Lecture 12 Time Series Model Estimation

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 • Lecture 12 Probabilistic Time Series.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 presence of a trend, i.e., use the Augmented Dickie-Fuller test =DF(Data Series, Trend, 0, No Differences) Trend = True for augmented DF Trend = False for regular DF

4. Augmented Dickie Fuller Test • Dickey-Fuller test indicates whether the data series used for the model, Di,t, is stationary w/o a trend adjustment D1,t = a + b1 D2,t DF test statistic is the stat for b1 and it needs to be more negative than -2.90 • Augmented DF test indicates whether the data series Di,t are stationary, if we added a trend to the model, so the OLS equation estimated becomes: Di,t=a + b1 Di,t-1 + b4 Tt

5. 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 different lags for 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

6. 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

7. Time Series Model Estimation in Simetar • An alternative to estimating the OLS regression model and having to forecast the model by hand we let Simetar do the work • Simetar time series function is driven by a menu

8. Time Series Model Estimation • Read Simetar output as a regression output • Beta coefficients are the OLS slope coefficients • S.E. 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)

9. Goodness of Fit Statistics • 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 • In the Restriction row, changing the number of lags also changes the MAPE and SD for residuals • Changing the number of differences is a quick check. If you like the new model completely re-estimate model using the new number of differences.

10. Forecasting a Time Series Model • If a series 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

11. Forecasting a D1 Times Series Model • What if D1,t was stationary? How do you forecast? • Let T represent the last know observation • Steps for the first period ahead forecast: 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

12. Forecasting A D1 Time Series Model • Second period ahead forecast for the D1 model 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

13. For Forecast Model D1,t = 4.019 + 0.42859 D1,T-1

14. Forecasting A D2 Time Series Model • First period ahead forecast if one lag and two differences D̂2,T = D2,T-1 - D2,T-2 D̂1,T+1 = a + b1 D̂2,T The OLS model estimated ŶT+1 = ŶT + D̂1,T+1 • Second Period ahead Forecast D̂2,T = D̂2,T - D2,T-1 D̂1,T+2 = a + b1 D̂2,T+1 ŶT+2 = ŶT+1 + D̂1,T+2

15. Forecasting A D2 Time Series Model • First period ahead forecast if two lags and two differences D̂2,T-1 = D2,T-2 - D2,T-3 D̂2,T = D2,T-1 - D2,T-2 D̂1,T+1 = a + b1 D̂2,T + b2 D̂2,T-1 The OLS model ŶT+1 = ŶT + D̂1,T+1 • Repeat the process to forecast ŶT+2 just rolling all values ahead by one period

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

17. 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 a short number of periods

18. 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

19. Simulation of a Time Series Model • Dynamic stochastic Simulation of a time series model

20. Deterministic Forecast • Look at the simulation in Lecture 12 Time Series.XLSX

21. Dynamic Stochastic Forecast • Dynamic stochastic simulation uses the previous random value to simulate the next period

22. 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 each 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)

23. VAR 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 and 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

24. VARModel 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

25. VARModel Estimation • Advertising and sales VAR model