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Elementary Forecasting Methods A Time Series is a set of regular observations Z t taken over time. By the term spot estimate we mean a forecast in a model that works under deterministic laws. Exponential Smoothing. This uses a recursively defined smoothed series S t and a doubly smoothed series S t [2]. Exponential smoothing requires very little memory and has a single parameter. For commercial applications, the value = 0.7 produces good results. Filter: S t = Z t + (1 - ) S t-1, [ 0, 1] = Z t + (1 - ) Z t-1 + (1 - ) 2 S t-2 S t [2] = S t + (1 - ) S t-1 [2] Forecast: Z T+m = {2 S T - S T [2] } + {S T - S T [2] } m / (1 - ) Example [ = 0.7] Time t Z t S t (66) S t [2] (66) Z 1983 = {2 (355.7) - 333} + { } (2) (0.7) / (0.3) = 484.3

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Moving Average Model. If the time series contains a seasonal component over n “seasons”, the Moving Average model can be used to generate deseasonalised forecasts. Filter:M t = X i / n = M t { Z t - Z t - n } / n M t [2] = M t / n Forecast: Z T + k = { 2 (M T - M T [2] ) } + { M T - M T [2] } 2 k / ( n - 1) Example. Time t Sp Su Au Wi Sp Su Au Wi Sp Su Au Wi Sp Su Au Wi Z T M T M T [2] The deseasonalised forecast for Sp 1992, which is 4 periods beyond the last observation, is Z T+4 = { 2 ( ) } + { } 2 (4) / 3 = In simple multiplicative models we assume that the components are Z t = T (trend) * S(seasonal factor) * R (residual term). The following example demonstrates how to extricate these components from a series.

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Time (1) Raw (2) Four Month (3) Centered (4) Moving (5) Detrended (6) Deseasonalised (7) Residual Data Moving Total Moving Total Average Data (1) / (4) Data (1)/(Seasonal) Series (6) / (4) t Z t =T*S*R T*R T T S*R T*R R Sp Su Au Wi Sp Su Au Wi Sp Su Au Wi Sp Su Au Wi

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The seasonal data is got by rearranging Sp Su Au Wi column (5). The seasonal factors are then reused in column (6) Due to round-off errors in the arithmetic, it is necessary to readjust the means, so Means that they add up to 400 (instead of ).Factors The diagram illustrates the components present in the data. In general when analysing time series data, it is important to remove these basic components before proceeding with more detailed analysis. Otherwise, these major components will dwarf the more subtle component, and will result in false readings. The reduced forecasts are multiplied by the appropriate trend and seasonal components, at the end of the analysis. The forecasts that result from the models above, are referred to as “spot estimates”. This is meant to convey the fact that sampling theory is not used in the analysis and so no confidence intervals are possible. Spot estimates are unreliable and should only be used to forecast a few time periods beyond the last observation in the time series Trend Raw Data

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Normal Linear Regression Model In the model with one independent variable, we assume that the true relationship is y = b 0 + b 1 x and that our observations (x 1, y 1 ), (x 2, y 2 ), …, (x n, y n ) is a random sample from the bivariate parent distribution, so that y = x +, where -> N( 0, ). If the sample statistics are calculated, as in the deterministic case, then 0, 1 and r are unbiased estimates for the true values, b 0, b 1 and, where r and are the correlation coefficients of the sample and parent distributions, respectively. Ify = x 0 is the estimate for y given the value x 0, then our estimate of 2 is s 2 = SSE / (n - 2) = ( y i - y i ) 2 / (n - 2) and VAR [ y] = s 2 { 1 + 1/n + (x 0 - x ) 2 / ( x i - x ) 2 }. The standardised variable derived from y has a t n - 2 distribution, so confidence intervals for the true value of y corresponding to x 0 is y 0 + t n - 2 s 1 + 1/n + (x 0 - x ) 2 / ( x i - x ) 2.

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Example. Consider our previous regression example: y = 23 / / 35 x x i ( 6 ) y i y i (y i - y i ) => ( y i - y i ) 2 = 1.774, s 2 = , s = 0.666, (x i - x ) 2 = 17,5, x = 2.5, t 4, 0.95 = 2.776, t 4, 0.95 (s) = Letf(x 0 ) = t 4, 0.9 s 1 + 1/n + (x 0 - x ) 2 / (x i - x ) 2. Thenx f(x 0 ) y 0 - f(x 0 ) y 0 + f(x 0 ) The diagram shows the danger of extrapolation. It is important in forecasting that the trend is initially removed from the data so that the slope of the regression line is kept as close to zero as possible. A description of the Box-Jenkins methodology and Spectral Analysis, which are the preferred techniques for forecasting commercial data, is to be found in standard text books Y X 6 95% Confidence Interval when x=6

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