Download Homework 5 answers - Nathaniel Higgins

Homework 5 answers
Nathaniel Higgins
[email protected], [email protected]
Assignment
The assignment was to do the following book problems:
• 10.1, 10.2, 10.7
• C10.1
10.1
Decide if you agree or disagree with each of the following statements and give a brief
explanation of your decision:
(i) Like cross-sectional observations, we can assume that most time series observations
are independently distributed.
Disagree. Time series processes are often identically but not independently distributed.
Things that happen in one period tend to influence things that happen in the next —
which is another way of saying that most time series processes are correlated over time,
which is a violation of the independence assumption.
(ii) The OLS estimator in a time series regression is unbiased under the first three GuassMarkov assumptions.
Agree. If the first three GM assumptions are met, then the estimator is unbiased.
(iii) A trending variable cannot be used as the dependent variable in multiple regression
analysis.
Disagree. We must take care to remove trends that might cause spurious correlation,
but there is no reason why we cannot use a trending variable as a dependent variable.
(iv) Seasonality is not an issue when using annual time series observations.
Agree. Mostly. Seasonality usually refers to inter-annual variations, but really, the
phenomenon is simply one of repeated patterns, which could occur over any period of
time. So technically, I think the best answer is “yes,” seasonality is not an issue when
using annual time series observations, but we still need to be aware of regular patterns
that could exist in the data. If you answered “disagree” but gave this as a reason, I give
you full credit.
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10.2
Let gGDPt denote the annual percentage change in gross domestic product and let intt
denote a short-term interest rate. Suppose that gGDPt is related to interest rates by
gGDPt = α0 + δ0 intt + δ1 intt−1 + ut ,
where ut is uncorrelated with intt , intt−1 , and all other past values of interest rates.
Suppose that the Federal Reserve follows the policy rule:
intt = γ0 + γ1 (gGDPt−1 − 3) + vt ,
where γ1 > 0. (When last year’s GDP growth is above 3%, the Fed increases interest
rates to prevent an “overheated” economy.) If vt is uncorrelated with all past values of
intt and ut , aruge that intt must be correlated with ut−1 . (Hint: Lag the first equation
for one time period and substitute for gGDPt in the second equation.) Which GuassMarkov assumption does this violate?
Following the hint, we lag the first equation one time period to obtain
gGDPt−1 = α0 + δ0 intt−1 + δ1 intt−2 + ut−1 ,
which can then be substituted into the second equation to obtain
intt = γ0 + γ1 (α0 + δ0 intt−1 + δ1 intt−2 + ut−1 − 3) + vt .
This makes pretty clear that intt must be correlated with ut−1 — the latter causes the
former. This violates the exogeneity assumption for the first equation. ut in the first
equation is correlated with intt+1 , i.e. the unobservables in the first equation are not
exogenous to all periods of all explanatory variables (Assumption TS.2).
10.7
In Example 10.4, we wrote the model that explicitly contains the long-run propensity
θ0 , as
gf rt = α0 + θ0 pet + δ1 (pet−1 − pet ) + δ2 (pet−2 − pet ) + ut ,
where we omit the other explanatory variables for simplicity. As always with multiple
regression analysis, θ0 should have a ceteris paribus interpretation. Namely, if pet increases by one (dollar) holding (pet−1 − pet ) and (pet−2 − pet ) fixed, gf rt should change
by θ0 .
(i) If (pet−1 − pet ) and (pet−2 − pet ) are held fixed but pet is increasing, what must be
true about changes in pet−1 and pet−2 ?
The only way that pet can increase while the difference terms stay the same is if pet−1
and pet−2 increase by the exact same amount that pet increases.
(ii) How does your answer in part (i) help you to interpret θ0 in the above equation as
the LRP?
In order for θ0 to be the effect of pet , holding all other regressors constant, it must be
the case that pet−1 and pet−2 are also increasing at the same rate. This enables us to
interpret θ0 as the total effect of a long-term (permanent) increase in pet .
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C10.1
In October 1979, the Federal Reserve changed its policy of targeting the money supply
and instead began to focus directly on short-term interest rates. Using the data in
INTDEF, define a dummy variable equal to 1 for years after 1979. Include this dummy
in equation (10.15) to see if there is a shift in the interest rate equation after 1979. What
do you conclude?
Including a dummy variable post79 in the regression results in a large and statistically
significant coefficient, leading us to conclude that the interest rate equation shifts higher
after 1979.
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