Time Series Model Estimation

1
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

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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

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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 diferences based on DF() test,
  say Di,t
• Determine the number of lags to use in the AR
  model based on
• AUTOCORR(), say
  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

4
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
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

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Time Series Model Estimation

• 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 lt -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

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
• YT1 a b1 YT
• Forecast the second period ahead as
• YT2 a b1 YT1
• Continue in this fashion for more periods
• This ONLY works if Y is stationary, based on the
  DF test for zero lags

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Time Series Model Forecasting

• What if D1,t was stationary? How do you
  forecast?
• First period ahead forecast is
• D1,T YT YT-1
• D1,T1 a b1 D1,T
• Add the calculated D1,T1 to YT
• YT1 YT D1,T1
• Second period ahead forecast is
• D1,T2 a b D1,T1
• YT2 YT1 D1,T2
• Repeat the process for period 3 and so on
• This is referred to as the chain rule of
  forecasting

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For Model D1,t 4.019 0.42859 D1,T-1
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Time Series Model Forecast
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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

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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

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Time Series Model Estimation

• Dynamic stochastic Simulation of a time series
  model

Lecture 6
14
Time Series Model Estimation

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

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Time Series Model Estimation

• Result of a dynamic stochastic simulation

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Vector Autoregressive (VAR) Models

• VAR models a time series models where two or more
  variables at 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)

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Time Series Model Estimation

• Take the example of advertising and sales
• ATi a b1DA1,T-1 b2 DA1,T-2
• c1DS1,T-1 c2 DS1,T-2
• STi 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

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Time Series Model Estimation

• Advertising and sales VAR model