Assuming equal variances, can estimate using regression. E.g. for - - PowerPoint PPT Presentation

Threshold Models

• A simple form of nonlinear model, basically a “switching”

AR(p): xt = α(j) + φ(j)

1 xt−1 + · · · + φ(j) p xt−p + σ(j)wt

if xt−1 ∈ Rj, where xt−1 = (xt−1, . . . , xt−p)′, R1, R2, . . . , Rr is a partition of Rp, and wt is white noise with variance 1.

• That is, the AR(p) parameters in the equation for xt change,

depending on the values of the previous p observations xt−1, . . . , xt−p.

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• Assuming equal variances, can estimate using regression.

E.g. for monthly pneumonia and influenza deaths:

flu = ts(scan("flu.dat")); dflu = diff(flu); a = dflu; for (l in 1:6) a = cbind(a, lag(dflu, -l)); a = cbind(a, lag(dflu, -1) > 0.05); a = data.frame(a); names(a) = c("x", paste("x", 1:6, sep = ""), "delta"); summary(lm(x ~ delta + x1*delta + x2*delta + x3*delta + x4*delta + x5*delta + x6*delta, data = a)); flu.l = lm(x ~ -1 + delta + x1*delta + x2*delta + x3*delta + x4*delta, data = a); summary(flu.l); flu.r = residuals(flu.l); delta = a$delta[4 + 1:length(flu.r)]; lapply(split(flu.r, delta), sd); acf(flu.r);

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• This is inefficient, and standard errors are invalid, if variances

are unequal; here F = 1.93, df = (17, 110), P = .022.

• We can also fit the model using two separate regressions:

flu.lF = lm(x ~ x1 + x2 + x3 + x4, data = a, subset = (delta == 0)); summary(flu.lF); flu.lT = lm(x ~ x1 + x2 + x3 + x4, data = a, subset = (delta == 1)); summary(flu.lT);

• Setting up a residual series:

flu.r01 = flu.r; flu.r01[!delta] = residuals(flu.lF) / sd(residuals(flu.lF)); flu.r01[delta] = residuals(flu.lT) / sd(residuals(flu.lT)); acf(flu.r01);

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Regression with Autocorrelated Errors

• Regression model

yt = β′zt + xt where xt has covariance matrix Γ.

• Generalized least squares (GLS) estimates for known Γ:

ˆ

β =

• Z′Γ−1Z

−1 Z′Γ−1y.

• For unknown Γ, plug in an estimate.

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• If xt is a stationary time series, can fit using OLS, then either:

– get estimated autocovariances ˆ γ(h) from the residuals, and plug in ˆ

Γ;

– or use Cochrane-Orcutt method.

• More generally, can use mixed model methods (SAS proc

mixed).

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• Cochrane and Orcutt suggested:

– fit using OLS to get initial estimate of β; – fit AR(p) to OLS residuals (wt is white noise): φ(B)xt = wt; – transform the regression to φ(B)yt = β′φ(B)zt + φ(B)xt = β′φ(B)zt + wt

• r

ut = β′vt + wt.

• Residuals are now white, so fit using OLS.

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• SAS proc arima offers better solution:

– Mortality and air pollution (Example 5.6): program and

• utput.

– Global temperature and cumulative CO2 emissions: pro- gram and output.

• In R, temperature (slightly different) and CO2:

arima(globtemp, order = c(1, 0, 0), xreg = co2w); arima(globtemp, order = c(4, 0, 0), xreg = co2w); arima(globtemp, order = c(0, 0, 4), xreg = co2w);

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