INTRODUCTION TO STOCHASTIC PROCESSES EXAM QUESTIONS AND CORRECT ANSWERS

INTRODUCTION TO STOCHASTIC PROCESSES EXAM
QUESTIONS AND CORRECT ANSWERS




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, 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

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