BS 1002 FINANCIAL STATISTICS
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FINANCIAL STATISTICS
(BS 1002)
1. Consider the so-called linear probability model
yi = β0 + β1xi + ui
where yi can be either 0 or 1. In particular, we assume that Pr(yi = 1 | xi) = β0 + β1xi .
(a) Show that E(ui | xi) = 0.
(b) Show that var(ui | xi) = (β0 + β1xi)(1 − (β0 + β1xi)).
(c) Comment on results (a) and (b) and on how the statistical properties of the OLS estimators for β0 , β 1 could be affected.
2. Suppose that we are given the return for N assets in January 2004 (ri,J04, i = 1, ..., N) as well as the return for the same assets in December 2004 (ri,D04, i = 1, ..., N).
Assume that when N = 24 we have that the standard deviation of the ri is equal to 0.5087 for Jan 04 and equal to 0.3645 for Dec 04. Moreover the correlation between the ri,J04 and ri,D04 is 0.8753.
(a) Are we able to estimate at least some of the parameters of the linear regression model
ri,D04 = α + βri,J04 + ∈i, i = 1, .., N?
(b) Can we test the hypothesis that
H0 : β = 1 against H1 : β < 1 ?
(c) Can we derive the R2 of the regression?
3. Let y1 , ..., yT , x1 , ..., xT be a random sample from the model
yi = βxi + ∈i ,
where E(∈i) = 0, var(∈i) = 1, E(xi) = µx 0, var(xi) = σx(2), where xi, ∈j inde- pendent from each other for any i, j .
Define the estimator
where = 1/T xt , y¯ = 1/T
β˜ = ,
t=1 yt .
(a) Recalling the asymptotic properties of the sample mean of a set of i.i.d. observations, show that β˜ converges in probability to β (Hint: g() is a consistent estimator of g(α) if g(.) is a continuous function and the vector →p α).
(b) Is β˜ as efficient as, or more than, or less than the OLS estimator for β? Derive the sample variance of β˜ and the variance of the OLS βˆ and comment on the results.
4. In the simple regression model
yt = α + βxt + ut, t = 1, ..., n,
where xt > 0 for all data points and ut ∼ NID(0, σ0(2)/xt(2)).
(a) Write down the OLS estimators , βˆ and show that they are unbiased. (b) Show that the variance of βˆ is
var(βˆ) = σ0(2) (xt − )2 /xt(2)
Is the OLS estimator BLUE? Discuss.
5. Assume that a random sample (x1 , ..., xn) is observed where each xi is the realiza- tion of a random variable with density:
f (x) =
(a) Derive the MLE of θ .
(b) State the asymptotic distribution of the MLE working out the asymptotic variance.
(c) Assume that we are interested in
γ = θ 2 .
Based on (b) derive the MLE of γ and its asymptotic distribution, spelling out its asymptotic variance.
6. (a) For the linear regression model
y = Xβ + u,
where β = (β1 , β2 , β3 )/ , derive the expression for R, r, stating the value of q (num- ber of restrictions) such that the following hypotheses
(i) H0 : β 1 = 2;
(ii) H0 : β 1 = β2 ;
(iii) H0 : β 1 − β2 = β3 ,
can each be written as
H0 : Rβ = r,
where R is a q × 3 matrix and r is a q × 1 vector.
(b) When the model
Yi = α + βWi + γ1X1i + γ2X2i + γ3X3i + γ4X4i + ui ,
is estimated using a sample of size 80, the residual sum of squares is 510. When, instead, Yi is regressed just on the intercept, on Wi and on SUMi = X1i + X2i + X3i − X4i, the residual sum of squares is 560. Test the hypothesis
H0 : γ1 = γ2 = γ3 = −γ4
at 5% level.
(c) If you wanted to test
H0 : γ1 = γ2 = γ3 = −γ4 = 10,
how would you modify the above result? (Hint: a numerical answer is not possible so simply describe how you would implement the test.)
7. Consider a regression with AR(1) disturbance term ut and one regressor (k = 1)
yt = βxt + ut
ut = φut-1 + ∈t , | φ |< 1, ∈i ∼ iid(0, σ2 ).
(a) Derive an expression for the covariance matrix σ Ω of the vector of disturbances2 u = (u1 , u2 , ..., uT )/ .
(b) Under which conditions on φ the OLS estimator βˆ is BLUE?
(c) Suppose that xt is independent of ∈ s for any t, s and that φ is known and not equal to zero. Derive the BLUE of β .
8. (a) Is the stochastic process, for | φ |≤ 1,
o
yt = µ + ut + φut-1 + φ2ut-2 + φ3ut-3 + ... = µ + φkut-k ,
k=0
where µ is constant and ut is i.i.d. with Eut = 0, var(ut) = σ2 , a linear process? Specify, if any, the conditions required on φ for yt to be a linear process.
(b) Derive the population mean, variance and autocovariance function when | φ |< 1.
(c) Does yt belong to some of the members of the ARMA family? Please discuss the analogies and state an exact result, if available.
2021-12-16