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Time Series Model Estimation

Time Series Model Estimation. Materials for this lecture Read Chapter 15 pages 30 to 37 Lecture 6 Time Series.XLSX Lecture 6 Vector Autoregression.XLSX. Time Series Model Estimation. Outline for this lecture Review the first times series lecture Discuss model estimation

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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 6 Time Series.XLSX • Lecture 6 Vector Autoregression.XLSX

  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 differences based on =DF() test, say Di,t • Determine the number of lags to use in the AR model based on =AUTOCORR() or =ARLAG() 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 (or use Simetar)

  4. Time Series Model Estimation • An alternative to estimating the differences and lag variables by hand and using an OLS 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. 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+ b4Tt • 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 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 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 D̂1,T+1 = a + b1D1,T Add the calculated D1,T+1 to YT ŶT+1 = YT + D̂1,T+1

  9. Time Series Model Forecasting Continued • 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

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

  11. Time Series Model Forecast

  12. 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

  13. 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

  14. Time Series Model Estimation • Dynamic stochastic Simulation of a time series model Lecture 6

  15. Time Series Model Estimation • Look at the simulation in Lecture 6 Time Series.XLS

  16. Vector Autoregressive (VAR) Models • VAR models a 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)

  17. Time Series Model Estimation • Result of a dynamic stochastic simulation

  18. 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 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

  19. Time Series Model Estimation • Advertising and sales VAR model

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