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Notes on Econometrics Applications Jorge Rojas Freedom Fighter Seattle, WA, USA January 25, 2012 Abstract
This is a summary containing the main ideas in the subject. This is not a summary of the lecture notes, this is a summary of ideas and basic basic concep concepts. ts. The mathematic mathematical al mac machin hinery ery is necess necessary ary,, but the principles are much more important.
1
Lin Linear ear Algeb lgebr ra
of Transpose Master your Properties semester with Scribd 1. (AT )T = A & The New York Times Special offer for students: Only 2. $4.99/month. (A + B )T =
AT + B T
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9. If A If A is a square matrix, then its eigenvalues are equal to the eigenvalues of its transpose. Notice that if A
(n m) ,
∈M
×
then AAT is always symmetric.
Properties of the Inverse
1. (A−1 )−1 = A 2. (kA) kA)−1 = k1 A−1
∀k ∈ R \ {0}
3. (AT )−1 = (A−1 )T 4. (AB) AB)−1 = B −1 A−1 5. det(A det(A−1 ) = [det(A [det(A)]−1 Properties of the trace
1. Definit Definition ion.. tr(A tr(A) =
n i=1
aii
2. tr(A tr(A + B ) = tr(A tr( A) + tr(B tr( B ) 3. tr(cA tr(cA)) = c tr(A tr(A)
·
∀c ∈ R
4. tr(AB tr(AB)) = tr(BA tr(BA)) 5. Similarit Similarity y invarian invariant: t: tr(P tr(P −1 AP ) tr(A) AP ) = tr(A
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Properties of the Kronecker Product
⊗ B)− 2. If A If A ∈ M
1
= (A−1
1
⊗ B− ) × and B ∈ M × , then: |A ⊗ B| = |A| |B| (A ⊗ B ) = A ⊗ B tr(A tr(A ⊗ B ) = tr(A)tr(B )tr(B ) 3. (A ⊗ B )(C )(C ⊗ D) = AC ⊗ BD 1. (A
(m m)
( n n)
n
T
m
T
T
Careful! it doesn’t distribute with respect to the usual multiplication
Properties of Determinants
only defined for A
∈ Mn×n
1. det(aA det(aA)) = an det(A det(A)
· ∀a ∈ R 2. det(−A) = (−1) · det(A det(A) 3. det(AB det(AB)) = det(A det(A) · det(B det(B ) n
4. det(I det(I n ) = 1 1 5. det(A det(A) = det(A
−1
)
6. det(BAB det(BAB −1with ) = det(A det(Scribd A) similarity transformation. Master your semester Read Free Foron 30this Days Sign up to vote title T 7. det(A det( det(A ) A) = det(A & The New York Times Useful Not useful Special offer for students: Only $4.99/month. ¯
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8. det(A) = det(A det(A) the bar represents complex conjugate.
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Differentiation of traces
1.
∂tr( ∂tr (AX) AX ) ∂X
2.
∂tr( ∂tr (AXB ) ∂X
3.
∂tr( ∂tr (AXBX T C ) ∂X
=
∂tr( ∂tr (XA) XA ) ∂X
=
= AT
∂tr( ∂tr (XB A) ∂X
=
= (BA) BA)T
∂tr( ∂tr (XB X T CA) CA ) ∂X
= (BX T C A)T + (CAXB (CAXB))
|X | = cofactor( 4. ∂ ∂X cofactor(X ) = det( det(X ) (X −1 )T
·
2
Proba Probabil bilit ity y Distri Distribu butio tions ns
Here, we could say that, starts the summary for Econometrics ECON581. Definition 2.1 Normal distribution distribution: wher where µ is the mean and σ 2 is the variance. 2
f ( f (x) =
1 √ e− σ 2π
(x−µ) 2σ 2
∀x ∈ R
If the mean is zero and the variance is one, then we have the standard normal (0, 1). distribution N (0,
The normal distribution has no closed form solution for its cumulative density function CDF.
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Definition 2.3 2.3 t-student t-student Distribution Distribution: We say say tha that t t(r) has r degrees of fre freedom dom.. The t-distribu t-distributio tion n has “fa “fatte tter” r” tails tails than than the standar standard normal normal distribution.
Z
Z ∼ N (0, ∼t (0, 1) ∧ A ∼ χ ∧ (Z and A are independent ) =⇒ T = A/r 2 (r)
(r)
E (T ) T ) = 0 and V ( V (T ) T ) = r−r 2 The t distribution is an “appropriate” ratio of a standard normal and a χ2(r) random variables. Definitio Definition n 2.4 F Distribu Distribution tion: We say say that that F ( degrees of F (r1 , r2 ) has r1 degrees freedom in the numerator and r2 degrees of freedom in the denominator.
A1
2 (r1 )
2 (r2 )
∼ χ ∧A ∼ χ ∧(A 2
1
⇒ F = AA /r ∼ F ( F (r , /r
and A2 are independent ) =
1
1
2
2
1
We use the F distribution to test whether two variances are the same or not after a structural break. For instance, H0: σ02 = σ12 against H1: σ02 > σ 12.
3
Prob Probab abil ilit ity y Defin Definit itio ions ns
Definition 3.1 The expected value of a continuos random variable is given by:
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Some useful properties:
1. E (a + bX + bX + cY ) cY ) = a + bE (X ) + cE (Y ) Y ) 2. V ( 2 bcCov((X, Y ) V (a + bX = cY ) cY ) = b2 V ( V (X ) + c2 V ( V (Y ) Y ) + 2bcCov Y ) 3. C ov( ov(a1 + b1 X + X + c1 Y, a2 + b2 X + X + c2 Y ) Y ) = ( b1 c2 + c1 b2)C ov( b1 b2 V ( V (X ) + c1 c2 V ( V (Y ) Y ) + (b ov(X, Y ) Y ) 4. If Z = h(X, Y ), )] Law of iterated expecY ), then E (Z ) = E X [E Y Y |X (Z X )] tations
|
4
Eco Econom omet etr rics ics
A random variable is a real-valued function defined over a Sample Space. The Sample Space (Ω) is the set of all possible outcomes. Before collecting the data (ex-ante) all our estimators are random variables. Once we have have realized realized the data(ex-post data(ex-post), ), we get a specific number number for our estimators. These numbers are what we called estimates. Remark 4.1 A simple Econometric Model: yi = µ + ei is not a regression model, but is an econometric one.
∀i = 1, . . . , n. This
In order to estimate µ we make the following assumptions:
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3. C ov( ov(ei , e j ) = E (ei e j ) = 0
∀i = j
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Definition 4.2 We say that an estimator is Unbiased if: E (µ ˆ) = µ In other words, if after infinitely sampling we are able to achieve the true population value.
For this particular estimator (ˆ µ) is easy to see that is indeed unbiased 1 2 and its variance is V ar( ˆ) = n σ , given the assumption that the draws are ar(µ iid. iid. Note 4.3 Linear combination of normal distribution is a normal distribution. Proposition 4.4 If µ ˆ
∼ N (µ,
σ2 ), n
then Z =
µ ˆ µ σ/ n
−√
∼ N (0, (0, 1) Standard normal values: Φ(z Φ(z ≥ 1.96) = 0. 0.025 and Φ(z Φ(z ≥ 1.64) = 0. 0.05
Note:
1. 2.
ei2
σ2 2 eˆi σ2
2 n
∼χ
2
σ = (n−σ1)ˆ χn2 −1 We lose one degree of freedom here because we 2 need to use one datum to estimate µ ˆ.
∼
3. When we we do not know know σ2 our standardise variable is Z =
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µ ˆ µ σ ˆ 2/ n
−√
For Sign to1vote title H0 is truRead e upFree H is ton r30 uthis eDays Useful ror Not useful Type ype I erro er r Cancel OKanytime. Reject H1 OK Type I I error
∼t
(n 1)
−
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ASSUMPTIONS
E (ei) = 0 i 2 2 E (ei ) = σ i E (ei ej) ej ) = 0 i=j ei N (0 N (0,, σ2 )
(4)
∀ ∀ ∀
∼
X variables are non-stochastic. There is NO exact linear relationship among X variables. If ei is not normal, we may apply the Central Limit Theorem (CLT). Howe Howeve ver, r, for this we need to have have a large large sample sample size. size. How How large large is large large enough? 30 (n (n K ) is one number, but it will depend on the problem.
−
OLS estimator results results from minimising minimising the SSE(error SSE(error sum of squares) squares) n
n
− ˆ β=( xx) xy
i i
i=1
1
(5)
i i
i=1
The above estimator is useful if we are in “Asymptopia”. In matrix notation we have: y = Xβ + e Master your semester with Scribd e ∼ iid N (0, (0, σ 2I Read Free Foron 30this Days Sign to vote title n ) up & The New York Times non-st -stoch ochast astic X is non icUseful Not useful Special offer for students: Only $4.99/month.
The OLS from the sample is:
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(6)
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Definition 4.6 The matrix matrix M X = I n symmetric ric and X (X X )−1 X is symmet T idempotent, i.e., M X = M X and M X M X = M X
−
×
In general, we can have M i = I n X i (X i X i )−1 X i . Thus, M i X j is interpreted as the residuals from regressing X j on X i .
−
Note 4.7 The following properties are important for demonstrations:
1. If A If A is a square matrix, then A = C ΛC −1 where Λ is a diagonal matrix with the eigenvalues of A, and C is the matrix of the eigenvectors in column form.
2. If A If A is symmetric, then C C = C C = I n and hence A = C ΛC
3. If A is symmetric and idempotent, then Λ is a diagonal matrix with either eigenvalues 1 or 0.
4. If A If A = C ΛC , then rank(A rank(A)=r )=r n where r = i=1 λi
Using this definition we get that
ˆ eˆ = e M X e e
− K ) Master your semester with Scribd Read Free Foron 30this Days Sign up to vote title Theorem 4.8 Gauss-Markov Theorem: & The New York Times Useful Not useful
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In a linear regression model in which the errors have expectation zero and are
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One page Summary Y βˆ E (βˆ) C ov( ov(βˆ) βˆ ˆ σ
2
= = = =
∼ =
2
Xβ + e (X X )−1 X Y β σ2 (X X )−1 N (β, σ2 (X X )−1 )
1 n
(ˆσ ) = σ E (ˆ 2
V ar(ˆ ar(ˆσ ) =
ee
(n
− K )ˆ )ˆσ
2
σ2
e
2
∼ N (0, (0, σ I ) n
∼ =
ˆ eˆ = e
− K
2
n
1
n
− K
e
2 i
i=1
2σ 4 (n K )
−
χ2(n)
ˆ eˆ e
2 (n K )
∼χ − ⇔ σe ∼ N (0, (0, I ) σ2
n
M X = I n − X (X X )−1 X Master your semester with Scribd Read Free Foron 30this Days Sign up to vote title ˆ = M X Y e & The New York Times Useful Not useful
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Theorem:
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