Time Series Model Estimation
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Materials for lecture 6
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 stationarity and no. of lags 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 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)
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
Time Series Model Estimation
• 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
0.144
0.143
1
1
S.E. of Coefficients
Sales
30.764
Restriction Matrix
Sales
1
Differences
1
Characteristics
Dickey-Fuller
Aug.Test
Dickey-Fuller
Schwarz Test
S.D. Residuals
MAPE
Sales
Forecast
-4.471
-4.271
Impulse Auto-
5.529
212.955
t-Statistic Partial
8.86
AIC
SIC
10.84 10.95345
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
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
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 + b1 D1,T
Add the forecasted D1,T+1 to YT to forecast ŶT+1
ŶT+1 = YT + D̂1,T+1
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
For estimated Model D1,t = 4.019 + 0.42859 D1,T-1
Year
History and
Forecast
ŶT+i
Change Forecast
Ŷ or
D1T+i
D̂1,T
Forecast ŶT+i
T-1
1387
T
1289
-98.0
-37.925 = 4.019 + 1251.1 = 1289 +
0.428*(-98)
(-37.925)
T+1
1251.1
-37.9
-12.224 = 4.019 + 1238.91 = 1251.11 +
0.428*(-37.9)
(-12.22)
T+2
1238.91
-12.19
-1.198 = 4.019 +
0.428*(-12.19)
T+3
1237.71
1237.71=1238.91 +
(-1.198)
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
0.129
0.000
1
0
S.E. of Coefficients
Sales
30.621
Restriction Matrix
Sales
1
Differences
1
Characteristics
Dickey-Fuller
Aug.Test
Dickey-Fuller
Schwarz Test
S.D. Residuals
MAPE
Sales
Forecast
-4.471
-4.271
Impulse Auto-
5.529 214.1866
t-Statistic Partial
9.06
AIC
10.81
t-Statistic
Response Correlation
(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
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
Impulse Response Function
1.200
1.000
0.800
0.600
0.400
0.200
0.000
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
Time Series Model Estimation
• Result of a dynamic stochastic simulation
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)
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
Time Series Model Estimation
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Advertising and sales VAR model
Highlight two columns
Specify number of lags
Specify number differences
Time Series Model Estimation
• Advertising and sales VAR model
Equilibrium Displacement Model Forecasting
• Elasticities of demand and supply can be used to
forecast changes in price, quantity demanded and
quantity supplied
• Small changes in the exogenous variables allows
one to forecast the dependent variable
• This method is simple and reliable for small
changes from equilibrium
• Information needed:
– A Baseline of equilibrium quantities and prices
– Own and cross elasticities for demand and supply
– Residuals from trend (or a structural model) for the
dependent variable if the forecast is to be stochastic
Equilibrium Displacement Model Forecasting
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Baseline prices and quantities are available from FAPRI, USDA, and some
private consulting firms
Here is an example of the corn S&U from the FAPRI March 2013 Baseline
Equilibrium Displacement Model Forecasting
• To forecast a price change given a change in
quantity supplied
Price1 = P0 * [1 + Price Flex *( Q1 - Q0) / Q0)] + ẽ
Price
?P1
P0
Demand
Q1 Q0
Q/UT
P0 and Q0 are baseline values Q1 is assumed change
Equilibrium Displacement Model Forecasting
• To forecast the Q Supplied given a change in price
Qt Supplied1 = Qs0 * [1 + Es *( P1 - P0) / P0)] + ẽ
Price
Supply
P1
P0
Q0 ?Q1
Q/UT
P0 and Q0 are baseline values P1 is assumed change
Equilibrium Displacement Model Forecasting
• We can expand the supply response equation by
including cross elasticities
Qt Supplied x = Qsx0 * (1 + [Exs *( Px1 - Px0) / Px0)]
+ [Ex,ys *( Py1 - Py0) / Py0)]
+ [Ex,zs *( Pz1 - Pz0) / Pz0)] ) + ẽ
Where x is the own crop (say, corn) and y is the price of soybeans, and z
is the price of wheat
The supply response equation can be expanded to
contain cross elasticities for all other crops
Note: Ex,ys is the elasticity of corn supply with respect to the price of
soybeans
Equilibrium Displacement Model Forecasting
• Elasticity of demand equation can be used to forecast
quantity exported, or quantity demanded for ethanol or
any other quantity demanded
All we need is the Baseline quantity demanded, baseline
price and the own and cross elasticity
QD1 = QD0 * (1 + ED for exports * ( P1 - P0) / P0)) + ẽ
Price
P0
P1
Demand for Exports
Q0 ?Q1
Q/UT
Equilibrium Displacement Model Forecasting
• This method of forecasting is widely used in
agricultural economics
• Particularly useful for policy analysis and
consulting
• This is why economists place so much emphasis
on estimating unbiased elasticities