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Forecasting Models With Linear Trend. Linear Trend Model. y-intercept. Random Error at Period t. Slope. Value of the time series at Period t. If a modeled is hypothesized that has only linear trend and random effects, it will be of the form: - PowerPoint PPT Presentation
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ForecastingForecasting
Models With Linear TrendModels With Linear Trend
Linear Trend ModelLinear 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 β1 ≠ 0.– Conclude β1 ≠ 0 if there is a low p-value for this test.
yt = β0 + β1t + εt
Value of the time series at Period t
y-intercept Slope Random Error at Period t
Forecasting Methods For Models Forecasting Methods For Models With Long Term TrendWith 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 MethodRegression Forecasting MethodBasic ApproachBasic 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 ForecastsRegression Forecasts
• Regression will return the best straight line that fits through the set of time series values: b0 + b1t.
Forecast for Period k
Fk = b0 + b1k
ExampleExample• 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 $millions.
Plot the Time SeriesPlot the Time Series Year End Assets
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0 2 4 6 8 10 12Year
Asse
ts ($
Mill
ion)
Long term linear trenddoes appear to be present
Verify using regression!
Regression OutputRegression Output
L0W p-valueCan conclude
LINEAR TREND
Forecasts for Periods 1 -13Forecasts for Periods 1 -13• 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 Performance Measures for Regression ApproachRegression 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 ApproachHolt’s ApproachBasic ConceptsBasic Concepts
• Smooths current point to a point Lt
• Re-evaluates the trend from one period to the next based on the new time series value, Tt
• Forecast for the next period, t+1, starts from the smoothed level Lt and changes by Tt(1) since the next period is one period into the future: Ft+1 = Lt + Tt– The forecast for k periods from period t is: Ft+k
= Lt + Tt(k)– Forecast changes when additional time series
data is observed.
Initial Values for Holt’s ApproachInitial Values for Holt’s Approach• Need some initial values for L2 and T2
• 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:
TT22 = y = y22 - y - y11
LL22 = y = y22
FF33 = L = L22 + T + T22
Holt’s ApproachHolt’s ApproachExponential smoothing is then done to
determine: Ft+1 = Lt + Tt
A representative value of where thetime series “should be” at time t
Exponential smoothing based on:Actual value at time t -- yt
Forecasted value for time t -- Ft
Lt = yt+ (1- )Ft
Tt = Trend = exponentially smoothedvalue for the slope for current period
A representative value of what theslope “should be” at time t
Exponential smoothing based on:Difference in last two levels Lt - Lt-1
Forecasted value for time t-1 – Tt-1
Tt = (Lt-Lt-1)+ (1- )Tt-1
Forecast for next period, t+1:Ft+1 = Lt + Tt
Lt = Level = exponentiallysmoothed value for current period
Excel: Holt’s ApproachExcel: Holt’s ApproachInitializationInitialization
=B3-B2
=C3+D3=B3
Excel: Holt’s ApproachExcel: Holt’s ApproachRecursive CalculationsRecursive 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:E4down to C11:E11
Excel Holt’s ApproachExcel Holt’s ApproachForecastsForecasts
=C11+D11
=E12+$D$11
Relativeaddressof last forecast
Absoluteaddress of last trend estimate
Drag E13 down to E14
ReviewReview
• 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