Forecasting ARMA Models

19 Pages·2009·3.71 MB·English

by  Robert Stine

Forecasting
ARMA Models
INSR 260, Spring 2009
Bob Stine
1
Over view
Review
Model selection criteria
Residual diagnostics
Prediction
Normality
Stationary vs non-stationary models
Calculations
Case study
2
Review
Autoregressive, moving average models
AR(p)
Y = δ + φ Y + φ Y + … + φ Y + a
t 1 t-1 2 t-2 p t-p t
MA(q)
Y = μ + a - θ a - θ a -... - θ a Watch negative signs
t t 1 t-1 2 t-2 q t-q!! !
ARMA(p,q)
Y = δ + φ Y + φ Y + a - θ a ! ! ! ARMA(2,1)
t 1 t-1 2 t-2 t 1 t-1
Common feature
Every stationary ARMA model specifies Yt as a weighted
sum of past error terms
!Y = a + w a + w a + w a +…
t t 1 t-1 2 t-2 3 t-3
e.g., AR(1) sets w = φj
j
ARMA models for non-stationary data
Differencing produces a stationary series.
These differences are a weighted average of prior errors.
3
Modeling Process
Initial steps
Before you work with data: think about context!! !
What do you expect to find in a model?
What do you need to get from a model? ARIMA = short-term forecasts
Set a baseline: What results have been obtained by other models?
Plot time series
Inspect SAC, SPAC
Estimation
Fit initial model, explore simpler & more complex models
Check residuals for problems
Ljung-Box test of residual autocorrelations
Residual plots show outliers, other anomalies
Forecasting
Check for normality
Extrapolate pattern implied by dependence
Compare to baseline estimates
4
Forecasting ARMA
Characteristics
Forecasts from stationary models revert to mean
Integrated models revert to trend (usually a line)
Accuracy deteriorates as extrapolate farther
Variance of prediction error grows
Prediction intervals at fixed coverage (e.g. 95%) get wider
Calculations
Fill in unknown values with predictions
Pretend estimated model is the true model
Example: ARMA (2,1) !
Y = δ + φ Y + φ Y + a - θ a
t 1 t-1 2 t-2 t 1 t-1
One-step ahead:!Ŷ = δ + φ Y + φ Y + (â =0) - θ â
n+1 1 n 2 n-1 n+1 1 n
Two! ! ! ! ! Ŷ = δ + φ Ŷ + φ Y + (â =0) - θ (â =0)
n+2 1 n+1 2 n n+2 1 n+1
Three!! ! ! ! Ŷ = δ + φ Ŷ + φ Ŷ + 0 + 0
n+3 1 n+2 2 n+1
AR gradually damp out, MA terms disappear (as in autocorrelations)
5
Accuracy of Forecasts
Assume
Estimated model is true model
Key fact
ARMA models represent Y as weighted sum of past errors
t
Theory: Forecasts omit unknown error terms
Y = μ + a + w a + w a + w a +…
n+1 n+1 1 n 2 n-1 3 n-2
“known”
Ŷ = μ + w a + w a + …
n+1 1 n 2 n-1
(cid:15482)
! ! ! ! ! ! ! ! ! ! !Y - Ŷ = a
n+1 n+1 n+1
Y = μ + a + w a + w a + w a +…
n+2 n+2 1 n+1 2 n 3 n-1
Ŷ = μ + w a + w a +…
n+2 2 n 3 n-1
(cid:15482)
! ! ! ! ! ! ! ! ! ! !Y - Ŷ = a + w a
n+2 n+2 n+2 1 n+1
Variance of forecast error grows as (a are iid)
t
! ! ! ! σ2(1 + w 2 + w 2 + …)
1 2
6
Example: ARMA(1,1)
Simulated data
Know that we’re fitting the right model
Compare forecasts to actual future values
δ is not the
mean of the
Estimated model
series
Parameter Estimates
δ
Constant
Term Estimate Estimate
AR1 0.7467 0.07988923
MA1 -0.7954
intercept = mean Intercept 0.3154
Reverse sign on
Forecasts moving average
estimates
n=200
ŷ =0.080+0.75(1.22)+0.80(-0.29)
201
ŷ =0.080+0.75(0.76)+0.80(0)
202
ŷ =0.080+0.75(0.65)
203
ŷ = δ + φy - θa
n+f n+f-1 n+f-1
SD(y) = 2.50 7
0.315
Forecasts
Forecasts revert quickly to series mean
Unless model is non-stationary or has very strong
autocorrelations
Prediction intervals open as extrapolate
Variance of prediction errors rapidly approaches series
variance
10.00
5.00
Y
0.00
-5.00
observed forecast
-10.00
196 198 200 202 204 206 208 210
Rows
8
Detailed Calculations
MA(2)!! ! Y = μ + a - θ a - θ a
t t 1 t-1 2 t-2
Forecasting
1-step!! ! ! ! ! Y = μ + a - θ a - θ a
n+1 n+1 1 n 2 n-1
Ŷ = μ - θ a - θ a
! ! ! ! ! ! ! n+1 1 n 2 n-1
Y - Ŷ = a
n+1 n+1 n+1
Var(Y - Ŷ ) = σ2
n+1 n+1
2-steps! ! ! ! ! Y = μ + a - θ a - θ a
n+2 n+2 1 n+1 2 n
Ŷ = μ - θ a
! ! ! ! ! ! ! n+2 2 n
Y - Ŷ = a - θ a
n+2 n+2 n+2 1 n+1
Var(Y - Ŷ ) = σ2(1+θ 2)
n+2 n+2 1
3-steps! ! ! ! ! Y = μ + a - θ a - θ a
n+3 n+3 1 n+2 2 n+1
Ŷ = μ
! ! ! ! ! ! ! n+3
Y - Ŷ = a + θ a + θ a
n+3 n+3 n+3 1 n+2 2 n+1
Var(Y - Ŷ ) = σ2(1+θ 2+θ 2)
n+3 n+3 1 2
9
Example: MA(2) w/Numbers
Predictions
Ŷ = μ - θ a - θ a
n+1 1 n 2 n-1
Model Summary
= 0.162 + 0.996(-.0464) + 0.380(.144)
DF 177.0000 Stable Yes !
Sum of Squared Errors 188.3783 Invertible Yes
!= 0.1705
Variance Estimate 1.0643
Standard Deviation 1.0316
Akaike's 'A' Information Criterion 526.0554 Ŷ = μ - θ â - θ a
n+2 1 n+1 2 n
Parameter Estimates
= 0.162 + 0.996(0) + 0.380(-.0464)
Constant !
Term Lag Estimate Std Error t Ratio Prob>|t| Estimate
!= 0.1443
MA1 1 -0.9962 0.0647622 -15.38 <.0001* 0.16199858
MA2 2 -0.3803 0.0590225 -6.44 <.0001*
Intercept 0 0.1620 0.1802599 0.90 0.3700 Ŷ = μ - θ â - θ â
n+3 1 n+2 2 n+1
= 0.046 + 0.996(0) + 0.380(0)
!
!= 0.162
SD of prediction error
1-step:
σ = 1.0316
2-steps
σ sqrt(1 + θ 2) = 1.0316(1+.9962)1/2
1
! ! ! ! ! = 1.456
3-steps
σ sqrt(1+θ 2+θ 2) = 1.0316(1+.9962+.3802)1/2
1 2
! ! ! ! ! = 1.508
10