Decision 411: Class 4

Transcription

1 Decision 411: Class 4 Non-seasonal averaging & smoothing models Simple moving average (SMA) model Simple exponential smoothing (SES) model Linear exponential smoothing (LES) model Combining seasonal adjustment with non-seasonal smoothing Winters seasonal smoothing model Guidelines for future HW writeups Presentation should stand on its own (SG files are mainly just for audit trail) What s the bottom line? (forecast, trend, key drivers?) Clearly define the variables (units, dates, transformations, etc.) used in the analysis Use bullet points for key observations & findings Use tables to present key numbers (forecasts & CI s) Embed the most important chart(s), with annotations Show where the numbers came from Explain your model s assumptions in layman s terms 1

2 Averaging & smoothing models Today s topics Later: ARIMA models We ll meet ARIMA later in the course, but briefly, an ARIMA (p,d,q) model is like a regression model in which the dependent variable is a d-order difference of the input variable, and the independent variables are p lagged values of the dependent variable (AR terms) and/or q lagged values of the forecast errors (MA terms), plus an optional constant term. Many of the averaging & smoothing models are special cases, e.g., an ARIMA(0,1,1) model is an SES model. p = # AR terms (lags of dependent variable) q = # MA terms (lags of errors) d = order of differencing of input variable 2

3 Averaging & smoothing models The problem: sometimes nonseasonal (or seasonally adjusted) data appears to be locally stationary with a time-varying mean The mean (constant) model doesn t track changes in the mean, has positively autocorrelated errors The random walk model may not perform well either in this situation: it oversteers, picks up too much noise in the data, and yields negatively correlated errors Example: series X Mean (constant) model yields positively autocorrelated errors... doesn t react to changes in the local mean...rmse = Constant mean = Autocorrelations Residual Autocorrelations for X Constant mean = lag No reaction to local changes in data Strong positive autocorrelation at lag 1 3

4 Example, continued Random walk model for series X yields negatively autocorrelated errors... overreacts to changes... RMSE=122 not any better! Random walk Autocorrelations Residual Autocorrelations for X Random walk lag Over-reaction to local changes in data (always 1 period too late) Strong negative autocorrelation at lag 1 A solution: Use a model that averages or smooths the recent data to filter out some of the noise and estimate the local mean, such as the Simple Moving Average (SMA) model: Yt 1 + Y t Y Ŷ t m t = m i.e., just average the last m observed values. 4

5 Properties of SMA model Average age of the data in the forecast is (m+1)/2 Yt 1 + Y t Y Ŷ t m t = m m=3 avg. age = 2 m=5 avg. age = 3 m=9 avg. age = 5, etc. (m+1)/2 is midway between 1 period old and m periods old hence it lags behind turning points by (m+1)/2 periods Properties of SMA, continued Long-term forecasts = horizontal straight line (=simple average of last few values) Confidence limits??? No theory!! Works well on highly irregular data: no data point receives more weight than others, so it s relatively robust against outliers Can also be tapered for even greater robustness 5

6 Example, continued SMA with m=3 (average age=2) yields RMSE=104 (significantly better!) and less negative autocorrelation (50% confidence limits are shown here, but don t trust them: they are based on the assumption of the mean remaining fixed at the latest value) Simple moving average of 3 terms Forecasts lag behind turning point by about 2 periods Autocorrelations % confidence limits (?) -1 Residual Autocorrelations for X Simple moving average of 3 terms lag No autocorrelation at lag 1 Example, continued SMA with m=5 (average age=3) yields RMSE=102 (very slightly better), smoother forecasts, slight positive autocorrelation in errors 50% confidence limits shown Simple moving average of 5 terms Autocorrelations Residual Autocorrelations for X Simple moving average of 5 terms lag Forecasts lag behind turning point by about 3 periods Slight positive autocorrelation at lag 1 6

7 Example, continued SMA with m=9 (average age=5) yields RMSE=104 (slightly worse), more positive autocorrelation in errors Simple moving average of 9 terms Forecasts lag behind turning point by about 5 periods Autocorrelations Residual Autocorrelations for X Simple moving average of 9 terms lag More positive autocorrelation at lag 1 Example, continued SMA with m=19 (average age=10) yields RMSE=118 (significantly worse), very smooth forecasts, much more positive autocorrelation Simple moving average of 19 terms Forecasts lag behind turning point by about 10 periods Autocorrelations Residual Autocorrelations for X Simple moving average of 19 terms lag Strong positive autocorrelation at lag 1 7

8 Smoothness vs. responsiveness Note that the more we smooth the data, the more clearly we see the signal stand out. But...greater clarity comes at the expense of getting the news later. If we want our forecasting model to respond quickly to changes, it will also pick up false alarms due to noise in the data. Conclusions For a time series with a randomly varying local mean, the SMA model may outperform both the mean model and the random walk model It allows us to strike a balance between averaging over too much past data or too little past data. However... 8

9 Shortcomings of SMA model It s hard to optimize the number of terms (m), because it is a discrete parameter... you must use trial and error. Intuitively, you should not equally weight the last m observations when computing the average... it would be better to discount the older data in a gradual fashion. These observations motivate... Brown s Simple Exponential Smoothing Let: α = smoothing constant S t = smoothed series at period t Recursive smoothing formula: S t = αy t + (1 α) S t-1 Forecast for next period = current smoothed value: Y ˆ = t+1 S t 9

10 ˆ t+ 1 Y Mathematically equivalent formulas for SES forecasts = αy Yˆ Yˆ 1 = + αe t + (1 α) Yˆ t forecast=interpolation between previous forecast and previous observation forecast=previous forecast plus t+ t t fraction α of previous error: et = Yt Yˆ t ˆ t+ 1 Y = Y (1 α) e t t forecast=previous observation minus fraction 1-α of previous error Yˆ Mathematically equivalent formulas for SES forecasts, continued Last but not least: 2 3 t+ t 1 = α[ Yt + (1 α) Yt 1 + (1 α) Yt 2 + (1 α) Y ] forecast = exponentially weighted moving average of all past observations or in other words, a discounted moving average with a discount factor of 1-α per period 10

11 Properties of SES model SES uses a smoothing parameter (α) which is continuously variable, so it is easily optimized by least squares If α = 1, SES random walk model If α = 0, SES constant model Average age of data in SES forecast is 1/α Examples: α = 0.5 avg. age = 2 α = 0.2 avg. age = 5 α = 0.1 avg. age = 10, etc. Properties of SES, continued For a given average age, SES is somewhat superior to SMA because it places relatively more weight on the most recent observation Hence it is slightly more "responsive" to changes occuring in the recent past. Caveat: it is also more sensitive to recent outliers than the SMA model-- not so good for messy data. 11

12 SMA (m=9) vs. SES (α=0.2) average age = 5 for both models SES weights are larger than SMA weights at first few lags, then gradually decline to zero Lag SMA weight SES weight SMA weights are 1/9 on first 9 lags of Y, zero afterward Average age is the center of mass ( balancing point ) of the weight distribution Properties of SES, continued Long-term forecasts from the basic SES model are a horizontal straight line (no trend, as in random walk and SMA) SES = ARIMA(0,1,1), i.e., random walk model (without drift) plus MA=1, which adds a multiple of lag-1 forecast error: ˆ t+ 1 Y = Y (1 α) e t t random walk lag-1 error 12

13 Properties of SES, continued Exact k-step ahead forecast standard error can be computed using ARIMA theory: SE 2 fcst( k) = 1+ ( k 1) α SE fcst(1) Note that it increases with k more slowly than for the random walk model, which is the special case α=1: SE = k SE fcst( k) fcst(1) Hence the SES model assumes the series is more predictable than a random walk Example, continued SES with optimal α=0.3 (average age=3.3) yields RMSE = 99 (best yet, by a small margin), no significant residual autocorrelations Simple exponential smoothing with alpha = % confidence limits shown Don t worry about an isolated spike at an oddball lag like lag 9 probably just due to a pair of large errors separated by 9 periods Autocorrelations Residual Autocorrelations for X Simple exponential smoothing with alpha = lag 13

14 SES with constant trend A constant linear trend can be added to an SES model by fitting it as an ARIMA(0,1,1) model with constant Alas, the ARIMA implementation of SES models can t be combined with seasonal adjustment in the Forecasting procedure in Statgraphics (although you could seasonally adjust and then fit an ARIMA model in two steps) SES with constant trend, continued A constant exponential trend can be added to SES by using the inflation adjustment option in Statgraphics The average percentage growth per period can be estimated from the slope coefficient of a linear trend model or ARIMA(0,1,0)+c model fitted with a natural log transformation See video clip #10 for examples 14

15 What if the series has a time-varying trend, as well as a time-varying mean? Evidently what is needed is an estimate of the local trend as well as the local mean This is the motivating idea behind Brown s Linear Exponential Smoothing (LES) model It s also sometimes called double exponential smoothing, because it involves a double application of exponential smoothing How LES works Apply SES once to get a singly-smoothed series S t that lags behind the current value by 1/α 1 periods.* Smooth the smoothed series (using same α) to get an even smoother series S t that lags behind by 2(1/α 1) periods To forecast the future, extrapolate a line between the two points (t (1/α 1), S t ) and (t 2(1/α 1), S t ) *Average age relative to next value is 1/α, so age relative to current value is 1/α -1 15

16 LES forecasts from t = 90, α=0.1* Draw a horizontal line extending 9 periods back in time from the current value of the singly-smoothed series Draw a horizontal line extending 18 periods back in time from the current value of the doubly-smoothed series X S' S'' 3. Extrapolate a line into the future through the left endpoints of the two horizontal lines *1/α = 10, so 1/α - 1 = 9 How LES works There are two equivalent sets of mathematical formulas for implementing the logic of the LES model One set of formulas (I) explicitly computes the current estimates of level and trend in each period The other set of formulas (II) merely computes the next forecast from the observed data and forecast errors in the last two periods 16

17 LES formulas: I 1. Compute singly smoothed series at period t: S' t = αy t + (1-α)S' t-1 2. Compute doubly smoothed series: S'' t = α S' t + (1-α) S'' t-1 3. Compute the estimated level at period t: Startup: S' 1 = S'' 1 = Y 1 L t = 2S' t S'' t 4. Compute the estimated trend at period t: T t = (α/(1-α))(s' t S'' t ) 5. Finally, the k-step ahead forecast is given by: Y ˆ = L + kt t+ k t t LES formulas: II Mathematically equivalent formula (requires fewer columns on a spreadsheet): Yˆ 2 t+ 1 = 2Yt Yt 1 2(1 α) et + (1 α) et 1 Very important start-up values: Yˆ ˆ 2 = Y1 = Y1, hence e1 = 0, e 2 = Y 2 Y 1 (If you don t use these start-up values, the early forecasts will gyrate wildly!) 17

18 Example, continued LES model is optimized at α=0.16, yielding RMSE=102 (about the same as SES) but the forecast plot shows a decreasing trend due to the local downward trend at end of series, confidence intervals also widen more rapidly due to assumption that trend may be varying Brown's linear exp. smoothing with alpha = Residual Autocorrelations for X Brown's linear exp. smoothing with alpha = % confidence limits shown Autocorrelations lag LES vs. SES SES assumes only a time-varying level (i.e., a local mean), while LES assumes a time-varying level and trend. SES assumes that the series is more predictable than a random walk, while LES is assumes it is less predictable. LES model is relatively unstable, hence it may be dangerous to extrapolate the local trend very far. There are fancier versions of LES that include a trend-dampening factor. 18

19 LES vs. SES, continued In both SES and LES, the smaller the value of α, the more smoothing (i.e., less response to the most recent observation) Remember that the average age is 1/α in SES model (amount of lag behind turning points). In LES model, forecast is based on what was happening between 1/α and 2/α periods ago. When fitted to the same series, LES usually has a smaller optimal α than SES. LES vs. SES, continued SES is the most widely used non-seasonal forecasting model. It has a sounder underlying theory than the SMA model, and it is computationally convenient to use on hundreds or thousands of parallel time series (e.g., for SKU-level forecasting). Its assumption of no trend is often unrealistic, but it is surprisingly robust in practice for short-term forecasts--often better than LES even for series that have trends. You can add an exponential trend via the inflation adjustment option. You can add a linear trend to an SES model by fitting it as an ARIMA(0,1,1) model with constant--but you can t combine ARIMA with seasonal adjustment in the Forecasting procedure. 19

20 Estimation issues Optimization of α is performed by nonlinear least squares (like Excel s nonlinear solver). Nonlinear estimation requires a search process whose solution is inexact and may depend on the starting value. In Statgraphics, you may notice that the optimal α varies slightly when the model is revisited, because it restarts the estimation from the previous optimum. Estimation issues, continued α is constrained to lie between and for SES and LES models. If the best SES model is actually a random walk model (α=1), then the estimation algorithm will converge to This will often happen if the series has a significant trend. Once α hits its upper bound (0.9999), the estimation may get stuck there. Try manually changing the initial value to (say) 0.5 before refitting the model if the data sample is changed. 20

21 Estimation issues, continued Because LES and SES use recursive formulas in which each forecast depends on prior errors, their estimation also depends on how they are initialized (i.e., on the prior errors that are assumed at the very beginning). The usual approach is to just assume that the first error is zero. A more sophisticated approach, available as an estimation option in Statgraphics, is to use backforecasting * to start up the model. *We ll discuss this in more detail later in the course. Holt s linear exponential smoothing Holt s model improves on LES by introducing separate smoothing constants for level and trend ( alpha and beta ) In theory, this allows it to perform more stable trend estimation while adapting to sudden jumps in level 21

22 Holt s model formulas 1. Updated level L t is an interpolation between the most recent data point and the previous forecast of the level: Lt = αyt + (1 α )( Lt 1+ Tt 1) Most recent data point Forecast of L t made at period t-1 Holt s model formulas 2. Updated trend T t is an interpolation between the change in the estimated level and the previous estimate of the trend: T ) t = β( Lt Lt 1) + ( 1 β Tt 1 Just-observed change in the level Previous trend estimate 22

23 Holt s model formulas 3. k-step ahead forecast from period t: Yˆt + k = L t + kt t Extrapolation of level and trend from period t Example, continued Holt s model is optimized at α=0.306, β=0.007 yielding RMSE = 100 (essentially same as SES & LES) but forecast plot shows a slightly increasing local trend at end of series, due to relatively heavy smoothing of trend! Holt's linear exp. smoothing with alpha = and beta = % confidence limits shown Autocorrelations Residual Autocorrelations for X Holt's linear exp. smoothing with alpha = and beta = lag 23

24 Model comparisons Models B-C-D-E hardly differ on error measures. Model choice should also depend on theoretical considerations, such as the reasonableness of the trend assumptions A cautionary word about trend extrapolation If you are forecasting more than one period ahead, it is especially important to estimate the trend correctly In general, trend assumptions and estimation should be based on everything you know about a time series, not just error statistics of one-period-ahead forecasts or t-stats of slope coefficients 24

25 A cautionary word about trend extrapolation Extrapolation of time-varying trends estimated by double smoothing can be dangerous Hence SES (perhaps with fixed trend) often works better in practice A trend dampening factor (0 < φ < 1) is often used in conjunction with LES or Holt s: ˆ 2 k Y + = L + ( φ + φ φ ) T t k t t Combining seasonal adjustment with a non-seasonal smoothing model Often a seasonally adjusted series looks like a good candidate for fitting with a smoothing or averaging model. Hence, you can forecast a seasonal series by a combination of seasonal adjustment and non-seasonal smoothing (or other non-seasonal model). This hybrid approach allows you to model the seasonal pattern explicitly, but it does not have a solid underlying statistical theory--confidence limits may be dubious. There is also some danger of overfitting the seasonal pattern if you don t have enough seasons of data. 25

26 Example of LES + seasonal adjustment on a spreadsheet The single-equation form of the LES model is easily implemented on a spreadsheet, and Solver can be used to find the value of α that minimizes RMSE. LES out-of-sample forecasts The LES model, like any other one-step-ahead forecasting model, can extrapolate its forecasts into the future by bootstrapping itself, i.e., by substituting the one-step-ahead forecast for the next data point and then forecasting the next period from there, and so on. 26

27 LES forecasts for seasonally adjusted data Note that LES lags behind turning points, like all smoothing models Seasonally adjusted LES forecast but it tracks the data pretty well during stretches where the trend is consistent and its out-of-sample forecasts extrapolate the most recent trend Dec-83 Dec-84 Dec-85 Dec-86 Dec-87 Dec-88 Dec-89 Dec-90 Dec-91 Dec-92 Dec-93 Dec-94 Re-seasonalized LES forecasts Dec-83 Jun-84 Dec-84 Jun-85 Dec-85 Jun-86 Dec-86 Jun-87 Dec-87 Jun-88 Dec-88 Jun-89 Dec-89 Jun-90 Dec-90 Jun-91 Dec-91 Jun-92 Dec-92 Jun-93 Dec-93 Jun-94 Dec-94 Jun-95 Original series Reseasonalized forecast 27

28 Example: housing starts Series displays strong seasonality as well as cyclicality Original data (not seasonally adjusted) 139 Time Series Plot for HousesNSA HousesNSA Note the last observation 39 1/83 1/87 1/91 1/95 1/99 1/03 New residential construction since

29 Seasonally adjusted data SADJUSTED Time Series Plot for SADJUSTED In seasonally adjusted terms, the last observation is abnormally large! How will different models react to it? (This abnormality was not so apparent on the unadjusted graph!) 54 1/83 1/87 1/91 1/95 1/99 1/03 After seasonal adjustment, variations in level and trend are clearer Nonseasonal forecasting model fitted to adjusted data: RW+drift Time Sequence Plot for SADJUSTED Random walk with drift = /83 1/88 1/93 1/98 1/03 1/08 actual forecast 50.0% limits This model extrapolates the long-term trend from the most recent (higher) level Depending on the kind of long-term trend assumptions we feel are appropriate, we could fit the seasonally adjusted series with a non-seasonal model such as a random walk with drift... 29

30 Nonseasonal forecasting model fitted to adjusted data: SES Time Sequence Plot for SADJUSTED Simple exponential smoothing with alpha = actual forecast 50.0% limits /83 1/88 1/93 1/98 1/03 1/08 This model extrapolates a flat trend from an exponentiallyweighted average of recent levels or a simple exponential smoothing model... Nonseasonal forecasting model fitted to adjusted data: Brown s LES Time Sequence Plot for SADJUSTED Brown's linear exp. smoothing with alpha = actual forecast 50.0% limits /83 1/88 1/93 1/98 1/03 1/08 This model tries to extrapolate the recent trend, which is jerked upward by the last observation or Brown s linear exponential smoothing model... 30

31 Nonseasonal forecasting model fitted to adjusted data: Holt s LES Time Sequence Plot for SADJUSTED Holt's linear exp. smoothing with alpha = and beta = actual forecast 50.0% limits /83 1/88 1/93 1/98 1/03 1/08 This model also tries to extrapolate the recent trend, but the trend estimate is more conservative due to small beta (heavy smoothing) or Holt s linear exponential smoothing model... Hybrid seasonal models in SG You can fit hybrid models in the Forecasting procedure in Statgraphics by selecting multiplicative seasonal adjustment in conjunction with a RW or SES or LES model type. The forecasts are automatically reseasonalized in the plots and model comparison statistics Be on guard against overfitting: seasonal adjustment adds many parameters to the model, and estimation period statistics may not be fully adjusted to correct for additional parameters. 31

32 Hybrid seasonal models RW + seasonal adjustment Time Sequence Plot for HousesNSA Random walk with drift = /83 1/88 1/93 1/98 1/03 1/08 actual forecast 50.0% Note limits sharply raised forecasts, driven by unusual seasonally adjusted value of last data point Here s the result of fitting the RW-with-drift model with multiplicative seasonal adjustment 32

33 SES + seasonal adjustment Time Sequence Plot for HousesNSA Simple exponential smoothing with alpha = /83 1/88 1/93 1/98 1/03 1/08 actual forecast 50.0% limits More conservative (though still raised) forecasts, tighter confidence limits Here s the result of fitting the SES model with multiplicative seasonal adjustment Brown s LES + seasonal adjustment Time Sequence Plot for HousesNSA Brown's linear exp. smoothing with alpha = /83 1/88 1/93 1/98 1/03 1/08 actual forecast 50.0% limits Forecasts march steeply upward, confidence limits are rather wide Here s the result of fitting the LES model with multiplicative seasonal adjustment 33

34 Holt s LES + seasonal adjustment Time Sequence Plot for HousesNSA Holt's linear exp. smoothing with alpha = and beta = actual 150 forecast 50.0% limits 50 1/83 1/88 1/93 1/98 1/03 1/08 Forecasts start from higher level but with flatter trend than LES, but confidence limits are rather optimistic Here s the result of fitting Holt s model with multiplicative seasonal adjustment Linear trend + seasonal adjustment (?) Time Sequence Plot for HousesNSA Linear trend = t actual forecast 50.0% limits /83 1/88 1/93 1/98 1/03 1/08 Obviously not appropriate! Just for fun, here s a linear trend model with multiplicative seasonal adjustment 34

35 Model comparison report shows that SES and Holt s do the best in estimation period, although RW model is slightly luckier in validation period (last 4 years of data were held out) Residual Plot for adjusted HousesNSA Simple exponential smoothing with alpha = Residual /83 1/87 1/91 1/95 1/99 1/03 Residual Autocorrelations for adjusted HousesNSA 1 Simple exponential smoothing with alpha = Residual plots for SES model show stable variance, no significant autocorrelation model appears OK Autocorrelations lag 35

36 proportion Residual Plot for adjusted HousesNSA Simple exponential smoothing with alpha = Even the (vertical) probability plot looks good.* This is a pane option behind the residual plots. *This result validates the use of normal distribution theory to compute the confidence intervals from the forecast standard errors. What s the best forecast? The main issue here is what to infer from the recent jump in seasonally adjusted housing starts. Our modeling results do not really answer this question for us they merely show the consequences of different assumptions we may wish to make. Ideally, domain knowledge should shed additional light on the appropriateness of the assumptions. The SES model is clearly the most conservative choice, because its forecasts are less radically affected by one recent observation. 36

37 Winter s Seasonal Smoothing The logic of Holt s model can be extended to recursively estimate time-varying seasonal indices as well as level and trend. Let L t, T t, and S t denote the estimated level, trend, and seasonal index at period t. Let s denote the number of periods in a season. Let α, β, and γ denote separate smoothing constants* for level, trend, and seasonality *numbers between 0 and 1: smaller values more smoothing Winters model formulas 1. Updated level L t is an interpolation between the seasonally adjusted value of the most recent data point and the previous forecast of the level: L t Yt = α + ( 1 α)( Lt 1 + Tt 1) S t s Seasonally adjusted value of Y t Forecast of L t made at period t-1 37

38 Winters model formulas 2. Updated trend T t is an interpolation between the change in the estimated level and the previous estimate of the trend: T ) t = β( Lt Lt 1) + ( 1 β Tt 1 Just-observed change in the level Previous trend estimate Winters model formulas 3. Updated seasonal index S t is an interpolation between the ratio of the data point to the estimated level and the previous estimate of the seasonal index: S t Y t = γ + (1 γ) Lt S t s Ratio to moving average of current data point Last estimate of seasonal index in the same season 38

39 Winters model formulas 4. k-step ahead forecast from period t: ˆ Y t+ k = ( Lt + ktt ) St s+ k Extrapolation of level and trend from period t Most recent estimate of the seasonal index for k th period in the future Estimation issues Estimation of Winters model is tricky, and not all software does it well: sometimes you get crazy results. There are three separate smoothing constants to be jointly estimated by nonlinear least squares (α, β, γ). Initialization is also tricky, especially for the seasonal indices. 39

40 Estimation issues Some common initialization schemes: Naïve approach: set initial level = 1st data point, trend = 0, seasonal indices = 1.0 More sophisticated: perform a seasonal decomposition to obtain initial seasonal indices & fit trend line to obtain initial trend Even more sophisticated: use backforecasting Calculation of confidence intervals is also complicated & not always done correctly. Winter s model fitted to housing starts Time Sequence Plot for HousesNSA Winter's exp. smoothing with alpha = , beta = , gamma = actual 150 forecast 50.0% limits /83 1/88 1/93 1/98 1/03 1/08 Results of fitting Winters model In this case, the Winters forecasts & confidence intervals look similar to those of the Holt s model with seasonal adjustment (alpha and beta are very similar as should be expected) 40

41 Model comparison report shows that Winters fits a little less well than SES or Holt s model, but is otherwise OK Winters model in practice The Winters model is popular in automatic forecasting software, because it has a little of everything (level, trend, seasonality). Sometimes it works well, but difficulties in initialization & estimation can lead to strange results in other cases. In principle it is similar to linear exponential smoothing and can produce similarly unstable long-term trend projections. 41

42 What really happened in last 5 years? Variables RW+drift SES LES HOLT WINTERS ACTUAL DATE All models overpredicted housing starts for the rest of 1992 and 1993, over-responding to the Feb. 02 jump, but later values were in the middle range of predictions until recent plunge Class 4 recap Averaging and smoothing models enable you to estimate time-varying levels and trends. SMA, SES, and LES models can be combined with seasonal adjustment to forecast seasonal data (...but beware of changing seasonal patterns and possibility of overfitting) Winters estimates time-varying seasonal indices. You need to exercise judgment in model selection in order to make appropriate assumptions about changing levels and trends & unusual events. 42