Time-Series Analysis and Forecasting – Part V To read at home

Analytical smoothing of time series (continued)

The second-order parabola smoothing: where b – speed of change of the levels of time series c – acceleration

The second-order parabola smoothing is done when the preliminary analysis shows that the second differences are approximately equal between each other - the first difference; - the second difference

To determine parameters Least Square Method is used:

Hyperbola smoothing is used when there is saturation in the development of time series

To determine the parameters least square method LSM is used:

Smoothing of time series with the help of power functions or exponent is used when the preliminary analysis shows: the level of time series is changing with approximately the same chain coefficients of growth. The coefficient b is interpreted as the average coefficient of growth

To determine the parameters the function is preliminarily changed to the linear form by taking the logarithm of the left and the right sides of the equation We do not find out a & b, we calculate lga & lgb

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Exponential Smoothing  A weighted moving average  Weights decline exponentially  Most recent observation weighted most  Used for smoothing and short term forecasting (often one or two periods into the future)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Exponential Smoothing  The weight (smoothing coefficient) is   Subjectively chosen  Range from 0 to 1  Smaller  gives more smoothing, larger  gives less smoothing  The weight is:  Close to 0 for smoothing out unwanted cyclical and irregular components  Close to 1 for forecasting (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Exponential Smoothing Model  Exponential smoothing model where: = exponentially smoothed value for period t = exponentially smoothed value already computed for period i - 1 x t = observed value in period t  = weight (smoothing coefficient), 0 <  < 1

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Exponential Smoothing Example  Suppose we use weight  =.2 Time Period (i) Sales (Y i ) Forecast from prior period (E i-1 ) Exponentially Smoothed Value for this period (E i ) etc etc etc. 23 (.2)(40)+(.8)(23)=26.4 (.2)(25)+(.8)(26.4)=26.12 (.2)(27)+(.8)(26.12)= (.2)(32)+(.8)(26.296)= (.2)(48)+(.8)(27.437)= (.2)(48)+(.8)(31.549)= (.2)(33)+(.8)(31.840)= (.2)(37)+(.8)(32.872)= (.2)(50)+(.8)(33.697)= etc. = x 1 since no prior information exists

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Sales vs. Smoothed Sales  Fluctuations have been smoothed  NOTE: the smoothed value in this case is generally a little low, since the trend is upward sloping and the weighting factor is only.2

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Forecasting Time Period (t + 1)  The smoothed value in the current period (t) is used as the forecast value for next period (t + 1)  At time n, we obtain the forecasts of future values, X n+h of the series

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Exponential Smoothing in Excel  Use tools / data analysis / exponential smoothing  The “damping factor” is (1 -  )

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap  To perform the Holt-Winters method of forecasting:  Obtain estimates of level and trend T t as  Where  and  are smoothing constants whose values are fixed between 0 and 1  Standing at time n, we obtain the forecasts of future values, X n+h of the series by Forecasting with the Holt-Winters Method: Nonseasonal Series

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap  Assume a seasonal time series of period s  The Holt-Winters method of forecasting uses a set of recursive estimates from historical series  These estimates utilize a level factor, , a trend factor, , and a multiplicative seasonal factor,  Forecasting with the Holt-Winters Method: Seasonal Series

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap  The recursive estimates are based on the following equations Forecasting with the Holt-Winters Method: Seasonal Series Where is the smoothed level of the series, T t is the smoothed trend of the series, and F t is the smoothed seasonal adjustment for the series (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap  After the initial procedures generate the level, trend, and seasonal factors from a historical series we can use the results to forecast future values h time periods ahead from the last observation X n in the historical series  The forecast equation is where the seasonal factor, F t, is the one generated for the most recent seasonal time period Forecasting with the Holt-Winters Method: Seasonal Series (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Autoregressive Models  Used for forecasting  Takes advantage of autocorrelation  1st order - correlation between consecutive values  2nd order - correlation between values 2 periods apart  p th order autoregressive model: Random Error

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Autoregressive Models  Let X t (t = 1, 2,..., n) be a time series  A model to represent that series is the autoregressive model of order p:  where  ,  1  2,...,  p are fixed parameters   t are random variables that have  mean 0  constant variance  and are uncorrelated with one another (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Autoregressive Models  The parameters of the autoregressive model are estimated through a least squares algorithm, as the values of ,  1  2,...,  p for which the sum of squares is a minimum (continued)

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Forecasting from Estimated Autoregressive Models  Consider time series observations x 1, x 2,..., x t  Suppose that an autoregressive model of order p has been fitted to these data:  Standing at time n, we obtain forecasts of future values of the series from  Where for j > 0, is the forecast of X t+j standing at time n and for j  0, is simply the observed value of X t+j

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Autoregressive Model: Example Year Units The Office Concept Corp. has acquired a number of office units (in thousands of square feet) over the last eight years. Develop the second order autoregressive model.

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Autoregressive Model: Example Solution Year x t x t-1 x t Excel Output  Develop the 2nd order table  Use Excel to estimate a regression model

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Autoregressive Model Example: Forecasting Use the second-order equation to forecast number of units for 2007:

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap Autoregressive Modeling Steps  Choose p  Form a series of “lagged predictor” variables x t-1, x t-2, …,x t-p  Run a regression model using all p variables  Test model for significance  Use model for forecasting

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