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 =
∞
X
αj xt−j + ηt = α(B)xt + ηt,
j=0
where the noise ηt is uncorrelated with the inputs.
• This generalizes regression with correlated errors by including lags, and specializes the frequency domain lagged regression by excluding future inputs.
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• Preliminary estimation of α0, α1, . . . often suggests a parsimonious model
δ(B)
α(B) = B d ×
,
ω(B)
where:
– d is the pure delay : α0 ≈ α1 ≈ · · · ≈ αd−1 ≈ 0 and αd 6= 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 
∞
X


αj xt+h−j + ηt+h xt
j=0
= αhvar (xt) ,
d (xt) provides an estimate of αh.
so γ̂y,x(h)/var
• 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 regression equation:
ỹt =
∞
X
αj wt−j + η̃t,
j=0
where ỹt = [φ(B)θ(B)−1]yt and η̃t = [φ(B)θ(B)−1]ηt.
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• Now the cross correlation γỹ,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 forcing 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 preindustrial levels, and the temperature response is called the
climate sensitivity. The IPCC states:
Analysis of models together with constraints from
observations suggest that the equilibrium climate sensitivity 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 using a grid search in R (with p = 4), gives an estimated value
of 1.85◦C and 95% confidence limits of 1.44◦C to 2.27◦C.
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1.5
2.0
2.5
-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
10
−306
−308
−310
ll2
−304
−302
-2 Log-Likelihood profile for climate sensitivity:
1.5
2.0
2.5
4.4 * theta
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−310 −308 −306 −304 −302 −300
ll2
-2 Log-Likelihood profile for decay factor:
0.4
0.5
0.6
0.7
0.8
0.9
lambda
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