]]>With V estimated as below, hatbeta=0.00448, with SE(hatbeta)=0.0022. An immediate implication is that there is evidence, not overwhelming, of a positive trend. We remark that there is in fact literature on alternative models (see Bloomfield (1992), Smith (1993) and Haslett (1997)), and given that, as we have seen from our model hatbeta/SE(hatbeta) is ‘borderline’ there is a real issue in addressing such alternatives.

<em>Residuals For the Linear Model With General Covariance Structure</em> John Haslett; Kevin Hayes Journal of the Royal Statistical Society. Series B (Statistical Methodology), Vol. 60, No. 1. (1998), pp. 201-215.

They use Jones and Briffa (1992) global temperature data, and fit a trend.

<blockquote>With V estimated as below, , with latex \hat{\beta}\SE(\hat{\beta}) $ is ‘borderline’ there is a real issue in addressing such alternatives. </blockquote>

It is all about how you model the covariance matrix of the noise term (V in the quote).. Gavin uses a bit naive model in the above, and IMO trend with CI is quite useless in such data where the generating model is almost completely unknown. ]]>

Bob

]]>In order to use LaTeX, you need to put the expressions between [tex ][/ tex] tags (except without the spaces) and not use the $$ commands.

Thus your test would be

[tex ](1-\alpha)F_{o}=\epsilon\sigma4T^4[/ tex]

which will render to

]]>$$(1-\alpha)F_{o}=\epsilon\sigma4T^4$$ ]]>

He shows that very significantly e(t) is not i.i.d,

Ooops, meant Y(t), but probably e(t) is neither,

[P,dw]=dwtest(res,X)

P =

0.0107977427341995

dw =

1.52406604865248

]]>and i.i.d Gaussian e(t). He shows that very significantly e(t) is not i.i.d, funny guy.

]]>I think what they are saying is that they regressed Y(t)=a+bt+e(t) where Y(t) is the temperature series and e(t) are the residuals, modeled e(t) = rho.e(t-1)+u(t), ie an AR1 process for e(t), computed a DW stat for u(t), and instead of using the tables based on T-2 degrees of freedom, computed the “effective degrees of freedom” concept, which is a formula involving rho.

With model Y(t) = a +bt + e(t), and unknown covariance matrix of e(t), where can you go?? Maybe they, for some reason, know that structure of that covariance matrix is the one of AR1 process, but still the variance and rho remains to be estimated. How to deal with the case b = 0? That yields completely different estimate of covariance matrix.

One answer to my question in #15 seems to be the exponential function, or integrated RW, cumsum(cumsum(white noise)). Hopefully we’ll see the exact code soon, maybe it makes sense, I’d be happy to learn new stuff.

]]>