How to choose forecasting models

# Data transformations and forecasting models: what to use and when

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`Transformation Properties            When to use               Points to keep in mind                        `
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`Deflation by   Converts data from    When data are measured    To generate a true forecast for the future    `
`CPI or         nominal dollars (or   in nominal dollars (or    in nominal terms, you will need to make an    `
`another price  other currency) to    other currency) and you   explicit forecast of the future value of the  `
`index          constant dollars;     want to explicitly show   price index--i.e., you will need to forecast  `
`               usually helps to      the effect of             the inflation rate (but this is easy if       `
`               stablilize variance   inflation--i.e., uncover  you're in a period of steady inflation)       `
`                                     "real growth"                                                           `
` `
`Deflation at   Merely applies a      When you only need to     When used with a zero-trend model like        `
`a fixed rate   constant discount     approximately model the   simple exponential smoothing or random walk   `
`               factor to past data   effect of past inflation  without growth, the assumed inflation rate    `
`                                     and/or you wish to        is precisely the percentage growth in the     `
`                                     impose an assumption      future forecasts.                             `
`                                     about the current and                                                   `
`                                     future inflation                                                        `
`                                     rate--you can twiddle                                                   `
`                                     the inflation rate to                                                   `
`                                     see what value does the                                                 `
`                                     best job of flattening                                                  `
`                                     out the trend and/or                                                    `
`                                     stabilizing the variance                                                `
` `
`Logarithm      Converts              When compound growth is   Logging is not the same as deflating:  it     `
`               multiplicative        not due to inflation      linearizes growth but does not remove a       `
`               patterns to additive  (e.g. when data is not    general upward trend; if logged data still    `
`               patterns and/or       measured in currency);    have a consistent upward trend, then you      `
`               linearizes            when you do not need to   should use a model that includes a trend      `
`               exponential growth;   separate inflation from   factor (e.g., random walk with growth,        `
`               converts absolute     real growth; when data    ARIMA, linear exponential smoothing).         `
`               changes to            distribution is positive                                                `
`               percentage changes;   and highly skewed (e.g.,                                                `
`               often stablizes the   exponential or                                                          `
`               variance of data      log-normal                                                              `
`               with compound         distribution); when                                                     `
`               growth, regardless    variables are                                                           `
`               of whether deflation  multiplicatively related  `
`               is also used                  `
` `
`First          Converts "levels" to  When you need to          Differencing is an explicit option in ARIMA   `
`difference     "changes"             stationarize a series     modeling and it is implicitly a part of       `
`                                     with a strong trend       random walk and exponential smoothing         `
`                                     and/or random-walk        models; therefore you would not manually      `
`                                     behavior (often useful    difference the input variable (using the      `
`                                     when fitting regression   DIFF function) when specifying model type as  `
`                                     models to time series     "random walk" or "exponential smoothing" or   `
`                                     data)                     "ARIMA"; first difference of LOG(Y) is the`
`                                                               percentage change in Y`
`      `
`Seasonal       Converts "levels" to  When you need to remove   Seasonal differencing is an explicit option   `
`difference     "seasonal changes"    the gross features of     in ARIMA modeling; you MUST include a         `
`                                     seasonality from a        seasonal difference (as a modeling option,    `
`                                     strongly seasonal series  not an SDIFF transformation of the input      `
`                                     without going to the      variable)  if the seasonal pattern is         `
`                                     trouble of estimating     consistent and you wish it to be maintained   `
`                                     seasonal indices          in long-term forecasts                        `
` `
`Seasonal       Removes a constant    When you wish to          Adds a lot of parameters to the model--one  `
`adjustment     seasonal pattern      separate out the          for each season of the year.          `
`               from a series         seasonal component of a   (In Statgraphics, the seasonal indices        `
`               (either               series and then fit       are not explicitly shown in the output      `
`               multiplicative or     what's left with a        of the Forecasting procedure--you must`
`               additive)             nonseasonal model         separately run the Descriptive Methods        `
`                                     (regression, smoothing,   procedure to display the seasonal indices.)                                   or trend line); normally                                                `
`                                     use the multiplicative                                                  `
`                                     version unless data has                                                 `
`                                     been logged                                                             `
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`Model type     Properties            When to use               Points to keep in mind                        `
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`Random walk    Predicts that "next   As a baseline against     Plot of forecasts looks exactly like a plot   `
`               period equals this    which to compare more     of the data, except lagged by one period      `
`               period" (perhaps      elaborate models; when    (and shifted slightly up or down if a growth  `
`               plus a constant);     applied to logged data,   term is included);  long term forecasts       `
`               a.k.a. ARIMA(0,1,0)   it is a "geometric"       follow a straight line (horizontal if no      `
`               model                 random walk--the default  growth term is included); confidence          `
`                                     model for stock market    intervals for long-term forecasts widen       `
`                                     data                      according to a square-root law (sideways-     `
`                                                               parabola shape); logically equivalent to      `
`                                                               MEAN model fitted to DIFF(Y)                  `
` `
`Linear trend   Regression of Y on    Rarely the best model     Forecasts follow a straight line whose slope  `
`               the time index        for forecasting--use      equals the average slope over the whole       `
`                                     only when you have very   estimation period but whose intercept is      `
`                                     few data points and no    anchored in the distant past;  short-term     `
`                                     obvious pattern in data   forecasts therefore may miss badly and        `
`                                     other than a slight       confidence intervals for long-term forecasts  `
`                                     trend; can be used in     are usually not reliable; other models that   `
`                                     conjunction with          extrapolate a linear trend into the future    `
`                                     seasonal adjustment--but  (random walk with growth, linear exponential  `
`                                     if you have enough data   smoothing, ARIMA models with 1                `
`                                     to seasonally adjust,     difference w/constant or 2 differences w/o      `
`                                     you probably should use   constant) often do a better job by            `
`                                     another model             "reanchoring" the trend line on recent data   `
` `
`Simple moving  Simple (equally       When data are in short    Primitive but relatively robust against       `
`average        weighted) average of  supply and/or highly      outliers and messy data; long-term forecasts  `
`               recent data           irregular                 are a horizontal line extrapolated from the   `
`                                                               most recent average; a long-term trend can    `
`                                                               be incorporated via fixed-rate deflation at   `
`                                                               an assumed interest rate                      `
` `
`Simple         Exponentially         When data are nonseasonal Long-term forecasts are a horizontal line     `
`exponential    weighted average of   (or deseasonalized) and   extrapolated from the most recent smoothed    `
`smoothing      recent data;          display a time-varying    value;  same as a random walk model without    `
`               "average age" of      mean without a            growth if alpha=0.9999; forecasts get         `
`               data in forecast      consistent trend          smoother and slower to respond to turning     `
`               (amount by which                                points as alpha approaches zero; confidence   `
`               forecasts lag behind                            intervals widen less rapidly than in the      `
`               turning points) is                              random walk model; a long-term trend can be   `
`               1/alpha; same as an                             incorporated via fixed-rate deflation at an   `
`               ARIMA(0,1,1) model                              assumed interest rate or by fitting an     `
`               without constant                                ARIMA(0,1,1) model with constant              `
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`Linear         Assumes a             When data are nonseasonal Long-term forecasts follow a straight line    `
`exponential    time-varying linear   (or deseasonalized) and   whose slope is the estimated local trend at   `
`smoothing      trend as well as a    display time-varying      the end of the series; confidence intervals   `
`(Brown's or    time-varying level    local trends (usually     for long-term forecasts widen rapidly--the    `
`Holt's)        (Brown's uses 1       applicable to data that   model assumes that the future is VERY         `
`               parameter, Holt's     are "smoother" in         uncertain because of time-varying trends;     `
`               uses separate         appearance--i.e., less    often does not outperform simple exponential  `
`               smoothing parameters  noisy--than what would    smoothing, even for data with trends,         `
`               for level and         be well fitted by simple  because extrapolation of time-varying trends  `
`               trend); essentially   exponential smoothing)    is risky                                      `
`               an ARIMA(0,2,2)                                                                               `
`               model without                                                                                 `
`               constant                                                                                      `
` `
`Seasonal       Predicts that "next   As a baseline against     Long-term forecasts have same seasonal        `
`random walk    period equals same    which to compare fancier  pattern as last year; long-term trend is      `
`               period last year"     seasonal models; as       equal to the average trend over whole past    `
`               (plus constant); an   foundation for seasonal   history of series; confidence intervals       `
`               ARIMA(0,0,0)x(0,1,0)  ARIMA models (e.g.,       widen slowly; slow to respond to cyclical     `
`               model with constant   (1,0,0)x(0,1,1))          upturns and downturns; logically equivalent   `
`                                                               to MEAN model fitted to SDIFF(Y,s)              `
` `
`Seasonal       Predicts that change  As a baseline against     Long-term forecasts have same seasonal        `
`random trend   from this period to   which to compare fancier  pattern as last year; long-term trend is      `
`               next period will be   seasonal models; as       equal to the most recently observed annual    `
`               the same as change    foundation for seasonal   trend; confidence intervals widen rapidly;    `
`               observed at this      ARIMA models (e.g.,       quick to respond to cyclical upturns and      `
`               time last year; an    (0,1,1)x(0,1,1) without   downturns; logically equivalent to MEAN       `
`               ARIMA(0,1,0)          constant)                 model fitted to DIFF(SDIFF(Y)) (with no       `
`               x(0,1,0) model                                  constant--i.e., mean is assumed to be zero)   `
`               without constant                                                                              `
` `
`Winter's       Assumes time-varying  When data are trended and Initialization of seasonal indices and joint  `
`seasonal       level, trend, and     seasonal and you wish to  estimation of three smoothing parameters is   `
`smoothing      seasonal indices      decompose it into local   sometimes tricky--watch to see that           `
`               (either               level/trend/seasonal      parameter estimates converge and that         `
`               multiplicative or     factors; normally you     forecasts and confidence intervals look       `
`               additive              use the multiplicative    reasonable; a popular choice for "automatic"  `
`               seasonality)          version unless data is    forecasting because it does a little of       `
`                                     logged                    everything, but has a lot of parameters and   `
`                                                               sometimes overfits the data or is unstable    `
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`Multiple       A general linear      When data are             Forecasts cannot be extrapolated into the     `
`regression     forecasting equation  correlated with other     future unless and until values are available  `
`               involving other       explanatory or causal     for the independent variables; for this       `
`               variables             variables (e.g., price,   reason the independent variables must often   `
`                                     advertising, promotions,  be lagged by one or more periods--but when    `
`                                     interest rates,           only lagged variables are used, a regression  `
`                                     indicators of general     model may fail to outperform a time series    `
`                                     economic activity,        model which relies only on the history of     `
`                                     etc.);  the key is to     the original series; regressions of           `
`                                     choose the right          nonstationary variables often have high       `
`                                     variables and the right   "R-squared" but poor performance compared to  `
`                                     transformations of those  time series models; it often helps to         `
`                                     variables to justify the  stationarize the dependent variable and/or    `
`                                     assumption of a linear    add lags of the dependent and independent     `
`                                     model and to take into    variables to the model; "automatic" model     `
`                                     account the time          selection techniques such as stepwise         `
`                                     dimension in the data     regression and all-possible regressions are   `
`                                                               available, but beware of overfitting; it is   `
`                                                               important to validate the model by testing    `
`                                                               it on hold-out data and by computing its      `
`                                                               "effective R-squared" (percent of variance    `
`                                                               explained)  relative to a random walk model   `
`                                                               or other appropriate time series model        `
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`ARIMA          A general class of    When data are relatively  ARIMA models are designed to squeeze all      `
`               models that includes  plentiful (4 seasons or   autocorrelation out of the original time      `
`               random walk, random   more) and can be          series; a systematic procedure exists for     `
`               trend, seasonal and   satisfactorily            identifying the best ARIMA model for any  `
`               non-seasonal          stationarized by          given time series; features of ARIMA   `
`               exponential           differencing and other    and multiple regression models can be         `
`               smoothing, and auto-  mathematical              combined in a natural way;  ARIMA models      `
`               regressive models;    transformations; when it  often provide a good fit to highly            `
`               forecasts for the     is not necessary to       aggregated, highly plentiful data; they may   `
`               stationarized         explicitly separate out   perform relatively less well on               `
`               dependent variable    the seasonal component    disaggregated, irregular, and/or sparse data  `
`               are a linear          (if any) in the form of                                                 `
`               function of lags of   seasonal indices                                                        `
`               the dependent                                                                                 `
`               variable and/or lags                                                                          `
`               of the errors                                                                                 `
` `