Download
lecture 12 time series model estimation n.
Skip this Video
Loading SlideShow in 5 Seconds..
Lecture 12 Time Series Model Estimation PowerPoint Presentation
Download Presentation
Lecture 12 Time Series Model Estimation

Lecture 12 Time Series Model Estimation

743 Vues Download Presentation
Télécharger la présentation

Lecture 12 Time Series Model Estimation

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

  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