Forecasting Models With Linear Trend

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