Transcript Quant_Chapter_03_math_c

Chapter 3: Part c – Parameter Estimation
We will be discussing
 Nonlinear Parameter Estimation
 Maximum Likelihood Parameter Estimation
(These topics are needed for Chapters 9, 12, 14 and 15)
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Slide 3c.1
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Why Do We Need Nonlinear Parameter Estimation?
With the Linear Model, y = X + e, we end up with a closed form, algebraic solution.
.
Sometimes there is no algebraic solution for the unknowns in a Marketing Model
Suppose the data depend in a nonlinear way on an unknown parameter , lets say
y = f() + e
To minimize e′e, we need to find the spot at which de′e/d = 0.
But if there is no way to get  by itself on one side of an equation and stuff that we
know on the other….
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Slide 3c.2
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Steps to the Algorithm of Nonlinear Estimation
1.
We take a stab at the unknown, inventing a starting value for it.
2.
We assess the derivative of the objective function at the current value of . If the
derivative is not zero, we modify  by moving it in the direction in which the derivative
getting closer to 0. We keep repeating this step until the derivative arrives at zero.
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Slide 3c.3
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A Picture of Nonlinear Estimation
f
2

1
If the derivative is positive, we should move to the left (go more negative)
If the derivative is negative, we should move to the right (go more positive)
This suggests the rule:
i 1  i  
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f
i
Slide 3c.4
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A Brief Introduction to Maximum Likelihood
 ML is an alternative philosophy to Least Squares.
 If ML estimators exist, they will be consistent
 If ML estimators exist they will be normally distributed.
 If ML estimators exist, they will be asymptotically efficient.
 ML leads to a Chi Square test of the model
 The Covariance Matrix for ML estimators can be calculated from the second
order derivatives.
 Marketing Scientists really like ML estimators.
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Slide 3c.5
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The Likelihood Principle
We wish to maximize the probability of the data given the model. We will
start with the example of estimating the population mean, .
1.0
Pr(x)
0

x
Assume we draw a sample of 3 values, x1 = 4, x2 = 5 and x3 = 6.
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Slide 3c.6
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The Likelihood of The Sample
What would be the likelihood of observing x1, x2 and x3 given that  = 212?
How about if  = 5?
With ML we choose an estimate for  that maximizes the likelihood of the
sample.
The sample that we observed was presumably more likely on average than the
samples that we did not observe. We should make its probability
as large as possible.
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Slide 3c.7
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Steps to ML Estimation
Derive the probability of an observation given the parameters, Pr(yi | ).
Derive the likelihood of the sample, which typically involves
multiplication when we assume independent sampling,
n
l 0   Pr( y i | θ)
i
Derive the likelihood under the general alternative that the data are
arbitrary.
n
l A   Pr( y i )
i
Pick elements of the unknown parameter vector  so that
ˆ 2  2 ln( l 0 l A ).
is as small as possible.
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Slide 3c.8
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