# Time Series analysis tsa¶

statsmodels.tsa contains model classes and functions that are useful for time series analysis. Basic models include univariate autoregressive models (AR), vector autoregressive models (VAR) and univariate autoregressive moving average models (ARMA). Non-linear models include Markov switching dynamic regression and autoregression. It also includes descriptive statistics for time series, for example autocorrelation, partial autocorrelation function and periodogram, as well as the corresponding theoretical properties of ARMA or related processes. It also includes methods to work with autoregressive and moving average lag-polynomials. Additionally, related statistical tests and some useful helper functions are available.

Estimation is either done by exact or conditional Maximum Likelihood or conditional least-squares, either using Kalman Filter or direct filters.

Currently, functions and classes have to be imported from the corresponding module, but the main classes will be made available in the statsmodels.tsa namespace. The module structure is within statsmodels.tsa is

• stattools : empirical properties and tests, acf, pacf, granger-causality, adf unit root test, kpss test, bds test, ljung-box test and others.
• ar_model : univariate autoregressive process, estimation with conditional and exact maximum likelihood and conditional least-squares
• arima_model : univariate ARMA process, estimation with conditional and exact maximum likelihood and conditional least-squares
• vector_ar, var : vector autoregressive process (VAR) estimation models, impulse response analysis, forecast error variance decompositions, and data visualization tools
• kalmanf : estimation classes for ARMA and other models with exact MLE using Kalman Filter
• arma_process : properties of arma processes with given parameters, this includes tools to convert between ARMA, MA and AR representation as well as acf, pacf, spectral density, impulse response function and similar
• sandbox.tsa.fftarma : similar to arma_process but working in frequency domain
• tsatools : additional helper functions, to create arrays of lagged variables, construct regressors for trend, detrend and similar.
• filters : helper function for filtering time series
• regime_switching : Markov switching dynamic regression and autoregression models

Some additional functions that are also useful for time series analysis are in other parts of statsmodels, for example additional statistical tests.

Some related functions are also available in matplotlib, nitime, and scikits.talkbox. Those functions are designed more for the use in signal processing where longer time series are available and work more often in the frequency domain.

## Descriptive Statistics and Tests¶

 stattools.acovf(x[, unbiased, demean, fft, …]) Autocovariance for 1D stattools.acf(x[, unbiased, nlags, qstat, …]) Autocorrelation function for 1d arrays. stattools.pacf(x[, nlags, method, alpha]) Partial autocorrelation estimated stattools.pacf_yw(x[, nlags, method]) Partial autocorrelation estimated with non-recursive yule_walker stattools.pacf_ols(x[, nlags]) Calculate partial autocorrelations stattools.ccovf(x, y[, unbiased, demean]) crosscovariance for 1D stattools.ccf(x, y[, unbiased]) cross-correlation function for 1d stattools.periodogram(X) Returns the periodogram for the natural frequency of X stattools.adfuller(x[, maxlag, regression, …]) Augmented Dickey-Fuller unit root test stattools.kpss(x[, regression, lags, store]) Kwiatkowski-Phillips-Schmidt-Shin test for stationarity. stattools.coint(y0, y1[, trend, method, …]) Test for no-cointegration of a univariate equation stattools.bds(x[, max_dim, epsilon, distance]) Calculate the BDS test statistic for independence of a time series stattools.q_stat(x, nobs[, type]) Return’s Ljung-Box Q Statistic stattools.grangercausalitytests(x, maxlag[, …]) four tests for granger non causality of 2 timeseries stattools.levinson_durbin(s[, nlags, isacov]) Levinson-Durbin recursion for autoregressive processes stattools.arma_order_select_ic(y[, max_ar, …]) Returns information criteria for many ARMA models x13.x13_arima_select_order(endog[, …]) Perform automatic seaonal ARIMA order identification using x12/x13 ARIMA. x13.x13_arima_analysis(endog[, maxorder, …]) Perform x13-arima analysis for monthly or quarterly data.

## Estimation¶

The following are the main estimation classes, which can be accessed through statsmodels.tsa.api and their result classes

### Univariate Autogressive Processes (AR)¶

 ar_model.AR(endog[, dates, freq, missing]) Autoregressive AR(p) model ar_model.ARResults(model, params[, …]) Class to hold results from fitting an AR model.

### Autogressive Moving-Average Processes (ARMA) and Kalman Filter¶

 arima_model.ARMA(endog, order[, exog, …]) Autoregressive Moving Average ARMA(p,q) Model arima_model.ARMAResults(model, params[, …]) Class to hold results from fitting an ARMA model. arima_model.ARIMA(endog, order[, exog, …]) Autoregressive Integrated Moving Average ARIMA(p,d,q) Model arima_model.ARIMAResults(model, params[, …]) kalmanf.kalmanfilter.KalmanFilter Kalman Filter code intended for use with the ARMA model.

### Exponential Smoothing¶

 holtwinters.ExponentialSmoothing(endog[, …]) Holt Winter’s Exponential Smoothing holtwinters.SimpleExpSmoothing(endog) Simple Exponential Smoothing wrapper(…) holtwinters.Holt(endog[, exponential, damped]) Holt’s Exponential Smoothing wrapper(…) holtwinters.HoltWintersResults(model, …) Holt Winter’s Exponential Smoothing Results

### Vector Autogressive Processes (VAR)¶

 vector_ar.var_model.VAR(endog[, exog, …]) Fit VAR(p) process and do lag order selection vector_ar.var_model.VARResults(endog, …[, …]) Estimate VAR(p) process with fixed number of lags vector_ar.dynamic.DynamicVAR(data[, …]) Estimates time-varying vector autoregression (VAR(p)) using equation-by-equation least squares

See also

tutorial VAR documentation

## Vector Autogressive Processes (VAR)¶

Besides estimation, several process properties and additional results after estimation are available for vector autoregressive processes.

 vector_ar.var_model.LagOrderResults(ics, …) Results class for choosing a model’s lag order. vector_ar.var_model.VAR(endog[, exog, …]) Fit VAR(p) process and do lag order selection vector_ar.var_model.VARProcess(coefs, …[, …]) Class represents a known VAR(p) process vector_ar.var_model.VARResults(endog, …[, …]) Estimate VAR(p) process with fixed number of lags vector_ar.irf.IRAnalysis(model[, P, …]) Impulse response analysis class. vector_ar.var_model.FEVD(model[, P, periods]) Compute and plot Forecast error variance decomposition and asymptotic standard errors vector_ar.hypothesis_test_results.HypothesisTestResults(…) Results class for hypothesis tests. vector_ar.hypothesis_test_results.CausalityTestResults(…) Results class for Granger-causality and instantaneous causality. vector_ar.hypothesis_test_results.NormalityTestResults(…) Results class for the Jarque-Bera-test for nonnormality. vector_ar.hypothesis_test_results.WhitenessTestResults(…) Results class for the Portmanteau-test for residual autocorrelation. vector_ar.dynamic.DynamicVAR(data[, …]) Estimates time-varying vector autoregression (VAR(p)) using equation-by-equation least squares

See also

tutorial VAR documentation

## Vector Error Correction Models (VECM)¶

 vector_ar.vecm.select_order(data, maxlags[, …]) Compute lag order selections based on each of the available information criteria. vector_ar.vecm.select_coint_rank(endog, …) Calculate the cointegration rank of a VECM. vector_ar.vecm.CointRankResults(rank, neqs, …) A class for holding the results from testing the cointegration rank. vector_ar.vecm.VECM(endog[, exog, …]) Class representing a Vector Error Correction Model (VECM). vector_ar.vecm.VECMResults(endog, exog, …) Class for holding estimation related results of a vector error correction model (VECM). vector_ar.vecm.coint_johansen(endog, …) Perform the Johansen cointegration test for determining the cointegration rank of a VECM.

## Regime switching models¶

 regime_switching.markov_regression.MarkovRegression(…) First-order k-regime Markov switching regression model regime_switching.markov_autoregression.MarkovAutoregression(…) Markov switching regression model

## ARMA Process¶

The following are tools to work with the theoretical properties of an ARMA process for given lag-polynomials.

 arima_process.ArmaProcess([ar, ma, nobs]) Theoretical properties of an ARMA process for specified lag-polynomials arima_process.ar2arma(ar_des, p, q[, n, …]) Find arma approximation to ar process arima_process.arma2ar(ar, ma[, lags]) Get the AR representation of an ARMA process arima_process.arma2ma(ar, ma[, lags]) Get the MA representation of an ARMA process arima_process.arma_acf(ar, ma[, lags]) Theoretical autocorrelation function of an ARMA process arima_process.arma_acovf(ar, ma[, nobs]) Theoretical autocovariance function of ARMA process arima_process.arma_generate_sample(ar, ma, …) Generate a random sample of an ARMA process arima_process.arma_impulse_response(ar, ma) Get the impulse response function (MA representation) for ARMA process arima_process.arma_pacf(ar, ma[, lags]) Partial autocorrelation function of an ARMA process arima_process.arma_periodogram(ar, ma[, …]) Periodogram for ARMA process given by lag-polynomials ar and ma arima_process.deconvolve(num, den[, n]) Deconvolves divisor out of signal, division of polynomials for n terms arima_process.index2lpol(coeffs, index) Expand coefficients to lag poly arima_process.lpol2index(ar) Remove zeros from lag polynomial arima_process.lpol_fiar(d[, n]) AR representation of fractional integration arima_process.lpol_fima(d[, n]) MA representation of fractional integration arima_process.lpol_sdiff(s) return coefficients for seasonal difference (1-L^s)
 sandbox.tsa.fftarma.ArmaFft(ar, ma, n) fft tools for arma processes

## Time Series Filters¶

 filters.bk_filter.bkfilter(X[, low, high, K]) Baxter-King bandpass filter filters.hp_filter.hpfilter(X[, lamb]) Hodrick-Prescott filter filters.cf_filter.cffilter(X[, low, high, drift]) Christiano Fitzgerald asymmetric, random walk filter filters.filtertools.convolution_filter(x, filt) Linear filtering via convolution. filters.filtertools.recursive_filter(x, ar_coeff) Autoregressive, or recursive, filtering. filters.filtertools.miso_lfilter(ar, ma, x) use nd convolution to merge inputs, then use lfilter to produce output filters.filtertools.fftconvolve3(in1[, in2, …]) Convolve two N-dimensional arrays using FFT. filters.filtertools.fftconvolveinv(in1, in2) Convolve two N-dimensional arrays using FFT. seasonal.seasonal_decompose(x[, model, …]) Seasonal decomposition using moving averages

## TSA Tools¶

 tsatools.add_trend(x[, trend, prepend, …]) Adds a trend and/or constant to an array. tsatools.detrend(x[, order, axis]) Detrend an array with a trend of given order along axis 0 or 1 tsatools.lagmat(x, maxlag[, trim, original, …]) Create 2d array of lags tsatools.lagmat2ds(x, maxlag0[, maxlagex, …]) Generate lagmatrix for 2d array, columns arranged by variables

## VARMA Process¶

 varma_process.VarmaPoly(ar[, ma]) class to keep track of Varma polynomial format

## Interpolation¶

 interp.denton.dentonm(indicator, benchmark) Modified Denton’s method to convert low-frequency to high-frequency data.