POWERPOINT SHOWING ECONOMICS AND ECONOMETRICS

ECONOMETRICS

1. Unit 5 – Time Series

2. Introduction Time Series consists of numerical data recorded at equal intervals of time and arranged in chronological order. It can be viewed as a sequence of a finite number of observations Y 1 , Y 2 ,…………. Y n of a variable Y made at equally spaced points in time. The restriction of equally spaced times is not a great one given that most business series are measured daily, monthly, quarterly or yearly and so will be equally spaced.

3. Introduction The values in a time series are the resultant of the interplay of large numbers of diverse economic, political, social and other factors. Time Series are a special case of the two variable situation found in regression and correlation in which one variable is always time. Time is therefore the independent variable i.e. Y = f(t) There are two types of time series – discrete (with observations taken at discrete intervals) and continuous (with observations taken at all points in time along a continuum of the real line) e.g. ECG.

4. Introduction The single most important thing to do when first exploring time series data is to visualize the data through graphs. The basic features of the data including patterns and unusual observations are most easily seen through graphs. Sometimes graphs also suggest possible explanations for some of the variation in the data.

5. Time Series Components Generally, time series can be decomposed into four distinct components: The Trend Component The Cyclical Component The Seasonal Component The Irregular Component

6. The Trend Component The Trend Component is the long term change in the level of the data and reflects the tendency of the series to grow or decline. If over an extended period of time, the time series moves upward, we say that the data shows a positive trend. If on the other hand, the data diminishes over time, we say that there is a negative trend. Data is considered stationary when it possesses neither a positive trend nor a negative trend. The trend is usually modelled by a smooth curve.

7. The Seasonal Component The Seasonal Component is defined as a short term recurring pattern within the data that completes itself within the period of a calendar year and then continues in a repetition of this basic pattern. In a manner of speaking, it is a regular variation in the level of the data that repeats itself at the same time each year. Causes are weather, customs and culture to name a few. Sales of products such as paint and alcoholic beverages exhibit this pattern.

8. The Cyclical Component The Cyclical Component is represented by a wavelike upward and downward movement of the data around the long term trend. These fluctuations are of longer duration and less regular than the seasonal variations. It is usually attributed to the ups and downs in the general level of business activity that are frequently referred to as ‘business cycles’. 

9. The Irregular Component The Irregular Component contains the fluctuations that are not part of the other three components; they are often called random fluctuations or residual variations. As such, they are most difficult to capture in a forecasting model. They are short in duration, erratic in nature and follow no regularly recurrent or other discernible pattern. They are called residual variations because they represent what is left over in an economic time series after trend, cyclical and seasonal variations have been accounted for. Causes are sporadic, unsystematic occurrences such as strikes, earthquakes, hurricanes, accidents and the like.

10. Time Graph The time graph or time plot is the most popular form of pictorial presentation of time series data Since it illustrates most clearly the relationship between the dependent variable and time. A time plot immediately reveals any trends over time, any regular seasonal behavior and other systematic features of the time series data.

11. Time Graph

12. Time Series Models Let T represent the Trend Component Let S represent the Seasonal Component Let C represent the Cyclical Component Let I represent the Irregular Component Three (3) models exist : Additive Model: Y = T + S + C + I  Multiplicative Model: Y = T x S x C x I Composite Model (i.e. Mixed Additive / Multiplicative)

13. Time Series Analysis Classical Time Series Analysis is aimed at disentangling these fluctuations so as to learn about the behaviour of the series and to use this knowledge as a basis for forecasting the series into the future. General Objectives of Time Series Analysis Descriptive Purposes : Plots of the data and simple descriptive measures (mean and variance) can give an idea of the main properties of the data. In some series, the variation may be dominated by cyclical rather than by trend or seasonal components. Plots of the data may also show up outliers or unusual patterns which may not correspond with the general pattern of the series. Explanative Purposes : With observations taken of two or more variables, it may be possible to use the variation in one or more time series to explain the variation or behaviour of another series. e.g. Variations in a time series on oil prices may be utilized to explain variations in government revenue and the deficit in the national budget.

14. Time Series Analysis Predictive Purposes : Understanding how a variable behaves in the future can often spell the difference between success and disaster for a firm or country. By studying the quantitative “history” of a variable one can forecast its likely trajectory in the future. e.g. Being able to forecast oil prices correctly is an important issue in national, regional and international circles. Control Purposes : This is really a combination of the first three objectives. Understanding how a variable behaves or is likely to behave allows the policy maker to devise rules and mechanisms to keep the variable on course in accordance with well defined targets. e.g. unemployment, inflation, foreign exchange risks.

15. Time Series Analysis Key elements of such time series analysis are: Moving averages to show the long term Seasonal judgements to provide control for seasonal variations Straight line graphs to emphasise the general direction of the trend and to remove the short term and cyclical fluctuations. If a time series has no seasonal component ( e.g. annual data series), the trend-cycle can be estimated by smoothing the series to reduce the random variation. A range of ‘smoothers’ are available; the oldest and simplest smoother is the moving average

16. MOVING AVERAGES Moving Averages ( abbreviated by MA ) provide a simple method of smoothing ‘past history’ data and constitute a fundamental building block in all methods for decomposing a time series into its components. They are in reality artificially created time series in which each respective datapoint at a known periodicity (e.g. monthly, quarterly) is replaced by an average of itself and values corresponding to a number of preceding and succeeding periods. Moving averages allow the user to substitute original data series by smoother series.

17. MOVING AVERAGES Table #1 – Original Time Series GRAPH #1 – The Original Time Series 1980Q1 1980Q2 1980Q3 1980Q4 1981Q1 1981Q2 1981Q3 1981Q4 11.7 9.1 8.1 8.1 5.4 8.2 6.8 5.7

18. MOVING AVERAGES The very name ‘moving average’ highlights the fact that each average is computed by dropping the oldest observation and including the next observation. A moving average may utilize an even number of data points in finding its averages; these are called even points moving averages . Odd points moving averages , on the other hand, utilize an odd number of data points. The number of points in the moving average affects the smoothness of the resulting estimate.

19. Computing an Odd Point MA : the 3-Point  MA t X t Moving Total Moving Avg 1980 Q1 11.7             1980 Q2 9.1 28.9 9.63         1980 Q3 8.1 25.3 8.43         1980 Q4 8.1 21.6 7.2         1981 Q1 5.4 21.7 7.23         1982 Q2 8.2 20.4 6.8         1981 Q3 6.8 20.7 6.9         1981 Q4 5.7    

20. Computing an Odd Point MA Graph #2

21. Computing an Odd Point MA Graph #2 Graph #1

22. Computing an Odd Point MA The averaging formula used in an odd points moving average above is given by Xt = 1 / 3 ( X t-1 + X t + X t+1 ) = 1 / 3 X t-1 + 1 / 3 X t + 1 / 3 X t+1 . The coefficients of X t-1 , X t and X t+1 are called the weights applied to the data points in the moving average. Note that these weights are equal for an odd points moving average.

23. Computing an Odd Point MA If the odd points moving average involves datapoints over k consecutive time periods: each average is located at the centre of the string of time periods i.e. the ½ (k + 1) th period no averages would be possible for the first ½ (k - 1) periods at the beginning of the time series no averages would be possible for the last ½ (k - 1) periods at the end of the time series. The weights assigned to the data points in the moving average are equal and sum to 1.

24. Computing an Even Point MA: a 4-Point MA T X t Moving Total Moving Avg 2 Pt Total 2x4 MA 1980 Q1 11.7                     1980 Q2 9.1             37 9.25     1980 Q3 8.1     16.925 8.46     30.7 7.675     1980 Q4 8.1     15.125 7.56     29.8 7.45     1981 Q1 5.4     14.575 7.29     28.5 7.125     1982 Q2 8.2     13.65 6.825     26.1 6.525     1981 Q3 6.8                     1981 Q4 5.7        

25. Computing an Even Point MA The 4-Point MA is displayed in the fourth column. Observe however that the entries in this column do not align with the original time series. A 2-point MA is therefore performed in columns #5 & #6 on the 4-point MA to align the MA with the original time series. The result is called a 2x4 Moving Average otherwise called a double moving average.

26. Computing an Even Point MA The averaging formula used in the first entry in the 4-point moving average column above is given by 1 / 4 ( X t-2 + X t-1 + X t + X t+1 ). This value is not aligned with the original time series: it is ‘off’ by one row. The averaging formula used in the second entry in the 4-point moving average column above is given by 1 / 4 ( X t-1 + X t + X t+1 + X t+2 ). This value is not aligned with the original time series: it is ‘off’ by one row.

27. Computing an Even Point MA In order to align them, we take a 2-point average of these two entries. The formula for the 2-point MA is ½ [ 1 / 4 ( X t-2 + X t-1 + X t + X t+1 ) + 1 / 4 ( X t-1 + X t + X t+1 + X t+2 ) ] or equivalently 1 / 8 X t-2 + 1 / 4 X t-1 + 1 / 4 X t + 1 / 4 X t+1 + 1 / 8 X t+2 This value is now aligned with the original time series. The weights are now 1 / 8 , 1 / 4 , 1 / 4 , 1 / 4 , 1 / 8 . These weights add to 1.

28. Computing an Even Point MA  GRAPH #3 GRAPH #1

29. Computing an Even Point MA  GRAPH #3: 4pt MA GRAPH #2: 3pt MA

30. Computing an Even Point MA Note for the even point moving average: no averages would be possible for the first ½ k periods at the beginning of the time series no averages would be possible for the last ½ k periods at the end of the time series the weights assigned to the data points in a double moving average are symmetric about the middle value and sum to 1. The same trend values are generated whether we smooth forward or backward in time.

31. Limitations of Moving Averages In computing MA’s, a large number of terms in the MA increases the likelihood that randomness will be eliminated . However, the longer the length of the MA, the greater the number of data points that will be lost at the beginning and at the end of the series. The terms lost at the beginning of the series are usually of little consequence (given the fact that you wish the more recent data to impact your forecast) but those lost in the end of the series are critical. The loss may be quite serious for model building and forecasting in the Caribbean given the small data sets that are usually available. 

32. Limitations of Moving Averages The largest weights are assigned to the middle values of the set of past data (in the Even Pt MA). The MA is not capable of objective projection into the future since it is not actually defined by some mathematical function i.e. the projection always stays within past levels.

33. Decomposition of Time Series Models Finding the Trend Component There exists a wide range of methods for analyzing trends in time series You may use Simple Linear Regression to get either Y = a + bT (linear trend) Log Y = log a + b log T ( exponential trend) Alternatively, use can also be made of Moving Averages of appropriate length.

34. Finding the Cyclical Component Cyclical Components in business and economic series tend to vary widely in both duration and amplitude from cycle to cycle. As such they do not lend themselves to fitting by simple periodic functions. Smoothing Methods can help to isolate the cyclical component. To obtain the cyclical component of a time series we need to work with series that contain no seasonal effects . Annual Time Series (i.e. series consisting of annual sales) contain no seasonal components. These lend themselves easily to the isolation of the cyclical component.

35. Finding the Cyclical Component For a Multiplicative Model :  If S is not present then Y = T x C x I  Then Y / T = C x I Divide each Y t by its corresponding Trend Value T t . The resulting series would be a combination of Cyclical and Irregular components.  The Irregular component can be easily separated from the combined C x I component by smoothing. Accordingly, a simple MA of length at least 2 will be sufficient. The Moving Average Series thus generated represents the Cyclical Component. For an Additive Model : Develop an appropriate decomposition

36. Finding The Seasonal Component Frequently this is the chief source of pronounced short term fluctuations in business and economic time series. The Ratio to Moving Average Method is commonly used to identify this component. Thus will be demonstrated in a later section.

37. Finding The Seasonal Component For the Multiplicative Model: Step 1 – Smooth the series to obtain the trend-cyclical component T x C Step 2 – Obtain the Seasonal – Irregular Component S x I by dividing the original series values of Y by the smoothed values T x C.  S x I = Y / ( T x C) Isolate the seasonal Component S by making use of a simple MA to eliminate the Irregular Component I. For the Additive Model: Develop an appropriate decomposition

38. Finding the Irregular Component For the Multiplicative Model: After obtaining the Trend-Cyclical and Seasonal components we can derive the Irregular component via: I = Y / ( S x T x C) For the Additive Model: Develop an appropriate decomposition

39. Simple Exponential Smoothing Recall that in simple moving averages the weights applied to the data points are equal. In double moving averages the weights are symmetric with the highest being associated with the middle data point. In forecasting however, the most recent observations (data points) will usually provide the best guide as to the future. Hence we want a weighting scheme that provides for decreasing weights as the observations get older. Simple Exponential Smoothing provides such a weighting scheme. As its name suggests, the weights decrease exponentially as the observations get older.

40. Simple Exponential Smoothing Suppose we wish to forecast the next value of our time series Y t which is yet to be observed. Our forecast is denoted by F t . When the observation Y t becomes available, there is most likely to be a difference between Y t and F t . We call the difference Y t - F t the forecast error. The method of simple exponential smoothing takes the forecast for the previous period and adjusts it using the forecast error. Thus the forecast for the next period is given by New Forecast = Old Forecast + a fraction of the error in the most recent forecast, or equivalently F t + 1 = F t +  ( Y t - F t ) where  is a constant between 0 and 1.  is called the smoothing constant or smoothing factor

41. Simple Exponential Smoothing F t + 1 = F t +  ( Y t - F t ) where  is a constant between 0 and 1. When  has a value close to 1, the new forecast will include a substantial adjustment for the error in the previous forecast. Consequently, when  is close to 0, the new forecast will include very little adjustment.   An alternate form of the forecast formula above is as follows: F t + 1 =  Y t + ( 1 -  ) F t This formula speaks to the forecast F t + 1 is based on weighting the most recent observation Y t with a weight of  and weighting the most recent forecast F t with a weight of ( 1 -  ).

42. Simple Exponential Smoothing It substantially reduces any storage problem because it is no longer necessary to store all of the historical data or a subset of them (as in the case of the moving average). Rather, only the most recent observation, the most recent forecast, and a value for  must be stored. Beginning with F t + 1 =  Y t + ( 1 -  ) F t . Replacing t by t – 1 gives F t =  Y t - 1 + ( 1 -  ) F t – 1. Substituting back for F t we get F t + 1 =  Y t + ( 1 -  )[  Y t - 1 + ( 1 -  ) F t – 1 ] i.e. F t + 1 =  Y t +  ( 1 -  ) Y t - 1 + ( 1 -  ) 2 F t – 1.

43. Simple Exponential Smoothing If this substitution process is repeated for F t – 1 and then for F t – 2 and so on, the result is F t + 1 =  Y t +  ( 1 -  ) Y t - 1 +  ( 1 -  ) 2 Y t – 2 +  ( 1 -  ) 3 Y t - 3 +  ( 1 -  ) 4 Y t – 4 + …………………+  ( 1 -  ) t - 1 Y 1 +  ( 1 -  ) t F 1. Our next forecast F t + 1 can also be considered as a weighted moving average of all past observations. The weights  ,  ( 1 -  ),  ( 1 -  ) 2 ,  ( 1 -  ) 3 ,  ( 1 -  ) 4 , ………… ,  ( 1 -  ) t - 1 and  ( 1 -  ) t can be seen to belong to a geometric progression in which the first term is  and the common ratio is ( 1 -  ). Are these weights getting larger or smaller?

44. Simple Exponential Smoothing The nearer  is to 1, very little smoothing is generated and the more sensitive our forecast becomes to current conditions. The nearer  is to 0, the last forecast plays a more prominent role than when a larger  is used. Considerable smoothing is generated and the more stable our forecast will be (i.e. the forecast will react less sensitively to current conditions).

45. Simple Exponential Smoothing Example:   The following data relates to the ATM traffic at a Bank Automatic Teller Facility along the East West Corridor. ATM Transactions (thousands) Using the recursion formula and a smoothing constant of 0.2, generate forecasts of the number of ATM transactions for the period 1998 Qtr 1 through 2001 Qtr 1. What differences (if any) would you observed if the smoothing constant were 0.8? Year Qtr 1 Qtr 2 Qtr 3 Qtr 4 1998 450 440 460 410 1999 380 400 370 360 2000 410 450 470 490

46. Simple Exponential Smoothing of ATM Transactions with  = 0.2 F 2 = 0.2 Y 1 + 0.8 F 1 = 0.2 (450) + 0.8 (450) = 450 F 3 = 0.2 Y 2 + 0.8 F 2 = 0.2 (440) + 0.8 (450) = 448 Quarter Data Point Transactions Forecast Forecast   Label   Label Value 1998 Q1 Y1 450 F1 450 1998 Q2 Y2 440 F2 450 1998 Q3 Y3 460 F3 448 1998 Q4 Y4 410 F4 450.4 1999 Q1 Y5 380 F5 442.32 1999 Q2 Y6 400 F6 429.86 1999 Q3 Y7 370 F7 423.89 1999 Q4 Y8 360 F8 413.11 2000 Q1 Y9 410 F9 402.49 2000 Q2 Y10 450 F10 403.99 2000 Q3 Y11 470 F11 413.19 2000 Q4 Y12 490 F12 424.55 2001 Q1 Y13 ?? F13 437.64

47. Simple Exponential Smoothing of ATM Transactions with  = 0.8 F 2 = 0.8 Y 1 + 0.2 F 1 = 0.8 (450) + 0.2 (450) = 450 F 3 = 0.8 Y 2 + 0.2 F 2 = 0.8 (440) + 0.2 (450) = 442 Quarter Data Point Transactions Forecast Forecast   Label   Label Value 1998 Q1 Y1 450 F1 450 1998 Q2 Y2 440 F2 450 1998 Q3 Y3 460 F3 442 1998 Q4 Y4 410 F4 456.4 1999 Q1 Y5 380 F5 419.28 1999 Q2 Y6 400 F6 429.86 1999 Q3 Y7 370 F7 387.86 1999 Q4 Y8 360 F8 375.51 2000 Q1 Y9 410 F9 363.10 2000 Q2 Y10 450 F10 400.62 2000 Q3 Y11 470 F11 440.12 2000 Q4 Y12 490 F12 464.02 2001 Q1 Y13 ?? F13 484.8

48. Advantages of Simple Exponential Smoothing Greater weight is given to more recent data in the series All past data are incorporated and no data point is cut off as in Simple MA The forecasts are all driven by the choice of the initial forecast F 1 and the value of  Less data needs to be stored than with longer period MA Whatever form is adopted, changing the model to suit changing conditions can simply be made by altering the value of  We can forecast for the time period immediately after the last period in the data series.

49. Limitations of Simple Exponential Smoothing The choice of an appropriate value for  (an option is to choose  such that it yields the smallest MSE value ) The choice of the forecast F 1 for the first time period. ( an option is to use the average of the first four or five data points in the series; another is to simply set F 1 = Y 1 )

50. Using an MA for Forecasting Quarter Data Point Transactions Forecast 3 Pt MA   Label   Label 1998 Q1 Y1 450 F1 1998 Q2 Y2 440 F2 1998 Q3 Y3 460 F3 1998 Q4 Y4 410 F4 450.00 1999 Q1 Y5 380 F5 436.67 1999 Q2 Y6 400 F6 416.67 1999 Q3 Y7 370 F7 396.67 1999 Q4 Y8 360 F8 383.33 2000 Q1 Y9 410 F9 376.67 2000 Q2 Y10 450 F10 380.00 2000 Q3 Y11 470 F11 406.67 2000 Q4 Y12 490 F12 443.33 2001 Q1 Y13 ?? F13 470.00

51. EVALUATING FORECASTS We come now to another fundamental concern i.e. how to compare alternate forecasts generated by means of applying moving averages of different lengths. In most forecasting situations, accuracy is treated as the overriding criterion for selecting a forecasting method. In many instances, ‘accuracy’ refers to ‘goodness of fit’, which in turn refers to how well the forecasting model is able to reproduce the data that are already known. To the user of the forecast, it is the accuracy of the future forecast that is most important.

52. EVALUATING FORECASTS Among the number of criteria that could be used to evaluate and compare forecasting methods, five common ones are: mean absolute deviation (MAD) mean percentage error (MPE) mean absolute percentage error (MAPE) mean-squared error (MSE) the root mean-squared error (RMSE).

53. Evaluating Forecasts The mean absolute deviation is calculated by the formula: The mean percentage error is calculated by the formula: The mean absolute percentage error is calculated by the formula:  

54. Evaluating Forecasts The mean –squared error (MSE) is calculated by the formula: The root mean –squared error (RMSE) is calculated by the formula: For all five measures, a lower value is preferred  

55. Example Yt Forecast 1 Forecast 2 50 45 49 25 10 32 30 9 25

56. Measurement of Seasonality and Making Seasonal Adjustments Seasonal variations/fluctuations are frequently of interest to researchers or users of time series data if only because there is frequently a need for these variations/fluctuations to be eliminated. A company, for example, might be interested in analysing its seasonal variations in sales of its core products to iron out variations in production, scheduling and personnel requirements and to disaggregate a predicted annual sales figure into monthly and/or quarterly targets based on seasonal variations in the past.  The term ‘Seasonality’ is used interchangeably with seasonal variation.

57. Measurement of Seasonality and Making Seasonal Adjustments Seasonal Adjustment is the name given to the method by which we generate a series that is free of the seasonal variations. The most widely used method for identifying the Seasonality or undertaking Seasonal Adjustment to a time series is the Ratio-to-Moving Average Method (R-to-MA) which we will present later. This method generates seasonality in the form of seasonal indices. It is helpful when acquiring an understanding of the rationale of the measurement of seasonal fluctuations/variations to begin with the final product – the seasonal indices.

58. Measurement of Seasonality and Making Seasonal Adjustments First an example of how these seasonal indices may be used. Suppose that $40,000,000 of sales of particular products was budgeted for the next year – an average of $10,000,000 per quarter. If the quarterly seasonal indices based on an observed stable seasonal pattern in the past were 97.0, 110.0, 85.0 and 108.0 respectively for the four quarters of the year, we interpret these indices as percentages. These percentages suggest that there is a decrease of 3% vis a vis the average for the year as a whole in the first quarter, an increase of 10% in the second, a decrease of 15% in the third quarter and an 8% increase in the fourth quarter.

59. Measurement of Seasonality and Making Seasonal Adjustments Accordingly, the amounts budgeted for each quarter would be as follows: 1 st quarter 0.97 x $10m = $ 9.7m 2 nd quarter 1.10 x $10m = $11.0m 3 rd quarter 0.85 x $10m = $ 8.5m 4 th quarter 1.08 x $10m = $10.8m Total = $40.0m

60. Measurement of Seasonality and Making Seasonal Adjustments Secondly, we note that the object of the R-to-MA computations when the raw data is quarterly in nature and demonstrates a stable or regular seasonal pattern is to obtain four seasonal indices , each one indicating the seasonal importance of a quarter to the entire year. The arithmetic mean of the four seasonal indices is 100.0 Thirdly, we note that the object of the R-to-MA computations when the raw data is monthly in nature and demonstrates a stable or regular seasonal pattern is to obtain twelve seasonal indices , each one indicating the seasonal importance of a month to the entire year. The arithmetic mean of the twelve seasonal indices will be 100.0 .

61. Measurement of Seasonality and Making Seasonal Adjustments The essential problem in the measurement of seasonal variations is that of eliminating from the original data series the non-seasonal elements in order to isolate the stable seasonal component. A practical method is as follows: Obtain a series of moving averages that roughly include the Trend-Cyclical components. Divide the original data by these moving average figures to eliminate the Trend and Cyclical elements thereby yielding a series of figures which contain only Seasonal and Irregular components. Average these figures by quarters (or months) to eliminate the irregular disturbances thereby isolating the seasonal factor.

62. Ratio-To-Moving Average Method Assume that the data is quarterly in nature and the model is multiplicative . The R-to-MA Method consists of the following steps :  Derive a 4-quarter moving average which contains the trend and cyclical components present in the original time series.  Perform a 2-period MA so as to align the moving average figures.  Divide the original data for each quarter by the corresponding double moving average figure. The quotients are called the ‘ratio-to-moving averages’.

63. Ratio-To-Moving Average Method Arrange these ratio-to-moving averages by quarter i.e. all the first quarters in one group, all the second quarters in another, etc. Average these ratio-to-moving averages for each quarter in an attempt to isolate the stable seasonal component for the quarter. It is recommended that a truncated mean of the ratio-to-moving averages be used – similar to a tri-mean . Make an adjustment to force the four truncated means to total to 400 and thus average to 100. The resulting four means are the seasonal indices for the original time series. EXAMPLE See the table on the next page for the application of the R-to-MA Method to a time series consisting of quarterly grain price indices for the period 1989 Qtr 1 to 1995 Qtr 2.

64. Qtr Grain 4-Pt 4-Pt 2 x 4 Pt Ratio-to- Seasonal Seasonally   Price Moving Moving Moving Moving Index Adjusted   Index Total Average Average Average   Grain -1 -2 -3 -4 -5 (2)/(5) -8.00 Price Index           as a %   (2)X100/(8) 1989 I 96         98.56 97.40                 II 103         102.37 100.61     391 97.75         III 100     97.375 102.70 102.72 97.35     388 97         IV 92     96.375 95.46 96.35 95.49     383 95.75         1990 I 93     95.25 97.64 98.56 94.36     379 94.75         II 98     94 104.26 102.37 95.73     373 93.25         III 96     93 103.23 102.72 93.46     371 92.75         IV 86     92.125 93.35 96.35 89.26     366 91.5         1991 I 91     91.625 99.32 98.56 92.33     367 91.75         II 93     92.625 100.40 102.37 90.85     374 93.5         III 97     93.75 103.47 102.72 94.43     376 94         IV 93     94.5 98.41 96.35 96.52     380 95        

65. 1992 I 93     94.875 98.02 98.56 94.36     379 94.75         II 97     94.75 102.37 102.37 94.75     379 94.75         III 96     95.25 100.79 102.72 93.46     383 95.75         IV 93     96.375 96.50 96.35 96.52     388 97         1993 I 97     98.25 98.73 98.56 98.42     398 99.5         II 102     100.125 101.87 102.37 99.64     403 100.75         III 106     101.125 104.82 102.72 103.19     406 101.5         IV 98     101.75 96.31 96.35 101.71     408 102         1994 I 100     101.625 98.40 98.56 101.46     405 101.25         II 104     101.625 102.34 102.37 101.59     408 102         III 103     102.875 100.12 102.72 100.27     415 103.75         IV 101     104.75 96.42 96.35 104.83     423 105.75         1995 I 107         98.56 108.56                 II 112         102.37 109.41

66. Ratio-to-Moving Averages by Quarters qtr r-to-ma qtr r-to-ma qtr r-to-ma qtr r-to-ma I 97.64 II 104.26 III 102.70 IV 95.46 I 99.32 II 100.40 III 103.23 IV 93.35 I 98.02 II 102.37 III 103.47 IV 98.41 I 98.73 II 101.87 III 100.79 IV 96.50 I 98.40 II 102.34 III 104.82 IV 96.31         III 100.12 IV 96.42

67. Tri-Means by Quarter qtr r-to-ma qtr r-to-ma qtr r-to-ma qtr r-to-ma 1992 I 98.02 II 102.37 III 102.70 IV 95.46 1993 I 98.73 II 101.87 III 103.23 IV 96.50 1994 I 98.40 II 102.34 III 103.47 IV 96.31         III 100.79 IV 96.42 Tri-Means 98.38   102.19   102.54   96.17

68. Tri-Means by Quarter qtr Tri - Mean I 98.38 II 102.19 III 102.54 IV 96.17 Sum of Tri-Means 399.30 Average of Tri-Means 99.82 Adjustment (100-avg) 0.18

69. Seasonal Index qtr Tri - Mean Adjustment Seasonal Index I 98.38 + 0.18 = 98.56 II 102.19 + 0.18 = 102.37 III 102.54 + 0.18 = 102.72 IV 96.17 + 0.18 = 96.35 Average = 100.00

70. Ratio-To-Moving Average Method EXERCISE Redefine the Ratio-to-Moving Average Method for 1. quarterly time series data in an additive model. 2. monthly time series data in a multiplicative model.

71. End of Lecture