This document defines and discusses key concepts in time series analysis. It begins by defining a time series as a sequence of data points measured at successive time intervals. Time series analysis involves extracting meaningful statistics and characteristics from time series data. Examples of time series include stock prices, exchange rates, GDP, and population growth measured over time. The document outlines properties of time series data including autoregressive, moving average, and seasonal processes. It also discusses the importance of stationarity and describes various tests to check for stationarity like the Dickey-Fuller test. Finally, it lists common univariate time series models like AR, MA, ARMA, ARIMA and SARIMA that are used to analyze time series data.

1. Time series ….
• A time series is a sequence of data points, measuring
typical a successive points in time spaced at uniform time
interval
• Time series analysis comprises methods for analyzing time
series data in order to extract meaningful statistics and
other characteristics of the data.
• Example of time series: stock price, exchange rate, interest
rate, inflation rate, national GDP, stock market prices,
population growth etc.
• Data points: daily, weekly, monthly, quarterly, annually
among others.

2. Types of time series data
• Single time series
– U.S. presidential approval, monthly (1978:1-2004:7)
– Number of militarized disputes in the world annually (1816-
2001)
– Changes in the monthly Dow Jones stock market value (1978:1-
2001:1)
• Pooled time series
– Dyad-year analyses of interstate conflict
– State-year analyses of welfare policies
– Country-year analyses of economic growth

3. Properties of Time Series Data
• Property #1: Time series data have autoregressive (AR),
moving average (MA), and seasonal dynamic processes.
• Because time series data are ordered in time, past values
influence future values.
• This often results in a violation of the assumption of no
serial correlation in the residuals of a standard OLS model.

4. Properties of Time Series Data…
• Property #2: Time series data often have time-dependent
moments (e.g. mean, variance, skewness, kurtosis).
• The mean or variance of many time series increases over
time. This is a property of time series data called
nonstationarity
• As Granger & Newbold (1974) demonstrated, if two
independent, nonstationary series are regressed on each
other, the chances for finding a spurious relationship are
very high.

5. Properties of Time Series Data…
0
50
100
150
1800 1850 1900 1950 2000
year
Number of Militarized Interstate Disputes (MIDs),
1816-2001
0
10
20
30
40
50
Number
of
Democratic
Countries
1800 1850 1900 1950 2000
Year
Number of Democracies, 1816-2001

6. Properties of Time Series Data…
• Nonstationarity in the Variance of a Series
– If the variance of a series is not constant over time, we can model
this heteroskedasticity using models like ARCH, GARCH, and
EGARCH.
Example: Changes in the monthly Consumer Price Index [CPI].
-1000
-500
0
500
1000
dowdf
1980jan 1985jan 1990jan 1995jan 2000jan
date

7. Properties of Time Series Data…
• Property #3: The sequential nature of time series data
allows for forecasting of future events.
• Property #4: Events in a time series can cause structural
breaks in the data series. We can estimate these changes
with intervention analysis, transfer function models,
regime switching/Markov models, etc.

8. Properties of Time Series Data…
• Property #5: Many time series are in an equilibrium
relationship over time, what we call cointegration. We
can model this relationship with error correction models
(ECM).
• Property #6: Many time series data are endogenously
related, which we can model with multi-equation time
series approaches, such as vector autoregression (VAR).
• Property #7: The effect of independent variables on a
dependent variable can vary over time; we can estimate

9. Why not estimate time series with OLS?
• OLS estimates are sensitive to outliers.
• OLS attempts to minimize the sum of squares for
errors; time series with a trend will result in OLS
placing greater weight on the first and last
observations.
• OLS treats the regression relationship as
deterministic, whereas time series have many
stochastic trends.
• We can do better modeling dynamics than treating
them as a nuisance.

10. Regression Example, Approval

11. Regression Example, Approval
Durbin's alternative test for autocorrelation
---------------------------------------------------------------------------
lags(p) | chi2 df Prob > chi2
-------------+-------------------------------------------------------------
1 | 1378.554 1 0.0000
---------------------------------------------------------------------------
H0: no serial correlation
The null hypothesis of no serial correlation is clearly violated. What if we
included lagged approval to deal with serial correlation?

12. Regression Example, Approval

13. Time series vs. cross-sectional and panel data
• A time series is a group of observations on a single entity
over time — e.g. the daily closing prices over one year for
a single financial security, or a single patient’s heart rate
measured every minute over a one-hour procedure.
• A cross-section is a group of observations of multiple
entities at a single time — e.g. today’s closing prices for
each of the S&P 500 companies, or the heart rates of 100
patients at the beginning of the same procedure.
• If your data is organized in both dimensions — e.g. daily
closing prices over one year for 500 companies — then
you have panel data.

14. Components of Time Series
• The change which are being in time series. They are
effected by economic, Natural, industrial and political
reasons. There reasons are called Components of Time
Series.
• These includes;
–Secular trend
–Seasonal variation
–Cyclical variation
–Irregular variation

15. Components of Time Series…
• Secular trend- The increase and decrease in the movements of a time
series.
• A time series data may show upward trend or download trend for a
period of years and this may e due to factors like:
– Increase in population
– Changes in technological progress
– Large scale shift in consumer demands
• For example:
• Population increases over a period of time, price increases over a
period of years among others. These are examples of upward trend.
• The sales of a commodlity may decrease over a period of time
because of better products coming to the market. This is an example
of declining trend or downward.

16. Components of Time Series…
• Seasonal variation are short-term fluctuations in a time series
which occurs periodically in a year.
• This continues to repeat year after year
–The major factors that are weather conditions and customs
of people.
–More woolen clothes are solid in winter than in the
seasons of summer.
–Each year more ice creams are slod in summer and very
little in winter season.
–The sales of in the departmental stores are more during
festive seasons that in the normal days.

17. Components of Time Series…
• Cyclical variation are recurrent upward and downward
movements in a time series but the period of cycle is
greater than a year. Also these variation are not regular as
seasonal variation.
• A business cycle showing these oscillatory movements has
to pass through four phrases-expansion, recession,
depression and recovery. In a business, these four phrase
are completed by passing one to another in this order

18. Components of Time Series…
–Irregular variation/random: these are fluctuations in time
series that are short in duration, erratic in nature and
follow no regularity in the occurrence pattern.
–These variations are also referred to as residual variations
since by definition they represent what is left out in a time
series after trend, cyclical and seasonal variations.
–Irregular fluctuations results due to the occurrence of
unforeseen events like: Floods, earthquakes, wars, famines

19. Stationary Vs Non-stationary Process
• When there is no trend, seasonal, or cyclical pattern
remains in series. Series is stationary or “white noise”
• When there is a trend, seasonal, or cyclical pattern
remains in series is said to be nonstationary or random
walk process or Unit root.
• There are two types of random walks:
– Random walk with drift [no constant or intercept]
– Random Walk without drift [constant term is present]

20. Why is stationarity important in a time
series?

21. Stationarity and unit roots

22. Stationarity and unit roots
Forms of Stationarity…
• A time series is weakly stationary if its mean and variance
are constant over time and the value of the covariance
between two periods depends only on the distance (or
lags) between the two periods.
• A time series if strongly stationary if for any values j1, j2,…jn,
the joint distribution of (Yt, Yt+j1, Yt+j2,…Yt+jn) depends only
on the intervals separating the dates (j1, j2,…,jn) and not on
the date itself (t).
• A weakly stationary series that is Gaussian (normal) is also
strictly stationary.
• This is why we often test for the normality of a time series.

23. Stationarity and unit roots
• Stationary vs. Nonstationary Series…
• Shocks to a stationary series are temporary; the series
reverts to its long run mean.
• For nonstationary series, shocks result in permanent moves
away from the long run mean of the series.
• Stationary series have a finite variance that is time
invariant; for nonstationary series, σ2 → ∞ as t → ∞.

24. Stationarity and unit roots

25. Stationarity and unit roots

26. Test for Stationarity
• Unit Root Testing
– Dickey Fuller Test [DF]
– Augmented Dickey Fuller Test [ADF]
– Phillips Perron Test [PP]
– Kwiatkowski-Phillips-Schmidt-Shin Test [KPSS]
– Ng-Perron Test

27. Dickey Fuller Test [DF]
• If you accept the null hypothesis, you conclude that the
time series has a unit root.
• In that case, you should first difference the series before
proceeding with analysis.
• NB: You can include a constant, trend, or both in the test.

28. Augmented Dickey Fuller Test [ADF]
• We can use this version if we suspect there is
autocorrelation in the residuals.
• This model is the same as the DF test, but includes lags of
the residuals too.

29. Phillips Perron Test [PP]
• Makes milder assumptions concerning the error term,
allowing for the εt to be weakly dependent and
heterogenously distributed.

30. Kwiatkowski-Phillips-Schmidt-Shin Test [KPSS]

31. Ng-Perron Test

32. Univariate Time Series Modeling Process
• Autoregressive models [AR]
• Moving averages model [MA]
• ARMA
• ARIMA
• SARIMA
• ARMAX

35. Techniques of time series analysis

36. Thank you