... given β
10
and β
20
, with β
20
> β
10
, the asymp-
totic distribution of n(
ˆ
β
1
− β
10
) is characterized by the density (10. 03) with
θ = (β
20
− β
10
)
−1
.
10. 3 Generate 10, 000 random samples ... suggested by Fisher (1925) and later stated in its
modern form by Cram´er (1946) and Rao (1945).
Copyright
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1999, Russell Davidson and James G. MacKinnon
10. 10 E...
... factor on the right-hand side tends to S
−1
X
X
as n → ∞, and
the second factor, which is just v, tends to a random vector distributed as
Copyright
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1999, Russell Davidson and James G. MacKinnon
5.3 ... may still seem strange that the lower and upper limits of (5.13) depend,
respectively, on the upper-tail and lower-tail quantiles of the t(n − k) distri-
bution. This actually...
... horizontally.
Similarly, X
11
and X
21
have the same number of columns, and also X
12
and
X
22
, as required for the submatrices to fit together vertically as well.
Copyright
c
1999, Russell Davidson and James ... of the error terms, if necessary spec-
ifying parameters such as its mean and variance;
• Use a random-number generator to generate the n successive and mutu-
ally...
... OC.
Copyright
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1999, Russell Davidson and James G. MacKinnon
2.4 The Frisch-Waugh-Lovell Theorem 69
by looking again at Figure 2.13, in which the constant ι plays the role of X
1
,
and the centered ... least squares esti-
mates, which we will refer to as the Frisch-Waugh-Lovell Theorem, or FWL
Theorem for short. It was introduced to econometricians by Frisch and Waugh
(1933), an...
... (3.19)
Copyright
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1999, Russell Davidson and James G. MacKinnon
3.6 Residuals and Error Terms 109
The consistency of
ˆ
β implies that
ˆ
u → u as n → ∞, but the finite-sample
properties of
ˆ
u ... itself.
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1999, Russell Davidson and James G. MacKinnon
108 The Statistical Properties of Ordinary Least Squares
has mean zero and is uncorrelated with
ˆ
β. The rando...
... provided in Davidson and MacKinnon (1993). How-
ever, it is impossible to understand large parts of econometrics without having
some idea of how asymptotic theory works and what we can learn ... the sum of m independent, squared, standard
normal random variables. From the definition of the chi-squared distribution,
Copyright
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1999, Russell Davidson and James G. MacKinnon
4.4 E...
... include Bard (1974), Gill, Murray, and Wright (1981), Quandt (1983),
Bates and Watts (1988), Seber and Wild (1989, Chapter 14), and Press et al.
(1992a, 1992b, Chapter 10) .
There are many algorithms ... (6.43)
Copyright
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1999, Russell Davidson and James G. MacKinnon
6 .10 Exercises 251
other extremum estimators: generalized least squares (Chapter 7), general-
ized ins...
... matrix-
weighted averages of the within-groups, or fixed-effects, estimator (7.85) and
Copyright
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1999, Russell Davidson and James G. MacKinnon
Chapter 7
Generalized Least Squares
and Related Topics
7.1 Introduction
If ... the
2
See Dufour, Gaudry, and Liem (1980) and Betancourt and Kelejian (1981).
Copyright
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1999, Russell Davidson and James G. MacKinnon
300 Gener...
... (1999), Donald and Newey
(2001), Hahn and Hausman (2002), Kleibergen (2002), and Stock, Wright,
and Yogo (2002). There remain many unsolved problems.
Copyright
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1999, Russell Davidson and James ... terms.
Copyright
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1999, Russell Davidson and James G. MacKinnon
8.3 Instrumental Variables Estimation 313
It can be seen from this solution that p
t
and q
t
will depend...
... σ
2
) = 0, (9 .106 )
where f (·) and F (·) are given by (9 .104 ), and Ω is given by (9 .105 ). By
explicitly performing the multiplications of partitioned matrices in (9 .106 ),
inverting Ω, and ignoring ... σ
2
) = 0 and ι
f
2
(µ, σ
2
) = 0. (9 .107 )
The solution to these two equations is ˆµ = ¯z and ˆσ
2
given by (9 .101 ). Curi-
ously, it appears that the explicit expressions...