Samenvatting les 4 colleges

LES 4


Hypothesis Testing on a single parameter




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,Hypothesis Testing on multiple parameters

1. Testing whether 2 parameters are equal to one another

β p=β q
Y¿ α + β 1 X 1 + β 2 X 2+ u
H 0 : β 1=β 2 ⇔ β 1−β 2=0
H a : β 1 ≠ β 2 ⇔ β 1−β 2 ≠0
Remember!!!
 T-test  Difference in parameters EXAMEN
( a ± b )2=a2 +b2 ± 2ab
 Difference of two ‘normal distributions’
^β − ^β −0 ^β − ^β −0
p q p q
t= = t (df =N −k )
√ σ^ 2
β p−β q √ σ^ 2
βp + σ^ 2β −2 σ^ β
p p βq


 F-test  Difference in terms (Linear Restriction)
 Y =α + β 1 X 1+ β 2 X 2 +u B2 vervangen door B1
Under de H0 kan ik
⇒Y =α + β1 X 1 + β 1 X 2 +u “under the Null”
een model schatten
⇔ Y =α + β1 ( X 1 + X 2) + u

If the Linear Restriction holds in reality; both models should perform the same… in other
words; same Sum of Squared Residuals = H0
Y =α + β 1 X 1+ β 2 X 2 +u ≅ Y =α + β1 ( X 1 + X 2 ) +u

F-test: comparing SSR
 F-tests are based on differences in the Sum of Squared Residuals
 Remember that SSR is actually the “unexplained part of the variance in the
dependent variable”
 SSR is minimized by OLS
 SSR is ‘lack of fit’
2 SSR
 R =1− , which is ‘Model Fit’
SST

 Basically, we compare how well two models fit the data
 Unrestricted model “as it is” SS Ru
 Restricted model “under the Null” SS Rr
m = number of restricted
SS Rr −SS R u parameters
m
F= F ( m; N−k )
SS Ru
N−k“too much” from each other, the Restriction will
 If the ‘Model Fits’ deviate
probably not be true  Reject H 0
 “too much” is determined using F-test: hoe extreme is het wat we hebben
berekend? Kijk waar de F-verdeling ongeveer op de curve ligt?
 A particular H0: all coefficients are equal to 0

Example




2

, Zijn de 2 waarden aan
mekaar gelijk?
P-waarde zeer klein dus
H0 verwerpen: zie je ook
heel duidelijk




2. Testing whether the sum (or difference) of 2 parameters is
equal to a specified value β p + β q=k

2-Parameter restriction

Cobb-Douglas Production Function
β1 β2 u
Y = β0 X 1 X 2 e ⇔ lnY =α + β 1 ln X 1+ β 2 ln X 2 +u
Constant Returns to Scale? H 0 : β 1+ β 2=1

 T-test Approach
^β p + β^ q −k ^β 1+ ^β 2−1
t= = t (df =N −k )
√ σ^ 2
βp −βq √ σ^ 2
β1
2
+ σ^ β +2 σ^ β
2 1 β2




 F-test  Difference in terms (Linear Restriction)
 lnY =α + β 1 ln X 1 + β 2 ln X 2 +u
⇒lnY =α + β 1 ln X 1+(1−β ¿¿ 1) ln X 2+u ¿ “under the Null”
⇔ lnY =α + β 1 ln X 1+ ln X 2−β 1 ln X 2 +u
⇔ lnY −ln X 2=α + β 1 ( ln X 1−ln X 2 ) +u (dit is het nieuwe restrictie model)

If the Linear Restriction holds in reality; same Sum of Squared Residuals




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