Robust tests for deterministic seasonality and seasonal mean shifts

1 Introduction

The ability to correctly specify the deterministic component in the econometric analysis of time series processes is crucial for delivering reliable policy modelling, prediction and forecasting. It is also important in the context of unit root testing; in particular, omitting deterministic components present in the underlying data generating process (DGP) can lead to non-similar and inconsistent unit root tests, while the inclusion of irrelevant deterministic components can effect significant efficiency losses, even in large samples.

Perron ( 1989) showed that an unmodelled broken trend in the DGP can bias standard (zero frequency) unit root tests toward non-rejection of the unit root null, while allowing for an unnecessary broken trend leads to a loss of power to reject the unit root null when the data are stochastically stationary (denoted, following standard convention, ). One could therefore envisage pre-testing for the presence of deterministic components prior to performing a unit root test. This is not straightforward, however. As discussed in Harvey et al. ( 2007), if the data are , then an optimal test for the presence of a linear time trend can be performed on the levels data, whereas if the data admit a zero frequency autoregressive unit root (denoted ), an optimal test involves testing for a nonzero mean in the first difference of the series. However, tests based on the first differences of the data exhibit poor power properties if the data are, in fact, , and the form of the limiting null distributions of tests based on levels data depend on whether the series is or . A circular testing problem therefore exists. There have accordingly been a number of papers that look to break this circularity by deriving tests for the presence of deterministic linear and broken linear trend components that are robust to whether the series contains a zero frequency unit root or not; see inter alia Vogelsang ( 1998), Bunzel and Vogelsang ( 2005), Harvey et al. ( 2007), Perron and Yabu ( 2009) and Sayginsoy and Vogelsang ( 2011).

The assumption that a time series can admit a unit root and deterministic components at only the zero frequency is likely to be too restrictive when we are dealing with seasonally observed data. Here it is natural to allow the deterministic component to vary across the seasons and also to allow autoregressive unit roots to potentially occur at both the zero and seasonal frequencies. Testing for the presence of deterministic components in seasonally observed data is considerably complicated by the fact that the performance of any test using the levels data will depend on the order of integration of the data at all of the zero and seasonal frequencies. Moreover, there is currently no test procedure available to practitioners that allows them to test for the presence of deterministic components in seasonally unadjusted data in such a way as to yield inference that is robust to whether the series contains unit roots at the zero and seasonal frequencies or not. In the absence of such tests, practitioners wishing to perform (zero and/or seasonal frequency) unit root tests on seasonally observed data need to make an ad hoc decision on what form of deterministic seasonality to allow for in their testing procedures, and where an incorrect choice is made, qualitatively similar problems to those seen in the non-seasonal case occur; see, for example, Franses and Vogelsang ( 1998) and Harvey et al. ( 2006).

One of the most common adjustments made to seasonally observed macroeconomic data is seasonal adjustment. Seasonal adjustment tends to be motivated by the desire to give a clearer picture of the underlying growth rate in the data. The quality and reliability of the resulting seasonally adjusted data are, therefore, heavily dependent on whether the underlying deterministic seasonal component is well specified, or at least well approximated. In relation to this point, the Office for National Statistics (ONS), one of the leading providers of seasonally adjusted data, specifically notes that the quality of its seasonally adjusted data can be negatively impacted by abrupt changes in the seasonal patterns. 1 Even when working with data that are not seasonally adjusted, the presence of deterministic seasonality is of interest in its own right, with deterministic seasonality and any shifts in deterministic seasonality important for both the identification of seasonal effects and the ability to correctly forecast a seasonally observed series; see, in particular, the extensive discussion on these issues in Miron ( 1996). Clearly, if a time series is subject to structural change in deterministic seasonality, then any forecasts produced from a model that does not account for such a break will be unreliable.

In the non-seasonal context, Vogelsang ( 1998) and Bunzel and Vogelsang ( 2005) show that an appropriately constructed test statistic for the presence of zero frequency deterministic trend components has a limiting distribution that depends on the order of integration of the data at the zero frequency. Based on this result, they apply a scaling factor that is a function of an auxiliary zero frequency unit root test statistic that ensures that, for a given significance level, the asymptotic size of the test procedure is controlled when the data are either integrated or stationary at the zero frequency. Likewise, Sayginsoy and Vogelsang ( 2011) employ a similar approach to test for breaks in the deterministic component at the zero frequency. We extend the approach of Vogelsang ( 1998), Bunzel and Vogelsang ( 2005) and Sayginsoy and Vogelsang ( 2011) to the seasonal context. Specifically, we propose tests based on data that have been filtered to remove potential unit roots at all but the frequency of the deterministic component(s) under test. We show that appropriate test statistics can be constructed from the filtered data such that they have well defined limiting distributions whose form depends only on the order of integration at the frequency corresponding to the deterministic component(s) under test. Consequently, and paralleling the approach taken in the non-seasonal case, we suggest modifying these statistics by the use of a scaling factor that is a function of an appropriate auxiliary seasonal unit root test statistic. This allows the asymptotic size of the modified tests to be controlled for a given significance level, irrespective of the order of integration of the data at each of the zero and seasonal frequencies.

The remainder of the paper is organized as follows. Section 2 outlines the seasonal model and the underlying assumptions we will work under. Our proposed test statistics for deterministic seasonality and for seasonal mean shifts are outlined in Section 3. Section 4 provides asymptotic critical values and scaling constants for implementing the proposed tests and discusses issues relating to optimal bandwidth choice when using kernel-based variance estimates. Section 5 discusses issues relating to the practical implementation of the proposed tests. Section 6 reports the results of an empirical application of the proposed tests to seasonally unadjusted quarterly UK gross domestic product (GDP) data. Section 7 concludes. The online Appendix details the results of a Monte Carlo study into local asymptotic power and finite sample properties of our proposed tests, and provides representations for their limiting distributions under both the null and the local alternatives.

2 The model and assumptions

Consider the univariate process , observed with constant seasonal periodicity, S, formed as the sum of a purely deterministic component, , and a purely stochastic process, ; viz.,

(2.1)

(2.2)

where is an Sth order autoregressive polynomial in the usual lag operator, L. For the purposes of this paper we will concentrate on the quarterly case, ; generalisations to an arbitrary S are entirely straightforward and only introduce additional notational complexity. Again to simplify notation, but with no loss of generality, we assume that , where N is the number of complete seasonal cycles within the data span. The initial conditions, , are taken to be of .

The shocks in ( 2.2) are taken to follow a zero-mean linear process driven by the innovations . Precise conditions are now detailed in Assumption 2.1.

The polynomial in ( 2.1) can be factorized as

(2.3)

where associates the parameter with the zero frequency , corresponds to the harmonic (annual) seasonal frequency and associates the parameter with the Nyquist (biannual) seasonal frequency . We can therefore permit to be either (stochastically) stationary or (near-) integrated at the zero and seasonal frequencies through the parameters , , and . When , , is stationary at the zero frequency and the Nyquist frequency, respectively, and when , is stationary at the annual frequency. Setting , , with finite constants, is (near-) integrated at the zero and Nyquist frequencies, respectively, with , , yielding an exact unit root at the zero and Nyquist frequencies, respectively. When , with a finite constant, and , the process is (near-) integrated at the annual frequency. Here yields a pair of complex conjugate exact unit roots at the annual frequency. As shorthand notation, in what follows we will denote a process that is stationary at frequency as , and a process that is (near-) integrated at frequency as , . In what follows, and where no confusion arises, the terms ‘near-integrated’ and ‘integrated’, the latter denoting the exact unit root case, will be used synonymously.

For the purposes of this paper, we will specify the deterministic component in ( 2.1) using the frequency-based representation

(2.4)

for , where is a (standard) zero frequency intercept, is a pair of annual frequency spectral intercepts and is a Nyquist frequency intercept. Moreover, in ( 2.4) is a level break dummy variable that takes the value 1 after some (deterministic) break date, . The vectors of associated spectral intercept and spectral level break coefficients are given by , where , and , where , respectively. The break fraction associated with the latter will be denoted . We assume throughout this paper that the putative break date, , is unknown to the practitioner but that it lies within the set , with the convention that and remain fixed constants as the sample size increases.

The deterministic component in ( 2.4) can be equivalently written in terms of standard seasonal indicator variables as

(2.5)

where , with a conventional seasonal indicator variable that takes the value 1 if t lies in season i and 0 otherwise, . Defining , the intercept (prior to any seasonal level breaks occurring) in season i is therefore given by , . The vector of break magnitudes, , entails that the intercept in season i switches from to at time , a break occurring in the level for season i if , . The representations in ( 2.5) and ( 2.4) are mathematically equivalent with and , where the matrix and where , . Notice that the columns of the matrix are mutually orthogonal.

3 Tests for deterministic seasonality and seasonal mean shifts

3.1 Preliminaries

Tests for the presence of deterministic seasonality or a shift in deterministic seasonality at a given seasonal frequency (or frequencies) cannot be based on the levels data, . This is because the form of the limiting null distribution of the resulting test statistics would depend on the order of integration of the data at both the zero and all of the seasonal frequencies. As we shall see, this problem can be circumvented by following an approach first used in Hylleberg et al. ( 1990) (HEGY, hereafter) and applying test procedures to transformations of the levels data that reduce the order of integration by 1 at all but the frequency under test. This reduces the problem down to the need to develop tests that are robust to whether or not the data are integrated at the particular seasonal frequency (or frequencies) at which one is testing, which can be readily solved by using a scale factor approach in the manner of Vogelsang ( 1998).

To that end, consider the linear transformations of the levels data,

(3.1)

(3.2)

(3.3)

where denotes the order of , . We will show that the filtered data can be used for testing hypotheses concerning the Nyquist frequency deterministic component of , while is the relevant filtered data to use for testing hypotheses concerning the annual frequency deterministic component. One could also use the filtered data to construct tests relating to the deterministic component at the zero frequency. We will not pursue this further here because the limiting distributions of test statistics constructed from would only depend on the order of integration of at the zero frequency, so one could, for example, simply apply test procedures of the form proposed in Vogelsang ( 1998) and Bunzel and Vogelsang ( 2005) to test for a zero frequency mean, and in Sayginsoy and Vogelsang ( 2011) to test for the presence of a zero frequency mean shift.

We will be able to base our tests on these filtered data because the transformations , , reduce the order of integration by 1 at each frequency , . From ( 2.1) and ( 2.2), for , we have that , , where . Consequently, , where , . The filtered error process , , might therefore contain moving average unit roots (equivalently, spectral zeroes), but only at the frequencies at which the order of integration has been reduced. For example, the filtered data appropriate for testing hypotheses concerning the Nyquist frequency deterministic component, , would contain moving average unit roots in at the zero () frequency if was , and at the annual () frequency if was , but would not contain moving average unit roots in at the Nyquist () frequency.

It is also important to notice that the filtered series , , will only contain deterministic components associated with frequency . However, where a level break occurs, the filtered series , , will also contain additional impulse dummy terms occurring at the dates immediately following the level shift. Specifically, the filtered series , , satisfy

(3.4)

where is an impulse dummy variable that takes the value 1 if and 0 otherwise, and where is the vector formed of those regressors that do not depend on the break date and is the vector formed of those regressors (other than the impulse dummies) whose form depends on the break date. In ( 3.4), and , where when and when ; that is, the filter applied to the data magnifies the coefficient on any deterministic components at the frequency under test. The coefficients on the impulse dummies are functions of the break magnitudes on any level breaks present at the zero and all of the seasonal frequencies. They are, therefore, uninformative in practice about the break magnitude(s) at any proper subset of the zero and seasonal frequencies.

In what follows we will consider two different classes of tests. The first, outlined in Section 3.2, focuses on testing whether deterministic seasonality is present at frequency , . We do so by considering the filtered data and testing for the presence of the deterministic components contained in in ( 3.4). These tests will, therefore, be based on estimating the regression by ordinary least squares (OLS),

(3.5)

setting and for the annual () and Nyquist () frequencies, respectively.

The second class of tests we consider, outlined in Section 3.3, test whether a level break is present at frequency , , by testing for the presence of the deterministic components contained in in ( 3.4). For a generic putative break date, , and associated break fraction, , these would be based on the estimated OLS regression

(3.6)

again setting for tests at the annual frequency and for tests at the Nyquist frequency. However, because the true break date is assumed unknown, we will construct estimates of the form in ( 3.6) over all possible break dates within the set . 2

3.2 Tests for seasonal spectral means

We first consider tests of the separate null hypotheses and in ( 3.4) (equivalently, and , respectively, in ( 2.4)). In each case we will work under the maintained hypothesis that no seasonal level break is present in ; that is, (equivalently, that in ( 2.4)). However, it should be clear that such tests would also be consistent against series that display level breaks at the seasonal frequency under test, and indeed many other more general deterministic seasonal patterns at that frequency. The first of these null hypotheses involves two linear restrictions (other than those imposed by the maintained hypothesis) that together entail there being no deterministic seasonal component present at the annual frequency. The second involves one linear restriction that entails there being no deterministic seasonal component present at the Nyquist frequency. Tests of the joint null hypothesis of no deterministic seasonal component, such that in ( 2.4), will be subsequently discussed in Section 3.5.

Following the approach taken for the non-seasonal case in Bunzel and Vogelsang ( 2005), we will test the null hypotheses , , using the heteroskedasticity autocorrelation (HAC) robust Wald-type statistics

(3.7)

where and are the OLS residuals and OLS estimator of , respectively, from ( 3.5), and where

(3.8)

is a spectral long run variance estimator where are the sample autocovariances. We make the following assumptions on the kernel, and bandwidth, M.

Assumption 3.1.The kernel function is continuous at and satisfies , , and . The function is also twice continuously differentiable everywhere with associated second derivative . We also define , where the bandwidth M is such that , denoting the integer part of its argument, with bandwidth fraction .

Remark 3.1.The kernel function is therefore assumed to satisfy the conditions given for a type 1 kernel in Sayginsoy and Vogelsang ( 2011). Like Sayginsoy and Vogelsang ( 2011), we found that among a range of popularly applied kernel functions, tests based on the Daniell kernel delivered the best finite sample performance. As a result, all of the numerical results reported in this paper pertain to the use of the Daniell kernel.

For each of , an alternative and closely related test statistic can be formed by replacing the HAC estimator in ( 3.7) with , where is a seasonal variance estimator, corresponding to frequency , based on the spectrally (at frequency ) cumulated OLS residuals from ( 3.5):

(3.9)

The resulting statistics, which we denote by , , can be viewed as the seasonal frequency analogues (aside from the scale factor, which we will subsequently discuss in Section 3.4) of the PSW test statistic outlined on p.131 of Vogelsang ( 1998) for the present testing problem. 3 Notice that in Vogelsang ( 1998), because the hypotheses being tested relate to a zero frequency deterministic component and robustness is required to the order of integration of the data at the zero frequency, the equivalent of would be calculated from standard (rather than spectrally) cumulated residuals; that is, standard partial sums of the residuals would be taken, not the seasonal partial sums required here.

3.3 Tests for seasonal spectral mean shifts

We next develop tests of the null hypotheses and in ( 3.4). The former involves two linear restrictions, and entails that there is no seasonal level break present at the annual frequency, . The latter imposes one linear restriction that entails there is no level break in the seasonal component at the Nyquist frequency, . 4 Joint frequency tests will again be discussed in Section 3.5.

If the true break date, , were known, then , , could be tested along the same lines considered for the tests for deterministic seasonality in Section 3.2 by using the HAC robust Wald statistics,

(3.10)

where , and, in each case for , is the OLS estimate of from ( 3.6) and . The HAC variance estimator is constructed exactly as for the statistic in Section 3.2 but using the OLS residuals, , from estimating ( 3.6) rather than . Again, an alternative statistic can be formed using the OLS-type variance estimator of the form given in ( 3.9), but again with replacing ; we will denote this estimator by and the resulting test statistic by .

In the case we are concerned with in this paper, where the true break date, , is taken to be unknown to the practitioner, we follow the approach of Sayginsoy and Vogelsang ( 2011) and base our tests on the supremum of the sequences of Wald statistics of the form given in ( 3.10), evaluated across all possible break dates in the search set . Our proposed test statistics for testing for a seasonal mean shift at frequency , (where we again recall that corresponds to the annual frequency, , and to the Nyquist frequency, ), are, therefore, given by

(3.11)

Remark 3.2.In addition to the supremum-based statistics in ( 3.11), we also considered the corresponding statistics based on taking the average across the sequence of Wald statistics, . However, and in accordance with the findings in Sayginsoy and Vogelsang ( 2011) for the problem of testing for a zero frequency mean shift, we found that tests based on supremum-based statistic delivered superior power properties to the corresponding average-based tests. Therefore, we only report results for tests based on the supremum statistic in what follows. Corresponding results for tests based on the average statistic can be obtained from the authors on request.

3.4 Scaled statistics

Although, as the results presented in Section 4 will show, the Wald-type statistics proposed in Sections 3.2 and 3.3 have well defined limiting null distributions, both when the data are stationary and when the data are integrated at the frequency under test, crucially these limiting distributions do not coincide. In particular, for a given statistic, the limiting null distribution in the integrated case is right-skewed relative to the corresponding limiting null distribution in the stationary case. Therefore, and in the same manner as is done in, inter alia, Vogelsang ( 1998), Bunzel and Vogelsang ( 2005) and Sayginsoy and Vogelsang ( 2011), we propose scaling the aforementioned Wald-type statistics for seasonal means and seasonal mean shifts by an exponential function of an auxiliary seasonal unit root test statistic such that, for a chosen significance level, the tests have asymptotically controlled size regardless of whether the data are stationary or integrated at the seasonal frequency under test (as well as to the order of integration of the data at the other spectral frequencies).

We use to generically denote either or and use to generically denote either or , in each case for (recall that relates to the annual frequency, , and relates to the Nyquist frequency, ). We will also use the notation UR to generically denote the unit root statistic used in the scale factor approach. The scaled statistics can, thus, be written in corresponding generic form as

(3.12)

(3.13)

In the context of ( 3.12) and ( 3.13), and are scaling constants while and are seasonal unit root test statistics that, respectively, allow for either spectral means or spectral means and spectral mean shifts in their modelled deterministic component. These unit root statistics will need to possess the properties that where is (i.e., stationary at the frequency under test), they converge to zero, and where is (i.e., integrated at the frequency under test), they have a well defined limiting distribution that does not depend on any unknown nuisance parameters, other than , under the null hypothesis being tested. As a result, where is , the exponential scaling factors in ( 3.12) and ( 3.13) will converge to unity as the sample size diverges, leaving the asymptotic distribution of the statistics unaffected. In the case where is , and selecting the ξ level critical value from the asymptotic null distribution of or appropriate for the stationary case, we can, therefore, for any candidate seasonal unit root test statistic, choose values of the scaling constants or such that the asymptotic sizes of the tests based on the resulting scaled statistics and this critical value do not exceed ξ% across a range of values of .

We consider three possible sets of candidate seasonal unit root statistics to use in the context of the scaling factors in ( 3.12) and ( 3.13). Consider first the spectral mean testing case in ( 3.12). The first set is motivated by the choice of statistic made in Vogelsang ( 1998) and is, therefore, based on a generalization of the zero frequency unit root statistics of Park and Choi ( 1988) and Park ( 1990) to the seasonal unit root testing context. To that end, we first estimate the regression by OLS:

(3.14)

We then construct the statistics , where denotes the sum of squared residuals from estimating ( 3.14) and denotes the sum of squared residuals from the OLS estimation of onto , and . We set for the Nyquist frequency unit root test statistic, , and set for the annual frequency unit root test statistic, , as we found these choices of m gave the best asymptotic power performance for the resulting tests for seasonal means at the Nyquist and annual frequencies, respectively. The scaled statistics in ( 3.12) based on will be denoted with the associated scaling constant denoted as , , in what follows.

The second set of unit root statistics to use in ( 3.12) are the seasonal variance ratio unit root test statistics proposed in Taylor ( 2005). For the Nyquist frequency, this is given by

(3.15)

while the variance ratio statistic for the annual frequency is given by

(3.16)

, , the OLS residuals from ( 3.5) and where . The scaled statistics in ( 3.12) based on will be denoted , with the associated scaling constant denoted as , , in what follows.

Harvey et al. ( 2007) modify the zero frequency linear trend test of Vogelsang ( 1998) by constructing the scaling factor using the reciprocal of the absolute value of a standard augmented Dickey–Fuller statistic. They show that this can improve the finite sample properties of the resulting trend test relative to the use of the Park and Choi ( 1988) statistic. In light of these findings, the final set of candidate seasonal unit root statistics we will consider for use in ( 3.12) are based on the HEGY regression-based seasonal unit root statistics. These are obtained from OLS estimation of the regression

(3.17)

where and the lag length, q, satisfies the usual rate condition that as . The HEGY test statistic for a Nyquist frequency unit root is given by the standard regression t-ratio for , say , in ( 3.17) and that for the (complex pair of) annual frequency unit roots is given by the regression F-statistic for in ( 3.17), say . In each case our candidate unit root statistic will be based on the reciprocal of the absolute value of the statistic. The scaled statistics in ( 3.12) using the appropriate frequency HEGY statistic will be denoted with the associated scaling constant denoted as , , in what follows.

Consider next the scaling factor used in connection with the seasonal spectral level shift tests in ( 3.13). The seasonal unit root statistics outlined above can be straightforwardly adapted to the case where a level break may occur at a known break date . For the and statistics one simply augments the test regression in ( 3.14) with the additional regressors , , and , . For tests based on HEGY statistics, we may use the reciprocals of HEGY-type test statistics obtained from the two-step procedure embodied in (4)–(6) of Franses and Vogelsang ( 1998), whereby the broken deterministic components of are first estimated, and a HEGY regression, using appropriate dummy variables, is applied to the residuals from this first-step regression. For tests based on the variance ratio statistics, these would be computed as in ( 3.15) and ( 3.16) but now using the OLS residuals and , respectively, from ( 3.6); we denote the resulting statistics as , .

In practice the putative trend break date is unknown, and here we follow the approach of Sayginsoy and Vogelsang ( 2011). To that end, for each frequency we will calculate a unit root test statistic that allows for spectral mean shifts described above for a generic possible break date, , and do so across all values of . As we treat the true break date as unknown, we then select the infimum of unit root statistics in the resulting sequence following Sayginsoy and Vogelsang ( 2011). Specifically, denoting a generic unit root test statistic at frequency , , which allows for a level break at time as , we would calculate for . In what follows we will consider level break tests based on the infimum of the sequence of statistics when testing for a seasonal mean shift at frequency ; that is,

(3.18)

The scaled statistics in ( 3.13) based on will be denoted , with the associated scaling constant denoted as , , in what follows. Tests based on the Park and Choi ( 1988)-type statistics and the HEGY-type statistics were found to deliver very poor finite sample behaviour and so will not be discussed further here.

3.5 Joint frequency tests

The test procedures outlined thus far are designed to test for a seasonal spectral mean or a shift in seasonal spectral mean at either the annual frequency, , or the Nyquist frequency, . In practice, one might wish to also consider joint tests for the presence of seasonal spectral means or level shifts in seasonal spectral means occurring at either or both frequencies, thereby yielding size controlled tests for deterministic seasonality in the former case and for a shift in deterministic seasonality in the latter case.

Joint tests are straightforward to develop because the limiting distributions of the test statistics constructed at frequency will turn out to be independent of the limit distributions of the statistics constructed at frequency , . Simulation results in the online Appendix show that the best overall performance when testing for individual frequency spectral means and spectral mean shifts are obtained using the and the statistics, respectively, and so we will focus attention in what follows on how joint tests for spectral means and spectral mean shifts can be constructed from these individual frequency statistics; however, the same principles could be applied to any of the individual frequency statistics discussed previously.

A natural basis to use for developing a joint test for the presence of seasonal means is the simple average of modified versions of the individual annual and Nyquist frequency statistics, and , respectively; that is,

(3.19)

where , , are the scaling constants appropriate for the individual frequency and tests that will be detailed in Section 4.1. 5 Note that if we took a simple average of the two individual frequency and test statistics, there would be many possible combinations of , , that would yield a robust joint test. As such, the appropriate scaling constants for the individual frequency tests are both multiplied by a further scaling constant , where can be chosen such that the asymptotic size of the resulting test does not exceed ξ% when the data are or , . This has the practical advantage of reducing the problem from choosing between multiple pairs of , , to selecting a single value of .

When testing for the presence of a shift in deterministic seasonality, we can form a joint test across both frequencies in the same manner. Specifically, a test for a break in deterministic seasonality at either or both of the annual and Nyquist frequencies can be constructed as

(3.20)

where , , are the scaling constants appropriate for the individual frequency statistics .

Relevant values of and that yield robust tests will be given in Section S.1.2 in the online Appendix.

4 Large sample results

The online Appendix provides representations for the limiting distributions of the and , , statistics proposed in Section 3, together with the corresponding joint frequency test statistics from Section 3.5, under both the null and local alternative hypotheses. The form of the latter depend on whether the data are stationary or integrated at the frequency (or frequencies in the case of the joint frequency tests) of interest. We first define these local alternatives formally. The relevant local alternatives when testing for seasonal spectral means or for seasonal spectral mean shifts are now given in Definitions 4.1 and 4.2, respectively.

Representations for the limiting distributions under the relevant local alternative for the annual frequency spectral mean statistics, and , are given in Theorems S.2 and S.3 for the case where the data are and , respectively. Corresponding results for the Nyquist frequency statistics, and , are given in Theorems S.4 and S.5 for the case where the data are and , respectively. The corresponding limiting distributions for the annual frequency mean shift statistics, and , are given in Theorems S.7 and S.8 for the case where the data are and , respectively. Finally, corresponding results for the Nyquist frequency mean shift statistics, and , are given in Theorems S.9 and S.10 for the case where the data are and , respectively.

The results in Theorems S.2–S.5 and S.7–S.10 show that for all of the statistics, where the data are stationary at the frequency under test, then so the limiting distributions of the statistics are unaffected by the choice of unit root statistic employed in the scale factor used in their construction. This is because, the unit root statistics converge in probability to zero in large samples, such that the multiplicative exponential functions used in the scale factors converges in probability to unity. Where the data are integrated at the frequency of interest, the limiting distributions of the statistics now also involve the limiting distribution of the seasonal unit root statistic used in their construction. In Section 4.1 we will discuss how appropriate values of the scaling factor constants can be chosen to yield tests that are robust to whether the data are stationary or integrated at the frequency under test.

The results in Theorems S.2–S.5 and S.7–S.10 also show that the limiting distributions of each of the annual frequency test statistics, either for spectral means or for spectral mean shifts, are independent of the limiting distributions of each of the corresponding Nyquist frequency test statistics. This is because the terms that feature in the limiting distributions of the statistics at the annual and Nyquist frequencies are formed from independent Brownian motions. An implication of this is that the limiting distributions of the joint frequency statistics in Section 3.5 are simply the averages of the limiting distributions of their two constituent statistics.

4.2 Asymptotic local power

We next explore how the asymptotic local power of the test statistics that use the variance estimate can be used to determine the optimal bandwidth to use for a given value of . The asymptotic distributions of the test statistics were simulated using the same Monte Carlo methods used for simulating the limiting null distributions and power for various local alternatives was computed. In all simulations we set and . We restricted our attention to this one parameter setting because of the intensive computational requirements when simulating the limiting distributions under local alternatives. 7 Using the 5% asymptotic critical values and scaling factors reported in Section S.1.2, asymptotic local power was computed for the case where the data were and for a range of values of .

For individual frequency , , tests using the variance estimator, local power depends on the bandwidth, b, and the kernel, , used to construct this variance estimator when the series is or , and additionally depends on when the series is . It is also the case that no single bandwidth maximizes asymptotic local power uniformly for all local alternatives for any individual test; in particular, the asymptotic local power curves for tests constructed using different values of the bandwidth fraction b for a given kernel cross one another. Consequently, we proceed in the same manner as Sayginsoy and Vogelsang ( 2011) to determine the optimal bandwidth fraction, b, to use both when the series are or by evaluating integrated asymptotic power.

To that end, let denote the limiting distribution of an individual frequency statistic constructed using the appropriate scaling constant for a ξ level test under the local alternative and where the kernel function and bandwidth fraction in ( 3.8) are given by and b, respectively. Asymptotic local power is given by , where is the ξ% asymptotic critical value of the test statistic when the data are . Consequently, the integrated asymptotic local power of the test is given by

(4.1)

which can be computed using numerical integration methods. For a given , the maximum value of is chosen such that the asymptotic local power of any given test is equal to at least 0.99 for at least one bandwidth fraction b. Using this method we find values of b, when using the Daniell kernel, such that integrated power is maximized for a given over a grid of values of selected such that the optimal bandwidth fraction for the largest value of is equal to the lowest bandwidth fraction considered of . The optimal bandwidth fraction for each test is larger for smaller values of , declining to 0.02 as increases. A similar analysis when the process is confirmed that is also optimal in this instance. Optimal bandwidth fractions for all test statistics using the variance estimate are given in Section S.1.1 in the online Appendix. 8

5 Additional practical implementation issues

5.1 Bandwidth selection

In the previous section we derived optimal bandwidth functions for tests based on the variance estimator using the Daniell kernel. These involve the non-centrality parameters , , which are unknown in practice and cannot be consistently estimated. We can, however, use a feasible data-dependent bandwidth rule to select the bandwidth.

Concentrating initially on tests at the Nyquist frequency, because we can propose a similar method to that used by Bunzel and Vogelsang ( 2005) and Sayginsoy and Vogelsang ( 2011) to obtain an estimate of . Specifically, estimate by OLS

(5.1)

where are the residuals from the OLS regression in ( 3.5) when testing for a Nyquist frequency mean, or in the case of the tests for a level break at the Nyquist frequency, the residuals from regression ( 3.6) evaluated at the break date , where is chosen such that it minimizes the sum of squared residuals . We then compute . The test can then be performed using the optimal bandwidth corresponding to the estimated value of .

The same approach can be used for selecting a bandwidth in connection with the annual frequency tests. Specifically, we first estimate the OLS regression

(5.2)

where are the residuals from the OLS regression in ( 3.5) when testing for an annual frequency mean, or for the annual frequency level break test, the residuals from regression ( 3.6) evaluated at the break date , where is chosen such that it minimizes the sum of squared residuals . If , we perform the test using the optimal bandwidth for when the data are stationary at the annual frequency; otherwise we compute . The test can then be performed using the optimal bandwidth for the estimated value of . 9

6 An empirical application to UK GDP

We now provide an empirical example using quarterly unadjusted real GDP data from 1997Q1–2015Q4 taken from the ONS website. Our aim is to investigate whether or not the deterministic seasonal pattern in these data is constant across the sample period. Visual inspection of the data in Figure 1 is suggestive of the presence of a segmented zero frequency linear trend in the data, characterized by a negative growth rate in real GDP during the financial crisis of 2007–2009 in a series otherwise exhibiting positive long run growth. A segmented zero frequency trend of this form would lead to zero frequency level shifts in the filtered data used to construct the seasonal frequency tests outlined in this paper. We therefore first detrend the data to allow for two trend breaks, with the break dates chosen so as to minimize the residual sum of squares from the detrending regression. The fitted zero frequency segmented trend is also graphed in Figure 1.

In Table 1 we report the results of applying the tests for the presence of a seasonal mean shift at the annual and Nyquist frequencies, and , respectively, together with the joint frequency seasonal mean shift test, , to the detrended data. All tests are run at the nominal asymptotic 5% level in each case using the Daniell kernel and the bandwidth selection rules outlined in Section 5.1. All of the tests reject the null hypothesis of no level break. The timing of the break, identified by the estimated break date that led to the largest Wald statistic in the construction of the test statistic, is estimated to be 2006Q4 for the Nyquist frequency deterministic component and 2009Q2 for the annual frequency deterministic component. Although these estimated break dates are at different points in the sample, they do, however, straddle the onset of the financial crisis in 2007–2008. 10

We next estimate the deterministic component in the detrended GDP series both for the case where we allow for the broken deterministic seasonal components identified by our test procedures, and for the case where it is assumed that there are no seasonal mean shifts. In the former case we include the impulse dummies after the break identified at frequency to eliminate the effect of any outliers in the filtered data caused by the break; cf. ( 3.4). Figure 2 plots the detrended GDP series against the fitted seasonal deterministic components for both cases. This graph suggests that both fitted series appear to track the detrended data reasonably well for the majority of the sample. However, the impact of failing to model the identified seasonal level breaks can be more clearly seen in Figure 3, which graphs the squared deviations of each fitted deterministic component from the detrended series. The model that does not allow for a break in deterministic seasonality at the time of the financial crisis is prone to making very large errors in the post-crisis period. For the period 2010Q1–2015Q4 (chosen such that the sample contains no indicator dummies to ensure a fair comparison) the mean square error (MSE) for the model where the seasonal level breaks are accounted for is 0.0000401, which is around 40% lower than the MSE of 0.0000658 for the model that does not allow for seasonal level breaks. It therefore seems likely that not allowing for the seasonal mean shifts identified by our proposed tests would have had an adverse affect on the quality of the seasonally adjusted data and any forecasts made in the post-crisis period.

7 Conclusions

We have proposed tests for the presence of deterministic seasonality and breaks in deterministic seasonality in seasonally observed time series that are designed to be asymptotically robust to the order of integration of the data at both the zero and the seasonal frequencies. Simulation results in the online Appendix show that our proposed tests have good size control in finite samples. Recommendations have been provided for how to perform these tests in practice. In an empirical example, we have shown that quarterly seasonally unadjusted UK GDP appears to have been subject to a significant shift in its deterministic seasonal pattern around the time of the financial crisis.