Hypothesis Testing Under

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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
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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

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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

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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

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Hypothesis Testing Under General Linear Model
known coefficients
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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
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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

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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
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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 ?

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Hypothesis Testing Under General Linear Model

• The 1st FOC
• Substitute the expression for ?/2 into the 1st
  FOC

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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
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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
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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.

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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

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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

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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

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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

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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

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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)

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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
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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

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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

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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)
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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

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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

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Hypothesis Testing Under General Linear Model

• Location of our calculated test statistic

PDF
F3,25
F
0.84
area 0.4848
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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

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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
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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)
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Hypothesis Testing Under General Linear Model
C22

• This implies var(G) is

C12
2C1C2