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    Lecture 4

    Uploaded by

    Dina Saad Eskandere
    1. The document discusses various aggregate response models that can be used to model marketing phenomena. These include linear, power series, fractional root, semi-log, exponential, modified exponential, logistic, and Adbudg models. 2. It also covers how to handle multiple marketing instruments using additive and multiplicative models. These models can be estimated using linear regression. 3. The document explains the method of least squares and how to determine the slope and intercept coefficients to fit a linear regression line that minimizes the squared errors between predicted and actual values. It provides an example of estimating an advertisement budget using historical sales and budget data.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Lecture 4

    Uploaded by

    Dina Saad Eskandere
    14 views27 pages
    1. The document discusses various aggregate response models that can be used to model marketing phenomena. These include linear, power series, fractional root, semi-log, exponential, modified exponential, logistic, and Adbudg models. 2. It also covers how to handle multiple marketing instruments using additive and multiplicative models. These models can be estimated using linear regression. 3. The document explains the method of least squares and how to determine the slope and intercept coefficients to fit a linear regression line that minimizes the squared errors between predicted and actual values. It provides an example of estimating an advertisement budget using historical sales and budget data.

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    lecture 4

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    1. The document discusses various aggregate response models that can be used to model marketing phenomena. These include linear, power series, fractional root, semi-log, exponential, modified exponential, logistic, and Adbudg models. 2. It also covers how to handle multiple marketing instruments using additive and multiplicative models. These models can be estimated using linear regression. 3. The document explains the method of least squares and how to determine the slope and intercept coefficients to fit a linear regression line that minimizes the squared errors between predicted and actual values. It provides an example of estimating an advertisement budget using historical sales and budget data.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    14 views27 pages

    Lecture 4

    Uploaded by

    Dina Saad Eskandere
    1. The document discusses various aggregate response models that can be used to model marketing phenomena. These include linear, power series, fractional root, semi-log, exponential, modified exponential, logistic, and Adbudg models. 2. It also covers how to handle multiple marketing instruments using additive and multiplicative models. These models can be estimated using linear regression. 3. The document explains the method of least squares and how to determine the slope and intercept coefficients to fit a linear regression line that minimizes the squared errors between predicted and actual values. It provides an example of estimating an advertisement budget using historical sales and budget data.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 27

    MARKETING (UNR471)

    Dr. Mohamed Sameh


    RESPONSE MODEL PHENOMENA

    P1: Through Origin P2: Linear

    Y Y

    X X

    P3: Decreasing Returns


    P4: Saturation
    (concave) —
    Q

    Y Y

    X X
    PHENOMENA

    P5: Increasing Returns P6: S-shape


    (convex)

    Y Y

    X X

    P7: Threshold P8: Super-saturation

    Y Y

    X X
    Aggregate Response Models:
    Linear Model

    Y = a + bX

    • Handles Linear/through origin (P1, P2).

    • Handles Saturation and threshold (P4, P7) (in ranges)


    Aggregate Response Models:
    Power Series Model

    Y = a + b X + c X2+ d X3 + e X4 + …
    • fits well within the range of the data but will.

    • behaves badly (becoming unbounded) outside the data range.

    • may be designed to handle phenomena P1, P2, P3, P5, P6, and P8
    Aggregate Response Models:
    Fractional Root Model

    c
    Y = a + bX
    • When c=1/2 the model is called the square root model.

    • When c= –1 it is called the reciprocal model.

    • c can be interpreted as elasticity when a = 0.

    • handles P1, P2, P3, P4, and P5.


    Aggregate Response Models:
    Semi-log Model

    Y = a+ b ln(X)

    • Handles situations in which constant increase in marketing result in constant

    absolute increases in sales.

    • handles P3 and P7.


    Aggregate Response Models:
    Exponential Model

    Y = ae bx; x > 0
    • Is widely used as a price-response function for b<0 (i.e., increasing returns to
    decreases in price).
    • handles phenomena P5 and, if b is negative, P4
    Aggregate Response Models:
    Modified Exponential Model

    Y = a(1- e – bx)+ c
    • has an upper bound at a+c and a lower bound of c
    • shows decreasing returns to scale.
    • handles phenomena P3 and P4 and
    • can accommodate P1 when c=0
    Aggregate Response Models:
    Logistic Model

    𝑎𝑎
    𝑌𝑌 = −(𝑏𝑏+𝑐𝑐𝑐𝑐)
    + 𝑑𝑑
    1 + 𝑒𝑒
    • the most common algorithm to handle S- Shaped models.
    • has a saturation level at a+d .
    • has a region of increasing returns followed by decreasing return.
    • Handles phenomena P4 and P6
    Aggregate Response Models:
    Adbudg Function

    𝑋𝑋 𝑐𝑐
    𝑌𝑌 = 𝑏𝑏 + (𝑎𝑎– 𝑏𝑏)
    𝑑𝑑 + 𝑋𝑋 𝑐𝑐
    • S-shaped for c>1 and concave for 0<c<1.

    • bounded between b (lower bound) and a (upper bound).

    • handles phenomena P1, P3, P4, and P6


    Aggregate Response Models:
    Multiple Instruments

    • Additive model for handling multiple marketing instruments

    Y = af (X1) + bg (X2)

    Easy to estimate using linear regression.


    Aggregate Response Models:
    Multiple Instruments cont’d
    • Multiplicative model for handling multiple marketing instruments

    Y = aXb1 X2c

    b and c are elasticities.

    Widely used in marketing.

    Can be estimated by linear regression.


    Regression
    The observed value yi corresponding to xi and the value α+βxi on the
    regression line y = a+ bx.
    CISE301_Topic4

    METHOD OF LEAST SQUARES

    We want to find a and b to minimize :


    n
    Φ ( a , b) = ∑ ( a + bxi − yi ) 2
    i =1

    How do we obtain a and b to minimize : Φ ( a, b) ?


    CISE301_Topic4

    DETERMINE THE UNKNOWNS

    Necessary condition for the minimum :

    ∂ Φ ( a, b)
    =0
    ∂a
    ∂ Φ ( a, b)
    =0
    ∂b
    DETERMINING THE UNKNOWNS

    ∂ Φ ( a , b) n
    = ∑ 2(a + bxi − yi ) = 0
    ∂a i =1

    ∂ Φ ( a , b) n
    = ∑ 2(a + bxi − yi )xi = 0
    ∂b i =1
    Estimation

     After some simple algebraic rearranging, we obtain:

    𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 − (∑ 𝑥𝑥𝑖𝑖 )(∑ 𝑦𝑦𝑖𝑖 )


    𝛽𝛽 = (slope)
    𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖2 − (∑ 𝑥𝑥𝑖𝑖 )2

    𝛼𝛼 = 𝑦𝑦�𝑛𝑛 − 𝛽𝛽 𝑥𝑥̅𝑛𝑛 (intercept)


    EXAMPLE

    • A company wants to launch a new product. They want the marketing


    department to make a proper estimation of the advertisement budget. Using
    historical sales data of similar products and the accompanying advertisement
    campaign sizes, estimate the budget if the company wants to sell at least 40000
    units of the product.

    Budget(1000 $) 11 12 11 15 8 10 11 12 17 11
    Sales (1000 units) 25 33 22 41 18 28 32 24 53 26
    FİRST STEP:

    � 𝑥𝑥𝑖𝑖 = 118

    � 𝑥𝑥𝑖𝑖2 = 1450

    � 𝑦𝑦𝑖𝑖 = 302

    � 𝑦𝑦𝑖𝑖2 = 10072

    � 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 = 3779


    SUM OF SQUARES:

    𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 − (∑ 𝑥𝑥𝑖𝑖 )(∑ 𝑦𝑦𝑖𝑖 )


    𝛽𝛽 = 2 2
    = 3.74
    𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖 − (∑ 𝑥𝑥𝑖𝑖 )

    𝛼𝛼 = 𝑦𝑦�𝑛𝑛 − 𝛽𝛽𝑥𝑥̅𝑛𝑛 = −13.93

    Therefore, 𝑦𝑦� = −13.93 + 3.74𝑥𝑥


    SUM OF SQUARES:

    𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 − (∑ 𝑥𝑥𝑖𝑖 )(∑ 𝑦𝑦𝑖𝑖 )


    𝛽𝛽 = 2 2
    = 3.74
    𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖 − (∑ 𝑥𝑥𝑖𝑖 )

    𝛼𝛼 = 𝑦𝑦�𝑛𝑛 − 𝛽𝛽𝑥𝑥̅𝑛𝑛 = −13.93

    Therefore, 𝑦𝑦� = −13.93 + 3.74𝑥𝑥


    LINEARIZATION OF NONLINEAR BEHAVIOR

    Linear regression is useful to represent a linear relationship.


    •If the relation is nonlinear either another technique can be used or the data can be
    transformed so that linear regression can still be used.
    The latter technique is frequently used to fit the following nonlinear equations to a set of
    data.
    • Exponential equation
    • Power equation
    • Saturation-growth rate equation
    LINEARIZATION OF NONLINEAR BEHAVIOR

    lnY = ln A + BX
    Y = Ae BX
    y= a + bx
    LINEARIZATION OF NONLINEAR BEHAVIOR

    •Calculate a= 6.25 and b= 0.841. Straight line is ln y = 6.25 + 0.841 x


    •Switch back to the original equation. A= ea= 518, B= b= 0.841.
    •Therefore, the exponential equation is y = 518 e0.841x.
    LINEARIZATION OF NONLINEAR BEHAVIOR

    logY = log A + B logX


    Y = AX B
    y= a + bx

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