Seasonality

Seasonality - ApEc 8212 Applied Econometrics - Lecture #25...

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ApEc 8212 Applied Econometrics -- Lecture #25 Time Series Analysis IV: Model Selection + Forecasts (Enders, Ch. 2, Sections 7-12) This lecture explains how to estimate ACFs and PACFs to choose the “best” ARMA(p, q) model for a time series. I. Sample Autocorrelations of Stationary Series The first step is to calculate ACF’s and PACF’s and their standard errors. The sample mean and variance of y are: y = (1/T) T t t y 2 ˆ σ = (1/T) T t 2 t ) y y ( The estimated ACF coefficients ρ s , denoted as r s , are: r s = = + = T 1 t 2 t T 1 s t s t t ) y y ( ) y y )( y y ( We want to check whether r s = 0. Box & Jenkins (1976, 1984) showed that, under the null hypothesis that ρ s = 0, and assuming that y t is stationary and ε t ~ N[0, σ ε 2 ]: 1
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Var(r s ) = 1/T for s = 1 = (1 + 2 Σ = 1 1 s j r j 2 )/T for s > 1 Asymptotically, r s will be normally distributed and have a mean of zero under the null that ρ s = 0. For PACF coefficients, under the null hypothesis φ ss = 0, the “approximate” variance of the estimate of φ ss is T -1 . Usually, if a ACF or PACF coefficient = 0, then all higher order coefficients also = 0. Don’t test each coefficient individually; even if all = 0, if your significance level is 5% there is a 1/20 chance that you reject the null. It is best to use a joint test. Box & Pierce proposed the ‘Q-statistic’: Q = T Σ = s k 1 r k 2 Under the null hypothesis that all s ACFs = 0, this statistic is asymptotically distributed as χ 2 (s). Ljung and Box modified this to give it better small sample performance: Q = T(T+2) Σ = s k 1 [r k 2 /(T-k)] Both tests can be applied to residuals from an estimated ARMA(p, q) model to see if they are “white noise”. 2
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t and y in the r s formula with the estimated residuals. Of course, autocorrelated residuals contradict the white noise assumption. The degrees of freedom for the χ 2 test is lower (s - p - q); the residuals are from an estimated model, this “uses up” p+q degrees of freedom. Model Selection Criteria When estimating ARMA processes maybe you should, to be cautious, allow for many non-zero a’s and β ’s. Yet allowing for unnecessary a’s and β ’s will reduce the precision of the estimated a’s and β ’s that really are 0, reducing the precision of your forecasts. Two ‘model selection’ criteria are the Akaike Information Criterion (AIC) and the Schwartz Bayesian Criterion (SBC) : AIC = T × ln(residual sum of squares) + 2n SBC = T × ln(residual sum of squares) + n × ln(T) T is “effective sample size”, i.e. subtracting observations deleted due to lags, and n = parameters estimated. Choose the model with the smallest AIC or SBC. Both are increasing in the parameters (n). SBC puts a bigger penalty (ln(T) > 2 for T > 7). SBC has better asymptotic properties; AIC may have better small sample properties. 3
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This note was uploaded on 02/24/2010 for the course ECON 570 taught by Professor Staff during the Fall '08 term at UNC.

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Seasonality - ApEc 8212 Applied Econometrics - Lecture #25...

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