An example of a discrete time series is the population of Korea as shown in [Table 13.1.1]. This data is from the census conducted every five years in Korea from 1925 to 2020 (except for 1944 and 1949).

As shown in the table above, it is not easy to understand the overall shape of the time series displayed in numbers. The first step in time series analysis is to observe the time series by drawing a time series plot with the X axis as time and the Y axis as time series values. For example, the time series plot of the total population in Korea is shown in <Figure 13.1.1>.

<Figure 13.1.1>Time Series of Korea Population

Observing this figure, Korea's population has an overall increasing trend, but the population decreased sharply in 1944-1949 due to World War II. It can be seen that the population expanded rapidly after the Korean war in 1953 and slowed since 1990. It can also be seen that the growth has slowed further in the last 10 years. By observing the time series in this way, trends, change points, and outliers can be observed, which is helpful in selecting an analysis model or method suitable for the data.

Time series that we frequently encounter include monthly sales of department stores and companies, daily composite stock index, annual crop production, yearly export and import time series, and yearly national income and economic growth rate, and so on.

[Table 13.1.2] shows the percent increase in monthly sales of the US toy/game industry for the past 6 years, and <Figure 13.1.2> is a plot of this time series. As it is the rate of change from the previous month, it can be observed that it is seasonal data showing a large increase in November and December every year, moving up and down based on 0. However, May 2020 is an extreme with an increase rate of 211% unlike other years. For time series, you can better examine the characteristics of the data by converting the raw time series into the rate of change.

<Figure 13.1.2>Percent Increase, Monthly Sales of Toy/Game in US(%)

Most time series have four components: trend, seasonal, cycle, and other irregular factors. Trend is a case in which a time series has a certain trend, such as a line or a curved shape as time elapses, and there are various types of trends. Trends can be understood as a consumption behavior, population variations, and inflation that appear in time series over a long period of time. Seasonal factors are short-term and regular fluctuation factors that exist quarterly, monthly, or by day of the week. Time series such as monthly rainfall, average temperature, and ice cream sales have seasonal factors. Seasonal factors generally have a short cycle, but fluctuations when the cycle occurs over a long period of time rather than due to the season is called a cycle factor. By observing these cyclical factors, it is possible to predict the boom or recession of a periodic economy. <Figure 13.1.3> shows the US S&P 500 Index from 1997 to 2016, and a six-year cycle can be observed.

<Figure 13.1.3>] US S&P500 Index (1997- 2016)

Other factors that cannot be explained by trend, season, or cyclical factors are called irregular or random factors, which refer to variable factors that appear due to random causes regardless of regular movement over time.

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The N-point centered moving average of a time series refers to the average of N data from a single point in time. For example, in crude oil price data, the value of the five-point moving average for a specific year is the average of the data for two years before the specific year, that year, and the data for the next two years. Expressed as an expression, if \(M_t\) is a moving average in time \(t\), the 5-point centered moving average is as follows: $$ M_t = \frac{Y_{t-2} \,+\, Y_{t-1} \,+\, Y_{t} \,+\, Y_{t+1} \,+\, Y_{t+2} } {5 } $$ For example, the 5-point centered moving average for 1989 is as follows.

\(\qquad M_{1989} \,=\, \frac {Y_{1987} + Y_{1988} +Y_{1989} + Y_{1990} + Y_{1991} } {5 } \)
\( \qquad \qquad \quad =\, \frac {16.74 + 17.12 + 21.84 + 28.48 + 19.15} {5} \,=\, 20.6660 \)

[Table 13.2.2] shows the values ​​of all 5-points centered moving averages obtained in this way and <Figure 13.2.1> is the graph of 5-points moving average. Note that the moving averages for the first two years and the last two years cannot be obtained here. It can be seen that the graph of the moving average is better for grasping the long-term trend than the graph of the original data because short-term fluctuations are removed.

The choice of a value N for the N-point moving average is important. A large value of N will provide a smoother moving average, but it has the disadvantage of losing more points at both ends and insensitive to detecting important trend changes. On the other hand, if you choose small N, you will lose less data at both ends, but you may not be able to get the smoothing effect because you will not sufficiently eliminate short-term fluctuations. In general, try a few values N​​ to reflect important changes that should not be missed, while achieving a smoothing effect and balancing the points not to lose too much at both ends.

If the value of N is an even number, there is a difficulty in obtaining a central moving average with the same number of data on both sides of the base year. For example, the center of the four-point moving average from 1987 to 1990 is between 1988 and 1989. If you denote this as \(M_{1988.5}\), it can be calculated as follows:

\( \qquad M_{1988.5} \,=\, \frac {Y_{1987} + Y_{1988} +Y_{1989} + Y_{1990} } {4 } \)
\( \qquad \qquad \quad \,=\, \frac {16.74 + 17.12 + 21.84 + 28.48 } {4} \,=\, 21.045 \)

The 4-point moving average obtained in this way is called a non-central 4-points moving average. In the case of this even number N, the non-central moving average does not match the observation year of the original data, which is inconvenient. In the case of this even number N, it is calculated as the average of the noncentral moving average values of two adjacent non-central moving averages. In other words, the central four-point moving average in 1989 is the average of \(M_{1988.5}\) and \(M_{1989.5}\) as follows:

\( \qquad M_{1989} \,=\, \frac {M_{1988.5} \,+\, M_{1989.5} } {2 } \)
\( \qquad \qquad \quad =\, \frac {21.0450 \,+\, 21.6475 } {2} \,=\, 21.3463 \)

If the time series is quarterly or monthly, a 4-point central moving average or a 12-point central moving average is an average of one year, so it is often used to observe data without seasonality.

3-point moving average can be considered the weighted average of three data with each weight \(\frac{1}{3}\) as follows: $$ M_t \,=\, \frac{Y_{t-1} \,+\, Y_{t} \,+\, Y_{t+1} } {3 } \,=\, \frac{1}{3}Y_{t-1} \,+\, \frac{1}{3}Y_{t} \,+\, \frac{1}{3}Y_{t+1} $$ When the weights are \(w_1 , w_2 , ... , w_n\), the weighted moving average \(M_t\) of the time series is defined as follows: $$ M_t \,=\, \sum_{i=1}^{n} w_i Y_{i} $$ where \(n\) is the number of data, \(w_i \ge 0\) and \(\sum_{i=1}^{n} w_i = 1 \).

Various weighted averages with different weights can be used depending on the purpose. Among them, a smoothing method that gives more weight to data closer to the present and smaller weights as it is farther from the present is called exponential smoothing. The exponential smoothing method is determined by an exponential smoothing constant \(\alpha\) that has a value between 0 and 1. The exponentially smoothed data \(E_t\) is calculated as follows:

\( \qquad E_{1} \,=\, \alpha \,Y_{1} \,+\, (1- \alpha)\, E_{0} \)
\( \qquad E_{2} \,=\, \alpha \,Y_{2} \,+\, (1- \alpha)\, E_{1} \)
\( \qquad E_{3} \,=\, \alpha \,Y_{3} \,+\, (1- \alpha)\, E_{2} \)
\( \qquad \cdots \)
\( \qquad E_{t} \,=\, \alpha \,Y_{t} \,+\, (1- \alpha)\, E_{t-1} \)

Here, an initial value \(E_{0}\) is required, and \(Y_1\) is usually used a lot, and the average value of the data can also be used. The exponentially smoothed value \(E_t\) at the point in time \(t\) gives weight \(\alpha\) to the current data, and the \(1-\alpha\) weight to the previous smoothed data is given. The exponentially smoothed value \(E_t\) can be represented with the original data \(Y_t\) as follows: $$ E_t \,=\, \alpha Y_t + \alpha (1-\alpha) Y_{t-1} + \alpha(1-\alpha)^2 Y_{t-2} + \cdots + \alpha(1-\alpha)^{t-2} Y_2 + \alpha(1-\alpha)^{t-1} Y_1 + (1-\alpha)^t E_0 $$ Therefore, the exponential smoothing method uses all data from the present and the past, but gives the current data the highest weight α, and gives a lower weight as the distance from the present time increases.

Exponential smoothing of the crude oil price in [Table 13.2.1] with the initial value \(E _{1986} = Y _{1987} \) and exponential smoothing constant \(\alpha\) = 0.3 is as follows.

\( \qquad E_{1986} \,=\, E_{1987} = 16.74 \)
\( \qquad E_{1987} \,=\, 0.3 \,Y_{1987} \,+\, (1- 0.3)\, E_{1986} \,=\, (0.3)(16.74)+(0.7)(16.74)=16.74 \)
\( \qquad E_{1988} \,=\, 0.3 \,Y_{1988} \,+\, (1- 0.3)\, E_{1987} \,=\, (0.3)(17.12)+(0.7)(16.74)=16.854 \)

All data exponentially smoothed with \(\alpha\) = 0.3 are given in [Table 13.2.2]. It can be seen that, in the exponential smoothing method, there is no loss of data at both ends, unlike the moving average method. The crude oil price time series and exponentially smoothed data are shown in <Figure 13.2.2>. It can be seen that the smoothed data are not significantly different from the original data. If the value of \(\alpha\) is small, more weight is given to the past data than to the present, making it less sensitive to sudden changes in the present data. Conversely, the closer the value of \(\alpha\) is to 1, that is, the more weight is given to the current data, the more the smoothed data resembles the original data, and the smoothing effect disappears.

<Figure 13.2.2> Price of Crude Oil and Exponential Smoothing with α=0.3

For example, the 1989 5-point central moving median for crude oil prices in [Table 13.2.3] is as follows:

\(\qquad MovingMedian_{1989} \,=\, median \{ Y_{1987} , Y_{1988} ,Y_{1989} , Y_{1990} , Y_{1991} \} \) \(\qquad \qquad \qquad \qquad \qquad \;\;=\, median \{16.74, 17.12 , 21.84 , 28.48,19.15 \} \,=\, 19.15 \)

[Table 13.2.3] and <Figure 13.2.3> show all the five-point moving median values obtained in this way and their graphs. Note that the moving median for the first two years and the last two years are not available here. Because the centered moving medians remove extreme values, it is called a filtering and the time series is much smoother than the original data.

<Figure 13.2.3> Price of Crude Oil and 5-point Centered Moving Median

If the value of N is an even number, there is a difficulty in obtaining the central moving median having the same number of data on both sides of the base year. For example, the center of the four-point moving median from 1987 to 1990 is between 1988 and 1989. If you denote this as \(Median_{1988.5}\), it can be calculated as follows:

\(\qquad MovingMedian_{1988.5} \,=\, median \{Y_{1987} , Y_{1988} , Y_{1989} , Y_{1990} \} \) \(\qquad \qquad \qquad \qquad \qquad \;=\, median \{16.74 , 17.12 , 21.84 , 28.48 \} \,=\, \frac {17.12 +21.84} {2} = 19.48 \)

The 4-point moving median obtained in this way is called the non-central 4-point moving median. As such, the non-central moving average in the case of this even number N does not match the observation year of the original data, which is inconvenient. In the case of this even number, it is calculated as the average of the values of the two non-central moving medians that are adjacent to each other. In other words, the central four-point moving median in 1989 is the mean of \(MovingMedian_{1988.5}\) and \(MovingMedian_{1989.5}\).

A. Percent Change

\(\qquad P_{2014} \,=\, \frac{19161.2 - 18742.1} {18742.1} \times 100 \,=\, 2.23 \)

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[Table 13.3.2] is a simple index for the number of houses in Korea from 2010 to 2020, with the base time being 2010. If you look at the figure for the index, you can see that in this case, there is no significant change from the original time series and trend. It can be seen that there is a 22.18% increase in the number of houses in 2020 compared to 2010.

\(\qquad Index_{2020} \,=\, \frac{Y _{2020}} {Y_{2010}} \times 100 \,=\, \frac{21673.5} {17738.8} \times 100 \,=\, 122.18 \)

<Figure 13.3.2> Simple Index of Number of Houses in Korea

For the composite index, a weighted composite index that is calculated by weighting the price of each product with the quantity consumed is often used. When calculating such a weighted composite index, the case where the quantity consumption at the base time is used as a weight is called the Laspeyres method, and the case where the quantity consumption at the current time is used as the weight is called the Paasche method. In general, the Laspeyres method of weighted composite index is widely used, and the consumer price index is a representative example. The price index of the Paasche method is used when the consumption of goods used as weights varies greatly over time, and can be used only when the consumption at each time point is known. It is expensive to examine the quantity consumption at each point in time.

Assuming that \(P_{1t} , \cdots , P_{kt} \) are the prices of \(k\) number of products at the time point \(t\), and \(Q_{1t_0} , \cdots , Q_{kt_0} \) are the quantities of each product consumpted at the base time, the formula for calculating each composite index is as follows:

\( \qquad \text{Laspeyres Index:} \quad Index_t \,=\, \frac { Q_{1t_0} P_{1t} + \cdots + Q_{kt_0} P_{kt} } {Q_{1t_0} P_{1t_0}+ \cdots + Q_{kt_0} P_{kt_0} } \times 100 \)
\( \qquad \text{Paasche Index:} \qquad Index_t \,=\, \frac { Q_{1t} P_{1t} + \cdots + Q_{kt} P_{kt} } {Q_{1t} P_{1t_0} + \cdots + Q_{kt} P_{kt_0} } \times 100 \)

The data in [Table 13.3.3] shows the price and quantity of three metals by month in 2020.

In [Table 13.3.3], the Laspeyres index for the data for February with January as the base time is as follows.

\( \qquad Index_t \,=\, \frac { Q_{1t_0} P_{1t} + \cdots + Q_{kt_0} P_{kt} } {Q_{1t_0} P_{1t_0}+ \cdots + Q_{kt_0} P_{kt_0} } \times 100 \)
\( \qquad \quad \,=\, \frac {(100.7)(1399.0)+(4311)(213)+(46.1)(520) } {(100.7)(1361.6)+(4311)(213)+(47.0)(530) } \,=\, 100 .31 \)

Similarly, Paasche index is as follows:

\( \qquad Index_t \,=\, \frac { Q_{1t} P_{1t} + \cdots + Q_{kt} P_{kt} } {Q_{1t} P_{1t_0} + \cdots + Q_{kt} P_{kt_0} } \times 100 \)
\( \qquad \quad \,=\, \frac {(95.1)(1399.0)+(4497)(213)+(47.0)(520) } {(95.1)(1361.6)+(4497)(213)+(47.0)(530) } \,=\, 100.28 \)

In [Table 13.3.3], it can be seen that the production quantity of iron and lead in the last 4 quarters is significantly different from the production quantity in January, which is the base time. In this way, when the quantity fluctuates greatly and the quantity at each time is known, the Pasche index can be said to be the best index because it appropriately reflects the price change at that time.

The correlation coefficient between the time lag data and the raw data is called the autocorrelation coefficient. If the average of time series is \(\small \overline Y\) , the k-lag autocorrelation \(r_k\) is defined as follows: $$ \small r_k = \frac { \sum_{t = k+1}^ n (Y_t - \overline Y ) (Y_{t-k} - \overline Y ) } { \sum _{t =1}^n (Y_t - \overline Y )^2 }, \quad k=0, 1, 2, ..., n-1 $$ \(r_1 , r_2 , ... , r_k \) are called an autocorrelation function and are used to determine a time series model.

[Table 13.3.4] shows the monthly consumer price index for the past two years and time lag 1 to 12 for this data, and the autocorrelation coefficients are shown in [Table 13.3.5]. <Figure 13.3.3> shows the original time series and the autocorrelation function.

<Figure 13.3.3> Autocorrelation Function Graph

<Figure 13.3.4> shows the first order differencing of [Table 13.3.4] time series and it becomes horizontal series.

<Figure 13.3.4> 1st Order Differencing of Consumer Price Index

\( \qquad \text{Log function } \qquad\qquad \qquad W = log(Y) \)
\( \qquad \text{Square root function} \qquad\quad W = \sqrt(Y) \)
\( \qquad \text{Square function } \qquad\quad \qquad W = Y^2 \)
\( \qquad \text{Box-Cox Transformation} \quad \; W = \frac {Y^p - 1 }{p} \) if p ≠ 0, else log(Y)

[Table 13.3.6] is a toy company's quarterly sales, and <Figure 13.3.5> is a diagram of this data. It is a seasonal data by quarter, but, as time goes on, dispersion of sales increases over time. It is not easy to apply a time series model to data with this increasing dispersion over time. In this case, log transformation \(W = log(Y)\) can reduce the dispersion as time increases, as shown in <Figure 13.3.5>, so that a model can be applied. After predicting by applying the model to log-transformed data, exponential transformation \(Y = exp(W)\) is performed again to predict the raw data.

<Figure 13.3.5> Log Transformation of Toy Sales

The square root transform is used for a similar purpose to the log transform, and the square transform can be used when the variance decreases with time. The Box-Cox transform is a general transformation.

When the estimated regression coefficients are \(\beta_0 , \beta_1\), the validity test of the linear regression model is the same as the method described in Chapter 12. The standard error of estimate and coefficient of determination are often used. In a linear trend model, \(\sigma\) represents the degree to which observations can be scattered around the estimated regression line at each time point. As an estimate of this \(\sigma\), the following standard error is used. $$ s \,=\, \sqrt { \frac{1} {n-2} \sum_{t=1}^n ( Y_t - {\hat Y}_t )^2 } $$ A smaller standard error \(s\) value indicates that the observed values are close to the estimated regression line, which means that the regression line model is well fitted.

The coefficient of determination is the ratio of the regression sum of squares, RSS, which is explained out of the total sum of squares, TSS. $$ R^2 \,=\, \frac{RSS}{TSS} $$ The value of the coefficient of determination is always between 0 and 1, and the closer the value is to 1, the more dense the samples are around the regression line, which means that the estimated regression equation explains the observations well.

As explained in Chapter 12, since it is difficult to determine the absolute criteria for adequacy of the standard error or the coefficient of determination, a hypothesis test is used to determine whether the trend parameter \(\beta_1\) is zero or not.

\(\small \qquad \text{Hypothesis: } \qquad \,\, H_0 : \beta_1 = 0, H_1: \beta_1 \ne 0\)
\(\small \qquad \text{Test statistic:} \qquad t_{obs} = \frac {{\hat \beta}_1 } { SE ({{\hat \beta}_1 }) } \) , Here, \( SE( \hat{ \beta_1} ) \,=\, \frac{ s} { \sqrt { \sum_{i=1}^ n (i - \overline t )^2 } } \)
\(\small \qquad \text{Rejection region:} \quad If \,\; |t_{obs}| \,>\, t_{n-2,\alpha/2}, \,\,reject\,\, H_0 \) with significance level \(\alpha\)

If the null hypothesis \(H_0\) is not rejected, the model cannot be considered valid.

The assumption for error \(\epsilon_t\) is tested using the residual, which is the difference between the observed time series value and the predicted value which is called residual analysis. Residual analysis usually examines whether assumptions about error terms such as independence and equal variance between errors are satisfied by drawing a scatter plot of the residuals over time or a scatter plot of the residuals and predicted values. In the scatterplots, if the residuals do not show a specific trend around 0 and appear randomly, it means that each assumption is valid. To examine the normality assumption of the error term, draw a normal probability plot of the residuals, and if the points on the figure show the shape of a straight line, it is judged that the assumption of the normal distribution is appropriate.

If the linear regression model is suitable, the predicted value \({\hat Y}_{t_0} \,=\, {\hat \beta}_0 \,+\, {\hat \beta}_1 \cdot t_0 \) at the time point \(t_0\) can be interpreted as a point estimate for the mean of the random variable \(Y_{t_0}\) at the time point, and the confidence interval for the mean of \({\hat Y}_{t_0}\) at this time \(t_0\) is as follows: $$\small {\hat Y}_{t_0} \,±\, t_{n-2,\alpha/2} \cdot SE ({\hat Y}_{t_0} ) \;\; where \;\, SE ( {\hat Y}_{t_0} ) \,=\, s \cdot \sqrt { \frac{1}{n} + \frac {(t_0 - \overline t )^2} { \sum_{i=1}^n (i - \overline t )^2 } } $$

If the trend is in the form of a quadratic, cubic or higher polynomial, the following multiple linear regression model can be assumed.

\( \qquad \text{Quadratic} \qquad Y_t = \beta_0 + \beta_1 \cdot t + \beta_2 \cdot t^2 + \epsilon _t \)
\( \qquad \text{Cubic} \qquad \qquad Y_t = \beta_0 + \beta_1 \cdot t + \beta_2 \cdot t^2 + \beta_3 \cdot t^3 + \epsilon _t \)

The prediction method is similar to the above simple linear regression model.

If the trend is not a polynomial model as above, the following model can also be considered.

\( \qquad \text{Square root} \qquad Y_t = \beta_0 + \beta_1 \cdot \sqrt{t} + \epsilon _t \)
\( \qquad \text{Log} \qquad \qquad \quad \; Y_t = \beta_0 + \beta_1 \cdot log(t) + \epsilon _t \)

These models are the same as the linear regression model if \(\sqrt{t}\) or \(log(t)\) are replaced with the independent variable X in the simple linear regression and the prediction method is similar.

In addition, the function types to which the linear regression model can be applied by transformation are as follows.

\( \qquad \text{Power} \qquad \qquad Y _{t} = \beta_{0} \cdot t ^{\beta _{1}} + \epsilon _{t} \)
\( \qquad \text{Exponential} \quad \;\; Y_t = \beta_0 \cdot e^ {( \beta_1 t)} + \epsilon _t \)

In the case of these two models, the parameters should be estimated using the nonlinear regression model, but if the error term is ignored, the linear model can be estimated approximately as follows:

\( \qquad \text{Power} \qquad \qquad log(Y_{t}) = log(\beta_{0}) +{\beta _{1}} \cdot log(t) \)
\( \qquad \text{Exponential} \quad \;\; log(Y_t) = log(\beta_0) + \beta_1 \cdot t \)

Korea's GDP from 1986 to 2021 is shown in [Table 13.4.1]. <Figure 13.4.1> shows the application of three regression models to this data. Among these models, the quadratic model has the largest value of \(r^2\) = 0.9591, so it can be said that the time series is the most suitable model. However, additional validation of the model is required.

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13.5 Exponential Smoothing Model and Forecasting

13.5.1 Stationary Time Series

A. Single Moving Average Model

In a stationary time series model, the estimated value of \(\mu\), \(\hat \mu\), is the mean of the data. . $$ {\hat \mu} \;=\; \frac{1}{T} \sum_{ i=1}^T Y_i $$ Using this model, the prediction after \(\tau\) time points ahead at the current time \(T\) denoted as \({\hat Y}_{T+\tau}\), is as follows: $$ {\hat Y}_{T+\tau} \;=\; \hat \mu, \quad \tau=1,2,... $$ It is called a simple average model.

The simple average model uses all observations until the current time. However, the unknown parameter \(\mu\) may shift slightly over time, so it would be reasonable to give more weight to recent data than to past data for prediction. If a weight \(\frac{1}{N}\) is given to only the most recent \(N\) observations at the present time \(T\) and the weight of the remaining observations is set to 0, the estimated value of \(\mu\) is as follows. $$ {\hat \mu} \;=\; \frac{1}{N} \sum_{ i=T-N+1}^T Y_i \;=\; \frac{1}{N} ( Y_{T-N+1} + Y_{T-N+2} + \cdots + Y_T ) $$ This is called a single moving average at the time point \(T\) and it is denoted by \(M_T\). The single moving average means the average of the \(N\) observations adjacent to the time point \(T\). Notice that \(Y_1, Y_2 , ... , Y_T\) are independent of each other by assumption, but \(M_1, M_2 , ... , M_T\) are not independent of each other, but are correlated.

The value of the single moving average varies depending on the size of \(N\). When the value of \(N\) is large, it becomes insensitive to the fluctuations of the original time series, so it changes gradually, and when the value of \(N\) is small, it becomes sensitive to fluctuations. Therefore, when the fluctuation of the original time series is small, it is common to set the small value of \(N\), and when the fluctuation is large, it is common to set the value of large \(N\).

Using the single moving average model at the time point \(T\), the predicted value and the mean and variance of the predicted value at the time point \(T+\tau\) are as follows.

\(\qquad {\hat Y}_{T+\tau} \;=\; M_T , \quad \tau=1,2, ... \)
\(\qquad E( {\hat Y}_{T+\tau} ) \;=\; E(M_T ) \;=\; \mu \)
\(\qquad Var({\hat Y}_{T+\tau} )\;=\; Var(M_t ) \;=\; \frac{ \sigma ^2} {N } \)

When the single moving average model is used, the 95% confidence interval estimation of the predicted value is approximately as follows.

\(\qquad {\hat{Y}} _{T+ \tau } \;±\; 1.96 \sqrt {Var( {\hat{Y}} _{T+ \tau } )} \)
\(\qquad M_{T} \;±\; 1.96 \sqrt{ \frac{MSE} {N} } \)

The monthly sales for the last two years of a furniture company are as shown in [Table 13.5.1], and the residual between the raw data and the predicted value of one point in time was calculated by obtaining a six-point moving average. <Figure 13.5.1> shows the time series for this. This time series fluctuates up and down based on approximately 95, and such a time series is called a stationary time series.

When \(N\) = 6, the moving average for the first 5 time points cannot be obtained. The moving average at time 6 is as follows: $$ M_6 \;=\; \frac{95+100+87+123+90+96} {6} \;=\; 98.50 $$ Therefore, one time prediction at time 6 becomes \({\hat Y}_{6+1} = 98.50 \) and the residual at time 7 is as follows: $$ e_{7} \;=\; Y_{7} \;-\; {\hat{Y}}_{6+1} \;=\; 75-98.50 \;=\; -23.50 $$ In the same way, the moving average of the remaining time points, the predicted values ​​after one time point, and the residuals are as shown in [Table 13.5.1], so the mean square error is as follows: $$ { MSE} \;=\; \frac { \sum_{ i=7}^{ 24} ( Y_i \;-\; {\hat Y}_i )^{2} } {18} \;=\;331.22 $$ Sales for the next three months are the last moving average \(M_{24}\), and the 95% confidence interval for the forecast is as follows:

\(\qquad {\hat Y}_{24+1} \;=\; {\hat Y}_{24+2} \;=\; {\hat Y}_{24+3} \;=\; M_{24} \;=\; 104.67 \)
\(\qquad M_T \;±\; 1.96 \sqrt \frac{MSE}{N } \)
\(\qquad 104.67 \;±\; 1.96 \sqrt \frac{331.22}{6} \)
\(\qquad [90.10, 119.23] \)

[Table 13.5.1] Montly Sales of a Furniture Company and 6-point Moving Average, One Time Forecast and Residuals

time \(t\)	Sales (unit million $) \(Y_t\)	6-pt Moving Average \(M_t\)	One Time Forecast \({\hat Y}_t\)	Residual \(e_t = Y_t - {\hat Y}_t\)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24	95 100 87 123 90 96 75 78 106 104 89 83 118 86 86 112 85 101 135 120 76 115 90 92	98.50 95.17 91.50 94.67 91.50 91.33 89.17 96.34 97.67 94.34 95.67 95.00 98.00 100.84 106.50 104.84 105.34 106.17 104.67	98.50 95.17 91.50 94.67 91.50 91.33 89.17 96.34 97.67 94.34 95.67 95.00 98.00 100.84 106.50 104.84 105.34 106.17	-23.50 -17.17 14.50 9.33 -2.50 -8.33 28.83 -10.33 -11.67 17.67 -10.67 6.00 37.00 19.17 -30.50 10.17 -15.33 -14.17

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<Figure 13.5.1> Montly Sales of a Furniture Company and 6-point Moving Average and One Time Forecast

※ Moving average at initial period

Since the \(N\)-point single moving average cannot be obtained before the time point \(N\), the prediction model cannot be applied. When there are many time series, this may not be a big problem, but when the number of data is small, it can affect the prediction. In order to solve this problem, the moving average at initial period can be obtained as follows until the time point \(N-1\).

\(\qquad t=1 \qquad \quad M_1 = Y_1 \)
\(\qquad t=2 \qquad \quad M_2 = \frac{Y_1 + Y_2 } {2} \)
\(\qquad \cdots \)
\(\qquad t=N-1 \quad M_{N-1} = \frac{Y_1 + Y_2 + \cdots + Y_{N-1} } {N - 1} \)

B. Single Exponential Smoothing Model

In the single moving average model, the same weight \(\frac{1}{N}\) is given to only the latest \(N\) observations, and the previous observations are completely ignored by setting the weight to 0. The single exponential smoothing method compensates for the shortcomings of the moving average model by assigning weights to all observations when predicting future values from past observations, but giving more weight to recent data. This single exponential smoothing model uses the value of the exponential smoothing method as the predicted value.

The single exponential smoothing model calculates the weighted average of the exponential smoothing estimator \({\hat \mu}_{t-1} \) at the immediately preceding time point and the observation value \(Y_t\) at the time point \(t\). Assuming that the exponential smoothing estimated value at the time point \(t\) is \(S_t \;=\; {\hat \mu}_t \) and \(\alpha\) is a real number between 0 and 1, the single exponential smoothing value \(S_t\) is defined as follows:

\(\qquad S_1 \;=\; \alpha \;Y_1 \;+\; (1-\alpha) S_0 \)
\(\qquad S_2 \;=\; \alpha \;Y_2 \;+\; (1-\alpha) S_1 \)
\(\qquad \cdots \)
\(\qquad S_t \;=\; \alpha \;Y_t \;+\; (1-\alpha) S_{t-1} \)

Here, \(\alpha\) is called the smoothing constant, and the single exponential smoothing value \(S_t\) is the weighted average value given the weight \(\alpha\) of the most recent observation \(Y_t\) and the weight (1-\(\alpha\)) of the exponential smoothing value \(S_{t-1}\) at the time \(t-1\). You can better understand the meaning of exponential smoothing if you write down the recursive equation as follows:

\(\qquad S_t \;=\; \alpha \;Y_t \;+\; (1-\alpha) S_{t-1} \)
\(\qquad \;=\; \alpha \; Y_t + (1- \alpha )\;( \alpha Y_{t-1} + (1- \alpha ) S_{t-2} ) \)
\(\qquad \;=\; \alpha Y_t + \alpha (1- \alpha ) Y_{t-1} + (1- \alpha )^2 S_{t-2} \)
\(\qquad \;=\; \alpha Y_t + \alpha (1- \alpha ) Y_{t-1} + \alpha(1- \alpha )^2 Y_{t-2} \;+\cdots\; + \alpha (1- \alpha )^{(t-1)} Y_1 +(1- \alpha )^t S_0 \)

In other words, for the single exponential smoothing value \(S_t\), the most recent observation \(Y_t\) is given a weight \(\alpha\), and the next most recent observation is given \(\alpha(1-\alpha)\), the next is \(\alpha(1-\alpha)^2\) and so on, a gradually smaller weight. Therefore, if the size of \(\alpha\) is small, the current observation value is given a small weight, and the exponential smoothing value is insensitive to the fluctuations of the time series. if the size of \(\alpha\) is large, the current observation value is given a large weight, and the exponential smoothing value is sensitive to the fluctuations of the time series. In general, a value between 0.1 and 0.3 is often used as the value of \(\alpha\).

In order to obtain a single exponential smoothing value, an initial smoothing value \(S_0\) is required, and the first observation value or the sample average of several initial data or the overall sample average can be used. The exponential smoothing method has the advantage of being less affected by extreme point or intervention than the ARIMA model and easy to use, although the selection of the smoothing constant is arbitrary and it is difficult to obtain a prediction interval.

The predicted value, average and variance of the predicted value at the time point \(T+\tau\) using the single exponential smoothing model are as follows:

\(\qquad {\hat Y}_{T+\tau} \;=\;S_T \)
\(\qquad E( {\hat Y}_{T+\tau} )\;=\;E(S_T )\;=\; \mu \)
\(\qquad Var( {\hat{Y}}_{T+ \tau } )\;=\; Var(S_{T} )\;=\; \frac{\alpha } {2- \alpha } \sigma^{2} \)

Therefore, when the single exponential smoothing model is used, the 95% interval estimation in the predicted value is approximately as follows.

\(\qquad {\hat Y}_{T+\tau} \;±\; 1.96 \sqrt { Var ({\hat Y}_{T+\tau})} \)
\(\qquad S_{T} \;±\;1.96\; \sqrt { \frac{\alpha } {2- \alpha } MSE} \)

To the data of [Table 13.5.1], predict sales for the next three months by a single exponential smoothing model with smoothing constant \(\alpha\) = 0.1. Lets use the first observed value for the initial value of exponential smoothing, that is \(S_0 = Y_1 = 95\). The exponential smoothing value for the first three time series are as follows:

\(\qquad S_{1} \;=\;0.1\; \times \;Y _{1} \;+\;(1-0.1\;)\; \times \;S _{0} \;=\;\;0.1\; \times \;95\;+\;0.9\; \times \;95\;=\;95 \)
\(\qquad S_{2} \;=\;0.1\; \times\;Y_2 \;+\; (1-0.1\;)\;\times\; S_1 \;=\; \;0.1\;\times\;100\;+\;0.9 \;\times \;95 \;=\; 95.50 \)
\(\qquad S_{3} \;=\;0.1\; \times \;Y _{3} \;+\;(1-0.1\;)\; \times \;S _{2} \;=\;\;0.1\; \times \;87\;+\;0.9\; \times \;95.5\;=\;94.65 \)
\(\qquad \cdots \)

At each time point, the prediction after one point in time is as follows:

\(\qquad {\hat Y}_{0+1} \;=\; S_0 \;=\;95.00 \)
\(\qquad {\hat Y}_{1+1} \;=\; S_1 \;=\;95.00 \)
\(\qquad {\hat Y}_{2+1} \;=\; S_2 \;=\;95.50 \)

Hence the residuals using the above estimated values are as follows:

\(\qquad e_1 \;=\; Y_1\;-\;{\hat Y}_{0+1} \;=\; 95.00 - 95.00 \;=\; 0 \)
\(\qquad e_2 \;=\; Y_2\;-\;{\hat Y}_{1+1} \;=\; 100.00 - 95.00 \;=\; 5.00 \)
\(\qquad e_3 \;=\; Y_3\;-\;{\hat Y}_{2+1} \;=\; 87.00-95.50\;=\;-8.50 \)

In the same way, the single exponential smoothing of the remaining time points, the predicted values after one time point, and the residuals are as shown in [Table 13.5.2]. Therefore, the mean square error is as follows:

\(\qquad {MSE} \;= \; \frac{1}{24} \sum_{i=1}^{24} \; ( Y_i \;-{\hat Y}_i )^{2} \;=\;269.72 \)

In terms of mean square error, the MSE of the 6-point single moving average model is 331.22, so it can be said that the exponential smoothing model has better fit.

Sales for the next three months are the last moving average \(S_{24}\), and the 95% confidence interval for the forecast is as follows:

\(\qquad {\hat Y}_{24+1} \;=\; {\hat Y}_{24+2} \;=\; {\hat Y}_{24+3} \;=\; S_{24} \;=\; 98.66 \)
\(\qquad S_{T} \;±\; 1.96\; \sqrt { \frac{\alpha } {2- \alpha} } MSE \)
\(\qquad 98.66 \;±\; 1.96 \; \sqrt { \frac{0.1} {2-0.1} 269.72 } \)
\(\qquad [65.27, 132.05] \)

[Table 13.5.2] summarizes the above equations, and <Figure 13.5.2> shows the prediction after one time point and the prediction for the next 3 months using the single exponential smoothing model with \(\alpha\) = 0.1.

[Table 13.5.2] Exponential Smoothing with α = 0.1, One Time Forecast and Residual

time \(t\)	Sales (unit million $) \(Y_t\)	Exponential Smoothing \(S_t\)	One Time Forecast \({\hat Y}_t\)	Residual \(e_t = Y_t - {\hat Y}_t\)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24	95 100 87 123 90 96 75 78 106 104 89 83 118 86 86 112 85 101 135 120 76 115 90 92	95.00 95.50 94.65 97.48 96.74 96.66 94.50 92.85 94.16 95.15 94.53 93.38 95.84 94.86 93.97 95.77 94.70 95.33 99.29 101.36 98.83 100.45 99.40 98.66	95.00 95.00 95.50 94.65 97.48 96.74 96.66 94.50 92.85 94.16 95.15 94.53 93.38 95.84 94.86 93.97 95.77 94.70 95.33 99.29 101.36 98.83 100.45 99.40	0.00 5.00 -8.50 28.35 -7.48 -0.74 -21.66 -16.50 13.15 9.84 -6.15 -11.53 24.62 -9.84 -8.86 18.03 -10.77 6.30 39.67 20.71 -25.36 16.17 -10.45 -7.40

<Figure 13.5.2> Exponential Smoothing with α = 0.1 and One Time Forecast

※ Initial value of exponential smoothing

Since the initial exponential smoothing value \(S_0\) at the time point \(t=1\) cannot be obtained, the following three methods are commonly used.

1) The first observation, i.e., \(S_0 \;=\; Y_1\)
2) Partial average using the initial \(n\) observation values, i.e.,
\(\qquad S_0 \;=\; \frac{1}{n} ({Y_1 + Y_2 + \cdots + Y_n }) \)
3) The mean up to the entire time point \(T\), i.e.,
\(\qquad S_0 \;=\; \frac{1}{T} ({Y_1 + Y_2 + \cdots + Y_T }) \)

※ Initial smoothing constant

The same smoothing constant \(\alpha\) can be applied to all time series, but the following method is also used to reduce the effect of the initial value \(S_0\).

\(\qquad \alpha_t \;=\; \frac{1}{t} , \quad until \; \alpha_t \; reaches\; \alpha \)

13.5.2 Linear Trend Time Series

A. Double Moving Average Model

In the previous section, we examined that the single moving average model can be applied to a stationary time series. What would happen if the single moving average model was applied to a time series with a linear trend? That is, for a time series with a linear trend \(Y_t \;=\; \beta_0 \;+\; \beta_1 \cdot t \;+\; \epsilon _t \), the \(N\)-point single moving average at time \(T\) is as follows: $$ M_T \;= \; \frac{1}{N} ( Y_{T-N+1} \;+\; Y_{T-N+2} \;+\; \cdots \;+\; Y_T ) $$ The expected value can be shown as follows: $$ E(M_T) \;=\; \beta_0 + \beta_1 T - \frac{N-1} {2} \beta_1 $$ That is, in the case of a linear trend model, it can be seen that the single moving average \(M_t\) is biased by \(\frac{N-1}{2} \beta_1 \). For example, if the consumer price index with a linear trend in [Table 13.5.3] is predicted after one point in time using the 5-point single moving average, it is as shown in <Figure 13.5.3>. It can be seen that the predicted value using \(M_t\) is under estimated value of the time series .

<Figure 13.5.3> 5-pt Moving Average of Consumer Price Index with Linear Trend

In the case of a linear trend, one way to eliminate the bias of the single moving average model is the double moving average, which obtains the moving average again for the single moving average. The \(N\)-point double moving average \(M_T^{(2)}\) at the time \(T\) and its expected value are as follows: $$ \begin{align} M_T^{(2)} &\;=\; \frac{1}{N} (M_T \;+\; M_{T-1} \;+\; \cdots \;+\; M_{T-N+1}) \\ E(M_T^{(2)}) &\;=\; \beta_0 \;+\; \beta_1 T \;-\; (N-1) \beta_1 \end{align} $$ Since \(E(M_T )\) and \(E(M_T ^{(2)} )\) have the same number of parameters, \(\beta_0 ,\;\beta_1 \) can be estimated by solving the system of two equations as follows: $$ \begin{align} {\hat{\beta }}_{1} &\;=\; \frac{2}{N-1} \; (M_{T} \;-\; M_{T}^{(2)} ) \\ {\hat{\beta }}_{0} &\;=\; 2M_{T} \;-\; M_{T}^{(2)} \;-\; {\hat{\beta }}_{1} \;T\; \end{align} $$ Therefore, the predicted value at the time point \(T+\tau\) using the double moving average at time \(T\) as follows: $$ {\hat Y}_{T+\tau} \;=\; 2 M_T \;-\; {M_T ^{(2)}} \;+\; \tau \;\; (\frac {2}{N-1} ) \;\; (M_T\;-\; M_T^{(2)}) $$ Such a double moving average model can be said to be a kind of heuristic method. That is, although logical, it is not based on any optimization such as least squares method. However, it can be approximated by the least-squares method, which we will omit in this book.

[Table 13.5.3] is a calculation table for predicting the consumer price index using the 5-point double moving average model. Note that the third column is a 5-point single moving average \(M_T\), but the single moving average cannot be calculated from time points 1 to 4. The fourth column is the calculation of the 5-point double moving average \(M_t^{(2)}\), but the double moving average cannot be calculated until 5 single moving averages have been calculated, that is, from time points 1 to 8. Using \(M_9\) and \(M_9^{(2)}\) to obtain the prediction after time 1 from time 9, \({\hat Y}_{9+1}\) is as follows: $$ \begin{align} {\hat Y}_{9+1} &\;=\; 2M_9 \;- M_9^{(2)} \;+ 1 \;\; (\frac{2}{5-1}) \;\;(M_9\;-\;M_9 ^{(2)}) \\ &\;=\; 2 \times 104.50\;-104.028\;+1\;\; ( \frac{2}{5-1} ) \;\;(104.50\;-104.028)\;=\;105.2080 \end{align} $$ The predicted values calculated in the same way are shown in the fifth column.

[Table 13.5.3] Double Moving Average of Consumer Price Index and One Time Forecast

time \(t\)	CPI \(Y_t\)	5-pt Single Moving Average \( M_t\)	5-pt Double Moving Average \( M_t ^(2)\)	One Time Forecast\({\hat Y}_{(t-1)+1}\)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24	102.3 102.8 103.7 104.1 104.0 104.3 104.5 104.9 104.8 104.8 104.2 104.4 105.0 105.5 106.1 106.7 107.1 107.0 106.7 107.4 108.0 107.7 107.8 108.3	103.38 103.78 104.12 104.36 104.50 104.66 104.64 104.62 104.64 104.78 105.04 105.54 106.08 106.48 106.72 106.98 107.24 107.36 107.52 107.84	104.028 104.284 104.456 104.556 104.612 104.668 104.744 104.924 105.216 105.584 105.972 106.36 106.70 106.956 107.164 107.388	105.2080 105.2240 104.9160 104.7160 104.6820 104.9480 105.4840 106.4640 107.3760 107.8240 107.8420 107.9100 108.0500 107.9660 108.0540

<Figure 13.5.4> Forecast using Double Moving Average of Consumer Price Index

B. Holt Double Exponential Smoothing Model

Holt proposed a model for a linear time series \(Y_t \;=\; \beta_0 \;+\; \beta_1 \cdot t \;+\; \epsilon_t \) which uses a smoothing constant for the level and a smoothing constant for the trend. This is called Holt's linear trend exponential smoothing model or two-parameters double exponential smoothing model. Let \({\hat b}_0 \) and \({\hat b}_1 \) be the initial values of the intercept and slope, and \(\alpha\) be the smoothing constant of the level and \(\gamma\) is the smoothing constant of the trend. The predicted values \({\hat Y}_t\), level \({\hat \beta}_0 (t)\) and trend \({\hat \beta}_1 (t)\) are as follows: $$ \begin{align} & \text{Predicted value:} \quad {\hat Y}_t \;=\; {\hat \beta}_0 (t-1) + {\hat \beta}_1 (t-1) \\ & \text{Level:} \qquad\qquad \; {\hat \beta}_{0}(t) \;=\; \alpha Y_t + (1-\alpha) {\hat Y}_t \\ & \text{Trend:} \qquad\qquad {\hat \beta}_{1}(t) \;=\; {\gamma} \{ {\hat \beta}_0 (t) - {\hat \beta}_0 (t-1) \} + (1-\gamma) {\hat \beta}_1 (t-1) \end{align} $$ That is, the level is the weighted average of the current observed value \(Y_t\) and the predicted value \({\hat Y}_t\), and the trend is the weighted average of the level difference \({\hat \beta}_0 (t) - {\hat \beta}_0 (t-1) \) between the time points \(t\) and \((t-1)\) and the trend \({\hat \beta}_1 (t-1) \) at the time point \((t-1)\). For this model, initial values of level \({\hat \beta}_0 (0) \) and slope \({\hat \beta}_1 (0) \) are required and the intercept and slope of the simple regression analysis of all observed values are widely used as initial estimates. Similar to the single exponential smoothing model, a value between 0.1 and 0.3 is often used to determine the smoothing constants \(\alpha\) and \(\gamma\).

The predicted values for the time point \(T+\tau\) at time \(T\) using the trend exponential smoothing model are as follows: $$ {\hat Y}_{T+\tau} \;=\; {\hat Y}_T + \tau {\hat \beta}_1 (T) $$ Such a trend exponential smoothing model is also a kind of heuristic method. That is, although logical, it is not based on any optimization such as least squares method.

The result of simple linear regression model to all data in [Table 13.5.4] is as follows: $$ {\hat{Y}}_{t} =102.574 \;+\; 0.2344 \;\;t $$ [Table 13.5.4] is a calculation table for predicting the consumer price index with the Holt double exponential smoothing model using this initial values. The third column is the predicted value of the level \({\hat \beta}_0 (t) \), the fourth column is the trend \({\hat \beta}_ (t) \), and the fifth column is the prediction \({\hat Y}_t = {\hat \beta}_0 (t-1) + {\hat \beta}_1 (t-1) \) obtained one time after each time point. Therefore, the forecast of the consumer price index for the next three months is as follows: $$ \begin{align} & t=25 \qquad {\hat{Y}} _{24+1} \;=\; {\hat{Y}} _{24} \;+\; 1\; \times {\hat{\beta}} _{1} (24)\;=\;108.19 +0.237\;=\;108.42 \\ & t=26 \qquad {\hat{Y}} _{24+2} \;=\; {\hat{Y}} _{24} \;+\; 2\; \times {\hat{\beta}} _{1} (24)\;=\;108.19 + 2 \times0.237\;=\;108.66 \\ & t=27 \qquad {\hat{Y}} _{24+3} \;=\; {\hat{Y}} _{24} \;+\; 3\; \times {\hat{\beta}} _{1} (24)\;=\;108.19 + 3 \times0.237\;=\;108.90 \end{align} $$

[Table 13.5.4] Forecasting using Holt Double Exponential Smoothing Model of Consumer Price Index

time \(t\)	CPI \(Y_t\)	Constant \( {\hat \beta}_0 (t)\)	Trend \( {\hat \beta}_1 (t)\)	One Time Forecast\({\hat Y}_{(t-1)+1}\)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24	102.3 102.8 103.7 104.1 104.0 104.3 104.5 104.9 104.8 104.8 104.2 104.4 105.0 105.5 106.1 106.7 107.1 107.0 106.7 107.4 108.0 107.7 107.8 108.3	102.57 102.76 102.97 103.25 103.54 103.80 104.07 104.33 104.61 104.85 105.07 105.20 105.33 105.50 105.70 105.93 106.20 106.49 106.75 106.96 107.21 107.50 107.73 107.95 108.20	0.234 0.229 0.227 0.232 0.239 0.241 0.243 0.245 0.249 0.248 0.245 0.234 0.224 0.218 0.216 0.218 0.223 0.230 0.233 0.230 0.232 0.238 0.237 0.236 0.237	102.81 102.99 103.20 103.48 103.78 104.04 104.31 104.58 104.86 105.10 105.31 105.44 105.56 105.72 105.91 106.15 106.43 106.72 106.98 107.19 107.44 107.73 107.97 108.19

<Figure 13.5.5> shows the predicted values using the Holt’s double exponential smoothing model.

<Figure 13.5.5> Forecasting using Holt Double Exponential Smoothing Model of CPI

13.6 Seasonal Model and Forecasting

As a seasonal time series model, a multiplicative model using a central moving average and a Holt-Winters model are introduced.

13.6.1 Seasonal Multiplicative Model

Assume that a time series \(Y_t\) with a seasonal period \(L\) can be expressed as the product of a trend (\(T\)), a seasonal (\(S\)), and an irregular component (\(I\)) as follows: $$ Y_t \;=\; T_t \cdot S_t \cdot I_t $$ The ratio to moving average method removes the trend and irregular components to obtain the seasonal component as follows

(Step 1) For the time series, find the \(L\)-point centered moving average. This moving average represents the trend component \(T_t\) after removing seasonal component and irregular component from the time series.

(Step 2) Divide the time series \(Y_t\) by the trend component \(T_t\) obtained in Step 1. This value implies the seasonal component and the irregular component \({S_t \cdot I_t}\), and is called the seasonal ratio.

$$ \frac{Y_t} {T_t } \;=\; S_t \cdot I_t $$ (Step 3) Calculate the trimmed average for each seasonal ratio obtained in Step 2. This is the seasonal index \(S_t\), but the normalization should be performed so that the sum of the seasonal indices is \(L\).

After obtaining the seasonal index as shown, dividing the original time series data by the seasonal index. The results is called a deseasonal time series.

\( \qquad \text{Deseasonal time series:} \qquad D_t \;=\; \frac{Y_t} {S_t } = T \cdot I \)

This deseasonal time series \(D_t\) implies \(T \cdot I\). An appropriate time series model is applied to this deseasonal data and predict the future vaules. Then multiply the corresponding seasonal index to obtain the final predicted value of the desired season.

[Table 13.6.1] shows a company's quarterly sales. Since the seasonal period is 4, the 4-point centered moving average is as shown in column 4 of the table. By dividing the original time series by a 4-point centered moving average, the seasonal ratio in column 5 can be calculated. $$ \frac{Y_t} {T_t } \;=\; S_t \cdot I_t $$

[Table 13.6.1] Quarterly Sales of a Company

	① Year Quarter	② Sales	③ 4-point MA	④ Centered 4-point MA	⑤ Seasonal Ratio	⑥ Deseasonal Data \(D_t\)
1 2 3 4 5 6 7 8 9 10 11 12 130 14 15 16	2018 Quarter 1 2018-Quarter 2 2018-Quarter 3 2018-Quarter 4 2019 Quarter 1 2019-Quarter 2 2019-Quarter 3 2019-Quarter 4 2020 Quarter 1 2020 Quarter 2 2020 Quarter 3 2020 Quarter 4 2021 Quarter 1 2021 Quarter 2 2021 Quarter 3 2021 Quarter 4	75 60 54 59 86 65 63 80 90 72 66 85 100 78 72 93	62.000 64.750 66.000 68.250 73.500 74.500 76.250 77.000 78.250 80.750 82.250 83.750 85.750	63.375 65.375 67.125 70.875 74.000 75.375 76.625 77.625 79.500 81.500 83.000 84.750	0.852 0.902 1.281 0.917 0.851 1.061 1.175 0.928 0.830 1.043 1.205 0.920	62.552 65.502 63.754 56.840 71.726 70.961 74.380 77.071 75.063 78.603 77.922 81.888 83.403 85.153 85.006 89.595

[Table 13.6.2] shows the seasonal ratio by year and quarter. If the maximum and minimum values are removed for each quarter and the average is obtained (trimmed average), it is the seasonal index in column 6. Since the sum of these values is 4.0197, the seasonal index in column is normalized as in column 7. $$ {\hat S}_1 \;=\; 1.199,\quad {\hat S}_2 \;=\; 0.916,\quad {\hat S}_3 \;=\; 0.847,\quad {\hat S}_4 \;=\; 1.038 $$

[Table 13.6.2] Seasonal Index

①       Year Quarter	② 2018	③ 2019	④ 2020	⑤ 2021	⑥ Trimmed Mean	⑦ Seasonal Index \(s_t\)
1st Quarter 2nd Quarter 3rd Quarter 4th Quarter	0.852 0.902	1.281 0.917 0.851 1.061	1.175 0.928 0.830 1.043	1.205 0.920	1.205 0.920 0.851 1.043	1.199 0.916 0.847 1.038
					sum 4.019

Column 6 of [Table 13.6.1] shows the non-seasonal data \(D_t\) obtained by dividing the original data by the seasonal index of each quarter. The linear regression line for this non-seasonal data is as follows (<Figure 13.6.1>)., $$ D_{t} \;=\; 59.335 \;+\; 1.839 \cdot t $$ Therefore, the forecast for the next one year is as follows. $$ \begin{align} & \text{Time 17 :} \quad{\hat Y}_{17} \;=\; (59.335 \;+\; 1.839 \times 17) \times 1.199 \;=\; 108.627 \\ & \text{Time 18 :} \quad{\hat Y}_{18} \;=\; (59.335 \;+\; 1.839 \times 18) \times 0.916 \;=\; 84.672 \\ & \text{Time 19 :} \quad{\hat Y}_{19} \;=\; (59.335 \;+\; 1.839 \times 19) \times 0.847 \;=\; 79.852 \\ & \text{Time 20 :} \quad{\hat Y}_{20} \;=\; (59.335 \;+\; 1.839 \times 20) \times 1.038 \;=\; 99.767 \\ \end{align} $$ <Figure 13.6.2> is a graph of seasonal forecasts.

[ : ]
      Y = () × () × ()

<Figure 13.6.1> Forcasting Model of Deseasonal Sales of a Company

13.6.2 Holt-Winters Model

Assume that a time series with a seasonal period \(L\) is observed over \(m\) cycles as follows:

	Season 1	Season 2	\(\cdots\)	Season \(L\)
Cycle 1 Cycle 2 \(\cdots\) Cycle \(m\)	\(Y_{1}\) \(Y_{L+1}\) \(\cdots\) \(Y_{(m-1)L+1}\)	\(Y_{2}\) \(Y_{L+2}\) \(\cdots\) \(Y_{(m-1)L+2}\)	\(\cdots\) \(\cdots\) \(\cdots\) \(\cdots\)	\(Y_{L}\) \(Y_{2L}\) \(\cdots\) \(Y_{mL}\)

The Holt-Winters model is an extension of Holt's linear double exponential smoothing method studied in the previous section to a seasonal model. It consists of a level component \(l_t\), a trend component \(b_t\), and a seasonal component \(s_t\). There are additive model and multiplicative model, but only the multiplication model is introduced here. $$ \begin{align} {\hat{Y}}_{t+h} &\;=\; (l_{t} \;+\; h \cdot b_{t} ) \cdot s_{t+h-m(k+1)} \\ l_{t} &\;=\; \alpha \frac{y_t}{s_{t-m} } + (1-\alpha)(l_{t-1} + b_{t-1} ) \\ b_{t} &\;=\; \beta (l_{t} - l_{t-1} )+(1- \beta ) b_{t-1} \\ s_{t} &\;=\; \gamma \;\; \frac{y_{t}}{l_{t-1} + b_{t-1}} \;+\; (1- \gamma ) s_{t-m} \end{align} $$ Here \(k\) is integer part of \((h-1)/m\). \(l_t\) is a time series level, which means the exponential smoothing of the current level (\( \frac{y_t}{s_{t-m} } \) ) with seasonality removed and the value predicted one time ago \((l_{t-1} + b_{t-1} ) \). \(b_t\) is the slope which is the exponential smoothing of the slope of the current time point \( (l_t - l_{t-1} )\) and the previous time point (\(b_{t-1}\)). \(s_t\) is a seasonal index which is the exponential smoothing of the current seasonal component (\(\frac {y_t}{l_{t-1} + b_{t-1} } \)) and the seasonal component of the previous season \(s_{t-m}\).

[Table 13.6.3] calculates exponential smoothing values of level, slope, and seasonal indices using the Holt-Winters model with α = 0.3, β = 0.3, γ = 0.3 to the quarterly sales of a company. The last column is one time prediction at time \(t-1\), \({\hat Y}_{(t-1)+1}\). The initial values \(l_0\) and \(b_0\) are the intercept and slope of the linear regression model for all data, and the initial values of the seasonal index are calculated by the model \(Y \;=\; T \times S \times I \).

[Table 13.6.3] Holt-Winters Forecasting Model of Quarterly Sales

time	Year Quarter	Sales \(Y_t\)	Level \(l_t\)	Slope \(b_t\)	Seasonal \(s_t\)	One Time Forecast \(\hat Y _{(t-1)+1}\)
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16	2018 Quarter 1 2018-Quarter 2 2018-Quarter 3 2018-Quarter 4 2019 Quarter 1 2019-Quarter 2 2019-Quarter 3 2019-Quarter 4 2020 Quarter 1 2020 Quarter 2 2020 Quarter 3 2020 Quarter 4 2021 Quarter 1 2021 Quarter 2 2021 Quarter 3 2021 Quarter 4	61.2 62.7312 64.6753 65.5795 63.9809 66.7081 68.7061 71.6766 75.7320 76.5997 78.2283 79.2477 81.6382 83.21580 84.7427 86.0501 88.4618	1.61 1.5863 1.6937 1.4568 0.5402 1.1963 1.4368 1.8969 2.5445 2.0414 1.9176 1.6481 1.8708 1.7829 1.7061 1.5865 1.8341	1.61 1.5863 1.6937 1.4568 0.5402 1.1963 1.4368 1.8969 2.5445 2.0414 1.9176 1.6481 1.8708 1.7829 1.7061 1.5865 1.8341	1.1991 0.9159 0.8472 1.0378 1.1976 0.9210 0.8371 0.9905 1.2382 0.9319 0.8555 1.0195 1.2117 0.9270 0.8459 1.0289 1.2074 0.9242 0.8420 1.0386	75.315 58.908 56.230 69.570 77.270 62.539 58.720 72.874 96.920 73.282 68.561 82.477 01.184 78.791 73.124 90.169

<Figure 13.6.2> is the Holt-Winters forecast for the next one year and is calculated as follows: $$ \begin{align} & \text{Time 17 :} \quad {\hat Y}_{16+1} \;=\;\; (l_{16} + 1 \times\;b_{16} ) \cdot s_{13} \;=\;(88.4618 + 1.8341)\times1.2074 \;=\;109.024 \\ & \text{Time 18 :} \quad {\hat Y}_{16+2} \;=\;\; (l_{16} + 2 \times\;b_{16} ) \cdot s_{14} \;=\;(88.4618+2 \times 1.8341) \times 0.9242\;=\;85.144 \\ & \text{Time 19 :} \quad {\hat Y}_{16+3} \;=\;\; (l_{16} + 3 \times\;b_{16} ) \cdot s_{15} \;=\;(88.4618+3 \times 1.8341) \times 0.9420\;=\;79.115 \\ & \text{Time 20 :} \quad {\hat Y}_{16+4} \;=\;\; (l_{16} + 4 \times\;b_{16} ) \cdot s_{16} \;=\;(88.4618+4 \times 1.8341) \times 1.0386\;=\;99.495 \\ \end{align} $$

[ : ]
      Y(t+h) = (L(t) + h × b(t)) × S(t+h-m(k+1))

<Figure 13.6.2> Holt-Winters Forecasting of Quarterly Sales

13.7 Exercise