BUSINESS STATISTICS (PART-37) TIME SERIES & FORECASTING (PART-2) ANALYSIS FOR SECULAR TREND-I (UNITS V & VI)

Transcription

1 BUSINESS STATISTICS (PART-37) TIME SERIES & FORECASTING (PART-2) ANALYSIS FOR SECULAR TREND-I (UNITS V & VI) 1. INTRODUCTION Hello viewers, we have been discussing the time series analysis and forecasting methods. In my last lecture we have seen that there are four different components of a time series namely: the secular trend, the cyclic fluctuations, the seasonal variations and the irregular or random variations. In a time series all the fours or any combination of these may be present. We know that the secular trend and the cyclic fluctuations are the long term movements whereas the seasonal fluctuations are the short term movements. Now the question is how to measure these different types of variations present in a time series? Now first; how to measure the secular trend present in the data? And how to isolate the effect of the secular trend in the data in order to study the effect of other factors present in the time series.

2 So, in this lecture we shall introduce the methods of studying secular trend or simply trend presented in a time series, and in particular discuss in detail one such method known as Moving Average method. 2. REASONS FOR STUDYING TREND Now what are the reasons for studying the trend? The first one is: 1. The study of secular trends allows us to describe a historical pattern. There are many instances when we can use a past trend to evaluate the success of a previous policy. The second reason is that: 2. Studying secular trends permits us to project past patterns, or trends, into the future. Knowledge of the past can tell us a great deal about the future. Examining the growth rate of the world s population, for example, can help us to estimate the population for some future time. So, this is the forecasting. That is using the past data we can forecast about the future trend or future estimates of the variable under study. Similarly like the consumer price index numbers we know that normally the consumer price index number shows an increasing trend over a long period of time. So measuring the secular trend will provide us an idea how can project the value of the CPI for a future year or for a future period of time.

3 Third is that: 3. In many situations, studying the secular trend a time series allows us to eliminate the trend component from the series. This makes it easier for us to study the other three components of the time series. If we want to determine the seasonal variation in umbrella sales, for example, eliminating the trend component gives us a more accurate idea of the seasonal component. Another example of the seasonal variation is, i.e., in the case of the sale of the ornaments. We know that the sale of the ornaments picks up during the marriage season. So, in order to study the effect, i.e. the seasonal effect, on the sales of the ornaments we have to isolate the other variations present in the data like the secular trend. So that is the third reason we would like to study the secular trend. 3. METHODS OF TREND ANALYSIS Let us see what are the methods of trend analysis? That is, how we can measure this secular trend. and what are the methods to be used to isolate the secular trend in order to study other components of time series? The first method is: The Moving Average method. The other one is: The Fitting of curves.

4 We are familiar about the fitting of the curves where we use the method of least squares. And by plotting the data on a graph paper we can know that whether this time series follows a linear trend or whether it depicts a curvilinear graph. In that case either we use: Or, The Linear Regression method. The Curvilinear Regression method. We know that the curvilinear regression method may include a parabola or an exponential curve. Then we also have another method of fitting of both curves. Of course this method does not use the method of least squares. So this we shall be discussing in my upcoming lectures. Here, we shall concentrate on the moving average method. The moving average method isolates the trend in the time series, in an effective manner because it eliminates the short term fluctuations. Or, we can say: Moving average is a series of arithmetic means of variate values of a sequence of fixed number of years known as the period. Now let us look at this example and see how the method of moving average works. In this table here in the first column different years are given i.e. 1994, 95, 96 and so on up to 1999.

5 Then in the second column the value of the response variable is given i.e. 2, 5, 2, 2, 7 and 6. Now here we take the moving average period as 3. This moving average period I denote by m. So the value of m is 3. So, it indicates that we have to take the average of the three values of the variable under study in a particular manner. That is first we pick up the values of the variables for first three years i.e. 1994, 1995 and And so that is the three years, as our m is 3. So we first take the average of these three figures which is (2+5+2)/3 give me the value 3. And this value will be placed against the mid year because here we are considering three years 1994, 1995 & So 1995 is a mid year so this first average we placed against the mid year Then now as a second step we ignore or delete the data of the first year and include in this place the data of the year And again we concentrate on three years i.e. 1995, 1996 and And the values of the variables are 5, 2 and 2. So the average of these three figures is again 3 and it is placed against the mid year Now in this case the mid year is 1996 because we are concentrating on the years 1995, 96 & 97. So this way again now we delete or omit the year 1995 also along with the And then we take the average of these three figures 2, 2 and 7. That is for the year 1996, 97 and 98. And the average of these comes out to be And this we place against the mid year 1997.

6 This way when we now delete the 1996 year also along with the 1994 and 1995 then we take the average of these three figures 2, 7 and 6. That comes out to be 5 and it is again placed against the mid year, 1998 in this case. So, these are the moving averages. And when these data i.e. when the original data and the moving averages are plotted then we can see that we get this type of graph. Here we can see that the original data are placed and join by the red lines whereas the moving averages are given in the blue colour. So the original data does not show any particular type of trend present in the data. But when we look at the values of the moving averages and; plot the graph of the moving averages, then it indicates that there is an increasing trend present in the data. 4. EXAMPLES OF MOVING AVERAGE METHOD Now let us consider one more example. In my last example we have taken the period of moving average as 3, i.e., an odd number. Now in this next example, we shall take the period of moving average as an even number. And this example is on gross capital formation. Here we are given the data on gross capital formation in crores of rupees and it starts from the year 1966 and goes up to the year So now look at this table the first column gives the years and the second column gives the gross capital formation in rupees.

7 Now, if I take the moving average period as 4 i.e. my m =4. Then in the first place I take the average of these four figures starting from the data of 1966 and up to That is I take the average of these four figures and this comes out to be Now the question is: what is the mid year? Because the mid year falls between the 1967 and 1968 so this figure we placed in between the year 1967 and 68. Now as a next step we ignore or we remove the first year data and now include the 1970 s data i.e So now we take the average of 20.9, 17.8, 16.1 and And the average of these four figures comes out to be Now this figure is placed between the year 1968 and 69, because the midyear falls between these two period. So this way by living the one year and including the succeeding year we go on taking the averages of the four years entries. And place them against the mid years. And in this case we have one more column or one more step in comparison to what we have in the last example. Now here this falls between 1967 and And, fall between 1968 and So again we take the average of this and of i.e. ( )/2. This gives me the value Now and we place this value again in between these two years and it comes out that it will be placed against the year So now we have the entry for the year 1968 in this way.

8 And again then we leave the first moving average and include the next moving average i.e and take the average of and And the average of this is now placed against the year So this way we complete this column. So this column gives you the moving averages centered against the particular year. So in the last example and in this example the main difference was their period of moving average was 3 i.e. an odd number. And in this case the period of moving average which is 4 is an even number. Now if I plot these data on a graph then we see that the original data are plotted and join by the straight lines and shown here with the blue color. And the moving averages given in the last column i.e. the column 4 are plotted and joined by this red line. Now we can see that if I look at the graph of the original data then it is difficult for us to make any prediction or to measure any trend present in the data. But once we compute the moving averages and when we look at the graph of the moving average then what it shows? It shows that there is an increasing trend present in the data. That is, the gross capital formation is increasing over the period of years. And also this graph shows that some cyclic fluctuations are present in the data. So even we can use some methods to isolate the cyclic fluctuations then it will gives us more clear picture about the presence of the secular trend in the data. We shall discuss in my upcoming lectures, that how to isolate the cyclic fluctuations present in the data.

9 5. EFFECTS OF MOVING AVERAGE METHOD In our last two examples we have seen that m denotes the period of moving average. And the value of m may be either an odd number as the case of the first example and the value of m may be an even number. So this m may be equal to (2k+1) i.e. an odd no or the m may be equal to 2k, where k takes different values 1, 2,.... If m=(2k+1) i.e. an odd no, moving averages are placed against the middle value of the time interval it covers. If m=(2k) i.e. an even no., it is placed between the two values of the time interval it covers. And then these moving averages are centered. So if I summarize this whole method then, what this method is? To obtain the moving average, take a few years data of the series, i.e. the period of moving average, in a group and find their average. Then delete or ignore the first year from this group of years and add succeeding year to this group. In this way the member of observed values for averaging remains the same. Enter the average values of each group against the mid year of that group. The location of midyear is straight forward in case when period of moving average is an odd number. In other case where period of moving average is an even number, location of midyear involves one more step i.e. centering the moving averages. Now the question arises: What should be the value of m=??

10 In the first example we have taken the value of m as 3. And in second example we have taken the value of m =4. So, what is the criterion? How to decide the value of m? The value of m should be one which gives the moving averages which completely eliminates the oscillatory movements. Thus, the value of m will be equal to the average period of short term cyclic variations or the average of the periods between the peaks. Now this we can understand by looking at this graph. We know that here the original data are joined by this blue color straight line. So they show the graph of the original data. And now we see the peaks. The first peak is given against the year 1967, and then the next low peak is given as for the So this way, we can find the difference between the periods of the peaks i.e and here is So this period is of 2 years. This way we can note down the periods of the peak and the difference between the periods of these peaks looking at this graph. And once we note down these figures we take the average of these and that will give the value of m. The moving average is considered to be a better method for finding secular trend provided (a) the trend is linear; (b) the cyclical variations are regular both in period and amplitude. So the merits of this method are: It is a very simple and flexible method for measuring trend. And

11 The greatest advantage of this method is that it reduces the effect of extreme values for respective years to a great extent. Whereas the demerits of this method are: 1. Few of the years in the beginning and at the end are such that the average values are not entered against them. This entails the loss of information. We have seen in our example, that for the first two years we did not have any moving average and also for the last two years we did not have any moving average. Where we have taken the period of moving average as 4 i.e. the value of m was This method is not suitable for the comparison of the two series as it is not fully mathematical. 3. The moving average method takes care of cyclic and short fluctuations very well. But there is subjectivity in deciding the number of years (periods) in group for the moving average. This can lead to any number of trend lines for the same time series data. 6. SUMMARY In today s lecture we have introduce the methods of measuring the trend. There are mainly two methods: one is the moving average method and the other one is the curve fitting method.

12 Now today we have discussed this moving average method in detail. And we have seen that how you can measure the effect of the trend present in the data. And how you can isolate the effect of the trend present in the data in order to study the other components of the time series? In my last lecture we shall continue with our discussions on the measure of secular trend. Thank You!