Lagged Regression again: Transfer Functions To forecast an output - - PowerPoint PPT Presentation

Lagged Regression again: Transfer Functions

• To forecast an output series yt given its own past and the

present and past of an input series xt, we might use yt =

∞

• j=0

αjxt−j + ηt = α(B)xt + ηt, where the noise ηt is uncorrelated with the inputs.

• This generalizes regression with correlated errors

by includ- ing lags, and specializes the frequency domain lagged regres- sion by excluding future inputs.

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• Preliminary estimation of α0, α1, . . . often suggests a parsi-

monious model α(B) = Bd × δ(B) ω(B), where: – d is the pure delay: α0 ≈ α1 ≈ · · · ≈ αd−1 ≈ 0 and αd = 0; – δ(B) and ω(B) are low-order polynomials: ω(B) is needed if the α’s decay exponentially, and δ(B) is needed if the first few nonzero α’s do not follow the decay.

• Preliminary estimates from frequency domain method, or a

similar time domain method.

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Time Domain Preliminary Estimates

• If the input series xt were white noise, the cross correlation

γy,x(h) = E

• yt+hxt
• = E

   

∞

• j=0

αjxt+h−j + ηt+h

  xt  

= αhvar (xt) , so ˆ γy,x(h)/ var (xt) provides an estimate of αh.

• Usually, xt is not white noise, but if it is a stationary time

series, we know how to make it white: fit an ARMA model.

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Prewhitening

• Suppose that xt is ARMA:

φ(B)xt = θ(B)wt, where wt is white noise.

• Apply the prewhitening filter φ(B)θ(B)−1 to the lagged re-

gression equation: ˜ yt =

∞

• j=0

αjwt−j + ˜ ηt, where ˜ yt = [φ(B)θ(B)−1]yt and ˜ ηt = [φ(B)θ(B)−1]ηt.

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• Now the cross correlation γ˜

y,w(h) provides an estimate of αh.

• You can use SAS’s proc arima to do this:

– first identify and estimate a model for xt; – then identify yt with xt as a crosscorr variable. At the second step, SAS uses the prewhitening filter from the first step to filter both xt and yt before calculating cross correlations.

• Note: SAS announces that both series have been “prewhitened”,

but the filter is designed to prewhiten only xt; yt is filtered, but typically not prewhitened.

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• Finally estimate the model for yt, specifying the input series,

in the form: input = (d$(L1,1, L1,2, . . . ) . . . (Lk,1, . . . ) /(Lk+1,1, . . . ) . . . (. . . )variable)

• E.g.

for Southern Oscillation and the fisheries recruitment series: program and output.

• E.g. for global temperature and an estimated historical forc-

ing series: program and output.

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Interpreting a Transfer Function

• For the global temperature case, we have

yt = 0.087917×(xt +0.79513xt−1 +0.795132xt−2 +. . . )+ηt.

• So the effect of an impulse in the forcing xt, say a dip due

to a volcanic eruption, is felt in the current year and several subsequent years, with a mean delay of 1/(1−0.79513) ≈ 4.9 years.

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• Also, the effect of a sustained change of +4.4W/m2 would

be 0.087917 × 4.4 × (1 + 0.79513 + 0.795132 + . . . ) = 0.087917 × 4.4/(1 − 0.79513) ≈ 1.9◦C.

• This is the expected forcing for a doubling of CO2 over pre-

industrial levels, and the temperature response is called the climate sensitivity. The IPCC states: Analysis of models together with constraints from

• bservations suggest that the equilibrium climate sen-

sitivity is likely to be in the range 2◦C to 4.5◦C, with a best estimate value of about 3◦C. It is very unlikely to be less than 1.5◦C.

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• Our estimate is at the low end of that range, but quantifying

its uncertainty is difficult using proc arima.

• The profile likelihood for climate sensitivity, constructed us-

ing a grid search in R (with p = 4), gives an estimated value

• f 1.85◦C and 95% confidence limits of 1.44◦C to 2.27◦C.

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• 2 Log-Likelihood contours for climate sensitivity (y-axis) and

decay factor (x-axis):

0.4 0.5 0.6 0.7 0.8 0.9 1.5 2.0 2.5

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• 2 Log-Likelihood profile for climate sensitivity:

1.5 2.0 2.5 −310 −308 −306 −304 −302 4.4 * theta ll2

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• 2 Log-Likelihood profile for decay factor:

0.4 0.5 0.6 0.7 0.8 0.9 −310 −308 −306 −304 −302 −300 lambda ll2

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