1 Hypothesis Testing Under General Linear Model  Previously we derived the sampling property results assuming normality:  Y = X  + e where e t ~N(0,  2 )  → Y~N(X ,  2 I T )   s =(X'X) -1 X'Y, E(  s )=   Cov(  s )=  β =  2 (X'X) -1   l ~N( ,  2 (X'X) -1 )  σ U 2 unbiased estimate of σ 2  An estimate of Cov(β s ) =  βs =σ U 2 (X'X) -1 e l = y - Xβ l

2 Hypothesis Testing Under General Linear Model  Single Parameter (β k,L ) Hypothesis Test  β k,l ~N(β k,Var(β k )) k th diagonal element of  βs  When σ 2 is known:  unknown true coeff.  When σ 2 not known:  Σ βs =σ u 2 (X'X) -1

3 Hypothesis Testing Under General Linear Model  Can obtain (1-  ) CI for β k :  There is a (1-α) probability that the true unknown value of β is within this range  Does this interval contain our hypothesized value? If it does, than we can not reject H 0

4 Hypothesis Testing Under General Linear Model  Testing More Than One Linear Combination of Estimated Coefficients  Assume we have a-priori information about the value of β  We can represent this information via a set of J-Linear hypotheses (or restrictions):  In matrix notation

5 Hypothesis Testing Under General Linear Model known coefficients

6 Hypothesis Testing Under General Linear Model  Assume we have a model with 5 parameters to be estimated  Joint hypotheses: β 1 =8 and β 2 =β 3  J=2, K=5 β 2 -β 3 =0

7 Hypothesis Testing Under General Linear Model  How do we obtain parameter estimates if J hypotheses are true?  Constrained (Restricted) Least Squares   R is β that minimizes: S=(Y-Xβ)'(Y-Xβ) s.t. Rβ=r = e'e s.t. Rβ=r e.g. we act as if H 0 are true  S*=(Y-Xβ)'(Y-Xβ)+λ'(r-Rβ)  λ is (J x1) Lagrangian multipliers associated with J-joint hypotheses  We want to choose β such that we minimize SSE but also satisfy the J constraints (hypotheses), β R

8 Hypothesis Testing Under General Linear Model  Min. S*=(Y-Xβ)'(Y-Xβ) + λ'(r-Rβ)  What and how many FOC’s?  K+J FOC’s K-FOC’s J-FOC’s

9 Hypothesis Testing Under General Linear Model  What are the FOC’s?  Substitute these FOC into 2 nd set ∂S * /∂λ = (r-Rβ R ) = 0 J → S*=(Y-Xβ)'(Y-Xβ)+λ'(r-Rβ) CRM βSβS

10 Hypothesis Testing Under General Linear Model  The 1 st FOC  Substitute the expression for λ/2 into the 1 st FOC:

11 Hypothesis Testing Under General Linear Model  β R is the restricted LS estimator of β as well as the restricted ML estimator  Properties of Restricted Least Squares EstimatorRestricted Least Squares  → E(  R )   if R   r  V(  R ) ≤ V(  S ) →[V(  S ) - V(  R )] is positive semi- definite  diag(V(  R )) ≤ diag(V(  S )) True but Unknown Value

12 Hypothesis Testing Under General Linear Model  From above, if Y is multivariate normal and H 0 is true  β l,R ~N(β,σ 2 M * (X'X) -1 M * ') ~N(β,σ 2 M * (X'X) -1 )  From previous results, if r-Rβ≠0 (e.g., not all H 0 true), estimate of β is biased if we continue to assume r-Rβ=0 ≠0

13 Hypothesis Testing Under General Linear Model  The variance is the same regardless of he correctness of the restrictions and the biasedness of β R  → β R has a variance that is smaller when compared to β s which only uses the sample information.

14 Hypothesis Testing Under General Linear Model  Beer Consumption Example :   q B ≡ quantity of beer purchased P B ≡ price of beer P L ≡ price of other alcoholic bev. P O ≡ price of other goods INC ≡ household income  Real Prices Matter? All prices and INC  by 10% β 1 + β 2 + β 3 + β 4 =0  Equal Price Impacts? Liquor and Other Goods β 2 =β 3  Unitary Income Elasticity? β 4 =1  Data used in the analysis Data

15  Given the above, what does the R-matrix and r vector look like for these joint tests?  Lets develop a test statistic to test these joint hypotheses  We are going to use the Likelihood Ratio (LR) to test the joint hypotheses Hypothesis Testing Under General Linear Model

16 Hypothesis Testing Under General Linear Model  LR=l U * /l R *  l U * =Max  [l(  |y 1,…,y T );  =(β, σ  )   ] = “unrestricted” maximum likelihood function  l R * =Max  [l(  |y 1,…,y T );  =(β, σ  )   ; Rβ=r] = “restricted” maximum likelihood function  Again, because we are possibly restricting the parameter space via our null hypotheses, LR≥1

17 Hypothesis Testing Under General Linear Model  If l U * is large relative to l R * →data shows evidence that the restrictions (hypotheses) are not true (e.g., reject null hypothesis)  How much should LR exceed 1 before we reject H 0 ?  We reject H 0 when LR ≥ LR C where LR C is a constant chosen on the basis of the relative cost of the Type I vs. Type II errors  When implementing the LR Test you need to know the PDF of the dependent variable which determines the density of the test statistic

18 Hypothesis Testing Under General Linear Model  For LR test, assume Y has a normal distribution  →e~N(0,σ  I T )  This implies the following LR test statistic (LR * )test statistic  What are the distributional characteristics of LR * ? Will address this in a bit

19 Hypothesis Testing Under General Linear Model  We can derive alternative specifications of LR test statistic  LR*=(SSE R -SSE U )/(J  2 U ) (ver. 1)  LR *=[(R  e -r)′[R(X′X) -1 R′] -1 (R  e -r)]/(J  2 U ) (ver. 2) (ver. 2)  LR*=[(  R -  e )′(X′X)(  R -  e )]/(J  2 U ) (ver. 3) β e =β S =β l  What are the Distributional Characteristics of LR* (JHGLL p. 255)Distributional Characteristics  LR* ~ F J,T-K  J = # of Hypotheses  K= # of Parameters (including intercept)

20 Hypothesis Testing Under General Linear Model  Proposed Test Procedure  Choose  = P(reject H 0 | H 0 true) = P(Type-I error)  Calculate the test statistic LR* based on sample information  Find the critical value LR crit in an F-table such that:  = P(F (J, T – K)  LR crit ), where α = P(reject H 0 | H 0 true) f(LR*) α LR crit α = P(F J,T-K ≥ LR crit )

21 Hypothesis Testing Under General Linear Model  Proposed Test Procedure  Choose  = P(reject H 0 | H 0 true) = P(Type-I error)  Calculate the test statistic LR* based on sample information  Find the critical value LR crit in an F- table such that:  = P(F (J, T – K)  LR crit ), where α = P(reject H 0 | H 0 true)  Reject H 0 if LR*  LR crit  Don’t reject H 0 if LR* < LR crit

22 Hypothesis Testing Under General Linear Model  Beer Consumption Example  Does the regression do a better job in explaining variation in beer consumption than if assumed the mean response across all obs.?  Remember SSE=(T-K)σ 2 U  Under H 0 : All slope coefficients=0  Under H 0, TSS=SSE given that that there is no RSS and TSS=RSS+SSE

23 Hypothesis Testing Under General Linear Model Log-Log Beer Consumption Model Unconstrained Model R2R Adj. R σUσU Obs30 Variable CoeffStd ErrorT-Stat Intercept lnP B lnP L lnP O ln(INC) Constrained Model σUσU SSE R = *29= CoeffStd ErrorT-Stat Intercept SSE = *25 = R 2 = / TSS=SSE R Mean of LN(Beer)

24 Hypothesis Testing Under General Linear Model  Results of our test of overall significance of regression modeloverall significance  Lets look at the following GAUSS CodeGAUSS Code  GAUSS command: CDFFC(29.544,4,25)=3.799e-009 CDFFC Computes the complement of the cdf of the F distribution (1- F df1,df2 ) Unlikely value of F if hypothesis is true, that is no impact of exogenous variables on beer consumption Reject the null hypothesis  An alternative look An alternative look

25 Hypothesis Testing Under General Linear Model  Beer Consumption Example  Three joint hypotheses examplejoint hypotheses example Sum of Price and Income Elasticities Sum to 0 (e.g., β 1 + β 2 + β 3 + β 4 =0) Other Liquor and Other Goods Price Elasticities are Equal (e.g., β 2 =β 3 ) Income Elasticity = 1 (e.g., β 4 =1) cdffc(0.84,3,25)=0.4848

26 Hypothesis Testing Under General Linear Model PDF F 3, area =  Location of our calculated test statistic F

27 Hypothesis Testing Under General Linear Model  A side note: How do you estimate the variance of an elasticity and therefore test H 0 about this elasticity?  Suppose you have the following model: FDX t = β 0 + β 1 Inc t + β 2 Inc 2 t + e t  FDX= food expenditure  Inc=household income  Want to estimate the impacts of a change in income on expenditures. Use an elasticity measure evaluated at mean of the data. That is:

28 Hypothesis Testing Under General Linear Model  Income Elasticity (Γ) is:  How do you calculate the variance of Γ?  We know that: Var(α′Z)= α′Var(Z)α  Z is a column vector of RV’s  α a column vector of constants  Treat β 0, β 1 and β 2 are RV’s. The α vector is: FDX t = β 0 + β 1 Inc t + β 2 Inc 2 t + e t Linear combination of Z

29 Hypothesis Testing Under General Linear Model  This implies var(Γ) is: (1 x 1) σ 2 (X'X) -1 (3 x 3) (1 x 3) (3 x 1) Due to 0 α value

30 Hypothesis Testing Under General Linear Model  This implies: var(Γ) is: C12C12 C22C22 2C 1 C 2