Lecture 2 - Lecture 2:Statistical inference and testing Xin...

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Lecture 2:Statistical inference and testing Xin T Tong Sunday 21 st August, 2016 Xin Tong Statistics 1 / 46

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Last time Your grader: KOU Han, email: kouhan[email protected] There will be webcasts. Asset return: time series with investment value. Risk and uncertainty: modeled by random variables Quantification: moments such as mean, variance, e.t.c. Two errors The portfolio formula should be R p,t = X w i P i,t P i,t - 1 - 1 = w i R i,t When X ∼ N ( μ X , Σ X ), Y = AX ∼ N ( X , A Σ X A T ). Diagonal A Σ X A T , components of Y are independent. Xin Tong Statistics 2 / 46
Last time Your grader: KOU Han, email: [email protected] There will be webcasts. Asset return: time series with investment value. Risk and uncertainty: modeled by random variables Quantification: moments such as mean, variance, e.t.c. Two errors The portfolio formula should be R p,t = X w i P i,t P i,t - 1 - 1 = w i R i,t When X ∼ N ( μ X , Σ X ), Y = AX ∼ N ( X , A Σ X A T ). Diagonal A Σ X A T , components of Y are independent. Xin Tong Statistics 2 / 46

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Last time Your grader: KOU Han, email: [email protected] There will be webcasts. Asset return: time series with investment value. Risk and uncertainty: modeled by random variables Quantification: moments such as mean, variance, e.t.c. Two errors The portfolio formula should be R p,t = X w i P i,t P i,t - 1 - 1 = w i R i,t When X ∼ N ( μ X , Σ X ), Y = AX ∼ N ( X , A Σ X A T ). Diagonal A Σ X A T , components of Y are independent. Xin Tong Statistics 2 / 46
Financial time series modeling Models can always be made, but leave important questions: How to find the model parameters. How accurate are these results. How to convince others why these results are reliable. These questions are answered by statistics. Xin Tong Statistics 3 / 46

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Financial time series modeling Models can always be made, but leave important questions: How to find the model parameters. How accurate are these results. How to convince others why these results are reliable. These questions are answered by statistics. Xin Tong Statistics 3 / 46
Population and samples Statistics terms Population : objects or people to be studied e.g. QF5210 students. Parameter (truth): a characteristic θ of population e.g. avg prior knowledge of stat. Sample : a subset with observation, X 1 , . . . , X T e.g. students did the first assessment. Statistics : a function of samples ˆ θ ( X 1 , X 2 , . . . X T ) e.g. sample average score of assessment. We use ˆ θ to estimate the true θ . This is called inference . In frequencists’ point of view, θ is of unknown but deterministic value. Xin Tong Statistics 4 / 46

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Random sampling In most applications, samples are drawn randomly. This makes the estimator a random variable, with random performance. Quantification of the confidence and testify the result. Simplest i.i.d. assumption: X 1 , X 2 , . . . , independently follow a population distribution μ.
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