Time Varying Parameters Slides

# Time Varying Parameters Slides - NBER Summer Institute...

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Lecture 2 - 1, July 21, 2008 NBER Summer Institute Minicourse – What’s New in Econometrics: Time Series Lecture 2: July 14, 2008 The Functional Central Limit Theorem and Testing for Time Varying Parameters

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Lecture 2 - 2, July 21, 2008 Outline 1. FCLT 2. Overview of TVP topics (Models, Testing, Estimation) 3. Testing problems 4. Tests
Lecture 2 - 3, July 21, 2008 1. FCLT The Functional Central Limit Theorem Problem: Suppose ε t ~ iid(0, 2 σ ) (or weakly correlated with long-run variance 2 ), x t = 1 t i i = , and we need to approximate the distribution of a function of ( x 1 , x 2 , x 3 , … x T ), say 2 1 T t t x = . Solution: Notice x t = 1 t i i = = x t 1 + t . This suggests an approximation based on a normally distributed (CLT for first equality) random walk (second equality). The tool used for the approximation is the Functional Central Limit Theorem.

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Lecture 2 - 4, July 21, 2008 Some familiar notions 1. Convergence in distribution or “weak convergence”: ξ T , T = 1, 2, … is a sequence of random variables. T d means that the probability distribution function (PDF) of T converges to the PDF of . As a practical matter this means that we can approximate the PDF of T using the PDF of when T is large. 2. Central Limit Theorem: Let ε t be a mds(0, 2 σ ) with 2+ δ moments and T = 1 1 T t t T = . Then T d ~ N(0, 2 ). 3. Continuous mapping theorem. Let g be a continuous function and T d , then g ( T ) d g ( ). (Example T is the usual t -statistic, and T d ~ N(0, 1), then 222 1 d T ξχ . These ideas can be extended to random functions:
Lecture 2 - 5, July 21, 2008 A Random Function: The Wiener Process, a continuous-time stochastic process sometimes called Standard Brownian Motion that will play the role of a “standard normal” in the relevant function space. Denote the process by W ( s ) defined on s א [0,1] with the following properties 1. W (0) = 0 2. For any dates 0 t 1 < t 2 < … < t k 1, W ( t 2 ) W ( t 1 ), W ( t 3 ) W ( t 4 ), … , W ( t k ) W ( t k 1 ) are independent normally distributed random variables with W ( t i ) W ( t i 1 ) ~ N(0, t i t i 1 ). 3. Realizations of W( s ) are continuous w.p. 1. From (1) and (2), note that W (1) ~ N(0,1).

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Lecture 2 - 6, July 21, 2008 Another Random Function: Suppose ε t ~ iidN(0,1), t = 1, … , T , and let ξ T ( s ) denote the function that linearly interpolates between the points T ( t/T ) = 1 1 t i i T = . Can we use W to approximate the probability law of T ( s ) if T is large? More generally, we want to know whether the probability distibution of a random function can be well approximated by the PDF of another (perhaps simpler, maybe Gaussian) function when T is large. Formally, we want to study weak convergence on function spaces. Useful References: Hall and Heyde (1980), Davidson (1994), Andrews (1994)
Lecture 2 - 7, July 21, 2008 Suppose we limit our attention to continuous functions on s א [0,1] (the space of such functions is denoted C[0,1]), and we define the distance between two functions, say x and y as d ( x,y ) = sup 0 s 1 | x ( s ) – y ( s )|.

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Time Varying Parameters Slides - NBER Summer Institute...

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