Introduction to Stochastic Processes UPDATED ACTUAL
Questions and CORRECT Answers
Time series analysis 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 Finance
economics -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 a fancy name for a sequence of random variables
When the sequence of random variables of a stochastic time series
process is indexed by a time subscript, we call it a
The term time series can also be used to describe the realization of the stochastic process
Economic time series are viewed as realizations of stochastic processes (i.e., of a sequence of random variables over
time which are typically not independent)
Idea of randomness -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)} {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 Process
-Estimated by taking sample averages
-Neater
Realization
-Estimated by taking ensemble averages at each point
-Messier
Two required concepts in time series analysis 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 E(X(t)) = mew for all t
if
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 its distribution does not depend on t:
F(x(t(1)),...x(t(n))(x(1),...,x(n)) = F(x(t1+j),...,x(tn+j)(x(1),...,x(n))
--> Joint distribution of two or more random variables in a sequence does not
depend on t
**Example: F(x100,x200)(a,b) = F(x900,x1000)(a,b)
Implications from stationarity -If a sequence is strictly stationary and the variance and covariances are finite,
then the sequence is also weakly stationary
-Special case: Gaussian process
- As the first two moments are sufficient to identify the normal distribution, for the
Gaussian process weak stationarity also implies strict stationarity
A stochastic process X(t) is trend stationary if the process is stationary after subtract a (usually linear) function of time t, which
called time trend
A stochastic process X(t) is difference stationary if the process is not stationary but tis first difference, X(t) - X(t-1) is stationary
**X(t) is also called integrated of order 1, I(1)-process
A stochastic process X(t) is ergodic if the dependencies between X(t) and X(t-j) get weaker and weaker over time
Questions and CORRECT Answers
Time series analysis 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 Finance
economics -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 a fancy name for a sequence of random variables
When the sequence of random variables of a stochastic time series
process is indexed by a time subscript, we call it a
The term time series can also be used to describe the realization of the stochastic process
Economic time series are viewed as realizations of stochastic processes (i.e., of a sequence of random variables over
time which are typically not independent)
Idea of randomness -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)} {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 Process
-Estimated by taking sample averages
-Neater
Realization
-Estimated by taking ensemble averages at each point
-Messier
Two required concepts in time series analysis 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 E(X(t)) = mew for all t
if
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 its distribution does not depend on t:
F(x(t(1)),...x(t(n))(x(1),...,x(n)) = F(x(t1+j),...,x(tn+j)(x(1),...,x(n))
--> Joint distribution of two or more random variables in a sequence does not
depend on t
**Example: F(x100,x200)(a,b) = F(x900,x1000)(a,b)
Implications from stationarity -If a sequence is strictly stationary and the variance and covariances are finite,
then the sequence is also weakly stationary
-Special case: Gaussian process
- As the first two moments are sufficient to identify the normal distribution, for the
Gaussian process weak stationarity also implies strict stationarity
A stochastic process X(t) is trend stationary if the process is stationary after subtract a (usually linear) function of time t, which
called time trend
A stochastic process X(t) is difference stationary if the process is not stationary but tis first difference, X(t) - X(t-1) is stationary
**X(t) is also called integrated of order 1, I(1)-process
A stochastic process X(t) is ergodic if the dependencies between X(t) and X(t-j) get weaker and weaker over time
