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## STAT 497 LECTURE NOTES 7

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**STAT 497LECTURE NOTES 7**FORECASTING**FORECASTING**• One of the most important objectives in time series analysis is to forecast its future values. It is the primary objective of modeling. • ESTIMATION (tahmin)the value of an estimator for a parameter. • PREDICTION (kestirim)the value of a r.v. when we use the estimates of the parameter. • FORECASTING (öngörü)the value of a future r.v. that is not observed by the sample.**FORECASTING FROM AN ARMA MODEL**THE MINIMUM MEAN SQUARED ERROR FORECASTS Observed time series, Y1, Y2,…,Yn. n: the forecast origin Observed sample Y1 Y2 ………………….. Yn Yn+1? Yn+2?**FORECASTING FROM AN ARMA MODEL**• The stationary ARMA model for Yt is or • Assume that we have data Y1, Y2, . . . , Ynand we want to forecast Yn+l (i.e., l steps ahead from forecast origin n). Then the actual value is**FORECASTING FROM AN ARMA MODEL**• Considering the Random Shock Form of the series**FORECASTING FROM AN ARMA MODEL**• Taking the expectation of Yn+l , we have where**FORECASTING FROM AN ARMA MODEL**• The forecast error: • The expectation of the forecast error: • So, the forecast in unbiased. • The variance of the forecast error:**FORECASTING FROM AN ARMA MODEL**• One step-ahead (l=1)**FORECASTING FROM AN ARMA MODEL**• Two step-ahead (l=2)**FORECASTING FROM AN ARMA MODEL**• Note that, • That’s why ARMA (or ARIMA) forecasting is useful only for short-term forecasting.**PREDICTION INTERVAL FOR Yn+l**• A 95% prediction interval for Yn+l (l steps ahead) is For one step-ahead the simplifies to For two step-ahead the simplifies to • When computing prediction intervals from data, we substitute estimates for parameters, giving approximate prediction intervals**REASONS NEEDING A LONG REALIZATION**• Estimate correlation structure (i.e., the ACF and PACF) functions and get accurate standard errors. • Estimate seasonal pattern (need at least 4 or 5 seasonal periods). • Approximate prediction intervals assume that parameters are known (good approximation if realization is large). • Fewer estimation problems (likelihood function better behaved). • Possible to check forecasts by withholding recent data . • Can check model stability by dividing data and analyzing both sides.**REASONS FOR USING A PARSIMONIOUS MODEL**• Fewer numerical problems in estimation. • Easier to understand the model. • With fewer parameters, forecasts less sensitive to deviations between parameters and estimates. • Model may applied more generally to similar processes. • Rapid real-time computations for control or other action. • Having a parsimonious model is less important if the realization is large.**EXAMPLES**• AR(1) • MA(1) • ARMA(1,1)**UPDATING THE FORECASTS**• Let’s say we have n observations at time t=n and find a good model for this series and obtain the forecast for Yn+1, Yn+2and so on. At t=n+1, we observe the value of Yn+1. Now, we want to update our forecasts using the original value of Yn+1 and the forecasted value of it.**UPDATING THE FORECASTS**The forecast error is We can also write this as**UPDATING THE FORECASTS**n=100**FORECASTS OF THE TRANSFORMED SERIES**• If you use variance stabilizing transformation, after the forecasting, you have to convert the forecasts for the original series. • If you use log-transformation, you have to consider the fact that**FORECASTS OF THE TRANSFORMED SERIES**• If X has a normal distribution with mean and variance 2, • Hence, the minimum mean square error forecast for the original series is given by**MOVING AVERAGE AND EXPONENTIAL SMOOTHING**• This is a forecasting procedure based on a simple updating equations to calculate forecasts using the underlying pattern of the series. Not based on ARIMA approach. • Recent observations are expected to have more power in forecasting values so a model can be constructed that places more weight on recent observations than older observations.**MOVING AVERAGE AND EXPONENTIAL SMOOTHING**• Smoothed curve (eliminate up-and-down movement) • Trend • Seasonality**SIMPLE MOVING AVERAGES**• 3 periods moving averages Yt = (Yt-1 + Yt-2 + Yt-3)/3 • Also, 5 periods MA can be considered.**SIMPLE MOVING AVERAGES**• One can impose weights and use weighted moving averages (WMA). EgYt = 0.6Yt-1+ 0.3Yt-2+ 0.1Yt-2 • How many periods to use is a question; more significant smoothing-out effect with longer lags. • Peaks and troughs (bottoms) are not predicted. • Events are being averaged out. • Since any moving average is serially correlated, any sequence of random numbers could appear to exhibit cyclical fluctuation.**SIMPLE MOVING AVERAGES**• Exchange Rates: Forecasts using the SMA(3) model**SIMPLE EXPONENTIAL SMOOTHING (SES)**• Suppressing short-run fluctuation by smoothing the series • Weighted averages of all previous values with more weights on recent values • No trend, No seasonality**SIMPLE EXPONENTIAL SMOOTHING (SES)**• Observed time series Y1, Y2, …, Yn • The equation for the model is where : the smoothing parameter, 0 1 Yt: the value of the observation at time t St: the value of the smoothed obs. at time t.**SIMPLE EXPONENTIAL SMOOTHING (SES)**• The equation can also be written as • Then, the forecast is**SIMPLE EXPONENTIAL SMOOTHING (SES)**• Why Exponential?: For the observed time series Y1,Y2,…,Yn, Yn+1 can be expressed as a weighted sum of previous observations. where ci’s are the weights. • Giving more weights to the recent observations, we can use the geometric weights (decreasing by a constant ratio for every unit increase in lag).**SIMPLE EXPONENTIAL SMOOTHING (SES)**• Then, St+1 St**SIMPLE EXPONENTIAL SMOOTHING (SES)**• Remarks on (smoothing parameter). • Choose between 0 and 1. • If = 1, it becomes a naive model; if is close to 1, more weights are put on recent values. The model fully utilizes forecast errors. • If is close to 0, distant values are given weights comparable to recent values. Choose close to 0 when there are big random variations in the data. • is often selected as to minimize the MSE.**SIMPLE EXPONENTIAL SMOOTHING (SES)**• Remarks on (smoothing parameter). • In empirical works, 0.05 0.3 commonly used. Values close to 1 are used rarely. • Numerical Minimization Process: • Take different values ranging between 0 and 1. • Calculate 1-step-ahead forecast errors for each . • Calculate MSE for each case. • Choose which has the min MSE.**SIMPLE EXPONENTIAL SMOOTHING (SES)**• EXAMPLE: • Calculate this for =0.2, 0.3,…,0.9, 1 and compare the MSEs. Choose with minimum MSE**SIMPLE EXPONENTIAL SMOOTHING (SES)**• Some softwares automatically chooses the optimal using the search method or non-linear optimization techniques. INITIAL VALUE PROBLEM • Setting S1to Y1 is one method of initialization. • Take the average of, say first 4 or 5 observations and use this as an initial value.**DOUBLE EXPONENTIAL SMOOTHING OR HOLT’S EXPONENTIAL**SMOOTHING • Introduce a Trend factor to the simple exponential smoothing method • Trend, but still no seasonality SES + Trend = DES • Two equations are needed now to handle the trend. Trend term is the expected increase or decrease per unit time period in the current level (mean level)**HOLT’S EXPONENTIAL SMOOTHING**• Twoparameters : = smoothingparameter = trend coefficient • h-step ahead forecast at time t is • Trend prediction is added in the h-step ahead forecast. Current level Current slope**HOLT’S EXPONENTIAL SMOOTHING**• Now, we have two updated equations. The first smoothing equation adjusts St directly for the trend of the previous period Tt-1 by adding it to the last smoothed value St-1. This helps to bring Stto the appropriate base of the current value. The second smoothing equation updates the trend which is expressed as the difference between last two values.**HOLT’S EXPONENTIAL SMOOTHING**• Initial value problem: • S1 is set to Y1 • T1=Y2Y1or (YnY1)/(n1) • and can be chosen as the value between 0.02< ,<0.2 or by minimizing the MSE as in SES.**HOLT’S EXPONENTIAL SMOOTHING**• Example: (use = 0.6, =0.7; S1= 4, T1= 1)**HOLT-WINTER’S EXPONENTIAL SMOOTHING**• Introduce both Trend and Seasonality factors • Seasonality can be added additively or multiplicatively. • Model (multiplicative):**HOLT-WINTER’S EXPONENTIAL SMOOTHING**Here, (Yt /St) captures seasonal effects. s = # of periods in the seasonal cycles (s = 4, for quarterly data) Three parameters : = smoothing parameter = trend coefficient = seasonality coefficient**HOLT-WINTER’S EXPONENTIAL SMOOTHING**• h-step ahead forecast • Seasonal factor is multiplied in the h-step ahead forecast • , and can be chosen as the value between 0.02< ,,<0.2 or by minimizing the MSE as in SES.**HOLT-WINTER’S EXPONENTIAL SMOOTHING**• To initialize Holt-Winter, we need at least one complete season’s data to determine the initial estimates of It-s. • Initial value:**HOLT-WINTER’S EXPONENTIAL SMOOTHING**• For the seasonal index, say we have 6 years and 4 quarter (s=4). STEPS TO FOLLOW STEP 1: Compute the averages of each of 6 years.**HOLT-WINTER’S EXPONENTIAL SMOOTHING**• STEP 2: Divide the observations by the appropriate yearly mean.**HOLT-WINTER’S EXPONENTIAL SMOOTHING**• STEP 3: The seasonal indices are formed by computing the average of each row such that