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Unformatted text preview: LECTURE 1
1. Let y = E (y x) be the conditional expectation of y given x. Prove that
ˆ
E {(y − y )2 } ≤ E {(y − π )2 }, where π = π (x) is any other function of x.
ˆ
Show that E x(y − y ) = 0 and give an interpretation of this condition.
ˆ
Demonstrate that, if E (y x) is a linear function of x, then we have
E (y x) = E (y ) + β x − E (x) ,
where β = C (x, y )/V (x).
2. A simple model of the economy comprises two equations in income y ,
consumption c and investment i:
y = c + i, and c = α + βy + ε. Also, s = y − c or s = i, where s is savings. Demonstrate that the
probability limit of the ordinary leastsquares estimate of β the marginal
propensity to consume, which is attained as the sample size increases, is
given by
σ 2 (1 − β )
βmii + σ 2
ˆ
plim β = β +
=
.
mii + σ 2
mii + σ 2
When could the bias in this estimator be safely ignored and under which
circumstances is it liable to become severe?
LECTURE 2
3. Let P = X (X X )−1 X . Show that the conditions P = P = P 2 , which
indicate that P is a symmetric idempotent matrix, are equivalent to the
condition that P (I − P ) = 0. Prove that, if Xb = P y , then y − Xb ≤
y − Xγ , where γ is any vector of the appropriate order.
Show that, if y ∼ (Xβ, σ 2 I ), then b = (X X )−1 X y has E (b) = β and
ˆ
D(b) = σ 2 (X X )−1 . Prove that, if Ay = β ∗ has E (β ∗ ) = β , then V (q β ) ≤
V (q β ∗ ), where q is an arbitrary non stochastic vector.
4. Derive the leastsquares estimator of β in the restricted regression model
(y ; Xβ Rβ = r, σ 2 I ) under the assumption that the matrix X has full
column rank. Show that the restricted estimator has the same structure
as the conditional expectation of the vector y given the value of a vector
x with which it is jointly distributed:
E (y x) = E (y ) + C (y, x)D−1 (x){x − E (x)}.
What possibilities are there for estimating β when X has less than full
column rank?
1 5. Find expressions for the ordinary leastsquares estimators b1 , b2 of β1 , β2
in the partitioned regression model (y ; X1 β1 + X2 β2 , σ 2 I ).
Find simpliﬁed forms for these estimators in the case where the columns
of X1 are orthogonal to those of X2 , and show that, if all of the columns of
X = [X1 , X2 ] are mutually orthogonal, then the k regression parameters
in β = [β1 , β2 ] can be estimated via k univariate ordinary leastsquares
regressions.
What do you understand by the omitted variables bias?
LECTURE 3
6. Derive the socalled Yule–Walker equations γ0 γ1
γ2 γ1
γ0
γ1 2 α0
σε
γ2 α1 = 0 ,
γ1
γ0
α2
0 which relate to the secondorder autoregressive process
α0 y (t) + α1 y (t − 1) + αy (t − 2) = ε(t).
Show how the equations may be transformed into a set of equations which
2
will determine the values of γ0 , γ1 and γ2 when α0 , α1 , α2 and σε are
known. How would the ensuing values {γ3 , γ4 , . . .} be found?
How do the values of the parameters α0 , α1 , α2 aﬀect the form of the
sequence of autocovariances?
7. What is meant by the periodogram of an empirical data series? How is it
calculated and what are its uses?
Figure 1 shows (a) the logarithm of a series of monthly observations on
the U.S. money supply over the period January 1960 to December 1970,
(b) the residuals after the removal of a quadratic trend and (c) the periodogram of the residuals. Describe and account for the main features of
this periodogram. What kind of model would you build in order to forecast
the series? 2 5.4
5.2
5.0
4.8
0 25 50 75 100 125 25 50 75 100 125 0.05
0.04
0.03
0.02
0.01
0.00
−0.01
−0.02
−0.03
0
1.5 1.0 5.0 0.0
0 π/4 π/2 3π/2 π Figure 1. Monthly observations on the U.S money stock January 1960 to December
1970, (a) top the logarithm of the money stock with a quadratic tend (b) middle the
residuals from ﬁtting the trend and (c) bottom the periodogram of the residuals. 3 6
4
2
0
−2
−4
−6
−8
0 25 50 75 100 1.00
0.75
0.50
0.25
0.00
−0.25
−0.50
−0.75
0 10
20 30 40 0 10 20 30 40 1.00
0.75
0.50
0.25
0.00
−0.25
−0.50
−0.75 Figure 2. The graph of 120 observations on a computersimulated ARMA
process together with the empirical autcorrelation function (middle) and
the empirical partial autcorrelation function (bottom). 4 LECTURE 4
8. Describe the main features of the methodology of Box and Jenkins for identifying autoregressive–moving average (ARMA) models. By what means
might a time series be reduced to stationarity before the attempt is made
to identify the orders of an ARMA model?
Figure 2. displays the time plot of a computergenerated ARMA series
together with the corresponding empirical autocorrelation and partial autocorrelation functions. Attempt to identify the orders of the underlying
model, giving reasons for your choice.
9. Show how the following function can be expanded to produce an inﬁinite
series in the powers of z :
µ0 + µ1 z
µ(z )
.
=
α(z )
α1 + α1 z + α2 z 2
Describe the nature of the forecast function of the ARMA process
α(z )y (t) = µ(z )ε(t) in the following cases:
(a) When α(z ) had one root of unity.
(b) When α(z ) had two roots of unity.
(c) When the roots of α(z ) are conugate complex numbers.
Let yt+ht denote the forecast of y for h periods ahead that is made at time
ˆ
t, and let yt+ht+1 be the forecast h − 1 periods ahead made at time t + 1.
How could the latter forecast be obtained from yt+ht in the light of the
ˆ
observations made at time t. LECTURE 5
10. Derive, from ﬁrst principles, the F statistic for testing the hypothesis that
β = β∗ in the classical linear regression model N (y, Xβ, σ 2 I ). What geometric interpretation can you give to this statistic?
How would you proceed if you were interested only in testing an hypothesis
relating to a subset of the parameters within β ?
11. Let x be the height of a father and let y be the height of his fullygrown
son. Derive a test statistic based on the diﬀerence y − x of the sample
¯¯
means of N pairs of observations for testing whether there has been any
signiﬁcant increase in the heights of adult males from one generation to
the next. 5 60
40
20
0
7.5 10 12.5 15 Figure 3. The percentage of boys who have tried smoking by a given age. LECTURE 6
12. Let y.j = αj ιT + Xj βj + ε.j represent T observations on the output of
the j th unit, which is subject to the k input variables in Xj . Using the
Kronecker product, where relevant, show how the observations on the M
units indexed by j = 1, . . . , M can be represented jointly in a single system.
Specialise this system of equations to reﬂect the following conditions:
(a) Xj = X for all j ,
(b) βj = β for all j ,
(c) βj = β and α = α for all j ,
Deﬁne the eﬃcient estimators for each these cases on the supposition that
the disturbances are distributed independently and identically.
How would you test the restrictions of (b) in the context of the original
equation, and how would you test the restrictions of (c) given your acceptance of the restriction of (b)?
LECTURE 7
13. Figure 3 shows the proportion of boys who have tried smoking by a given
age. The data were gathered in 1992. How would you ﬁt a growth curve to
this series and how would you forecast the prevalence of smoking amongst
this group in later years?
In 1992, men in the unskilled manual group were three times more likely to
smoke than those in the professional group. There were similar, although
less marked, diﬀerences amongst women. There is a clear link between
children’s smoking habits and those of their parents. In view of these facts
and of other real or imagined facts, how would you propose to predict the
development of childhood smoking in Britain over a 25year period from
1992?
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This note was uploaded on 03/02/2012 for the course EC 3062 taught by Professor D.s.g.pollock during the Spring '12 term at Queen Mary, University of London.
 Spring '12
 D.S.G.Pollock

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