Title: Hypothesis Testing Under
1Hypothesis 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
2Hypothesis 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.
3Hypothesis 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
4Hypothesis 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
5Hypothesis Testing Under General Linear Model
known coefficients
6Hypothesis 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
7Hypothesis 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
8Hypothesis Testing Under General Linear Model
- Min. S(Y-Xß)'(Y-Xß) ?'(r-Rß)
- What and how many FOCs?
- KJ FOCs
K-FOCs
J-FOCs
9Hypothesis Testing Under General Linear Model
S(Y-Xß)'(Y-Xß)?'(r-Rß)
CRM
ßS
- Substitute these FOC into 2nd set
- ?S/?? (r-RßR) 0J ?
10Hypothesis Testing Under General Linear Model
- The 1st FOC
- Substitute the expression for ?/2 into the 1st
FOC
11Hypothesis 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
12Hypothesis 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
13Hypothesis 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.
14Hypothesis 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
15Hypothesis 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
16Hypothesis 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
17Hypothesis 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
18Hypothesis 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
19Hypothesis 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)
20Hypothesis 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
21Hypothesis 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
22Hypothesis 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
23Hypothesis 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)
24Hypothesis 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
25Hypothesis 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
26Hypothesis Testing Under General Linear Model
- Location of our calculated test statistic
PDF
F3,25
F
0.84
area 0.4848
27Hypothesis 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
28Hypothesis 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
29Hypothesis Testing Under General Linear Model
(1 x 3)
s2(X'X)-1 (3 x 3)
(3 x 1)
Due to 0 a value
(1 x 1)
30Hypothesis Testing Under General Linear Model
C22
C12
2C1C2