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)

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

Slide 6.3 Linear Hypotheses MathematicalMarketing What of the Distribution of the Estimator? As normal Central Limit Property of Linear Combinations

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)

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!

Slide 6.6 Linear Hypotheses MathematicalMarketing Lets Think About the Denominator where d ii are diagonal elements of D = (XX) -1 = {d ij }

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

Slide 6.8 Linear Hypotheses MathematicalMarketing A Confidence Interval for  i A 1 -  confidence interval for  i is given by which simply means that

Slide 6.9 Linear Hypotheses MathematicalMarketing Graphic of Confidence Interval ii 

Slide 6.10 Linear Hypotheses MathematicalMarketing Statistical Hypothesis Testing: Step One H 0 :  i = c H A :  i ≠ c Generate two mutually exclusive hypotheses:

Slide 6.11 Linear Hypotheses MathematicalMarketing Statistical Hypothesis Testing Step Two Summarize the evidence with respect to H 0 :

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

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

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 :

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

Slide 6.16 Linear Hypotheses MathematicalMarketing Multiple Degree of Freedom Hypotheses

Slide 6.17 Linear Hypotheses MathematicalMarketing Examples of Multiple df Hypotheses tests H 0 :  2 =  3 = 0 tests H 0 :  1 =  2 =  3

Slide 6.18 Linear Hypotheses MathematicalMarketing Testing Multiple df Hypotheses

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:

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

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

Slide 6.22 Linear Hypotheses MathematicalMarketing SS Error for GLS with

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

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

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′

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

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

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

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?

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

Slide 6.31 Linear Hypotheses MathematicalMarketing In Addition to Being Idempotent…