An analysis of the trade balance for OECD countries using periodic integration and cointegration

We analyze imbalances in external accounts that have historically a¤ected most developed countries. The purpose of this study is to shed some light on the sustainability of the current account for a group of OECD countries by merging the popular Husted (1992) testing procedure with recent econometric analysis dealing with seasonality. A necessary condition for current account sustainability is that exports and imports are cointegrated. Following previous empirical studies (Husted (1992), Arize (2002) and Hamori (2009)) we analyze the long-run relationship linking exports and imports, using quarterly data. In contrast to these studies, we explicitly deal with seasonal e¤ects through the use of periodic integration and cointegration and nd a long-run relationship for the majority of the countries.

Italy, Netherlands, Finland and Spain. We use periodic integration and cointegration to deal with the seasonality and nonstationarity present in our time series and to analyze the long-run relationship between exports and imports. Finally in the case of France, Italy, Netherlands, Finland and Spain a clear change in the mean associated to the change of currencies it is observed in the data, hence we extend the test of periodic integration proposed by Boswijk and Franses (1996) allowing for a change in the mean.
VERY PRELIMINARY VERSION

Motivation
This paper analyzes the recent imbalances in external accounts that have historically a¤ected most of the developed countries from the point of view of trade balance. The relevance of the disequilibria in the external balances during the last decade has renewed the academic interest for this issue. In particular, the case of the EMU countries deserves special attention as the union's net external position is close to equilibrium but many of their members present very large and persistent de…cits or surpluses. The approach of this paper follows the traditional theory that postulates the trade channel as the external adjustment mechanism. For this reason, the variables of interest are exports and imports of the countries analyzed.
Previous empirical studies, such as Husted (1992), Arize (2002) and, more recently, Hamori(2009) have analyzed the long-run relationship linking exports and imports using the cointegration methodology. Although the majority of the empirical evidence is based on annual or quarterly data and the latter can be a¤ected by seasonal e¤ects, to the best of our knowledge, the empirical literature has neglected the presence of seasonal non-stationary components giving rise to instabilities in the long-run relationships.
A simple way to deal with the instability of the relationships between elements of the current account, without assuming the existence of unobserved components, is the use of seasonal and periodical cointegration techniques. Seasonality is a phenomenon that has not received su¢ cient attention in the economic literature in general. The standard treatment is either to assume that the seasonality that appears in the time series is deterministic or, alternatively, to use a method to remove the seasonal component of the variables and estimate the models using seasonally-adjusted variables.
When it is assumed that the seasonality is deterministic, the normal practice is to use seasonal dummies, which implicitly assumes that seasonality is a deterministic phenomenon. In this case, the methods of seasonal adjustment commonly used are variants of the X-11 procedure (X-12 ARIMA and X-13 ARIMA SEATS) of the U.S. Census Bureau as well as procedures based on ARIMA-SEATS SPAN models developed at the Bank of Spain. However, these procedures usually corrupt the stochastic structure of the variables. Ghysels (1990), Ghysels and Perron (1993) and del Barrio Castro et (2002) show that the removal of seasonality with X-11 and SEATS standard proce-dures introduces excessive persistence in the series, which reduces the power of unit root tests. Maravall (1993) shows how seasonal adjustment procedures induce non inverted moving average processes in the …ltered series, invalidating the inference made in most of the unit root and cointegration tests. Olekalns (1994) extends this result to cases in which dummies or band-pass …lters are used to remove seasonality. Abeysinghe (1994) shows that treatment of seasonal stochastic dummies leads to a spurious regression problem. In order to avoid these problems we intend to use a seasonal treatment that includes seasonal unit roots (Hylleberg, 1990, Hylleberg et al, 1995, Rodrigues and Taylor 2007 as well as periodical integration tests (Boswijk and Franses 1995, del Barrio Castro and Osborn, 2010) to determine the type of seasonality present in the non-stationary series analyzed. As shown in Ghysels and Osborn (2001), this point is crucial as it determines the type of cointegration between the set of variables analyzed. Speci…cally, if the series are seasonally integrated, longterm relationships can occur at each frequency, that is, "seasonal cointegration", (Lee, 1992 andJohansen andSchaumburg, 1999) or between the seasonal components of the series, namely "periodic cointegration" (Boswijk and Franses, 1995). However, if the series are periodically integrated, they can only be periodically cointegrated (del Barrio Castro and Osborn 2008). Moreover, if one does not take into account all the above mentioned and ignores the univariate properties of the series analyzed, it may originate problems of spurious correlations and unstable parameterization. Therefore, an important part of the instability observed in the estimates of traditional export-import relationships could be due to the omission of the above phenomena. Finally, another factor to consider is the modeling process where there is seasonal or periodical cointegration in the context of error correction models because the role that both types of cointegration can perform in improving the quality of the estimates and the stability of the parameters can be very relevant. The omission of common trends in seasonal frequencies or shared by the seasons of the series analyzed can lead to problems of omitted variables and instability in the estimated models. Therefore, in this paper the econometric analysis consists of …rst determining the order of integration of the trade ‡ows and then, if nonstationary, to test and estimate the existence of a long-run relationship between a country's exports and imports. Researchers confronted with nonstationary seasonal time series have two alternatives methods to deal with non-stationary seasonality Seasonal integration (SI) and or Periodic Integration (P I). Periodic Integration is more attractive than Seasonal Integration for the following reasons, …rst P I can arise naturally from the application of economic theory when the underlying economic driving forces, such as preferences or technologies, vary seasonally, as shown by Gersovitz and McKinnon (1978), Osborn (1988) and Hansen and Sargent (1993). Secondly, from an econometric perspective, P I is attractive because it implies that the seasons of the year are cointegrated with each other (Osborn (1991), Franses (1994)), and hence ensures that the patterns associated with the various seasons are linked in the long-run.
Finally we also extend the test of periodic integration proposed by Boswijk and Franses (1996) allowing for a changing mean in order to obtain the results for France, Italy, Netherlands, Finland and Spain, where a change in the mean in observed for exports and imports relative to the gross domestic product (see pictures 9 to 13) The rest of the paper is organized as follows. Section 2 presents the theoretical background while section 3 states a review of the most relevant empirical literature. The econometric tests and the empirical results are reported in section 4 for the countries not a¤ected by a change in the mean. Section 5 present the extension of the test for periodic integration allowing for a change in the mean, and also the empirical results for France, Italy, Netherlands, Finland and Spain. Finally, section 6 concludes.

Theoretical model
In this paper we follow Huster (1992) who presents a simple theoretical model of a small open economy with no government where there is a representative consumer. This economy produces and exports a composite good. The consumer can borrow and lend in the international markets using one-period instruments. His resources are output and pro…ts from …rms. that are used for consumption and savings. The consumer's budget constraint in the current period is: where C 0 is current consumption; Y 0 is output, I 0 is investment, r 0 is the one period world interest rate, B 0 is international borrowing that can be positive or negative, whereas (1+r 0 )B t 1 is the stock of debt by the agent (or the country's external debt). The budget constraint must hold for every period. Therefore, they can be combined to obtain the intertemporal budget constraint by iterating (1) forward: where represents the trade balance in period t (that is, income minus absorption), X t are exports, M t imports, 0 = 1 (1+r0) and t is the discount factor (the product of the …rst t values of . When the last term in equation (2) equals zero, the amount that a country borrows (lends) in international markets equals the present value of the future trade surpluses (de…cits). Assuming that the world interest rate is stationary, Husted(1992) expresses (1) as: where Z t = M t + (r t r)B t 1 . Solving forward as Hakkio and Rush (1991) do the next expression is obtained: where = 1 (1+r) . The left-hand side consists of spending on imports and interest payments (receipts) on net foreign debt (assets). If we substract X t from both sides and multiply by minus one, the left hand side becomes the economy's current account. Assuming that both Z t and X t are I(1), (4) can be rewritten as: where M M t = M t + r t B t 1 :Assuming that the limit term equals zero, (5) we can obtain a testable equation: where under the null hypothesis that the economy satis…es its intertemporal budget constraint, we expect b = 1 and e t is stationary. Thus, if both variables are I(1), under the null, they are cointegrated, with a cointegrating vector (1; 1). We have also assumed earlier than the world interest rate is stationary. Therefore, the term r t B t 1 would also be stationary. In practice, we can test for cointegration between exports and imports when we believe that the adjustment works essentially through the trade channel. Alternative theories, such as Gourinchas and Rey (2007) consider that changes in assets valuations have been very important in the last twenty years. If this is the case, we should also account for valuation e¤ects and our regression would su¤er from an omitted variables bias.

Literature review
There are a few empirical studies that, in the last twenty years, have analyzed the trade channel adjustment of the external accounts. A summary is presented in the table below.
The evidence on cointegration is mixed. For a large group of countries there is cointegration between exports and imports, as in Hamori(2009) and Nayaran and Nayaran (2005), although the vector found is not frequently (1; 1). The papers use either quarterly or annual data, in nominal and in real terms. Other papers analyze relative exports and imports over GDP. In none of the papers the authors consider the issue of seasonality, with the exception of Irandoust and Ericsson (2002), that use seasonally adjusted variables in their analysis. Note that in the case France, Italy, Netherlands, Finland and Spain we clearly observe a level shift (or change in the mean) that start in 1999Q1 associated with the change of the national currency for the Euro, hence we are going to analyze the evolution of these countries in a separate section as we need to deal with a structural break or a change in the mean From pictures 1 to 8 we can observe that the ratios exp/gdp and imp/gdp show clear seasonal variation but without huge seasonal oscillations. Note also, that from the evolution of the time series presented in pictures 1 to 8, we do not observe a deterministic trending behavior in our data, hence we are going to consider only seasonal dummies in the deterministic part.
Researchers confronted with apparently nonstationary seasonal time series require methods of analysis that concurrently deal with the seasonal and nonstationary features of their data. Particularly within an economic context, the concept of Periodic Integration (P I) often provides a useful framework for such analysis for two reasons. Firstly, as shown by Gersovitz and McKinnon (1978), Osborn (1988) and Hansen and Sargent (1993), P I can arise naturally from the application of economic theory when the underlying economic driving forces, such as preferences or technologies, vary seasonally. Secondly, from an econometric perspective, P I can be attractive because it implies that the seasons of the year are cointegrated with each other (Osborn (1991), Franses (1994)), and hence ensures that the patterns associated with the various seasons are linked in the long-run. Indeed, the conventional class of integrated, or I(1), time series form a special case of P I processes where the cointegrating vectors between adjacent (seasonal) observations have the form (1, -1).
Taking into account the previous arguments and the evolution of the ratios exp/gdp and imp/dgp for each country (…gures 1 to 8), we are going to focus on periodic integration as a possible source of non-stationarity in our data. In order to explicitly recognize the role of seasonality, it is often convenient to represent a univariate time series as y s , where the …rst subscript refers to the season (s) and the second subscript to the year ( ), as we have quarterly data s = 1; 2; 3; 4. For simplicity of exposition, we assume that data are available for precisely N years, so that the total sample size is T = 4N . Note that, throughout the paper, it is understood that y s k; = y 4 s+k; 1 for s k 0.
Applications of periodic processes within economics have focused on the autoregressive case, with the zero-mean p th order periodic autoregressive, or P AR(p) process, de…ned by y s = s + 1s y s 1; + 2s y s 2; + + ps y s p; + e s ; s = 1; 2; 3; 4 (7) where e s is white noise. In (7) we only consider seasonal intercepts s due to the ratio nature of the analyzed data. Note that all the coe¢ cients in this process may vary over seasons s = 1; :::; 4. The conventional (nonperiodic) AR(p) process is a special case with is = i (s = 1; 2; 3; 4) for all i = 1; 2; : : : ; p. However, in the presence of seasonality, it is important to consider the possibility that the process may be periodic, with at least some AR coe¢ cients in (7) varying over the year. Under the assumption that y s is integrated of order 1, and using a similar notation to Boswijk and Franses (1996), (7) can also be written as (y s ' s y s 1; ) = s + + 1s y s 1; ' s 1 y s 2; + + + p 1;s y s p+1; ' s p+1 y s p; + e s where In the special case ' s = 1 (s = 1; 2; 3; 4), (8) may be a periodic I(1) process, such that the …rst di¤erence is a stationary P AR(p 1) process. On the other hand, when Y S s=1 ' s = 1 but not all ' s = 1 (s = 1; 2; 3; 4), (8) is a periodically integrated, or P I(1), process with the quasi-di¤erence y s ' s y s 1; being stationary; see Ghysels and Osborn (2001, pp.153-155) for further discussion of these possibilities. In the latter case y s ' s y s 1; may have constant coe¢ cients over seasons, although for convenience we refer to it as a stationary PAR process. Boswijk and Franses (1996) analyze the distribution of the Likelihood Ratio test statistic for the null of periodic integration (8), with this statistic de…ned by where RSS 0 and RSS 1 denote the residual sum of squares under the null hypothesis and from the unrestricted form (7), respectively. Under the null hypothesis of a P I(1) or I(1) process, they show that this statistic has the same asymptotic distribution as the squared Dickey-Fuller t statistic for a conventional (nonperiodic) I(1) process.
In order to implement the previous test (31) we need to determine the order p for the unrestricted and restricted models (7) and (8). To do that we follow Franses and Paap (2004) and use the AIC criteria to determine p using 5 as the maximin value. Franses and Boswijk (1997) also proposed a F-type statistic F per to test the null of non periodic variation in the coe¢ cients of (7) H 0 : js = j for j = 1; p. The results of these tests are reported in tables 1.a to 8.a. Note that models (7) and (8) tend to have a lot of parameters, and also that in order to …t model (8) we will need non linear methods of estimation. Recently del Barrio Castro and Osborn (2011) have proposed two non-parametric tests (based on the Breitung (2002) and Stock (1999) unit roots tests) that allow us to circumvent the limitations of the Boswijk and Franses (1996) test. They propose to compute a variance ratio statistic for a given season s as whereÛ s is the season-speci…c partial sumû s1 +û s2 + +û s ; withû s obtained as the OLS residualsû s = y s b 0 s z from a regression of observations for season s, y s ( = 1; :::; N ), on z that collects the deterministic part, in our case z = 1. In order to test the P I(1)=I(1) null hypothesis, they use the average variance ratio statistic where each V R s is de…ned in (10). Additionally, based on Perron and Ng (1996) and Stock (1999) they propose to apply for a single season s, the corresponding season-speci…c M SB test statistic: which requires an appropriate long-run variance estimator b sl for the annual di¤erence u s = u s u s; 1 relating to season s. b sl is obtained based on sample autocovariances using the Barlett and quadratic spectral kernels, following Newey and West (1994, equations (3.8) to (3.15) and Table 1) datadependent bandwidth procedure.
As in the previous case they propose the use of the average M SB statistic del Barrio Castro and Osborn (2011) show that the distributions of the V RT (11) and M SB (13) is the same as those reported for the original tests proposed by Breitung (2002) and Stock (1999) respectively. The results obtained for these tests are also reported in tables 1.a to 8.a. Finally, M SB b and M SB q denote the statistic M SB with the Barlett and quadratic spectral kernels, respectively. From the results of the F per test we …nd clear evidence of periodicity in both exp/gdp and imp/gdp ratios as well as in their natural logs for the majority of the countries. Exceptions are the case of Norway for exp/gpd and ln(exp/gdp), Switzerland for imp/gdp and ln(imp/gdp), and Japan for ln(imp/gdp). For Canada, Sweden and Japan all the periodic integration tests (LR, M SB and V RT ) do not reject the null of periodic integration. In the case of Australia we do no reject the null of periodic integration with the M SB and V RT tests. Concerning the LR test, we do not reject the null of periodic integration for imp/gdp and ln(imp/gdp) but we do reject the null with the LR for both exp/gdp and ln(exp/gdp). In the case of Denmark we only reject the null of periodic integration with the LR test for exp/gdp imp/gdp and ln(imp/gdp). In the UK the null is rejected only for imp/gdp with the LR and at 5% level of signi…cance and with the M SB q at 10% and for ln(imp/gdp) also at 10% for the LR and the M SB tests. For Norway we only reject the null with the M SB tests at 10% for exp/gdp and imp/gdp. Finally, in the case of Switzerland the null is rejected with the V RT test for exp/gdp and imp/gdp at 10% level of signi…cance and for the natural logarithms of the variables at 10%. Overall we can conclude that we have found reasonable empirical evidence in favour that both ratios follow periodically integrated processes for all the counties.
As shown in Ghysels and Osborn (2001, pp.168-171) and del Barrio Castro and Osborn (2008), when the series follow P I processes, the only cointegration possibilities are periodic cointegration or nonperiodic cointegration, with cointegration for any one season implying cointegration for all seasons, that is, full cointegration. They also show that in order to have full nonperiodic cointegration the involved processes must share the same ' s coe¢ cients in (8). Note that full nonperiodic cointegration is equivalent to conventional cointegration. Hence if exp/gdp and imp/gdp or their natural logs are cointegrated with a (1; 1) vector both processes must share the same ' s coe¢ cients in (8). In tables 1.b to 8.b we report these coe¢ cients for the analyzed time series for all the countries. In tables 1.a to 8.a we also report the results obtained with the LR, M SB and V RT when applied to the di¤erence between exp/gdp and imp/gdp , that is labelled dif, as well as to the di¤erence between ln(exp/gdp) and ln(imp/gdp), that we label difln.
Finally, del Barrio Castro and Osborn (2008) propose a residual based LR to test the null of not full periodic cointegration between periodically integrated processes and obtain their asymptotic distribution, in particular they show that the LR CR statistics follow the squared distribution reported by Phillips and Ouliaris (1988) for the residual based ADF cointegration test. The results for the former test are also reported in tables 1.b to 8.b.
In the case of Canada and the UK we …nd clear evidence of cointegration with a (1; 1) vector for the levels and the logs. Note that the coe¢ cients in tables 2.b and 5.b are quite similar. Moreover, with the LR CR test we also …nd evidence of full periodic cointegration as expected. In the case of Australia the results point to cointegration with vector (1; 1) except for the variable dif with the V RT test. As in the previous cases, the coe¢ cients in table 1.b are quite similar in levels and natural logs, and as expected we also …nd evidence of full periodic cointegration with the LR CR test. In the case of Norway there is no evidence in favour of (1; 1) cointegration but we detect full nonperiodic cointegration at a 10% level. Also note that in this case the coe¢ cients ' s are quite di¤erent. For Japan we …nd weak evidence of (1; 1) cointegration, but strong full nonperiodic cointegration. Finally for Denmark, Sweden and Switzerland we do not …nd nonperiodic cointegration with vector (1; 1) nor full periodic cointegration.

Testing for periodic integration in time series with a changing mean
In this section we extend the Periodic integration test proposed by Boswijk and Franses (1996) to the case where we allow for a change in the mean in the deterministic part of the periodic autorregressive process. In particular we consider the following four cases, that are the periodic counterpart of the case considered by Perron(1990) and Perron and Vogelsang (1992a) under the null hypothesis of periodic integration. Maekawa (1997) consider structural breaks in a periodically integrated processes but he only pay attention to the PAR(1) model and do not consider case the following model: is the date of break and we are going to assume that N B = N , where is the fraction of break. As it is pointed out in Boswijk and Franses (1996) and In Ghysels and Osborn (2003) the key to explore the long run properties of PI processes is the vector of quarters representation and in particular the vector moving-average (VMA) representation: where B is the annual lag operator. Following the lines of Boswijk and Franses (1996) from (14) it is possible to write: with: Which is the common trend representation of the PI process without considering deterministic terms, to obtain a equivalent representation to our case we only have to replace E in (15) Note that in the previous expression the term C (1) D (N B) plays a role equivalent to the correction added by Perron and Vogelsang (1992a,1992b) to the initial analysis developed by Perron (1990) when testing for unit roots with a changing mean Finally it is possible to summarize the main stochastic characteristics of a PI(1) process in the following Lemma due to Boswijk and Franses (1996).

The test.
In this section we present the test for PI that allows the presence of structural breaks in the deterministic part. Hence we are testing the null hypothesis ' 1 ' 2 ' 3 ' 4 = 1 against the alternative hypothesis ' 1 ' 2 ' 3 ' 4 < 1, we have the following PAR(p) unrestricted models: Under the alternative the time series follows a stationary PAR(p) process. And the restricted models: ' s jỹs j 1; + " s (19) with the restriction ' 1 ' 2 ' 3 ' 4 = 1 imposed, but js unrestricted, with the estimation achieved using nonlinear least squares. We employ the test Likelihood Ratio test proposed by Boswijk and Franses (1996): In the following proposition we present the distribution of the test is the four di¤erent cases.
Proposition 1 Under the null hypothesis of periodic integration the distribution of the likelihood ratio test statistic obtained from (18)/(19) is the following: Note results (20) is the square of the distribution reported in Perron and Vogelsang (1992a,1992b) In table 9.a we report the empirical quantiles of the LR io ( ) test based on 20:000 replications and for a sample size of = 1000 with S = 4. For = 0:2; 0:3; 0:4; 0:5; 0:6; 0:7 and 0:8. It is clear that the quantiles associated (20) is equivalent to the square of the quantiles reported in Perron (1988).
In order to check size and power performance of the LR io ( ) test we run a small monte-carlo experiment based in the following data generating process: y s = ' s y s 1; + u s s = 1; 2; 3; 4 (21) with : a) ' 1 = 0:9 ' 2 = 1 ' 3 = 1:25 with the combination of parameters a) we are under the null of periodic integration, hence we will measure the empirical size of the test, and with the combinations b) and c) we are under the alternative and we will measure the empirical power of the test. We consider 3 alternative possibilities for u s : The results are obtained for a sample size of 4N = 200 and based on 5000 replications, we report the results obtained when the order of the …tted models (18)/(19) goes form 1 to 5. The results obtained for i), ii) and iii) are collected in tables 9.b, 9.c and 9.d respectively. Clearly the best performance in terms of size and power it is obtained with and P AR(1) for i), with a P AR(5) for ii) and with a P AR(2) for iii). Note that we obtain a reasonable performance in terms of empirical size and power in small sample for the data generating process (21) without the presence of a structural break as in Perron and Vogelsang (1992) we also consider a monte carlo experiment with a change in the mean using the following data generating process: x s = 0:975 0:42DU s + y s s = 1; 2; 3; 4 (22) y s = ' s y s 1; + u s u s = 0:5u s 1; + " s " s ~N iid(0; 1) with : a) ' 1 = 0:9 ' 2 = 1 ' 3 = 1:25 The results for this case are collected in table 9.e. Note that in this case we also obtain the best performance in terms of empirical size and power when the correct order of augmentation it is used, that is when a P AR(2) order it is used to …t models (18)/ (19). In this case we also obtain a reasonable performance in terms of empirical size and power. Remark 2 Following Carrion-i-Silvestre and Berenguer (2010), Zivot and Andrews (1992) ,Gregory and Hansen (1996) and Perron (1997) based on the results of Proposition 1 it is possible to establish for LR io = sup 2 :LR io ( ) : where is closed subset of the interval (0; 1).
In this case using the supremum (sup) it is also possible to obtain a test that do not depend on and hence it is possible to use the critical values reported in

Empirical results
In this subsection we collect the empirical results for France, Italy, Netherlands, Finland and Spain. It is clear for pictures 10 to 14 that for all the mentioned countries we observe a clear change in the mean associated with the change of the national currencies by the Euro. Hence in this section we use LR io ( ) as we know the break date. We determine the order on the unrestricted and restricted P AR(p) models (18)/ (19) with the AIC and BIC criteria starting for a maximum order of p = 5. In this case we also report the result obtained with the statistic F per to test the null of non periodic variation in the coe¢ cients in the model (18) H 0 : js = j for j = 1; p. We also report the results obtained for the LR io ( ) test described in the previous subsection and …nally assuming that there is a co-break we report the results obtained with the LR CR to test the null of the absence of cointegration between the ratios exp/gdp and imp/gdp. These results could be found in tables 10 to 14. Note …rst that with the F per test we obtain evidence against the null for Italy, Finland and in levels for Netherlands. We only clearly reject the null of periodic integration with the LR io ( ) test in the case of France. We …nd evidence of cointegration between the exp/gdp and imp/gdp in logs for the case of Netherlands and Finland, and reject the null of no cointegration for Spain but only at the 10%.
Finally it will be interesting to extend the approach of Gregory and Hansen (1996) to the case of periodic cointegration, in order to allow for breaks in the cointegration vector, but the this is part our future agenda...   3,4386** 9,5601** 0,1663** 0,1708** 0,0208 difln 2,6053** 13,0961** 0,1826** 0,1792** 0,0095**                      (14) it is possible to write: with u = (L) 1 e , them we have that: Replace E by ( D (N B) + u ), hence we have: As in Perron and Vogelsang (1992a) it is possible to write forỹ s = y s ^ s ^ s DU s , where y a Additionally we de…nẽ y s as the residuals from a projection ofỹ s on D (N B) s; in the case where we do not have serial correlation and as the the residuals from a projection of y s on D (N B) s; and its p 1 lags, assume for simplicity the absence of serial correlation, hence: Following the lines of the proof of Theorem 1 in Boswijk and Franses (1996) it is convenient to write (18)/(19) using conventional time subscripts and seasonal dummy variable notation (D st taking the value unity when observation t falls in season s and zero otherwise). Employing this notation yields the representation (see Boswijk and Franses, 1996, p. 238): where the restrictions ' 1 ' 2 ' 3 ' 4 = 1 is imposed. Note that since the deterministic terms enter unrestrictedly thenỹ t are the residuals as de…ned in (25) Note that for z 1 t we have that (a 4 S 1 ) + o p (1) : From lemma 1 in Boswijk and Franses (1996) it is possible to establish: The distribution of the LR test is established by Boswijk and Franses (1996) using where Y N = diag N I 4 ; N 1=2 I 4(p 1) , Y 1 N Q Y 1 N 11 is the …rst element of the principal diagonal of the inverse matrix Y 1 N q , and q and Q are the score and negative of the Hessian matrix, respectively, formulated in terms of . Note that, as in Boswijk and Franses (1996), From (29), (??) and (30) it is easy to see that where : Therefore, where : S (r) = 1 (K 0 K) 1 K 0 E (r) : Now, partitioning K = K 1 . . .K 2 to focus on the …rst element of Y 1 N Q Y 1 where : In Boswijk and Franses (1996) it is shown that S 1 (r) = (K 0 1 M 2 K 1 ) 1=2 w (r) hence we have: note also that: Then …nally substituting (35) and (36) into (31) the required result is easily obtained.