Time Series Model Estimation

• 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

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

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 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

Time Series Model Forecasting

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

• First period ahead forecast isD1,T = YT – YT-1

DF1,T+1 = a + b1 D1,T

Add the forecasted D1,T+1 to YT to forecast ŶT+1

ŶT+1 = YT + DF1,T+1

Time Series Model Forecasting Continued

• 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 estimated 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

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

0.000

0.200

0.400

0.600

0.800

1.000

1.200

Impulse Response Function

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 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

Time Series Model Estimation

• 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

• 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

P0 and Q0 are baseline values Q1 is assumed change

Price

Q/UTQ0

Q1

P0

?P1

Demand

Equilibrium Displacement Model Forecasting

• To forecast the Q Supplied given a change in price

Qt Supplied1 = Qs0 * [1 + Es *( P1 - P0) / P0)] + ẽ

P0 and Q0 are baseline values P1 is assumed change

Price

Q/UTQ0

?Q1

P0

P1

Supply

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

Q/UTQ0

?Q1

P0

P1

Demand for Exports

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