrelations and the response of a variable to a shock take much longer time to disappear than in a standard (d = 0) stationary case. If d [0.5, 1) the series is no longer covariance stationary but is still mean reverting, with the effect of the shocks dying away in the long run. Thus, the fractional differencing parameter d plays a crucial role for our understanding of the economy and the macro dynamics. The majority of the applications of fractional integration in economic time series are based on univariate models, and we find among others the papers of [START_REF] Diebold | Long memory and persistence in aggregate output[END_REF], Baillie and Bollerslev (1994) and Gil-Alana and Robinson (1997). 4 In this section we present a novel approach that permits us to consider a structural fractional VAR model from its reduced form and then obtain the impulse response functions. We derive a simple method in a multivariate fractional integration framework, which lets the data determine simultaneously the response of one variable over the other(s). This method presents some advantages with respect to previous approaches. First, the fractional integration approach allows to discern the order of integration of a given variable without the econometrician to choose between zero or one. The order of integration may be zero, a fraction of one, one or it could be even above one. Second, this approach is agnostic with respect to the order of integration of F o r P e e r R e v i e w 8 the variables before including them in a vector autoregressive (VAR) framework. As a result, pre-tests of the orders of integration of the variables are not required. Third, there is no disagreement between the responses of the variables in levels or in first differences as the responses in first differences are exactly the same as those implied by the variables in levels by construction.

The starting point is the following structural model:

... , 2 , 1 , = = t u y D A t t (2) , ... , 2 , 1 , 1 = + = t v u G u t t t (3)
where A is a (nxn) matrix of parameters; D is an (nxn) diagonal matrix of form:

n d d L L ) 1 ( 0 0 0 0 0 0 ) 1 ( 1 ,
where d 1 , d 2 , …, d n can be real values; y t is a (nx1) vector of the observable variables; u t is a (nx1) vector, which is assumed to be I(0) 5 ; G is another (nxn) matrix of parameters, and v t is a (nx1) structural error vector with zero mean and diagonal variancecovariance matrix V. Substituting (2) into (3), we obtain

... , 2 , 1 , 1 = + = t v y D A G y D A t t t (4) implying that ... , 2 , 1 , 1 1 1 = + = t v A y D A G A y D t t t .
(5)

Using now the lag-operator (i.e., Ly t = y t-1 ): In a multivariate system the number of procedures for fractional integration is very limited. Gil-Alana (2003a,b) proposed an extension of the univariate tests of [START_REF] Robinson | Efficient tests of nonstationary hypotheses[END_REF] in the frequency domain, while [START_REF] Nielsen | Multivariate Lagrange Multiplier tests for fractional integration[END_REF] developed time domain versions of Gil-Alana's tests. 6 These methods allow us to estimate a reduced-form model of form:

[ ] ..., , 2 , 1 , 1 1 = = t v A y D L A G A I t t we get [ ] ..., , 2 , 1 , 1 1 1 1 = = t v A L A G A I D y t t (6)
... , 2 , 1 , = = t y D t t (7) ... , 2 , 1 , 1 = + = t w F t t t , ( 8
)
where t is a (nx1) vector of the d-differenced variables; F is a (nxn) matrix of parameters, and w t is an I(0) vector with variance-covariance matrix W. Substituting now ( 7) into (8), ..., ,

= + = t w y D F y D t t t (9) implying that [ ] , ... , 2 , 1 , = = t w y D L F I t t and then [ ] , ... , 2 , 1 , 1 1 = = t w L F I D y t t (10) 2 , 1 , 1
which is the reduced-form MA( ) representation of y t .

Note that the structural model in (6) has 2n 2 +2n parameters to estimate: n corresponding to the fractional differencing parameters in D; 2n 2 of the two matrices A and G; and the n variances in V. On the other hand, the reduced-form MA( ) representation in (10) contains n+n 2 +n(n+1)/2 parameters: the n d-parameters in D; n 2 in F, and n(n+1)/2 parameters of the variance-covariance matrix W. Therefore, in order to F o r P e e r R e v i e w 10 identify the system we need to impose (n/2)(n+1) restrictions in the structural model. N restrictions can be obtained by imposing a 1-unit variance in the variance-covariance matrix of v t in (3), V. However, (n 2 -n)/2 restrictions will still be required. Here, there are two possibilities: one is to impose triangularity in the A matrix in (2) -this would imply that the contemporaneous and the future effects of some of the variables on the others will be zero, which may be a relatively strong assumption in some cases. The second approach uses the [START_REF] Blanchard | The dynamics effects of aggregate demand and supply disturbances[END_REF] decomposition, which implies that in the long run some variables have no effect on the others. 7, 8 3.

The data

We look at the relationship between trade and tourism by focussing on the interdependencies between the exports of Spanish wines to Germany and the number of German travellers to Spain.

The raw data were obtained from two different Eurostat databases. First, exports of Spanish wine (without sparkling wine) and of sparkling wine to Germany are taken from COMEXT "EU trade since 1995 by CN6" database. The arrivals of Germans in Spanish collective accommodation establishments are obtained from the "TOUR_OCC _NINRMW = Nights spent by non-residents -world geographical breakdownmonthly data" database.

The empirical work

The first thing we do in this section is to model individually the two series, which are the total number of arrivals of Germany in Spanish collective accommodation 7 A full description of these methods in presented in Appendix B. 8 Alternative identification strategies also involving restrictions on the long run behaviour of the series are given in [START_REF] King | Stochastic trends and economic fluctuations[END_REF]. Other possibilites include restrictions on the sign and/or

INSERT FIGURE 1 ABOUT HERE

A visual inspection at the series (in Figure 1) clearly shows that the two series present a seasonal component, which is changing over time. Dealing with seasonality is a matter that is still controversial. Deterministic approaches based on seasonal dummy variables are discouraged in this case in view of the changing seasonal patterns. A standard approach here is to perform a test of seasonal unit roots against the alternative of stochastic stationary behaviour. The most commonly-used method when dealing with monthly data is the one proposed by [START_REF] Beaulieu | Seasonal unit roots in aggregate US data[END_REF], which is basically an extension of the Hylleberg, Engle, Granger and Yoo (HEGY, 1990) method to the monthly case. A drawback of this approach is that it is restricted to the case of I(1) and I(0) specifications and thus, it does not take into account fractional alternatives.

Therefore, we also perform an alternative method [START_REF] Robinson | Efficient tests of nonstationary hypotheses[END_REF] that is nested in the fractional seasonal model of the form:

... , 2 , 1 , ) 1 ( 12 = = t u y L t t d
, where the (seasonal) unit root corresponds to the case of d = 1. 9 Though we do not report the results in the paper, we perform both [START_REF] Beaulieu | Seasonal unit roots in aggregate US data[END_REF] and [START_REF] Robinson | Efficient tests of nonstationary hypotheses[END_REF] approaches, and we found in both cases strong evidence of unit roots with respect to the two series. Thus, since the two series are based on logarithm shape of the impulse responses [START_REF] Faust | The robustness of identified VAR conclusions about money[END_REF]; via heteroskedasticity [START_REF] Rigobon | Identification through heteroskedasticity[END_REF]; or the use of highfrequency data [START_REF] Faust | Identifying VARs based on high frequency futures data[END_REF]). 9 See Gil-Alana and [START_REF] Gil-Alana | Testing seasonal fractional integration in the UK and Japansese consumption and income[END_REF] and [START_REF] Gil-Alana | Seasonal long memory in the aggregate output[END_REF], 2005) for descriptions of seasonal fractional models.

Univariate results

First we examine individually each series to check if they are truly stationary I(0). Here we employ a simple version of [START_REF] Robinson | Efficient tests of nonstationary hypotheses[END_REF]

: = d H o , (13)
while d = 0 corresponds to the stationary I(0) case. This method has some advantages compared with other more classic approaches of testing unit roots [START_REF] Dickey | Distributions of the estimators for autoregressive time series with a unit root[END_REF][START_REF] Phillips | Testing for a unit root in a time series regression[END_REF]; or any of its recent developments, [START_REF] Elliot | Efficient tests of an autoregressive unit root[END_REF]Ng and Perron, 2001, etc.). The most obvious one is clearly the fact that the latter approaches are too restrictive in relation with the order of integration since only I(0) and in ( 14), and leading to a non-standard limit distribution, unlike what happens in [START_REF] Robinson | Efficient tests of nonstationary hypotheses[END_REF] where the limit distribution is standard normal. Fractional and AR departures from ( 13) and ( 15) have very different long run implications. In ( 11), y t is nonstationary but non-explosive for all d 0.5. As d increases beyond 0.5 and through 1, y t can be viewed as becoming "more nonstationary", but it does so gradually, unlike in case of ( 15) around ( 14). The dramatic long-run change in ( 14) around = 1 has the attractive implication that rejection of ( 15) can be interpreted as evidence of either stationarity or explosivity. However, rejection of the null does not necessarily warrant acceptance of any particular alternative and they can be consistent against many of the numerous other types of departure [START_REF] Robinson | Highly insignificant F-ratios[END_REF]. On the other hand, the approach employed here applies equally to any real null hypothesized value of d and the same standard, null and local limit distribution theory obtains. This is also in sharp contrast to asymptotic theory for statistics directed against AR alternatives, where, for example, different null theory obtains for I(2) than for I(1) processes.

We use [START_REF] Robinson | Efficient tests of nonstationary hypotheses[END_REF] cannot be rejected at the 5% level, for the two series assuming that u t in ( 11) is first white noise, and then allowing for some type of weak dependence structure, in particular, AR(1) and Bloomfield-type disturbances. 10 Moreover, we also permit the inclusion of an intercept and/or a linear time trend, and thus, we report the results for the three cases of no deterministic terms, an intercept and an intercept with a linear trend.

INSERT TABLE 1 ABOUT HERE

Starting with the arrivals, the first thing we observe in Table 1 is that the two null hypotheses of d = 0 and d = 1 are both rejected in practically all cases, and the nonrejection values of d are constrained between these two values. We also display in the table the value of d producing the lowest statistic (in absolute value). That value should be an approximation to the maximum likelihood estimate of d since [START_REF] Robinson | Efficient tests of nonstationary hypotheses[END_REF] method is based on the Whittle function, which is an approximation to the likelihood function. We observe that d is equal to 0.54 in case of white noise disturbances, and it is slightly higher for autocorrelated u t . However, a very different picture is obtained for the export series. Thus, if u t is white noise, d seems to be slightly above 0, and if autocorrelation is permitted, the null of I(0) stationarity cannot be rejected. Thus, it seems clear that the arrivals present a stronger degree of association across time and thus, a higher degree of persistency than the corresponding export series.

The results presented so far may be biased because of the presence of a structural break in the data. (See, again Figure 2). [START_REF] Gourieroux | Memory and infrequent breaks[END_REF], [START_REF] Diebold | Long memory and regime switching[END_REF], [START_REF] Granger | Occasional structural breaks and long memory with an application to the S&P 500 absolute stock returns[END_REF] and others showed that I(d) models and structural change are issues which are highly connected. Thus, we also perform another recent procedure (Gil-Alana, 2007) that permits us to estimate the fractional differencing parameters and the coefficients associated to the linear trend, along with the time of the structural break in a model given by:

b t t d t t T t u x L x t y , ... , 1 , ) 1 ( ; 1 1 1 = = + + =
(16) 10 The [START_REF] Bloomfield | An exponential model in the spectrum of a scalar time series[END_REF] model is a non-parametric approach of modelling the I(0) disturbances that produces autocorrelations decaying exponential as in the AR(MA) case.

F

+ = = + + = ( 17
)
where the 's and the 's are the coefficients corresponding to the intercept and the linear trend; d 1 and d 2 may be real values, u t is again I(0) and T b is the time of the break that is supposed to be unknown. Note that the model in ( 16) and ( 17) can also be written as:

, , ... , 1 , ) ( ) ( 1 ) 1 ( 1 1 1 1 1 b t t t t d T t u d t d y L = + + = (18) , , ... , 1 , ) ( ) ( 1 ) 1 ( 2 2 2 2 2 T T t u d t d y L b t t t t d + = + + = (19) where , 1 ) 1 ( ) ( 1 ~i d i t L d = and , t ) L 1 ( ) d ( t ~i d i t = i = 1, 2.
The procedure employed here is based on the least square principle and is similar to the one proposed by [START_REF] Bai | Estimating and testing linear models with multiple structural changes[END_REF] , we estimate the 's and the 's by minimizing the sum of squared residuals,

2 1 ) 1 ( 2 2 ) 1 ( 2 2 ) 1 ( 2 } 2 , 1 , 2 , 1 { . . . 2 1 ) 1 ( 1 1 ) 1 ( 1 1 ) 1 ( 1 ) ( ) ( 1 ) 1 ( ) ( ) ( 1 ) 1 ( min + + = = T b T t

=

, where the minimization is taken over all partitions T 1 , T 2 , …, T m , such that T i -T i-1 | T|.

Then, the regression parameter estimates are the associated least-squares estimates of the estimated k-partition, i.e., }), T ({

ˆk i i = }), T ({ ˆk i i =
and their corresponding differencing parameters are }), T ({ d d ˆk i i = for i = 1 and 2. Several Monte Carlo experiments conducted in [START_REF] Gil-Alana | Fractional integration and structural breaks at unknown points in time[END_REF] show that this procedure performs relatively well even with small samples. This procedure can be easily extended to the case of multiple breaks. However, for the validity of the type of long-memory (fractional integration) model we use in this application it is necessary that the data span a sufficiently long period of time to detect the dependence across time of the observations; given the sample size of the series employed here, the inclusion of two or more breaks would result in relatively short sub-samples, therefore invalidating the analysis based on fractional integration.

INSERT TABLE 2 ABOUT HERE

The results based on the above approach are displayed in Table 2. Starting with the arrivals, we observe that the break date takes place at 2001M1, which is surprisingly a few months earlier than the September 11 th attack in the U.S., and this happens for the two cases of white noise and AR(1) disturbances. If u t is white noise, the orders of integration are 0.58 and 0.21 respectively for the first and second sub-samples, and the I(0) hypothesis (i.e., d = 0) is rejected in the two cases. If u t is AR(1) the orders of integration are again statistically higher than 0, and are slightly higher in the two subsamples though again decreasing after the time break. If we concentrate now on the exports we observe that the break date occurs two months later than in the previous case In sum, the results presented so far indicate that the two series display different orders of integration, independently of the inclusion or not of a structural break. In what follows we consider a bivariate set-up that solves the potential problems of unbalanced orders of integration in standard time series regression frameworks as is the case in the present work.

Multivariate results

In this section we look at the multivariate model. This is important if we want to determine the effect of structural shocks on the dynamic path of travelling and export trade variables. The bivariate model is estimated following the procedure in Gil-Alana (2003a). This method is briefly described in Appendix A and present two main advantages. First, it is an extension of the univariate tests of [START_REF] Robinson | Efficient tests of nonstationary hypotheses[END_REF] to the multivariate case and thus, similarly to the univariate case, we do not need to impose a priori any assumption about the orders of integration of the series as is the case with other approaches (e.g., [START_REF] Nielsen | Efficient inference in multivariate fractionally integrated time series models[END_REF] or with standard VAR models. Second, with respect to the univariate case the orders of integration are estimated more efficiently since it makes use of additional information from the cross dependencies between the variables.

We estimate the orders of integration from the reduced form model ( 7), which, in this bivariate case, becomes:

... , 2 , 1 , ) 1 ( 0 0 ) 1 ( , 2 , 1 , 2 , 1 2 1 = = t y y L L t t t t d d (20)
where y 1,t refers to the arrivals monthly growth series and y 2,t is the monthly growth rate of exports, and test the null hypothesis: , ) , ( ) , ( :

2 1 2 1 o T o o T o d d d d d d H = (21)
in ( 20) for (d 1o , d 2o )-values from -1 to 2, with 0.01 increments, with t = ( 1,t , 2,t ) T assumed to be first a white noise vector process, and later a VAR(1) specification. 11 In order to avoid the inclusion of deterministic terms in (20) both series are meansubstracted before the implementation of the procedure.

INSERT FIGURES 3 AND 4 ABOUT HERE

Figures 3 and 4 display the region of (d 1o , d 2o )-values where the null hypothesis ( 21) cannot be rejected at the 5% level, letting the residuals to follow first (in Figure 3) a white noise vector process, and, in Figure 4, a stationary VAR(1) specification.

Starting with the white noise model, we observe that the results are completely in line with the univariate ones. Thus, d 1 , the order of integration of the number of arrivals is constrained between 0.5 and 0.7, while d 2 , the order of integration of the export series is

INSERT TABLE 4 ABOUT HERE

The next step in our analysis is to report the associated impulse response function of exports to a shock in the travelling series. For this purpose we employ the two identification strategies described in Section 2, which, for the bivariate case is fully presented in Appendix B.

INSERT TABLE 5 AND FIGURE 5 ABOUT HERE

Table 5 and Figure 5 display the cross impulse responses of the effect of 1-unit shock in the growth rate of travelling on the growth rate of exports according to the two specifications described above, that is, the one based on white noise and the VAR(1) model. In both cases, for identification purposes, we first assume that c = 0 in equation (B1) (Appendix B), implying that there are nor contemporaneous neither future effects 12 Similar results were obtained when using higher VAR orders for the disturbance term. Nevertheless, in the final part of this section, we also conduct the analysis using an additional identification schedule based on the [START_REF] Blanchard | The dynamics effects of aggregate demand and supply disturbances[END_REF] decomposition. 14 We observe that according to the white noise specification, an increase in travelling produces a negative (though statistically insignificant) initial impact though it becomes positive for the following period and then, start decreasing fast. On the other hand, if we employ the VAR(1) specification, which seems to be a much more realistic assumption, the initial impact is positive (and significant) and then decreases at a lower rate than in the previous case of white noise disturbances. 15

INSERT FIGURE 6 ABOUT HERE

In the final part of this section we focus on the identification using the decomposition method proposed by [START_REF] Blanchard | The dynamics effects of aggregate demand and supply disturbances[END_REF]. The impulse response functions, for the two cases of white noise and VAR(1) disturbances, are displayed in Figure 6. These confidence intervals were computed from 1,000 Monte Carlo draws of the estimated model using normally distributed errors. The results are fairly similar to those displayed in Figure 5 and based on the triangular identification. Thus, assuming white noise errors, the initial impact is negative though insignificantly different from zero, and using VAR disturbances, the effect is positive and significant for the initial periods, decreasing then fast to zero, implying that travelling produces a positive effect 13 Standard F-tests were conducted in the VAR(1) specification and we could not reject the null hypothesis of no serial correlation. A Likelihood Ratio test also produced evidence for the VAR(1) against the VAR(2) model. 14 Furthermore, and following the recommendations in [START_REF] Sims | An autoregressive index model for the US: 1948-1975[END_REF] (see, also Lutkepohl and Breitung, 1996) we also checked whether the results were robust to the ordering of the variables. Altering the order of the variables the results for the impulse responses were practically identical to those reported in Figure 4. 15 The confidence intervals here were obtained based on the asymptotic theory for multivariate fractional processes. (See, [START_REF] Nielsen | Efficient inference in multivariate fractionally integrated time series models[END_REF].

Comparison of the results to those of other studies

Our results imply that, for the more realistic case of the VAR(1), in the event of a shock in travelling almost 10% of its effect on exports remains in the following period, though then decreases to 2.76%, 0.79%, 0.33% etc. (see Table 5). Overall, the total effect sums up to 14.7% over the tracked 20 months. However, 90% of this total effect is already realised after four months, and 95% of it after 9 months. These results seem to be in line with those obtained by [START_REF] Fischer | The Nature of the Relationship Between International Tourism and International Trade: the Case of German Imports of Spanish Wine[END_REF] who find (using long memory regressions and very similar monthly growth rate data) that the tourism-trade effect lasts two months (in the case of white noise disturbances) and 9 months in the case of Bloomfield (p = 1) disturbances. However, the present study outperforms Fischer and Gil-Alana ( 2007) not only in the longer time period examined but also in the methodology employed that permits us to examine the cross impulse responses from a multivariate model.

Concluding comments

In this article we have examined the relationship between international trade and travelling. For this purpose we have focussed on a particular case study, analyzing the connections between the exports of Spanish wines to Germany with the total number of German travellers to Spain. We have employed a methodology based on fractional vector autoregressive models, which is more general than the standard VAR approach in the sense that we do not restrict the series to be I(0) or I(1) but I(d) for any real value d.

Starting from a structural model we have derived the conditions to identify the The series under analysis were the exports of Spanish wine to Germany and the total number of arrivals of Germans to Spain, monthly, from 1995M1 to 2006M7, both in logarithm form. Due to the nonstationary seasonal nature observed in the two series, first seasonal differences were adopted, working then with the monthly growth rates.

The univariate work showed that the two series display different orders of integration.

Thus, the arrivals present an order of integration which is constrained between 0.5 and 1, while the degree of integration of exports is slightly above 0. The multivariate work confirms that result and the impulse response analysis suggests that a positive shock in the arrivals tends to increase the exports to a certain level, decreasing then slowly the effect in the long run.

There are several contributions of the present work. In terms of new results, we provide further evidence of the relationship between Spanish wine exports to Germany and German travellers to Spain, noting that there are no previous results based on multivariate (fractional) impulse response functions. The obtained results are in line with earlier studies and thus add to the literature dealing with the tourism-trade relationship. While econometric findings in themselves cannot incontestably prove that a positive relationship between international travel activities and resulting trade flows exists in reality, the increasing body of empirical results clearly suggest that the existence of such a link may not be unlikely, at least for some products and in some countries.

Methodological innovations are the following: first, we extend fractional integration to the multivariate case, which has not yet been implemented in the A number of conditions are required on A and f when deriving the test statistic; their practical implications being that though t is capable of exhibiting a much stronger degree of autocorrelation than multiple ARMA processes, its spectral density matrix must be finite, with eigenvalues bounded and bounded away from zero. In Gil-Alana (2003a) it is shown that a Lagrange Multiplier (LM) test of H o (A1) in ( 7) takes the form:

[ ] , 1 1 b D E D C b T S T T = (A4)
where T is the sample size and where T* is a compact subset of q-dimensional Euclidean space. Extending the conditions in [START_REF] Robinson | Efficient tests of nonstationary hypotheses[END_REF]), Gil-Alana (2003a)

( ) ( ) , c a bc ad 1 2 2 2 22 + = ) ( ) ( ). 1 2 12 ab dc bc ad + = )
The new system of equations is still not identified, as there are only three equations for four unknowns. One possibility is to assume that one of the coefficients (a, b, c or d) is equal to 0. For example, b = 0 implies, according to (B6), that a structural shock to y 2t (u 2t ) has no effect on y 1t neither contemporaneously nor in the long run. Similarly, if c = 0, a shock to y 1t will have no effect on y 2t . This is a plausible

= = t v v L C L C L C L C u u t t t t (B13)
where C ij (L), i, j = 1, 2 are polynomials of infinite order in L. From (B2) and (B13):

= + + = t t t t t d t d v L C v L C v L C v L C a c b d bc ad y L y L , 2 22 , 1 21 , 2 12 , 1 11 , 2 2 , 1 1 ) ( ) ( ) ( ) ( 1 ) 1 ( ) 1 ( = + + + t t t t t t t t t t w w v L C a v L C a v L C c v L C c v L C b v L C b v L C d v L C d bc ad , 2 , 1 , 2 22 , 1 21 , 2 12 , 1 11 , 2 22 , 1 21 , 2 12 , 1 11 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 . (
+ = = = 0 , 2 ) 2 , 1 ( 0 , 1 ) 1 , 1 ( , 1 , j j t j j j t j t v v y (B16) + = = = 0 , 2 ) 2 , 2 ( 0 , 1 ) 1 , 2 ( , 2 , j j t j j j t j t v v y (B17)
where the impulse response functions are: The large sample standard error under the null hypothesis of no autocorrelation is 1/+T or roughly 0.08 for the series used in this application.

( ) ; bc ad ) L ( C b ) L ( C d 21 11 ) 1 ( j ) 1 , 1 ( j # = ( ) ; bc ad ) L ( C b ) L ( C d 22 12 ) 1 ( j ) 2 , 1 ( j # = (B18) ( ) ; bc ad ) L ( C a ) L ( C c 21 11 ) 2 ( j ) 1 , 2 ( j + # = ( ) . ) ( ) ( 22 12 ) 2 ( ) 2 , 2 ( bc ad L C a L C c j j + = # (
The periodograms are computed based on the discrete frequencies j = 2 j/T. The large sample standard error under the null hypothesis of no autocorrelation is 1/+T or roughly 0.08 for the series used in this application.

The periodograms are computed based on the discrete frequencies j = 2 j/T.

  production (in value terms) being exported worldwide in 2001

  structural MA( ) representation of y t .

  the total Spanish wine exports (including sparkling wine) toGermany, monthly, from 1995M1 to 2006M7.

  Figure 2 displays the two monthly differenced log-transformed series with their

  for the case of stationary I(0) processes. First we choose a grid for the values of the fractionally differencing parameters d 1 and d 2 , for example, d io = 0, 0.01, 0.02, …, 1, i = 1, 2. Then, for a given partition {T b } and given initial d 1 , d 2 -values,

  and the orders of integration are 0.26 and -0.04 with uncorrelated errors and 0.31 and 0.02 with AR(1) disturbances. Here, the I(0) hypothesis is rejected in the first subsample but cannot be rejected after the break.The finding of the break in the arrivals at 2001M1 may be explained by the fact that during 2000 the German economy slowed down significantly after the bursting of the technology bubble at the end of 1999. Thus, from the beginning of 2001 there was a significant decrease in German international travel activities, in particular to Spain, the most important travel destination of Germans (Provincial Tourist Board of the Costa delSol, 2003).

  widely ranging between 0.05 and 0.45. If we permit a VAR specification on the differenced series, the values of d 1 range between 0.5 and 1.2, while those of d 2 are constrained between -0.1 and 0.3.12

  least for the case of the Spanish wine exports to Germany with respect to the German travellers to Spain.

  reduced form model, which is basically a generalization of the standard I(0)/I(1) VAR case. The impulse response functions are then immediately obtained.

  Second, the use of a structural fractional VAR model and the detailed description of the restrictions required for the identification of the system is the other methodological contribution of the paper. Finally, compared with[START_REF] Fischer | The Nature of the Relationship Between International Tourism and International Trade: the Case of German Imports of Spanish Wine[END_REF] the superiority of the presented technique is evident from the facts that first we extend the time period examined both backwards and forwards. Second, we use multivariate rather than univariate models with the incorporation of additional information through the variance-covariance matrices, and third we display impulse response functions that, in the context of multivariate systems permits us to examine the influence of one of the variables over the other.Nevertheless, as argued above, there might be other factors that could have an influence in the relationship between German travellers to Spain and Spanish wine exports to Germany (for instance, a depreciation of the Spanish peseta vis-à-vis the German mark in the pre-EMU period would facilitate Spanish wine exports). These factors may have altered the relationship between the two variables and noting that fractional integration and structural breaks are issues which are intimately related, the inclusion of a structural break in our fractional VAR set-up seems mandatory in future work. Further work also needs to aim at establishing a theoretical economic framework (perhaps along the lines of[START_REF] Fischer | The influence of immigration and international tourism on the import demand for consumer goods -a theoretical model[END_REF], which can contribute to explain the existence of the tourism-trade phenomenon. means the complex-conjugate transpose of w.

  Figures and tables

  FIGURE 2Monthly growth rate series with their corresponding correlograms and periodogramsMonthly growth rate of ArrivalsMonthly growth rate of Exports

  FIGURE 3

  FIGURE 5

  approach, testing H o (12) in model (11) for d o -values from -1 to 2 with 0.01 increments. Table 1 presents the values of d o where H o (12)

Table 4

 4 identifies the estimates of the orders of integration of the two series, which are the values of d 1 and d 2 producing the lowest statistics in the multivariate procedure. We observe that if u t is white noise, the values are 0.54 and 0.18 respectively

	for the arrivals and exports. Imposing a VAR(1) specification, the values become 0.70
	and 0.04. 13

  shows that:

			F o r F o r
			P P
			e e
			e r e r
				R e v R e v
						i e i e
	)	=	+				w w
			~2 % S n d &	as	T	%	.	(A5)

TABLE 4 Estimates of the orders of integration in a multivariate framework

 4

	Disturbances / Order of Int.	d 1 (Arrivals)	d 2 (Exports)
	White noise	0.54	0.18
	VAR (1)	0.70	0.04
		TABLE 5
	Cross impulse responses of travelling on exports
	Time periods	White noise	VAR (1)
	1	-0.0248	0.0979
	2	0.0204	0.0276
	3	0.0018	0.0079
	4	0.0007	0.0033
	5	0.0004	0.0018
	6	0.0002	0.0013
	7	0.0002	0.0010
	8	0.0001	0.0009
	9	0.0001	0.0007
	10	0.0001	0.0007
	11	0.0001	0.0006
	12	0.0001	0.0005
	13	0.0000	0.0005
	14	0.0000	0.0005
	15	0.0000	0.0004
	16	0.0000	0.0004
	17	0.0000	0.0004
	18	0.0000	0.0004
	19	0.0000	0.0003
	20	0.0000	0.0003

The condition x t = 0, t 0 is required for the Type II definition of fractional integration. For an alternative definition (Type I) see[START_REF] Marinucci | Alternative forms of fractional Brownian motion[END_REF].

Models with d ranging between -0.5 and 0 are short memory and have been addressed as anti-persistent by[START_REF] Mandelbrot | Fractals, form, chance and dimension[END_REF], because the spectral density function is dominated by high frequency components.

See also[START_REF] Baillie | Long memory processes and fractional integration in econometrics[END_REF] for a complete review of I(d) processes. Other recent surveys are those of[START_REF] Doukhan | Theory and applications of long range dependence[END_REF],[START_REF] Robinson | Long memory time series[END_REF] and[START_REF] Gil-Alana | Fractional integration and cointegration. An overview and an empirical application[END_REF].

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

An I(0) vector process is defined as a covariance stationary process with spectral density matrix that is positive definite.

In another paper,[START_REF] Nielsen | Efficient inference in multivariate fractionally integrated time series models[END_REF] develops a likelihood approach in a fractionally integrated multivariate setting.

Note that a non-diagonal matrix in (20) would lead to the analysis of fractional cointegration, which is an area of research that has received increasing attention in recent years. (See Gil-Alana and Hualde, 2008, for a review).

* The author gratefully acknowledges financial support from the Ministerio de Ciencia y Tecnologia (SEJ2005-07657, Spain). ** The author gratefully acknowledges financial support from the H. Wilhelm Schaumann Stiftung,

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Appendix A

A simple version of the procedure proposed in Gil-Alana (2003a,b) for any real vector d o , in the model given by ( 7) where t is supposed to be an I(0) vector process with spectral density function F( ) that is positive definite. Thus t may be white noise but it also allows us to include VAR structures. To allow for some degree of generality, let us suppose that t in ( 7) is generated by a parametric model of form:

where w t is white noise and W is the unknown variance-covariance matrix of w t . The spectral density matrix of t is then

Appendix B

We consider the following structural bivariate model:

where, initially, u 1,t and u 2,t are assumed to be serially uncorrelated, mutually orthogonal structural disturbances, whose variances are normalized to unity. Note that this model can be expressed as: ( )

and using the Binomial expansions in the fractional differencing polynomials in the lefthand-side of (B1), we obtain

and (x) stands for the Gamma function and d i , i = 1, 2 are the orders of integration of the two series. Substituting (B3) into (B4):

where the impulse response coefficients are:

Appendix B.1: Identification in a pure vector fractional model

From the reduced-form system:

we can obtain the estimates of d 1 and d 2 under the assumption that t is a white noise vector process. Using now (B2) and (B7): assumption in some cases. Alternatively, [START_REF] Blanchard | The dynamics effects of aggregate demand and supply disturbances[END_REF] we can impose the restriction:

Combining the previous expression with (B11) the system is now completely identified and the impulse response functions can easily be obtained.

Appendix B.2 A (2x1) vector fractionally autoregressive model

Here, we extend the structural model (B1) to the case of weak parametric autocorrelation in u t . In particular, we consider the case of a VAR(1) system for u t .

Thus, the structural model is now (B1) with , ... ,

where v 1,t and v 2,t are serially uncorrelated and mutually orthogonal with unit variance (i.e., 1

and v 12 ) = 0) and with all the roots lying outside the unit circle.

First, we describe the impulse response functions. Assuming that u t is stationary, (B12) can be written as: