INTRODUCTION
Since, Fuller (1969) introduced the concept of spline or grafted polynomial,
many researchers have utilized it to make ex-post and ex-ante forecast
of economic time series data beyond the estimation period. Such studies
include those by Phillip (1990), Rahman and Damisa (1999), Nmadu and Amos
(2002), Nmadu and Phillp (2001) and Nmadu et al. (2004). For example
Bormann et al. (2002) estimated lactation stage, age at milking,
previous days open and days pregnant using quadratic polynomials by fitting
joint points. Meyer (2005) successfully modeled growth of Australian Angus
cattle using the spline function. Some other researchers have made innovations
to the original model. Those include Fox and Grafton (2000), Parsons and
Hunt (1981) and Marsh (1986). Fox and Grafton (2000) used capital and
model selection criteria rather than trend to determine appropriate break
points.
The concept is based on the visual examination of the scatter diagram
of the available data series against trend in order to divide the data
into sub-periods and to suggest suitable joint points to capture all the
sub-periods into a single model (Fuller, 1969; Phillip, 1990; Meyer, 2005;
Pierre et al., 1987). Since, the eventual model estimated is subject
to the visual examination of the base data by the researcher, it therefore
means that the appropriateness of the eventual estimated function and
the forecast based on it is accurate to the extent of the accuracy of
the researcher`s visualization. In this circumstance, the same data can
be modeled along different lines depending on the researcher. Hence, there
is need to find out if the quality of the model resulting from this modeling
is affected by the type of function and other factors. The main objective
of this study is to investigate whether the choice of joint points in
a spline function and the type of model selected affects the forecasting
ability of the resulting estimated coefficients.
MATERIALS AND METHODS
The data used in this research were mainly secondary data sourced from
Earth Trend (2006). The data included cereal grains production in Nigeria
in metric tones between 1961 and 2005, percent agriculture contribution
to Nigerian GDP between 1965 and 2004 and aggregate fertilizer consumption
by Nigerian farmers in metric tones between 1961 and 2001.
Normally, the available data is plotted against trend in order to divide
the series into segments based on visual examination. Traditionally, the
data is usually divided into three sub-periods and no attempt was made
in this study to go beyond that.
There are two commonly used models, that is, Linear-Quadratic-Linear
and Quadratic-Quadratic-Linear models. These two are preferred because
it is normal to have linear portion as the terminal (Fuller, 1969; Phillip,
1990) as that enhances forecasting which is the main objective of using
the system. However, an attempt was made to explore all possible models
in order to show if the eventual model is acceptable for forecasting.
Therefore, Linear-Quadratic-Quadratic, Linear-Linear-Quadratic, Linear-Linear-Linear
and Quadratic-Quadratic-Quadratic models were tried. Linear-Linear-Quadratic
was dropped because some of the coefficients were over-identified while
Linear-Linear-Linear was dropped because the variables were over-identified
and any of the linear regression models can be applied to a data series
that is linear over the entire trend and there will be no need to divide
it into sub-periods. In the case of Quadratic-Quadratic-Quadratic, the
model was dropped because the variables were over-identified and the data
series with this type of behavior is better estimated with higher polynomial
instead of dividing into sub-periods.
The details of the models and the mean equation are shown below for the
three models left. The detail of how the mean equations were obtained
is shown for one of the models in Appendix.
Linear Quadratic Linear
A graphical examination of the data may show that it can be divided
into three segments; hence the following trend function was suggested:
|
 |
(1) |
|
 |
(2) |
|
 |
(3) |
| Yt |
= |
Data series in year t |
| t |
= |
Trend |
| α's, β's and φ |
= |
Structural parameters to be estimated |
| JP1 and JP2 |
= |
Joint point 1 and 2, respectively |
Equation 1-3 are then reworked
as shown below:
|
 |
(4) |
|
 |
(5) |
|
 |
(6) |
Equation 4-6, are then formed into
a single equation for estimation as follows:
|
 |
(7) |
| Zo |
= |
 |
| Z1 |
= |
 |
| Z2 |
= |
 |
|
= |
 |
|
= |
 |
| Ut |
= |
Error term assumed to be well behaved |
Quadratic Quadratic Linear
A graphical examination of a data series may reveal that it can be divided
into different segments as the trend equation below:
|
 |
(8) |
|
 |
(9) |
|
 |
(10) |
| Qt |
= |
Data series in year t |
| t |
= |
Trend |
| α's, β's and φ |
= |
Structural parameters to be estimated |
| JP1 and JP2 |
= |
Joint point 1 and 2, respectively |
Equation 8-10 are then reworked
as shown below:
|
 |
(11) |
|
 |
(12) |
|
 |
(13) |
Equation 11-13, are then formed
into a single equation for estimation as follows:
|
 |
(14) |
| Zo |
= |
 |
| Z1 |
= |
 |
| Z2 |
= |
 |
|
= |
 |
|
= |
 |
| Z3 |
= |
 |
|
= |
 |
|
= |
 |
| μ's |
= |
Structural parameters to be estimated |
| Ut |
= |
Error term assumed to be well behaved |
Linear Quadratic Quadratic
A graphical examination of a data series may reveal that it can be divided
into different segments as the trend equation below:
|
 |
(15) |
|
 |
(16) |
|
 |
(17) |
| GDt |
= |
Data series in year t |
| t |
= |
Trend |
| α's, β's and φ |
= |
Structural parameters to be estimated |
| JP1 and JP2 |
= |
Joint point 1 and 2, respectively |
Equation 15-17, are then reworked
as shown below:
|
 |
(18) |
|
 |
(19) |
|
 |
(20) |
Equation 18-20 are then formed
into a single equation for estimation as follows:
|
 |
(21) |
| Zo |
= |
 |
| Z1 |
= |
 |
| Z2 |
= |
 |
|
= |
 |
|
= |
 |
| Z3 |
= |
 |
|
= |
|
|
= |
|
| Z4 |
= |
 |
|
= |
 |
|
= |
 |
| μ's |
= |
Structural parameters to be estimated |
| Ut |
= |
Error term assumed to be well behaved |
Equations 7, 14 and 21
are the mean equations; they are continuous with the various restrictions
relating to each model.
The three final equations were applied to cereal grains production in
Nigeria in metric tones between 1961 and 2005 and percent agriculture
contribution to Nigerian GDP between 1965 and 2004. In addition, the models
were applied to cereal grains production when either GDP or aggregate
fertilizer consumption by Nigerian farmers in metric tones between 1961
and 2001 or both are added as explanatory variables. Ex-post forecast
of the trend was then made for estimation period while ex-ante forecast
was made to year 2020 and the forecasts compared with the observed data.
The data were obtained from EarthTrend (2006). After the estimation of
the models, the forecasting ability of each of them was assessed using
Mean Square Error (MSE). MSE is given as:
| MSE |
= |
Mean Square Error |
| Yt |
= |
Observed value |
| yt |
= |
Estimated value |
| n |
= |
Sample size |
The model with the least MSE is adjudged better than the other.
RESULTS AND DISCUSSION
The estimates of the various explanatory variables using the different
models are presented in Table 1-3
while the ex-post and ex-ante estimates of the data series are shows in
Fig. 1-4. Table 4
gives the MSE for all the models.
The result in Table 1 (Linear-Quadratic-Linear) shows that all variables
are significant in the trend of cereal grains production during the period
under study and the estimates of the coefficients of GDP or fertilizer
or both were not significant as explanatory variables in the trend of
cereal production. Table 1 also shows that the trend of GDP was not well
explained by the variables included in the model. The result in Table 2 (Quadratic-Quadratic-Linear) shows that all the variables were significant
in the trend of cereal production and GDP but the estimates of the coefficients
of GDP was not a significant explanatory variable in cereal production.
| Table 1: |
Estimates of the coefficient for Linear Quadratic Linear
model |
 |
| (1) JP1 = 1965, JP2 = 1987; (2)
JP1 = 1980, JP2 = 1988; Values in parenthesis
are SE; *p<0.10; **p<0.05; ***p<0.01, ns: Not significant |
| Table 2: |
Estimates of the coefficient for the Quadratic Quadratic
Linear model |
 |
| (1) JP1 = 1973, JP2 = 1987; (2)
JP1 = 1988, JP2 = 1998; Values in parenthesis
are SE; *p<0.10; **p<0.05; ***p<0.01; ns: Not significant |
| Table 3: |
Estimates of the coefficient for the Linear Quadratic
Quadratic model |
 |
| (1) JP1 = 1965, JP2 = 1987; 2:
JP1 = 1980, JP2 = 1992; Values in parenthesis
are SE; *p<0.10; **p<0.05; ***p<0.01; ns: Not significant |
| Table 4: |
Estimates of MSE for all the models |
 |
|
| Fig. 1: |
Ex-post and ex-ante forecast of cereal grains using
the Linear Quadratic Linear model |
|
| Fig. 2: |
Ex-post and ex-ante forecast of cereal grains using
the Quadratic Quadratic Linear model |
|
| Fig. 3: |
Ex-post and ex-ante forecast of cereal grains using
the Linear Quadratic Quadratic model |
|
| Fig. 4: |
Ex-post and ex-ante forecast of per cent contribution
of agriculture to GDP using various models |
The results shown in Table 3 indicate that the variables
of the Linear-Quadratic-Quadratic model captured the trend in cereal production
and GDP significantly. However, the estimates of the coefficient of GDP
and fertilizer when added as explanatory variables are not significant
in explaining the trend in cereal grains production. The non-significance
of added explanatory variables in the trend equation is quite contrary
to what Nmadu and Phillip (2001) and Nmadu et al. (2004) found
in the case of sorghum. The result would seem to indicate that there is
universality as to the appropriateness of the grafted model used. But
that is sharply contrasted with the result of the ex-post and ex-ante
forecast shown in Fig. 1-4, respectively.
It would be noticed that similar results were obtained with models that
have linear terminal but the result with Quadratic terminal is kinked
at the joint points which is against one of the major requirement of the
spline system (Fuller, 1969; Philip, 1990) even though the ex-ante forecast
from the model compares favourably with the other models for cereal grains
production and also compares favourably with results obtained from other
series (Rahman and Damisa, 1999; Nmadu and Amos, 2002; Nmadu and Philip,
2001; Nmadu et al., 2004). The ex-post and ex-ante forecast for
the GDP from the three models show some interesting results. While the
Linear-Quadratic-Linear model show a slow upward trend, the Quadratic-Quadratic-Linear
show a sliding trend and Linear-Quadratic-Quadratic show a rapid upward
trend. However, the forecast from the Quadratic-Quadratic-Linear is most
consistent with the observed trend. Given such scenario, it would seem
that the choice of the spline model to use is not based solely on visual
examination, but it will also depend on the nature of data series involved
and the use to which the forecast would be put. While any of the models
could be cautiously used for cereal grains production forecasting, other
factors would have to be considered in choosing a model for forecasting
GDP. In that regard, it is advised that all possible spline models should
be tried and the one that gives best result should be utilized for further
studies. For example, the result of the MSE in Table 4
shows that the best model is not uniform across. Different models may
be recommended if type of spline system or number of explanatory variables
in the various systems is considered. While QQL seams to be a better model
with GDP and Cereal based splines; with regard to number of number of
variables, the choice is a mixed bag. Changing of joint points has not
shown any significant effect of the output of the models.
CONCLUSION
The effect of changing joint points and the type of spline function was
investigated in this research. The result obtained show that the there
was no universality as to the effect of the model and joint points chosen.
Therefore, attempts should be made to model the data series with as many
models as possible. The choice of the most acceptable should be based
on the conformity of the ex-post and ex-ante forecasts to the observed
data and economic sense.
APPENDIX
Full Details of the Grafting of Linear Quadratic Quadratic Model
A graphical examination of a data series may reveal that it can be
divided into different segments as the trend equation below:
|
 |
(1) |
|
 |
(2) |
|
 |
(3) |
| GDt |
= |
Data series in year t |
| t |
= |
Trend |
| α's, β's and φ |
= |
Structural parameters to be estimated |
| JP1 and JP2 |
= |
Joint point 1 and 2, respectively |
The restrictions (Fuller, 1969) on the Eq. 1-3
are:
|
αo + βoK1=α1
+ β1K1 + φ1K12( |
(4) |
|
α1 + β1K2
+ φ1K22 = α2
+ β2 K2 + φ2K22 |
(5) |
|
β1 + 2φ1k2
= β2 + 2φ2k2 |
(7) |
From Eq. 1-3, there are eight parameters
with four restrictions as shown in Eq. 4-7,
therefore, four parameters were estimated. We retain the terminal parameters
being the most recent, hence α2, β2, φ2
and φ2-φ1 were estimated while α1
, β1 , αo and βo were
dropped. φ2-φ1 was estimated in order
to study the transition from one phase to another in the data series.
Equation 4-7 are now redefined in
favour of the dropped parameters viz.:
By inspecting Eq. 4-7, it is obvious
that it is better to start from Eq. 7, respectively
because they have only one term, which we intend to drop i.e.,
(D1) β1 = β2
+ 2φ2k2 - 2φ1k2
= β2 + 2k2(φ2-φ1) |
From Eq. 6, substituting (D1), we obtain
(D2) βo = β2
+ 2K2(φ2-φ1) + 2φ1K2 |
We now estimate α1 from Eq. 5 substituting
(D1) i.e.,
α1 = α2 + β2
K2 + φ2K22 - K2
{β2 + 2φ2k2 - 2φ1k2
= β2 + 2k2(φ2 - φ1)}
- φ1K22 |
(D3) α1 = α2
- K22 (φ2 - φ1) |
Finally we estimate αo by making use of (D1),
(D2) and (D3)
αo = α1 + K1
{β2 + 2φ2k2 - 2φ1k2
= β2 + 2k2(φ2 - φ1)}
+ φ1K12 - K1 {
β2 + 2K2(φ2-φ1)
+ 2φ1K2} |
(D4) αo = α2
- K22 (φ2 - φ1)-
K12φ1 |
|
αo=α2 - K22
(φ2 - φ1)- K12φ1 |
(8) |
|
βo=β2 + 2K2(φ2-φ1)
+ 2φ1K2 |
(9) |
|
α1=α2 - K22
(φ2 - φ1) |
(10) |
|
β1=β2 + 2φ2k2
- 2φ1k2 = β2 + 2k2(φ2
- φ1) |
(11) |
The mean equation can now be obtained by substituting α1
, β1 , αo and βo in Eq.
1-3. From Eq. 1, substituting
for αo and βo
| |
Gdt = α2-K22
(φ2-φ1)- K12φ1
+ t{ β2 + 2K2(φ2-φ1)
+ 2φ1K2} |
| |
(E1)GDt = α2 +
β2 t + (2K2t - K22)
(φ2- φ1) (2K1t - K12)
φ1, t <= K1 |
From Eq. 2, substituting for α1 and
β1,
| |
GDt= α2 - K22
(φ2 - φ1 + t{β2 +
2k2(φ2 - φ1)} + φ2t2 |
| |
(E2)GDt = α2 +
β2 t + (2K2t - K22)
(φ2- φ1)+ φ2t2
, K1< t ≤ K2 |
From Eq. 3, all coefficients were retained for forecasting
purposes.
| |
(E3)GDt = α2 +
β2t + φ2t2, t>K2 |
The grafted Eq. 12-14, are then
formed by inspection of (E1), (E2) and (E3)
above. The mean equation is continuous on the data set:
|
 |
(12) |
|
 |
(13) |
|
 |
(14) |
Equation 12-14, are then formed
into a single equation for estimation as follows:
|
GDt=μoZo + μ1Z1
+ μ2Z2 + μ3Z3
+ μ3Z3+ μ4Z4+
Ut |
(15) |
| Zo |
= |
|
| Z1 |
= |
|
| Z2 |
= |
|
|
= |
|
|
= |
|
| Z3 |
= |
|
|
= |
 |
|
= |
|
| Z4 |
= |
|
|
= |
 |
|
= |
 |
| μ's |
= |
Structural parameters to be estimated |
| Ut |
= |
Error term assumed to be well behaved |
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