slides_arma_v220171111092323.pdf

slides_arma_v220171111092323.pdf - Univariate time-series...

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Univariate time-series analysis Carlo Favero and Celso Brunetti Carlo Favero and Celso Brunetti Univariate time-series analysis 1 / 47
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Time-Series Time-series is a sequence { x 1 , x 2 , ..., x T } or { x t } , t = 1, ..., T , where t is an index denoting the period in time in which x occurs. We shall treat x t as a random variable; hence, a time-series is a sequence of random variables ordered in time. Such a sequence is known as a stochastic process. The probability structure of a sequence of random variables is determined by the joint distribution of a stochastic process. The simplest possible probability model for such a joint distribution is: x t = α + e t , e t n . i . d . ( 0, σ 2 e ) , i.e., x t is normally independently distributed over time with constant variance and mean equal to α . In other words, x t is the sum of a constant and a white-noise process. If a white-noise process were a proper model for financial time-series, forecasting would not be very interesting as the best forecast for the moments of the relevant time series would be their unconditional moments. Carlo Favero and Celso Brunetti Univariate time-series analysis 2 / 47
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Better models The model: x t = α + e t , e t n . i . d . ( 0, σ 2 e ) , ˆ α = 1 T T i = t x t , ˆ σ 2 e = T i = t 1 T ( x t - ˆ α ) 2 Reflect the traditional approach to portfolio allocation, but it does not reflect the data. At high frequency (daily, intra-day) the variance is not constant and predictable, at low frequency returns are persistent and predictable. To construct more realistic models, we concentrate on univariate models first to consider then multivariate models. Carlo Favero and Celso Brunetti Univariate time-series analysis 3 / 47
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Better models 1960 1970 1980 1990 2000 2010 -0.20 -0.05 0.10 US 1-month nominal stock market returns time return actual returns white noize 1960 1970 1980 1990 2000 2010 -1 1 2 3 4 5 US 10-year nominal stock market returns time return actual returns white noize While the CER gives a plausible representation for the 1-month returns, the behaviour over time of the YTM of the 10-Year returns does not resemble at all that of the simulated data. Carlo Favero and Celso Brunetti Univariate time-series analysis 4 / 47
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ARMA modelling A more general and more flexible class of models emerges when combinations of e t are used to model x t . We concentrate on a class of models created by taking linear combinations of the white noise, the autoregressive moving average (ARMA) models: AR ( 1 ) : x t = ρ x t - 1 + e t , MA ( 1 ) : x t = e t + θe t - 1 , AR ( p ) : x t = ρ 1 x t - 1 + ρ 2 x t - 2 + ... + ρ p x t - p + e t , MA ( q ) : x t = e t + θ 1 e t - 1 + ... + θ q e t - q , ARMA ( p , q ) : x t = ρ 1 x t - 1 + ... + ρ p x t - p + θ 1 e t - 1 + ... + θ q e t - q . Carlo Favero and Celso Brunetti Univariate time-series analysis 5 / 47
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An Illustration 1960 1970 1980 1990 2000 2010 0 1 2 3 4 5 US 10-year nominal return time return 1960 1980 2000 -5 0 5 10 AR(1) with phi = 0.99 time return 1960 1980 2000 -3 -2 -1 0 1 2 AR(1) with phi = 0 time return Carlo Favero and Celso Brunetti Univariate time-series analysis 6 / 47
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Analysing time-series models To illustrate empirically all fundamentals we consider a specific member of the ARMA family, the AR model with drift, x t = ρ 0 + ρ 1 x t - 1 + e t , (1) e t
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