32 ✦ Chapter 2: Introduction  diagnostic statistics to help judge the adequacy of the model including the following: – Akaike’s information criterion (AIC) – Schwarz’s Bayesian criterion (SBC or BIC) – Box-Ljung chi-square test statistics for white-noise residuals – autocorrelation function of residuals – partial autocorrelation function of residuals – inverse autocorrelation function of residuals – automatic outlier detection Vector Time Series Analysis The VARMAX procedure enables you to model the dynamic relationship both between the dependent variables and between the dependent and independent variables. The VARMAX procedure includes the following features:  several modeling features: – vector autoregressive model – vector autoregressive model with exogenous variables – vector autoregressive and moving-average model – Bayesian vector autoregressive model – vector error correction model – Bayesian vector error correction model – GARCH-type multivariate conditional heteroscedasticity models  criteria for automatically determining AR and MA orders: – Akaike information criterion (AIC) – corrected AIC (AICC) – Hannan-Quinn (HQ) criterion – final prediction error (FPE) – Schwarz Bayesian criterion (SBC), also known as Bayesian information criterion (BIC)  AR order identification aids: – partial cross-correlations – Yule-Walker estimates – partial autoregressive coefficients – partial canonical correlations Vector Time Series Analysis ✦ 33  testing the presence of unit roots and cointegration: – Dickey-Fuller tests – Johansen cointegration test for nonstationary vector processes of integrated order one – Stock-Watson common trends test for the possibility of cointegration among nonstation- ary vector processes of integrated order one – Johansen cointegration test for nonstationary vector processes of integrated order two  model parameter estimation methods: – least squares (LS) – maximum likelihood (ML)  model checks and residual analysis using the following tests: – Durbin-Watson (DW) statistics – F test for autoregressive conditional heteroscedastic (ARCH) disturbance – F test for AR disturbance – Jarque-Bera normality test – Portmanteau test  seasonal deterministic terms  subset models  multiple regression with distributed lags  dead-start model that does not have present values of the exogenous variables  Granger-causal relationships between two distinct groups of variables  infinite order AR representation  impulse response function (or infinite order MA representation)  decomposition of the predicted error covariances  roots of the characteristic functions for both the AR and MA parts to evaluate the proximity of the roots to the unit circle  contemporaneous relationships among the components of the vector time series  forecasts  conditional covariances for GARCH models 34 ✦ Chapter 2: Introduction State Space Modeling and Forecasting The STATESPACE procedure provides automatic model selection, parameter estimation, and fore- casting of state space models. (State space models encompass an alternative general formulation of multivariate ARIMA models.) The STATESPACE procedure includes the following features:  multivariate ARIMA modeling by using the general state space representation of the stochastic process  automatic model selection using Akaike’s information criterion (AIC)  user-specified state space models including restrictions  transfer function models with random inputs  any combination of simple and seasonal differencing; input series can be differenced to any order for any lag lengths  forecasts with confidence limits  ability to save selected and fitted model in a data set and reuse for forecasting  wide range of output options including the ability to print any statistics concerning the data and their covariance structure, the model selection process, and the final model fit Spectral Analysis The SPECTRA procedure provides spectral analysis and cross-spectral analysis of time series. The SPECTRA procedure includes the following features:  efficient calculation of periodogram and smoothed periodogram using fast finite Fourier transform and Chirp-Z algorithms  multiple spectral analysis, including raw and smoothed spectral and cross-spectral function estimates, with user-specified window weights  choice of kernel for smoothing  output of the following spectral estimates to a SAS data set: – Fourier sine and cosine coefficients – periodogram – smoothed periodogram – cospectrum – quadrature spectrum Seasonal Adjustment ✦ 35 – amplitude – phase spectrum – squared coherency  Fisher’s Kappa and Bartlett’s Kolmogorov-Smirnov test statistic for testing a null hypothesis of white noise Seasonal Adjustment The X11 procedure provides seasonal adjustment of time series by using the Census X-11 or X-11 ARIMA method. The X11 procedure is based on the U.S. Bureau of the Census X-11 seasonal adjustment program and also supports the X-11 ARIMA method developed by Statistics Canada. The X11 procedure includes the following features:  decomposition of monthly or quarterly series into seasonal, trend, trading day, and irregular components  both multiplicative and additive form of the decomposition  all the features of the Census Bureau program  support of the X-11 ARIMA method  support of sliding spans analysis  processing of any number of variables at once with no maximum length for a series  computation of tests for stable, moving, and combined seasonality  optional printing or storing in SAS data sets of the individual X11 tables that show the various components at different stages of the computation; full control over what is printed or output  ability to project seasonal component one year ahead, which enables reintroduction of seasonal factors for an extrapolated series The X12 procedure provides seasonal adjustment of time series using the X-12 ARIMA method. The X12 procedure is based on the U.S. Bureau of the Census X-12 ARIMA seasonal adjustment program (version 0.3). It also supports the X-11 ARIMA method developed by Statistics Canada and the previous X-11 method of the U.S. Census Bureau. The X12 procedure includes the following features:  decomposition of monthly or quarterly series into seasonal, trend, trading day, and irregular components  support of multiplicative, additive, pseudo-additive, and log additive forms of decomposition  support of the X-12 ARIMA method 36 ✦ Chapter 2: Introduction  support of regARIMA modeling  automatic identification of outliers  support of TRAMO-based automatic model selection  use of regressors to process missing values within the span of the series  processing of any number of variables at once with no maximum length for a series  computation of tests for stable, moving, and combined seasonality  spectral analysis of original, seasonally adjusted, and irregular series  optional printing or storing in a SAS data set of the individual X11 tables that show the various components at different stages of the decomposition; full control over what is printed or output  optional projection of seasonal component one year ahead, which enables reintroduction of seasonal factors for an extrapolated series Structural Time Series Modeling and Forecasting The UCM procedure provides a flexible environment for analyzing time series data using structural time series models, also called unobserved components models (UCM). These models represent the observed series as a sum of suitably chosen components such as trend, seasonal, cyclical, and regression effects. You can use the UCM procedure to formulate comprehensive models that bring out all the salient features of the series under consideration. Structural models are applicable in the same situations where Box-Jenkins ARIMA models are applicable; however, the structural models tend to be more informative about the underlying stochastic structure of the series. The UCM procedure includes the following features:  general unobserved components modeling where the models can include trend, multiple seasons and cycles, and regression effects  maximum-likelihood estimation of the model parameters  model diagnostics that include a variety of goodness-of-fit statistics, and extensive graphical diagnosis of the model residuals  forecasts and confidence limits for the series and all the model components  Model-based seasonal decomposition  extensive plotting capability that includes the following: – forecast and confidence interval plots for the series and model components such as trend, cycles, and seasons – diagnostic plots such as residual plot, residual autocorrelation plots, and so on Time Series Cross-Sectional Regression Analysis ✦ 37 – seasonal decomposition plots such as trend, trend plus cycles, trend plus cycles plus seasons, and so on  model-based interpolation of series missing values  full sample (also called smoothed) estimates of the model components Time Series Cross-Sectional Regression Analysis The TSCSREG procedure provides combined time series cross-sectional regression analysis. The TSCSREG procedure includes the following features:  estimation of the regression parameters under several common error structures: – Fuller and Battese method (variance component model) – Wansbeek-Kapteyn method – Parks method (autoregressive model) – Da Silva method (mixed variance component moving-average model) – one-way fixed effects – two-way fixed effects – one-way random effects – two-way random effects  any number of model specifications  unbalanced panel data for the fixed or random-effects models  variety of estimates and statistics including the following: – underlying error components estimates – regression parameter estimates – standard errors of estimates – t-tests – R-square statistic – correlation matrix of estimates – covariance matrix of estimates – autoregressive parameter estimate – cross-sectional components estimates – autocovariance estimates – F tests of linear hypotheses about the regression parameters – specification tests 38 ✦ Chapter 2: Introduction Automatic Time Series Forecasting The ESM procedure provides a quick way to generate forecasts for many time series or transactional data in one step by using exponential smoothing methods. All parameters associated with the forecasting model are optimized based on the data. You can use the following smoothing models:  simple  double  linear  damped trend  seasonal  Winters method (additive and multiplicative) Additionally, PROC ESM can transform the data before applying the smoothing methods using any of these transformations:  log  square root  logistic  Box-Cox In addition to forecasting, the ESM procedure can also produce graphic output. The ESM procedure can forecast both time series data, whose observations are equally spaced at a specific time interval (for example, monthly, weekly), or transactional data, whose observations are not spaced with respect to any particular time interval. (Internet, inventory, sales, and similar data are typical examples of transactional data. For transactional data, the data are accumulated based on a specified time interval to form a time series.) The ESM procedure is a replacement for the older FORECAST procedure. ESM is often more convenient to use than PROC FORECAST but it supports only exponential smoothing models. The FORECAST procedure provides forecasting of univariate time series using automatic trend extrapolation. PROC FORECAST is an easy-to-use procedure for automatic forecasting and uses simple popular methods that do not require statistical modeling of the time series, such as exponential smoothing, time trend with autoregressive errors, and the Holt-Winters method. The FORECAST procedure supplements the powerful forecasting capabilities of the econometric and time series analysis procedures described previously. You can use PROC FORECAST when you Time Series Interpolation and Frequency Conversion ✦ 39 have many series to forecast and you want to extrapolate trends without developing a model for each series. The FORECAST procedure includes the following features:  choice of the following forecasting methods: – EXPO method—exponential smoothing: single, double, triple, or Holt two-parameter smoothing – exponential smoothing as an ARIMA Model – WINTERS method—using updating equations similar to exponential smoothing to fit model parameters – ADDWINTERS method—like the WINTERS method except that the seasonal parame- ters are added to the trend instead of multiplied with the trend – STEPAR method—stepwise autoregressive models with constant, linear, or quadratic trend and autoregressive errors to any order – Holt-Winters forecasting method with constant, linear, or quadratic trend – additive variant of the Holt-Winters method  support for up to three levels of seasonality for Holt-Winters method: time-of-year, day-of- week, or time-of-day  ability to forecast any number of variables at once  forecast confidence limits for all methods Time Series Interpolation and Frequency Conversion The EXPAND procedure provides time interval conversion and missing value interpolation for time series. The EXPAND procedure includes the following features:  conversion of time series frequency; for example, constructing quarterly estimates from annual series or aggregating quarterly values to annual values  conversion of irregular observations to periodic observations  interpolation of missing values in time series  conversion of observation types; for example, estimate stocks from flows and vice versa. All possible conversions are supported between any of the following: – beginning of period – end of period – period midpoint – period total 40 ✦ Chapter 2: Introduction – period average  conversion of time series phase shift; for example, conversion between fiscal years and calendar years  identifying observations including the following: – identification of the time interval of the input values – validation of the input data set observations – computation of the ID values for the observations in the output data set  choice of four interpolation methods: – cubic splines – linear splines – step functions – simple aggregation  ability to perform extrapolation by a linear projection of the trend of the cubic spline curve fit to the input data  ability to transform series before and after interpolation (or without interpolation) by using any of the following: – constant shift or scale – sign change or absolute value – logarithm, exponential, square root, square, logistic, inverse logistic – lags, leads, differences – classical decomposition – bounds, trims, reverse series – centered moving, cumulative, or backward moving average – centered moving, cumulative, or backward moving range – centered moving, cumulative, or backward moving geometric mean – centered moving, cumulative, or backward moving maximum – centered moving, cumulative, or backward moving median – centered moving, cumulative, or backward moving minimum – centered moving, cumulative, or backward moving product – centered moving, cumulative, or backward moving corrected sum of squares – centered moving, cumulative, or backward moving uncorrected sum of squares – centered moving, cumulative, or backward moving rank – centered moving, cumulative, or backward moving standard deviation – centered moving, cumulative, or backward moving sum – centered moving, cumulative, or backward moving median Trend and Seasonal Analysis on Transaction Databases ✦ 41 – centered moving, cumulative, or backward moving t-value – centered moving, cumulative, or backward moving variance  support for a wide range of time series frequencies: – YEAR – SEMIYEAR – QUARTER – MONTH – SEMIMONTH – TENDAY – WEEK – WEEKDAY – DAY – HOUR – MINUTE – SECOND  support for repeating of shifting the basic interval types to define a great variety of different frequencies, such as fiscal years, biennial periods, work shifts, and so forth Refer to Chapter 3, “Working with Time Series Data,” and Chapter 4, “Date Intervals, Formats, and Functions,” for more information about time series data transformations. Trend and Seasonal Analysis on Transaction Databases The TIMESERIES procedure can accumulate transactional data to time series and perform trend and seasonal analysis on the accumulated time series. Time series analyses performed by the TIMESERIES procedure include the follows:  descriptive statistics relevant for time series data  seasonal decomposition and seasonal adjustment analysis  correlation analysis  cross-correlation analysis The TIMESERIES procedure includes the following features:  ability to process large amounts of time-stamped transactional data . coefficients – periodogram – smoothed periodogram – cospectrum – quadrature spectrum Seasonal Adjustment ✦ 35 – amplitude – phase spectrum – squared coherency  Fisher’s Kappa and Bartlett’s Kolmogorov-Smirnov. previously. You can use PROC FORECAST when you Time Series Interpolation and Frequency Conversion ✦ 39 have many series to forecast and you want to extrapolate trends without developing a model for