1. Financial Econometrics Lecture Notes 2
University of Piraeus
Antypas Antonios

2. Contents Time Series Models Forecasting
Stationarity VS Non Stationarity
ARIMA(p,d,q) Models
Forecasting
In Sample & Out of Sample
Modeling The Dynamics of The Variance
Conditional Heteroskedasticity Models - GARCH(p,q)
Re-specifying our initial model

3. Data
Time Series
We analyze how a variable of interest evolves over time
Example: Collect the historic values of X1,t: “Advertisement Expenditures “ and Yt: “Sales of Product A “ realized by Company “ABC” to examine the success of advertisement campaigns of this company
Subscript “t” refers to the value of a variable at date t, e.g. Y2000 refers to the sales that Company “ABC” realized during the year 2000

4. Time Series Analysis Modeling time series Definitions
Exogenous Models: Describe a time series as a function of other time series
Time Series Models: Models aiming to describe the dynamics of the series
Definitions
First Order Stationarity
E(yt)=μ is constant
Second Order Stationarity ( or weakly stationarity )
var(yt)=σ2 is constant
cov(yt,yt+s)=f(s) is a function of the distance “s”
Second Order Stationarity Implies First Order Stationarity

5. Series is Second Order Stationary
Time Series Analysis
Examples of stationarity
Series is Second Order Stationary

6. Time Series Analysis Examples of stationarity
Series is First Order Stationary and not Second Order Stationary
(the variance changes at some time)

7. Series is Not Stationary
Time Series Analysis
Examples of stationarity
Series is Not Stationary

8. Time Series Analysis Stationarity and Empirical Stylized Facts
Prices of an Asset are not Stationary
Returns of an Asset are Stationary

9. Time Series Analysis
Autoregressive models of order p are used to model time dependence in our series using past realizations
First Order Autoregressive Model - AR(1) – is the simplest case
We regress Yt on itself (auto regressive), one lag back (first order)
gretl command for estimating an AR(1) Model
ols Y const Y(-1) or arma 1 0 ; y or arima ; y
gretl command for estimating an AR(3) Model
ols y const y(-1) y(-2) y(-3) or arma 3 0 ; y or arima ; y
arma and arima functions will be explained in more detail later
Behavior of Yt depends on the value of ρ

10. Time Series Analysis First Order Autoregressive Model - AR(1)
Example 1: ρ=0 Observe that series is mean-reverting fast - Independent

11. Time Series Analysis First Order Autoregressive Model - AR(1)
Example 2: ρ=0.9 – Observe stochastic cycles: Slow Mean Reversion - Dependent

12. Time Series Analysis First Order Autoregressive Model - AR(1)
Properties
Case 1: |ρ|<1 – Series is stationary
Mean is Constant
Variance is Constant
Covariance is a function of i. As i increases, since |ρ|<1, covariance (therefore correlation )will be decreasing

13. gretl output (right click series and select Correlogram)
Time Series Analysis
First Order Autoregressive Model - AR(1)
Properties
Case 1: |ρ|<1 – Series is stationary
If ρ is positive, autocorrelation will decrease monotically. (why?). Therefore, if we assume that a series can be modeled through an AR(1) model, with a positive ρ, the autocorrelation function should look like here
gretl output (right click series and select Correlogram)

14. gretl output (right click series and select Correlogram)
Time Series Analysis
First Order Autoregressive Model - AR(1)
Properties
Case 1: |ρ|<1 – Series is stationary
If ρ is negative, autocorrelation will decrease non-monotically (why?). Therefore, if we assume that a series can be modeled through an AR(1) model, with a negative ρ, the autocorrelation function should look like here
gretl output (right click series and select Correlogram)

15. Econometric Analysis Time Series Analysis
First Order Autoregressive Model - AR(1)
Case 2: ρ=1 – Series is not stationary!

16. series name=y-y(-1) or series name=diff(y)
Time Series Analysis
First Order Autoregressive Model - AR(1)
Case 1: ρ=1 – Series is not stationary!
In this case we say that our series has (at least) one unit root.
We also say that our series contains stochastic trend
If this is the case we can transform our non-stationary series into stationary by taking the first differences, that is creating a new series as:
gretl Command for Creating this new series:
series name=y-y(-1) or series name=diff(y)
As noted before, prices of assets are non-stationary. If we take first differences of prices, we generate a new series which will be stationary. BUT, we prefer taking first differences at the log prices since the generated stationary series will represent the returns of the asset.

17. Time Series Analysis First Order Autoregressive Model - AR(1)
Case 1: ρ=1 – Series is not stationary!
It is possible that our series is non-stationary because it evolves naturally with time, that is it contains deterministic trend.
If this is the case, we will model our series including in the explanatory factors the time trend.
We need to test the Null Hypothesis of a Unit Root in our series in order to decide for the appropriate method of de-trending our series (Check the following slides)

18. Time Series Analysis Autoregressive Model of order p - AR(p)
We regress Yt on itself (auto regressive), p lags back (p order)
Case 1: Σ ρ i < 1 – Series is stationary
Case 2: Σ ρ i = 1 – Series has a unit root
Analysis of correlation function remains the same

19. Econometric Analysis Time Series Analysis
Testing for a Unit Root = Testing for stationarity in the Mean
Popular tests
Augmented Dickey - Fuller Test
Phillips - Perron Test
If we reject the null hypothesis of a unit root then our series is stationary (|ρ|<1) and we can move on modeling
If we do not reject the null hypothesis of a unit root, then our series is non-stationary. We take first differences and test the new series for stationarity. We keep taking first differences to the generated series until we reject the null.

20. Econometric Analysis Time Series Analysis
How to perform a Unit Root Test for series yt(example refers to Y1t from dataset2.xlsx)
Select series Y1 (click on it)
Go Variable -> Unit Root Tests and
select the preferred statistic
e.g. Augmented Dickey-Fuller
Consider the other choices as
discussed in class
Press OK

21. Econometric Analysis Time Series Analysis
How to perform a Unit Root Test for series yt(example refers to Y1t from dataset2.xlsx)
Examine the results of the hypothesis testing
Since the Probability is over 0.05 and the Null Hypothesis is Y has a unit root, then
we do not reject that
our series is
non-stationary
gretl output

22. Time Series Analysis
Testing for a Unit Root = Testing for stationarity in the Mean
If we reject the Null Hypothesis, we say that our series is integrated of order 0 and write yt – I(0)
If we do not reject the Null Hypothesis, we take the first differences dyt and test the new series for a unit root. If the new series dyt is stationary then we say that our series is integrated of order 1, that is we need to differentiate once to transform our non-stationary series to stationary and write yt – I(1) (also dyt – I(0))

23. Time Series Analysis
Moving Average model of order q are used to model time dependence in our series using past values of the error term
First Order Moving Average Model - MA(1) – is, as before, the simplest case
We regress Yt on the error term one lag back (first order)
Behavior of Yt depends on the value of θ

24. Time Series Analysis First Order Moving Average Model - MA(1)
Properties
Case 1: |ρ|<1 – Series is stationary
Mean is Constant
Variance is Constant
Only covariance of today (t) with yesterday (t-1) is non-zero.

25. Time Series Analysis First Order Moving Average Model - MA(1)
If we expect that a series can be modeled through an MA(1) model, the autocorrelation function should look like here
gretl output

26. Time Series Analysis Moving Average of order q – MA(q)
Analysis of correlation function remains the same
How to estimate a MA(1) model in gretl
arma 0 1 ; y or arima ; y
How to estimate a MA(3) model in gretl
arma 0 3 ; y or arima ; y

27. Time Series Analysis
We can combine the discussed models into one, creating the Autoregressive Moving Average models of order p and q
If we add the information of the order of integration, that is the number of times we have to take first differences in order to result to a stationary series, we define ARIMA (p,d,q) where d is the order of integration, p & q as before
Example: If we write that yt –ARIMA(2,1,1) this means that we have to generate dyt=yt-yt-1 and use the following model:

28. Time Series Analysis
How to determine the appropriate ARIMA(p,d,q) model to describe a time series
Step 1: Perform Unit Root Testing to determine the order of integration (d). If d>=1 then go to step 2, else go to step 3
Step 2: Based on step 1, generate from original (non-stationary) series a new stationary series
Step 3: Estimate alternative competitive models and use model selection criteria (Akaike, Hannan Quinn, Swhatch) to select the best model
Step 4: Run misspecification test procedures to make sure no dependency is left out of the models

29. Time Series Analysis How to estimate a ARMA (p,q) model in gretl
arma p q ; y or arima p 0 q ; y
How to estimate a ARIMA (p,d,q) model in gretl
arima p d q ; y