321_09_slides16

# 321_09_slides16 - Introduction to Time Series Chapters 10...

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Introduction to Time Series Chapters 10 + 11 + 12 Part III

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Which Type of Series are we dealing with? Need to model dependence to find out whether weakly dependent or instead highly persistent time series
Modelling Dependence with an AR(1) Specification: E( ε t | y t-1 , y t-2 , …)=0 The AR(1) model expresses what we don’t know in terms of what we do know (based on past). 3 t 0 1 t 1 t AR(1) : Y Y T 1 T 0 1 T T 1 T 0 1 T E Y |Y Y E |Y Y

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Example: Efficient Market Hypothesis In a context of a large sample y t is the weekly percentage return on the New York Stock Exchange composite index. EMH: information observable to the market prior to week t should not help to predict the return during week t: The market (sellers and buyers) has already taken this past information into account when trading and the price on the market reflects that information E(y t | y t-1 y t-2 ...) = E(y t ) 4
Example: Efficient Market Hypothesis To test for this, specify an AR(1) as the alternative model and test whether H 0 : β 1 = 0 against H a : β 1 ≠ 0 Use NYSE.RAW 5

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Example: Efficient Market Hypothesis 6 . regress return return_1 Source | SS df MS Number of obs = 689 -------------+------------------------------ F( 1, 687) = 2.40 Model | 10.6866231 1 10.6866231 Prob > F = 0.1218 Residual | 3059.73817 687 4.45376735 R-squared = 0.0035 -------------+------------------------------ Adj R-squared = 0.0020 Total | 3070.42479 688 4.46282673 Root MSE = 2.1104 ------------------------------------------------------------------------------ return | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- return_1 | .0588984 .0380231 1.55 0.122 -.0157569 .1335538 _cons | .179634 .0807419 2.22 0.026 .0211034 .3381646 ------------------------------------------------------------------------------
Random Walks The case of 1 = 1 deserves special attention because of it's importance in economic data series. A random walk is an AR(1) model with 1 = 1 t 1 t o t Y Y ε β

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Random Walks The intercept, 0 , is called the drift parameter for the random walk. Let's first consider the case of 0 = 0. This called a random walk with zero drift: t 1 t t Y Y ε A random walk with zero drift “meanders” around zero with no particular trend. However, it can take very long “excursion” away from zero. These “excursions” can look like trends until the series turns back toward 0.
Random Walks The random “walk” get its name from the idea of a random walker on the number line. A random walker is someone who has an equal chance of taking a step forward or a step backward. The size of the steps are random as well. To see this, it is very useful to re-express the

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321_09_slides16 - Introduction to Time Series Chapters 10...

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