[1]. Graphical Analysis
[2]. Autocorrelation Function (AFC) and
Correlogram
[3].The unit root test

Using US_price.xls

One simple test of stationary is based on the
so-called autocorrelation function (AFC). The
AFC at lag k, denoted by
k
Y  Y  Y



0 
t
n
2
  Yt  Y 
k
k 
0
n
t k
 Y

In the time series data a large proportion of
correlation between Yt and Yt-k may be due
to the correlations they have with the
interventing lags Yt-1 Yt-2 Yt-3,…..Yt-k+1. The
partial correlation removes the influence of
these intervening variables.

One way of accomplishing this is to consider
the ACF and PACF and the associated
correlograms of a selected number of ARMA
processes, such as AR(1), AR(2), MA(1),
MA(2), ARMA(1,1), ARMA (2,2), and so on.
Type of model
Typical pattern of
ACF
Typical pattern of
PACF
AR(p)
Decays
exponentially or
with damped sine
wave pattern or
both
Significant spikes
through lags p
MA(q)
Significant spikes
through lags q
Declines
exponentially
ARMA(p,q)
Exponential decay
Exponential decay
Yt   Yt 1  t
Yt     Yt 1   t
Yt     T   Yt 1   t





H0: Unit root
H1: no unit root
Decision rule
*
If   ADF critical value >> not reject null
hypothesis, i.e. unit root exists.
If  *  ADF critical value >> reject null
hypothesis, i.e. unit root exists.
This graph shows the series has a constant mean
and constant variance which implies the first
difference series of “CPI” achieves stationary

Using hk_gdp.xls
From the graph
-deterministic upward
-seasonal cycle
Therefore, Non-stationary

The ACFs are
suffered from
linear decline
and there are
two spikes of
PACFs in 1 and 5

The firstdifference
series perform
with a nonconstant
variance.

The residuals are
not white noise.
Since there are
seasonal cycles
existing in the
correlogram, we
can try the fourperiod differences.


There are two
spikes of PACFs
and three spikes
of ACFs, thus we
can have the idea
to examine
whether there is
AR(2) and MA(3)
of the series.
ARIMA(2,1,3)
To judge which is a best fit ARIMA model
from different trials and errors, we have to
base on some criteria, such as the smallest
Schwartz criterion (BIC), Standard Error of
Regression (SEE), the highest adjusted R2
and the invertiability condition and
significant of AR and MA root, to determine
the best fitted model. Also, the residuals of
the selected BEST model must be white
noise.
Using the exchange
Rate data
The graph of
correlogram
suggests that
ARIMA (p,d,q)?

Since there is no
significant spikes of
ACFs and PACFs, it
means that the
residuals of this
selected ARIMA
model are white
noise , so that there
is no other
significant patterns
left in the time
series, then we can
stop at here .

If the 6- month TB
rate was higher than
the 3 month TB rate
more expected a
priori in the last
month, this month it
will be reduced by
0.2 percentage
points to restore the
long-run relationship
between the two
interest rates.

Using Problems 22.16 p. 866

Using the data on PCE and PDI given in table
21.1, develop a bivariate VAR model for the
period 1970-I to 1990-IV. Use this model to
forecast the value of these variables for the
four quarter of 1991 and compare the
forecast values with the actual values given
in table 21.1

22.18 estimate the impulse response function
for a period of up 8 lags for the VAR model
that you developed in exercise 22.16


The IRF traces out the response of the
dependence variable in the VAR system to
shocks in the error term.
Suppose the error term in equation 1 (PCE)
increases by a value of one standard
deviation. Such a shock or change will change
PCE in the current as well as future periods.
But since M1 appears in the PDI, the change
in error term in equation 1 will also have the
impact on PDI.