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 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

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


• 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)


• An alternative to estimating the differences and lag variables by hand and using an OLS regression package, we will use Simetar

• Simetar time series function is driven by a menu


• Read output as a regression output – Beta coefficients are OLS slope

coefficients– 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 test restricting out lags (variables)

AR Series Analysis Results for 2 Lags & 1 Difference, 2/27/2012 8:12:25 PM

Constant SalesL1 SalesL2

Sales 3.393 0.476 -0.107

S.E. of Coefficients

Sales 30.764 0.144 0.143

Restriction Matrix

Sales 1 1 1

Differences 1

CharacteristicsDickey-Fuller TestAug. Dickey-Fuller TestSchwarz S.D. ResidualsMAPE AIC SIC

Sales -4.471 -4.271 5.529 212.955 8.86 10.84 10.95345

Forecast Impulse Auto- t-Statistic Partial t-Statistic

Response Correlation(AutoCorr.)AutoCorrelation(Part.AutoCorr.)Period

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+ 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

Forecasting a Time Series Model

• If we assume a series that is stationary and has T observations of data we estimate the model as an AR(0 difference, 1 lag)

• Forecast the first period ahead asŶT+1 = a + b1 YT

• Forecast the second period ahead asŶT+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

Forecasting a Times Series Model

• What if D1,t was stationary? How do you forecast?

• First period ahead forecast isRecall that D1,T = YT – YT-1

So the regression lets us: DF1,T+1 = a + b1 D1,T

Next add the forecasted D1,T+1 to YT to forecast ŶT+1 as follows:

ŶT+1 = YT + DF1,T+1

Forecasting A Time Series Model

• Second period ahead forecast is

DF1,T+2 = a + b DF

1,T+1

ŶT+2 = ŶT+1 + DF1,T+2

• Repeat the process for period 3 and so on

• This is referred to as the chain rule of forecasting

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

Year History andForecast ŶT+i

Change Ŷ or DF

1,T

Forecast D1T+i

Forecast ŶT+i

T-1 1387

T 1289 -98.0 -37.925 = 4.019 + 0.428*(-98)

1251.1 = 1289 + (-37.925)

T+1 1251.1 -37.9 -12.224 = 4.019 + 0.428*(-37.9)

1238.91 = 1251.11 + (-12.22)

T+2 1238.91 -12.19 -1.198 = 4.019 + 0.428*(-12.19)

1237.71=1238.91 + (-1.198)

T+3 1237.71

Time Series Model Forecast

AR Series Analysis Results for 2 Lags & 1 Difference, 2/27/2012 8:12:25 PM

Constant SalesL1 SalesL2

Sales 3.028 0.430 0.000

S.E. of Coefficients

Sales 30.621 0.129 0.000

Restriction Matrix

Sales 1 1 0

Differences 1

CharacteristicsDickey-Fuller TestAug. Dickey-Fuller TestSchwarz S.D. ResidualsMAPE AIC

Sales -4.471 -4.271 5.529 214.1866 9.06 10.81

Forecast Impulse Auto- t-Statistic Partial t-Statistic

ResponseCorrelation(AutoCorr.)AutoCorrelation(Part.AutoCorr.)Period

1,249.849 1.000 0.427042 3.108914 0.427042 3.108914 1

1,236.027 0.431 0.096647 0.602284 -0.10484 -0.76322 2

1,233.106 0.186 0.073858 0.457153 0.09031 0.657466 3

1,234.877 0.080 0.109992 0.678136 0.062501 0.455016 4

1,238.667 0.034 0.033193 0.202893 -0.05185 -0.37748 5


• 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

0.000

0.200

0.400

0.600

0.800

1.000

1.200

Impulse Response Function


• 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


• Dynamic stochastic Simulation of a time series model

Lecture 6


• Look at the simulation in Lecture 12 Time Series.XLS


• Result of a dynamic stochastic simulation

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 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)

VAR Time Series Model Estimation

• Take the example of advertising and salesAT+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


• Advertising and sales VAR model• Highlight two columns • Specify number of lags• Specify number differences


• Advertising and sales VAR model

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.

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