Forecasting General problem: predict x n + m given x n , x n 1 , . - - PowerPoint PPT Presentation

Forecasting

• General problem: predict xn+m given xn, xn−1, . . . , x1.
• General solution: the (conditional) distribution of xn+m given

xn, xn−1, . . . , x1.

• In particular, the conditional mean is the best predictor (i.e.

minimum mean squared error).

• Special case: if {xt} is Gaussian, the conditional distribution

is also Gaussian, with a conditional mean that is a linear function of xn, xn−1, . . . , x1 and a conditional variance that does not depend on xn, xn−1, . . . , x1.

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Linear Forecasting

• What if xt is not Gaussian?
• Use the best linear predictor: xn

n+m.

• Not the best possible predictor, but computable.

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One-step Prediction

• The hard way: suppose

xn

n+1 = φn,1xn + φn,2xn−1 + · · · + φn,nx1.

• Choose φn,1, φn,2, . . . , φn,n to minimize the mean squared pre-

diction error E

• xn+1 − xn

n+1

2

.

• Differentiate and equate to zero: n linear equations in the n

unknowns.

• Solve recursively (in n) using the Durbin-Levinson algorithm.

Incidentally, the PACF is φn,n.

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One-step Prediction for an ARMA Model

• The easy way: suppose we can write

xn+1 = some linear combination of xn, xn−1, . . . , x1 + something uncorrelated with xn, xn−1, . . . , x1.

• Then the first part is the best linear predictor, and the second

part is the prediction error.

• E.g. AR(p), p ≤ n:

xn+1 = φ1xn + φ2xn−1 + · · · + φpxn+1−p

• first part

+ wn+1

second part

.

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General ARMA case

• Now

xn+1 =φ1xn + φ2xn−1 + · · · + φpxn+1−p + θ1wn + θ2wn−1 + · · · + θqwn+1−q + wn+1.

• First part on the right hand side is a linear combination of

xn, xn−1, . . . , x1.

• Last part, wn+1, is uncorrelated with xn, xn−1, . . . , x1.

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• Middle part?

If the model is invertible, wt is a linear com- bination of xt, xt−1, . . . , so if n is large, we can truncate the sum at x1, and wn, wn−1, . . . , wn+1−q are all (approximately) linear combinations of xn, xn−1, . . . , x1.

• So the middle part is also approximately a linear combination
• f xn, xn−1, . . . , x1, whence

xn

n+1 =φ1xn + φ2xn−1 + · · · + φpxn+1−p

+ θ1wn + θ2wn−1 + · · · + θqwn+1−q and wn+1 is the prediction error, xn+1 − xn

n+1.

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Multi-step Prediction

• The easy way: build on one-step prediction. E.g. two-step:

xn+2 =φ1xn+1 + φ2xn + · · · + φpxn+2−p + θ1wn+1 + θ2wn + · · · + θqwn+2−q + wn+2.

• Replace xn+1 by xn

n+1 + wn+1:

xn+2 =φ1xn

n+1 + φ2xn + · · · + φpxn+2−p

+ θ2wn + · · · + θqwn+2−q + wn+2 + (φ1 + θ1) wn+1.

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• The first two parts are again (approximately) linear combi-

nations of xn, xn−1, . . . , x1, and the last is uncorrelated with xn, xn−1, . . . , x1. So xn

n+2 =φ1xn n+1 + φ2xn + · · · + φpxn+2−p

+ θ2wn + · · · + θqwn+2−q and the prediction error is xn+2 − xn

n+2 = wn+2 + (φ1 + θ1) wn+1.

• Note that the mean squared prediction error is

σ2

w

• 1 + (φ1 + θ1)2

≥ σ2

w.

Mean squared prediction error increases as we predict further into the future.

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Forecasting with proc arima

• E.g. the fishery recruitment data.
• proc arima program

and output.

• Note that predictions approach the series mean, and “std

errors” approach the series standard deviation.

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• The “autocorrelation test for residuals” is borderline, largely

because of residual autocorrelations at lags 12, 24, . . . .

• Spectrum analysis shows that these are caused by seasonal

means, which can be removed: proc arima program and

• utput.

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