Modelling Seasonality of Gross Domestic Product in Belgium

Transcription

1 University of Vienna Univ.-Prof. Dipl.-Ing. Dr. Robert M. Kunst Course: Econometric Analysis of Seasonal Time Series (Oekonometrie der Saison) Working Paper Modelling Seasonality of Gross Domestic Product in Belgium Mag. Thomas Reininger Matr. Nr Mag. Uwe Fingerlos Matr. Nr Vienna, July 2007

2 List of Figures Contents Contents 2 List of Figures 2 List of Tables 3 1. Introduction and aim of the paper 4 2. Data 4 3. Descriptive analysis 4 4. Testing for non-seasonal unit root 7 5. Simple seasonal deterministic model Dummy variable representation Trigonometric representation Testing for seasonal integration DHF test HEGY test Two possible types of seasonal models for GDP in Belgium SARIMA models Seasonal deterministic models Results of model selection for both types of seasonal model by using AIC and SC Conclusions 20 References 21 A. Appendix: Testing strategy for (non-seasonal) unit roots 22 List of Figures 1. Line graphs: Natural log of Belgian real GDP in levels and in first differences 5 2. Franses graph of Belgian real GDP and Belgian real GDP by season Franses graph of first differenced Belgian real GDP and First differenced Belgian real GDP by season Line graph in first differences after de-trending Trigonometric Representation of first differenced Belgian real GDP

3 List of Tables 6. Testing strategy for (non-seasonal) unit roots List of Tables 1. Simple seasonal deterministic model: Dummy Variable Representation DHF test results HEGY test results SARIMA (4,1,1) as the preferred SARIMA specification Seasonal deterministic model with two seasonal AR terms as the preferred specification

4 3 DESCRIPTIVE ANALYSIS 1. Introduction and aim of the paper This paper is our research output within the seminar Econometric Analysis of Seasonal Time Series (Oekonometrie der Saison). Our goal is to apply the econometric theory of seasonality to a typical seasonal time series. We explicitly do not want to explain the data by a multivariate set of explanatory variables, but rather to try to fit a univariate model that explains the seasonal characteristics of the underlying data. The econometrics software package E-Views 5 is used to carry out the descriptive and econometric analysis. As our work is mainly based on the book by Ghysels and Osborn (2001), all of the appearing formulas can be found in this book. The first part of the paper starts with a brief description of the data set we use and some necessary descriptive analysis. Thereafter a section with testing for non-seasonal unit roots is provided. After fitting and explaining the simple seasonal deterministic model by the means of the convenient dummy variable representation and the corresponding trigonometric representation, we carry out a Dickey-Hasza-Fuller (DHF) test and a Hylleberg- Engle-Granger-Yoo (HEGY) test in order to examine whether or not our data corresponds to a seasonally integrated process. We conclude with a discussion of two possible types of models which we expect to be the best explanation of the studied data. 2. Data The data set under examination comes from the Eurostat database 1 and reflects quarterly Belgian real GDP in millions of national currency (at constant prices of 1995), i.e. first Belgian Francs and then Euro from 1 January 1999 onwards. It ranges from the first quarter of 1980 to the fourth quarter of For convenience, subsequent analysis is accomplished with logarithmic real GDP, i.e. after having taken the natural log of the basic data. We concentrate on Belgian GDP due to the fact that many other national statistical offices do not provide seasonally unadjusted data. Furthermore, Belgian GDP data exhibits clear seasonal patterns that open a wide range for analysis. 3. Descriptive analysis Figure 1(a) on page 5 shows a line graph of logarithmic Belgian real GDP in levels (LNGDP ) from the first quarter of 1980 to the fourth quarter of Overall, the line graph shows a clear dominance of a long-term upward trend, suggesting a non-stationary time series in levels. A change in this trend may be discovered for 1993, when Belgium was hit by the European-wide recession following the ERM crisis. The second main feature in this graph is a persistent pattern of short-term volatility around the trend over the whole range of our data set. This short term volatility seems to be related to seasonal factors. A closer look at the data reveals that there are two seasonal 1 See (Download: ) 4

5 3 DESCRIPTIVE ANALYSIS peaks in nearly every year. Taking the natural log of the data dampens the magnitudes of seasonal fluctuations, especially when GDP increases. Hence, in the later years of our time series, differences between high and low seasonal peaks are larger in absolute terms of the original time series in levels than in the earlier years. Already this fact may indicate some kind of seasonal random walk behaviour. The graph in Figure 1(b) rearranges Figure 1(a), as it shows the first differences of the underlying data, with the aim to remove the growth trend from the series: DLNGDP t = LNGDP t LNGDP t 1 (1) This procedure appears to yield a relatively mean-stable output, i.e. a relatively constant overall mean quarter-on-quarter growth rate. Moreover, the variance of this time series appears to be quite constant, apart from a short period of lower variance of q-o-q growth that was probably related to the European recession in Thus, the time series in levels is very probably difference stationary. At the same time, the feature of pronounced seasonal volatility is better visible now, with (a) Line graph in levels: natural log of real GDP; millions of national currency at constant 1995 prices (b) Line graph in first differences Figure 1: Line graphs: Natural log of Belgian real GDP in levels and in first differences positive q-o-q growth rates in the second and in particular in the fourth quarter as well as negative changes in particular in the first quarter and to a lesser extent also in the third quarter. Examining the Franses graphs both in levels and first differences (see Figures 2(a) to 3(b) below) clarifies and specifies our initial observations of seasonal behaviour of Belgian GDP. As can be seen from Figure 2(a) on page 6, the growth trend dominates clearly. It may 5

6 3 DESCRIPTIVE ANALYSIS be assumed that annual growth (year-on-year change in real GDP) is mostly similar for all of the four quarters and the majority of the observations. This would suggest a seasonal integrated process of order 1, SI(1), with seasonal differences as a stationary, invertible ARMA process. As the four seasonal trends do not seem to diverge, the process might be described in particular as a seasonal random walk with drift (SRW+drift) that consists of four independent seasonal random walks that have an identical drift. Moreover, it is obvious now that the two seasonal peaks in every year observed earlier are the second and the fourth quarter. As expected, the fourth quarter always shows the highest level of real GDP, followed by the second quarter. This probably stems from increased consumption due to pre-christmas purchases. The second quarter always shows higher levels of output than the first and the third quarter, probably due to increasing output of the construction sector and vacations in July and August (with the booking of vacations taking place mostly already before the third quarter). Quarters one and three yield approximately the same amount of output and their lines intersect each other quite frequently. Maybe there can be seen some kind of convergence of the levels of these two quarters in the last years of our series. The difference between the four seasons can clearly be seen also from Figure 2(b) which shows logarithmic Belgian real GDP (LN GDP ) by season. The seasonal means in quarter one and three are lowest, while the mean for quarter two is in the middle and that of quarter four is highest. A closer look at the data shows that the second-quarter peak is usually still below the level attained in previous year s fourth quarter. Moreover, it can clearly be seen that assuming a time constant mean in levels for each of the four quarters from 1980 to 2006 does not seem to be appropriate. This means that a seasonal deterministic model for the data in levels would not fit. (a) Franses graph of Belgian real GDP (b) Belgian real GDP by season Figure 2: Franses graph of Belgian real GDP and Belgian real GDP by season 6

7 4 TESTING FOR NON-SEASONAL UNIT ROOT Figure 3(a) and 3(b) depict the first-order differenced Belgian real GDP by season. The Franses graph in Figure 3(a) clearly exhibits the seasonal behaviour of the series. There is also some kind of drifting apart between quarters four and two, especially after 1992/1993, implying accelerating quarter-on-quarter growth in the fourth and decelerating quarter-onquarter growth in the second quarter. Overall, it is remarkable that there are rather few crossings between the seasonal lines (and these are nearly exclusively between the third and the first quarter). Moreover, the seasonal pattern seems to be dominated by a strong semi-annual cycle, with peaks of positive q-o-q growth rates in Q2 and Q4 and lows of positive q-o-q growth rates in Q1 and Q3. Figure 3(b) reproduces Figure 2(b) with first-order differenced data from 1980Q1 to (a) Franses graph of first differenced Belgian real GDP (b) First differenced Belgian real GDP by season Figure 3: Franses graph of first differenced Belgian real GDP and First differenced Belgian real GDP by season 2006Q4. Here, first differencing removes the growth trend from the time series. The quarterly means of first-order differenced data might be assumed to be stationary, especially when comparing the magnitude of fluctuations within each season with the difference in the growth levels between the seasons. However, as in Figure 3(a), also Figure 3(b) indicates a small increase of q-o-q growth in Q4 and a slight decrease in Q2. Nevertheless, a seasonal deterministic model for the first-order differenced data, which implies constant seasonal means (of quarter-on-quarter growth rates) might be appropriate. 4. Testing for non-seasonal unit root Given the observation of the dominant trend in the GDP level time series, we have to examine whether this time series is difference stationary (implying a non-seasonal unit 7

8 4 TESTING FOR NON-SEASONAL UNIT ROOT root and the need to take first order differences to render the series stationary) or trend stationary (implying a deterministic time-dependent trend and the need to de-trend) or both (as a random walk with drift and deterministic trend that implies the need to take first order differences and to de-trend). In our examination, we followed an approach by Mosconi (1998), who provides a testing strategy for (non-seasonal) unit roots. 2 Thus, we started from a model for the time series that included both a deterministic trend and an intercept. Applying the Augmented Dickey-Fuller test to this model, the null of a unit root could not be rejected. In a next step, we set up a constrained model by including both a unit root and no trend (zero coefficient for the deterministic trend variable) and applied an F-test by using the specially tabulated critical F-values of the ADF test. As the null joint hypothesis of unit root and no trend had to be rejected at the 5% significance level (but not at the 10% significance level), our result indicated an I(1) plus deterministic trend. Therefore, we de-trended (by subtracting the deterministic trend variable) the time series and then took the first differences in order to get a stationary time series. The resulting stationary time series can be seen in Figure 4. In fact, as the deterministic trend is rather weak, albeit significant, the de-trended differenced time series is very similar to the differenced one, which can be seen also by comparing Figure 1(b) and Figure 4. Figure 4: Line graph in first differences after de-trending 2 An overview of the underlying testing strategy is shown in the appendix. See Figure 6 on page 22. 8

9 5 SIMPLE SEASONAL DETERMINISTIC MODEL 5. Simple seasonal deterministic model Given the above result of a non-seasonal unit root, we first aimed at fitting a seasonal deterministic model for the first-order differenced and de-trended GDP time series. The further analysis provides two representations of the seasonal deterministic model: The Dummy Variable Representation and the Trigonometric Representation Dummy variable representation This representation takes the form shown in Equation 2, with t = 1,..., T, γ s being the seasonal mean of y t for any season s and δ st (s = 1,..., S) being the seasonal dummy variables (D1 to D4 in 1 on page 10). z t is a weakly stationary stochastic process with zero mean. y t = S γ s δ st + z t (2) s=1 Equation 2 yields an overall unconditional mean for all the s seasons of E(y t ) = µ = 1 S S s=1 γ s and therefore m s = γ s µ. This model may be extended by including a separate trend component that is constant over the seasons: S y t = µ 0 + µ 1 t + (m 0s + m 1s t)δ st + z t (3) s=1 When applying the seasonal deterministic model to our time series, we used the first-order differenced and de-trended Belgian real GDP (i.e. DLNGDP DET REND in Table 1 on page 10) as the dependent variable y t. Thus, no trend variable was included anymore. In effect, Equation 2 rather than Equation 3 was applied. Estimating Equation 2 yields the results shown in Table 1. All the four dummies δ st, s = 1,..., 4, i.e. D1 to D4, are highly significant and show the expected signs, like in Figures 3(a) and 3(b) Trigonometric representation As shown in Equation 4, the Trigonometric Representation of the seasonal deterministic model in Equation 2 may be de-composed into the following terms: S/2 y t = µ + k=1 [α k cos( 2πkt S ) + β ksin( 2πkt S )] + z t (4) Since our data is of quarterly structure, the parameters of Equation 4 and Equation 2 exhibit the following relationships: γ 1 = µ + β 1 α 2 γ 2 = µ α 1 + α 2 γ 3 = µ β 1 α 2 γ 4 = µ + α 1 + α 2 (5) 9

10 5 SIMPLE SEASONAL DETERMINISTIC MODEL Dependent Variable: DLN GDP DET REN D Method: Least Squares Date: 06/30/07 Time: 15:22 Sample (adjusted): 1980Q2 2006Q4 Included observations: 107 after adjustments Variable Coefficient Std. Error t-statistic Prob. D D D D Statistics: R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Table 1: Simple seasonal deterministic model: Dummy Variable Representation Thus, the matrix (vector) Γ = γ 1 γ 2 γ 3 γ 4 = (6) of the seasonal means (coefficients) in the dummy variable representation and the matrix (vector) µ B = α 1 β 1 (7) α 2 of the parameters in the trigonometric representation are related via the matrix R = (8) For our time series of first-order differenced and de-trended Belgian real GDP, this yields 10

11 5 SIMPLE SEASONAL DETERMINISTIC MODEL the following values for the parameters: µ B = = R 1 Γ = α 1 β 1 α = (9) This result can be checked by inserting the results for our parameters into Equation 5 on page 9, which yields γ 1 = µ + β 1 α 2 = = γ 2 = µ α 1 + α 2 = = γ 3 = µ β 1 α 2 = = γ 4 = µ + α 1 + α 2 = = (10) The calculation of µ as part of the matrix multiplication corresponds to calculating the expected value of y t, E(y t ) = µ = 1 S S s=1 γ s, which becomes E(y t ) = s=1 γ s = 1 ( ) = (11) 4 Using the relationship m s = γ s µ, yields m 1 = γ 1 µ = = m 2 = γ 2 µ = = m 3 = γ 3 µ = = m 4 = γ 4 µ = = , (12) with 4 s=1 m s = = 0. The values for m s can be applied to calculate α 1, α 2, and β 1 via α 1 = m s cos( sπ 2 ) = 1 2 ( m 2 + m 4 ) s=1 = 1 ( ) = (13) 2 α 2 = 1 4 m s cos(sπ) = ( m 1 + m 2 m 3 + m 4 ) s=1 = 1 ( ) = (14) 4 β 1 = m s sin( sπ 2 ) = 1 2 (m 1 m 3 ) s=1 = 1 ( ) = (15) 2 11

12 6 TESTING FOR SEASONAL INTEGRATION Figure 5 on page 13 shows the trigonometric representation of first differenced (logarithmic) Belgian real GDP over a time period of 12 quarters, i.e. three years. In the plot, the blue dotted lines depict the components µ, α 1, α 2, and β 1, each multiplied by the corresponding cosine or sine term (according to Equation 4), while the sum of these products, which is equal to the time series of seasonal means Y t and serves as a simulation of the actual time series y t of the first-order differenced original data, is shown by the red solid line. For convenience, the calculations are shown again in the legend on the left. As we plot first differences, the graph represents simulated changes in Belgian real GDP from one quarter to the next. Obviously, α 2, which represents the amplitude of the semi-annual cycle, dominates the seasonal pattern, as opposed to the amplitudes of the annual cycles, α 1 and β 1. Since, in the original data set, (almost every) quarter 4 exhibits the highest real GDP level, while (almost every) quarter 1 exhibits the lowest level, the change from each quarter 4 (and quarter 8, 12,...) to each quarter 1 (and quarter 5, 9, 13,...) is most negative. Analogously, from each quarter 3 (and quarter 7, 11,...) to each quarter 4 (and quarter 8, 12,...) the highest positive change takes place. Thus, in terms of first differences, the time series changes from its highest positive value in quarter 4 to its most negative value in quarter 1. This is exactly the interpretation of the line for y t between quarter 4 and quarter 5, i.e. quarter 1 of the following year. Analogously, from each quarter 3 (and quarter 7, 11,...) to each quarter 4 (and quarter 8, 12,...) we can see the highest positive change in differences. This is perfectly in line with our findings in the visual inspection of Figures 3(a) and 3(b). 6. Testing for seasonal integration A non-stationary stochastic process y t is called seasonally integrated process of order d, y t SI(d), if it fulfils the condition that d S y t is a stationary, invertible ARMA process. SI (1) processes of quarterly frequency are described by the general model φ(b) 4 y t = γ + θ(b)ɛ t. (16) This general form includes ARMA terms of lag orders p and q (i.e. lagged seasonal differences) as well as a constant that serves as the drift term. The main item is the seasonal difference of y t, i.e. the year-on-year difference between the period (quarter) in the current year and the corresponding period (quarter) in the previous year. The seasonal difference operator 4 can be written as (1 L 4 ), which, in turn, can be decomposed into (1 L)(1 + L)(1 + L 2 ), with L as the Lag operator. It follows that the seasonal difference and thus the SI(1) process imply 4 unit roots. The term (1 L) yields the non-seasonal unit root at +1, while (1 + L) yields the seasonal unit root at 1 and (1 + L 2 ) gives the two complex unit roots at ±i. In the following two subsections, a Dickey-Hasza-Fuller Test (DHF test) and a Hylleberg- Engle-Granger-Yoo Test (HEGY Test) are conducted to verify whether our GDP data finds an adequate representation in a SARIMA model specification, or whether a (simple or a 12

13 6 TESTING FOR SEASONAL INTEGRATION Figure 5: Trigonometric Representation of first differenced Belgian real GDP 13

14 6 TESTING FOR SEASONAL INTEGRATION bit more sophisticated) seasonal deterministic model is more appropriate. While the DHF test is constructed similar to the ADF test and, thus, allows only to test for the existence of all 4 unit roots implied by the seasonal difference simultaneously, the HEGY test allows for individual testing of the 4 unit roots as a condition for an SI(1) process DHF test The Dickey-Hasza-Fuller test (DHF test) is based on the idea to test the null hypothesis H 0 of y t SI(d). Starting point is a seasonal AR(1) process, y t = φ S y t S + ɛ t or, equivalently, S y t = α S y t S +ɛ t with ɛ t = iid(0, σ 2 ). In the first of the two equations, the H 0 of seasonal integration of order 1 (φ S = 1) is tested against the alternative H 1 that y t is stationary (φ s < 1). In the second one, H 0 of seasonal integration (α S = 0 = φ S 1) is tested against H 1, i.e. y t is a stationary stochastic seasonal process (α S < 0). The t-ratio of the estimated value of α S, becomes t(ˆα S ) = ˆα T S ŝe(ˆα S ) = t=1 y t S S y t σ[ T t=1 y2 t S ], 1 2 and under the null hypothesis H 0 : α S = 0, the t statistic becomes T t=1 t(ˆα S ) = y t Sɛ t σ[ T t=1 y2 t S ]. 1 2 Table 2 on page 15 shows the result of the DHF test for our quarterly GDP time series, with the test equation extended with deterministic regressors (dummy variables) and augmented by lagged stationary seasonal differences 4 y t. Both the Akaike info criterion (AIC) and the Schwarz criterion (SC) recommended five lags. Obviously, we cannot reject the Null of simultaneous 4 unit roots, as the very small absolute value of the t-statistic for the seasonally lagged variable in levels, y t 4, is smaller by far than the critical t-value of the DHF test statistic, which is equal to that of the ADF test statistic. Thus, our GDP time series in levels is well described by an SI(1) process. This means that a variant of the SARIMA model will be appropriate according to the DHF test HEGY test Hylleberg, Engle, Granger, and Yoo motivate their test by the aforementioned factorization of 4 = 1 L 4 = (1 L)(1 + L)(1 + L 2 ) and a test regression of the form 4 y t = π 1 y (1) t 1 π 2 y (2) t 1 π 3 y (3) t 2 π 4 y (3) t 1 + ɛ t, (17) with y (1) t y (2) t y (3) t y (4) t = y t 3 y t 2 y t 1 y t. (18) 14

15 6 TESTING FOR SEASONAL INTEGRATION Dependent Variable: D4LN GDP Method: Least Squares Date: 06/30/07 Time: 15:22 Sample (adjusted): 1982Q2 2006Q4 Included observations: 99 after adjustments Variable Coefficient Std. Error t-statistic Prob. D D D D LN GDP ( 4) D4LN GDP ( 1) D4LN GDP ( 2) D4LN GDP ( 3) D4LN GDP ( 4) D4LN GDP ( 5) Statistics: R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Table 2: DHF test results This test regression should be extended by adding seasonal dummies to improve test properties and to capture purely deterministic seasonality without (some) unit roots. Moreover, it should be augmented by lags of the dependent variable 4 y t in order to get white noise errors. The H 0 of y t SI(1) implies π 1 = π 2 = π 3 = π 4 = 0 and, what can be directly seen from Equation 17, 4 y t = ɛ t. Thus, similar to the DHF test, it would be possible to test the joint hypothesis of the simultaneous existence of all 4 unit roots by means of an F-statistic. Alternatively, the coefficients of the test regression may be evaluated separately, with H 0 : π 1 = 0 testing for a non-seasonal unit root at +1 by means of a t-test using the critical t-value of the ADF test. Similarly, H 0 : π 2 = 0 tests for a seasonal unit root at 1, i.e. for the semi-annual cycle, by means of a t-test using the critical t-value of the ADF test. Finally, the H 0 : π 3 = π 4 = 0 tests for the two complex seasonal unit roots at ±i, i.e. for the annual seasonal cycle, by means of a F-test using specially tabulated critical F-values (similar to the special critical F-values in the ADF test). Table 3 on page 16 shows the results of the HEGY test for our quarterly GDP time series, with the test equation extended with deterministic regressors (dummy variables) and augmented by lagged stationary seasonal differences 4 y t. Both the AIC and the SC rec- 15

16 6 TESTING FOR SEASONAL INTEGRATION ommended three lags. The variable y (1) t 1 (etc.) of the test regression given in Equation 17 are denoted here as LN GDP H1( 1) (etc.). Dependent Variable: D4LN GDP Method: Least Squares Date: 06/30/07 Time: 15:22 Sample (adjusted): 1981Q4 2006Q4 Included observations: 99 after adjustments Variable Coefficient Std. Error t-statistic Prob. D D D D LN GDP H1( 1) LN GDP H2( 1) LN GDP H3( 2) LN GDP H3( 1) D4LN GDP ( 1) D4LN GDP ( 2) D4LN GDP ( 3) Statistics: R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Table 3: HEGY test results Obviously, we cannot reject the Null of a non-seasonal unit root at +1 (based on the ADF tau statistic). This is in line with our results of the ADF test for a non-seasonal unit root. Also, we cannot reject the Null of seasonal complex roots at ±i (based on the F statistic of the HEGY test, which is similar to that of the ADF test). However, we have to reject the Null of a seasonal unit root at 1 (based on the ADF tau statistic). This is particularly striking, as the semi-annual cycle is so dominant in the seasonal pattern. On the one hand, not all four unit roots exist according to the HEGY test results. This suggests that our GDP level time series is not best described by an SI(1) process. This stands in contrast to the result of the DHF test. On the other hand, the Null of insignificance of the seasonal dummies included in the HEGY test regression for explaining the seasonally differenced time series cannot be rejected. (In this situation, it would be interesting to test the joint hypothesis of the simultaneous existence of all 4 unit roots by means of an F-statistic. However, the null distribution of this F-statistic is not an F-distribution but 16

17 7 TWO POSSIBLE TYPES OF SEASONAL MODELS FOR GDP IN BELGIUM depends on integrals over four Brownian motion terms, which seems to us to be quite a difficult task.) Overall, we conclude from the HEGY test results that rather a seasonal deterministic model (possibly with seasonal stationary patterns) of the first-order differenced time series (hence, making use of the confirmed non-seasonal unit root) rather than a SARIMA model should be considered as most appropriate to fit the data. 7. Two possible types of seasonal models for GDP in Belgium Given the contradicting results of the DHF test and the HEGY test (with the latter rejecting only the Null of the seasonal unit root at 1), we consider two types of models: 1. Seasaonal ARIMA (SARIMA) models 2. Seasonal deterministic models 7.1. SARIMA models These models have as the dependent variable the seasonally differenced GDP level time series, denoted as 4 y t. As explanatory variables, we have considered the following variants: 1. Constant only (thus SRW with drift) 2. Constant plus ARMA terms in the form of various lags (i.e. one or more quarter lags) of the dependent variable and the error term Moreover, we consider as a testing alternative for comparison reasons, a SRW with seasonal dummies (again without and with ARMA terms in various lags), a hybrid model that implies divergent seasonal drifts. We expect this model to be inferior Seasonal deterministic models These models have as the dependent variable the first-order differenced GDP level time series, denoted as y t. (Here, as opposed to the above section on the simple deterministic model, we have not taken the de-trended version. In fact, the estimation results of the simple deterministic model are very similar for both the first-order differenced series and the de-trended and first-order differenced series.) As explanatory variables, we have considered the following variants: 1. Seasonal dummies only (simple seasonal deterministic model) 17

18 7 TWO POSSIBLE TYPES OF SEASONAL MODELS FOR GDP IN BELGIUM 2. Seasonal dummies plus ARMA terms in the form of one or two seasonal lags (i.e. one or two 4-quarter lags) of the dependent variable and the error term for possible seasonal stationary patterns Moreover, we consider as a testing alternative for comparison reasons a non-seasonal ARIMA (0,1,0) model of y t, i.e. a non-seasonal ARMA model for y t, either with constant only (thus RW with drift) or with constant plus ARMA terms in the form of one or two seasonal lags (i.e. one or two 4-quarter lags) of the dependent variable and the error term for possible seasonal stationary patterns. We expect this model to be inferior Results of model selection for both types of seasonal model by using AIC and SC Among the SARIMA models, both AIC and SC prefer adding ARMA terms over the pure SRW with drift. According to the Schwarz criterion, the preferred SARIMA specification is SARIMA (4,1,1), which is shown in Table 4 on page 19. According to the Akaike criterion, the preferred specification would be SARIMA (8,1,1) with four seasonal dummies. However, in this model the estimated MA process is noninvertible. Taking into account this implication, the principle of parsimony and our a-priori expectation of no divergence in the seasonal trends, we settle with the SARIMA (4,1,1) specification, which received the smallest SC value and roughly the second-smallest AIC value. With respect to the seasonal deterministic models, both AIC and SC clearly indicate as expected that such models are superior to the non-seasonal ARMA models (for the data in first differences). Among the seasonal deterministic models, only AIC generally prefers adding seasonal ARMA terms over the simple seasonal deterministic model (i.e. without seasonal ARMA terms). However, according to both AIC and SC, the preferred seasonal deterministic model specification contains two seasonal AR terms (and no seasonal MA term). This specification is shown in Table 5 on page

19 7 TWO POSSIBLE TYPES OF SEASONAL MODELS FOR GDP IN BELGIUM Dependent Variable: D4LN GDP Method: Least Squares Date: 06/30/07 Time: 15:22 Sample (adjusted): 1982Q1 2006Q4 Included observations: 100 after adjustments Convergence achieved: after 9 iterations Backcast: 1981Q4 Variable Coefficient Std. Error t-statistic Prob. C D4LN GDP ( 1) D4LN GDP ( 2) D4LN GDP ( 3) D4LN GDP ( 4) M A(1) Statistics: R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted MA Roots: -.98 Table 4: SARIMA (4,1,1) as the preferred SARIMA specification Dependent Variable: DLN GDP Method: Least Squares Date: 06/30/07 Time: 15:22 Sample (adjusted): 1982Q2 2006Q4 Included observations: 99 after adjustments Variable Coefficient Std. Error t-statistic Prob. D D D D DLN GDP ( 4) DLN GDP ( 8) Statistics: R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Durbin-Watson stat Table 5: Seasonal deterministic model with two seasonal AR terms as the preferred specification 19

20 8 CONCLUSIONS 8. Conclusions In this paper, we applied the econometric theory of seasonality to explain the seasonal characteristics of quarterly Belgian real GDP level time series from 1980 Q1 to 2006 Q4 by fitting a univariate model. This time series exhibits both a dominant long-term upward trend and a persistent pattern of short-term volatility around the trend that shows two seasonal peaks in the second and the fourth quarter. The Franses graph of the level data shows four seasonal trends that do not seem to diverge, a feature that might be captured by an SI(1) process with an identical drift for all seasons. The Franses graph of the firstorder differenced data shows few crossings between the seasonal lines and peaks of positive q-o-q growth rates in Q2 and Q4. The quarterly means of first-order differenced data seem to be stationary and, hence, a seasonal deterministic model for the first-order differenced data, which implies constant seasonal means (of quarter-on-quarter growth rates), might be appropriate. As expected from the visual inspection, the ADF test indicates that the GDP level time series is not stationary, but follows an I(1) process plus a significant, albeit weak, deterministic trend. Therefore, we estimated a seasonal deterministic model for the first-order differenced data. In the corresponding trigonometric representation, α 2, which represents the amplitude of the semi-annual cycle, dominates the seasonal pattern, as opposed to the amplitudes of the annual cycles, α 1 and β 1. In a next step, we tested for seasonal integration by applying a DHF test and a HEGY Test. While the DHF test results indicated that the GDP level time series is well described as an SI(1) process, the HEGY test results suggested rather a seasonal deterministic model of the first-order differenced time series that makes use of the confirmed non-seasonal unit root. Given the contradicting results of the DHF test and the HEGY test (with the latter rejecting only the Null of the seasonal unit root at 1), we considered two types of models: Among the SARIMA models, both AIC and SC prefer adding ARMA terms over the pure SRW with drift and we arrived at the SARIMA (4,1,1) as the preferred specification. Among the seasonal deterministic models of the first-order differenced time series, both AIC and SC recommend the specification that contains two seasonal AR terms (and no seasonal MA term). 20

21 REFERENCES References Ghysels, E. & Osborn, D. R. (2001). The Econometric Analysis of Seasonal Time Series. Themes in Modern Econometrics. Cambridge: Cambridge University Press. Kunst, R. M. (2007). Econometric analysis of seasonal time series. Slide show: Oekonometrie der Saison. Mosconi, R. (1998). MALCOLM version 2.0: The Theory and Practice of Cointegration Analysis in RATS. Cafoscarina. 21

22 A APPENDIX: TESTING STRATEGY FOR (NON-SEASONAL) UNIT ROOTS A. Appendix: Testing strategy for (non-seasonal) unit roots Testing strategy for unit roots Determination of the optimal lag for models (1), (2) and (3) Estimation of model (1), with trend and constant Test of H0 : p=0 yes H0 rejected? not no Test of Fischer of the joint hypothesis H0 :(µ, ß, p)=( µ, 0, 0) Test of H0 : ß=0 no H0 rejected? not yes H0 rejected? not no Series is I(1) with linear trend Series is TS Estimation of model (2), with constant Test of H0 : p=0 yes H0 rejected? not no Test of Fischer of the joint hypothesis H0 :(µ,, p)=( 0, 0) Test of H0 : µ=0 no H0 rejected? not yes H0 rejected? not no Series is I(1) with constant Series is I(0) with constant Estimation of model (3) Test of H0 : p=0 Series is I(1) without constant yes H0 rejected? not no Series is I(0) without constant Figure 6: Testing strategy for (non-seasonal) unit roots. Source: Mosconi (1998) 22