# Decomposition Method.

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Decomposition Method

Types of Data Time series data: a sequence of observations measured over time (usually at equally spaced intervals, e.g., weekly, monthly and annually). Examples of time series data include: Gross Domestic Product each quarter; annual rainfall; daily stock market index Cross sectional data: data on one or more variables collected at the same point in time

Time Series vs Causal Modeling
Causal (regression) models: the investigator specifies some behavioural relationship and estimates the parameters using regression techniques; Time series models: the investigator uses the past data of the target variable to forecast the present and future values of the variable

Time Series vs Causal Modeling
On the other hand, there are many cases when one cannot, or one prefers not to, build causal models: insufficient information is known about the behavioural relationship; lack of, or conflicting, theories; insufficient data on explanatory variables; expertise may be unavailable; time series models may be more accurate

Time Series vs Causal Modeling
Direct benefits of using time series models: Little storage capacity is needed; some time series models are automatic in that user intervention is not required to update the forecasts each period; some time series models are evolutionary in that the models adapt as new information is received;

Classical Decomposition of Time Series
Trend – does not necessarily imply a monotonically increasing or decreasing series but simply a lack of constant mean, though in practice, we often use a linear or quadratic function to predict the trend; Cycle – refers to patterns or waves in the data that are repeated after approximately equal intervals with approximately equal intensity. For example, some economists believe that “business cycles” repeat themselves every 4 or 5 years;

Classical Decomposition of Time Series
Seasonal – refers to a cycle of one year duration; Random (irregular) – refers to the (unpredictable) variation not covered by the above

Decomposition Method Multiplicative Models Additive Models
Find the estimates of these four components.

Multiplicative Decomposition
Examples: (1) US Retail and Food Services Sales from Q1 to 2008 Q1 Figure 2.1 (2) Quarterly Number of Visitor Arrivals in Hong Kong from 2002 Q1 to 2008 Q1 Figure 2.2

Figure 2.1 US Retail Sales Back

Figure 2.2 Visitor Arrivals

Cycles are often difficult to identify with a short time series.
Classical decomposition typically combines cycles and trend as one entity:

Illustration : Consider the following 4-year quarterly time series on sales volume:
Period (t) Year Quarter Sales 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 72 110 117 172 76 112 130 194 78 119 128 201 81 134 141 216

Figure 2.3

Step 1 : Estimation of seasonal component (SNt)
Yt = TCt  SNt  IRt Moving Average for periods 1 – 4 for periods 2 – 5

Period (t) Year Quarter Sales MA (t) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 72 110 117 172 76 112 130 194 78 119 128 201 81 134 141 216 117.75 118.75 119.25 122.5 128.5 130.25 129.75 131.5 132.25 136 139.25 143

Assuming the average of the observations is also the median of the observations, the MA for periods 1 – 4, 2 – 5, 3 – 6 are centered at positions 2.5, 3.5 and 4.5 respectively.

To get an average centered at periods 3, 4, 5 etc
To get an average centered at periods 3, 4, 5 etc. the means of two consecutive moving averages are calculated: Centered Moving Average for period 3 Average for period 4

Period (t) Year Quarter Sales MA (t) CMA(t) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 72 110 117 172 76 112 130 194 78 119 128 201 81 134 141 216 117.75 118.75 119.25 122.5 128.5 130.25 129.75 131.5 132.25 136 139.25 143 118.25 125.25 128.25

Because the CMAt contains no seasonality and irregularity, the seasonal component may be estimated by

Period (t) Year Quarter Sales MA (t) CMA(t) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 72 110 117 172 76 112 130 194 78 119 128 201 81 134 141 216 117.75 118.75 119.25 122.5 128.5 130.25 129.75 131.5 132.25 136 139.25 143 118.25 125.25 128.25 0.6

After all have been computed, they are further averaged to eliminate irregularities in the series. We also adjust the seasonal indices so that they sum to the number of seasons in a year (i.e., 4 for quarterly data, 12 for monthly data). Why?)

Quarter Average 1 ( )/3= 2 ( )/3= 3 ( )/3= 4 ( )/3= Sum =

Step 2 : Estimation of Trend/Cycle
Define deseasonalized (or seasonally adjusted) series as for example, D1 = 72/ =

TCt may be estimated by regression using a linear trend:
where b0 and b1 are least squares estimates of 0 and 1 respectively.

EXCEL regression output :

For example,

Step 3 : Computation of fitted values and out-of-sample forecasts

Out of sample forecast :

Figure 2.4

Measuring Forecast Accuracy :
1) Mean Squared Error Mean Absolute Deviation

Method A Method B et = – 2 – 4 Method A : MSE = 2.43 MAD = 1.46 Method B : MSE = 3.742 MAD = 1.34

Naive Prediction Theil’s u Statistics
if U = 1  Forecasts produced are no better than naive forecast U = 0  Forecasts produced perfect fit The smaller the value of U, the better the forecasts.

MSE = MAD = Theil’s U =

Out-of-Sample Forecasts
Expost forecast Prediction for the period in which actual observations are available Exante forecast Prediction for the period in which actual observations are not available.

T1 T2 T3 Ex-post forecast Ex-ante forecast in-sample simulation
estimation period (today) Time “back” casting in-sample simulation Ex-post forecast Ex-ante forecast

Yt Yt Trend Trend (Multiplicative Seasonality) Time (Additive Seasonality) Time

Multiplicative decomposition is used when the time series exhibits increasing or decreasing seasonal variation (Yt=TCt  SNt  IRt) TCt SNt Yt Yt – Yt-1 Yr 1 Q1 Q2 Q3 Q4 11.5 13 14.5 16 1.5 0.5 0.8 1.2 17.25 6.5 11.6 19.2 –10.75 5.1 7.6 Yr 2 17.5 19 20.5 22 26.25 9.5 16.4 26.4 –16.75 6.9 10

Additive decomposition is used when the time series exhibits constant seasonal variation (Yt=TCt + SNt + IRt) TCt SNt Yt Yt – Yt-1 Yr 1 Q1 Q2 Q3 Q4 11.5 13 14.5 16 1.8 –1 –1.5 0.7 13.3 12 16.7 –1.3 1 3.7 Yr 2 17.5 19 20.5 22 19.3 18 22.7

Step 1 : Estimation of seasonal component (SNt)
Calculation of MAt and CMAt is the same as per multiplicative decomposition Initial seasonal component may be estimated by For example,

Seasonal indices are averaged and adjusted so that they sum to zero (Why?)

Step 2 : Estimation of Trend/Cycle
Deseasonalized series is defined as TCt may be estimated by regression as per multiplicative decomposition

i.e., Dt = o + 1t + t and Multiplicative decomposition

So, and For example,