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

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

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 D i,t Determine the number of lags to use in the AR model based on =AUTOCORR() D i,t =a + b 1 D i,t-1 + b 2 D i,t-2 +b 3 D i,t-3 + b 4 D i,t-4 Create all of the data lags and estimate the model using OLS (or use Simetar)

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 Lecture 6

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

Before You Estimate TS Model (Review) Dickey-Fuller test indicates whether the data series used for the model, D i,t, is stationary and if the model is D 2,t = a + b 1 D 1,t the DF it indicates that t stat for b 1 is < 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 D i,t =a + b 1 D i,t-1 + b 2 D i,t-2 +b 3 D i,t-3 + b 4 T t 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

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 Ŷ T+1 = a + b 1 Y T Forecast the second period ahead as Ŷ T+2 = a + b 1 Ŷ T+1 Continue in this fashion for more periods This ONLY works if Y is stationary, based on the DF test for zero lags

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

For Model D 1,t = D 1,T-1 YearHistory and Forecast Ŷ T+i Change Ŷ or D̂ 1,T Forecast D 1T+i Forecast Ŷ T+i T T = *(-98) = ( ) T = *(-37.9) = (-12.22) T = *(-12.19) = (-1.198) T

Time Series Model Forecast

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

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

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

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

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 Y t = f(Y t-i and Z t-i )

Time Series Model Estimation Result of a dynamic stochastic simulation

VAR Time Series Model Estimation Take the example of advertising and sales A T+i = a +b 1 DA 1,T-1 + b 2 DA 1,T-2 + c 1 DS 1,T-1 + c 2 DS 1,T-2 S T+i = a +b 1 DS 1,T-1 + b 2 DS 1,T-2 + c 1 DA 1,T-1 + c 2 DA 1,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

Time Series Model Estimation Advertising and sales VAR model