Temp - Revision history

AS: AS moved page Monte-Carlo to Temp: AH request

2017-10-06T12:07:27Z

AS moved page Monte-Carlo to Temp: AH request

2013-02-04T15:54:12Z

‎On a notion of conditionality

2013-02-04T15:53:34Z

‎OLS properties of AR(1) process estimator

2013-02-04T15:50:51Z

‎OLS properties of AR(1) process estimator

2013-02-04T15:50:18Z

‎OLS properties of AR(1) process estimator

2013-02-04T15:49:19Z

‎OLS properties of AR(1) process estimator

2013-02-04T15:47:59Z

‎OLS properties of AR(1) process estimator

2013-02-04T15:46:54Z

‎OLS properties of AR(1) process estimator

2013-02-04T15:45:33Z

‎OLS properties of AR(1) process estimator

2013-02-04T15:43:32Z

‎OLS properties of AR(1) process estimator

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 One outcome that contradicts our “intuition” is the fact that bias in absolute value ''decreases'' after some point. This point depends on the sample size. The intuition for this behavior lies in the notion of non-stationarity. As <math>\alpha</math> further and further increases, stationary series start to behave more and more like non-stationary ones. For non-stationary time series <math>X'X/T^2=\sum y_t^2 /T^2 \Rightarrow C(B(t))</math>, while <math>X'e/T = \sum y_{t-1}e_t/T \Rightarrow C'(B(t))</math>, where <math>C(B(t)),C'(B(t))</math> – finite functionals of Brownian motions. Thus, for <math>|\rho|\rightarrow 1</math>, <math>(X'X)^{-1}</math> will dominate <math>X'y</math> and the bias will decrease. Moreover, for <math>\rho=1</math>, <math>\hat\alpha</math> is a consistent estimator of <math>\alpha</math> even in the case of <math>Cov(x_t,e_t)\ne 0</math>.
-= On a notion of conditionality =
+= On a notion of Conditional Distribution =
 Repeat simulation using a slightly modified version of the main code, i.e.

@@ Line 81: / Line 81: @@
 ## Draw <math>T+T_0</math> observations from <math>N(0,1)</math> with <math>T_0=200</math>, save them in a vector <math>e</math> and assume <math>z_0=0</math>
 ## Construct <math>z_t</math> for <math>t=1</math>: <math>z_t=\alpha z_{t-1}+e_t</math>
-## Repeat 1.a) for <math>t=2,3,\ldots,T+T_0</math>
+## Repeat 1.b) for <math>t=2,3,\ldots,T+T_0</math>
 ## Save the last T observations as vector <math>y</math>

← Older revision		Revision as of 15:50, 4 February 2013
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	[[Image:bias.png\|image\|thumb\|center\|upright=3.0\|alt=Bias of AR(1) Estimation.\|Dependence of Bias on $\alpha$.]]		[[Image:bias.png\|image\|thumb\|center\|upright=3.0\|alt=Bias of AR(1) Estimation.\|Dependence of Bias on $\alpha$.]]

−	[[Image:std.png\|image\|thumb\|center\|upright=4.0\|alt=~~Bias~~ of AR(1) Estimation.\|Dependence of ~~Bias~~ on $\alpha$.]]	+	[[Image:std.png\|image\|thumb\|center\|upright=4.0\|alt=St. deviation of AR(1) Estimation.\|Dependence of std on $\alpha$.]]

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 The next two figures are generated using the code above. To create a smoother picture, we ran simulations for <math>N=10000</math>, <math>\alpha=0,0.05,0.1,0,15,\ldots,0.95,0.99</math>
-[[Image:bias.png|image|thumb|center|upright=4.0|alt=Bias of AR(1) Estimation.|Dependence of Bias on $\alpha$.]]
+[[Image:bias.png|image|thumb|center|upright=3.0|alt=Bias of AR(1) Estimation.|Dependence of Bias on $\alpha$.]]
 [[Image:std.png|image|thumb|center|upright=4.0|alt=Bias of AR(1) Estimation.|Dependence of Bias on $\alpha$.]]

@@ Line 148: / Line 148: @@
 [[Image:bias.png|image|thumb|center|upright=4.0|alt=Bias of AR(1) Estimation.|Dependence of Bias on $\alpha$.]]
-[[Image:std.png|image]]
+[[Image:std.png|image|thumb|center|upright=4.0|alt=Bias of AR(1) Estimation.|Dependence of Bias on $\alpha$.]]
 Some of the findings coincide with our predictions, some of them do not. In particular,

← Older revision	Revision as of 12:07, 6 October 2017
(No difference)