CHAPTER 1: Decomposition Methods

Transcription

1 CHAPTER 1: Decomposition Methods Prof. Alan Wan 1 / 48

2 Table of contents 1. Data Types and Causal vs.time Series Models 2 / 48

3 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: Quarterly Gross Domestic Product (GDP), Annual rainfall volume, daily stock market index, etc. 3 / 48

4 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: Quarterly Gross Domestic Product (GDP), Annual rainfall volume, daily stock market index, etc. Cross sectional data: data on one or more variables collected at the same point in time. 3 / 48

5 Causal vs. Time Series Models Causal (regression) models: the investigator specifies some behaviourial relationship and estimates the unknown parameters using regression techniques. 4 / 48

6 Causal vs. Time Series Models Causal (regression) models: the investigator specifies some behaviourial relationship and estimates the unknown parameters using regression techniques. Time series models: the investigator uses past data of the target variable to forecast the present and future values of the variable. 4 / 48

7 Causal vs. Time Series Models Causal models provide information on the causal relationship between the target variable and its determinants (the regressors). 5 / 48

8 Causal vs. Time Series Models Causal models provide information on the causal relationship between the target variable and its determinants (the regressors). On the other hand, there are many instances when one cannot, or prefers not to, construct causal models due to reasons such as 1. insufficient information on the behaviourial relationship 2. lack of, or conflicting, theories 3. insufficient data on the explanatory variables 4. superior forecasts produced by time series models 5 / 48

9 Causal vs. Time Series Models Here are some of the 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 6 / 48

10 Causal vs. Time Series Models Here are some of the 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 This course is mainly concerned with forecasting using time series models 6 / 48

11 Classical Decomposition of Time Series Trend (TC) - does not necessarily imply a monotonically increasing or decreasing series but simply a lack of constant mean, although in practice, we often use a linear or quadratic function to predict the trend. 7 / 48

12 Classical Decomposition of Time Series Trend (TC) - does not necessarily imply a monotonically increasing or decreasing series but simply a lack of constant mean, although in practice, we often use a linear or quadratic function to predict the trend. Cycle (CL) - 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. 7 / 48

13 Classical Decomposition of Time Series Trend (TC) - does not necessarily imply a monotonically increasing or decreasing series but simply a lack of constant mean, although in practice, we often use a linear or quadratic function to predict the trend. Cycle (CL) - 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. Seasonal (SN) - refers to a cycle of one year s duration. 7 / 48

14 Classical Decomposition of Time Series Trend (TC) - does not necessarily imply a monotonically increasing or decreasing series but simply a lack of constant mean, although in practice, we often use a linear or quadratic function to predict the trend. Cycle (CL) - 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. Seasonal (SN) - refers to a cycle of one year s duration. Random (Irregular) (IR) - refers to the (unpredictable) variations not covered by the three components above. 7 / 48

15 Classical Decomposition of Time Series We are concerned with two types of Decomposition Models: Multiplicative Model: Y t = TC t SN t CL t IR t 8 / 48

16 Classical Decomposition of Time Series We are concerned with two types of Decomposition Models: Multiplicative Model: Y t = TC t SN t CL t IR t Additive Model: Y t = TC t + SN t + CL t + IR t 8 / 48

17 Classical Decomposition of Time Series We are concerned with two types of Decomposition Models: Multiplicative Model: Y t = TC t SN t CL t IR t Additive Model: Y t = TC t + SN t + CL t + IR t The goal is to find estimates of the four components. 8 / 48

18 Multiplicative Decomposition Model Example 1: U.S. Retail and Food Services Sales from 1996Q1 ro 2008Q1: US Retail & Food Services Sales 500, ,000 Sales Y(t) (in MN US$) 400, , , , , , ,000 50,000 0 Q1-96 Q3-96 Q1-97 Q3-97 Q1-98 Q3-98 Q1-99 Q3-99 Q1-00 Q3-00 Q1-01 Q3-01 Q1-02 Q3-02 Q1-03 Q3-03 Q1-04 Q3-04 Q1-05 Q3-05 Q1-06 Q3-06 Q1-07 Q3-07 Q1-08 Ti m e 9 / 48

19 Multiplicative Decomposition Model Example 2: Quarterly Number of Visitor Arrivals in Hong Kong from 2002Q1 to 2008Q1: Number of Visitor Arrivals in Hong Kong Number of Visitors Y(t) Q1-02 Q3-02 Q1-03 Q3-03 Q1-04 Q3-04 Q1-05 Q3-05 Q1-06 Q3-06 Q1-07 Q3-07 Q1-08 Time 10 / 48

20 Multiplicative Decomposition Model Cycles are often difficult to identify with a short time series. Classical decomposition typically combines cycles and trend as one entity, that is, Y t = TC t SN t IR t 11 / 48

21 Multiplicative Decomposition Model Consider the following 4-year quarterly time series on sales volume: 12 / 48

22 Multiplicative Decomposition Model A plot of the series reveals the following pattern: 13 / 48

23 Multiplicative Decomposition Model We first estimate the seasonal component (SN t ). 14 / 48

24 Multiplicative Decomposition Model We first estimate the seasonal component (SN t ). Note that Y t = TC t SN t IR t SN t = Yt TC t IR t 14 / 48

25 Multiplicative Decomposition Model We first estimate the seasonal component (SN t ). Note that Y t = TC t SN t IR t SN t = Yt TC t IR t Moving Average for periods 1 4 = = Moving Average for periods 2 5 = = / 48

26 Multiplicative Decomposition Model Assuming that the average of the observations is also the median of the observations, the moving average (MA) for periods 1-4, 2-5, 3-6 are centered at t = 2.5, 3.5 and 4.5 respectively. 15 / 48

27 Multiplicative Decomposition Model To obtain averages that center at periods 3, 4, 5, etc. we calculate the mean of every two consecutive moving averages as follows: 16 / 48

28 Multiplicative Decomposition Model To obtain averages that center at periods 3, 4, 5, etc. we calculate the mean of every two consecutive moving averages as follows: Centered Moving Average for period 3 = = Centered Moving Average for period 4 = = / 48

29 Multiplicative Decomposition Model 17 / 48

30 Multiplicative Decomposition Model Because the centered moving average (CMA) contains no seasonality and no or little irregularity, the seasonal component may be estimated by SN t = Yt CMA t 18 / 48

31 Multiplicative Decomposition Model Because the centered moving average (CMA) contains no seasonality and no or little irregularity, the seasonal component may be estimated by SN t = Yt CMA t For example, SN 3 = = SN 4 = = / 48

32 Multiplicative Decomposition Model 19 / 48

33 Multiplicative Decomposition Model After all the SN t s 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?) 20 / 48

34 Multiplicative Decomposition Model J~...., ~r r :-... -~::.~;if~":~ "';i;:; -~;''.;~1:}~{'.' ~ I\ Period (t) Year Quarter Sales MA(t) CMA(t) SN(t) SN(t) A. Quarter Average Final SN(t) 1 ( )/3 = f ~ = ( )/3 = ( )/3 = Sum= Normalizing Factor: 4/ = / 48

35 Multiplicative Decomposition Model We next estimate the trend (TC t ). 22 / 48

36 Multiplicative Decomposition Model We next estimate the trend (TC t ). Define the deseasonalized or seasonally adjusted series as: D t = Yt ŜN t For example, D 1 = = / 48

37 Multiplicative Decomposition Model 23 / 48

38 Multiplicative Decomposition Model Plot of D(t) against t / 48

39 Multiplicative Decomposition Model TC t may be estimated by regression based on a linear trend. Write D t = β 0 + β 1 t + ε, t = 1, 2,, n. 25 / 48

40 Multiplicative Decomposition Model TC t may be estimated by regression based on a linear trend. Write D t = β 0 + β 1 t + ε, t = 1, 2,, n. Then the estimated trend is TC t = ˆD t = b 0 + b 1 t, where b 0 and b 1 are the least squares estimators of β 0 and β 1 respectively. 25 / 48

41 Multiplicative Decomposition Model For this data set, TC t = t 26 / 48

42 Multiplicative Decomposition Model The predicted values of TC may be computed by substituting the relevant values of t into the estimated trend equation. For example, TC 1 = (1) = TC 2 = (2) = / 48

43 Multiplicative Decomposition Model 28 / 48

44 Multiplicative Decomposition Model One can then compute the forecasted values of Y t by: Ŷ t = TC t ŜN t 29 / 48

45 Multiplicative Decomposition Model One can then compute the forecasted values of Y t by: Ŷ t = TC t ŜN t In-sample fitted values: Ŷ 1 = = Ŷ16 = = / 48

46 Multiplicative Decomposition Model Out-of-sample forecasts: Ŷ 17 = TC 17 ŜN 17 = [ (17)] = = Ŷ 18 = TC 18 ŜN 18 = [ (18)] = = / 48

47 Multiplicative Decomposition Model 31 / 48

48 Multiplicative Decomposition Model 32 / 48

49 Measuring Forecast Accuracy Let e t = Y t Ŷt be the forecast error. 33 / 48

50 Measuring Forecast Accuracy Let e t = Y t Ŷt be the forecast error. Mean Squared Error (MSE) MSE = n t=1 e2 t /n RMSE = MSE 33 / 48

51 Measuring Forecast Accuracy Let e t = Y t Ŷt be the forecast error. Mean Squared Error (MSE) MSE = n t=1 e2 t /n RMSE = MSE Mean Absolute Deviation (MAD) MAD = n t=1 e t /n RMAD = MAD 33 / 48

52 Measuring Forecast Accuracy Method A Method B e t = Method A: MSE = 2.43, MAD = 1.46 Method B: MSE = 3.742, MAD = / 48

53 Measuring Forecast Accuracy Naive prediction - use the last period actual value to predict the next period s unknown, i.e.,use Y t 1 to predict Y t. 35 / 48

54 Measuring Forecast Accuracy Naive prediction - use the last period actual value to predict the next period s unknown, i.e.,use Y t 1 to predict Y t. Theil s U Statistic: U = (Yt Ŷt)2 /n (Yt Y t 1 ) 2 /n 35 / 48

55 Measuring Forecast Accuracy Naive prediction - use the last period actual value to predict the next period s unknown, i.e.,use Y t 1 to predict Y t. Theil s U Statistic: U = (Yt Ŷt)2 /n (Yt Y t 1 ) 2 /n if U = 1 forecasts produced are no better than naive forecasts; if U = 0 forecasts produced perfect fit 35 / 48

56 Measuring Forecast Accuracy Naive prediction - use the last period actual value to predict the next period s unknown, i.e.,use Y t 1 to predict Y t. Theil s U Statistic: U = (Yt Ŷt)2 /n (Yt Y t 1 ) 2 /n if U = 1 forecasts produced are no better than naive forecasts; if U = 0 forecasts produced perfect fit U is expected to lie between 0 and 1 - the smaller the value of U, the better the forecasts 35 / 48

57 Measuring Forecast Accuracy For the model used in our last example, MSE = , MAD = and U = / 48

58 Types of Forecasts Expost forecast - Prediction for the period in which the actual observation is available Exante forecast - Prediction for the period in which the actual observation is not available 37 / 48

59 Types of Forecasts 38 / 48

60 Additive Decomposition Model The diagrams in the top and bottom panels depict situations of multiplicative seasonality and additive seasonality respectively. 39 / 48

61 Additive Decomposition Model Multiplicative decomposition (Y t = TC t SN t IR t ) is used when the time series exhibits seasonal variations that follow the trend (multiplicative seasonality). For example, 40 / 48

62 Additive Decomposition Model Additive decomposition (Y t = TC t + SN t + IR t ) is used when the time series exhibits seasonal variations that are constant and do not follow the trend (additive seasonality). For example, 41 / 48

63 Additive Decomposition Model To construct the additive model, we first calculate MA t and CMA t as per multiplicative decomposition. 42 / 48

64 Additive Decomposition Model To construct the additive model, we first calculate MA t and CMA t as per multiplicative decomposition. The initial seasonal component may be estimated by SN t = Y t CMA t. For example, using our previous data set, SN 3 = = 1.25 SN 4 = = / 48

65 Additive Decomposition Model The initial seasonal indices are then averaged and adjusted so that they sum to zero (Why?) 43 / 48

66 Additive Decomposition Model 44 / 48

67 Additive Decomposition Model The seasonally adjusted series is D t = Y t ŜN t. 45 / 48

68 Additive Decomposition Model The seasonally adjusted series is D t = Y t ŜN t. TC t may be estimated by regression as per multiplicative decomposition, i.e., D t = β 0 + β 1 t + ε, t = 1, 2,, n. and TC t = ˆD t = b 0 + b 1 t 45 / 48

69 Additive Decomposition Model 46 / 48

70 Additive Decomposition Model So, TC t = t and Ŷ t = TC t + ŜN t For example, TC 1 = (1) = and Ŷ 1 = = / 48

71 Additive Decomposition Model MSE = and MAD = / 48