- Email: [email protected]
A different look at the Census X-11 filter
Economics Letters 79 (2003) 1–6 www.elsevier.com / locate / econbase
A different look at the Census X-11 filter Carlos Lenz* ¨ Basel, Institut f ur ¨ Volkswirtschaft, Petersgraben 51, CH-4003 Basel, Switzerland Universitat Received 19 January 2002; received in revised form 30 May 2002; accepted 28 June 2002
Abstract A band-pass filter approximating the ideal seasonal filter proposed by Sims [Journal of the American Statistical Association 69 (1974) 618] is constructed and its properties are compared to those of the Census X-11 filter. This highlights some properties of the X-11 filter and shows that the proposed band-pass filter is simpler and more flexible. 2002 Elsevier Science B.V. All rights reserved. Keywords: Seasonal adjustment; Band-pass filtering; Census X-11 JEL classification: C22; C4
1. Introduction A large part of the literature on seasonal adjustment of economic time series has been devoted to discuss the properties of data filtered with the Census X-11 method. Prominent examples include Sims (1974) and Wallis (1974) who study theoretically the effects of seasonal adjustment on relations between variables. Extending these results Ghysels and Perron (1993) and Olekalns (1994) show that the power of unit root tests is reduced when using X-11 adjusted data. Other studies show that X-11 adjustment can produce spurious seasonality in nonseasonal series (Auerbach and Rutner, 1978), alter the trending properties of data (Raveh, 1984), or the persistence of shocks (Jaeger and Kunst, 1990), Even though these studies focus almost exclusively on the Census X-11 method, their results hold for any (seasonal) filter based on moving averages. In spite of the fact that the properties of the X-11 filter have been extensively studied it is not fully understood how this procedure affects the data. The main reason is that the X-11 filter is non-linear. Therefore, it is untractable to derive its exact frequency-domain properties analytically. There are * Tel.: 141-61-267-3371; fax: 141-61-2671-236. E-mail address: [email protected] (C. Lenz). 0165-1765 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0165-1765(02)00248-3
C. Lenz / Economics Letters 79 (2003) 1–6
2
basically two alternative approaches to deal with this problem. The first one was pioneered by Nerlove (1964) and consists of comparing estimated spectra of unadjusted data to X-11 adjusted data. This approach yields important insights about some undesirable properties of the filter, in particular the presence of strong phase shifts at low frequencies. However, the approach suffers from the usual problems inherent in the empirical estimation of spectra. The second approach goes back to Young (1968) and amounts to analyzing a linear approximation to the X-11 filter. The frequency domain properties of this linear approximation are well-understood but recent work by Ghysels et al. (1995) suggests that the linear version of the X-11 filter is far from being a good approximation to the actual X-11 procedure. My analysis extends this work and shows how an optimal approximation to the seasonal filter put forward by Sims (1974) can be obtained. Then I demonstrate that under certain conditions this filter corresponds very closely to the linear Census X-11 filter. This highlights the main finding of the paper: within the class of linear seasonal adjustment filters, the linear approximation of the Census X-11 filter can be viewed as approximately optimal. In addition, the paper shows that the rather obscure and complicated X-11 procedure of taking successive moving averages can be replicated using a simple and well-defined filter-design procedure. The rest of the paper is organized as follows: Section 2 presents the derivation of an approximate ideal seasonal band-pass filter, Section 3 compares the properties of this filter to those of a linear version of the Census X-11 filter, and Section 4 draws some conclusions.
2. An approximate seasonal band-pass filter In the present context, seasonal adjustment of an economic time series amounts to applying a linear filter to the series. That is, the adjusted series x˜ t is a moving average of the original series x t and the filtering process is given by x˜ t 5 a(L)x t , where a(L) is a polynomial in the lag-operator L. I concentrate on symmetric filters, that is on filters of the form a(L) 5 ? ? ? 1 a 2 L 22 1 a 1 L 21 1 a 0 1 a 1 L 1 a 2 L 2 1 ? ? ? . The properties of such filters are conveniently summarized in the frequency domain by the frequency response function a (v ): 5 a(e 2i v ). Following Sims (1974) I define an ideal seasonal filter as a filter which passes only frequencies within a band of width ep around the seasonal frequencies. That is, its frequency response function is given by:
aIS (v ) 5
5
0: uv u [ [s] h ,s¯ h ]
P h 5 1,2, . . . ,] 2
(1)
1: else,
where ]s h 5 (2h /P 2 ´)p, s¯ h 5 (2h /P 1 ´)p, and P is the number of observations per year. A filter with the above characteristics is called band-pass filter as it passes only the frequencies within a certain band. In order to construct this ideal seasonal filter I exploit the fact that a filter with the frequency response function (1) can be constructed from low-pass filters that is, from filters which pass only frequencies in a given interval around zero. The frequency response function of the ideal seasonal filter can be written in terms of the frequency response functions of low-pass filters:
C. Lenz / Economics Letters 79 (2003) 1–6
O a (v) 2 O a (v) P/ 2
aIS (v ) 5
3
P / 221
s ]h
h 51
h 51
(2)
s¯ h
where az (v ) is the frequency response function of a low-pass filter which passes frequencies in the interval [2z,z]. The time domain representation of the ideal seasonal filter is now easily found as the sum of the inverse fourier transforms of the low-pass filters which have a particularly simple form. The weights of the linear filter with frequency response function aIS (v ) are given as: a 0 5 1 2 (P 2 1)´
F
O
G
P / 221 sin(´p j) 2h a j 5 2 ]]] cos(p j) 1 2 cos( ]p j) pj P h 51
j 5 1,2, . . .
(3)
it is immediately verified that these weights indeed imply a symmetric filter as a 2j 5a j . It is well known that a finite filter cannot have the characteristics of aIS (v ). However, for practical purposes a finite filter is needed and therefore I construct an approximation to the ideal filter using the approach proposed by Baxter and King (1999). This approach is based on the minimization of the squared distance between the frequency response functions of the ideal and approximate filters under a set of restrictions. In the present case the frequency response function of the approximate filter is restricted to equal unity at frequency zero and to equal zero at the seasonal frequencies. These restrictions ensure that the approximate filter eliminates exact seasonal cycles without altering the long-run properties of the series. Formally, I solve the problem: min Q 5 bj
E
p
2p
subject to:
ua (v ) 2 bK (v )u 2 dv
bK (0) 5 1
bK
2h SU] pUD 5 0 P
P h 5 1,2, . . . ,]. 2
(4)
where the b j ’s represent the weights of the approximate filter b K (L) and bK (v ) its frequency response function 1 K is the number of leads and lags. It is worth mentioning, that the unrestricted version of the above problem has a simple solution inasmuch as the optimal approximate filter sets b j 5 a j for j 5 0,1, . . . ,K and b j 5 0 for j . K. The solution of the restricted version of the problem implies that the approximate optimal filter weights are of the form: b j 5 a j 1 fj
j 5 0,1, . . . K
where the fj ’s are complicated functions 2 of a j , K and P.
1 2
Note that the requirement bK (0) 5 1 implies that the sum of the filter weights (bK (1)) equals one. Their derivation is shown in an appendix available from the author.
(5)
C. Lenz / Economics Letters 79 (2003) 1–6
4
3. Comparison to the linear Census X-11 filter The Census X-11 filter procedure consists of applying sequentially moving averages to the data in order to separate the seasonal from the trend-cycle and irregular components. In addition, some supplementary steps need to be taken in order to adjust for trading-day and holiday components as well as for outliers 3 . The representation of the X-11 procedure as a linear filter amounts to using the actual X-11 program with standard options and results in a symmetric filter with K 5 84 leads and lags in the monthly case. The exact derivation of this linear approximation is presented in Ghysels and Perron (1993). The seasonal band-pass filter presented in the preceding section is similar to the linear version of the Census X-11 filter: it is also a symmetric filter with filter weights that sum to one. However, its construction rests on frequency domain considerations and allows to choose the number of terms to include and the band around the seasonal frequencies for which the frequency response is close to zero. It is therefore of interest to compare the two filters and to investigate their similarities and differences. From the symmetry of both filters and the fact that their frequency response equals one at the zero and zero at the seasonal frequencies three properties follow directly: (1) the filters extract a non-trending seasonal component from the data, (2) the filters include the polynomial (1 1 L 1 L 2 1 ? ? ? 1 L P 21 ) and therefore remove seasonal unit-roots from the data and (3) the filters produce seasonally adjusted data in which annual totals differ from those of the unadjusted data, that is, P consecutive seasonal factors do not sum to zero. The band-pass and the Census X-11 filters differ, however, in the exact form of their frequency response functions. Fig. 1 depicts the frequency response functions of the monthly seasonal band-pass and the approximate Census X-11 filters. The linear X-11 filter is computed with K 5 84 4 , for the seasonal band-pass filter the cases of K 5 12, 36, and 84 are considered. Regarding the choice of ´, I take into account that Sims (1974) recommends a value of ´ 5 1 / 8 for quarterly data as ´ 5 1 /q implies that the correlation between annual seasonal patterns becomes small within 2q /P years. I set ´ 5 1 / 24, which corresponds to Sims’ recommendation for the monthly case and ´ 5 1 / 48 to obtain a narrower band which is more in line with the X-11 filter. It is not surprising that increasing K improves the approximation of the seasonal band-pass filter. Setting K 5 12 yields a filter whose frequency response is quite far from that of the ideal seasonal band-pass filter but with K 5 36 the approximation looks reasonable and with K 5 84 it is very close to that of the ideal filter. In contrast to the linear X-11 filter whose frequency response is very flat at low frequencies and has some wiggles at higher frequencies, the frequency response of the band-pass filter shows similar wiggles at all frequencies. The probably most interesting feature of Fig. 1 is the close resemblance of the frequency responses of the band-pass filter with ´ 5 1 / 48 and K 5 36 to the X-11 filter. On the one hand, this shows that an excellent approximation to the X-11 filter can be obtained with a band-pass filter including less than one half of the terms. On the other hand, it shows that the frequency response function of the X-11 filter has dips which are too narrow, at least with
3
A detailed description of the Census X-11 method can be found in Shiskin et al. (1967). Depending on the relative contributions of the trend-cycle and irregular components to the variability of the series the number of leads and lags in the Census X-11 filter is set to 82, 84, or 89. See Wallis (1974) for details. 4
C. Lenz / Economics Letters 79 (2003) 1–6
5
Fig. 1. Frequency response functions, band-pass filter (bold line) vs. census X-11 filter (thin line).
respect to Sims’ recommendation, which implies that it assumes a relatively constant seasonal pattern. This does not mean, however, that the application of the X-11 filter is always suboptimal. It rather reflects the fact that the X-11 filter is tailor-made for a certain type of seasonality.
4. Conclusions This paper shows how to construct an optimal approximation to the seasonal filter proposed by Sims (1974). The resulting band-pass filter has very similar characteristics to those of the linear version of the Census X-11 filter which shows that the latter can be viewed as approximately optimal within the class of linear seasonal adjustment filters. In addition, the band-pass filter has less than half of the terms compared to the Census X-11 filter. This makes its use preferable from a practical point of view as it achieves seasonal adjustment more efficiently. Finally, the band-pass filter can be constructed to fit the needs of a researcher dealing with a particular data set as it offers more flexibility.
6
C. Lenz / Economics Letters 79 (2003) 1–6
Acknowledgements ¨ For helpful comments and discussions I thank Klaus Neusser, Tobias Rotheli, and Stephen Pollock. I also thank an anonymous referee for useful suggestions. Financial support was provided by the Swiss National Foundation through grant No. 12-40498.94
References Auerbach, R.D., Rutner, J.L., 1978. The misspecification of a nonseasonal cycle as a seasonal by the X-11 seasonal adjustment program. Review of Economics & Statistics 60, 601–603. Baxter, M., King, R.G., 1999. Measuring business cycles: Approximate band-pass filters for economic time series. Review of Economics & Statistics 81, 575–593. Ghysels, E., Granger, C.W., Siklos, P.L., 1995. Is seasonal adjustment a linear or nonlinear data filtering process. Journal of Business & Economic Statistics 14, 374–386. Ghysels, E., Perron, P., 1993. The effect of seasonal adjustment filters on tests for a unit root. Journal of Econometrics 55, 57–98. Jaeger, A., Kunst, R.M., 1990. Seasonal adjustment and measuring persistence in output. Journal of Applied Econometrics 5, 47–58. Nerlove, M., 1964. Spectral analysis of seasonal adjustment procedures. Econometrica 32, 241–286. Olekalns, N., 1994. Testing for unit roots in seasonally adjusted data. Economics Letters 45, 273–279. Raveh, A., 1984. Comments on some properties of X-11. Review of Economics & Statistics 66, 343–348. Shiskin, J., Young, A.H., Musgrave, J., 1967. The X-11 variant of the Census method II seasonal adjustment program. Technical Paper 15, U.S. Bureau of the Census. Sims, C.A., 1974. Seasonality in regression. Journal of the American Statistical Association 69, 618–626. Wallis, K.F., 1974. Seasonal adjustment and relations between variables. Journal of the American Statistical Association 69, 18–31. Young, A.H., 1968. Linear approximations to the Census and BLS seasonal adjustment methods. Journal of the American Statistical Association 63, 445–471.
A different look at the Census X-11 filter Carlos Lenz* ¨ Basel, Institut f ur ¨ Volkswirtschaft, Petersgraben 51, CH-4003 Basel, Switzerland Universitat Received 19 January 2002; received in revised form 30 May 2002; accepted 28 June 2002
Abstract A band-pass filter approximating the ideal seasonal filter proposed by Sims [Journal of the American Statistical Association 69 (1974) 618] is constructed and its properties are compared to those of the Census X-11 filter. This highlights some properties of the X-11 filter and shows that the proposed band-pass filter is simpler and more flexible. 2002 Elsevier Science B.V. All rights reserved. Keywords: Seasonal adjustment; Band-pass filtering; Census X-11 JEL classification: C22; C4
1. Introduction A large part of the literature on seasonal adjustment of economic time series has been devoted to discuss the properties of data filtered with the Census X-11 method. Prominent examples include Sims (1974) and Wallis (1974) who study theoretically the effects of seasonal adjustment on relations between variables. Extending these results Ghysels and Perron (1993) and Olekalns (1994) show that the power of unit root tests is reduced when using X-11 adjusted data. Other studies show that X-11 adjustment can produce spurious seasonality in nonseasonal series (Auerbach and Rutner, 1978), alter the trending properties of data (Raveh, 1984), or the persistence of shocks (Jaeger and Kunst, 1990), Even though these studies focus almost exclusively on the Census X-11 method, their results hold for any (seasonal) filter based on moving averages. In spite of the fact that the properties of the X-11 filter have been extensively studied it is not fully understood how this procedure affects the data. The main reason is that the X-11 filter is non-linear. Therefore, it is untractable to derive its exact frequency-domain properties analytically. There are * Tel.: 141-61-267-3371; fax: 141-61-2671-236. E-mail address: [email protected] (C. Lenz). 0165-1765 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0165-1765(02)00248-3
C. Lenz / Economics Letters 79 (2003) 1–6
2
basically two alternative approaches to deal with this problem. The first one was pioneered by Nerlove (1964) and consists of comparing estimated spectra of unadjusted data to X-11 adjusted data. This approach yields important insights about some undesirable properties of the filter, in particular the presence of strong phase shifts at low frequencies. However, the approach suffers from the usual problems inherent in the empirical estimation of spectra. The second approach goes back to Young (1968) and amounts to analyzing a linear approximation to the X-11 filter. The frequency domain properties of this linear approximation are well-understood but recent work by Ghysels et al. (1995) suggests that the linear version of the X-11 filter is far from being a good approximation to the actual X-11 procedure. My analysis extends this work and shows how an optimal approximation to the seasonal filter put forward by Sims (1974) can be obtained. Then I demonstrate that under certain conditions this filter corresponds very closely to the linear Census X-11 filter. This highlights the main finding of the paper: within the class of linear seasonal adjustment filters, the linear approximation of the Census X-11 filter can be viewed as approximately optimal. In addition, the paper shows that the rather obscure and complicated X-11 procedure of taking successive moving averages can be replicated using a simple and well-defined filter-design procedure. The rest of the paper is organized as follows: Section 2 presents the derivation of an approximate ideal seasonal band-pass filter, Section 3 compares the properties of this filter to those of a linear version of the Census X-11 filter, and Section 4 draws some conclusions.
2. An approximate seasonal band-pass filter In the present context, seasonal adjustment of an economic time series amounts to applying a linear filter to the series. That is, the adjusted series x˜ t is a moving average of the original series x t and the filtering process is given by x˜ t 5 a(L)x t , where a(L) is a polynomial in the lag-operator L. I concentrate on symmetric filters, that is on filters of the form a(L) 5 ? ? ? 1 a 2 L 22 1 a 1 L 21 1 a 0 1 a 1 L 1 a 2 L 2 1 ? ? ? . The properties of such filters are conveniently summarized in the frequency domain by the frequency response function a (v ): 5 a(e 2i v ). Following Sims (1974) I define an ideal seasonal filter as a filter which passes only frequencies within a band of width ep around the seasonal frequencies. That is, its frequency response function is given by:
aIS (v ) 5
5
0: uv u [ [s] h ,s¯ h ]
P h 5 1,2, . . . ,] 2
(1)
1: else,
where ]s h 5 (2h /P 2 ´)p, s¯ h 5 (2h /P 1 ´)p, and P is the number of observations per year. A filter with the above characteristics is called band-pass filter as it passes only the frequencies within a certain band. In order to construct this ideal seasonal filter I exploit the fact that a filter with the frequency response function (1) can be constructed from low-pass filters that is, from filters which pass only frequencies in a given interval around zero. The frequency response function of the ideal seasonal filter can be written in terms of the frequency response functions of low-pass filters:
C. Lenz / Economics Letters 79 (2003) 1–6
O a (v) 2 O a (v) P/ 2
aIS (v ) 5
3
P / 221
s ]h
h 51
h 51
(2)
s¯ h
where az (v ) is the frequency response function of a low-pass filter which passes frequencies in the interval [2z,z]. The time domain representation of the ideal seasonal filter is now easily found as the sum of the inverse fourier transforms of the low-pass filters which have a particularly simple form. The weights of the linear filter with frequency response function aIS (v ) are given as: a 0 5 1 2 (P 2 1)´
F
O
G
P / 221 sin(´p j) 2h a j 5 2 ]]] cos(p j) 1 2 cos( ]p j) pj P h 51
j 5 1,2, . . .
(3)
it is immediately verified that these weights indeed imply a symmetric filter as a 2j 5a j . It is well known that a finite filter cannot have the characteristics of aIS (v ). However, for practical purposes a finite filter is needed and therefore I construct an approximation to the ideal filter using the approach proposed by Baxter and King (1999). This approach is based on the minimization of the squared distance between the frequency response functions of the ideal and approximate filters under a set of restrictions. In the present case the frequency response function of the approximate filter is restricted to equal unity at frequency zero and to equal zero at the seasonal frequencies. These restrictions ensure that the approximate filter eliminates exact seasonal cycles without altering the long-run properties of the series. Formally, I solve the problem: min Q 5 bj
E
p
2p
subject to:
ua (v ) 2 bK (v )u 2 dv
bK (0) 5 1
bK
2h SU] pUD 5 0 P
P h 5 1,2, . . . ,]. 2
(4)
where the b j ’s represent the weights of the approximate filter b K (L) and bK (v ) its frequency response function 1 K is the number of leads and lags. It is worth mentioning, that the unrestricted version of the above problem has a simple solution inasmuch as the optimal approximate filter sets b j 5 a j for j 5 0,1, . . . ,K and b j 5 0 for j . K. The solution of the restricted version of the problem implies that the approximate optimal filter weights are of the form: b j 5 a j 1 fj
j 5 0,1, . . . K
where the fj ’s are complicated functions 2 of a j , K and P.
1 2
Note that the requirement bK (0) 5 1 implies that the sum of the filter weights (bK (1)) equals one. Their derivation is shown in an appendix available from the author.
(5)
C. Lenz / Economics Letters 79 (2003) 1–6
4
3. Comparison to the linear Census X-11 filter The Census X-11 filter procedure consists of applying sequentially moving averages to the data in order to separate the seasonal from the trend-cycle and irregular components. In addition, some supplementary steps need to be taken in order to adjust for trading-day and holiday components as well as for outliers 3 . The representation of the X-11 procedure as a linear filter amounts to using the actual X-11 program with standard options and results in a symmetric filter with K 5 84 leads and lags in the monthly case. The exact derivation of this linear approximation is presented in Ghysels and Perron (1993). The seasonal band-pass filter presented in the preceding section is similar to the linear version of the Census X-11 filter: it is also a symmetric filter with filter weights that sum to one. However, its construction rests on frequency domain considerations and allows to choose the number of terms to include and the band around the seasonal frequencies for which the frequency response is close to zero. It is therefore of interest to compare the two filters and to investigate their similarities and differences. From the symmetry of both filters and the fact that their frequency response equals one at the zero and zero at the seasonal frequencies three properties follow directly: (1) the filters extract a non-trending seasonal component from the data, (2) the filters include the polynomial (1 1 L 1 L 2 1 ? ? ? 1 L P 21 ) and therefore remove seasonal unit-roots from the data and (3) the filters produce seasonally adjusted data in which annual totals differ from those of the unadjusted data, that is, P consecutive seasonal factors do not sum to zero. The band-pass and the Census X-11 filters differ, however, in the exact form of their frequency response functions. Fig. 1 depicts the frequency response functions of the monthly seasonal band-pass and the approximate Census X-11 filters. The linear X-11 filter is computed with K 5 84 4 , for the seasonal band-pass filter the cases of K 5 12, 36, and 84 are considered. Regarding the choice of ´, I take into account that Sims (1974) recommends a value of ´ 5 1 / 8 for quarterly data as ´ 5 1 /q implies that the correlation between annual seasonal patterns becomes small within 2q /P years. I set ´ 5 1 / 24, which corresponds to Sims’ recommendation for the monthly case and ´ 5 1 / 48 to obtain a narrower band which is more in line with the X-11 filter. It is not surprising that increasing K improves the approximation of the seasonal band-pass filter. Setting K 5 12 yields a filter whose frequency response is quite far from that of the ideal seasonal band-pass filter but with K 5 36 the approximation looks reasonable and with K 5 84 it is very close to that of the ideal filter. In contrast to the linear X-11 filter whose frequency response is very flat at low frequencies and has some wiggles at higher frequencies, the frequency response of the band-pass filter shows similar wiggles at all frequencies. The probably most interesting feature of Fig. 1 is the close resemblance of the frequency responses of the band-pass filter with ´ 5 1 / 48 and K 5 36 to the X-11 filter. On the one hand, this shows that an excellent approximation to the X-11 filter can be obtained with a band-pass filter including less than one half of the terms. On the other hand, it shows that the frequency response function of the X-11 filter has dips which are too narrow, at least with
3
A detailed description of the Census X-11 method can be found in Shiskin et al. (1967). Depending on the relative contributions of the trend-cycle and irregular components to the variability of the series the number of leads and lags in the Census X-11 filter is set to 82, 84, or 89. See Wallis (1974) for details. 4
C. Lenz / Economics Letters 79 (2003) 1–6
5
Fig. 1. Frequency response functions, band-pass filter (bold line) vs. census X-11 filter (thin line).
respect to Sims’ recommendation, which implies that it assumes a relatively constant seasonal pattern. This does not mean, however, that the application of the X-11 filter is always suboptimal. It rather reflects the fact that the X-11 filter is tailor-made for a certain type of seasonality.
4. Conclusions This paper shows how to construct an optimal approximation to the seasonal filter proposed by Sims (1974). The resulting band-pass filter has very similar characteristics to those of the linear version of the Census X-11 filter which shows that the latter can be viewed as approximately optimal within the class of linear seasonal adjustment filters. In addition, the band-pass filter has less than half of the terms compared to the Census X-11 filter. This makes its use preferable from a practical point of view as it achieves seasonal adjustment more efficiently. Finally, the band-pass filter can be constructed to fit the needs of a researcher dealing with a particular data set as it offers more flexibility.
6
C. Lenz / Economics Letters 79 (2003) 1–6
Acknowledgements ¨ For helpful comments and discussions I thank Klaus Neusser, Tobias Rotheli, and Stephen Pollock. I also thank an anonymous referee for useful suggestions. Financial support was provided by the Swiss National Foundation through grant No. 12-40498.94
References Auerbach, R.D., Rutner, J.L., 1978. The misspecification of a nonseasonal cycle as a seasonal by the X-11 seasonal adjustment program. Review of Economics & Statistics 60, 601–603. Baxter, M., King, R.G., 1999. Measuring business cycles: Approximate band-pass filters for economic time series. Review of Economics & Statistics 81, 575–593. Ghysels, E., Granger, C.W., Siklos, P.L., 1995. Is seasonal adjustment a linear or nonlinear data filtering process. Journal of Business & Economic Statistics 14, 374–386. Ghysels, E., Perron, P., 1993. The effect of seasonal adjustment filters on tests for a unit root. Journal of Econometrics 55, 57–98. Jaeger, A., Kunst, R.M., 1990. Seasonal adjustment and measuring persistence in output. Journal of Applied Econometrics 5, 47–58. Nerlove, M., 1964. Spectral analysis of seasonal adjustment procedures. Econometrica 32, 241–286. Olekalns, N., 1994. Testing for unit roots in seasonally adjusted data. Economics Letters 45, 273–279. Raveh, A., 1984. Comments on some properties of X-11. Review of Economics & Statistics 66, 343–348. Shiskin, J., Young, A.H., Musgrave, J., 1967. The X-11 variant of the Census method II seasonal adjustment program. Technical Paper 15, U.S. Bureau of the Census. Sims, C.A., 1974. Seasonality in regression. Journal of the American Statistical Association 69, 618–626. Wallis, K.F., 1974. Seasonal adjustment and relations between variables. Journal of the American Statistical Association 69, 18–31. Young, A.H., 1968. Linear approximations to the Census and BLS seasonal adjustment methods. Journal of the American Statistical Association 63, 445–471.