Spurious Persistence and Unit Roots due to Seasonal Differencing: The Case of Inflation Rates

Spurious Persistence and Unit Roots due to Seasonal Differencing: The Case of Inflation Rates Uwe Hassler Matei Demetrescu Goethe-University Frankfurt Revised version: July 30, 2004 Keywords annual growth rates; nonstationarity; Dickey-Fuller test; loss of power JEL classification: C22, E31 Abstract Studying annual growth rates (seasonal differences) in case of seasonal data produces much more persistence, autocorrelation and stronger evidence in favour of a unit root than analyzing seasonal growth rates (ordinary differences). First, this statement is quantified theoretically. Second, it is supported experimentally with simulations, and, finally, it is empirically illustrated with quarterly GDP deflators from 7 European economies. To be presented at the meeting of the European Economic Association in Madrid 2004. We are grateful to Jürgen Wolters for drawing our attention to the problem treated here and wish to thank Adina-Ioana Tarcolea for helpful comments. Statistics and Econometric Methods, Goethe-University Frankfurt, Gräfstr. 78, D-60054 Frankfurt, Germany, Tel: +49.69.798.23660, Fax: +49.69.798.23662, email: hassler@wiwi.uni-frankfurt.de, deme@wiwi.uni-frankfurt.de. 1

1 Introduction There is continued interest in measuring inflation convergence across countries and over time in particular for the EMS economies, see e.g. Caporale and Pittis (1993), Thom (1995), Siklos and Wohar (1997), and most recently Holmes (2002) and Mentz and Sebastian (2003). The appropriate method to measure convergence depends on the stationarity or nonstationarity of inflation rates. This motivates an ongoing debate in the applied literature whether inflation rate series should be modelled as nonstationary (integrated of order 1) or not. For controversial evidence see amongst others Hassler and Wolters (1995), Culver and Papell (1997) and Lee and Wu (2001). While the first of those papers allows for fractional integration in order to reconcile contradicting results, the second and third use panel data methods and account for structural breaks. In this paper we stress that the results of unit root tests for nonstationarity also depend on how inflation is measured. More precisely, we emphasize that the outcome of unit root test (augmented Dickey-Fuller [ADF] tests, see Dickey and Fuller, 1979, and Said and Dickey, 1984) with seasonal data crucially depends on whether annual or rather seasonal growth rates are considered. A similar but less concise point has been made previously in the literature. Ghysels (1990) studied the effect of seasonal adjustment filters (like Census X-11) on the ADF test 1. He pointed out that seasonal filters use smoothing, which [...] most likely introduces higher persistence in the series and higher first-order autocorrelation [...], Ghysels (1990, p.145), so that it will [...] be more difficult to reject a unit-root hypothesis [...]. Similarly, Jaeger and Kunst (1990) examined the effect of seasonal filters on measures of persistence with real and simulated data. Franses (1991) focussed on the application of the seasonal moving average filter and demonstrated how it may lower the power of the test for seasonal unit roots proposed by Hylleberg, Engle, Granger, and Yoo (1990). 1 His empirical and experimental evidence was supported by theoretical arguments in Ghysels and Perron (1993). Moreover, the effects of regressing on seasonal dummy variables and of a frequency domain filter were investigated by Olekalns (1994). 2

Our contribution goes beyond this work with respect to the following aspects. In Section 2 we quantify theoretically how much the seasonal moving average filter (implied by seasonal differencing, see (5) below) increases the persistence and autocorrelation of a stationary time series. In the third section we explain why the ADF test has low power in case of seasonal differencing when no seasonal unit roots are present. The theoretical argument is supplemented by computer experimental evidence for quarterly and monthly data that reveals that the loss of power grows with the number of seasons per year. Finally, we confront our theoretical results in Section 4 with quarterly inflation rates from 7 European countries. It is observed that the often reported evidence in favour of nonstationary inflation is strong if annual rates are tested but not in case of quarterly inflation, which is in accordance with our theoretical findings. The final section contains a more detailed nontechnical summary of the relevant aspects for applied work. 2 Spurious persistence Let Y t denote a seasonal time series with S seasons, for instance S = 4 or S = 12. The series to be analyzed is the logarithm, y t = log Y t. Annual and seasonal growth rates are defined as, S y t = log Y t log Y t S and y t = log Y t log Y t 1, respectively. In this section we first review how to measure persistence of growth rates and then present a theoretical result. Let us consider a simple seasonal time series model with S seasons over N complete years and S + 1 starting values, y t = S δ s D s,t + x t, t = S,..., 0, 1,..., T = SN, (1) s=1 with the conventional dummy variables D s,t (1 if t falls in season s, and 0 otherwise). The stochastic component x t is assumed to be an integrated 3

process of order 1, x t = t e j, (2) driven by a stationary and invertible zero mean ARMA(p,q) process, j=1 A(L) e t = B(L) ε t. (3) Here, L denotes the usual lag operator, L k y t = y t k, and the lag polynomials A(L) and B(L) are of order p and q, respectively. They have no common roots, and all their roots lie outside the unit circle. Moreover, the innovations are white noise, E(ε t ) = 0, V ar(ε t ) = σ 2, Cov(ε t, ε t+h ) = 0, h 0. Given the assumed invertibility of A(L), the MA( ) representation of e t is e t = C(L) ε t = c j ε t j, c 0 = 1, j=0 with C(L) = B(L) A(L) = c j L j, where the coefficients c j are given by series expansion. Often, they are called impulse responses measuring the effect of a shock ε on e after k periods: c k = j=0 e t ε t k. The impulse response function is widely used to investigate persistence: The more slowly c k converges to zero, the more persistent is e t. In order to obtain a unique one-dimensional measure of persistence, it is common practice to consider the cumulated impulse response function (see Campbell and Mankiw, 1987): CIR(e) := B(1) A(1) = C(1) = c j. 4 j=0

Alternative measures of persistence may be constructed from the autocorrelation function, ρ e (h). Cochrane (1988) proposed the so-called variance ratio with the limiting version being defined as VR(e) := ρ e (0) + 2 ρ e (h). h=1 But this simply is the normalized spectral density at frequency zero, which is a multiple of the squared CIR(e). Therefore, we will not consider VR(e) in what follows. If an economic time series y t is generated by (1) through (3), then it is a seasonal, integrated process (but without seasonal unit roots) 2. By assumption, the first differences ( y t = y t y t 1 ) are covariance stationary and invertible, y t = S γ s D s,t + e t, t = S + 1,..., 0, 1,..., T, (4) s=1 with seasonally varying means γ s where γ 1 = δ 1 δ S, γ s = δ s δ s 1, s = 2,..., S. If y t is the logarithm of a variable, economists often compute annual growth rates using seasonal differences, S = 1 L S = (1 L) (1 + L +... + L S 1 ) = (1 L) S(L) (5) where S(L) stands for the seasonal moving average filter. In addition to the nice interpretation of seasonal differences as annual growth rates, the popularity of seasonal differencing in economic time series analysis can be traced back at least to Box and Jenkins (1970). Now, consider the seasonally differenced model (1) with (2) and (3): S y t = (1 + L +... + L S 1 ) e t = S(L) e t. (6) 2 Just as well y t might be integrated with drift allowing for a linear trend µ t in (1). 5

Clearly, because of the factorization in (5), seasonal differencing removes the stochastic trend x t, and the seasonally varying means δ s. At the same time, a noninvertible MA(S 1) component is introduced in (6). Notice the resemblance between this component, S(L)e t = t j=t S+1 and the I(1) component x t from (2). The larger S gets, the more S y t mimics an integrated process! Therefore, the seasonal differences are characterized by higher persistence compared to the adequately differenced series y t. In fact, the degree of spurious persistence and autocorrelation can be quantified. e j, Proposition: Let y t be given by (1) with (2) and (3). a) Then it holds for the cumulated impulse responses of S y t : CIR( S y) S = CIR(e). b) Assuming for e t the particular seasonal AR process, e t = ρ e t S + ε t, ρ < 1, (7) then it holds for the autocorrelation function ρ S y of the seasonal differences from (6) for h = gs + s with s = 1,..., S and g = 0, 1,...: ρ S y(h) = ρ g ρ g s (1 ρ). S For the case where ρ = 0, this becomes: { 1 h ρ S y(h) =, h < S, S 0, otherwise. The proof of a) is obvious from (6): S y t = (1 + L + + L S 1 ) c j ε t j. j=0 6

The proof of b) is not difficult but cumbersome, see the Appendix. Proposition b) covers only the special case of the seasonal AR process in (7). This is the price we have to pay in order to be able to quantify precisely the effect of spurious autocorrelation. The autocorrelation of S y t from the Proposition should be compared with that of the adequately differenced series y t from (4). The latter equals that of e t, { ρ ρ y (h) = ρ e (h) = g, h = gs, g = 0, 1,... 0, otherwise, (8) where with time-varying mean, E( y t ) = S s=1 γ sd s,t, we define ρ y (h) = E [( y t E( y t )) ( y t+h E( y t+h ))] E [ ( y t E( y t )) 2] E [ ( y t+h E( y t+h )) 2] In particular, we observe that serial correlation at seasonal lags remains unchanged after seasonal differencing: ρ S y(s) = ρ y (S), ρ S y(2s) = ρ y (2S), and so on. At the same time, seasonal differencing introduces completely spurious correlation at nonseasonal lags, ρ S y(gs + s) ρ y (gs + s) = 0, s = 1,..., S 1. In particular, at lag 1, ρ S y(1) = 1 1 ρ S, the autocorrelation may be close to 1 for large S and ρ > 0. This reflects persistence in S y t that is spurious and only due to seasonal differencing. Moreover, values of ρ S y(1) close to 1 explain why tests may hardly reject the unit root hypothesis for the seasonal differences although S y t is stationary. To illustrate Proposition b) we plot for S = 4 and S = 12 with ρ = 0.5 and ρ = 0.8 the autocorrelogram of y t and S y t. From Figures 1 and 2 we learn that in this example seasonal differencing introduces strong nonseasonal 7

ρ = 0.5. Left: y t ; Right: 4 y t. ρ = 0.8. Left: y t ; Right: 4 y t. Figure 1: Theoretical autocorrelograms for S = 4. persistence that is purely spurious. This effect is stronger in the monthly case than with quarterly data. Given the spurious persistence and in particular the spurious autocorrelation at lag one, it may easily happen that empirical workers find that S y t has to be differenced to obtain stationarity, S y t = S e t, t = 1, 2,..., T = SN, (9) although in fact y t from (4) is not integrated of order 1. This is examined more closely in the next section. 3 Spurious unit roots We maintain the model from the previous section, in particular (1) through (3). Motivated by the observation on the first autocorrelation coefficient of S y t in Proposition b), we now study the augmented Dickey-Fuller test when 8

ρ = 0.5. Left: y t ; Right: 12 y t. ρ = 0.8. Left: y t ; Right: 12 y t. Figure 2: Theoretical autocorrelograms for S = 12. applied to seasonal differences 3, S y t = ĉ + φ S y t 1 + K α k S y t k + ε t, t = K + 1,..., T. (10) k=1 The null hypothesis of a unit root (nonstationarity) is parameterized as φ = 0. The corresponding theoretical model is available from (9) and (3): A(L) S y t = S B(L) ε t. (11) The augmentation proposed by Said and Dickey (1984) tries to approximate MA terms in (11) by AR(K) models with K growing to infinity, but growing more slowly than the number of observations T. By assumption, B(L) is invertible and hence there exists a valid approximation by autoregressions of 3 Though Demetrescu and Hassler (2004) advocate the inclusion of seasonal dummies when testing seasonal time series, they are not needed here because seasonal differences remove seasonally varying means. 9

growing order. The filter S(L), however, is not invertible, i.e. the expansion of the inverse is not summable: 1 S(L) = 1 L 1 L = (1 L) S = ( L ks L ks+1). k=0 Therefore, K in (10) has to be rather large relative to T to capture most of the autocorrelation of the noninvertible MA term S(L) ε t. With growing K, however, the number of parameters in (10) increases while the effective sample shrinks, which will lead to tests with little power. In order to assess this theoretical claim, a small Monte Carlo study was performed. We simulated model (1) with (2) and (7) and iid innovations ε t drawn from a pseudo standard normal distribution. Since S y t and S y t are not affected by seasonal means we restricted δ 1 =... = δ S = 0. Our experiment relies on 25000 replications performed by means of GAUSS. We simulated quarterly and monthly data over N = 20 and N = 50 years, S = 4, T {80, 200}, S = 12, T {240, 600}. Throughout we applied the asymptotic 5% critical value by MacKinnon (1991): -2.8621. The seasonal autocorrelation from (7) was chosen as ρ {0.25, 0.5, 0.75}. The choice of K is data driven: Given a maximum lag length M (M = 2S for N = 20 and M = 3S for N = 50) it was tested sequentially, i = 0, 1,..., at the 5% level whether S y t M+i have a significant contribution, and S y t K is the first significant lag. Our experimental study also includes ADF tests from ordinary differences, where seasonal dummy variables are included. It is well known from Dickey, Bell and Miller (1986, p.25) that the inclusion of seasonal dummy variables 10 k=0 L ks

does not affect the limiting distribution under the null hypothesis of unit root. Hence, the regression considered is 2 y t = S δ s D s,t + φ y t 1 + s=1 K k=1 α k 2 y t k + ε t, (12) where 2 = (1 L) 2 denotes second differences, with lag length chosen as above. Again, the true seasonal means are set to zero, δ 1 =... = δ S = 0. Almost identical results were obtained but are not reported for the case of different seasonal patterns. Table 1: Power of ADF tests, 5% N = 20 N = 50 ρ = 0.25 0.5 0.75 0.25 0.5 0.75 S = 4 ADF 4 38.44 32.48 15.95 69.44 58.99 38.67 ADF 1 86.27 47.69 14.98 97.78 93.43 50.85 S = 12 ADF 12 14.12 13.23 10.75 61.20 44.15 23.77 ADF 1 76.92 40.08 12.81 97.44 88.60 44.26 ADF 1 denotes the 5% rejection relying on the t-statistic from (12), while ADF 4 and ADF 12 stem from testing seasonal differences, (10). The true model is that y t from (4) is stationary. Table 1 reports the rejection frequencies testing at the 5% level. The true data generating process is (4), so that y t is not integrated. Hence, when testing y t for a unit root with ADF 1 from (12), the power should be high. It is of course growing with the number of observations. At the same time power is decreasing with growing ρ, because for ρ = 1 the differences y t indeed have a (seasonal) unit root. The power of ADF S from (10) in case of tests from seasonal differences, S y t, drops dramatically, and it drops more for monthly data (S = 12) than in the quarterly case (S = 4). This is not surprising given the higher degree of spurious persistence introduced into the series due to seasonal differencing with S = 12. 11

4 Inflation rates For empirical purposes, we use quarterly deflators for GDP at market prices Y t for 7 European countries. The data is taken from OECD Business Sector Data Base. The primary goal of this section is not to decide whether inflation rates are integrated of order 1, but rather to illustrate the effect seasonal differencing has on the outcome of unit root tests. Hence, we compare quarterly with annual inflation, qπ t = log Y t, aπ t = 4 log Y t, t = 1975.1,..., 1998.4. The sample begins 1975 and ends with the introduction of the Euro. The countries investigated are Austria (AUT), Spain (ESP), Finland (FIN), France (FRA), Germany (GER), Italy (ITA) and the Netherlands (NLD). The inflation rates are displayed in Figures 3 and 4. Clearly, the annual inflation looks much smoother and more persistent than qπ t. Table 2: Results of ADF tests annual K quarterly K trend AUT -1.24 8-3.23* 4 no ESP -4.87** 5-4.72** 8 yes FIN -0.74 12-5.48** 5 yes FRA -1.24 6-4.32** 4 yes GER -2.62(*) 8-3.94** 3 no ITA -2.58 5-3.86* 0 yes NLD -3.00 4-5.81** 2 yes Dickey-Fuller test statistics from (10) for annual rates and from (12) for quarterly rates. It is further reported whether a linear time trend has been included or not, and the number of lags, K. Significance at the 1, 5, and 10 % level is denoted by **, *, and (*), respectively. Let us look more closely at the German series. The respective autocorrelograms are depicted in Figure 5. The autocorrelation at lag one of annual inflation is more than twice as large than that of quarterly inflation! To test annual inflation the maximum lag length was specified as M = 12. The 8 th 12

Figure 3: Quarterly (left) and annual (right) inflation rates Austria Spain Finland France 13

Figure 4: Quarterly (left) and annual (right) inflation rates Germany Italy The Netherlands 14

Figure 5: Autocorrelograms of quarterly (left) and annual (right) inflation rates for Germany. lag was the first significant one. The inclusion of all lags from 1 to 8 in (10) lead to a value of the Dickey-Fuller statistic of 2.57. This, however, is not the value reported in Table 2. In order to increase power we eliminated insignificant lags of order lower than 8, namely lag 1 and lag 5. Then we rerun the test regression and observed that all lags are significant, while the Dickey-Fuller statistic became 2.62 - only marginally more significant. Similarly, we specified all other regressions as parsimoniously as possible. Moreover, for all countries but Austria and Germany a linear time trend was included in the regression of annual as well as quarterly rates. We decided to do so upon visual inspection of the series. But this trend is typically not crucial for the outcomes reported in Table 2. In Table 2, Spain is clearly the exception with two respects. It is the only country where more lags are required when testing quarterly inflation, and it is the only country where the ADF statistic strongly rejects the null hypothesis of a stochastic trend for annual inflation rates. Given the inclusion of a linear trend, this indicates that Spanish inflation converged as a trend stationary process to meet the criteria of the European Monetary Union. The same explanation holds for the quarterly inflation in Finland, France, Italy and the Netherlands. The data for Austria and Germany did not seem to require the inclusion of a deterministic trend, which we attribute mainly 15

to their central banks succeeding in controlling the inflation (long) before the introduction of the Euro. All other countries than Spain have more significant lags in the regression of annual inflation, which can be explained by the noninvertible moving average component due to seasonal differences, and, more importantly, all countries except for Spain indicate nonstationary annual inflation while quarterly inflation is stationary at least at the 5% level. Those results were to be expected in light of Sections 2 and 3. 5 Conclusions Inflation convergence within the EMS is still a much debated empirical question. Adequate convergence measurement depends on stochastic properties of inflation series such as stationarity or nonstationarity. For that reason increasingly sophisticated methods have been employed in the literature to determine whether inflation fluctuations are transitory or not. In this paper it is argued that the outcome of the augmented Dickey-Fuller test for a unit root crucially depends on whether annual or seasonal inflation rates are studied. Of course our results hold for any growth rates and can be put more precisely as follows: If the seasonal (monthly or quarterly) growth rate, log Y t log Y t 1, of some variable Y t is not integrated, then annual growth rate, log Y t log Y t S, is also not integrated but displays spurious persistence and autocorrelation due to the statistically inappropriate seasonal differencing. Seasonal differencing introduces a noninvertible moving average component in annual growth rate, which is responsible for a loss of power of unit root tests. The purpose of our empirical study is not to decide whether European inflation is stationary or not. An answer to this question requires nowadays more refined tools than just augmented Dickey-Fuller tests and has to address e.g. the topic of parameter stability (structural change). But we can say at least the following. The examination of annual inflation (seasonal differences) produces much more evidence in favour of nonstationarity than seasonal inflation (ordinary differences). This statement was quantified the- 16

oretically, it was supported experimentally with computer simulations, and it was illustrated with quarterly GDP deflators from 7 European economies. Appendix: Proof of Proposition b) The proof is not difficult but requires some notational care. Let γ z (h) denote the autocovariance function of some stationary variable z t, and ρ z (h) = γ z (h)/γ z (0). Because of (8) we obtain for S y t = S(L) e t from (6): The autocovariances γ S y(0) = S γ e (0). γ S y(gs + s) = E [ (e t + + e t S+1 ) (e t+gs+s + + e t+(g 1)S+s+1 ) ] = E [(A) (B)] can be determined by indexing systematically: A = e t + + e t S+1 = e t + + e t S+s+1 + e t S+s + + e t S+1, B = e t+gs+s + + e t+(g 1)S+s+1 = e t+(g+1)s S+s + + e t+(g+1)s S+1 + e t+gs + + e t+(g 1)S+s+1. Due to the fact that e t correlates only at lags that are multiples of S, (8) therefore yields γ S y(gs + s) = (S s) γ e (0) ρ g + s γ e (0) ρ g+1. Hence, (S s) ρ S y(gs + s) = ρ g + s S S ρg+1, as required. For ρ = 0, the proof is elementary. 17

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