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ECO 374
Jul 9, 2023
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1/12 Lec01 Intro to Time Series Feature of time series Trend - dependence Cycle - regular patterns appearing repeatedly - recession and boom Components Decomposition: Y = S + T + R Moving Avg Smoothing (MAS) - local avg - Typically used for seasonal adj, i.e. Filtering out seasonal variations to est trend-cycle component - Each avg has the same weight (1/m ), no special considerations to specific periods - From -k to k (so m=2k+1 ), goes k periods back and k periods forward - Not suitable for forecasting looking forward, more weights should be giving to recent information Simple exponential smoothing (SES) - Assigns diff weights to diff obs most weights to recent data - L_t = ·y_t + (1- )·L_t-1 , where L is the cycle-trend component and the smoothing constant (0,1) - Can est L_0 using MAS w/ a small fraction of data (e.g., first 10%) and obtain subsequent l recursively w/ SES - Optimal is determined by minimizing SSE for in-sample errors of one-period-ahead forecasts - is assumed constant in this model SES forecasts - Only a smoother not an assump based model, so only average data instead of capturing any pattern - Flat forecast func, limited use beyond very short-term forecasts, but the backbone of dynamic forecasts SES weights - Exponentially decreasing weights as going backward in time - As , the weight curve gets flatter Original form (above) Exponential form Error correction form of SES - rearrange the func into: L_T = L_T-1 + ( y_T - yhat_T|T-1 ) - last one-period ahead forecast error Forecasters' obj: - information set interval/density/point forecast - in this course: Variance for density forecast + point forecast interval Using past data to develop model for future forecasts Stochastic process / time series process - Def., a collection of random variables {Y_t} indexed by time - Each var associated w/ a density func conditional on time t - (Assume some mechanisms and shift it into the future to create forecasts) Time series {y_t} : Outcome of stochastic - One observation (y) for each random variable - Impose assumptions to est E(Y) and Var(Y) Strong Stationarity - Def., if Y for all t has the same density func - Unrealistic Weak Stationarity - More realistic but never perfectly satisfied (Closer to stationarity more accurate forecasts) - First order: same mean - Second order: same mean, var and covariances not dependent on t (?)
Non-stationarity Transformations In most cases, Non-stationary Yt -> first order stationary Y -> second order stationary logY Autocorrelation func (ACF) - The mapping of k to the correlation b/w Y_t and Y_t-k ( = cov/s.d.*s.d.) - For stationary, ACF can be simplified into: - Partial autocorrelation In-sample Model Fit - AIC, BIC - Mostly for cross-sectional Out-of-sample Forecast Error RSM336, RSM434, RSM439, 2*ECO300+ (ECO364,
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