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Statistical Time Series Analysis version 2.0 An O-Matrix Toolbox for Analyzing Time-Dependent Observations

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What Is STSA? STSA is a collection of O-Matrix function files that can help in the analysis of many kinds of time- dependent observations. It includes functions for filtering, smoothing and forecasting of time series, spectral analysis, estimation of many different models, diagnostic tests and a variety of other useful statistical functions not directly available in the basic O-Matrix distribution. The STSA functions can be easily incorporated into the users own programs or functions and they will speed-up the process of a variety of analyses for time series data. STSA is a collection of O-Matrix function files that can help in the analysis of many kinds of time- dependent observations. It includes functions for filtering, smoothing and forecasting of time series, spectral analysis, estimation of many different models, diagnostic tests and a variety of other useful statistical functions not directly available in the basic O-Matrix distribution. The STSA functions can be easily incorporated into the users own programs or functions and they will speed-up the process of a variety of analyses for time series data.

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There are 8 subdirectories in the main STSA directory: 1. 1.ARMA: functions for identification, estimation, testing and forecasting of time series based on the popular class of AutoRegressive Moving Average univariate models. It also has functions for the analysis of bivariate models using transfer functions and multivariate models using Vector AutoRegressive models BAYES: functions for the identification and forecasting of structural time series based on an underlying Bayesian methodology FILTER: functions for filtering and forecasting of univariate time series NONLIN: functions for estimation and forecasting of univariate time series models based on nonlinear and nonparametric models OPTIMIZE: functions for nonlinear optimization not available in the main O-Matrix distribution RNG: function for generating random numbers from various statistical distributions SPECTRAL: functions for spectral analysis of univariate and bivariate time series STATS: various statistical functions that aid in the analysis of time series data. This subdirectory on its own greatly extends the statistical capabilities of the main distribution of O-Matrix. There are 8 subdirectories in the main STSA directory: 1. 1.ARMA: functions for identification, estimation, testing and forecasting of time series based on the popular class of AutoRegressive Moving Average univariate models. It also has functions for the analysis of bivariate models using transfer functions and multivariate models using Vector AutoRegressive models BAYES: functions for the identification and forecasting of structural time series based on an underlying Bayesian methodology FILTER: functions for filtering and forecasting of univariate time series NONLIN: functions for estimation and forecasting of univariate time series models based on nonlinear and nonparametric models OPTIMIZE: functions for nonlinear optimization not available in the main O-Matrix distribution RNG: function for generating random numbers from various statistical distributions SPECTRAL: functions for spectral analysis of univariate and bivariate time series STATS: various statistical functions that aid in the analysis of time series data. This subdirectory on its own greatly extends the statistical capabilities of the main distribution of O-Matrix. STSA function categories

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A partial list of what the functions in the ARMA subdirectory can do: Simulate a Gaussian ARMA model. Compute the theoretical autocovariances of an ARMA model. Compute and plot the sample autocorrelation, partial correlation and cross- correlation functions. Compute best linear predictors and innovations using theoretical autocovariances and the Durbin-Levinson-Whittle algorithm. Estimate the parameters (possibly with constraints) of an ARMA model using nonlinear least squares. Compute the roots of the AR and MA polynomials based on the estimated coefficients. Compute a variety of residual diagnostic statistics for model adequacy. Forecast using the estimates from an ARMA model. Perform the Diebold-Mariano test for competing forecasting models. Compute Granger causality tests. Estimate the parameters of a bivariate transfer function model. Forecast using the estimates from a transfer function model. Estimate the parameters of a multivariate VAR model. Forecast using the estimates from a VAR model. A partial list of what the functions in the ARMA subdirectory can do: Simulate a Gaussian ARMA model. Compute the theoretical autocovariances of an ARMA model. Compute and plot the sample autocorrelation, partial correlation and cross- correlation functions. Compute best linear predictors and innovations using theoretical autocovariances and the Durbin-Levinson-Whittle algorithm. Estimate the parameters (possibly with constraints) of an ARMA model using nonlinear least squares. Compute the roots of the AR and MA polynomials based on the estimated coefficients. Compute a variety of residual diagnostic statistics for model adequacy. Forecast using the estimates from an ARMA model. Perform the Diebold-Mariano test for competing forecasting models. Compute Granger causality tests. Estimate the parameters of a bivariate transfer function model. Forecast using the estimates from a transfer function model. Estimate the parameters of a multivariate VAR model. Forecast using the estimates from a VAR model. The ARMA subdirectory

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A partial list of what the functions in the BAYES subdirectory can do: Simulate a first order polynomial Dynamic Linear Model. Fit a first order polynomial DLM. Fit a regression-through-the-origin DLM. Fit a generic time series DLM. Forecast from all the above models. Forecast using a combination of first order polynomial DLM. Compute forecast intervals from any of the above models. Construct the trend and seasonal matrices for the generic DLM. Compute appropriate reference priors for the generic DLM. A partial list of what the functions in the BAYES subdirectory can do: Simulate a first order polynomial Dynamic Linear Model. Fit a first order polynomial DLM. Fit a regression-through-the-origin DLM. Fit a generic time series DLM. Forecast from all the above models. Forecast using a combination of first order polynomial DLM. Compute forecast intervals from any of the above models. Construct the trend and seasonal matrices for the generic DLM. Compute appropriate reference priors for the generic DLM. The BAYES subdirectory

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A partial list of what the functions in the FILTER subdirectory can do: Smoothing a time series using simple and exponential moving averages. Smoothing and forecasting a time series using the Holt-Winters recursions. Filter a time series using a generic finite impulse response filter. Filter a time series using the Savitzky-Golay filter. Model, filter and forecast a time series using any time-invariant state- space model with the Kalman filter. Estimate, filter and forecast a time series based on a trend+cyclical structural model (uses the previous function). Estimate, filter and forecast a time series based on a trend+cyclical structural model (uses polynomial trends and sines / cosines). A partial list of what the functions in the FILTER subdirectory can do: Smoothing a time series using simple and exponential moving averages. Smoothing and forecasting a time series using the Holt-Winters recursions. Filter a time series using a generic finite impulse response filter. Filter a time series using the Savitzky-Golay filter. Model, filter and forecast a time series using any time-invariant state- space model with the Kalman filter. Estimate, filter and forecast a time series based on a trend+cyclical structural model (uses the previous function). Estimate, filter and forecast a time series based on a trend+cyclical structural model (uses polynomial trends and sines / cosines). The FILTER subdirectory

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A partial list of what the functions in the NONLIN subdirectory can do: Bootstrap a time series using the maximum entropy bootstrap. Compute the empirical probability density and cumulative density of a time series. Compute a Kolmogorov-Smirnov type test for the equality of distribution between two time series. Compute a regression-based F-test for linearity of a time series. Compute a nonparametric regression. Forecast using a nonparametric autoregressive model. Model selection, estimation and forecasting using a Self-Exciting Threshold AutoRegressive model. Model selection, estimation and forecasting using a Functional Coefficient AutoRegressive model. Simulation and Estimation of ARMA models with Generalized AutoRegressive Conditional Heteroskedasticity (ARMA-GARCH models). A partial list of what the functions in the NONLIN subdirectory can do: Bootstrap a time series using the maximum entropy bootstrap. Compute the empirical probability density and cumulative density of a time series. Compute a Kolmogorov-Smirnov type test for the equality of distribution between two time series. Compute a regression-based F-test for linearity of a time series. Compute a nonparametric regression. Forecast using a nonparametric autoregressive model. Model selection, estimation and forecasting using a Self-Exciting Threshold AutoRegressive model. Model selection, estimation and forecasting using a Functional Coefficient AutoRegressive model. Simulation and Estimation of ARMA models with Generalized AutoRegressive Conditional Heteroskedasticity (ARMA-GARCH models). The NONLIN subdirectory

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The OPTIMIZE subdirectory In this subdirectory we have three alternative optimization methods that are well suited for nonlinear estimation problems. For nonlinear maximum likelihood problems we have the BHHH algorithm. For nonlinear least squares problems we have the Gauss- Newton algorithm. For any of the above problems or generic optimization we have the Quasi-Newton algorithm with optional rank 2 symmetric update (this is the BFGS method). All the above functions provide optional formatted screen output, which makes them very convenient for getting immediately visible results after estimation. In this subdirectory we have three alternative optimization methods that are well suited for nonlinear estimation problems. For nonlinear maximum likelihood problems we have the BHHH algorithm. For nonlinear least squares problems we have the Gauss- Newton algorithm. For any of the above problems or generic optimization we have the Quasi-Newton algorithm with optional rank 2 symmetric update (this is the BFGS method). All the above functions provide optional formatted screen output, which makes them very convenient for getting immediately visible results after estimation.

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The RNG subdirectory In this subdirectory we have a collection of functions for generating random numbers from various statistical distributions. The random numbers are generated using the inverse distribution function method. In the subdirectory the corresponding cumulative distributions are also provided that can be used in statistical tests. Random numbers are provided for the following distributions: Cauchy distribution Exponential distribution Gaussian distribution Gumbel (Extreme value) distribution Logistic distribution t (Student) distribution Uniform distribution In this subdirectory we have a collection of functions for generating random numbers from various statistical distributions. The random numbers are generated using the inverse distribution function method. In the subdirectory the corresponding cumulative distributions are also provided that can be used in statistical tests. Random numbers are provided for the following distributions: Cauchy distribution Exponential distribution Gaussian distribution Gumbel (Extreme value) distribution Logistic distribution t (Student) distribution Uniform distribution

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A partial list of what the functions in the SPECTRAL subdirectory can do: Simulate a fractional Gaussian noise time series. Simulate a fractionally differenced time series. Compute and plot the Fourier transform and periodogram of a time series. Compute and plot the power spectrum of a time series using the smoothed periodogram, an autoregressive approximation or the autocovariances. Compute the cross-spectrum, the squared coherency, the amplitude and phase of two time series. Compute the impulse response coefficients between two time series. Fractionally difference a time series. Estimate the fractional order (Hurst exponent) of a time series (GPH regression and Whittle likelihood methods). Estimate and forecast using a fractional ARMA model (ARFIMA). A partial list of what the functions in the SPECTRAL subdirectory can do: Simulate a fractional Gaussian noise time series. Simulate a fractionally differenced time series. Compute and plot the Fourier transform and periodogram of a time series. Compute and plot the power spectrum of a time series using the smoothed periodogram, an autoregressive approximation or the autocovariances. Compute the cross-spectrum, the squared coherency, the amplitude and phase of two time series. Compute the impulse response coefficients between two time series. Fractionally difference a time series. Estimate the fractional order (Hurst exponent) of a time series (GPH regression and Whittle likelihood methods). Estimate and forecast using a fractional ARMA model (ARFIMA). The SPECTRAL subdirectory

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A partial list of what the functions in the STATS subdirectory can do: Compute sample central moments of any order. Compute a complete set of descriptive statistics. Compute the optimal Box-Cox transformation to near Gaussianity. Transform a time series using the optimal Box-Cox exponent. Compute empirical percentiles. Test for Gaussianity using sample moments. Test for Gaussianity using the QQ correlation coefficient. Compute and draw a scatter plot. Draw a fast, formatted plot of a time series. Estimate a regression model using linear least squares and rolling linear least squares. Estimate a regression model using least absolute deviations. Perform principal components analysis (PCA). Perform factor analysis based on principal components. Estimate polynomial and exponential trend functions. Protect your data and reduce the size of your data files by saving and loading them in O-Matrix binary format. A partial list of what the functions in the STATS subdirectory can do: Compute sample central moments of any order. Compute a complete set of descriptive statistics. Compute the optimal Box-Cox transformation to near Gaussianity. Transform a time series using the optimal Box-Cox exponent. Compute empirical percentiles. Test for Gaussianity using sample moments. Test for Gaussianity using the QQ correlation coefficient. Compute and draw a scatter plot. Draw a fast, formatted plot of a time series. Estimate a regression model using linear least squares and rolling linear least squares. Estimate a regression model using least absolute deviations. Perform principal components analysis (PCA). Perform factor analysis based on principal components. Estimate polynomial and exponential trend functions. Protect your data and reduce the size of your data files by saving and loading them in O-Matrix binary format. The STATS subdirectory

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Examples The subdirectory EXAMPLES contains a large number of worked-out examples that can be used as templates to speed-up the time needed to develop your own programs and applications.

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Let us know of your needs! We are constantly revising and updating the contents of the STSA toolbox. User feedback is a valuable contribution both to the breadth and quality of the provided functions. Let us know if you do not see a model class or a particular function that is useful for your work and we will incorporate it in the next update of the toolbox. If you require a custom solution for time series analysis also let us know and take advantage of the combined power of O- Matrix and STSA! We are constantly revising and updating the contents of the STSA toolbox. User feedback is a valuable contribution both to the breadth and quality of the provided functions. Let us know if you do not see a model class or a particular function that is useful for your work and we will incorporate it in the next update of the toolbox. If you require a custom solution for time series analysis also let us know and take advantage of the combined power of O- Matrix and STSA!

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Contact Harmonic for more information Harmonic Software Inc. PO Box 7365 Breckenridge, CO Ask about the STSA toolbox… Harmonic Software Inc. PO Box 7365 Breckenridge, CO Ask about the STSA toolbox…

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