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Hypothesis Testing Under

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Title: Hypothesis Testing Under


1
Hypothesis Testing Under General Linear Model
  • Previously we derived the sampling property
    results assuming normality
  • Y Xb e where etN(0,s2)
  • ? YN(Xb,s2IT)
  • bs(X'X)-1X'Y, E(bs)b
  • Cov(bs) ?ß s2(X'X)-1
  • blN(b, s2(X'X)-1)
  • sU2 unbiased estimate of s2
  • An estimate of Cov(ßs) ?ßssU2(X'X)-1

el y - Xßl
2
Hypothesis Testing Under General Linear Model
  • Single Parameter (ßk,L) Hypothesis Test
  • ßk,lN(ßk,Var(ßk))

Sßssu2(X'X)-1
kth diagonal element of ?ßs
unknown true coeff.
  • When s2 is known
  • When s2 not known

3
Hypothesis Testing Under General Linear Model
  • Can obtain (1-?) CI for ßk
  • There is a (1-a) probability that the true
    unknown value of ß is within this range
  • Does this interval contain our hypothesized
    value?
  • If it does, than we can not reject H0

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 ß
  • 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 ß18 and ß2ß3
  • J2, K5

ß2-ß30
7
Hypothesis Testing Under General Linear Model
  • How do we obtain parameter estimates if J
    hypotheses are true?
  • Constrained (Restricted) Least Squares
  • bR is ß that minimizes
  • S(Y-Xß)'(Y-Xß) s.t. Rßr
  • e'e s.t. Rßr
  • e.g. we act as if H0 are true
  • S(Y-Xß)'(Y-Xß)?'(r-Rß)
  • ? is (J x1) Lagrangian multipliers
  • associated with J-joint hypotheses
  • We want to choose ß such that we minimize SSE but
    also satisfy the J constraints (hypotheses), ßR

8
Hypothesis Testing Under General Linear Model
  • Min. S(Y-Xß)'(Y-Xß) ?'(r-Rß)
  • What and how many FOCs?
  • KJ FOCs

K-FOCs
J-FOCs
9
Hypothesis Testing Under General Linear Model
S(Y-Xß)'(Y-Xß)?'(r-Rß)
  • What are the FOCs?

CRM
ßS
  • Substitute these FOC into 2nd set
  • ?S/?? (r-RßR) 0J ?

10
Hypothesis Testing Under General Linear Model
  • The 1st FOC
  • Substitute the expression for ?/2 into the 1st
    FOC

11
Hypothesis Testing Under General Linear Model
  • ßR is the restricted LS estimator of ß as well as
    the restricted ML estimator
  • Properties of Restricted Least Squares Estimator
  • ?E(bR) ? b if Rb ? r
  • V(bR) V(bS)
  • ?V(bS) - V(bR) is positive semi-definite
  • diag(V(bR)) diag(V(bS))

True but Unknown Value
12
Hypothesis Testing Under General Linear Model
  • From above, if Y is multivariate normal and H0 is
    true
  • ßl,RN(ß,s2M(X'X)-1M')
  • N(ß,s2M(X'X)-1)
  • From previous results, if r-Rß?0 (e.g.,
  • not all H0 true), estimate of ß is biased
  • if we continue to assume r-Rß0

?0
13
Hypothesis Testing Under General Linear Model
  • The variance is the same regardless of he
    correctness of the restrictions and the
    biasedness of ßR
  • ? ßR has a variance that is smaller when
    compared to ßs which only uses the sample
    information.

14
Hypothesis Testing Under General Linear Model
  • Beer Consumption Example
  • qB quantity of beer purchased
  • PB price of beer
  • PL price of other alcoholic bev.
  • PO price of other goods
  • INC household income
  • Real Prices Matter?
  • All prices and INC ? by 10
  • ß1 ß2 ß3 ß40
  • Equal Price Impacts?
  • Liquor and Other Goods
  • ß2ß3
  • Unitary Income Elasticity?
  • ß41
  • Data used in the analysis

15
Hypothesis Testing Under General Linear Model
  • 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

16
Hypothesis Testing Under General Linear Model
  • LRlU/lR
  • lUMax? l(?y1,,yT)
  • ?(ß, s?)? ?
  • unrestricted maximum likelihood function
  • lRMax? l(?y1,,yT)
  • ?(ß, s?)?? Rßr
  • restricted maximum likelihood function
  • Again, because we are possibly restricting the
    parameter space via our null hypotheses, LR1

17
Hypothesis Testing Under General Linear Model
  • If lU is large relative to lR?data shows
    evidence that the restrictions (hypotheses) are
    not true (e.g., reject null hypothesis)
  • How much should LR exceed 1 before we reject H0?
  • We reject H0 when LR LRC where LRC 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
  • ?eN(0,s?IT)
  • This implies the following LR test statistic
    (LR)
  • 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(SSER-SSEU)/(Js2U)
  • (ver. 1)
  • LR(Rbe-r)'R(X'X)-1R'-1(Rbe-r)/(Js2U) (ver
    . 2)
  • LR(bR-be)'(X'X)(bR-be)/(Js2U)
  • (ver. 3)

ße ßSßl
  • What are the Distributional Characteristics of
    LR (JHGLL p. 255)
  • LR FJ,T-K
  • J of Hypotheses
  • K of Parameters
  • (including intercept)

20
Hypothesis Testing Under General Linear Model
  • Proposed Test Procedure
  • Choose a P(reject H0 H0 true) P(Type-I
    error)
  • Calculate the test statistic LR based on sample
    information
  • Find the critical value LRcrit in an F-table such
    that
  • a P(F(J, T K) ³ LRcrit), where a P(reject
    H0 H0 true)

f(LR)
a P(FJ,T-K LRcrit)
LRcrit
a
21
Hypothesis Testing Under General Linear Model
  • Proposed Test Procedure
  • Choose a P(reject H0 H0 true) P(Type-I
    error)
  • Calculate the test statistic LR based on sample
    information
  • Find the critical value LRcrit in an F-table such
    that
  • a P(F(J, T K) ³ LRcrit), where a P(reject
    H0 H0 true)
  • Reject H0 if LR ³ LRcrit
  • Dont reject H0 if LR lt LRcrit

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)s2U
  • Under H0 All slope coefficients0
  • Under H0, TSSSSE given that that there is no RSS
    and TSSRSSSSE

23
Hypothesis Testing Under General Linear Model
Log-Log Beer Consumption Model Log-Log Beer Consumption Model Log-Log Beer Consumption Model Log-Log Beer Consumption Model
Unconstrained Model Unconstrained Model Unconstrained Model Unconstrained Model
R2 0.8254
Adj. R2 0.7975
sU 0.05997
Obs 30
Variable  Coeff Std Error T-Stat
Intercept -3.243 3.743 -0.87
lnPB -1.020 0.239 -4.27
lnPL -0.583 0.560 -1.04
lnPO 0.210 0.080 2.63
ln(INC) 0.923 0.416 2.22
Constrained Model Constrained Model Constrained Model Constrained Model
sU 0.13326 SSER0.133262290.51497 SSER0.133262290.51497
  Coeff Std Error T-Stat
Intercept 4.019 0.0243 165.17
SSE 0.059972 25 0.08992 R21-
0.08992/0.51497
TSSSSER
Mean of LN(Beer)
24
Hypothesis Testing Under General Linear Model
  • Results of our test of overall significance of
    regression model
  • Lets look at the following GAUSS Code
  • GAUSS command
  • CDFFC(29.544,4,25)3.799e-009
  • CDFFC Computes the complement of the cdf of the F
    distribution (1-Fdf1,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

25
Hypothesis Testing Under General Linear Model
  • Beer Consumption Example
  • Three joint hypotheses example
  • Sum of Price and Income Elasticities Sum to 0
  • (e.g., ß1 ß2 ß3 ß40)
  • Other Liquor and Other Goods Price Elasticities
    are Equal (e.g., ß2ß3)
  • Income Elasticity 1 (e.g., ß41)
  • cdffc(0.84,3,25)0.4848

26
Hypothesis Testing Under General Linear Model
  • Location of our calculated test statistic

PDF
F3,25
F
0.84
area 0.4848
27
Hypothesis Testing Under General Linear Model
  • A side note How do you estimate the variance of
    an elasticity and therefore test H0 about this
    elasticity?
  • Suppose you have the following model
  • FDXt ß0 ß1Inct ß2 Inc2t et
  • FDX food expenditure
  • Inchousehold 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
FDXt ß0 ß1Inct ß2 Inc2t et
  • Income Elasticity (G) is
  • How do you calculate the variance of G?
  • We know that Var(a'Z) a'Var(Z)a
  • Z is a column vector of RVs
  • a a column vector of constants
  • Treat ß0, ß1 and ß2 are RVs. The a vector is

Linear combination of Z
29
Hypothesis Testing Under General Linear Model
  • This implies var(G) is

(1 x 3)
s2(X'X)-1 (3 x 3)
(3 x 1)
Due to 0 a value
(1 x 1)
30
Hypothesis Testing Under General Linear Model
C22
  • This implies var(G) is

C12
2C1C2
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