Essentials of Stochastic Processes (3rd Edition,
2016) – Solutions Manual – by Durrett
Time series analysis - answer- Collection of observations indexed by the date of each
observation
**Examples:
-Macroeconomic variables like income, consumption, interest rates, unemployment rates, etc.
-Financial data like stock returns and exchange rates
Applications of time series techniques in finance and economics - answer- Finance
-Predictability of returns
-Testing and estimating asset price models
-Properties of price formation processes
Economics
-Properties of macroeconomic time series
-Persistence of macro shocks
-Testing economic theories
-Transmission of monetary policy
Stochastic processes are - answer- a fancy name for a sequence of random variables
When the sequence of random variables of a stochastic process is indexed by a time subscript,
1
,we call it a - answer- time series
The term time series can also be used to describe the - answer- realization of the stochastic
process
Economic time series are viewed as - answer- realizations of stochastic processes (i.e., of a
sequence of random variables over time which are typically not independent)
Idea of randomness - answer- -Draws from distributions, no certain numbers - not deterministic
but stochastic
**Observe only one (possible) realization of the stochastic process in question (thus important
to distinguish between realization and stochastic process)
{X(t)} vs {x(t)} - answer- {X(t)} = stochastic process or sequence of random variables
{x(t)} = realization of the stochastic process or sequence of real numbers (that we do observe)
**Due to the dependencies between the random variables {...X(t-2),X(t-1),...} we have a more
complex structure than in the cross-sectional case with independent random variables {X(1),
X(2),...}
Stochastic process vs realization - answer- Process
-Estimated by taking sample averages
-Neater
2
, Realization
-Estimated by taking ensemble averages at each point
-Messier
Two required concepts in time series analysis - answer- 1. Stationarity: distribution doesn't
change over time/what matters is the relative position in the sequence but the moments
remain the same across time
2. Ergodicity: there might be dependencies of the random variables over time, but these
dependencies get smaller and smaller for larger time lags
A stochastic process X(t) is weakly/covariance stationary if - answer- E(X(t)) = mew for all t
Var(X(t)) = sigma squared for all t
Cov(X(t),X(t-j)) = Y(j) for all t
--> The mean, variance, and autocovariances do not depend on t
**The autocovariances only depend on the distance j
**Example: Cov(x(3), x(5)) = Cov(x(98),x(100))
A stochastic process X(t) is strictly stationary if - answer- its distribution does not depend on t:
3
2016) – Solutions Manual – by Durrett
Time series analysis - answer- Collection of observations indexed by the date of each
observation
**Examples:
-Macroeconomic variables like income, consumption, interest rates, unemployment rates, etc.
-Financial data like stock returns and exchange rates
Applications of time series techniques in finance and economics - answer- Finance
-Predictability of returns
-Testing and estimating asset price models
-Properties of price formation processes
Economics
-Properties of macroeconomic time series
-Persistence of macro shocks
-Testing economic theories
-Transmission of monetary policy
Stochastic processes are - answer- a fancy name for a sequence of random variables
When the sequence of random variables of a stochastic process is indexed by a time subscript,
1
,we call it a - answer- time series
The term time series can also be used to describe the - answer- realization of the stochastic
process
Economic time series are viewed as - answer- realizations of stochastic processes (i.e., of a
sequence of random variables over time which are typically not independent)
Idea of randomness - answer- -Draws from distributions, no certain numbers - not deterministic
but stochastic
**Observe only one (possible) realization of the stochastic process in question (thus important
to distinguish between realization and stochastic process)
{X(t)} vs {x(t)} - answer- {X(t)} = stochastic process or sequence of random variables
{x(t)} = realization of the stochastic process or sequence of real numbers (that we do observe)
**Due to the dependencies between the random variables {...X(t-2),X(t-1),...} we have a more
complex structure than in the cross-sectional case with independent random variables {X(1),
X(2),...}
Stochastic process vs realization - answer- Process
-Estimated by taking sample averages
-Neater
2
, Realization
-Estimated by taking ensemble averages at each point
-Messier
Two required concepts in time series analysis - answer- 1. Stationarity: distribution doesn't
change over time/what matters is the relative position in the sequence but the moments
remain the same across time
2. Ergodicity: there might be dependencies of the random variables over time, but these
dependencies get smaller and smaller for larger time lags
A stochastic process X(t) is weakly/covariance stationary if - answer- E(X(t)) = mew for all t
Var(X(t)) = sigma squared for all t
Cov(X(t),X(t-j)) = Y(j) for all t
--> The mean, variance, and autocovariances do not depend on t
**The autocovariances only depend on the distance j
**Example: Cov(x(3), x(5)) = Cov(x(98),x(100))
A stochastic process X(t) is strictly stationary if - answer- its distribution does not depend on t:
3
