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Model specification (identification) We already know about the sample autocorrelation function (SAC): Properties: Not unbiased (since a ratio between two random variables) Bias decreases with n Variance complicated, common to use general large-sample results

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Large-sample results (asymptotics): For large n the random vector has an approximate multivariate normal distribution with zero mean vector and covariance matrix ( c ij ) where This gives that Var (r k ) 0 as n does not diminish as n

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Hence, the distribution of r k will depend on the correlation structure of Y t and accordingly on the model behind (i.e. if it is and AR(1), an ARMA(2,1) etc.) For an AR(1), i.e. Y t = Y t – 1 + e t For an MA(q) i.e. not dependent on k after the qth lag i.e. not dependent on k for large lags For white noise

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Partial autocorrelation function Describes the “specific” part of the correlation between Y t and Y t – k that is not due to successive serial correlations between the variables Y t – 1, Y t – 2, …, Y t – k. Partial correlations are used for other types of data as well (for instance in linear models of cross-sectional data. Patterns For an AR(p)-process, k cuts off after lag p (i.e. the same type of behaviour like k has for an MA(q)-process For an MA(q)-process k shows approximately the same pattern as does k for an AR(p)-process

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Estimation from data, Sample Partial Autocorrelation function (SPAC): No explicit formula, estimation has to be made recursively Properties of SPAC: More involved, but for an AR(p)-process SPAC-values for lags greater than p are approximately normally distributed with zero mean and variance 1/n

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Extended Autocorrelation function (EACF) One (of several) tool to improve the choice of orders of ARMA(p, q)-processes. Very clear as a theoretical function, but noisy when estimated on series not too long. AR, MA or ARMA?

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No pattern at all?

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EACF table for Y AR/MA o o o o o o o o o o o o o o 1 o o o o o o o o o o o o o o 2 x o o o o o o o o o o o o o 3 o o x o o o o o o o o o o o 4 x o o o o o o o o o o o o o 5 x x o o o o o o o o o o o o 6 o x o o o x o o o o o o o o 7 o o o o o x o o o o o o o o ARMA(0,0) or ARMA(1,0)? True process: Y t = Y t – 1 + e t – 0.1 e t – 1

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Model selection from more analytical tools Dickey-Fuller Unit-Root test H 0 : The process Y t is difference non-stationary ( Y t is stationary) H a : The process Y t is stationary Augmented Dickey-Fuller test statistic (ADF): If =1 (difference non-stationary)

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Fit the model and test H 0 : a = 0 (difference non-stationary) vs. H a : a < 0 (stationary) using the test statistic However, not t-distributed under H 0. Another sampling distributions has been derived and tables (programmed in R)

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Akaike’s criteria For an ARMA(p,q)-process, let k = p + q + 1 and find the values of the parameters p, q, 1, …, p, 1, …, q of the model that minimizes –2 log{max L(p, q, 1, …, p, 1, …, q )} + 2 k AIC [ Akaike’s Information Criterion]. Works well when the true process have (at least one) infinite order –2 log{max L(p, q, 1, …, p, 1, …, q )} + k log(n) BIC [ (Schwarz) Bayesian Information Criterion]. Works well when we “know” that the true process is a finite-order ARMA(p,q)

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