Forecasting Models With Linear Trend

Linear Trend Model If a modeled is hypothesized that has only linear trend and random effects, it will be of the form: To check if this model is appropriate run a regression analysis and check to see if you can conclude that β 1 ≠ 0. –Conclude β 1 ≠ 0 if there is a low p-value for this test. y t = β 0 + β 1 t + ε t Value of the time series at Period t y-intercept Slope Random Error at Period t

Forecasting Methods For Models With Long Term Trend Regression –Places equal weight on all observations in determining a “best straight line”. Holt’s approach –Rather than calculate one straight line, this approach uses exponential smoothing twice (once to update the smoothed “level” and once to update the estimate for the slope. –It places more weight on the more recent time series values.

Regression Forecasting Method Basic Approach Construct the regression equation based on the historical data available for n periods using –Y (dependent variable) -- time series values –X (independent variable) – period values (1, 2, etc.) Extend the regression line into the future to generate future forecasts –Since regression is only technically valid within the observed values of the independent variable (periods 1 through n) the forecast should not be extrapolated too far into the future (beyond period n).

Regression Forecasts Regression will return the best straight line that fits through the set of time series values: b 0 + b 1 t. Forecast for Period k F k = b 0 + b 1 k

Example Standard and Poor’s (S&P) is a bond rating firm and is conducting an analysis of American Family Products Corp. (AFP). They need to forecast of year-end current assets for years 11, 12 and 13, based on time series data for the previous 10 years given below in $millions.

Plot the Time Series Long term linear trend does appear to be present Verify using regression!

Regression Output L0W p-value Can conclude LINEAR TREND

Forecasts for Periods Enter 11, 12 and 13 in cells A12, A13 and A14 =$I$17+$I$18*A2 Drag C2 down to cells C3:C14 Forecasts for Years 11,12 and 13

Performance Measures for Regression Approach =B2-C2=D2^2 =ABS(D2) =ABS(D2)/B2 Drag D2:G2 to D11:G11 = MAX(F2:F11) = AVERAGE(F2:F11) = AVERAGE(G2:G11) = AVERAGE(E2:E11)

Holt’s Approach Basic Concepts Smooths current point to a point L t Re-evaluates the trend from one period to the next based on the new time series value, T t Forecast for the next period, t+1, starts from the smoothed level L t and changes by T t (1) since the next period is one period into the future: F t+1 = L t + T t –The forecast for k periods from period t is: F t+k = L t + T t (k) –Forecast changes when additional time series data is observed.

Initial Values for Holt’s Approach Need some initial values for L 2 and T 2 Conventional starting values: –Since this is a “trend” model, 2 points are needed to get started –The initial “smoothed” trend at time 2 is just the observed trend that did occur between periods 1 and 2: –The initial level, the level at period 2 is set to the actual time series value at period 2: First forecast is for period 3: T 2 = y 2 - y 1 L 2 = y 2 F 3 = L 2 + T 2

Holt’s Approach Exponential smoothing is then done to determine: F t+1 = L t + T t A representative value of where the time series “should be” at time t Exponential smoothing based on:  Actual value at time t -- y t  Forecasted value for time t -- F t L t =  y t + (1-  )F t T t = Trend = exponentially smoothed value for the slope for current period A representative value of what the slope “should be” at time t Exponential smoothing based on:  Difference in last two levels L t - L t-1  Forecasted value for time t-1 – T t-1 T t =  (L t -L t-1 )+ (1-  )T t-1 Forecast for next period, t+1: F t+1 = L t + T t L t = Level = exponentially smoothed value for current period

Excel: Holt’s Approach Initialization =B3-B2 =C3+D3 =B3

Excel: Holt’s Approach Recursive Calculations Smoothing constant for the level: α =.1 Smoothing constant for the trend:  =.2 =.1*B4+.9*E4=.2*(C4-C3)+.8*D3 Drag C4:E4 down to C11:E11

Excel Holt’s Approach Forecasts =C11+D11 =E12+$D$11 Relative address of last forecast Absolute address of last trend estimate Drag E13 down to E14

Review Scatterplot to observe trend Regression to verify linear trend –Low p-value for t-test for  1 Models with Trend and Random Effects Only –Linear Regression –Holt’s Technique Use of Performance Measures to do comparisons