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A composite model for deterministic and stochastic trends
International Journal of Forecasting 6 (1990) 175-186 North-Holland
175
A composite model for deterministic and stochastic trends Heejoon KANG * Graduate School of Business, Indiana
University,
Bloomington,
IN 47405, USA
Abstract: Nonstationary time series are typically either detrended or differenced to achieve stationarity in forecasting analyses. Detrending assumes the presence of a deterministic trend and differencing assumes the presence of a stochastic trend. A composite model, which is needed when a time series contains both trends, is a special case of a transfer function analysis. A transfer function with linear or quadratic trend variables as inputs can be used to forecast nonstationary time series with trends. Forecasts of 14 macroeconomic series for the United States show that these series are better represented and forecastable by these transfer function models than by a univariate ARIMA analysis. Comparisons between forecasts from a univariate analysis through detrending or differencing and those from a transfer function analysis for artificially generated data in a series of Monte Carlo experiments show that either detrending or differencing alone does not effectively account for the presence of trend as well as the transfer function model does. Keywords: Univariate Detrending.
analysis,
Transfer
Function
1. Introduction Most macroeconomic time series such as gross national product and the consumer price index are nonstationary and appear to have a time trend. The mean and sometimes the variance of these time series change over time. Nonstationary series are typically either detrended or differenced to induce stationarity. Though exceptions are numerous, time series are frequently detrended in regression analyses, whereas they are differenced in most types of time series analyses. The choice between differencing and detrending has largely been made quite arbitrarily. Since the appearance of the work by Nelson and Plosser (1982) many studies (e.g. Stulz and Wasserfallen, 1985; and Mankiw and Shapiro, 1985) have also investigated whether most macroeconomic time series are to be differenced or * The author acknowledges helpful comments and suggestions from two anonymous referees. 0169-2070/90/$3.50
analysis,
Trend-stationarity,
Difference-stationarity,
detrended and the implications of the distinction between the two. In the terminology of Nelson and Plosser (1982), a series needs detrending to achieve stationarity if it is trend-stationary (TS) and differencing if it is difference-stationary (DS). The empirical evidences in the cited references generally tend to find that most, if not all, macroeconomic variables are DS. A test to determine whether a given series is of the DS or the TS class is used by Nelson and Plosser (1982) and others mentioned above. The test nests both hypotheses into a composite model which is a special case of a transfer function model with a time trend as input. The composite model includes both deterministic and stochastic trends. Although DS is preferred to TS for most macroeconomic variables, the test statistics show that TS is not necessarily false. That is, DS is preferred to TS not because TS is statistically insignificant, but rather because DS is statistically more significant. The results of these tests strongly suggest that those macroeconomic variables might
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
116
H. Kang / A composite model
be better represented with transfer functions than with either DS or TS as a forecasting model. Recently, Kang (1986) has shown that if there are alternative ways to forecast a given series, forecasts obtained from a composite model tend to be better than forecasts obtained from the individual alternative models. Moreover, forecasts from the composite model tend to be better than the combinations of forecasts obtained from the individual alternative models. Alternative models and the resultant composite model in that study are all regression analyses, whereas time series analyses are used here. The objective of the study is to establish that better forecasts for many macroeconomic time series can be obtained by using this composite model for the trend than by assuming that the trend is either TS or DS alone. The transfer function (TF) analysis is applied to 14 economic series of the United States. The results show that the TF analysis generally provides better forecasts than the univariate autoregressive integrated moving average (ARIMA) technique using DS or TS alone for these 14 variables. This therefore suggests that these variables are generated by a TF process, rather than by a DS or a TS process. A series of Monte Carlo experiments are conducted to investigate the forecast performance of DS and TS for time series that are generated by TF. The results show that a composite model for the trend tends to outperform ARIMA models for such series, indicating that DS and TS do not adequately stationarize the data. The structure of the paper is as follows. Section 2 discusses the differences between TS and DS models, and the results for most macroeconomic series in the literature which strongly suggest the use of TF as a forecasting model. Section 3 reports on a series of Monte Carlo experiments which compare TF and univariate forecasts. Four different models are tried, and 30 replications are made for each model. In section 4 the univariate and TF analyses of 14 macroeconomic series for the United States are compared. The results indicate that TF represents the data better than TS or DS alone. Section 5 provides some concluding remarks. 2. Detrending
versus
differencing
Following Nelson and Plosser (1982, pp. 141142), the linear TS model has the form
for deterministic y, =
and stochastic trends
LY +
j?t
+
c,,
+rs(B)c,=8,s(B)u,;
u,=i.i.d.
(0, u,‘),
(I)
where (Y and p are fixed parameters, B is the backward shift operator (i.e., Bky, =yrAk), and $+s( B) and 8,,(B) are respectively autoregressive (AR) and moving average (MA) polynomials in B that satisfy the stationarity and invertibility conditions. The two expressions above can be combined as Y,=~+B+j---&p;
h.(B)
u, e i.i.d. (0, ~2).
(2)
However, it is given separately as in (1) to indicate the typical practice of detrending analysis: a series is detrended before the detrended series is further modeled. The stationarity condition is satisfied when all the roots of $rS( B) lie outside the unit circle, and the invertibility condition is satisfied if all the roots of 8rS( B) lie outside the unit circle. On the other hand, the DS process has the form, (1 - B)Y,
= Y + d,,
$,,(
= 8,,(
B)d,
B)c,;
et = i.i.d. (0, o:),
(3)
where +ns( B) and 8,,(B) are AR and MA polynomials that satisfy the stationarity and invertibility conditions. As before, two equations in (3) can be combined into one: (1-B)y,=y+mo’,;
&s(B)
e,=i.i.d.(O,a:).
DS
(4) As before, it is given separately as in (3) to indicate the typical sequence of the analysis. In (3) y is a parameter and if $DS( B) = 8,,(B) = 1 then y, is a random walk with drift parameter y to be denoted ARIMA (0, 1, 0). It should be noted that the linear trend in (1) can be easily extended to a higher order polynomial in time and the first order differencing in (3) can be easily extended to a higher order of differencing. It should also be noted that y, in (1) and (3) may have been transformed, for example, by using the Box-Cox family of transformations. In fact, most macroeconomic variables are transformed logarithmically in practice. The transformation, or course, does not change any argument in the paper. Dickey and Fuller (1979, 1981) and Hasza and Fuller (1982) have developed a procedure and a
H. Kang / A composite model for deterministic
test statistic for choosing between model (1) and model (3). Under certain assumptions, DS and TS are combined into a single equation. Assuming the error term in (1) or (3) has only AR(p) process without MA part, the composite model becomes y, = p, + P*t + ‘yy,_, + 6,w,_, + 6,w,_, + . . . + Q&
+ u,,
(5)
where w, = (1 - B) y,. Eq. (5) is a particular case of a transfer function given below. Dickey and Fuller provide a statistic to test whether or not (Y= 1 and p, = 0. If the hypothesis is not rejected, DS is preferred to TS. Using this test procedure, Nelson and Plosser (1982) found that DS is a better specification than TS for many macroeconomic variables in the U.S. Stulz and Wasserfallen (1985) also found that most macroeconomic series for various countries are better described as DS than TS. Although DS is preferred to TS, the rejection of TS is not overwhelming. That is, the time trend in a composite specification, like (5), is not entirely insignificant. This strongly suggests that the use of the composite model discussed below may produce better forecasts. When the coefficient of a variable in a multiple regression has a t-statistic greater than unity in absolute value, the variable helps in forecasting the dependent variable because it generally reduces the standard error of the regression. As Dickey and Fuller (1978, 1981) indicated, direct t-statistics in (5) do not have Student t distributions, and, therefore, they cannot be used directly. However, the values of f-statistics for /3i are substantially greater than unity in all the empirical studies cited. This strongly suggests that the time trend may indeed reduce the standard error of regression, which in turn implies that the inclusion of the time trend will increase the forecasting accuracy. In fact, the evidence that differencing does not fully account for the presence of the time trend is given in Stock and Watson (1987), whose money supply series for the U.S. is modeled by TF in their causality investigation. A theoretical possibility of the presence of both deterministic and stochastic trends is also discussed in Froyen and Waud (1988). The most parsimonious combination of (1) and (3) is a transfer function (TF)
and stochastic trends
model, where the variable follows: y,=a+Bt+9(B)u,.
t’(B)
177
y, is related
u,=i.i.d.
to time as
(0, a,‘).
(6)
The MA polynomial 0(B) is invertible, but the AR polynomial Q(B) may not statisfy the stationary conditions. The AR polynomial Q(B) in (6) can be written as @(B) = (1 - B)$( B),where e(B) is stationary with its roots outside the unit circle and d is the degree of homogeneous nonstationarity. The series y, is stationary only when both /?=O and d=O. Assume now that a series, say y,, is generated by the process in (6). One important issue is how effectively detrending or differencing would account for the trend in y,. That is, the forecast performances of the univariate analysis with detrending or differencing should be evaluated for series that are generated by TF in (6). This performance comparison is important because such an inadequate analysis will still appear to be acceptable in practice. A transfer function analysis between y, and t will identify and estimate the model as
8,,(B) Y,=Q-C”+
@TF(B)~TFT
where the subscript TF indicates that the parameters and the residuals are from the transfer function analysis. If the process y, in (6) is differenced once and then analyzed through the univariate ARIMA analysis, we have
(1-B)y,=b+
~uv~~))e,v, uv
and theoretically, 8,,,(B) = e(B) and Guv( B) = (1 -B)-‘@(B). The subscript UV stands for the univariate analysis. Differencing once will produce a stationary series if Q(B) in (6) is in the form of @(B) = (1 - B)+(B). Otherwise, differencing of y, will not produce a stationary series. When y, is generated as in (6) it is not the presence of the trend term that should suggest the differencing. It is rather the homogeneous nonstationarity in Q(B) that dictates the differencing. If Q(B) in (6) is Q(B) = (1 - B)$( B),differencing achieves stationarity and (8) will then effectively account for the time trend in (6) as a byproduct.
178
H. Kang / A composite model for deterministic
The linear trend term disappears when a series is differenced once. But it does not mean that one should difference the series in order to remove the trend. Whether or not to difference a series should depend on the error structure - difference only if @(B) is homogeneous nonstationary - and not on the presence of the trend term. If the differencing is made to remove the time trend in TS, the differenced series will have noninvertible error terms. Suppose a series, y,, is TS as in (2). The difference of y, would yield
(’-GB)$fB)u,_ TS
(1 -B)y,=P+
(9)
A univariate ARIMA analysis of (9) of such a series would contain a noninvertible MA because 8,,(B) = (1 - B)B,,( B).Since 8,,(B) is invertible by assumption, 8,,(B) is noninvertible due to (1 - B). A TF model in (6) with the time trend can be and should be dealt with as such. It is not uncommon to read a statement in the literature saying that a series y, is differenced because of the trend. This unfortunate practice is due to a common belief that differencing would generally convert many, if not all, nonstationary series into stationary ones. Box and Jenkins (1976, p. 92) said, “In general, however, we may wish to include a deterministic function of time f(t) in the [univariate ARIMA] model” (emphasis is original). And they continued, “In practice, automatic allowance for a deterministic polynomial trend, of degree d, can be made by permitting [a constant term] to be nonzero [once the series is differenced d times].” The statement is misleading because it appears to suggest that the deterministic polynomial trend can be automatically analyzed with differenced series and that therefore the purpose of differencing is to effectively remove the trend. It is inappropriate to detrend the series in (6) unless the disturbance term in homogeneous nonstationary. Suppose y, in (6) is first detrended and the detrended series is analyzed by the univariate ARIMA analysis. The variable y, will be regressed on time as in y,=a+bt+u^
,,
(10)
and stochastic trends
where 6, are the residuals, which will subsequently be modeled as
;r=
eDT(
‘1
G,,,(B)
eDT3
(11)
where the subscript DT indicates that the parameters and the residuals are from the detrending analysis. The parameters eDT( B) and ~~r( B) in (11) however, will not be the same as 8,, and G&(B) in (6). As Nelson and Kang (1981, 1984) have shown, even if the series y, in (6) is a random walk, detrending (8) appears to be very acceptable with a significant b and, furthermore, with stationary +DT( B) in (11). Yet, such detrending of a random walk series will artificially create some cyclical patterns that are not in the original series. Inasmuch as the differencing is dictated by the homogeneous nonstationarity of the true model, detrending must also be dictated by the model. The importance of the difference between modelling with stochastic trends and with deterministic trends in forecasting is well documented by Dickey, Bell, and Miller (1986). If a model is of TF as in (6) and has a trend variable, needless to say, one has to analyze (6) as the stands. It is not appropriate to detrend y, and then use the detrended series in a subsequent ARIMA analysis. To see this, suppose that the true model of y, is given by: Y, =&I + b,Y,-,
+b,y,_,+dt+u,,
(12)
where u, = i.i.d.(O, u,‘), and that one detrends with yr = a + bt + e, and obtains the detrended series j( =_y, - (a + bt). The use of j$ in a univariate ARIMA analysis will not generally yield the same model as in (12) not to mention the same parameter values. The univariate ARIMA analysis on the detrended yr cannot substitute for the TF analysis with time as an explicit input. In summary, differencing or detrending in a univariate analysis is not a substitute for the TF analysis with time, because whether to difference or to detrend should depend on the error structure in TF, not on the mere presence of time trends. The very fact that the test procedure in Dickey and Fuller (1979, 1981) and Hasza and Fuller (1982) involves a TF equation suggests that TF may produce better forecasts than either DT or UV alone. Two important questions arise here: first, we have to investigate if actual time series
H. Kang / A composite model&r
are indeed generated by TF; second, we have to know the consequences of the use of DT and UV when the appropriate analysis is TF. That is, we want to know if the distortions in DT and UV are significant enough to make their forecasts inferior to those of TF.
3. Monte Carlo study To investigate the differences between a univariate ARIMA ( p, d, q) analysis and a TF analysis with time as an explanatory variable the following Monte Carlo experiments were performed. Four models assumed to be known are: ModelA:
y,=a+bl+(l/(l-0.9B))e,,
ModelB:
~=a+& + ((1 - 0.2%?)/(1
ModelC:
~~,=a+&+(1
Model D:
y, = a,+ (l/(1
- O.SB))e,,
+OSB)e,, - 0.9B))e,,
(13)
where a = 1.00 and b = 0.01 were used in the data generation. First, independently and identically distributed normal deviates were generated using the subroutine GGNML of the International Mathematical and Statistical Library. The variance of e, was set to be 0.02, and 100 data points of yt in (13) were generated. For each model, the generation was repeated 30 times, making each statistically independent from the others by generating a new series of e,. Altogether there were 120 different series of yt. Models A, B, and C include a linear trend with, autoregressive, autoregressive and respectively, moving average, and moving average parameters. Model D is a pure autoregressive form. They are simple cases of the combination of linear trends and stochastic error terms. They are designed to see whether or not differencing or detrending would forecast as well as a TF analysis does when the true model contains a linear trend. For each model and for each replications, three different time series forecasting analyses were made. First, a univariate ARIMA technique was used for the differenced series for each model with the exception of Model D for which no differencing was taken to achieve stationarity. This differencing was suggested from the detailed analysis of the first five replications for each model. The
deterministic
and stochastic trends
179
remaining 25 replications were assumed to be similar to the first five in taking differences. Second, the series y< was first detrended by regressing yt on time. The residuals from this detrending regression were used in a UV analysis. This method will be referred as DT. Third, a TF analysis was used with linear time as an input without differencing the series. The initial identification of a TF model was made from the autocorrelations and partial autocorrelations of the detrended series. A model that is conspicuously missing in the experiment is one with both stochastic and deterministic trends explicitly as in (1 - B)y, = a + There are two reasons for bf + (@(Q/+(Qe,. this omission. First, such a model is a generalization of either TS in (1) or DS in (3). Inappropriate differencing and inappropriate detrending discussed in section 1 will directly apply to such a model. Second, and more importantly, Monte Carlo experiments are designed in such a way that TF does not have obvious advantages over DT and UV. That is, models studied are stacked against TF to show the superiority of TF against DT and UV, if indeed there is any. To facilitate the analysis, the same starting specification for the ARIMA and the same starting initial guess values for the parameters in the nonlinear estimation were used for each model. For TF, the initial specification used was the same as the true one, and the initial parameter values used were the same as the true ones, except for b = 0.5 for all the models. Initial specifications and guess parameter values for UV and DT were 8, = 0.1 (which is roughly for the model (1 B)/(l - 0.9s)) for Model A; 9, = 0.2 and 8, = 0.25 (which are roughly for the model (1 - B)(l O.ZSS)/(l - 0.88)) for Model B; 6, = 0.5 and 8, = 0.4 (which are roughly for the model (1 - B)(l + 0.5B) with invertibility condition imposed) for Model C; and +i = 0.9 for Model D. Initial guess values for the constant term were 1.00 for all the models with UV, and the constant term was suppressed for all the models with DT. When the residuals appeared to be white noise from insignificant individual autocorrelations up to 12th order and from insignificant Ljung-Box statistic, and if estimated AR and/or MA parameters were significant at the 5% level, then the initial model tried was accepted. Otherwise, proper models were identified and estimated afresh for the particular series.
180
H. Kang
/ A composite
model
for determmistic
A brief summary of the comparison among the three analyses is as follows. The TF analysis most accurately identified Model A, whereas in many cases not only the trend but also (1 - 0.9B) disappeared in UV of the differenced series. For the same Model A, DT had a single +i in most of the cases. For Model B, UV had quite a variation in the identified models, DT lost 8, in most cases, while UV lost 4, in one third of these cases. For Model C, UV had two MA parameters while DT had only 8, in about 70% of cases, and TF came up with the correct specification in about two thirds of the cases. For Model D, UV had +i and a and DT had $~i in about two thirds of the cases, whereas TF had 21 cases with +i and a with mixed w0 in their significance. The residual series from the above analysis are the within-sample one step ahead forecasting errors. The accuracy of these forecasts was investigated through the comparisons of the variances of the residual series. The residuals from UV are denoted by eUv and the residuals from TF by err. The hypothesis that both are white noise with zero means cannot be rejected at the 5% significance
and stochastic
trends
level. It should be noted that the two models are not nested, which implies that both sets of residuals are correlated. The significance of the difference in the variances in such cases can be most effectively tested by a regression: (evv-e,r),=a+
(14)
b(e~~+e,~),+u,~
as given by Lehmann (1986, pp. 267-268) and as applied in Granger and Newbold (1986, p. 279). If b is positive (negative), Var(e,,) is greater (smaller) than Var( err), where Var( X) is the variance of the series X. The significance of b in (14) can be used to see whether or not the difference between the two variances is statistically significant. Since eUv and err are constructed to be white noise, u, in (14) can safely be assumed to be white noise. Standard t statistics can be used to test the significance of b by further assuming that eUv and err are normally distributed. The residuals obtained from each method were analyzed. They are denoted as euv, eDT, and err. Pairwise comparisons were made for the size of the variance of the residuals. The t statistics of the
Table 1 Monte Carlo results on UV, DT and TF. a Method
Significance (absolute t)
Model A
Model B
Model C
Model D
Var(euv) >Var(+)
t 21.96 1.96 >
1 19
5 17
10
I
3 20 4
0 3 23
t r 1.00
1.OO>t20
Var(eu”)
Var(euv)
< Var(+)
~Var(enr)
t 11.96
0
0
3
1.96 > t z 1.00 l.OO>t20
0
0
0
0
1
0
0 1 3
2 19 9
6 17 5
2 20 2
0 5 20
t > 1.96 1.96 > t r1.00
0 0
1.OO>t20
0
0 0 2
2 1 3
0 1 4
t 21.96
0
1.96 z t ~1.00
0
28
0 3 22
3 1 1
0 1 22
1.96> t >l.OO 1.OOzt>o
0 1 1
0 2 3
0 7 18
1 3 3
t 21.96
0
1.96 > t > 0
5
0 4
0 0
1 11
t 21.96 1.96> t 21.00
l.oo>t~O Var(eu,)
Var(e,r)
>Var(er,)
l.oort20 Var(eo,)
c Var(er,)
Residuals, euv, on time
t 21.96
a Entries show number of cases in which t statistics attain given values. The value of difference in the two forecast error variances.
t statistic shows the significance of the
H. Kang / A composite model for deterministic
regression coefficient, b, are shown in table 1. The first section describes the significance to show the difference between Var( e”,,) and Var(e,,), the second section between Var( euv ) and Var( eoT), and the third section between Var(e,,) and Var(eT,). Generally, Var(e,v) is greater than Var(e,,), Var(e,v) is greater than Var(e,,), and except for Model C, Var(e,,) is greater than Var(e,,). That is, TF yielded the most accurate and UV the least accurate forecasts among the three analyses. Between DT and TF, one central message from table 1 is that TF should be used instead of DT when there is a linear trend. DT appears to be the best one for Model C, in which there are no AR parameters. Without AR parameters, there would be little difference in the results between the one analysis with the explicit trend and the other in which the detrended data are analyzed, because there is no bias problem discussed in section 2 with eq. (12). Moreover, since most, if not all, variables in business and economics have very large values of the first order AR, the fact that DT is marginally better than TF for Model C can be safely discounted. Perhaps a more important comparison is one between UV and TF. Except for Model D, in which TF is only marginally better than UV, TF is generally better than UV with absolute t greater than one. If a few cases, the hypothesis that Var(e,,) is smaller than Var(e,,) can be solidly rejected at the 5% level by having t greater than 1.96. The improvement of the forecasting accuracy of TF over UV, though not substantial, is real, nonnegligible, and persistent. The marginal improvement in the accuracy from TF in Model D suggests that even if time trend is not present in the true model, TF with time trend does not deteriorate the accuracy of UV. This result may appear to be inconsistent with what Nelson and Kang (1981,1984) have found. According to them, inappropriate detrending artificially creates some autocorrelations in the data. Therefore, if trend is not present in the true model, the data should not be detrended. The conflict is only apparent, however, because TF with time trend as input is different from detrending the data in a regression analysis. At the bottom of table 1, figures show the significance of the coefficient for the time variable
and stochastic trends
181
when the residuals from UV were regressed on time. Since the hypothesis that the residuals are white noise is not rejected, they should be unrelated to time. In the ARIMA analysis, however, the time trend is not explicitly accounted for as a possible source of explanatory variables. It is therefore possible that the residuals could potentially be correlated with time, albeit not strongly. In fact, in five cases out of 30 in Model A, the residuals from UV show some marginally significant correlations with time indicated by their absolute t being greater than one. There were 11 such cases with Model D. For one case in Model D, the time variable turned out to be significant even at the 5% level. The significance of the trend that is discussed in Nelson and Kang (1981, 1984) when +I is very large could be the reason why the trend played an important role in the residuals of UV and the TF analysis. The analysis also suggests that analysts should check whether the trend Table 2 Macroeconomic time series data. Quarterly series from I 1960 to IV I984 (IO0 observarions) (1) GNP, gross national product, seasonally adjusted, in billions of dollars. (2) INV, gross private domestic investment, seasonally adjusted, in billions of dollars. (3) PCE, personal consumption expenditures, seasonally adjusted, in billions of dollars. (4) GOV, government purchases of goods and services, seasonally adjusted, in billions of dollars. (5) DEF, implicit price deflator of personal comsumption expenditures, seasonally adjusted, 1967 = 1.00. (6) MDNS, total mortgage debt outstanding, seasonally not adjusted, in billions of dollars. (7) TPNS, net transfer payments of foreigners by the Federal Government, seasonally not adjusted, in billions of dollars. Monthly series from Januaty I960 to December I984 (300 observalions) (8) CPI, consumer price index for all items for urban workers, seasonally adjusted, 1967 = 1.00. (9) LF, civilian labor force, seasonally adjusted, in millions. (10) Ml, money stock, MI, seasonally adjusted, in billions of dollars. (11) INT, market yield on U.S. Government 3-month bills, seasonally not adjusted, in annualized percentages. (12) UNE, unemployment rate, seasonally adjusted, in percentages. (13) ESNS, total employment in services, seasonally not adjusted, in thousands. (14) DTNS, money supply in small denomination time deposits, seasonally not adjusted, in billions of dollars.
182
H. Kang / A composite model for deterministic
Table
and stochastic trends
3
Univariate
ARIMA
analyses
for 14 macroeconomic
series. a
(1) GNP (l-
B)’
log GNP = (l-0.7318
-0.230B5
(11.60) Q, =11.8
+0.337B11
(3.02)
-0.301B’2)e~;P (3.20)
(3.40)
SER = 0.0094141
(2) INV (l-0.255B
+0.223B4
+0.244B5
+0.237B8
(2.32)
(2.59)
(2.51)
(2.85) Q, = 2.1
+0.234B=)(l-
B) log INV = 0.0344+
er;
(4.95)
(2.44)
SER = 0.049426
(3) PCE (1+0.2798*)(1-
E)’
log PCE = 0.000151
(2.69)
+(l-0.9208
(2.28)
Q, = 5.2
+0.119B8
(31.39)
(3.30)
-0.169B’4)eL5E (5.18)
SER = 0.0070931
(4) GOV (l-0.2838
- B) log GOV = 0.0214 + ez:”
+0.297B8X1
(2.96)
(2.92)
Q10 = 7.2
(7.10)
SER = 0.011630
(5) DEF (l-
B)’
log DEF = (1 -
0.3268
+0.272B3)egtF
(3.56) Q,, = 11.0
(2.97)
SER = 0.0031451
(6) MDNS (1 - 0.383B - 0.217B2)(1 (3.77) Qs = 7.2
(2.14)
- 0.277E4)(1
- B) log MDNS
= 0.00683 + egcNS
(2.66)
(3.64)
SER = 0.0063316
(7) TPNS (l-0.2168 (2.39) Q, = 12.1
+0.187B4 (2.02)
-0.521B”)(l-
B4) log TPNS = 0.0359+
(5.46)
e’u’v””
(2.34)
SER = 0.13626
(8) CPI (1 - B*) log CPI = (1 - 0.6498
- 0.170B3
(14.03) Q, = 6.8 (9) LF (1+0.2618 (4.74) Q, = 6.5
+ 0.238B9 - 0.100E’4)e~$
(3.51)
(5.96)
(2.46)
SER = 0.0020516
+0.128B5 (2.44)
-0.159@)(1-
B) log LF=
(3.04)
0.00208+
eb;
(8.86)
SER = 0.0030602
(10) Ml (1+0,35lB)(l-
B)’
log Ml = 0.0000204+(1-0.3698
(2.47) Q6 =11.7 (11) INT (l-0.3708 (6.66) Q, = 3.0
(4.01)
(2.83)
-0.444B2
-0.199B4
(4.29)
(3.53)
(3.73) Q, = 3.6
(3.07)
SER = 0.0039136
+0.153B2
+0.184B6
(2.69)
(3.62)
-0.143B9)(1 (2.78)
- B) log INT = (1 +O.l55B”)e:N,T (2.77)
SER = 0.065542
(12) UNE (1-0.196B’
-0.165B12
-0.110B3
-0.274B4
(2.18)
(5.22)
SER = 0.030331
+0.226B12)(1 (4.55)
- B) log UNE = e”u”v”
+0.183B’4)e;; (3.90)
H. Kang / A composite model for deterministic
and stochastic trends
183
Table 3 (continued) (13) ESNS (1 -O.l26E)(lB)(l - 8”) log ESNS = (1 +0.138B8 +0.154Bv) (2.38) (2.65) (2.13) SER = 0.0024718 Q, = 9.7 (14) DTNS (l -0.562B)(lB) log DTNS = 0.00613 +(1+0.280B)(1+0.362B’* (3.49) (3.67) (9.10) (8.57) SER = 0.0089986 Q, = 6.3
(1 -0.630B’2)e~.Ns (13.74)
+0.158BZ4 +0.635R36)e~~s (3.64) (16.88)
a Absolute values of t statistics are given in parentheses below coefficients. SER stands for the standard error of regression. Qk statistics reported are asymptotically distributed as x2 with degrees of freedom k. Q statistics, Ljung-Box statistics for a lag of 12, show that in each case the hypothesis of white noise residuals is not rejected at the 5% significance level.
could potentially explain any part of the residuals from UV, whereas such exercise is not necessary for TF by design. Parsimonious models were identified and estimated throughout. Other than the uniform starting specification schemes discussed earlier in this section, each series was analyzed individually in a conventional way in order to reflect the results as analysts would typically use similar methods in practice. The results repeatedly show that, though not overwhelmingly, TF would improve the forecasting accuracy over UV or DT for most cases. UV and DT do not represent a series very well when it contains a deterministic trend and autoregressive and moving average parameters, certainly not as well as TF does.
4. Macroeconomic with time trends
series in a transfer function
Transfer function (TF) models with polynomials in time as trend variables were used to analyze 14 macroeconomic series - seven quarterly and seven monthly - in the United States. Of the 14, ten were seasonally adjusted and four were not seasonally adjusted series. All the series were obtained from the Data Resources, Inc. data base. Table 2 describes the series in detail. All series were transformed by a logarithmic transformation. Each of the 14 series was analyzed two different ways. First, the univariate ARIMA (p, d, q) model was used for each series, and second, the TF with either a linear trend or a quadratic trend was used. An ARIMA model for each series was identified and estimated. The appropriateness of the models was investigated using standard di-
agnostic techniques. Models identified and estimated are reported in table 3. TF analyses were made first with a linear trend term as an input. For each series, the time trend was significant. As before, parsimonious models were identified and estimated and their appropriateness checked using the standard diagnostic techniques. Second, a series of TF analyses were made with both a linear and a quadratic trend term. The quadratic trend term was not significant at the 5% level of INV, MDNS, INT, UNE, ESNS, and DTNS. Consequently, these six variables were modelled with only the linear trend. The results are presented in table 4 where only the test model for each series is shown. For all the series shown in tables 3 and 4, the residuals appear to be white noise at the 5% level using the Ljung-Box statistic. This may sound inconsistent in that, strictly speaking, if the error term in UV is truly white noise, then that in TF cannot be white noise. In practice, however, the analysts would not reject the hypothesis that the residuals are white noise in two different analyses, since best models are identified and estimated to make the residuals a white noise series. In general, transfer function models are more difficult to identify and estimate than invariate ARIMA models. However, since the only input considered is the deterministic trend, the complexity of TF is about the same as that of UV. The orders of AR and MA in the two analyses given in tables 3 and 4 are about the same. The residual series from UV and TF analyses were used in eq. (11) to compare their variances. The t statistics of the regression coefficient b in (11) are shown in table 4 for each TF analysis under t,. For GNP, for example, the variance of
184
H. Kang / A composite model for deterministic
Table
and stochastic trends
4
Transfer
function
analyses with trends for 14 macroeconomic
(1) GNP with quadratic (1-
trend
B) log GNP = 0.00786+0.502~ (2.29)
Qs = 5.36
series
lo-%
-0.389x
(3.23)
+(1+0.255B
lo-‘t2
(2.62)
-0.2308s
(2.76)
+0.269B”)e$!;
(2.44)
(2.72)
tb = 1.06
SER = 0.0090466
(2) INV with linear trend log INV = 4.228 +(0.00694/(1-0.684B))r (71.38) Q, = 9.04
(3.38)
SER = 0.048698
(3) PCE with quadratic (1-
(4) GOV with quadratic
(5) DEF with quadratic
-0.550B’3)/(1
-0.476B9
(5.42)
(7.23)
-0.854))e$”
(7.52)
(15.65)
t* =1.51
trend 0.448~
10-3t
-0.329~
lo-st*
+((l
+0.330B3)/(1
(3.01)
SER = 0.0030514
-0.694B))eTF
(3.13)
(9.02)
Ib = 0.99
with linear trend
log MDNS
= 5.127+0.0250t
Q, = 6.96
(82.98) (24.48) SER = 0.0063654
(7) TPNS with quadratic (1-
10m4t2 +((l-0.299Bs
(3.62)
(5.13)
(6) MDNS
+ ezE
t* = 0.15
SER = 0.010750
Q,, = 10.6
-0.380X10-StZ (3.95)
+0.317X
(19.61)
B) log DEF=
(11.48)
trend
log GOV = 4.565 +O.O180t
(l-
10-3t
(5.02)
SER = 0.0070042
Qs = 9.37
-0.795B))e;;”
(2.46)
trend
(3.30)
(259.15)
-0.251B5)/(l
(2.53)
tb = 0.31
E) log PCE = 0.00740+0.509~
Q,, = 10.4
+((1+0.281B
(7.40)
B4)IogTPNS=
+(l/((l-
1.269B +0.331B”)(l
(23.04) tb = 0.99
trend -0.984X10-3t
+0.318X10-4t2+(1/(1+0.275B4
(1.02) Q,, = 12.5
(2.64)
SER = 0.13241
(8) CPI with quadratic
log LF=
-0.895B))eGE’ (23.56)
-0.103Bs
(5.21)
+O.l17B’)/(l-0.997B))ekF
(1.85)
(2.09)
(2092.7)
lb = 0.74
trend
log Ml = 4.871 +O.O0316t Q, = 8.04
X 10W4tZ +((1+0,29OB
SER = 0.0030433
(240.04)
(5.11)
t* = 0.21
(4.36)
(10) Ml with quadratic
+0.238B9)/(1
(11.12)
trend
0.01521 -0.146
Qs = 5.01
X 10m6t2 +((l -0.641B
(2.33)
SER = 0.0020493
(8.88)
(3.59)
trend
(3.89)
(9) LE with quadratic
-0.372B’3))eTFNS (2.77)
1, =1.35
(1 - E) log CPI = 0.575 X10W4t -0.143 Qs = 10.6
-0.298B4)))eTNS (2.65)
(5.87)
(13.32)
+0.568X
lo-‘t*
+((1+0.2458
SER = 0.0037988
-0.205B4
(4.29)
(8.96)
-0.242B’*
(3.66)
(4.20)
-0.173Bi3)/(1 (2.98)
-0.954B))e&! (61.95)
I/)= 1.74
(11) INT with linear trend log INT = 1.018 +(0.906x (6.18) Q, = 5.45
10m3/(1
-0.806B))t
+((1+0.3798
(6.58)
(1.60)
SER = 0.065160
(6.87)
-0.116B6 (1.92)
- 0.143B7+0.143B9)/(1 (2.37)
(2.61)
-0.944B))e\y (43.09)
tb = 0.64
(12) UNE with linear trend log UNE = 1.304+0.00261t (6.30) Qe = 7.57
+ ((1-0.254B
(2.57)
SER = 0.029166
(4.21)
tb =1.36
-0.2208’
+0.231B4
(3.42)
(4.12)
-0.299B12)/(1 (5.45)
- 1.226B +0.236B4))eypE (42.73)
(8.59)
Ii. Kang / A composite model&r
deterministic
and stochastic trends
185
Table 4 (cantinzted) (13) ESNS with linear trend log ESNS = 2.240+0.00319t +((l +0.141B8 +0.125B9)/((1 (3.26) (2.71) (2.29) (2.04) Qg = 7.41 SER = 0.0024709 tb = 0.41 (14) DTNS with linear trend (1- B) log DTNS = 0.0447-0.137~10-~t (3.88) (2.99)
Qe = 9.15
S&X = 0.0081704
- 1.081B +0.114B3)(1 (26.87) (2.85)
-0.970B*2)))e~~NS (76.66)
+(((l i-0.311B -0.170B6 +0.171B9)(1 -0.7778”)) (4.30) (3.04) (2.98) (18.12) /((l - 0.545B)(l- 0.877Bt2)))e,, DTNS (8.77) (70.21)
fb =I.92
a See notes to table 3. The values in tb are t statistics. Positive values indicate that forecasting error variances from TF in table 4 are smaller than those from UV analyses in table 3.
the forecasting errors from the UV was greater than that from the TF with a quadratic trend as indicated by a positive t statistic (1.06). For all the series, Var(e,v) is greater than Var( erF)_ That is, TF outperforms UV in forecasting. The difference between the variances is not significant. None of the values indicates that Var(e,v) is greater than Var(e,,) at, say, the 5% significance level. However, in eight out of 14 variables, t statistics are very close to one or larger than one, indicating that the difference in the variance is somewhat significant at about the 30% level. The improvement of the forecasting accuracy of TF over UV is not substantial, Nevertheless the improvement is reai and cannot be easily dismissed, especially because the time variable is available to any analyst without a cost and without decreasing degrees of freedom in the estimation. While TF has an additional coefficient to be estimated, UV loses one observation if a differencing is needed. Since the TF analysis with the deterministic time trend is about the same as the UV analysis in its complexities, the improvement of the forecasts can be viewed as something forecasters can obtain virtually without any additional cost. In summary, TF appears to better represent the 14 macroeconomic time series than UV. More importantly and contrary to common practice, differencing does not appear to account effectively for the time trend that exists in the time series. Many time series obtain both deterministic and stochastic trends.
5. Concluding remarks Transfer function anaIyses with linear (or quadratic) trend terms as inputs generally produce better forecasts with smaller variance than univariate analyses. A Monte Carlo experiment shows that a univariate ARIMA or detrending analysis does not represent well a series with a deterministic trend. Fourteen macroeconomic series in the United States are better represented by transfer functions with deterministic trends. Due to the fact that the time variable is available at no cost, the transfer function analysis appears to be a good alternative to the univariate analysis. With the deterministic time trend, the complexities of the two analyses are about the same, which is an additional important factor in practice. The main conclusion of the paper is that transfer function analysis with trend variables as inputs is a good technique to forecast many variables, since detrending of differencing alone may not account for the presence of this particular form of the trend very well.
References Box, G.E.P. and G.M. Jenkins, 1976, Time Series Anaiysis: Forecasting and Controi, rev. ed. (Holder&Day, San Francisco). Dickey, D.A., W.R. Bell, and R.B. Miller, 1986, “Unit roots in time series models: Test and implications”, The American Statistician, 40, 12-26. Dickey, D.A. and W.A. Fuller, 1979, “Distribution of the estimators for autoregressive time series with a unit root”,
186
H. Kong / A composite model for deterministic
Journal of the American Statistical Association, 14, Q-431. Dickey, D.A. and W.A. Fuller, 1981, “Likelihood ratio statistics for autoregressive time series with a unit root”, Econometrica, 49, 1057-1072. Froyen, R.T. and R.N. Waud, 1988, “Real business cycles and the Lucas paradigm”, Economic Inquiry, 26, 183-201. Granger, C.W.J. and P. Newbold, 1986, Forecasting Economic Tfme Series, 2nd ed. (Academic Press, New York). Hasza, D.P. and W.A. Fuller, 1982, “Testing for nonstationary parameter specifications in seasonal time series models”, The Annals of Statistics, 10, 1209-1216. Kang, H., 1986, “Unstable weights in the combination of forecasts”, Management Science, 32, 683-695. Lehmann, E.L., 1986, Testing Statistical Hypotheses, 2nd ed. (Wiley, New York). Mankiw, N.G. and M.D. Shapiro, 1985, “Trends, random walks, and tests of the permanent income hypothesis”, Journal of Monetary Economics, 16, 165-174. Nelson, C.R. and H. Kang, 1981, “Spurious periodicity in inappropriately detrended time series”, Econometrica, 49, 741-751. Nelson, C.R. and H. Kang, 1984, “Pitfalls in the use of time as an explanatory variable in regression”, Journal of Business and Economic Statistics, 2, 13-82. Nelson, C.R. and C.I. Plosser, 1982, “Trends and random ,
and stochastic trends
walks in macroeconomic time series: Some evidence and implications”, Journal of Monetary Economics, 10, 139-162. Stock, J.H. and M.W. Watson, 1987, “Interpreting the evidence on money-income causality”, NBER working paper no. 2228, April. Stulz, R.M. and W. Wasserfallen, 1985, “Macroeconomic time series, business cycles and macroeconomic policies”, in: Karl Brunner and Allan H. Meltzer, eds., Carnegie-Rochester Conference Series in Public Policy, Vol. 22 (North-Holland, Amsterdam) 9-54.
Biography: Heejoon KANG is an Associate Professor of Business Economics & Public Policy at the Indiana University School of Business. He received a B.S. from Seoul National University in Korea. His Ph.D. in Electrical Engineering was from the University of South Carolina in 1974 and his Ph.D. in Economics was from the University of Washington in 1980. He has taught macroecnomics, business and economic forecasting, and econometrics. His puliblications include articles in Journal of Economic Dynamics and Control, Journal of International Money and Finance, Biometrika, Management Science, Journal of Business and Eonomic Statistics, Review of Economics and Statistics, Review of Economic Studies, and Econometrica.
175
A composite model for deterministic and stochastic trends Heejoon KANG * Graduate School of Business, Indiana
University,
Bloomington,
IN 47405, USA
Abstract: Nonstationary time series are typically either detrended or differenced to achieve stationarity in forecasting analyses. Detrending assumes the presence of a deterministic trend and differencing assumes the presence of a stochastic trend. A composite model, which is needed when a time series contains both trends, is a special case of a transfer function analysis. A transfer function with linear or quadratic trend variables as inputs can be used to forecast nonstationary time series with trends. Forecasts of 14 macroeconomic series for the United States show that these series are better represented and forecastable by these transfer function models than by a univariate ARIMA analysis. Comparisons between forecasts from a univariate analysis through detrending or differencing and those from a transfer function analysis for artificially generated data in a series of Monte Carlo experiments show that either detrending or differencing alone does not effectively account for the presence of trend as well as the transfer function model does. Keywords: Univariate Detrending.
analysis,
Transfer
Function
1. Introduction Most macroeconomic time series such as gross national product and the consumer price index are nonstationary and appear to have a time trend. The mean and sometimes the variance of these time series change over time. Nonstationary series are typically either detrended or differenced to induce stationarity. Though exceptions are numerous, time series are frequently detrended in regression analyses, whereas they are differenced in most types of time series analyses. The choice between differencing and detrending has largely been made quite arbitrarily. Since the appearance of the work by Nelson and Plosser (1982) many studies (e.g. Stulz and Wasserfallen, 1985; and Mankiw and Shapiro, 1985) have also investigated whether most macroeconomic time series are to be differenced or * The author acknowledges helpful comments and suggestions from two anonymous referees. 0169-2070/90/$3.50
analysis,
Trend-stationarity,
Difference-stationarity,
detrended and the implications of the distinction between the two. In the terminology of Nelson and Plosser (1982), a series needs detrending to achieve stationarity if it is trend-stationary (TS) and differencing if it is difference-stationary (DS). The empirical evidences in the cited references generally tend to find that most, if not all, macroeconomic variables are DS. A test to determine whether a given series is of the DS or the TS class is used by Nelson and Plosser (1982) and others mentioned above. The test nests both hypotheses into a composite model which is a special case of a transfer function model with a time trend as input. The composite model includes both deterministic and stochastic trends. Although DS is preferred to TS for most macroeconomic variables, the test statistics show that TS is not necessarily false. That is, DS is preferred to TS not because TS is statistically insignificant, but rather because DS is statistically more significant. The results of these tests strongly suggest that those macroeconomic variables might
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
116
H. Kang / A composite model
be better represented with transfer functions than with either DS or TS as a forecasting model. Recently, Kang (1986) has shown that if there are alternative ways to forecast a given series, forecasts obtained from a composite model tend to be better than forecasts obtained from the individual alternative models. Moreover, forecasts from the composite model tend to be better than the combinations of forecasts obtained from the individual alternative models. Alternative models and the resultant composite model in that study are all regression analyses, whereas time series analyses are used here. The objective of the study is to establish that better forecasts for many macroeconomic time series can be obtained by using this composite model for the trend than by assuming that the trend is either TS or DS alone. The transfer function (TF) analysis is applied to 14 economic series of the United States. The results show that the TF analysis generally provides better forecasts than the univariate autoregressive integrated moving average (ARIMA) technique using DS or TS alone for these 14 variables. This therefore suggests that these variables are generated by a TF process, rather than by a DS or a TS process. A series of Monte Carlo experiments are conducted to investigate the forecast performance of DS and TS for time series that are generated by TF. The results show that a composite model for the trend tends to outperform ARIMA models for such series, indicating that DS and TS do not adequately stationarize the data. The structure of the paper is as follows. Section 2 discusses the differences between TS and DS models, and the results for most macroeconomic series in the literature which strongly suggest the use of TF as a forecasting model. Section 3 reports on a series of Monte Carlo experiments which compare TF and univariate forecasts. Four different models are tried, and 30 replications are made for each model. In section 4 the univariate and TF analyses of 14 macroeconomic series for the United States are compared. The results indicate that TF represents the data better than TS or DS alone. Section 5 provides some concluding remarks. 2. Detrending
versus
differencing
Following Nelson and Plosser (1982, pp. 141142), the linear TS model has the form
for deterministic y, =
and stochastic trends
LY +
j?t
+
c,,
+rs(B)c,=8,s(B)u,;
u,=i.i.d.
(0, u,‘),
(I)
where (Y and p are fixed parameters, B is the backward shift operator (i.e., Bky, =yrAk), and $+s( B) and 8,,(B) are respectively autoregressive (AR) and moving average (MA) polynomials in B that satisfy the stationarity and invertibility conditions. The two expressions above can be combined as Y,=~+B+j---&p;
h.(B)
u, e i.i.d. (0, ~2).
(2)
However, it is given separately as in (1) to indicate the typical practice of detrending analysis: a series is detrended before the detrended series is further modeled. The stationarity condition is satisfied when all the roots of $rS( B) lie outside the unit circle, and the invertibility condition is satisfied if all the roots of 8rS( B) lie outside the unit circle. On the other hand, the DS process has the form, (1 - B)Y,
= Y + d,,
$,,(
= 8,,(
B)d,
B)c,;
et = i.i.d. (0, o:),
(3)
where +ns( B) and 8,,(B) are AR and MA polynomials that satisfy the stationarity and invertibility conditions. As before, two equations in (3) can be combined into one: (1-B)y,=y+mo’,;
&s(B)
e,=i.i.d.(O,a:).
DS
(4) As before, it is given separately as in (3) to indicate the typical sequence of the analysis. In (3) y is a parameter and if $DS( B) = 8,,(B) = 1 then y, is a random walk with drift parameter y to be denoted ARIMA (0, 1, 0). It should be noted that the linear trend in (1) can be easily extended to a higher order polynomial in time and the first order differencing in (3) can be easily extended to a higher order of differencing. It should also be noted that y, in (1) and (3) may have been transformed, for example, by using the Box-Cox family of transformations. In fact, most macroeconomic variables are transformed logarithmically in practice. The transformation, or course, does not change any argument in the paper. Dickey and Fuller (1979, 1981) and Hasza and Fuller (1982) have developed a procedure and a
H. Kang / A composite model for deterministic
test statistic for choosing between model (1) and model (3). Under certain assumptions, DS and TS are combined into a single equation. Assuming the error term in (1) or (3) has only AR(p) process without MA part, the composite model becomes y, = p, + P*t + ‘yy,_, + 6,w,_, + 6,w,_, + . . . + Q&
+ u,,
(5)
where w, = (1 - B) y,. Eq. (5) is a particular case of a transfer function given below. Dickey and Fuller provide a statistic to test whether or not (Y= 1 and p, = 0. If the hypothesis is not rejected, DS is preferred to TS. Using this test procedure, Nelson and Plosser (1982) found that DS is a better specification than TS for many macroeconomic variables in the U.S. Stulz and Wasserfallen (1985) also found that most macroeconomic series for various countries are better described as DS than TS. Although DS is preferred to TS, the rejection of TS is not overwhelming. That is, the time trend in a composite specification, like (5), is not entirely insignificant. This strongly suggests that the use of the composite model discussed below may produce better forecasts. When the coefficient of a variable in a multiple regression has a t-statistic greater than unity in absolute value, the variable helps in forecasting the dependent variable because it generally reduces the standard error of the regression. As Dickey and Fuller (1978, 1981) indicated, direct t-statistics in (5) do not have Student t distributions, and, therefore, they cannot be used directly. However, the values of f-statistics for /3i are substantially greater than unity in all the empirical studies cited. This strongly suggests that the time trend may indeed reduce the standard error of regression, which in turn implies that the inclusion of the time trend will increase the forecasting accuracy. In fact, the evidence that differencing does not fully account for the presence of the time trend is given in Stock and Watson (1987), whose money supply series for the U.S. is modeled by TF in their causality investigation. A theoretical possibility of the presence of both deterministic and stochastic trends is also discussed in Froyen and Waud (1988). The most parsimonious combination of (1) and (3) is a transfer function (TF)
and stochastic trends
model, where the variable follows: y,=a+Bt+9(B)u,.
t’(B)
177
y, is related
u,=i.i.d.
to time as
(0, a,‘).
(6)
The MA polynomial 0(B) is invertible, but the AR polynomial Q(B) may not statisfy the stationary conditions. The AR polynomial Q(B) in (6) can be written as @(B) = (1 - B)$( B),where e(B) is stationary with its roots outside the unit circle and d is the degree of homogeneous nonstationarity. The series y, is stationary only when both /?=O and d=O. Assume now that a series, say y,, is generated by the process in (6). One important issue is how effectively detrending or differencing would account for the trend in y,. That is, the forecast performances of the univariate analysis with detrending or differencing should be evaluated for series that are generated by TF in (6). This performance comparison is important because such an inadequate analysis will still appear to be acceptable in practice. A transfer function analysis between y, and t will identify and estimate the model as
8,,(B) Y,=Q-C”+
@TF(B)~TFT
where the subscript TF indicates that the parameters and the residuals are from the transfer function analysis. If the process y, in (6) is differenced once and then analyzed through the univariate ARIMA analysis, we have
(1-B)y,=b+
~uv~~))e,v, uv
and theoretically, 8,,,(B) = e(B) and Guv( B) = (1 -B)-‘@(B). The subscript UV stands for the univariate analysis. Differencing once will produce a stationary series if Q(B) in (6) is in the form of @(B) = (1 - B)+(B). Otherwise, differencing of y, will not produce a stationary series. When y, is generated as in (6) it is not the presence of the trend term that should suggest the differencing. It is rather the homogeneous nonstationarity in Q(B) that dictates the differencing. If Q(B) in (6) is Q(B) = (1 - B)$( B),differencing achieves stationarity and (8) will then effectively account for the time trend in (6) as a byproduct.
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H. Kang / A composite model for deterministic
The linear trend term disappears when a series is differenced once. But it does not mean that one should difference the series in order to remove the trend. Whether or not to difference a series should depend on the error structure - difference only if @(B) is homogeneous nonstationary - and not on the presence of the trend term. If the differencing is made to remove the time trend in TS, the differenced series will have noninvertible error terms. Suppose a series, y,, is TS as in (2). The difference of y, would yield
(’-GB)$fB)u,_ TS
(1 -B)y,=P+
(9)
A univariate ARIMA analysis of (9) of such a series would contain a noninvertible MA because 8,,(B) = (1 - B)B,,( B).Since 8,,(B) is invertible by assumption, 8,,(B) is noninvertible due to (1 - B). A TF model in (6) with the time trend can be and should be dealt with as such. It is not uncommon to read a statement in the literature saying that a series y, is differenced because of the trend. This unfortunate practice is due to a common belief that differencing would generally convert many, if not all, nonstationary series into stationary ones. Box and Jenkins (1976, p. 92) said, “In general, however, we may wish to include a deterministic function of time f(t) in the [univariate ARIMA] model” (emphasis is original). And they continued, “In practice, automatic allowance for a deterministic polynomial trend, of degree d, can be made by permitting [a constant term] to be nonzero [once the series is differenced d times].” The statement is misleading because it appears to suggest that the deterministic polynomial trend can be automatically analyzed with differenced series and that therefore the purpose of differencing is to effectively remove the trend. It is inappropriate to detrend the series in (6) unless the disturbance term in homogeneous nonstationary. Suppose y, in (6) is first detrended and the detrended series is analyzed by the univariate ARIMA analysis. The variable y, will be regressed on time as in y,=a+bt+u^
,,
(10)
and stochastic trends
where 6, are the residuals, which will subsequently be modeled as
;r=
eDT(
‘1
G,,,(B)
eDT3
(11)
where the subscript DT indicates that the parameters and the residuals are from the detrending analysis. The parameters eDT( B) and ~~r( B) in (11) however, will not be the same as 8,, and G&(B) in (6). As Nelson and Kang (1981, 1984) have shown, even if the series y, in (6) is a random walk, detrending (8) appears to be very acceptable with a significant b and, furthermore, with stationary +DT( B) in (11). Yet, such detrending of a random walk series will artificially create some cyclical patterns that are not in the original series. Inasmuch as the differencing is dictated by the homogeneous nonstationarity of the true model, detrending must also be dictated by the model. The importance of the difference between modelling with stochastic trends and with deterministic trends in forecasting is well documented by Dickey, Bell, and Miller (1986). If a model is of TF as in (6) and has a trend variable, needless to say, one has to analyze (6) as the stands. It is not appropriate to detrend y, and then use the detrended series in a subsequent ARIMA analysis. To see this, suppose that the true model of y, is given by: Y, =&I + b,Y,-,
+b,y,_,+dt+u,,
(12)
where u, = i.i.d.(O, u,‘), and that one detrends with yr = a + bt + e, and obtains the detrended series j( =_y, - (a + bt). The use of j$ in a univariate ARIMA analysis will not generally yield the same model as in (12) not to mention the same parameter values. The univariate ARIMA analysis on the detrended yr cannot substitute for the TF analysis with time as an explicit input. In summary, differencing or detrending in a univariate analysis is not a substitute for the TF analysis with time, because whether to difference or to detrend should depend on the error structure in TF, not on the mere presence of time trends. The very fact that the test procedure in Dickey and Fuller (1979, 1981) and Hasza and Fuller (1982) involves a TF equation suggests that TF may produce better forecasts than either DT or UV alone. Two important questions arise here: first, we have to investigate if actual time series
H. Kang / A composite model&r
are indeed generated by TF; second, we have to know the consequences of the use of DT and UV when the appropriate analysis is TF. That is, we want to know if the distortions in DT and UV are significant enough to make their forecasts inferior to those of TF.
3. Monte Carlo study To investigate the differences between a univariate ARIMA ( p, d, q) analysis and a TF analysis with time as an explanatory variable the following Monte Carlo experiments were performed. Four models assumed to be known are: ModelA:
y,=a+bl+(l/(l-0.9B))e,,
ModelB:
~=a+& + ((1 - 0.2%?)/(1
ModelC:
~~,=a+&+(1
Model D:
y, = a,+ (l/(1
- O.SB))e,,
+OSB)e,, - 0.9B))e,,
(13)
where a = 1.00 and b = 0.01 were used in the data generation. First, independently and identically distributed normal deviates were generated using the subroutine GGNML of the International Mathematical and Statistical Library. The variance of e, was set to be 0.02, and 100 data points of yt in (13) were generated. For each model, the generation was repeated 30 times, making each statistically independent from the others by generating a new series of e,. Altogether there were 120 different series of yt. Models A, B, and C include a linear trend with, autoregressive, autoregressive and respectively, moving average, and moving average parameters. Model D is a pure autoregressive form. They are simple cases of the combination of linear trends and stochastic error terms. They are designed to see whether or not differencing or detrending would forecast as well as a TF analysis does when the true model contains a linear trend. For each model and for each replications, three different time series forecasting analyses were made. First, a univariate ARIMA technique was used for the differenced series for each model with the exception of Model D for which no differencing was taken to achieve stationarity. This differencing was suggested from the detailed analysis of the first five replications for each model. The
deterministic
and stochastic trends
179
remaining 25 replications were assumed to be similar to the first five in taking differences. Second, the series y< was first detrended by regressing yt on time. The residuals from this detrending regression were used in a UV analysis. This method will be referred as DT. Third, a TF analysis was used with linear time as an input without differencing the series. The initial identification of a TF model was made from the autocorrelations and partial autocorrelations of the detrended series. A model that is conspicuously missing in the experiment is one with both stochastic and deterministic trends explicitly as in (1 - B)y, = a + There are two reasons for bf + (@(Q/+(Qe,. this omission. First, such a model is a generalization of either TS in (1) or DS in (3). Inappropriate differencing and inappropriate detrending discussed in section 1 will directly apply to such a model. Second, and more importantly, Monte Carlo experiments are designed in such a way that TF does not have obvious advantages over DT and UV. That is, models studied are stacked against TF to show the superiority of TF against DT and UV, if indeed there is any. To facilitate the analysis, the same starting specification for the ARIMA and the same starting initial guess values for the parameters in the nonlinear estimation were used for each model. For TF, the initial specification used was the same as the true one, and the initial parameter values used were the same as the true ones, except for b = 0.5 for all the models. Initial specifications and guess parameter values for UV and DT were 8, = 0.1 (which is roughly for the model (1 B)/(l - 0.9s)) for Model A; 9, = 0.2 and 8, = 0.25 (which are roughly for the model (1 - B)(l O.ZSS)/(l - 0.88)) for Model B; 6, = 0.5 and 8, = 0.4 (which are roughly for the model (1 - B)(l + 0.5B) with invertibility condition imposed) for Model C; and +i = 0.9 for Model D. Initial guess values for the constant term were 1.00 for all the models with UV, and the constant term was suppressed for all the models with DT. When the residuals appeared to be white noise from insignificant individual autocorrelations up to 12th order and from insignificant Ljung-Box statistic, and if estimated AR and/or MA parameters were significant at the 5% level, then the initial model tried was accepted. Otherwise, proper models were identified and estimated afresh for the particular series.
180
H. Kang
/ A composite
model
for determmistic
A brief summary of the comparison among the three analyses is as follows. The TF analysis most accurately identified Model A, whereas in many cases not only the trend but also (1 - 0.9B) disappeared in UV of the differenced series. For the same Model A, DT had a single +i in most of the cases. For Model B, UV had quite a variation in the identified models, DT lost 8, in most cases, while UV lost 4, in one third of these cases. For Model C, UV had two MA parameters while DT had only 8, in about 70% of cases, and TF came up with the correct specification in about two thirds of the cases. For Model D, UV had +i and a and DT had $~i in about two thirds of the cases, whereas TF had 21 cases with +i and a with mixed w0 in their significance. The residual series from the above analysis are the within-sample one step ahead forecasting errors. The accuracy of these forecasts was investigated through the comparisons of the variances of the residual series. The residuals from UV are denoted by eUv and the residuals from TF by err. The hypothesis that both are white noise with zero means cannot be rejected at the 5% significance
and stochastic
trends
level. It should be noted that the two models are not nested, which implies that both sets of residuals are correlated. The significance of the difference in the variances in such cases can be most effectively tested by a regression: (evv-e,r),=a+
(14)
b(e~~+e,~),+u,~
as given by Lehmann (1986, pp. 267-268) and as applied in Granger and Newbold (1986, p. 279). If b is positive (negative), Var(e,,) is greater (smaller) than Var( err), where Var( X) is the variance of the series X. The significance of b in (14) can be used to see whether or not the difference between the two variances is statistically significant. Since eUv and err are constructed to be white noise, u, in (14) can safely be assumed to be white noise. Standard t statistics can be used to test the significance of b by further assuming that eUv and err are normally distributed. The residuals obtained from each method were analyzed. They are denoted as euv, eDT, and err. Pairwise comparisons were made for the size of the variance of the residuals. The t statistics of the
Table 1 Monte Carlo results on UV, DT and TF. a Method
Significance (absolute t)
Model A
Model B
Model C
Model D
Var(euv) >Var(+)
t 21.96 1.96 >
1 19
5 17
10
I
3 20 4
0 3 23
t r 1.00
1.OO>t20
Var(eu”)
Var(euv)
< Var(+)
~Var(enr)
t 11.96
0
0
3
1.96 > t z 1.00 l.OO>t20
0
0
0
0
1
0
0 1 3
2 19 9
6 17 5
2 20 2
0 5 20
t > 1.96 1.96 > t r1.00
0 0
1.OO>t20
0
0 0 2
2 1 3
0 1 4
t 21.96
0
1.96 z t ~1.00
0
28
0 3 22
3 1 1
0 1 22
1.96> t >l.OO 1.OOzt>o
0 1 1
0 2 3
0 7 18
1 3 3
t 21.96
0
1.96 > t > 0
5
0 4
0 0
1 11
t 21.96 1.96> t 21.00
l.oo>t~O Var(eu,)
Var(e,r)
>Var(er,)
l.oort20 Var(eo,)
c Var(er,)
Residuals, euv, on time
t 21.96
a Entries show number of cases in which t statistics attain given values. The value of difference in the two forecast error variances.
t statistic shows the significance of the
H. Kang / A composite model for deterministic
regression coefficient, b, are shown in table 1. The first section describes the significance to show the difference between Var( e”,,) and Var(e,,), the second section between Var( euv ) and Var( eoT), and the third section between Var(e,,) and Var(eT,). Generally, Var(e,v) is greater than Var(e,,), Var(e,v) is greater than Var(e,,), and except for Model C, Var(e,,) is greater than Var(e,,). That is, TF yielded the most accurate and UV the least accurate forecasts among the three analyses. Between DT and TF, one central message from table 1 is that TF should be used instead of DT when there is a linear trend. DT appears to be the best one for Model C, in which there are no AR parameters. Without AR parameters, there would be little difference in the results between the one analysis with the explicit trend and the other in which the detrended data are analyzed, because there is no bias problem discussed in section 2 with eq. (12). Moreover, since most, if not all, variables in business and economics have very large values of the first order AR, the fact that DT is marginally better than TF for Model C can be safely discounted. Perhaps a more important comparison is one between UV and TF. Except for Model D, in which TF is only marginally better than UV, TF is generally better than UV with absolute t greater than one. If a few cases, the hypothesis that Var(e,,) is smaller than Var(e,,) can be solidly rejected at the 5% level by having t greater than 1.96. The improvement of the forecasting accuracy of TF over UV, though not substantial, is real, nonnegligible, and persistent. The marginal improvement in the accuracy from TF in Model D suggests that even if time trend is not present in the true model, TF with time trend does not deteriorate the accuracy of UV. This result may appear to be inconsistent with what Nelson and Kang (1981,1984) have found. According to them, inappropriate detrending artificially creates some autocorrelations in the data. Therefore, if trend is not present in the true model, the data should not be detrended. The conflict is only apparent, however, because TF with time trend as input is different from detrending the data in a regression analysis. At the bottom of table 1, figures show the significance of the coefficient for the time variable
and stochastic trends
181
when the residuals from UV were regressed on time. Since the hypothesis that the residuals are white noise is not rejected, they should be unrelated to time. In the ARIMA analysis, however, the time trend is not explicitly accounted for as a possible source of explanatory variables. It is therefore possible that the residuals could potentially be correlated with time, albeit not strongly. In fact, in five cases out of 30 in Model A, the residuals from UV show some marginally significant correlations with time indicated by their absolute t being greater than one. There were 11 such cases with Model D. For one case in Model D, the time variable turned out to be significant even at the 5% level. The significance of the trend that is discussed in Nelson and Kang (1981, 1984) when +I is very large could be the reason why the trend played an important role in the residuals of UV and the TF analysis. The analysis also suggests that analysts should check whether the trend Table 2 Macroeconomic time series data. Quarterly series from I 1960 to IV I984 (IO0 observarions) (1) GNP, gross national product, seasonally adjusted, in billions of dollars. (2) INV, gross private domestic investment, seasonally adjusted, in billions of dollars. (3) PCE, personal consumption expenditures, seasonally adjusted, in billions of dollars. (4) GOV, government purchases of goods and services, seasonally adjusted, in billions of dollars. (5) DEF, implicit price deflator of personal comsumption expenditures, seasonally adjusted, 1967 = 1.00. (6) MDNS, total mortgage debt outstanding, seasonally not adjusted, in billions of dollars. (7) TPNS, net transfer payments of foreigners by the Federal Government, seasonally not adjusted, in billions of dollars. Monthly series from Januaty I960 to December I984 (300 observalions) (8) CPI, consumer price index for all items for urban workers, seasonally adjusted, 1967 = 1.00. (9) LF, civilian labor force, seasonally adjusted, in millions. (10) Ml, money stock, MI, seasonally adjusted, in billions of dollars. (11) INT, market yield on U.S. Government 3-month bills, seasonally not adjusted, in annualized percentages. (12) UNE, unemployment rate, seasonally adjusted, in percentages. (13) ESNS, total employment in services, seasonally not adjusted, in thousands. (14) DTNS, money supply in small denomination time deposits, seasonally not adjusted, in billions of dollars.
182
H. Kang / A composite model for deterministic
Table
and stochastic trends
3
Univariate
ARIMA
analyses
for 14 macroeconomic
series. a
(1) GNP (l-
B)’
log GNP = (l-0.7318
-0.230B5
(11.60) Q, =11.8
+0.337B11
(3.02)
-0.301B’2)e~;P (3.20)
(3.40)
SER = 0.0094141
(2) INV (l-0.255B
+0.223B4
+0.244B5
+0.237B8
(2.32)
(2.59)
(2.51)
(2.85) Q, = 2.1
+0.234B=)(l-
B) log INV = 0.0344+
er;
(4.95)
(2.44)
SER = 0.049426
(3) PCE (1+0.2798*)(1-
E)’
log PCE = 0.000151
(2.69)
+(l-0.9208
(2.28)
Q, = 5.2
+0.119B8
(31.39)
(3.30)
-0.169B’4)eL5E (5.18)
SER = 0.0070931
(4) GOV (l-0.2838
- B) log GOV = 0.0214 + ez:”
+0.297B8X1
(2.96)
(2.92)
Q10 = 7.2
(7.10)
SER = 0.011630
(5) DEF (l-
B)’
log DEF = (1 -
0.3268
+0.272B3)egtF
(3.56) Q,, = 11.0
(2.97)
SER = 0.0031451
(6) MDNS (1 - 0.383B - 0.217B2)(1 (3.77) Qs = 7.2
(2.14)
- 0.277E4)(1
- B) log MDNS
= 0.00683 + egcNS
(2.66)
(3.64)
SER = 0.0063316
(7) TPNS (l-0.2168 (2.39) Q, = 12.1
+0.187B4 (2.02)
-0.521B”)(l-
B4) log TPNS = 0.0359+
(5.46)
e’u’v””
(2.34)
SER = 0.13626
(8) CPI (1 - B*) log CPI = (1 - 0.6498
- 0.170B3
(14.03) Q, = 6.8 (9) LF (1+0.2618 (4.74) Q, = 6.5
+ 0.238B9 - 0.100E’4)e~$
(3.51)
(5.96)
(2.46)
SER = 0.0020516
+0.128B5 (2.44)
-0.159@)(1-
B) log LF=
(3.04)
0.00208+
eb;
(8.86)
SER = 0.0030602
(10) Ml (1+0,35lB)(l-
B)’
log Ml = 0.0000204+(1-0.3698
(2.47) Q6 =11.7 (11) INT (l-0.3708 (6.66) Q, = 3.0
(4.01)
(2.83)
-0.444B2
-0.199B4
(4.29)
(3.53)
(3.73) Q, = 3.6
(3.07)
SER = 0.0039136
+0.153B2
+0.184B6
(2.69)
(3.62)
-0.143B9)(1 (2.78)
- B) log INT = (1 +O.l55B”)e:N,T (2.77)
SER = 0.065542
(12) UNE (1-0.196B’
-0.165B12
-0.110B3
-0.274B4
(2.18)
(5.22)
SER = 0.030331
+0.226B12)(1 (4.55)
- B) log UNE = e”u”v”
+0.183B’4)e;; (3.90)
H. Kang / A composite model for deterministic
and stochastic trends
183
Table 3 (continued) (13) ESNS (1 -O.l26E)(lB)(l - 8”) log ESNS = (1 +0.138B8 +0.154Bv) (2.38) (2.65) (2.13) SER = 0.0024718 Q, = 9.7 (14) DTNS (l -0.562B)(lB) log DTNS = 0.00613 +(1+0.280B)(1+0.362B’* (3.49) (3.67) (9.10) (8.57) SER = 0.0089986 Q, = 6.3
(1 -0.630B’2)e~.Ns (13.74)
+0.158BZ4 +0.635R36)e~~s (3.64) (16.88)
a Absolute values of t statistics are given in parentheses below coefficients. SER stands for the standard error of regression. Qk statistics reported are asymptotically distributed as x2 with degrees of freedom k. Q statistics, Ljung-Box statistics for a lag of 12, show that in each case the hypothesis of white noise residuals is not rejected at the 5% significance level.
could potentially explain any part of the residuals from UV, whereas such exercise is not necessary for TF by design. Parsimonious models were identified and estimated throughout. Other than the uniform starting specification schemes discussed earlier in this section, each series was analyzed individually in a conventional way in order to reflect the results as analysts would typically use similar methods in practice. The results repeatedly show that, though not overwhelmingly, TF would improve the forecasting accuracy over UV or DT for most cases. UV and DT do not represent a series very well when it contains a deterministic trend and autoregressive and moving average parameters, certainly not as well as TF does.
4. Macroeconomic with time trends
series in a transfer function
Transfer function (TF) models with polynomials in time as trend variables were used to analyze 14 macroeconomic series - seven quarterly and seven monthly - in the United States. Of the 14, ten were seasonally adjusted and four were not seasonally adjusted series. All the series were obtained from the Data Resources, Inc. data base. Table 2 describes the series in detail. All series were transformed by a logarithmic transformation. Each of the 14 series was analyzed two different ways. First, the univariate ARIMA (p, d, q) model was used for each series, and second, the TF with either a linear trend or a quadratic trend was used. An ARIMA model for each series was identified and estimated. The appropriateness of the models was investigated using standard di-
agnostic techniques. Models identified and estimated are reported in table 3. TF analyses were made first with a linear trend term as an input. For each series, the time trend was significant. As before, parsimonious models were identified and estimated and their appropriateness checked using the standard diagnostic techniques. Second, a series of TF analyses were made with both a linear and a quadratic trend term. The quadratic trend term was not significant at the 5% level of INV, MDNS, INT, UNE, ESNS, and DTNS. Consequently, these six variables were modelled with only the linear trend. The results are presented in table 4 where only the test model for each series is shown. For all the series shown in tables 3 and 4, the residuals appear to be white noise at the 5% level using the Ljung-Box statistic. This may sound inconsistent in that, strictly speaking, if the error term in UV is truly white noise, then that in TF cannot be white noise. In practice, however, the analysts would not reject the hypothesis that the residuals are white noise in two different analyses, since best models are identified and estimated to make the residuals a white noise series. In general, transfer function models are more difficult to identify and estimate than invariate ARIMA models. However, since the only input considered is the deterministic trend, the complexity of TF is about the same as that of UV. The orders of AR and MA in the two analyses given in tables 3 and 4 are about the same. The residual series from UV and TF analyses were used in eq. (11) to compare their variances. The t statistics of the regression coefficient b in (11) are shown in table 4 for each TF analysis under t,. For GNP, for example, the variance of
184
H. Kang / A composite model for deterministic
Table
and stochastic trends
4
Transfer
function
analyses with trends for 14 macroeconomic
(1) GNP with quadratic (1-
trend
B) log GNP = 0.00786+0.502~ (2.29)
Qs = 5.36
series
lo-%
-0.389x
(3.23)
+(1+0.255B
lo-‘t2
(2.62)
-0.2308s
(2.76)
+0.269B”)e$!;
(2.44)
(2.72)
tb = 1.06
SER = 0.0090466
(2) INV with linear trend log INV = 4.228 +(0.00694/(1-0.684B))r (71.38) Q, = 9.04
(3.38)
SER = 0.048698
(3) PCE with quadratic (1-
(4) GOV with quadratic
(5) DEF with quadratic
-0.550B’3)/(1
-0.476B9
(5.42)
(7.23)
-0.854))e$”
(7.52)
(15.65)
t* =1.51
trend 0.448~
10-3t
-0.329~
lo-st*
+((l
+0.330B3)/(1
(3.01)
SER = 0.0030514
-0.694B))eTF
(3.13)
(9.02)
Ib = 0.99
with linear trend
log MDNS
= 5.127+0.0250t
Q, = 6.96
(82.98) (24.48) SER = 0.0063654
(7) TPNS with quadratic (1-
10m4t2 +((l-0.299Bs
(3.62)
(5.13)
(6) MDNS
+ ezE
t* = 0.15
SER = 0.010750
Q,, = 10.6
-0.380X10-StZ (3.95)
+0.317X
(19.61)
B) log DEF=
(11.48)
trend
log GOV = 4.565 +O.O180t
(l-
10-3t
(5.02)
SER = 0.0070042
Qs = 9.37
-0.795B))e;;”
(2.46)
trend
(3.30)
(259.15)
-0.251B5)/(l
(2.53)
tb = 0.31
E) log PCE = 0.00740+0.509~
Q,, = 10.4
+((1+0.281B
(7.40)
B4)IogTPNS=
+(l/((l-
1.269B +0.331B”)(l
(23.04) tb = 0.99
trend -0.984X10-3t
+0.318X10-4t2+(1/(1+0.275B4
(1.02) Q,, = 12.5
(2.64)
SER = 0.13241
(8) CPI with quadratic
log LF=
-0.895B))eGE’ (23.56)
-0.103Bs
(5.21)
+O.l17B’)/(l-0.997B))ekF
(1.85)
(2.09)
(2092.7)
lb = 0.74
trend
log Ml = 4.871 +O.O0316t Q, = 8.04
X 10W4tZ +((1+0,29OB
SER = 0.0030433
(240.04)
(5.11)
t* = 0.21
(4.36)
(10) Ml with quadratic
+0.238B9)/(1
(11.12)
trend
0.01521 -0.146
Qs = 5.01
X 10m6t2 +((l -0.641B
(2.33)
SER = 0.0020493
(8.88)
(3.59)
trend
(3.89)
(9) LE with quadratic
-0.372B’3))eTFNS (2.77)
1, =1.35
(1 - E) log CPI = 0.575 X10W4t -0.143 Qs = 10.6
-0.298B4)))eTNS (2.65)
(5.87)
(13.32)
+0.568X
lo-‘t*
+((1+0.2458
SER = 0.0037988
-0.205B4
(4.29)
(8.96)
-0.242B’*
(3.66)
(4.20)
-0.173Bi3)/(1 (2.98)
-0.954B))e&! (61.95)
I/)= 1.74
(11) INT with linear trend log INT = 1.018 +(0.906x (6.18) Q, = 5.45
10m3/(1
-0.806B))t
+((1+0.3798
(6.58)
(1.60)
SER = 0.065160
(6.87)
-0.116B6 (1.92)
- 0.143B7+0.143B9)/(1 (2.37)
(2.61)
-0.944B))e\y (43.09)
tb = 0.64
(12) UNE with linear trend log UNE = 1.304+0.00261t (6.30) Qe = 7.57
+ ((1-0.254B
(2.57)
SER = 0.029166
(4.21)
tb =1.36
-0.2208’
+0.231B4
(3.42)
(4.12)
-0.299B12)/(1 (5.45)
- 1.226B +0.236B4))eypE (42.73)
(8.59)
Ii. Kang / A composite model&r
deterministic
and stochastic trends
185
Table 4 (cantinzted) (13) ESNS with linear trend log ESNS = 2.240+0.00319t +((l +0.141B8 +0.125B9)/((1 (3.26) (2.71) (2.29) (2.04) Qg = 7.41 SER = 0.0024709 tb = 0.41 (14) DTNS with linear trend (1- B) log DTNS = 0.0447-0.137~10-~t (3.88) (2.99)
Qe = 9.15
S&X = 0.0081704
- 1.081B +0.114B3)(1 (26.87) (2.85)
-0.970B*2)))e~~NS (76.66)
+(((l i-0.311B -0.170B6 +0.171B9)(1 -0.7778”)) (4.30) (3.04) (2.98) (18.12) /((l - 0.545B)(l- 0.877Bt2)))e,, DTNS (8.77) (70.21)
fb =I.92
a See notes to table 3. The values in tb are t statistics. Positive values indicate that forecasting error variances from TF in table 4 are smaller than those from UV analyses in table 3.
the forecasting errors from the UV was greater than that from the TF with a quadratic trend as indicated by a positive t statistic (1.06). For all the series, Var(e,v) is greater than Var( erF)_ That is, TF outperforms UV in forecasting. The difference between the variances is not significant. None of the values indicates that Var(e,v) is greater than Var(e,,) at, say, the 5% significance level. However, in eight out of 14 variables, t statistics are very close to one or larger than one, indicating that the difference in the variance is somewhat significant at about the 30% level. The improvement of the forecasting accuracy of TF over UV is not substantial, Nevertheless the improvement is reai and cannot be easily dismissed, especially because the time variable is available to any analyst without a cost and without decreasing degrees of freedom in the estimation. While TF has an additional coefficient to be estimated, UV loses one observation if a differencing is needed. Since the TF analysis with the deterministic time trend is about the same as the UV analysis in its complexities, the improvement of the forecasts can be viewed as something forecasters can obtain virtually without any additional cost. In summary, TF appears to better represent the 14 macroeconomic time series than UV. More importantly and contrary to common practice, differencing does not appear to account effectively for the time trend that exists in the time series. Many time series obtain both deterministic and stochastic trends.
5. Concluding remarks Transfer function anaIyses with linear (or quadratic) trend terms as inputs generally produce better forecasts with smaller variance than univariate analyses. A Monte Carlo experiment shows that a univariate ARIMA or detrending analysis does not represent well a series with a deterministic trend. Fourteen macroeconomic series in the United States are better represented by transfer functions with deterministic trends. Due to the fact that the time variable is available at no cost, the transfer function analysis appears to be a good alternative to the univariate analysis. With the deterministic time trend, the complexities of the two analyses are about the same, which is an additional important factor in practice. The main conclusion of the paper is that transfer function analysis with trend variables as inputs is a good technique to forecast many variables, since detrending of differencing alone may not account for the presence of this particular form of the trend very well.
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Biography: Heejoon KANG is an Associate Professor of Business Economics & Public Policy at the Indiana University School of Business. He received a B.S. from Seoul National University in Korea. His Ph.D. in Electrical Engineering was from the University of South Carolina in 1974 and his Ph.D. in Economics was from the University of Washington in 1980. He has taught macroecnomics, business and economic forecasting, and econometrics. His puliblications include articles in Journal of Economic Dynamics and Control, Journal of International Money and Finance, Biometrika, Management Science, Journal of Business and Eonomic Statistics, Review of Economics and Statistics, Review of Economic Studies, and Econometrica.