(Covariance and correlation) Suppose that the annual revenues

| November 24, 2016

Problem set 1
October 16, 2013

1. (Covariance and correlation) Suppose that the annual revenues of the

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world’s two top oil producers have a covariance of 1,735,492.

(a) Based on the covariance, the claim is made that the revenues are
very strongly positively related. Evaluate the claim.

(b) Suppose instead that, again based on the covariance, the claim is
made that the revenues are positively related. Evaluate the claim.

(c) Suppose you learn that the revenues have a correlation of 0.93. In
light of that new information, reevaluate the claims in parts a and b.

2. ( Properties of loss function) State whether the following potential loss
functions meet the criteria introduced in the text and, if so, whether they
are symmetric or asymmetric:
(a)

L(e) = e2 + e

(b)

L(e) = e4 + 2e2

(c)

L(e) = 3e2 + 1

e, e > 0
L(e) =
|e|, e ≤ 0

(d)

3. (Calculating forecasts from trend models) You work for the International

Monetary Fund in Washington, D.C., monitoring Singapore’s real con-

is

sumption expenditures. Using a sample of real consumption data ( mea-

yt ,t= 1990:Q1,…,2006:Q4, you
yt = β0 + β1 T IM Et + εt ,
ˆ
estimates β0 = 0.51, β1 = 2.30, and

where

2

t

σ = 16.

∼ N (0, σ 2 ),

obtaining the

Based on your estimated trend model, construct feasible point ,

interval, and density forecasts for 2010:Q1.

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sured in billions of 2005 Singapore dollars),

estimate the linear consumption trend model,

4. (Selecting forecasting models involving calendar eects) You’re sure that a
series you want to forecast is trending and that a linear trend is adequate,
but you’re not sure whether seasonality is important . To be safe, you t
a forecasting model with both trend and seasonal dummies,

s

yt = β1 T IM Et +

γi Dit + εt
i=1

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(a) The hypothesis of no seasonality, in which case you could drop the
seasonal dummies, corresponds to equal seasonal coecients across
seasons, which is a set of

s−1

linear restrictions:

γ1 = γ2 , γ3 = γ4 , …, γs−1 = γs
How would you perform an F-test of the hypothesis? What assumptions are you implicitly making about the regression’s disturbance
term?
(b) Alternatively, how would you use forecast model selection criteria to
decide whether to include the seasonal dummies?

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(c) What would you do in the event that the results of the hypothesis
testing and model selection approaches disagree?

(d) How, if at all, would your answers change if instead of considering
whether to include seasonal dummies you were considering whether
to include holiday dummies? Trading-day dummies?

5. (Interpreting dummy variables) You t a purely seasonal model with a
full set of standard monthly dummy variables to a monthly series of em-

ployee hours worked. Discuss how the estimated dummy variable coe-

cients γ1 , γ2 , …would change if you change the rst dummy variableD1 =
ˆ ˆ
(1, 0, 0, 0, …) ( with all the other dummy variables remaining the same) to
(a)

D1 = (2, 0, 0, 0, …)

(b)

D1 = (−10, 0, 0, 0, …)

(c)

D1 = (1, 1, 0, 0, …)

6. (Lag operator expression1) Rewrite the following expressions without using the lag operator.
(a)

(Lτ )yt =

(b)

yt = ( 2+5L+0.8L )
L−0.6L3

is

(c)

t

2

yt = 2(1 +

t

L3
L ) t

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7. (Lag operator expressions 2) Rewrite the following expressions in lag operator form.

yt +yt−1 +…+yt−N = α +εt +εt−1 +…+

(b)

yt = εt−2 + εt−1 +

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(a)

t−N , where

α is a constant

t

8. (Autocorrelation functions of covariance stationary series) While interviewing at a top investment bank, your interviewer is impressed by the
fact that you have taken a course on time series forecasting. She decides
to test your knowledge of the autocovariance structure of covariance stationary series and lists four autocovariance functions:
(a)

γ(t, τ ) = α

2

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(b)

γ(t, τ ) = e−τ

(c)

γ(t, τ ) = ατ

(d)

γ(t, τ ) =

Where

α

α
τ

is a positive constant.

Which autocovariance functions(s) are

consistent with covariance stationarity, and which are not? Why?
9. ( Conditional and unconditional means) As head of sales of the leading
technology and innovation magazine publisher TECGIT, your bonus is
dependent on the rm’s revenue. Revenue changes from season to season,
as subscriptions and advertising deals are entered or renewed. From your
experience in the publishing business, you know that the revenue in a

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season is a function of the number of magazines sold in the previous season
and can be described as
residuals

y

εt ∼ N (0, 1000),

is revenue and

x

yt = 1000 + 0.9xt−1 + εt ,

with uncorrelated

where

is the number of magazines sold.

(a) What is the expected revenue for next season conditional on total
sales of 6340 this season?

(b) What is unconditionally expected revenue if unconditionally expected
sales are 8500?

(c) A rival publisher oers you a contract identical to your current contract ( same base pay and bonus). Based on a condential interview,

you know that the same revenue model with identical coecients is
appropriate for your rival.

The rival has sold and average of 900

magazines in previous seasons but only 5650 this season. Will you
accept the oer? Why or Why not?

10. (Empirical questions) Reproduce all the gures and tables in part 4 of
Chapter6, Application: Forecasting Housing Starts.(Please refer to Page

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104, Elements of forecast by Francis x. Diebold)

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