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Time Series Model Estimation

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**1. **Time Series Model Estimation Materials for this lecture
Read Chapter 15 pages 30 to 37
Lecture 7 Time Series.XLS
Lecture 7 Vector Autoregression.XLS

**2. **Time Series Model Estimation Outline for this lecture
Review the first times series lecture
Discuss model estimation
Demonstrate how to estimate Time Series (AR) models with Simetar
Interpretation of model results
How you forecast the results for an AR model

**3. **Time Series Model Estimation Plot the data to see what kind of series you are analyzing
Make the series stationary by determining the optimal number of diferences based on =DF() test, say Di,t
Determine the number of lags to use in the AR model based on
=AUTOCORR(), say Di,t =a + b1 Di,t-1 + b2 Di,t-2 +b3 Di,t-3+ b4 Di,t-4
Create all of the data lags and estimate the model using OLS

**4. **Time Series Model Estimation An alternative to estimating the differences and lag variables by hand and using a regression package, use Simetar
Simetar time series function is driven by a menu

**5. **Time Series Model Estimation Read the results like a regression
Beta coefficients are provided like OLS
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 restrict out variables

**6. **Time Series Model Estimation 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 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

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

**8. **Time Series Model Forecasting What if D1,t was stationary? How do you forecast?
First period ahead forecast is
D1,T = YT – YT-1
D^1,T+1 = a + b1 D1,T
Add the calculated D1,T+1 to YT
YT+1 = YT + D^1,T+1
Second period ahead forecast is
D^1,T+2 = a + b D^1,T+1
YT+2 = YT+1 + D^1,T+2
Repeat the process for period 3 and so on
This is referred to as the chain rule of forecasting

**9. **For Model D1,t = 4.019 + 0.42859 D1,T-1

**10. **Time Series Model Forecast

**11. **Time Series Model Estimation Impulse Response Function
Shows the impact of a 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

**12. **Time Series Model Estimation Impulse Response Function will die slowly if the model has to many lags
Same data series fit with 1 lag and a 6 lag model

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

**14. **Time Series Model Estimation Look at the simulation in Lecture 6 Time Series.XLS

**15. **Time Series Model Estimation Result of a dynamic stochastic simulation

**16. **Vector Autoregressive (VAR) Models VAR models a time series models where two or more variables at 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)

**17. **Time Series Model Estimation Take the example of advertising and sales
AT+i = a +b1DA1,T-1 + b2 DA1,T-2 +
c1DS1,T-1 + c2 DS1,T-2
ST+i = a +b1DS1,T-1 + b2 DS1,T-2 + 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

**18. **Time Series Model Estimation Advertising and sales VAR model