Dr. Harry M. Karamujic
Univariate Analysis of Seasonal
Variations in Building Approvals for
New Houses: Evidence from Australia
Objectives
• The paper examines the impact of seasonal influences on Australian
housing approvals, represented by the State of Victoria (Australia)
building approvals for new houses (BANHs).
• The paper focuses on BANHs as they are seen as a leading indicator of
investment and as such the general level of economic activity and
employment .
• In particular, the paper seeks to cast some additional light on modelling
the seasonal behaviour of BANHs by: (i) establishing the presence, or
otherwise, of seasonality in Victorian BANHs; (ii) if present, ascertaining
weather it is deterministic or stochastic; (iii) determining out of sample
forecasting capabilities of the considered modelling specifications; and (iv)
speculating on possible interpretation of results.
BANHs
•
BANHs denote a number of new houses building work approved.
According to the ABS, statistics of building work approved are
compiled from:
“- permits issued by local government authorities and other principal certifying authorities, and
- contracts let or day labour work authorised by commonwealth, state, semi-government and local
government authorities”.
•
In Australia, a new house (a building which previously did not exist) is
defined as the construction of a detached building that is primarily used
for long term residential purposes. From July 1990, the statistics
includes all approved new residential building valued at $10,000 or
more.
Seasonality
•
The focus of this study is not on modelling the behaviour of time series
in terms of explanatory variables (the conventional modelling
approach). Instead, this study uses a univariate structural time series
modelling approach (allows modelling both stochastic and
deterministic trend and seasonality) and as such shows that
conventional assumptions of deterministic trend and seasonality are not
always applicable.
•
The conventional modelling approach assumes that the behaviour of
the trend and seasonality can be effectively captured by a conventional
regression equation that assumes deterministic trend and seasonality.
•
The paper utilises a basic structural time series model of Harwey
(1989). Compared to the conventional procedure, Harvey’s (1989)
structural time series model involves an explicit modelling of
seasonality as an unobserved component.
Methodology
- Within a structural time series approach, the term ”structural”
implies that a time series (in this paper, BANHs) is observed as a set of
components not observable directly. The approach allows the selected
time series, including intervention variables, to be modelled
simultaneously with the unobserved components. The intention is to
decompose the selected time series in terms of its respective
components and to understand how these components relate to the
underlying
forces
that
shape
its
evolution.
- The empirical analysis uses the model as presented in Harvey (1985,
1990), whereby time series are modelled in terms of their components.
The
model
can
be
written
as:
rt = µt + γt + εt
(1)
where rt represents the actual value of the series at time t, µt is the trend component of
the series, γt is the seasonal component and εt is the irregular component (assumed to
be ‘white noise’).
Methodology
- The major reason for selecting the structural time series modeling
approach is that it allows for both stochastic and deterministic
seasonality .
- Conventional dynamic modeling with a deterministic seasonality
approach totally ignores the likely possibility of stochastic
seasonality (manifested as changing seasonal factors over the
sample period).
- Evidently, a problem with the conventional procedure is that
deterministic seasonality is imposed as a constraint, when in fact it
should be a testable hypothesis .
Results and Discussion
• The structural time series model represented by (1) is applied to seasonally
unadjusted monthly BANHs data for Victoria, between 2000:06 and 2009:05.
• The data have been sourced from the Australian Bureau of Statistics. For
consistency, the sample for each variable is standardised to start with the first
available July observation and end with the latest available June observation.
• As shown in Table 1, The paper considers the following three modelling
specifications:
- Model 1 (Stochastic Trend and Stochastic Seasonality)
- Model 2 (Stochastic Trend and Deterministic Seasonality)
- Model 3 (Deterministic Trend and Deterministic Seasonality)
Results and Discussion
Table 1:Estimated Coefficients of Final State Vector
State
Variable/Test
Statistic
Model 1
(Stochastic Trend and
Stochastic Seasonality)
2533.30
(17.28)
1.17
(0.05)
159.39
(3.87)
242.47
(5.79)
47.67
(1.39)
-97.10
(-2.82)
-113.7
(-3.45)
89.34
(2.71)
134.15
(4.13)
63.14
(1.95)
13.54
(0.42)
5.16
(0.16)
-21.80
(-0.96)
277.23
Model 2
(Stochastic Trend and
Deterministic Seasonality)
2533.30
(17.28)
1.17
(0.05)
288.62
(3.53)
154.24
(1.89)
266.56
(3.28)
31.38
(0.39)
94.81
(1.17)
100.83
(1.25)
-390.32
(-4.82)
-657.19
(-8.12)
-50.03
(-0.62)
20.75
(0.26)
-78.88
(-0.97)
274.39
Model 3
(Deterministic Trend and
Deterministic Seasonality)
2530.80
(34.04)
-0.74
(-0.61)
244.62
(1.98)
119.13
(0.96)
240.09
(1.94)
13.27
(0.11)
84.78
(0.69)
98.63
(0.80)
-384.97
(-3.12)
-644.56
(-5.22)
-30.38
(-0.25)
47.13
(0.38)
-46.02
(-0.37)
364.48
Rs2
0.30
0.31
-0.21
Rd2
0.58
0.59
0.33
DW
2.09
2.09
0.80
Q
9.64
15.08
125.71
t
t
1
2
3
4
5
6
7
8
9
 10
 11
t
N
3.45
6.37
1.791
H
0.37
0.41
0.29
AIC
BIC
11.55
11.94
11.51
11.88
12.04
12.36
Results and Discussion
• With respect to the goodness of fit, all models are relativelly well defined.
• Overall, the diagnostic tests are also predominately passed. The only
exception is the test for serial correlation (Q), for the Model two (which is
slightly above the statistically acceptable level) and Model three (significantly
above the statistically acceptable level). The Q statistics for Model three
indicate that the model suffers from serial correlation, implying a misspecified
model. In all cases the slope is insignificant and the level is significant.
• As shown in Table 1, out of three modelling specifications, Model two has the
highest R2s and lowest ε. On the other hand, Model three (deterministic trend
and deterministic seasonality) with negative R2s implies that the model is badly
determined i.e. the model is worst then a seasonal random walk model.
• Overall, all of goodness of fit measures imply that Model three is significantly
inferior to Models one and two, and that Model two is somewhat better then
Model one.
Results and Discussion
400
400
200
200
09
20
08
20
07
20
06
20
05
20
20
20
09
04
20
08
20
20
07
03
20
06
20
05
02
04
20
20
20
03
01
20
02
20
20
-200
01
20
20
00
00
0
0
-200
-400
-400
-600
-600
-800
-800
Figure 1: Model 1 - Seasonal Component
Figure 2: Model 2 - Seasonal Component
400
400
300
200
June
200
July
100
August
0
September
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
20
09
20
08
20
07
20
06
20
05
20
04
20
03
20
02
20
01
20
00
-100
0
November
-200
-200
October
December
-300
January
-400
-400
February
March
-500
April
-600
-600
May
-700
-800
Figure 3: Model 3 - Seasonal Component
Figure 4: Individual Seasonals
Results and Discussion
• Figures 1, 2 and 3 provide a visual interpretation of the seasonal elements
for each considered modeling specification. The seasonal components
evidenced in each of the figures show a constant repetitive pattern over the
sample period, providing an additional evidence of the deterministic nature of
the seasonal component (fixed seasonal components) in the number of new
dwellings approved in Victoria.
• Figure 4 shows this even more clearly with individual monthly seasonals
represented by horizontal lines, implying an unchanging seasonal effect
across the whole sample period.
• In summary, the analysis points out that the behaviour of BANHs exhibits
stochastic trend and deterministic seasonality. As a result, any regression
model based on assumptions of deterministic trend and seasonality is bound
to be misspecified
Results and Discussion
• Consequently, the interpretation of the modeling results focuses on the
Model two. Out of the eleven seasonal factors relating to the Model two,
presented in Table 1, factors corresponding to June, April, December and
November are found to be significant at 5% level.
• A possible explanation for the observed statistically significant reduction in
BANHs during December and November is the reduction of the level of
activity caused by the ‘summer holidays’ season. (The ‘summer holidays’ season
typically covers the period from the second half of November to the end of January. It is the period of
summer school holiday, several public/religious holidays and the time when most people take annual
leaves.
• On the other hand season-related increases during June and April may be
explained by a spike in the level of activity during ‘the end of financial year’
season and preparation for a surge in contraction activity during the ‘spring’
season (‘The end of financial year’ season typically starts by the end of April or the beginning of May,
and finishes at the end of the first week in July.)
Conclusion
• The modeling focus has been to (i) establishing the presence, or
otherwise, of seasonality in Victorian BANHs, (ii) if present,
ascertaining is it deterministic or stochastic, (iii) determining out of
sample forecasting capabilities of the considered models and (iv)
speculating on possible interpretation of results.
• This is done by estimating three modelling specifications comprised of
stochastic and deterministic trend and seasonal components. The
goodness of fit measures and the diagnostic test statistics indicate that
Model two, which is comprised out of stochastic trend and deterministic
seasonality, is superior to the other two specifications. Furthermore, the
analysis of the three presented modelling specifications evidently
indicates that the conventional modelling approach, characterised by
assumptions of deterministic trend and deterministic seasonality, would
not identify seasonal behaviour of time series characterised by stochastic
trend and/or seasonality.
Conclusion
• The examination of the out-of-sample forecasting power of the three
models clearly shows that the seasonality apparent in the actual data is
well picked up by specifications entailing deterministic seasonal factor,
corroborating the earlier finding that the seasonal pattern in the number
of dwelling units approved in Victoria is deterministic and not stochastic.
• Finally, the analysis of Model two points out that the behaviour of
BANHs exhibits statistically significant seasonal components. A
possible explanation for the observed statistically significant reduction in
BANHs during December and November is the reduction of the level
of activity caused by approaching to the ‘summer holidays’ season,
while the season-related increases during June and April may be
explained by a spike in the level of activity during ‘the end of financial
year’ season and preparation for a surge in contraction activity during
the ‘spring’ season.
Questions