1/12 Lec01 Intro to Time Series
Feature of time series
Trend  dependence
Cycle  regular patterns appearing repeatedly  recession and boom
Components
Decomposition: Y = S + T + R
Moving Avg Smoothing (MAS)  local avg

Typically used for seasonal adj, i.e. Filtering out seasonal variations to est trendcycle component

Each avg has the same weight (1/m
), no special considerations to specific periods

From k to k (so m=2k+1
), goes k periods back and k periods forward

Not suitable for forecasting
looking forward, more weights should be giving to recent information
Simple exponential smoothing (SES)

Assigns diff weights to diff obs
most weights to recent data

L_t =
·y_t + (1
)·L_t1
, where L is the cycletrend component and
the smoothing constant (0,1)

Can est L_0 using MAS w/ a small fraction of data (e.g., first 10%) and obtain subsequent l recursively
w/ SES

Optimal
is determined by minimizing SSE
for insample errors of oneperiodahead forecasts

is assumed constant
in this model
SES forecasts

Only a smoother not an assump based model, so only average data instead of capturing any pattern

Flat forecast func, limited use beyond very shortterm forecasts, but the backbone of dynamic
forecasts
SES weights

Exponentially decreasing weights as going backward in time

As
, the weight curve gets flatter
Original form (above)
Exponential form
Error correction form of SES

rearrange the func into:
L_T = L_T1 +
(
y_T  yhat_TT1
)

last oneperiod ahead forecast error
Forecasters' obj:

information set
interval/density/point forecast

in this course: Variance for density forecast + point forecast
interval
Using past data to develop model for future forecasts
Stochastic process / time series process

Def., a collection of random variables
{Y_t}
indexed by time

Each var associated w/ a density func conditional on time t

(Assume some mechanisms and shift it into the future to create forecasts)
Time series
{y_t}
: Outcome of stochastic

One observation
(y)
for each random variable

Impose assumptions to est E(Y) and Var(Y)
Strong Stationarity

Def., if Y for all t has the same density func

Unrealistic
Weak Stationarity

More realistic but never perfectly satisfied (Closer to stationarity
more accurate forecasts)

First order: same mean

Second order: same mean, var and covariances not dependent on t (?)