Cyclical Effect, and Measuring Irregular Effect

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1 Paper:15, Quantitative Techniques for Management Decisions Module- 37 Forecasting & Time series Analysis: Measuring- Seasonal Effect, Cyclical Effect, and Measuring Irregular Effect Principal Investigator Co-Principal Investigator Paper Coordinator Prof. S P Bansal Vice Chancellor Maharaja Agrasen University, Baddi Prof. YoginderVerma Pro Vice Chancellor Central University of Himachal Pradesh. Kangra. H.P. Prof. Pankaj Madan Dean- FMS Gurukul Kangri Vishwavidyalaya, Haridwar Content Writer Prof. Pankaj Madan Dean- FMS Gurukul Kangri Vishwavidyalaya, Haridwar

2 Items Subject Name Paper Name Module Title Description of Module Management Quantitative Techniques for Management Decisions Forecasting & Time series Analysis: Measuring- Seasonal Effect, Cyclical Effect, and Measuring Irregular Effect Module Id 37 Pre- Requisites Objectives Keywords Basic mathematical operations Time series models of forecasting Decomposition of a time series Secular trend, cyclical movements, random movements Methods to analysis of seasonality Two general aspects of time series patterns Trend analysis and trend Projections General model of seasonality Self-check exercise Summary Time series, seasonal Effect, cyclical effects Module-37 Forecasting & Time series Analysis: Measuring- Seasonal Effect, Cyclical Effect, and Measuring Irregular Effect Introduction of Time series Analysis... 4 Two Main Goals of Time Series Analysis... 4 Time series models of forecasting... 5 Decomposition of a time series... 5 Two General Aspects of Time Series Patterns... 7 Trend and... 7 Seasonality Trend Analysis... 9 Smoothing... 9

3 Fitting a function Trend Projections Graphic presentation Seasonal variations Graphic presentation... Error! Bookmark not defined. Methods to analysis of Seasonality General model Additive and multiplicative seasonality Additive and multiplicative trend-cycle... Error! Bookmark not defined. Moving average Ratios or differences Seasonal components Seasonally adjusted series Trend-cycle component Random or irregular component Self Check Exercise... 15

4 Quadrant-I Forecasting & Time series Analysis: Measuring- Seasonal Effect, Cyclical Effect, and Measuring Irregular Effect Learning Objectives: After the completion of this module the student will understand: Introduction of Time series Analysis Seasonal Effect, Cyclical Effect, and Measuring Irregular Effect Introduction of Time series Analysis A set of data depending on the time A series of values over a period of time Collection of magnitudes belonging to different time periods of some variable or composite of variables such as production of steel, per capita income, gross national income, price of tobacco, index of industrial production. Time is act as a device to set of common stable reference point. In time series, time act as an independent variable to estimate dependent variables Two Main Goals of Time Series Analysis There are two main goals of time series analysis: (a) identifying the nature of the phenomenon represented by the sequence of observations, and (b) forecasting (predicting future values of the time series variable). Both of these goals require that the pattern of observed time series data is identified and more or less formally described. Once the pattern is established, we can interpret and integrate it with other data (i.e., use it in our theory of the investigated phenomenon, e.g., sesonal commodity prices). Regardless of the depth of our understanding and the validity of our

5 interpretation (theory) of the phenomenon, we can extrapolate the identified pattern to predict future events. Identifying Patterns in Time Series Data Systematic pattern and random noise Two general aspects of time series patterns Trend Analysis Time series models of forecasting Time series models use time-based data. A time series is a collection of readings belonging to different time periods, usually equally spaced, which may be months, weeks or years, of some economic variable. It may be sales of HP personal computers, production of steel, per capita income, price of a share on National Stock Exchange and so on. Thus, it is a series in which the data are put in a chronological order. Forecasting time series data implies that predictions about the future values are made only from the past data, and other factors, no matter how important they might potentially be, are not considered. We will first review techniques used to identify patterns in time series data (such as smoothing and curve fitting techniques and autocorrelations), then we will introduce a general class of models that can be used to represent time series data and generate predictions (autoregressive and moving average models). Finally, we will review some simple but commonly used modeling and forecasting techniques based on linear regression. Decomposition of a time series A time series is the result of a number of movements which are caused by numerous economic, political, natural and other factors. The analysis of time series

6 means decomposing the past data into components and then projecting them forward. A time series typically has four components, though on occasions only one or two of these may eclipse the others. (i) Secular trend Over a long period, a time series will have an overall tendency either to move upwards or downwards or sometimes a blend thereof, though the actual movement will not be regular. This tendency of the time series is known as the secular trend. For example, a glance at the sales of a popular soft drink manufacturer is likely to reveal an increasing trend. (ii) Seasonal variations The fluctuations that occur periodically the movements recurring within a definite period, may be every week or month or quarter with reasonably high degree of predictability are called seasonal variations. For a soft drink manufacturer, while the yearly sales may be on the increase, the sales are likely to be high every summer and low every winter. (iii) Cyclical movements They are caused by business cycles or trade cycles. These are oscillatory movements a series of repeated sequences-superimposed on the original data. These movements are of longer duration than a year. The sales of a company, for example, may be high because the level of economic activity may be high. Similarly, the sales may be low due to overall subdued economic activity. (iv)random movements Random movements are residual, or erratic, movements that do not have any set pattern and are usually caused by some unpredictable reason, like wild-cat strike, earthquake, fire and so on. We will review techniques that are useful for analyzing time series data, that is, sequences of measurements that follow non-random orders. Unlike the analyses of random samples of observations that are discussed in the context of most other statistics, the analysis of time series

7 is based on the assumption that successive values in the data file represent consecutive measurements taken at equally spaced time intervals. The time series models first deals with projections of values without decomposition and then relating to trend, also making adjustment for the seasonal variations. The three time series models discussed in this chapter include moving averages, exponential smoothing and trend projections. Two General Aspects of Time Series Patterns Most time series patterns can be described in terms of two basic classes of components: Trend Seasonality The former represents a general systematic linear or (most often) nonlinear component that changes over time and does not repeat or at least does not repeat within the time range captured by our data (e.g., a plateau followed by a period of exponential growth). The latter may have a formally similar nature (e.g., a plateau followed by a period of exponential growth), however, it repeats itself in systematic intervals over time. Those two general classes of time series components may coexist in real-life data. For example, sales of a company can rapidly grow over years but they still follow consistent seasonal patterns (e.g., as much as 25% of yearly sales each year are made in December, whereas only 4% in August).

8 This general pattern is well illustrated in a "classic" Series G data set (Box and Jenkins, 1976, p. 531) representing monthly international airline passenger totals (measured in thousands) in twelve consecutive years from 1949 to 1960 (see example data file G.sta and graph above). If you plot the successive observations (months) of airline passenger totals, a clear, almost linear trend emerges, indicating that the airline industry enjoyed a steady growth over the years (approximately 4 times more passengers traveled in 1960 than in 1949). At the same time, the monthly figures will follow an almost identical pattern each year (e.g., more people travel during holidays then during any other time of the year). This example data file also illustrates a very common general type of pattern in time series data, where the amplitude of the seasonal changes increases with the overall trend (i.e., the variance is correlated with the mean over the segments of the series). This pattern which is called multiplicative seasonality indicates that the relative amplitude of seasonal changes is constant over time, thus it is related to the trend.

9 Trend Analysis There are no proven "automatic" techniques to identify trend components in the time series data; however, as long as the trend is monotonous (consistently increasing or decreasing) that part of data analysis is typically not very difficult. If the time series data contain considerable error, then the first step in the process of trend identification is smoothing. Smoothing. Smoothing always involves some form of local averaging of data such that the nonsystematic components of individual observations cancel each other out. The most common technique is moving average smoothing which replaces each element of the series by either the simple or weighted average of n surrounding elements, where n is the width of the smoothing "window" (see Box & Jenkins, 1976; Velleman & Hoaglin, 1981). Medians can be used instead of means. The main advantage of median as compared to moving average smoothing is that its results are less biased by outliers (within the smoothing window). Thus, if there are outliers in the data (e.g., due to measurement errors), median smoothing typically produces smoother or at least more "reliable" curves than moving average based on the same window width. The main disadvantage of median smoothing is that in the absence of clear outliers it may produce more "jagged" curves than moving average and it does not allow for weighting. In the relatively less common cases (in time series data), when the measurement error is very large, the distance weighted least squares smoothing or negative exponentially weighted smoothing techniques can be used. All those methods will filter out the noise and convert the data into a smooth curve that is relatively unbiased by outliers (see the respective sections on each of those methods for more details). Series with relatively few and systematically distributed points can be smoothed with bicubic splines.

10 Fitting a function. Many monotonous time series data can be adequately approximated by a linear function; if there is a clear monotonous nonlinear component, the data first need to be transformed to remove the nonlinearity. Usually a logarithmic, exponential, or (less often) polynomial function can be used. Trend Projections We now consider the last of time-series forecasting techniques, in the form of trend projections. For this method, a trend line is fitted to the given time series data and then projections are made into future using this line. The trend line may be linear (straight line) or curvi-linear in nature. There is a wide variety of curvi-linear trend lines possible to draw, but our focus in this module is on linear trend only. For obtaining the trend line, the given historical data are first plotted on the graph, representing time scale on the X-axis. Then a line is drawn through these points in such a way that (i) the sum of deviations above the line is equal to the sum of deviations below the line so that the sum of deviations is equal to zero, and (ii) the sum of squares of these (vertical) deviations is the minimum. In essence, the trend line is drawn on the basis of the principle of least squares. Such a line, like any other straight line, is represented by the equation: V = a + bx where V = the trend value (which is to be predicted) a = the V-axis intercept b = slope of the trend line X = the independent variable, the time Graphic presentation The given data and the trend line are shown in Figure The projections for the years 2001 and 2002 using the trend line are also shown. These projections have the same values as obtained earlier.

11 Years: Fig : Trend Line and Trend Projections Seasonal variations: The trend projections of time series data discussed earlier involves focusing at the trend (long term tendency) of the data over a series of time periods. Thus, if yearly sales data are given, we can obtain forecasts for likely yearly sales using the trend line. However, if projections are to be made on, say, monthly or quarterly basis, then recurring variations at certain seasons of the year make a seasonal adjustment in the trend forecast necessary. Sales of soft drinks, woollen garments, cooking gas and so on are certainly not expected to be the same in all months of the year. A consideration of monthly or quarterly data makes it easy to study seasonal patterns. Seasonal indices, which are used to study seasonal variations, can be obtained by applying a number of methods. Here, we demonstrate the method of simple averages. a=y-fcx=89-2x3.5=82 Accordingly, the trend equation is: Y, = X, with origin in Using this equation, trend values for various years are obtained (by substituting the corresponding X-values in the equation) and shown in the sixth column of the table. The next column gives the deviations of V-values

12 from the corresponding ^-values. Their sum is found to be equal to zero. The last column of the table contains the squares of deviations which sum upto 142. Forecasting K2001 = x 8 = Rs 98 million (for year 2001, X = 8) ^2002 = x 9 = Rs 100 million Methods to analysis of Seasonality ARIMA (Box & Jenkins) and Autocorrelations Interrupted Time Series Exponential Smoothing Seasonal Decomposition (Census I) X-11 Census method II seasonal adjustment X-11 Census method II result tables Distributed Lags Analysis Single Spectrum (Fourier) Analysis Cross-spectrum Analysis Basic Notations and Principles Fast Fourier Transformations General model The general idea of seasonal decomposition is straightforward. In general, a time series like the one described above can be thought of as consisting of four different components: (1) A seasonal component (denoted as St, where t stands for the particular point in time) (2) a trend component (Tt), (3) a cyclical component (Ct), and (4) a random, error, or irregular component (It). The difference between a cyclical and a seasonal component is that the latter occurs at regular (seasonal) intervals, while cyclical factors have usually a longer duration that varies from cycle to cycle. In the Census I method, the trend and cyclical components are customarily combined into a trend-cycle component (TCt). The specific functional relationship between these

13 components can assume different forms. However, two straightforward possibilities are that they combine in an additive or a multiplicative fashion: Additive model: Xt = TCt + St + It Multiplicative model: Xt = Tt*Ct*St*It Here Xt stands for the observed value of the time series at time t. Given some a priori knowledge about the cyclical factors affecting the series (e.g., business cycles), the estimates for the different components can be used to compute forecasts for future observations. (However, the Exponential smoothing method, which can also incorporate seasonality and trend components, is the preferred technique for forecasting purposes.) Additive and multiplicative seasonality. Let us consider the difference between an additive and multiplicative seasonal component in an example: The annual sales of toys will probably peak in the months of November and December, and perhaps during the summer (with a much smaller peak) when children are on their summer break. This seasonal pattern will likely repeat every year. Seasonal components can be additive or multiplicative in nature. For example, during the month of December the sales for a particular toy may increase by 3 million dollars every year. Thus, we could add to our forecasts for every December the amount of 3 million to account for this seasonal fluctuation. In this case, the seasonality is additive. when the sales for the toy are generally weak, then the absolute (dollar) increase in sales during December will be relatively weak (but the percentage will be constant); if the sales of the toy are strong, then the absolute (dollar) increase in sales will be proportionately greater.

14 Moving average. First a moving average is computed for the series, with the moving average window width equal to the length of one season. If the length of the season is even, then the user can choose to use either equal weights for the moving average or unequal weights can be used, where the first and last observation in the moving average window are averaged. Ratios or differences. In the moving average series, all seasonal (within-season) variability will be eliminated; thus, the differences (in additive models) or ratios (in multiplicative models) of the observed and smoothed series will isolate the seasonal component (plus irregular component). Specifically, the moving average is subtracted from the observed series (for additive models) or the observed series is divided by the moving average values (for multiplicative models). Seasonal components. The seasonal component is then computed as the average (for additive models) or medial average (for multiplicative models) for each point in the season. (The medial average of a set of values is the mean after the smallest and largest values are excluded). The resulting values represent the (average) seasonal component of the series.

15 Seasonally adjusted series. The original series can be adjusted by subtracting from it (additive models) or dividing it by (multiplicative models) the seasonal component. The resulting series is the seasonally adjusted series (i.e., the seasonal component will be removed). Trend-cycle component. Remember that the cyclical component is different from the seasonal component in that it is usually longer than one season, and different cycles can be of different lengths. The combined trend and cyclical component can be approximated by applying to the seasonally adjusted series a 5 point (centered) weighed moving average smoothing transformation with the weights of 1, 2, 3, 2, 1. Random or irregular component. Finally, the random or irregular (error) component can be isolated by subtracting from the seasonally adjusted series (additive models) or dividing the adjusted series by (multiplicative models) the trend-cycle component. Self-Check Exercise Question 1: Month Demand Month Demand Month Demand Month Demand Calculate (i) 3-monthly and (ii) 4-monthly moving averages. What is the forecast for month 16 for each one?

16 The given data is reproduced in Table TABLE 18.1 Forecasting Using Moving Averages Month Demand (Y) 3-Monthly Moving 4-Monthly Moving 3-Monthly Weighted The three-monthly moving averages [i.e /3 = 278] is shown as the first figure in third column and four-monthly moving averages [i.e /4 = 282.3] is shown as the first figure in the fourth column of the table. Consequently, finding three months moving averages, in the third column, the forecast for month 16 th month is 313 units, while taking 4- monthly moving average, in the fourth column, yields a forecast of 318 units for 16 th month. Weighted moving averages A careful analysis of the moving average method reveals that the moving average with a base of n periods is in fact an equal-weighted average with a weightage of 1/n to each of the preceding n values and a zero weightage to all the previous values. Thus, in a 3-monthly forecast, the immediately last three months' values are given a weightage of 1/3 each and the remaining values a weightage of zero. However, one may like to forecast values by giving differential weights to the values entering into moving average calculation. For example,

17 a 3-monthly moving average may be calculated by assigning weightage of 3, 2 and 1 respectively for the three months' values. Question 2: The sales data with the break-up of the yearly sales on a quarterly basis. You are required to (a) obtain seasonal indices, and (b) forecast the sales for the years 2001 and 2002 on a quarterly basis. Year Quarter Total I II III IV The given data are reproduced in Table To obtain seasonal indices, we first calculate average sales for each of the four quarters. The overall mean of these quarterly averages is calculated, which here works out to be ( )/4= 89/4 = Each of the quarterly values is then expressed as a percentage of this overall mean. For example, for Quarter I, we have (20.375/22.25) x 100 = These percentages are termed as the seasonal index numbers. TABLE 18.6 Calculation Seasonal idices V^or of Quarter In Total icdr I II in IV

18 Total Average Seasonal index /4 = Estimation of Quarterly Sales: The sales for the years 2001 and 2002 are estimated to be Rs 98 million and Rs 100 million respectively, which yield a corresponding average sales of Rs 24.5 million and Rs 25 million. Based on the seasonal indices, the quarterly sales are estimated on next page: Quarter Year 2001 Year 2002 I 24.5 x = Rs m 25 x = Rs m II 24.5 x =Rs 31.80m 25 x = Rs m III 24.5 x =Rs 26.56m 25 x =Rs m IV 24.5 x =Rs 17.20m 25 x =Rs m

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24 Summary This module focused on forecasting and time series analysis that helps in forecasting (predicting future values of the time series variable). In this, we have described the process of projecting past trend and seasonal variation into the future. There are four kinds of changes or variation involved in time-series analysis, which includes (a) secular trend, (b) Cyclical fluctuations (c) Seasonal variation (d) irregular variation. Although the irregular and cyclical components do affect the future, they are unreliable and difficult to use in forecasting. This module also concluded that the time series models first deals with projections of values without decomposition and then relating to trend make adjustment for the seasonal variations. For a problem that uses all four components of time series, we first need to deseasonalize the time series, then develop a trend line and then adjust the cyclical variation around the trend line to predict the future pattern.