Univariate Time Series Models
M RAMACHANDRAN
Professor
Department of Economics
Pondicherry University
Why univariate time-series?
An appropriate economic theory to the
relationship between series may not be
available and hence one considers only the
statistical relationship of the given series with
its past values
It may not be possible to obtain the entire set
(or even a subset) of such variables required to
estimate a regression model and one would
then have to use only a single series of the
dependent variable to forecast the future values
Autoregressive Model
yt depends only on its own past values yt −1 , yt −2 , yt −3 ,...
AR (1)
yt = 1 yt −1 + ut
(1 − 1L) yt = ut where ut wn (0, 2 )
AR (2)
yt = 1 yt −1 + 2 yt −2 + ut
(1 − 1L − 2 L2 ) yt = ut
AR (p) p
yt = y
i =1
i t −i + ut
(1 − 1L − 2 L2 − ... − p Lp ) yt = ut
Moving Average Model
yt depends only on past error terms ut −1 , ut −2 , ut −3 ,...
MA (1) yt = 1ut −1 + ut
yt = (1 + 1L)ut
MA (2) yt = 1ut −1 + 2ut − 2 + ut
yt = (1 + 1L + 2 L2 )ut
MA (q) y t = i u t −i + u t
i =1
y t = (1 + 1 L + 2 L2 + ... + q Lq )u t
Autoregressive Moving Average Model
The time-series may be represented as a mix of both AR
and MA model referred as ARMA (p, q)
yt = 1 yt −1 + φ1ut −1 + ut
ARMA (1, 1)
(1 − 1L) yt = (1 + φ1L)ut
ARMA (2, 2) y = y + y + φ u + φ u + u
t 1 t −1 2 t −2 1 t −1 2 t −2 t
(1 − 1L − 2 L2 ) yt = (1 + φ1L + φ2 L2 )ut
p q
ARMA (p, q) yt =
i =1
i yt −i + φ j ut − j + ut
j =1
(1 − 1L − 2 L2 − ... − p Lp ) yt = (1 + φ1L + φ2 L2 + ... + φ p Lp )ut
Integrated Process and ARIMA Models
• ARMA model can be fit for y t only if it is a stationary process.
• If y t is an integrated series of order (1) then ARIMA (1, 1, 1)
model can be defined as
yt = 1yt −1 + φut −1 + ut
(1 − 1L)yt = (1 + φ1L)ut
• Hence, ARIMA (p, d, q)
p q
d y = d y + φ u +u
t i t −i j t− j t
i =1 j =1
p p
(1 − L − L2 − ... − L ) d y = (1 + φ L + φ L2 + ... + φ L )u
1 2 p t 1 2 p t
Box-Jenkins decision tree
Steps in ARIMA Model-building.
Identify model.
Estimate
parameters.
Check bad
bad Revise
diagnostics. model.
good
Forecast.
Identifying the ARIMA Structure
Autocorrelation coefficient
T
((Y t − Y )(Yt − k − Yt − k )) /(T − K )
k = t = K +1
T
t =1
(Yt − Y ) 2 / T
Partial autocorrelation coefficient
Yt = 0 + 1Yt −1 + ... + k −1Yt −( k −1) + k Yt − k + t
Partial autocorrelation between yt and yt − k is k
Ljung-Box Q-statistics
k
2j
QLB = T (T + 2) T − j
j =1
Where j is j th autocorrelation.
Under the null hypothesis of no autocorrelation against the
alternative of at least one of the autocorrelation coefficient is
non-zero, Q is asymptotically distributed as 2 with degrees
of freedom equal to the number of autocorrelations.
Correlogram (Z)
Identification: Summary
Process ACF PACF
White noise No significant Spikes No significant
Spikes
Non Stationary Spikes damp out very
slowly
AR(p) Damps out Spikes cut off at lag
p
MA(q) Spikes cut off at lag q Damps out
ARMA(p,q) Damps out Damps out
Estimating ARIMA Model in Eviews
ARIMA (1, 0, 0) model (yt = 0 + 1 yt −1 + ut)
Go to Quick/Estimate equation and type in the equation box
y c ar(1)
yt = −0.29 + 0.39 yt −1 + ut
Estimating ARIMA Model in Eviews
ARIMA (0, 0, 1) model ( yt = 0 + 1ut −1 + ut )
Go to Quick/Estimate equation and type in the equation box
y c ma(1)
yt = −0.27 + 0.49ut −1 + ut
Estimating ARIMA Model in Eviews
ARIMA (1, 0, 1) model ( yt = 0 + 1 yt −1 + φ1ut −1 + ut )
Go to Quick/Estimate equation and type in the equation box
y c ar(1) ma(1)
yt = 0.86 + 0.95 yt −1 − 0.98ut −1 + ut
Estimating ARIMA Model in Eviews
ARIMA (1, 1, 1) model (yt = 0 + 1yt −1 + φ1ut −1 + ut )
Go to Quick/Estimate equation and type in the equation box
d(y) c ar(1) ma(1)
yt = 2.41 + 0.35yt −1 + 0.67ut −1 + ut
Diagnostics
Step 1:
Go to equation output window and click view/residual test/correlogram-Q statistics
Comparing two models
ARIMA (1,0,1) ARIMA (2,0,1)
Rules for best model: significance of the coefficient; (ii) highest R 2 ; (iii)
highest log likelihood; (iv) lowest AIC, SC and HQ values.
Forecasting
Consider the ARIMA(1, 1, 1) model yt = 0 + 1yt −1 + φ1ut −1 + ut
The forecast of yt for periodt + 1 is
E (yˆ ) = ˆ + ˆ yˆ + φ uˆ
t +1 0 1 t 1 t
ARIMA (1,1,1) estimates of real
GDP at factor cost for the period
1951 – 2000.
yt = 7189.638 + 1.091yt −1 − 0.997ut −1 + ut
Out of sample forecast from 2001 – 2008
If you click forecast from output window you will get following dialog
box. Change the sample period for which you need forecasted values.
The dialog box indicates that the forecast of GDP is made for the period
2001 – 2008 and the forecasted values are stored in the variable gdpf.
Out of sample forecast from 2001 – 2008
Comparing observed with forecasted GDP
Time Series Data Generation Add-in
