Basic regression with time series data
- 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.