1. Lecture 12 Time Series Model Estimation
Materials for lecture 12
Read Chapter 15 pages 30 to 37
Lecture 12 Time Series.XLSX
Lecture 12 Vector Autoregression.XLSX

2. Time Series Model Estimation
Outline for this lecture
Review stationarity and no. of lags
Discuss model estimation
Demonstrate how to estimate Time Series (AR) models with Simetar
Interpretation of model results
How to forecast the results for an AR model

3. Time Series Model Estimation -- Stationarity
Plot the data to see what kind of series you are analyzing
Make the series stationary by determining the optimal number of differences based on Dickie Fuller test, say Di,t
Student t statistic smaller than -2.90
May need to test for the need for a trend, i.e., Augmented Dickie-Fuller test
=DF(Data Series, Trend, 0, No Differences)
Trend = True for augmented DF
Trend = False for regular DF

4. Time Series Model Estimation – Number of Lags
Determine the number of lags to use in the AR model based on
=AUTOCORR() or =ARLAG()
Manually this is a series of regressions testing all different lags of the differenced data. To test for 4 lags use this regression
Di,t =a + b1 Di,t-1 + b2 Di,t-2 +b3 Di,t-3+ b4 Di,t-4
Student t statistic for the last lagged value
Simetar provided two functions to bypass the need to develop numerous Di series and run individual regressions

5. Time Series Model Estimation
Once you have determined the number of differences to make the series stationary and the number of lags to use
Then you estimate an OLS regression to estimate the predictive equation
For a series with “1” difference and four lags estimate coefficients for this regression
D1,t =a + b1 D1,t-1 + b2 D1,t-2 + b3 D1,t-3+ b4 D1,t-4
This regression will forecast the D1 which you use to forecast the ŶT+i

6. Time Series Model Estimation in Simetar
An alternative to estimating the differences and lag variables by hand and using an OLS regression package, we will use Simetar
Simetar time series function is driven by a menu

7. Time Series Model Estimation
Read output as a regression output
Beta coefficients are OLS slope coefficients
SE of Coef. used to calculate t ratios to determine which lags are significant
For goodness of fit refer to AIC, SIC and MAPE
Can test restricting out lags (variables)

8. Before You Estimate TS Model (Review)
Dickey-Fuller test indicates whether the data series used for the model, Di,t , is stationary and if the model is D2,t = a + b1 D1,t the DF it indicates that t stat for b1 is < -2.90
Augmented DF test indicates whether the data series Di,t are stationary, if we added a trend to the model and one or more lags
Di,t =a + b1 Di,t-1 + b2 Di,t-2 +b3 Di,t-3+ b4 Tt
SIC indicates the value of the Schwarz Information Criteria for the number lags and differences used in estimation
Change the number of lags and observe the SIC change
AIC indicates the value of the Aikia Information Criteria for the number lags used in estimation
Change the number of lags and observe the AIC change
Best number of lags is where AIC is minimized
Changing number of lags also changes the MAPE and SD residuals

9. Forecasting a Time Series Model
If we assume a series that is stationary and has T observations of data we estimate the model as an AR(0 difference, 1 lag)
Forecast the first period ahead as
ŶT+1 = a + b1 YT
Forecast the second period ahead as
ŶT+2 = a + b1 ŶT+1
Continue in this fashion for more periods
This ONLY works if Y is stationary, based on the DF test for zero differences

10. Forecasting a Times Series Model
What if D1,t was stationary? How do you forecast?
Let T represent the last know observation
First period ahead forecast is
Recall that D1,T = YT – YT-1
So the time series OLS regression is: D̂1,T+1 = a + b1 D1,T
Next add the forecasted D̂1,T+1 to YT to forecast ŶT+1 as follows:
ŶT+1 = YT + D̂1,T+1

11. Forecasting A Time Series Model
Second period ahead forecast is
D̂1,T+2 = a + b D̂1,T+1
ŶT+2 = ŶT+1 + D̂1,T+2
Repeat the process for period 3 and so on
This is referred to as the chain rule of forecasting

12. For Forecast Model D1,t = 4.019 + 0.42859 D1,T-1
Year
History and
Forecast ŶT+i
Change Ŷ or D̂1,T
Forecast
D1T+i
T-1
1387 T
1289
-98.0
= *(-98)
= ( )
T+1
1251.1
-37.9
= *(-37.9)
= (-12.22)
T+2
-12.19
= *(-12.19)
= (-1.198)
T+3

13. Time Series Model Forecast – Note that this Model Restricted Out the Second Lag

14. Time Series Model Estimation
Impulse Response Function
Shows the impact of a 1 time 1 unit change in YT on the forecast values of Y over time
Good model is one where impacts decline to zero in short number of periods

15. Time Series Model Estimation
Impulse Response Function will die slowly if the model has to many lags; they feed on themselves
Same data series fit with 1 lag and a 6 lag model

16. Time Series Model Estimation
Dynamic stochastic Simulation of a time series model

17. Time Series Model Estimation
Look at the simulation in Lecture 12 Time Series.XLSX

18. Time Series Model Estimation
Result of a dynamic stochastic simulation

19. Vector Autoregressive (VAR) Models
VAR models are time series models where two or more variables are thought to be correlated and together they explain more than one variable by itself
For example forecasting
Sales and Advertising
Money supply and interest rate
Supply and Price
We are assuming that
Yt = f(Yt-i and Zt-i)

20. VAR Time Series Model Estimation
Take the example of advertising and sales
AT+i = a +b1DA1,T-1 + b2 DA1,T-2 +
c1DS1,T c2 DS1,T-2
ST+i = a +b1DS1,T-1 + b2 DS1,T c1DA1,T-1 + c2 DA1,T-2
Where A is advertising and S is sales
DA is the difference for A
DS is the difference for S
In this model we fit A and S at the same time and A is affected by its lag differences and the lagged differences for S
The same is true for S affected by its own lags and those of A

21. Time Series Model Estimation
Advertising and sales VAR model
Highlight two columns
Data in columns B and C
Specify number of lags
Max lags for two variables
Specify number differences
Max for the two variables

22. Time Series Model Estimation
Advertising and sales VAR model