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Slide 6.1 Linear Hypotheses MathematicalMarketing In This Chapter We Will Cover Deductions we can make about even though it is not observed. These include Confidence Intervals Hypotheses of the form H 0 : i = c Hypotheses of the form H 0 : i c Hypotheses of the form H 0 : a′ = c Hypotheses of the form A = c We also cover deductions when V(e) 2 I (Generalized Least Squares)
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Slide 6.2 Linear Hypotheses MathematicalMarketing The Variance of the Estimator V(y) = V(X + e) = V(e) = 2 I From these two raw ingredients and a theorem: we conclude
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Slide 6.3 Linear Hypotheses MathematicalMarketing What of the Distribution of the Estimator? As normal Central Limit Property of Linear Combinations
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Slide 6.4 Linear Hypotheses MathematicalMarketing So What Can We Conclude About the Estimator? From the Central Limit Theorem From the V(linear combo) + assumptions about e From Ch 5- E(linear combo)
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Slide 6.5 Linear Hypotheses MathematicalMarketing Steps Towards Inference About In general In particular (X′X) -1 X′y But note the hat on the V!
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Slide 6.6 Linear Hypotheses MathematicalMarketing Lets Think About the Denominator where d ii are diagonal elements of D = (XX) -1 = {d ij }
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Slide 6.7 Linear Hypotheses MathematicalMarketing Putting It All Together Now that we have a t, we can use it for two types of inference about : Confidence Intervals Hypothesis Testing
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Slide 6.8 Linear Hypotheses MathematicalMarketing A Confidence Interval for i A 1 - confidence interval for i is given by which simply means that
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Slide 6.9 Linear Hypotheses MathematicalMarketing Graphic of Confidence Interval ii 1.0 0 1 -
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Slide 6.10 Linear Hypotheses MathematicalMarketing Statistical Hypothesis Testing: Step One H 0 : i = c H A : i ≠ c Generate two mutually exclusive hypotheses:
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Slide 6.11 Linear Hypotheses MathematicalMarketing Statistical Hypothesis Testing Step Two Summarize the evidence with respect to H 0 :
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Slide 6.12 Linear Hypotheses MathematicalMarketing Statistical Hypothesis Testing Step Three reject H 0 if the probability of the evidence given H 0 is small
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Slide 6.13 Linear Hypotheses MathematicalMarketing One Tailed Hypotheses Our theories should give us a sign for Step One in which case we might have H 0 : i c H A : i < c In that case we reject H 0 if
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Slide 6.14 Linear Hypotheses MathematicalMarketing A More General Formulation Consider a hypothesis of the form H 0 : a´ = c so if c = 0… tests H 0 : 1 = 2 tests H 0 : 1 + 2 = 0 tests H 0 :
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Slide 6.15 Linear Hypotheses MathematicalMarketing A t test for This More Complex Hypothesis We need to derive the denominator of the t using the variance of a linear combination which leads to
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Slide 6.16 Linear Hypotheses MathematicalMarketing Multiple Degree of Freedom Hypotheses
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Slide 6.17 Linear Hypotheses MathematicalMarketing Examples of Multiple df Hypotheses tests H 0 : 2 = 3 = 0 tests H 0 : 1 = 2 = 3
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Slide 6.18 Linear Hypotheses MathematicalMarketing Testing Multiple df Hypotheses
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Slide 6.19 Linear Hypotheses MathematicalMarketing Another Way to Think About SS H We could calculate the SS H by running two versions of the model: the full model and a model restricted to just 1 SS H = SS Error (Restricted Model) – SS Error (Full Model) so F is Assume we have an A matrix as below:
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Slide 6.20 Linear Hypotheses MathematicalMarketing A Hypothesis That All ’s Are Zero If our hypothesis is Then the F would be Which suggests a summary for the model
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Slide 6.21 Linear Hypotheses MathematicalMarketing Generalized Least Squares f = eV -1 e When we cannot make the Gauss-Markov Assumption that V(e) = 2 I Suppose that V(e) = 2 V. Our objective function becomes
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Slide 6.22 Linear Hypotheses MathematicalMarketing SS Error for GLS with
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Slide 6.23 Linear Hypotheses MathematicalMarketing GLS Hypothesis Testing H 0 : i = 0where d ii is the ith diagonal element of (XV -1 X) -1 H 0 : a = c H 0 : A - c = 0
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Slide 6.24 Linear Hypotheses MathematicalMarketing Accounting for the Sum of Squares of the Dependent Variable e′e = y′y - y′X(X′X) -1 X′y SS Error = SS Total - SS Predictable y′y = y′X(X′X) -1 X′y + e ′ e SS Total = SS Predictable + SS Error
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Slide 6.25 Linear Hypotheses MathematicalMarketing SS Predicted and SS Total Are a Quadratic Forms And SS Total yy = yIy SS Predicted is Here we have defined P = X(X′X) -1 X′
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Slide 6.26 Linear Hypotheses MathematicalMarketing The SS Error is a Quadratic Form Having defined P = X(XX) -1 X, now define M = I – P, i. e. I - X(XX) -1 X. The formula for SS Error then becomes
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Slide 6.27 Linear Hypotheses MathematicalMarketing Putting These Three Quadratic Forms Together SS Total = SS Predictable + SS Error yIy = yPy + yMy I = P + M here we note that
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Slide 6.28 Linear Hypotheses MathematicalMarketing M and P Are Linear Transforms of y = Py and e = My so looking at the linear model: and again we see that I = P + M Iy = Py + My
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Slide 6.29 Linear Hypotheses MathematicalMarketing The Amazing M and P Matrices = Py and = SS Predicted = y′Py e = My and = SS Error = y′My What does this imply about M and P?
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Slide 6.30 Linear Hypotheses MathematicalMarketing The Amazing M and P Matrices = Py and = SS Predicted = y′Py e = My and = SS Error = y′My PP = P MM = M
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Slide 6.31 Linear Hypotheses MathematicalMarketing In Addition to Being Idempotent…
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