- Time series data involves observations across time that are related as part of a stochastic or random process. We can only observe one realization of this process. - Static regression models assume the dependent variable is related only to independent variables in the same time period. Finite distributed lag (FDL) models allow for lags where past values of independent variables may affect current dependent values. - In FDL models, coefficients measure the impact of temporary or permanent changes in independent variables on dependent variables over time, showing the "memory" or lag effects. Coefficients of permanent changes sum to indicate total long-run impact.

1. Basic Regression Analysis with
Time Series Data

2. The Nature of Time Series Data
How should we think of sample randomness in time series data?
We are still dealing with random variables: we don’t know what
GDP for 2007 will be in Georgia. However, knowledge of GDP
for 2006 might help us to make a better guess about GDP
2007 => GDP2007 is not independent from GDP 2006…
The trick in time series is that we treat observations across time
as an outcome of a sequence of random variables.
A sequence of random variables indexed by time is called
stochastic (=random) process or a time series process.

3. We can only see one realization of the stochastic process – the
reality. We cannot go back in time and start the process over
again.
However, if certain conditions in history had been different, we
would obtain a different realization of the stochastic process
(a different time path) .
The set of all possible time realizations (time paths) plays the
role of the population.
The sample size for a time series data set is the number of time
periods over which we have observations.

4. Working with time-series data
In the time-series data if the model is:
And if t=2005 is the year of interest, we would have
Meaning that the inflation rate (in the given country) in 2005 depended
on the unemployment rate in that country in 2005 and some other
factors that were affecting inflation in 2005 (high level of
expenditures, inflationary expectations).
This is an example of a static model, since it assumes that inflation this
year only depends on things that take place this year, but not last year,
for example.
ttt uxy ++= 10 ββ
20052005102005 uUnempINF ++= ββ

5. Static Model
A model is called contemporaneous or static when observations
on the dependent variable are related to the observations on
x-variables for the same time period t:
Usually this model is used when a change in x at time t has an
immediate effect on y:
It is very easy to follow changes in y from change in x using
graphs. There will be 2 types of changes that we will discuss: a
temporary change in x and a permanent change in x.
ttt uxy ++= 10 ββ
0,1 =∆∆=∆ ttt uxy β

6. Suppose x=c for all time periods before 2003. In 2004 x goes up
by one unit to c+1, and then goes back to c from 2005 on =>
we look at a temporary change in x by 1 unit.
y
y04
β1
y03
01 02 03 04 05 06 07 Year
The immediate change in y due to a temporary 1-unit increase in
x in 2004 is impact propensity or impact multiplier.
However, in real life there are many situations where change in
one variable (x) needs time (more than current time period)
to fully affect a variable of interest (y).

7. Finite Distributed Lag Models
In a Finite Distributed Lag (FDL) Model one or more regressors
affect y with a lag. Here is an example of FDL of order one:
Example:
Gfr=general fertility rate
Pe=personal tax exemption
tttt upepegfr +++= −1100 δδα
tttt uxxy +++= −1100 δδα

8. Temporary Change in x in 1-lag
Model:
Let’s track the change in y over time that will be
caused by a temporary change in x in 2004:
 


 
0506
04
05
03
04
0203
21006
21005
21004
21003
)1(
)1(
xx
x
x
x
x
xx
ccy
ccy
ccy
ccy
βββ
βββ
βββ
βββ
++=
+++=
+++=
++=


tttt uxxy +++= −1210 βββ

9. Suppose x=c for all time periods before 2003. In 2004 x goes up
by one unit to c+1, and then goes back to c from 2005 on.
y y04
y05
β1 β2
y03
01 02 03 04 05 06 07 Year
As you can see, this picture allows for the temporary change in x today to
still have effect tomorrow. But the day after tomorrow things are back
to what they used to be before the change.

10. • So, in our one lag model, β1 measures an immediate
(same year) impact from one unit temporary change in x.
• β2 measures next time period (next year) impact from one
unit temporary change in x. It can also be looked at as current
effect on y from temporary change in x last year.
• There is no effect on y two periods after the temporary
change in x. Or, temporary change in x that took place more
than one period ago is not doing anything to current y.
• So, the “memory” in one-lag model is only one period long.

11. Temporary Change in x in 2-lag
Model:
Let’s track the change in y over time that will be
caused by a temporary change in x in 2004:
  
 
 
 
  
050607
04
0506
03
04
05
0203
04
010203
321007
321006
221005
321004
321003
)1(
)1(
)1(
xxx
x
xx
x
x
x
xx
x
xxx
cccy
cccy
cccy
cccy
cccy
ββββ
ββββ
ββββ
ββββ
ββββ
+++=
++++=
++++=
++++=
+++=



ttttt uxxxy ++++= −− 231210 ββββ

12. Suppose x=c for all time periods before 2003. In 2004 x goes up
by one unit to c+1, and then goes back to c from 2005 on.
y y04
y05
β1 β2
y03 β3
01 02 03 04 05 06 07 Year
As you can see, this picture allows for the temporary change in x
today to still have effect tomorrow and the day after
tomorrow. But after that things are back to what they used to
be before the change.

13. • So, in two lag model, β1 measures an immediate (same year)
impact from one unit temporary change in x.
• β2 measures next time period (next year) impact from one
unit temporary change in x. It can also be looked at as current
effect on y from temporary change in x last year.
• β3 measures two time period later impact from one unit
temporary change in x. It can also be looked at as current
effect on y from temporary change in x two years ago.
• There is no effect on y three periods after the temporary
change in x. Or, temporary change in x that took place more
than two period ago is not doing anything to current y.
• So, the “memory” in two-lag model is only two periods long.

14. What is the change in y due to a permanent change in x? Let’s
look at the static case first. Suppose in 2004 x goes up by one
unit, and stays there.
y
y04
β1
y03
01 02 03 04 05 06 07 Year
As before, β1 is the immediate change in y due to a permanent
1-unit increase in x at t = impact propensity.
Following the same argumentation as for temporary changes in
x we will try to track the permanent change in x in an FDL
case.

15. Permanent Change in x in 1-lag
Model:
Let’s track the change in y over time that will be
caused by a permanent change in x in 2004:
 




0506
0405
03
04
0203
)1()1(
)1()1(
)1(
21006
21005
21004
21003
xx
xx
xx
xx
ccy
ccy
ccy
ccy
++++=
++++=
+++=
++=
βββ
βββ
βββ
βββ
tttt uxxy +++= −1210 βββ

16. Suppose x=c for all time periods before 2003. In 2004 x goes up
by one unit to c+1, and stays there forever.
y y05
y04 β2 β1+β2
β1
y03
01 02 03 04 05 06 07 Year
As you can see, this picture allows for the permanent change in x
today to affect y in two steps.

17. Permanent Change in x in 2-lag
Model:
How about the change in y in 2-lag model when
x is up by 1 unit from 2004–on.
  
 





050607
040506
03
0405
0203
04
010203
)1()1()1(
)1()1()1(
)1()1(
)1(
321007
321006
221005
321004
321003
xxx
xxx
xxx
xxx
xxx
cccy
cccy
cccy
cccy
cccy
++++++=
++++++=
+++++=
++++=
+++=
ββββ
ββββ
ββββ
ββββ
ββββ
ttttt uxxxy ++++= −− 231210 ββββ

18. • So,β1 still measures an immediate (same year)
impact from one unit of permanent change in x.
• (β1+β2) measures the overall change in y in the next time
period (next year). It can also be looked at as current effect
on y from permanent change in x last year.
• The effect on y two and more periods later is still (β1+β2). In
1-lag model permanent change in x takes 1 period after the
change to fully affect y.
• In this model (β1+β2) measures the long-term impact (long-
run multiplier) of on y from a permanent change in x .

19. • β1 measures an immediate (same year) impact from
one unit permanent change in x.
• (β1+β2) measures the overall change in y in the next time
period (next year) after permanent change in x. It can also be
looked at as current effect on y from permanent change in x
last year.
• (β1+β2+β3) measures two time period later impact from one
unit permanent change in x. It is also the overall change in y
from permanent change in x two years ago.
• The overall effect on y after2 and more periods is (β1+β2+β3),
this is the long-run propensity (multiplier).

20. Suppose x=c for all time periods before 2003. In 2004 x goes up
by one unit to c+1, and stays there forever.
y y05 β3
y04 β2 β1+β2+β3
β1
y03
01 02 03 04 05 06 07 Year
As you can see, this picture allows for the permanent change in x
today to affect y in three steps.