1. Lecturer Dr. Veronika Alhanaqtah
ECONOMETRICS
Lecturer
Dr. Veronika Alhanaqtah

2. Topic 5. Time Series Time series and lags in economic models
Stationary and non-stationary time series
AR, MA, ARMA, ARIMA processes

3. 1. Time series and lags in economic models
When we analyze economic parameters we often use annual, quarterly, monthly and daily data.
For example:
annual GDP data
annual net export data
annual inflation rate
monthly data on trading volumes
daily data on production output
In order to fulfil economic analyses rationally we need to systematize moments of receiving of corresponding statistical data. In this case we have to put in order our data in terms of time. Thus we get time series.

4. 1. Time series and lags in economic models
One-dimensional time series: we observe the dynamics of just one random variable in time.
For example, only currency exchange rate or inflation rate
Year
GNP (mln JD)
2004
7320.8
2005
8285.1
2006
9163.9
2007
2008
2009
2010
2011
2012
2013
2014
2015
Source:

5. 1. Time series and lags in economic models
Multidimensional time series: we observe several variables in different moments of time, put in order.
For example:
Year
Population (thousands)
Unemployment rate (%)
2010
7.4
2011
6.5
2012
2013
5.5
Source:

6. 1. Time series and lags in economic models
Definition:
A time series is a sequence of data points, typically consisting of successive measurements made over a time interval. 
Comment: Time series: random data plus trend, with best-fit line and different applied filters.

7. 1. Time series and lags in economic models
Mathematical notation
We analyze a certain economic parameter Y.
Its value in current moment of time t (period) is yt.
Values of Y in succeeding (next) moments of time are yt+1, yt+2, …, yt+k, …
Values of Y in previous (past) moments of time are yt-1, yt-2, …, yt-k, …
When we study dynamics and relationship between such parameters, it is quite obvious that we can use as explainable variables (regressors) not only current values of variables, but also some past values, as well as time itself. We call such models as dynamic models.
Variables, that are characterized by certain delay, we call lag variables.

8. 1. Time series and lags in economic models

9. 1. Time series and lags in economic models
In many cases affecting one factors by the others doesn’t happen immediately, but with some delay – time lag.
Why lags do exist in economics:
Psychological reasons which expressed in people’s behavior.
Technological reasons.
Institutional reasons.
Mechanisms of formation of economic indicators.

10. 2. Stationary and non-stationary time series
One-dimensional time series: we observe the dynamics of just one random variable in time. Multidimensional time series: we observe several variables in different moments of time, put in order.
Year
GNP (mln JD)
2004
7320.8
2005
8285.1
2006
9163.9
2007
2008
2009
2010
2011
2012
2013
2014
2015
Year
Population (thousands)
Unemployment rate (%)
2010
7.4
2011
6.5
2012
2013
5.5

11. 2. Stationary and non-stationary time series
Simulating of one-dimensional time series
We observe a sequence of random variables (time series):
y1, y2, y3, …, yt.
In order to simulate we have to make assumptions.
Without assumptions we can’t make conclusions.
Basic assumption is about stationarity of series
(its invariability in time).
It doesn’t mean that y1=y2=…=yt.
But it means that y1, y2, y3, …, yt have common characteristics in time.

12. 2. Stationary and non-stationary time series

13. 2. Stationary and non-stationary time series

14. 2. Stationary and non-stationary time series
Recall:
Stationary process is a stochastic process the joint probability distribution of which does not change when shifted in time.
Consequently, parameters such as the mean and variance also do not change over time and do not follow any trends.
If a variable intersects its mean n times, we may say that the variable is stationary.
Non-stationary process is a process which shifts in time.

15. 2. Stationary and non-stationary time series
Stationary process.
“White noise”
Example. Assume
and are independent.
A random vector is a white noise vector (white random vector) if its components have a probability distribution with the zero mean and constant finite variance, and are statistically independent.
A necessary (but, in general, not sufficient) condition for statistical independence of two variables is that they are statistically uncorrelated (their covariance is zero):

16. 2. Stationary and non-stationary time series
Non-stationary processes:
(1) Process with a deterministic trend
(2) Process of random walk (drift)
Why this series are non-stationary?
The main problem is with dispersion (variance):
The further this process goes in time, the less we are confident that it will stay in the vicinity of a zero point.

17. Non-stationary time series
Process with a deterministic (stochastic) trend
Process of random walk (drift)
Mean is also not constant.

18. Random walk. Financial theory of random walk
The basic idea of a random walk is that the value of the series tomorrow is its value today plus an unpredictable change. Because the path followed by Y(t) consists of random "steps" ε(t) that path is a "random" walk.
The variance of a random walk increases over time, so the distribution of Y(t) changes over time. Because the variance of a random walk increases without bound, its population autocorrelations are not defined. However, the feature of a random walk is that its sample autocorrelations tend to be very close to 1.
Random walk with drift is the extension of the random walk model to include a tendency to move (or "drift"), in one direction or the other.
The financial theory of random walk is the theory, according to which stock prices, currencies and futures prices change randomly and cannot be predicted on the basis of past market data (the opposite to the technical analysis). This theory was proposed in the early 20th century and remained popular until the 1970s. Theory of random walk is a special case of the weak form of efficient markets hypothesis.

19. 3. AR, MA, ARMA, ARIMA processes

20. 3. AR, MA, ARMA, ARIMA processes

21. 3. AR, MA, ARMA, ARIMA processes

22. 3. AR, MA, ARMA, ARIMA processes

23. Economic time series
Economic time series are recommended to analyze via computation their logarithms.
100∆ ln⁡Y(t) is the percentage change of a time series Y(t) between periods (t-1) and t.
∆ Y(t)=Y(t)-⁡Y(t-1) is the first difference of a series.

24. 3. AR, MA, ARMA, ARIMA processes
Comment: When two out of the three terms are equal to zero, the model may be referred to based on the non-zero parameter, dropping "AR", "I" or "MA" from the acronym describing the model.
For example:
ARIMA (1,0,0) is AR(1)
ARIMA(0,1,0) is I(1)
ARIMA(0,0,1) is MA(1)
With the help of ARMA(p,q) we can compactly and precisely describe any stationary process.
In ARMA-models coefficients are difficult to interpret but they may well be used for prediction.

25. Time series Lecture summary
Time series can be stationary and non-stationary.
OLS is applied for stationary data; for non-stationary data OLS can’t be applied.
Stationary time series are simulated by ARMA-models.
Non-stationary time series (ARIMA) are reduced to stationary time series with the help of transformations (shift from absolute values to increments).
It is reasonable to use AR, MA, ARMA processes when we already know the set of regressors and the functional form of a model but there is the problem of autocorrelation.
In the regression analysis it is important to find long-term relationships. In case of autocorrelation we can also do this via the correction of standard errors (robust regression, i.e. model of correction of an equilibrium).