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1 Decomposition Method

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2 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

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3 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

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4 Time Series vs Causal Modeling On the other hand, there are many cases when one cannot, or one prefers not to, build causal models: 1.insufficient information is known about the behavioural relationship; 2.lack of, or conflicting, theories; 3.insufficient data on explanatory variables; 4.expertise may be unavailable; 5.time series models may be more accurate

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5 Time Series vs Causal Modeling Direct benefits of using time series models: 1.Little storage capacity is needed; 2.some time series models are automatic in that user intervention is not required to update the forecasts each period; 3.some time series models are evolutionary in that the models adapt as new information is received;

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6 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;

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7 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

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

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9 Examples: (1)US Retail and Food Services Sales from 1996 Q1 to 2008 Q1 Multiplicative Decomposition (2)Quarterly Number of Visitor Arrivals in Hong Kong from 2002 Q1 to 2008 Q1 Figure 2.1 Figure 2.2

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10 Figure 2.1US Retail Sales Back

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11 Figure 2.2Visitor Arrivals

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

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13 Illustration : Consider the following 4-year quarterly time series on sales volume: Period (t)YearQuarterSales 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 12341234 12341234123412341234123412341234 72 110 117 172 76 112 130 194 78 119 128 201 81 134 141 216

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14 Figure 2.3

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15 Step 1 : Estimation of seasonal component (SN t ) Y t = TC t SN t IR t Moving Average for periods 1 – 4 Moving Average for periods 2 – 5

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16 Period (t)YearQuarterSalesMA (t) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 12341234 12341234123412341234123412341234 72 110 117 172 76 112 130 194 78 119 128 201 81 134 141 216 117.75 118.75 119.25 122.5 128 128.5 130.25 129.75 131.5 132.25 136 139.25 143

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17 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.

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18 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 Centered Moving Average for period 4

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19 Period (t)YearQuarterSalesMA (t)CMA(t) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 12341234 12341234123412341234123412341234 72 110 117 172 76 112 130 194 78 119 128 201 81 134 141 216 117.75 118.75 119.25 122.5 128 128.5 130.25 129.75 131.5 132.25 136 139.25 143 118.25 119 120.875 125.25 128.25 129.375 130 130.625 131.875 134.125 137.625 141.125

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20 Because the CMA t contains no seasonality and irregularity, the seasonal component may be estimated by

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21 Period (t)YearQuarterSalesMA (t)CMA(t) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 12341234 12341234123412341234123412341234 72 110 117 172 76 112 130 194 78 119 128 201 81 134 141 216 117.75 118.75 119.25 122.5 128 128.5 130.25 129.75 131.5 132.25 136 139.25 143 118.25 119 120.875 125.25 128.25 129.375 130 130.625 131.875 134.125 137.625 141.125 0.989429175 1.445378151 0.628748707 0.894211577 1.013645224 1.499516908 0.6 0.911004785 0.970616114 1.49860205 0.588555858 0.949512843

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22 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?)

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23 Quarter Average 1 (0.628748707 + 0.6 + 0.588555858)/3= 2 (0.894211577 + 0.911004785 + 0.949512843)/3= 3 (0.989429175 + 1.013645224 + 0.970616114)/3= 4 (1.445378151 + 1.499516908 + 1.49860205)/3= Sum =

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24 Step 2 : Estimation of Trend/Cycle Define deseasonalized (or seasonally adjusted) series as for example, D 1 = 72/0.6063 = 118.7506

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26 TC t may be estimated by regression using a linear trend: where b 0 and b 1 are least squares estimates of 0 and 1 respectively.

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27 EXCEL regression output :

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28 For example,

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

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31 Out of sample forecast :

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33 Figure 2.4

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34 Measuring Forecast Accuracy : 1)Mean Squared Error 2)Mean Absolute Deviation

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35 Method AMethod B e t = – 2 – 4 1.50.7 –10.5 2.11.4 0.70.1 Method A:MSE=2.43 MAD=1.46 Method B:MSE=3.742 MAD=1.34

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36 Naive Prediction 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. Theil’s u Statistics

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37 MSE = 11.932MAD = 2.892Theil’s U = 0.0546

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38 Out-of-Sample Forecasts 1)Expost forecast Prediction for the period in which actual observations are available 2)Exante forecast Prediction for the period in which actual observations are not available.

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39 T1T1 T2T2 T3T3 estimation period (today) Time “back” casting in-sample simulation Ex-post forecast Ex-ante forecast

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40 Additive Decomposition Time Trend YtYt YtYt (Multiplicative Seasonality) (Additive Seasonality)

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41 Multiplicative decomposition is used when the time series exhibits increasing or decreasing seasonal variation (Y t =TC t SN t IR t ) TC t SN t YtYt Y t – Y t-1 Yr 1Q1 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 2Q1 Q2 Q3 Q4 17.5 19 20.5 22 1.5 0.5 0.8 1.2 26.25 9.5 16.4 26.4 –16.75 6.9 10

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

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43 Step 1 : Estimation of seasonal component (SN t ) Calculation of MA t and CMA t is the same as per multiplicative decomposition Initial seasonal component may be estimated by For example,

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

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

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47 i.e.,D t = o + 1 t + t and Multiplicative decomposition

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48 So, and For example, and

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49 MSE = 27.911 MAD = 4.477

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