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# SlidesChapter4 - Time Series Models 1 TIME SERIES MODELS...

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Time Series Models 1 TIME SERIES MODELS Time Series time series = sequence of observations Example: daily returns on a stock multivariate time series is a sequence of vectors of observations Example returns from set of stocks. statistical models for univariate times series widely used in finance to model asset prices in OR to model the output of simulations in business for forecasting

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Time Series Models 2 Stationary Processes often a time series has same type of random behavior from one time period to the next outside temperature: each summer is similar to the past summers interest rates and returns on equities stationary stochastic processes are probability models for such series process stationary if behavior unchanged by shifts in time
Time Series Models 3 a process is weakly stationary if its mean, variance, and covariance are unchanged by time shifts thus X 1 , X 2 , . . . is a weakly stationary process if E ( X i ) = μ (a constant) for all i Var( X i ) = σ 2 (a constant) for all i Corr( X i , X j ) = ρ ( | i - j | ) for all i and j for some function ρ

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Time Series Models 4 the correlation between two observations depends only on the time distance between them (called the lag ) example: correlation between X 2 and X 5 = correlation between X 7 and X 10
Time Series Models 5 ρ is the correlation function Note that ρ ( h ) = ρ ( - h ) covariance between X t and X t + h is denoted by γ ( h ) γ ( · ) is called the autocovariance function Note that γ ( h ) = σ 2 ρ ( h ) and that γ (0) = σ 2 since ρ (0) = 1 many financial time series not stationary but the changes in these time series may be stationary

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Time Series Models 6 Weak White Noise simplest example of stationary process no correlation X 1 , X 2 , . . . is WN( μ, σ 2 ) if E ( X i ) = μ for all i Var( X i ) = σ 2 (a constant) for all i Corr( X i , X j ) = 0 for all i 6 = j if X 1 , X 2 . . . IID normal then process is Gaussian white noise process
Time Series Models 7 weak white noise process is weakly stationary with ρ (0) = 1 ρ ( t ) = 0 if t 6 = 0 so that γ (0) = σ 2 γ ( t ) = 0 if t 6 = 0

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Time Series Models 8 White noise WN is uninteresting in itself but is the building block of important models It is interesting to know if a financial time series, e.g., of net returns, is WN.
Time Series Models 9 Estimating parameters of a stationary process observe y 1 , . . . , y n estimate μ and σ 2 with Y and s 2 estimate autocovariance with b γ ( h ) = n - 1 n - h X j =1 ( y j + h - y )( y j - y ) estimate ρ ( · ) with b ρ ( h ) = b γ ( h ) b γ (0) , h = 1 , 2 , . . . infinite number of parameters (bad)

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Time Series Models 10 AR(1) processes time series models with correlation built from WN in AR processes y t is modeled as a weighted average of past observations plus a white noise “error” AR(1) is simplest AR process ² 1 , ² 2 , . . . are WN(0, σ 2 ² ) y 1 , y 2 , . . . is an AR(1) process if y t - μ = φ ( y t - 1 - μ ) + ² t (1) for all t
Time Series Models 11 From previous page: y t - μ = φ ( y t - 1 - μ ) + ² t Only three parameters: μ – mean

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SlidesChapter4 - Time Series Models 1 TIME SERIES MODELS...

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