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In [[econometrics]], '''Prais–Winsten estimation''' is a procedure meant to take care of the [[Autocorrelation|serial correlation]] of type [[Autoregressive model#Example: An AR.281.29 process|AR(1)]] in a [[linear model]]. It is a modification of [[Cochrane–Orcutt estimation]] in the sense that it does not lose the first observation and leads to more [[efficiency (statistics)|efficiency]] as a result.
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==Theory==
Consider the model
 
:<math>y_t = \alpha + X_t \beta+\varepsilon_t,\,</math>
 
where <math>y_{t}</math> is the [[time series]] of interest at time ''t'', <math>\beta</math> is a [[Vector (geometry)|vector]] of coefficients, <math>X_{t}</math> is a matrix of [[explanatory variable]]s, and <math>\varepsilon_t</math> is the [[error term]]. The error term can be [[serial correlation|serially correlated]] over time: <math>\varepsilon_t =\rho \varepsilon_{t-1}+e_t,\ |\rho| <1 </math> and <math> e_t </math> is a white noise. In addition to the Cochrane–Orcutt procedure transformation, which is
 
:<math>y_t - \rho y_{t-1} = \alpha(1-\rho)+\beta(X_t - \rho X_{t-1}) + e_t. \,</math>
 
for t=2,3,...,T, Prais-Winsten procedure makes a reasonable transformation for t=1 in the following form
 
:<math>\sqrt{1-\rho^2}y_1 = \alpha\sqrt{1-\rho^2}+\left(\sqrt{1-\rho^2}X_1\right)\beta + \sqrt{1-\rho^2}\varepsilon_1. \,</math>
 
Then the usual [[Ordinary least squares|least squares]] estimation is done.
 
==Estimation procedure==
To do the estimation in a compact way it is directive to look at the auto-covariance function of the error term considered in the model above:
 
: <math>\mathrm{cov}(\varepsilon_t,\varepsilon_{t+h})=\frac{\rho^h}{1-\rho^2}, \text{ for } h=0,\pm 1, \pm 2, \dots \, . </math>
 
Now is easy to see that the variance-covariance, <math>\mathbf{\Omega} </math>, of the model is
 
: <math>\mathbf{\Omega} = \begin{bmatrix}
\frac{1}{1-\rho^2} & \frac{\rho}{1-\rho^2} & \frac{\rho^2}{1-\rho^2} & \cdots & \frac{\rho^{T-1}}{1-\rho^2} \\[8pt]
\frac{\rho}{1-\rho^2} & \frac{1}{1-\rho^2} & \frac{\rho}{1-\rho^2} & \cdots & \frac{\rho^{T-2}}{1-\rho^2} \\[8pt]
\frac{\rho^2}{1-\rho^2} & \frac{\rho}{1-\rho^2}  & \frac{1}{1-\rho^2} & \cdots &  \frac{\rho^{T-2}}{1-\rho^2} \\[8pt]
\vdots & \vdots & \vdots & \ddots & \vdots \\[8pt]
\frac{\rho^{T-1}}{1-\rho^2} & \frac{\rho^{T-2}}{1-\rho^2} & \frac{\rho^{T-3}}{1-\rho^2} & \cdots & \frac{1}{1-\rho^2}
\end{bmatrix}.</math>
Now having <math>\rho </math> (or an estimate of it), we see that,
 
: <math>\hat{\Theta}=(\mathbf{Z}'\mathbf{\Omega}^{-1}\mathbf{Z})^{-1}(\mathbf{Z}'\mathbf{\Omega}^{-1}\mathbf{Y}), \, </math>
 
where <math>\mathbf{Z}</math> is a matrix of observations on the independent variable (''X''<sub>''t''</sub>, ''t''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;...,&nbsp;''T'') including a vector of ones, <math>\mathbf{Y}</math> is a vector stacking the observations on the dependent variable (''X''<sub>''t''</sub>, ''t''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;...,&nbsp;''T'') and <math>\hat{\Theta} </math> includes the model parameters.
 
==Note==
To see why the initial observation assumption stated by Prais-Winsten (1954) is reasonable, considering the mechanics of general least square estimation procedure sketched above is helpful. The inverse of <math>\mathbf{\Omega} </math> can be decomposed as <math>\mathbf{\Omega}^{-1}=\mathbf{G}'\mathbf{G} </math> with
 
: <math>\mathbf{G} = \begin{bmatrix}
\sqrt{1-\rho^2} & 0 & 0 & \cdots & 0 \\
-\rho & 1 & 0 & \cdots & 0 \\
0 & -\rho & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1
\end{bmatrix}.</math>
A pre-multiplication of model in a matrix notation with this matrix gives the transformed model of Prais-Winsten.
 
==Restrictions==
 
The [[errors and residuals in statistics|error term]] is still restricted to be of an AR(1) type. If <math>\rho</math> is not known, a recursive procedure may be used to make the estimation feasible. See [[Cochrane–Orcutt estimation]].
 
{{More footnotes|date=November 2010}}
 
==References==
* {{Citation | last1=Prais | first1=S. J. |  last2=Winsten| first2=C. B. | title=Trend Estimators and Serial Correlation | year=1954}} [http://cowles.econ.yale.edu/P/ccdp/st/s-0383.pdf]
* Wooldridge, J. (2008) ''Introductory Econometrics: A Modern Approach'', 4th Edition, South-Western Pub. ISBN 0-324-66054-5 (p.&nbsp;435)
 
{{DEFAULTSORT:Prais-Winsten Transformation}}
[[Category:Econometrics]]
[[Category:Regression with time series structure]]

Latest revision as of 02:56, 10 February 2014

Welcome to Rada Cutlery exactly where you will find the finest kitchen knives you can acquire at the best price tag. If you have a finite quantity of revenue, it is far improved to use it to get the most effective chef's and paring knife you can, rather than spread that money more than 6 (or even 8!!) knives of lesser high quality. I've attempted Wustorf, Viking, and International as well, and they're all good as well...while I truly did not like the lightweight International knives. This is before the advent and craze for Japanese knives. They are now a favorite for absolutely everyone in our kitchen.

But some blades, like stain-prone higher-carbon ones, demand special care, says Jonathan Benno of Lincoln Ristorante You can remove rust and other spots with a sabitoru (or eraser) from Korin ($8) Benno also recommends periodically treating high-carbon knives with tsubaki oil ($11 for three.8 ounces ) to stop corrosion. The Osaka-made knives at Brushstroke are rather difficult to source ask for a related style at Korin. There are other that are superb knives.

For the uninitiated, Shun are some of the most well-known kitchen knife forgers in the globe and have years of heritage they even applied to make samurai swords! They have also made a number of diverse steak knives, but none pretty as fine as this set - technically to be referred to as the Shun Elite Steak Knives. It is not only the appearance of these knives that is impressive. Forged from SG-2 steel, these knives are extremely sharp.

My 30+ year old Chicago Cutlery knives are still in fantastic shape, but a Christmas present gave me the chance to get some "high-end" knives. Honestly, I was genuinely skeptical because I really like my CC knives. Here is more on very best cutting boards for well being take a look at our site. But, not being one to look a present horse in the mouth, I researched and eventually bought some excellent knives. I was incredibly picky due to the fact if I was going to spend more on a knife, it had greater be the very best darned knife in the world. The Wusthof knives are heavy, but completely balanced.

One can question the want for designing a kitchen when there are far greater locations in the home which can be produced even a lot more attractive and appealing. Having said that, getting a classy kitchen is as crucial as getting a effectively created lounge or bedroom. Even so, segregating both the areas through helpful use of space will make the kitchen appear additional appealing and will also assist in reducing the mess.

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