Constant process Separate signal & noise Smooth the data: Backward smoother: At any give T, replace the observation yt by a combination of observations at & before T Simple smoother : replace the current observation with the best estimate for How to obtain it? Use the least squares criterion Smoothing

Not Constant process Can we use ? this smoother accumulates more and more data points and gains some sort of inertia So it cannot react to the change Obviously, if the process change, earlier data do not carry the information about the it Use a smoother that disregards the old values and reacts faster to the change Simple moving average

Choice of span N: -If N small :reacts faster to the change - large N:constant process The moving averages will be autocorrelated since 2 successive moving averages contain the same N-1 observations

First order exponential smoothing Idea: give geometrically decreasing weights to the past observations Obtain a smoother that reacts faster to the change However, Adjust the smoother, by multiplying Simple or first order exponential smoother

First-order exponential smoothing: linear combination of the current observation & the smoothed observation at the previous time unit Discount factor : weight on the last observation : weight on the smoothed value of the previous observations

The initial value

2 commonly used estimates of If the changes in the process are expected to occur early & fast If the process at the beginning is constant

The value of As gets closer to 1, & more emphasis is given in the last observations : the smoothed values follow the original values more closely The smoothed values equal to a constant The least smoothed version of the original time series Values recommended Choice of the smoothing constant: 1Subjective: depending of willingness to have fast adaptivity or more rigidity. 2Choice advocated by Brown (inventor of the method): λ = 0.7 3Objective: constant chosen to minimize the sum of squared forecast errors

Use exponential smoothers for model estimation General class of models e.g. constant process If not all observations have equal influence on the sum : introduce weights that geometrically decrease in time Take the derivative The solution For large T

Second order exponential smoothing Uncorrelated with mean 0 & constant variance Use single exponential smoothing: underestimate the actual values Linear trend model Given Simple exponential smoother:biased estimate for the linear trend model Why? bias

Solutions: a) Use a large value Smoothed values very close to the observed: fails to satisfactorily smooth the data b) Use a method based on adaptive updating of the discount factor c) Use second order exponential smoothing

Apply simple exponential smoothing on Second order exponential smoothing Unbiased estimate of y T 2 main issues: choice of initial values for the smoothers& the discount factors Initial values Estimate parameters through least squares

Holt’s Method L t =λy t +(1−α)(L t−1 +T t−1 ) T t =β(L t +L t-1 )+(1−β)T t-1 F t+1 =L t +T t Divide time series into Level and Trend yt = actual value in time t λ = constant-process smoothing constant β = trend-smoothing constant Lt = smoothed constant-process value for period t Tt = smoothed trend value for period t F= forecast value for period t + 1 t = current time period

Use exponential smoothers for forecasting At time T, we wish to forecast observation at time T+1, or at time Constant Process Can be estimated by the first-order exponential smoother Forecast : At time T+1 One-step-ahead forecast error

Our forecast for the next observation is our previous forecast for the current observation plus a fraction of the forecast error we made in forecasting the current observation

Choice of Sum of the squared one-step-ahead forecast errors Calculate for various values of Pick the one that gives the smallest sum of squared forecast errors

Prediction Intervals Constant process first-order exponential smoother 100(1-a/2) percentile of the standard normal distribution Estimate of the standard deviation of the forecast errors

Linear Trend Process In terms of the exponential smoothers

Parameter Estimates 100(1-α/2) prediction intervals for any lead time τ

Estimation of the variance of the forecast errors, σ e 2 Assumptions: model correct & constant in time Apply the model to the historic data & obtain the forecast errors Update the variance of the forecast errors when have more data

Define mean absolute deviation Δ Assuming that the model is correct

Model assessment Check if the forecast errors are uncorrelated Sample autocorrelation function of the forecast errors

Exponential Smoothing for Seasonal data Seasonal Time Series L t : linear trend component S t : seasonal adjustment s: the period of the length of cycles ε t : uncorrelated with mean zero & constant variance σ ε 2 Constraint: Additive Seasonal Model

Forecasting - Start from current observation y T - Update the estimate L T - Update the estimate of β 1 - Update the estimate of S t - τ- step-ahead foreceast

Estimate the initial values of the smoother Use least-squares

Exponential Smoothing for Seasonal data Seasonal Time Series L t : linear trend component S t : seasonal adjustment s: the period of the length of cycles ε t : uncorrelated with mean zero & constant variance σ ε 2 Constraint: Multiplicative Seasonal Model

Forecasting - Start from current observation y T - Update the estimate L T - Update the estimate of β 1 - Update the estimate of S t - τ- step-ahead foreceast

Estimate the initial values of the smoother From the historical data, obtain the initial values