Econometrics: CH11 Further Issues in Using OLS with Time
Series Data || with 100% Error-free Solutions.
What is a stationary time series process? correct answers One whose probability distributions are
stable over time - if we take any collection of random variables in the sequence and then shift
that sequence ahead by h time periods, the joint probability remains unchanged
When is a stochastic process stationary? correct answers a stochastic process is stationary if for
every collection of indices is greater than 1 and the previous time period, the joint distribution of
the x values is the same as the joint distribution of x values plus h time periods.
What does stationarity require? correct answers that the nature of any correlation between
adjacent terms is the same across all time periods.
What do you call a stochastic process that is not stationary? correct answers a nonstationary
process
Discuss the restrictive nature of the assumptions that we have used so far correct answers Strict
exogeneity, homoskedasticity, and no serial correlation are very demanding requirements,
especially in the time series context
Statistical inference rests on the validity of the normality assumption
Much weaker assumptions are needed if the sample size is large
A key requirement for large sample analysis of time series is that the time series in question are
stationary and weakly dependent
What is a covariance stationary process? correct answers a stochastic process with a finite second
moment is covariance stationary its its expected value, variance and its covariances are constant
over time
- depends only on h and not on t
- focuses only on the first two moments of the stochastic process
Which is a stronger covariance stationary or stationary and what do we call the stronger one?
correct answers Stationarity is stronger and we call it strict stationarity
What is weak dependence? correct answers Weak dependence places restrictions on how
strongly related the random variables xt and x(t + h) can be as time distance between them (h)
gets large
When is a stationary time series process said to be weakly dependent? correct answers if xt and
x(t + h) are almost independent as h increases without bound
, When is a non stationary time series process said to be weakly dependent? correct answers if xt
and x(t + h) are almost independent as h increases without bound but we must assume that the
concept of being almost independent does not depend on the starting point, t.
What is an implication of weak dependence? correct answers correlation between xt and x(t + h)
must converge to zero if h grows to infinity.
- as the variables get farther apart in time, the correlation between them becomes smaller and
smaller
What do we call covariance stationary sequences where correlation between xt and x(t + h) must
converge to zero if h grows to infinity? correct answers asymptotically uncorrelated
Why is weak dependence important for regression analysis? correct answers it replaces the
assumption of random sampling in implying that the law of large numbers and the central limit
theorem hold
What do we need for the LLN and CLM to hold? correct answers need the individual
observations must not be too strongly related to each other, in particular their relation must
become weaker the farther they are apart
What is the simplest example of weakly dependent variable? correct answers independent
identically distributed sequence
What is a moving average process of order 1 [MA(1)]? correct answers et is an independent and
identical distributed (i.i.d) sequence, with mean 0 and variance 𝜎𝑒^2
What are the characteristics of MA(1)? correct answers An MA(1) is a stationary, weakly
dependent sequence, and the LLN and CLT can be applied to it.
What is the more popular example for weakly dependent time series? correct answers
Autoregressive process of Order One [AR(1)]
What is the autoregressive process of order one? correct answers et is an independent and
identical distribud (i.i.d) sequence with mean 0 and variance 𝜎𝑒^2, and the starting point of a
time series process is y0 at t=0. Assume et is independent of y0 and E(y0 )=0
Time series model whose current value depends on its most recent value plus an unpredictable
disturbance
What is the crucial assumption for weak dependence of AR(1)? correct answers Stability
condition
If the stability condition holds, why is the process weakly dependent? correct answers process is
weakly dependent because serial correlation
converges to zero as the distance between observations grows to infinity
Series Data || with 100% Error-free Solutions.
What is a stationary time series process? correct answers One whose probability distributions are
stable over time - if we take any collection of random variables in the sequence and then shift
that sequence ahead by h time periods, the joint probability remains unchanged
When is a stochastic process stationary? correct answers a stochastic process is stationary if for
every collection of indices is greater than 1 and the previous time period, the joint distribution of
the x values is the same as the joint distribution of x values plus h time periods.
What does stationarity require? correct answers that the nature of any correlation between
adjacent terms is the same across all time periods.
What do you call a stochastic process that is not stationary? correct answers a nonstationary
process
Discuss the restrictive nature of the assumptions that we have used so far correct answers Strict
exogeneity, homoskedasticity, and no serial correlation are very demanding requirements,
especially in the time series context
Statistical inference rests on the validity of the normality assumption
Much weaker assumptions are needed if the sample size is large
A key requirement for large sample analysis of time series is that the time series in question are
stationary and weakly dependent
What is a covariance stationary process? correct answers a stochastic process with a finite second
moment is covariance stationary its its expected value, variance and its covariances are constant
over time
- depends only on h and not on t
- focuses only on the first two moments of the stochastic process
Which is a stronger covariance stationary or stationary and what do we call the stronger one?
correct answers Stationarity is stronger and we call it strict stationarity
What is weak dependence? correct answers Weak dependence places restrictions on how
strongly related the random variables xt and x(t + h) can be as time distance between them (h)
gets large
When is a stationary time series process said to be weakly dependent? correct answers if xt and
x(t + h) are almost independent as h increases without bound
, When is a non stationary time series process said to be weakly dependent? correct answers if xt
and x(t + h) are almost independent as h increases without bound but we must assume that the
concept of being almost independent does not depend on the starting point, t.
What is an implication of weak dependence? correct answers correlation between xt and x(t + h)
must converge to zero if h grows to infinity.
- as the variables get farther apart in time, the correlation between them becomes smaller and
smaller
What do we call covariance stationary sequences where correlation between xt and x(t + h) must
converge to zero if h grows to infinity? correct answers asymptotically uncorrelated
Why is weak dependence important for regression analysis? correct answers it replaces the
assumption of random sampling in implying that the law of large numbers and the central limit
theorem hold
What do we need for the LLN and CLM to hold? correct answers need the individual
observations must not be too strongly related to each other, in particular their relation must
become weaker the farther they are apart
What is the simplest example of weakly dependent variable? correct answers independent
identically distributed sequence
What is a moving average process of order 1 [MA(1)]? correct answers et is an independent and
identical distributed (i.i.d) sequence, with mean 0 and variance 𝜎𝑒^2
What are the characteristics of MA(1)? correct answers An MA(1) is a stationary, weakly
dependent sequence, and the LLN and CLT can be applied to it.
What is the more popular example for weakly dependent time series? correct answers
Autoregressive process of Order One [AR(1)]
What is the autoregressive process of order one? correct answers et is an independent and
identical distribud (i.i.d) sequence with mean 0 and variance 𝜎𝑒^2, and the starting point of a
time series process is y0 at t=0. Assume et is independent of y0 and E(y0 )=0
Time series model whose current value depends on its most recent value plus an unpredictable
disturbance
What is the crucial assumption for weak dependence of AR(1)? correct answers Stability
condition
If the stability condition holds, why is the process weakly dependent? correct answers process is
weakly dependent because serial correlation
converges to zero as the distance between observations grows to infinity
