Linear Trend Lines Yt = b0 + b1 Xt

Where Yt is the dependent variable being forecasted

Xt is the independent variable being used to explain Y. In Linear Trend Lines, Xt is assumed to be t.

b1 is the slope of the line, determined by Excel

b0 is the y intercept of the line, determined by Excel

Tools Data Analysis Regression

Coefficient of Determination: R-square Proportion of variation in Y around

its mean that is accounted for by the regression model

0 <= R2 <= 1 Will always increase as add more

independent variables into regression model. Use adjusted R2 to compare when more than one independent variable is used

Standard Error of the line: Se

The standard deviation of estimation errors

The measure of amount of scatter around the regression line

Can be used as a rough rule of thumb for predicting level of accuracy.

Excel’s Trend Function

=trend(known y-range, known x-range, new x) Where known y-range are the cells that

hold known values for the y variable Where known x-range are the cells that

hold known values for the x variable Where new x is the cell or value for

which the y variable is to be forecasted

Multiplicative Seasonal Effects

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Tim e Pe riod

Additive Seasonal Effects

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Tim e Pe riod

Stationary Seasonal Effects

Text Use of Multiplicative Seasonal Indices (pg. 532)

1. Create a trend model and calculate the estimated value for each observation

2. Calculate the ratio of the actual value to the predicted value for each observation

3. Use the average of the values for each seasonal period to compute the seasonal index

4. Multiply any forecast produced by the trend model by the appropriate seasonal index

Use Solver to Identify Seasonal Indices and Trendline

1. Program linear trendline formula for trend forecast, referring to input data cells for b0 and b1

2. Program seasonal adjustment formula, referring to input data cells for seasonal indices

3. Program MAPE or MSE calculations4. Program Solver to Min MAPE/MSE

By Changing seasonal indices, b0 and b1

Subject to average seasonal index = 100% and seasonal indices>=0

Forecasting periods 37 and 38 for the Vintage Case

Y37 = 185.8 + .372*37 = 199.63 Seasonal forecast for 37 = seasonal

index for 37 * Y37

=1.44* 199.63 = 288.4 Y38 = 185.8 + .372*38 = 200 Seasonal forecast for 38 = 1.29*200 = 259

Simple Linear Regression: Example

You want to examine the linear dependency of the annual sales of produce stores on their size in square footage. Sample data for seven stores were obtained. Find the equation of the straight line that fits the data best.

Annual Store Square Sales

Feet ($1000)

1 1,726 3,681

2 1,542 3,395

3 2,816 6,653

4 5,555 9,543

5 1,292 3,318

6 2,208 5,563

7 1,313 3,760

Scatter Diagram: Example

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Excel Output

Equation for the Sample Regression Line: Example

0 1ˆ

1636.415 1.487i i

i

Y b b X

X

From Excel Printout:

CoefficientsIntercept 1636.414726X Variable 1 1.486633657

Graph of the Sample Regression Line: Example

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Y i = 1636.415 +1.487X i

Interpretation of Results: Example

The slope of 1.487 means that for each increase of one unit in X, we predict the average of Y to increase by an estimated 1.487 units.

The model estimates that for each increase of one square foot in the size of the store, the expected annual sales are predicted to increase by $1487.

ˆ 1636.415 1.487i iY X