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Graduate Program in Business Information Systems. FORECASTING Regression Analysis Aslı Sencer. Regression in Causal Models. Regression analysis can make forecasts with with a non-time independent variable. A simple regression employs a straight line. Ŷ ( X ) = a + bX

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Graduate Program in Business Information Systems

FORECASTINGRegression AnalysisAslı Sencer


Regression in causal models
Regression in Causal Models

  • Regression analysis can make forecasts with with a non-time independent variable.

  • A simple regression employs a straight line.

    Ŷ(X) = a + bX

    • The dependent variable is not time periods, such as:

      • store size

      • order amount

      • weight

  • For 10 rail shipments, the transportation time Y was forecast for specific distance X.



Linear regression equations
Linear Regression Equations

Equation:

Slope:

Y-Intercept:


Interpretation of coefficients
Interpretation of Coefficients

  • Slope (b)

    • Estimated Y changes by b for each 1 unit increase in X

      • If b = 2, then transportation time (Y) is expected to increase by 2 for each 1 unit increase in distance (X)

  • Y-intercept (a)

    • Average value of Y when X = 0

      • If a = 4, then transportation time (Y) is expected to be 4 when the distance (X) is 0


Least squares assumptions
Least Squares Assumptions

  • Relationship is assumed to be linear.

  • Relationship is assumed to hold only within or slightly outside data range.

  • Do not attempt to predict time periods far beyond the range of the data base.


Random error variation
Random Error Variation

  • Variation of actual Y from predicted

  • Measured by standard error of estimate, SY,X

  • Affects several factors

    • Parameter significance

    • Prediction accuracy



Assumptions on error terms
Assumptions on Error Terms

  • The mean of errors for each x is zero.

  • Standard deviation of error terms , SY,X

    • is the same for each x.

  • Errors are independent of each other.

  • Errors are normally distributed with mean=0 and

    • variance= SY,X. for each x.


Correlation
Correlation

  • Answers: ‘how strongis the linearrelationship between the variables?’

  • Correlation coefficient, r

    • Values range from -1 to +1

    • Measures degree of association

  • Used mainly for understanding



Coefficient of correlation and regression model

r = 1

r = -1

Y

Y

^

Y

=

a

+

b

X

i

i

^

Y

=

a

+

b

X

i

i

X

X

r = .89

r = 0

Y

Y

^

^

Y

=

a

+

b

X

Y

=

a

+

b

X

X

X

i

i

i

i

Coefficient of Correlation and Regression Model


Coefficient of determination
Coefficient of Determination

If we do not use any regression model, total sum of square of errors, SST

If we use a regression model, sum of squares of errors

Then sum of squares of errors due to regression

We define coef. of determination


  • Coefficient of determination r2 is the variation in y

  • that is explained and hence recovered/eliminated

  • by the regression equation !

  • Correlation coeficient r can also be found by using


Multiple regression in forecasting
Multiple Regression in Forecasting

  • Regression fits data employing a multiple regression equation with several predictors:

    Ŷ = a + b1X1 + b2X2

  • Floorspace X1 and advertising expense X2 make forecasts of hardware outlet sales Y:

    Ŷ = -22,979 + 11.42X1 + 23.41X2

  • The above was obtained in a computer run using 10 data points.

  • Forecast with X1 =2,500 sq.ft. and X2=$750:

    Ŷ = -22,979+11.42(2,500)+23.41(750)=$23,129


Guidelines for Selecting Forecasting Model

  • You want to achieve:

    • No pattern or direction in forecast error

      • Error = (Yi - Yi) = (Actual - Forecast)

      • Seen in plots of errors over time

    • Smallest forecast error

      • Mean square error (MSE)

      • Mean absolute deviation (MAD)


Trend Not Fully Accounted for

Desired Pattern

Error

Error

0

0

Time (Years)

Time (Years)

Pattern of Forecast Error


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