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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 H0: i = c Hypotheses of the form H0: i c Hypotheses of the form H0: a′ = c Hypotheses of the form A = c We also cover deductions when V(e) 2I (Generalized Least Squares) Mathematical Marketing Slide 6.1 Linear Hypotheses The Variance of the Estimator From these two raw ingredients and a theorem: βˆ ( XX) 1 Xy. V(y) = V(X + e) = V(e) = 2I we conclude V(βˆ ) [( XX) 1 X] 2I [( XX) 1 X] 2 ( XX) 1 XIX( XX) 1 2 ( XX) 1 Mathematical Marketing Slide 6.2 Linear Hypotheses What of the Distribution of the Estimator? As n 1 bn n a1 normal Central Limit Property of Linear Combinations Mathematical Marketing Slide 6.3 Linear Hypotheses So What Can We Conclude About the Estimator? From the V(linear combo) + assumptions about e From the Central Limit Theorem βˆ ( XX) 1 Xy ~ N[β, 2 ( XX) 1 ] From Ch 5- E(linear combo) Mathematical Marketing Slide 6.4 Linear Hypotheses Steps Towards Inference About In general q E (q ) V̂ (q ) ~ t df In particular (X′X)-1X′y ˆ i i ~ t n k ˆ V̂ ( ) But note the hat on the V! i Mathematical Marketing Slide 6.5 Linear Hypotheses Lets Think About the Denominator V(ˆ i ) 2 d ii where dii are diagonal elements of D = (XX)-1 = {dij} n e i2 SS ˆ 2 s 2 Error i nk nk so that V̂(ˆ i ) s 2 d ii Mathematical Marketing Slide 6.6 Linear Hypotheses Putting It All Together ˆ i i ŝ 2 d ii ~ t n k Now that we have a t, we can use it for two types of inference about : Confidence Intervals Hypothesis Testing Mathematical Marketing Slide 6.7 Linear Hypotheses A Confidence Interval for i A 1 - confidence interval for i is given by ˆ i t / 2,n k s 2 d ii which simply means that Pr ˆ i t / 2,n k s 2 d ii i ˆ i t / 2, n k s 2 d ii 1 Mathematical Marketing Slide 6.8 Linear Hypotheses Graphic of Confidence Interval 1- 1.0 Pr(ˆ i ) 0 ˆ i t / 2,n k s 2 d ii Mathematical Marketing i ˆ i t / 2,n k s 2 d ii Slide 6.9 Linear Hypotheses Statistical Hypothesis Testing: Step One Generate two mutually exclusive hypotheses: H0: i = c HA: i ≠ c Mathematical Marketing Slide 6.10 Linear Hypotheses Statistical Hypothesis Testing Step Two Summarize the evidence with respect to H0: ˆ ˆ ˆt i i i c s 2 d ii V̂(ˆ i ) Mathematical Marketing Slide 6.11 Linear Hypotheses Statistical Hypothesis Testing Step Three reject H0 if the probability of the evidence given H0 is small | tˆ| t /2,n-k , Mathematical Marketing Slide 6.12 Linear Hypotheses One Tailed Hypotheses Our theories should give us a sign for Step One in which case we might have H0: i c HA: i < c In that case we reject H0 if tˆ t , n-k Mathematical Marketing Slide 6.13 Linear Hypotheses A More General Formulation Consider a hypothesis of the form H0: a´ = c so if c = 0… a 0 1 1 0 0 a 0 1 1 0 0 1 1 a 0 1 0 2 2 Mathematical Marketing tests H0: 1= 2 tests H0: 1 + 2 = 0 tests H0: 1 2 3 2 Slide 6.14 Linear Hypotheses A t test for This More Complex Hypothesis We need to derive the denominator of the t using the variance of a linear combination V(aβˆ ) aV(βˆ ) a 2a( XX) 1 a which leads to tˆ Mathematical Marketing aβˆ c . s 2a( XX) 1 a Slide 6.15 Linear Hypotheses Multiple Degree of Freedom Hypotheses H 0 : Aβ q c1 a1. c1 a c 2. 2 H0 : β aq . cq Mathematical Marketing Slide 6.16 Linear Hypotheses Examples of Multiple df Hypotheses Mathematical Marketing 0 0 1 0 H0 : 0 0 0 1 0 0 1 2 0 3 tests H0: 2 = 3 = 0 0 1 1 0 H0 : 0 1 0 1 0 0 1 2 0 3 tests H0: 1 = 2 = 3 Slide 6.17 Linear Hypotheses Testing Multiple df Hypotheses 1 SSH ( Aβˆ c)A( XX) 1 A ( Aβˆ c) SSH / q ~ Fq ,n k SSError / n k SSError yy yX( XX) 1 Xy Mathematical Marketing Slide 6.18 Linear Hypotheses Another Way to Think About SSH Assume we have an A matrix as below: 0 0 1 0 H0 : 0 0 0 1 0 0 1 2 0 3 We could calculate the SSH by running two versions of the model: the full model and a model restricted to just 1 SSH = SSError (Restricted Model) – SSError (Full Model) so F is F̂ Mathematical Marketing SSError (Restricted ) SSError (Full ) / 2 SSError (Full ) / n k Slide 6.19 Linear Hypotheses A Hypothesis That All ’s Are Zero If our hypothesis is H 0 : 1 2 k* 0 Then the F would be F̂ SSError (Restricted to 0 ) SSError (Full) / k* SSError (Full) / n k Which suggests a summary for the model R2 Mathematical Marketing SSError (Re stricted to 0 ) SSError (Full ) SSError (Re stricted to 0 ) Slide 6.20 Linear Hypotheses Generalized Least Squares When we cannot make the Gauss-Markov Assumption that V(e) = 2I Suppose that V(e) = 2V. Our objective function becomes f = eV-1e βˆ [ XV 1 X]1 XV 1y Mathematical Marketing Slide 6.21 Linear Hypotheses SSError for GLS s2 SSError nk with SSError (y Xβˆ )V1 (y Xβˆ ) Mathematical Marketing Slide 6.22 Linear Hypotheses GLS Hypothesis Testing H0: i = 0 H0: a = c H0: A - c = 0 Mathematical Marketing t̂ tˆ ˆ i c s 2d ii where dii is the ith diagonal element of (XV-1X)-1 aβˆ c s 2a( XV 1 X) 1 a SSH / q ~ Fq ,n k SSError / n k SS H ( Aβˆ c)[ A( XV 1 X) 1 A]1 ( Aβˆ c) SS Error (y Xβˆ )(y Xβˆ ) Slide 6.23 Linear Hypotheses Accounting for the Sum of Squares of the Dependent Variable e′e = y′y - y′X(X′X)-1X′y SSError = SSTotal - SSPredictable y′y = y′X(X′X)-1X′y + e′e SSTotal = SSPredictable + SSError Mathematical Marketing Slide 6.24 Linear Hypotheses SSPredicted and SSTotal Are a Quadratic Forms SSPredicted is And SSTotal yX(XX) 1 Xy yPy yy = yIy Here we have defined P = X(X′X)-1X′ Mathematical Marketing Slide 6.25 Linear Hypotheses The SSError is a Quadratic Form Having defined P = X(XX)-1 X, now define M = I – P, i. e. I - X(XX)-1X. The formula for SSError then becomes ee y y y X( XX) 1 Xy y Iy y Py y [I P] y y My. Mathematical Marketing Slide 6.26 Linear Hypotheses Putting These Three Quadratic Forms Together SSTotal = SSPredictable + SSError yIy = yPy + yMy here we note that I=P+M Mathematical Marketing Slide 6.27 Linear Hypotheses M and P Are Linear Transforms of y ŷ = Py and e = My so looking at the linear model: y yˆ e Iy = Py + My and again we see that I=P+M Mathematical Marketing Slide 6.28 Linear Hypotheses The Amazing M and P Matrices ŷ = Py and yˆ yˆ = SSPredicted = y′Py What does this imply about M and P? e = My and = SSError = y′My Mathematical Marketing Slide 6.29 Linear Hypotheses The Amazing M and P Matrices Mathematical Marketing ŷ = Py and yˆ yˆ = SSPredicted = y′Py PP = P e = My and = SSError = y′My MM = M Slide 6.30 Linear Hypotheses In Addition to Being Idempotent… 1 1n n M n 1 0n 1 1n n Pn 1 0n PM n 0 n. Mathematical Marketing Slide 6.31 Linear Hypotheses