Part 3: Time Series
Forecasting: science and art to predict future event with degree of accuracy.
Science: statistical and mathematical methods to discover patterns in
historical data, time dependence is key to making statements about future
(time series).
Art: statistical methods depend on set of assumptions, models are limited
representations of economic and business environments: experience and
judgment of forecaster plays role.
Accuracy: don’t expect forecast to be exactly accurate in mathematical
sense (hitting exact future value of variable of interest), forecasts offer
measure of uncertainty of prediction (interval).
Types of forecast:
1. Event forecast: refers to future occurrence of outcome and/or timing of
such occurrence.
2. Time series forecast: use of time series information in prediction of
variable of interest.
a. Time series: sequence of numerical values ordered according to
time, exhibit one or several of these three categories:
i. Trend: when time series evolves slowly and smoothly over
time.
ii. Cycles: when time series exhibits periodic fluctuations.
iii. Seasonality: cycle, when specific fluctuations occur within
calendar year (activities that peak in summer months).
Order is important: variable is growing/decreasing in time: use past
to predict future. In cross sectional order isn’t important (doesn’t
matter which observation first). Important to tell Stata you’re
working with time series (order matters).
Basic notations:
Object to Time series { yt }
analyze
Value at Known value yt
present of series
time t
Future at Random Y t +h If you are in 1993,
time t+ h variable the value in 1994 is
a random variable.
Value at Unknown y t +h
future time value of
t+ h random
variable
Collection of Univariate It = { y1 , y2 , … , yt } y 1 is value linked
information information to first point in time,
set I t = { y 1 , y 2 , … , y t , x1 , x 2 , … , xt } y t is value linked
Multivariate to last point in time.
information
set
Final Forecast
objective 1-step f t ,1
ahead f t ,h
h -step
, ahead
Uncertainty Forecast error e t , h= y t +h−f t , h
Density forecast: conditional probability
Interval forecast
Density forecast
density function of
yt Yt h
0 1 2
. .
. ……………..
t
.
Point forecast
t+h
f
time
t ,h
Information set
Forecasting: science and art to predict future event with degree of accuracy.
Science: statistical and mathematical methods to discover patterns in
historical data, time dependence is key to making statements about future
(time series).
Art: statistical methods depend on set of assumptions, models are limited
representations of economic and business environments: experience and
judgment of forecaster plays role.
Accuracy: don’t expect forecast to be exactly accurate in mathematical
sense (hitting exact future value of variable of interest), forecasts offer
measure of uncertainty of prediction (interval).
Types of forecast:
1. Event forecast: refers to future occurrence of outcome and/or timing of
such occurrence.
2. Time series forecast: use of time series information in prediction of
variable of interest.
a. Time series: sequence of numerical values ordered according to
time, exhibit one or several of these three categories:
i. Trend: when time series evolves slowly and smoothly over
time.
ii. Cycles: when time series exhibits periodic fluctuations.
iii. Seasonality: cycle, when specific fluctuations occur within
calendar year (activities that peak in summer months).
Order is important: variable is growing/decreasing in time: use past
to predict future. In cross sectional order isn’t important (doesn’t
matter which observation first). Important to tell Stata you’re
working with time series (order matters).
Basic notations:
Object to Time series { yt }
analyze
Value at Known value yt
present of series
time t
Future at Random Y t +h If you are in 1993,
time t+ h variable the value in 1994 is
a random variable.
Value at Unknown y t +h
future time value of
t+ h random
variable
Collection of Univariate It = { y1 , y2 , … , yt } y 1 is value linked
information information to first point in time,
set I t = { y 1 , y 2 , … , y t , x1 , x 2 , … , xt } y t is value linked
Multivariate to last point in time.
information
set
Final Forecast
objective 1-step f t ,1
ahead f t ,h
h -step
, ahead
Uncertainty Forecast error e t , h= y t +h−f t , h
Density forecast: conditional probability
Interval forecast
Density forecast
density function of
yt Yt h
0 1 2
. .
. ……………..
t
.
Point forecast
t+h
f
time
t ,h
Information set
