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Ka-fu WongUniversity of Hong Kong
Modeling and Forecasting Seasonality
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Unobserved components model of time series
According to the unobserved components model of a time series, the series yt has three components
yt = Tt + St + Ct
Time trend
Seasonal component
Cyclical component
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Seaonality
Seasonality refers to the annual cyclical variation in a time series, which may due to weather patterns, holiday patterns, school calendar patterns, etc. Cyclical peaks in U.S. retail sales and
employment during the last quarter of each calendar year due to the holiday shopping season
Cyclical troughs in U.S. housing starts during the winter months of each calendar year due to weather patterns.
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Gasoline sales in the United States
People tend to travel more in the summer.Sales = quantity x price.
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Liquor Sales in the United States
People tend to drink more liquor during holiday seasons, e.g., Christmas.
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Durable goods sales in the United States
People tend to spend on non-durables and less on durables during holiday seasons.
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Electricity consumption (Terajoule)
0
4000
8000
12000
16000
20000
1970 1975 1980 1985 1990 1995 2000 2005
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Electricity consumption (Terajoule)
8000
10000
12000
14000
16000
18000
20000
2000 2001 2002 2003 2004 2005 2006
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Local Consumption of Gas
0
400
800
1200
1600
2000
2400
2800
1975 1980 1985 1990 1995 2000 2005
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Total Restaurant receipts (millions HK$)
2000
4000
6000
8000
10000
12000
14000
16000
18000
84 86 88 90 92 94 96 98 00 02 04 06
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Hong Kong interbank rate (Overnight and Six Month)
0
1
2
3
4
5
6
7
8
2000 2001 2002 2003 2004 2005 2006
OVERNIGHT SIXMTH
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(Overnight – 6 month)
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2000 2001 2002 2003 2004 2005 2006
13
-8
-6
-4
-2
0
2
4
6
82 84 86 88 90 92 94 96 98 00 02 04 06
(Overnight – 6 month)
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Trade weighted effective exchange rate (January 2000 = 100)
92
94
96
98
100
102
104
106
108
2001 2002 2003 2004 2005 2006
TRADEW IMPORTW EXPORTW
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Hong Kong’s unadjusted Composite CPI
96
98
100
102
104
106
108
2000 2001 2002 2003 2004 2005 2006
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Hong Kong’s unadjusted Composite CPI
50
60
70
80
90
100
110
120
1990 1992 1994 1996 1998 2000 2002 2004 2006
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Hong Kong’s unadjusted Composite CPI
0
20
40
60
80
100
120
1975 1980 1985 1990 1995 2000 2005
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Seasonal Adjustment
Most of the time series that you have selected for your study have been seasonally adjusted. That is,
tttSA Syy ˆ,
an estimate of the seasonal component of yt.
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Why use seasonally adjusted data
Typically, our interest in a macroeconomic time series is in the information it provides about the overall state of the economy and the direction the economy is heading.
Suppose we did not seasonally adjust our data we observe a huge increase in retail sales during the
fourth quarter of 2005. Should we interpret this as a sign that the economy is suddenly booming? No, unless retail sales are growing by more than
normal for that part of the year. we observe a huge increase in the unemployment rate
during May/June of 2006 (after schools let out and there is a large temporary increase in the economy’s labor force). Should I interpret this as a sign that the economy and its labor market are suddenly deteriorating?No, unless the unemployment rate is increasing by
more than normal for that part of the year.
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Why use seasonally adjusted data
Seasonally adjusted data are meant to smooth out the data, i.e., to remove the regular ups and downs that are associated with the seasonal cycle.
So, if seasonally adjusted retail sales increase during the fourth quarter or if the seasonally adjusted unemployment rate increases during May/June we can interpret these as movements beyond the movements that are part of the normal seasonal cycle.
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Should use seasonally un-adjusted data?
There are many business forecasting settings where the seasonal component of the series is fundamentally important and seasonally adjusted data would be inadequate and inappropriate.
Some forecasters will be interested in predicting housing starts including the seasonal component. A bank may be interested in forecasting housing starts in
its area (or, if the bank is large enough, at a national level) in order to anticipate the demand for mortgage loans.
A business that provides building supplies to homebuilders may need to forecast housing starts to anticipate the demand for its products and make current inventory decisions.
In these cases, you would want to use seasonally unadjusted data, model the seasonality and forecast it, along with forecasts of the trend and cyclical components.
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Should use seasonally un-adjusted data?
Would anyone be interested in predicting the number of birth by Hong Kong residents? Immigration Department Hospital Authority Private hospitals Restaurant owners
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Modeling seasonal components as deterministic
Two approaches to modeling seasonality – deterministic seasonality and stochastic seasonality
The two approaches differ according to whether St is perfectly predictable or is subject to random disturbances.
We will assume that the seasonal component is deterministic.
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Modeling seasonality with dummies
A straightforward and commonly used approach to modeling seasonality (which is, however, not the method government agencies typically use to seasonally adjust data) is to specify a seasonal dummy model.
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Modeling quarterly seasonality with dummies
Suppose you are working with quarterly data and want to allow each quarter to have a distinct seasonal effect on the series.
ConsiderSt = 1D1t + 2D2t + 3D3t + 4D4t
where D1t = 1 if t = quarter 1, and 0 otherwise
D2t = 1 if t = quarter 2, and 0 otherwise
D3t = 1 if t = quarter 3, and 0 otherwise
D4t = 1 if t = quarter 4, and 0 otherwise
So, St = 1 if t = quarter 1,…, St = 4 if t = quarter 4.
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Modeling quarterly seasonality with dummies
An equivalent model – St = 1 + 2D2t + 3D3t + 4D4t
where D2t = 1 if t = quarter 2, and 0 otherwise
D3t = 1 if t = quarter 3, and 0 otherwise
D4t = 1 if t = quarter 4, and 0 otherwise
So, St = 1 if t = quarter 1,…, St = 1 + 4 if t = quarter 4.
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Modeling seasonality with dummies
Suppose you are working with monthly data and want to allow each month to have a distinct seasonal effect on the series.
Consider
St = 1D1t + 2D2t + … + 12D12t
where
Dit = 1 if t = month i, and 0 otherwise
{Or St = 1+ 2D2t + … + 12D12t}
For example, St = 1 if t = January (month 1),…, St = 12 if t = December (month 12).
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Modeling seasonality with dummies
Suppose you are working with monthly data and want to allow each quarter to have a distinct seasonal effect on the series.
ConsiderSt = 1D1t + 2D2t + 3D3t + 4D4t
where Dit = 1 if month t is in i, and 0 otherwise
{Or St = 1 + 2D2t + 3D3t + 4D4t}
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Modeling seasonality with dummies
Suppose you are working with data, say monthly data, and you think that there is only one month that is different from the other months. Say, for example, that the seasonality is the same for months 1-11 but is different from the seasonality for month 12-
St = 1D1t + 2D2t (or, St = 1 + 2D2t)where D1t = 1 if t is in month 1,2,…,11, and 0 if t is in month 12andD2t = 1 if t is in month 12 and 0 if t is in month 1,2,…,11
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Modeling seasonality with dummies
Many monetary and financial time series are weekly or even daily (or hourly or…). if we are working with weekly or daily data we normally would not specify a separate seasonal for each week or day. We could, however, still allow for monthly or quarter seasonal variation in our weekly data. For example to allow for a monthly seasonal pattern in our weekly series:
St = 1D1t + 2D2t + … + 12D12t
Dit = 1 if week t falls in the month i, and 0 otherwise
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Modeling seasonality with dummies
More generally, regardless of the frequency of the data (quarterly, monthly, weekly, …) we can divide the year up into s seasons and define the seasonal component according to
St = γ1D1t + γ2D2t + … + γsDst
Dit = 1 if observation t falls in annual period i, and 0 otherwise
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Extensions
Suppose we are working with monthly data and we are allowing for quarterly seasonal variation but also want allow for a December effect – St = 1D1t + 2D2t + … + 4D4t + sDDECt
where DDECt =1 if t is a December, 0 otherwise.
Then, St = 1 for t = January, February, March
St = 2 for t = April, May, June
St = 3 for t = July, August, September
St = 4 for t = October and November
St = 4+ 5 for December.
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Extensions
Dummies can be use to capture other seasonal variations holiday variation
Tourist/hotel occupancies trading-day variation
opening hours of the trading day closing hour of the trading dayJapanese lunch time in modeling forex movements
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Trend and Seasonal components together
To estimate the seasonal model and the seasonal component of yt it is most straightforward if we detrend and deseasonalize simultaneously –
Let Tt = β0 + β1t + … + βptp
and St = 1 + 2D2t … + sDs,t
so that
yt = β0 + β1t + … + βptp + 1 + 2D2t … + sDs,t + εt
where εt is the cyclical component of yt.
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Trend and Seasonal components together
Note that the model yt = β0 + β1t + … + βptp + 1 + 2D2t … + sDs,t + εt
has a redundant parameter, since it has two constants, β0
and γ1.
According to this model of trend and seasonality, the nature of the seasonality is that the intercept of the trend line differs for each season.
Eliminating 1 from the model has no consequence since the intercept for season 1 is accounted for by β0. So the model can be written
yt = β0 + β1t + … + βptp + 2D2t … + sDs,t + εt
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Trend and Seasonal components together
Consider three models with trend and seasonal dummies:(1) yt = β0 + β1t + 1D1t+ 2D2t + εt
(2) yt = β0 + β1t + 2D2t + εt
(3) yt = β1t + 1D1t+ 2D2t + εt
Where D1t =1 when t falls in the first half of the year and 0 otherwise. D2t =1 when t falls in the second half of the year and 0 otherwise.
The expected seasonal effects from the three models are(1) (2) (3)
First half of the year β0 +1 β0 1
Second half of the year
β0 +2 β0 +2 2
Model (1) needs three parameters to obtain the seasonal effects of the two seasons.
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Trend and Seasonal components together
Consider three models with trend and seasonal dummies:(1) yt = β0 + β1t + 1D1t+ 2D2t + εt
(2) yt = β0 + β1t + 2D2t + εt
(3) yt = β1t + 1D1t+ 2D2t + εt
Where D1t =1 when t falls in the first half of the year and 0 otherwise. D2t =1 when t falls in the second half of the year and 0 otherwise.
In estimation of (1), all statistical programs will return error (say, singular matrix encountered) because the variables included are perfectly colinear, or, some set of variables are linearly dependent.
Rewrite(1) yt = β0*1 + β1t + 1D1t+ 2D2t + εt
One can easily show that 1 = D1t + D2t, showing that the three numbers (1,D1t,D2t) really contain two pieces of independent information. Similar to “using two equations to solve three variables is impossible,” we cannot identify all the parameters in (1).
So, one of them has to be dropped – i.e., (2) or (3).
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Trend and Seasonal components together
We can estimate the parameters of this model by OLS, regressing y on a constant, t,…,tp, D2t,…,Dst, for t = 1,…,T yielding:
tststp
pt DDtty ˆˆ...ˆˆ...ˆˆ2210
where the β-hats are the OLS estimates of the β’s and the ε-hats are the OLS residuals.
)ˆ...ˆˆ...ˆˆ(ˆ 2210 ststp
ptt DDtty and form the estimated cyclical component of the yt’s .
Question – After we have estimated this model, how would we test whether there is any seasonality in the original y series?
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Forecasting yT+h
According to our model,yT+h = β0 + β1(T+h) + … + βp(T+h)p + 1 + 2D2,T+h +… + sDs,T+h + εT+h
and so our forecast of yT+h formed at time T will be:
ThThTsshTp
pThT DDhThTy ,,,2210, ˆˆ...ˆ)(ˆ...)(ˆˆˆ
where is our forecast of εT+h formed at time T.ThT ,ˆ
If the ε’s are i.i.d. with mean zero, then = 0 andThT ,ˆ
hTsshTp
pThT DDhThTy ,,2210, ˆ...ˆ)(ˆ...)(ˆˆˆ
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Forecasting yT+h
If the ε’s are i.i.d. N(0,σ2) then, ignoring parameter uncertainty,
)1,0(ˆ
ˆ , Nyy ThThT
T
ttspT 1
22 ˆ)(
1ˆ
96.1ˆ , ThTy is a 95% forecast interval for yT+h
Hence:
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Example: Forecasting Housing Starts
Housing Starts of the United States, 1946:01 – 1994:11
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Example: Forecasting Housing Starts
Housing Starts of the United States, 1990:01 – 1994:11
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Model monthly seasonality
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Check the residuals
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Estimated seasonal factors
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Comparison of forecast and realization 1994:01 to 1994:11
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Example: Forecasting Hong Kong’s port cargo throughput (thousand tonnes)
10000
20000
30000
40000
50000
60000
70000
97 98 99 00 01 02 03 04 05 06
LOADED DISCHARGED TOTAL
Data: http://www.censtatd.gov.hk/hong_kong_statistics/statistics_by_subject/index.jsp
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Linear Trend only
Variable Coefficient Std. Error t-Statistic Prob.
C 38344.37 928.837 41.28213 0
TIME 517.6807 46.9733 11.02074 0
R-squared 0.786348 Mean dependent var 47144.94
Adjusted R-squared 0.779874 S.D. dependent var 5982.039
S.E. of regression 2806.63 Akaike info criteri 18.7728
Sum squared resid 2.60E+08 Schwarz criterion 18.86168
Log likelihood -326.524 F-statistic 121.4568
Durbin-Watson stat 1.027401 Prob(F-statistic) 0
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Forecast based on a linear trend
35000
40000
45000
50000
55000
60000
65000
97 98 99 00 01 02 03 04 05 06
TOTALYHAT1
UPPER1LOWER1
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Residual plot
-8000
-4000
0
4000
8000
35000
40000
45000
50000
55000
60000
1997 1998 1999 2000 2001 2002 2003 2004 2005
Residual Actual Fitted
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Seasonal variation only
Variable Coefficient Std. Error t-Statistic Prob.
C 44395.89 2003.134 22.16322 0
D2 4081.333 2832.859 1.440712 0.1597
D3 3934.667 2832.859 1.388938 0.1747
D4 3009.111 2920.044 1.030502 0.3107
R-squared 0.079875 Mean dependent var 47144.94
Adjusted R-squared -0.00917 S.D. dependent var 5982.039
S.E. of regression 6009.402 Akaike info criteri 20.34725
Sum squared resid 1.12E+09 Schwarz criterion 20.525
Log likelihood -352.077 F-statistic 0.897025
Durbin-Watson stat 0.103033 Prob(F-statistic) 0.45375
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Residual plot
-8000
-4000
0
4000
8000
12000
35000
40000
45000
50000
55000
60000
97 98 99 00 01 02 03 04 05
Residual Actual Fitted
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Forecast based on seasonal variations
30000
35000
40000
45000
50000
55000
60000
65000
97 98 99 00 01 02 03 04 05 06
TOTALYHAT3
UPPER3LOWER3
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Linear time trend and seasonal variations
Variable Coefficient Std. Error t-Statistic Prob.
C 36230.54 1089.544 33.25294 0
TIME 510.3345 42.71352 11.94784 0
D2 3570.999 1200.8 2.973851 0.0058
D3 2913.998 1203.076 2.422122 0.0217
D4 2498.777 1237.71 2.018871 0.0525
R-squared 0.840211 Mean dependent var 47144.94
Adjusted R-squared 0.818905 S.D. dependent var 5982.039
S.E. of regression 2545.669 Akaike info criteri 18.65374
Sum squared resid 1.94E+08 Schwarz criterion 18.87593
Log likelihood -321.44 F-statistic 39.43682
Durbin-Watson stat 0.48952 Prob(F-statistic) 0
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Residual plots
-6000
-4000
-2000
0
2000
4000
6000
36000
40000
44000
48000
52000
56000
60000
1997 1998 1999 2000 2001 2002 2003 2004 2005
Residual Actual Fitted
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Forecast based on linear time trend and seasonal variations
35000
40000
45000
50000
55000
60000
65000
97 98 99 00 01 02 03 04 05 06
TOTALYHAT2
UPPER2LOWER2
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End
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