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<h1 id="time-series-analysis-by-state-space-methods-statespace"><span id="statespace"></span><span id="module-statsmodels.tsa.statespace"></span>Time Series Analysis by State Space Methods <code class="xref py py-mod docutils literal notranslate"><span class="pre">statespace</span></code><a class="headerlink" href="#time-series-analysis-by-state-space-methods-statespace" title="Link to this heading">¶</a></h1>
<p><a class="reference internal" href="#module-statsmodels.tsa.statespace" title="statsmodels.tsa.statespace: Statespace models for time-series analysis"><code class="xref py py-mod docutils literal notranslate"><span class="pre">statsmodels.tsa.statespace</span></code></a> contains classes and functions that are
useful for time series analysis using state space methods.</p>
<p>A general state space model is of the form</p>
<div class="math notranslate nohighlight">
\[\begin{split}y_t &amp; = Z_t \alpha_t + d_t + \varepsilon_t \\
\alpha_{t+1} &amp; = T_t \alpha_t + c_t + R_t \eta_t \\\end{split}\]</div>
<p>where <span class="math notranslate nohighlight">\(y_t\)</span> refers to the observation vector at time <span class="math notranslate nohighlight">\(t\)</span>,
<span class="math notranslate nohighlight">\(\alpha_t\)</span> refers to the (unobserved) state vector at time
<span class="math notranslate nohighlight">\(t\)</span>, and where the irregular components are defined as</p>
<div class="math notranslate nohighlight">
\[\begin{split}\varepsilon_t \sim N(0, H_t) \\
\eta_t \sim N(0, Q_t) \\\end{split}\]</div>
<p>The remaining variables (<span class="math notranslate nohighlight">\(Z_t, d_t, H_t, T_t, c_t, R_t, Q_t\)</span>) in the
equations are matrices describing the process. Their variable names and
dimensions are as follows</p>
<p>Z : <cite>design</cite>          <span class="math notranslate nohighlight">\((k\_endog \times k\_states \times nobs)\)</span></p>
<p>d : <cite>obs_intercept</cite>   <span class="math notranslate nohighlight">\((k\_endog \times nobs)\)</span></p>
<p>H : <cite>obs_cov</cite>         <span class="math notranslate nohighlight">\((k\_endog \times k\_endog \times nobs)\)</span></p>
<p>T : <cite>transition</cite>      <span class="math notranslate nohighlight">\((k\_states \times k\_states \times nobs)\)</span></p>
<p>c : <cite>state_intercept</cite> <span class="math notranslate nohighlight">\((k\_states \times nobs)\)</span></p>
<p>R : <cite>selection</cite>       <span class="math notranslate nohighlight">\((k\_states \times k\_posdef \times nobs)\)</span></p>
<p>Q : <cite>state_cov</cite>       <span class="math notranslate nohighlight">\((k\_posdef \times k\_posdef \times nobs)\)</span></p>
<p>In the case that one of the matrices is time-invariant (so that, for
example, <span class="math notranslate nohighlight">\(Z_t = Z_{t+1} ~ \forall ~ t\)</span>), its last dimension may
be of size <span class="math notranslate nohighlight">\(1\)</span> rather than size <cite>nobs</cite>.</p>
<p>This generic form encapsulates many of the most popular linear time series
models (see below) and is very flexible, allowing estimation with missing
observations, forecasting, impulse response functions, and much more.</p>
<p><strong>Example: AR(2) model</strong></p>
<p>An autoregressive model is a good introductory example to putting models in
state space form. Recall that an AR(2) model is often written as:</p>
<div class="math notranslate nohighlight">
\[y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \epsilon_t,
\quad \epsilon_t \sim N(0, \sigma^2)\]</div>
<p>This can be put into state space form in the following way:</p>
<div class="math notranslate nohighlight">
\[\begin{split}y_t &amp; = \begin{bmatrix} 1 &amp; 0 \end{bmatrix} \alpha_t \\
\alpha_{t+1} &amp; = \begin{bmatrix}
   \phi_1 &amp; \phi_2 \\
        1 &amp;      0
\end{bmatrix} \alpha_t + \begin{bmatrix} 1 \\ 0 \end{bmatrix} \eta_t\end{split}\]</div>
<p>Where</p>
<div class="math notranslate nohighlight">
\[Z_t \equiv Z = \begin{bmatrix} 1 &amp; 0 \end{bmatrix}\]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[\begin{split}T_t \equiv T &amp; = \begin{bmatrix}
   \phi_1 &amp; \phi_2 \\
        1 &amp;      0
\end{bmatrix} \\
R_t \equiv R &amp; = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\
\eta_t \equiv \epsilon_{t+1} &amp; \sim N(0, \sigma^2)\end{split}\]</div>
<p>There are three unknown parameters in this model:
<span class="math notranslate nohighlight">\(\phi_1, \phi_2, \sigma^2\)</span>.</p>
<h2 id="models-and-estimation">Models and Estimation<a class="headerlink" href="#models-and-estimation" title="Link to this heading">¶</a></h2>
<p>The following are the main estimation classes, which can be accessed through
<cite>statsmodels.tsa.statespace.api</cite> and their result classes.</p>
<h3 id="seasonal-autoregressive-integrated-moving-average-with-exogenous-regressors-sarimax">Seasonal Autoregressive Integrated Moving-Average with eXogenous regressors (SARIMAX)<a class="headerlink" href="#seasonal-autoregressive-integrated-moving-average-with-exogenous-regressors-sarimax" title="Link to this heading">¶</a></h3>
<p>The <cite>SARIMAX</cite> class is an example of a fully fledged model created using the
statespace backend for estimation. <cite>SARIMAX</cite> can be used very similarly to
<a class="reference internal" href="tsa.html#tsa"><span class="std std-ref">tsa</span></a> models, but works on a wider range of models by adding the
estimation of additive and multiplicative seasonal effects, as well as
arbitrary trend polynomials.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.sarimax.SARIMAX.html#statsmodels.tsa.statespace.sarimax.SARIMAX" title="statsmodels.tsa.statespace.sarimax.SARIMAX (Python class) — Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors model"><code class="xref py py-obj docutils literal notranslate"><span class="pre">sarimax.SARIMAX</span></code></a>(endog[, exog, order, ...])</p></td>
<td><p>Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors model</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.sarimax.SARIMAXResults.html#statsmodels.tsa.statespace.sarimax.SARIMAXResults" title="statsmodels.tsa.statespace.sarimax.SARIMAXResults (Python class) — Class to hold results from fitting an SARIMAX model."><code class="xref py py-obj docutils literal notranslate"><span class="pre">sarimax.SARIMAXResults</span></code></a>(model, params, ...[, ...])</p></td>
<td><p>Class to hold results from fitting an SARIMAX model.</p></td>
</tr>
</tbody>
</table>
<p>For an example of the use of this model, see the
<a class="reference internal" href="examples/notebooks/generated/statespace_sarimax_stata.html"><span class="doc">SARIMAX example notebook</span></a>
or the very brief code snippet below:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><code><span class="c1"># Load the statsmodels api</span>
<span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>

<span class="c1"># Load your dataset</span>
<span class="n">endog</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;your/dataset/here.csv&#39;</span><span class="p">)</span>

<span class="c1"># We could fit an AR(2) model, described above</span>
<span class="n">mod_ar2</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">SARIMAX</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span>
<span class="c1"># Note that mod_ar2 is an instance of the SARIMAX class</span>

<span class="c1"># Fit the model via maximum likelihood</span>
<span class="n">res_ar2</span> <span class="o">=</span> <span class="n">mod_ar2</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="c1"># Note that res_ar2 is an instance of the SARIMAXResults class</span>

<span class="c1"># Show the summary of results</span>
<span class="nb">print</span><span class="p">(</span><span class="n">res_ar2</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>

<span class="c1"># We could also fit a more complicated model with seasonal components.</span>
<span class="c1"># As an example, here is an SARIMA(1,1,1) x (0,1,1,4):</span>
<span class="n">mod_sarimax</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">SARIMAX</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span>
                             <span class="n">seasonal_order</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">4</span><span class="p">))</span>
<span class="n">res_sarimax</span> <span class="o">=</span> <span class="n">mod_sarimax</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>

<span class="c1"># Show the summary of results</span>
<span class="nb">print</span><span class="p">(</span><span class="n">res_sarimax</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
</code></pre></div>
</div>
<p>The results object has many of the attributes and methods you would expect from
other statsmodels results objects, including standard errors, z-statistics,
and prediction / forecasting.</p>
<p>Behind the scenes, the <cite>SARIMAX</cite> model creates the design and transition
matrices (and sometimes some of the other matrices) based on the model
specification.</p>
<h3 id="unobserved-components">Unobserved Components<a class="headerlink" href="#unobserved-components" title="Link to this heading">¶</a></h3>
<p>The <cite>UnobservedComponents</cite> class is another example of a statespace model.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.structural.UnobservedComponents.html#statsmodels.tsa.statespace.structural.UnobservedComponents" title="statsmodels.tsa.statespace.structural.UnobservedComponents (Python class) — Univariate unobserved components time series model"><code class="xref py py-obj docutils literal notranslate"><span class="pre">structural.UnobservedComponents</span></code></a>(endog[, ...])</p></td>
<td><p>Univariate unobserved components time series model</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.structural.UnobservedComponentsResults.html#statsmodels.tsa.statespace.structural.UnobservedComponentsResults" title="statsmodels.tsa.statespace.structural.UnobservedComponentsResults (Python class) — Class to hold results from fitting an unobserved components model."><code class="xref py py-obj docutils literal notranslate"><span class="pre">structural.UnobservedComponentsResults</span></code></a>(...)</p></td>
<td><p>Class to hold results from fitting an unobserved components model.</p></td>
</tr>
</tbody>
</table>
<p>For examples of the use of this model, see the <a class="reference internal" href="examples/notebooks/generated/statespace_structural_harvey_jaeger.html"><span class="doc">example notebook</span></a> or a notebook on using the unobserved components model to <a class="reference internal" href="examples/notebooks/generated/statespace_cycles.html"><span class="doc">decompose a time series into a trend and cycle</span></a> or the very brief code snippet below:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><code><span class="c1"># Load the statsmodels api</span>
<span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>

<span class="c1"># Load your dataset</span>
<span class="n">endog</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;your/dataset/here.csv&#39;</span><span class="p">)</span>

<span class="c1"># Fit a local level model</span>
<span class="n">mod_ll</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">UnobservedComponents</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="s1">&#39;local level&#39;</span><span class="p">)</span>
<span class="c1"># Note that mod_ll is an instance of the UnobservedComponents class</span>

<span class="c1"># Fit the model via maximum likelihood</span>
<span class="n">res_ll</span> <span class="o">=</span> <span class="n">mod_ll</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="c1"># Note that res_ll is an instance of the UnobservedComponentsResults class</span>

<span class="c1"># Show the summary of results</span>
<span class="nb">print</span><span class="p">(</span><span class="n">res_ll</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>

<span class="c1"># Show a plot of the estimated level and trend component series</span>
<span class="n">fig_ll</span> <span class="o">=</span> <span class="n">res_ll</span><span class="o">.</span><span class="n">plot_components</span><span class="p">()</span>

<span class="c1"># We could further add a damped stochastic cycle as follows</span>
<span class="n">mod_cycle</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">UnobservedComponents</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="s1">&#39;local level&#39;</span><span class="p">,</span> <span class="n">cycle</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
                                        <span class="n">damped_cycle</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
                                        <span class="n">stochastic_cycle</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">res_cycle</span> <span class="o">=</span> <span class="n">mod_cycle</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>

<span class="c1"># Show the summary of results</span>
<span class="nb">print</span><span class="p">(</span><span class="n">res_cycle</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>

<span class="c1"># Show a plot of the estimated level, trend, and cycle component series</span>
<span class="n">fig_cycle</span> <span class="o">=</span> <span class="n">res_cycle</span><span class="o">.</span><span class="n">plot_components</span><span class="p">()</span>
</code></pre></div>
</div>
<h3 id="vector-autoregressive-moving-average-with-exogenous-regressors-varmax">Vector Autoregressive Moving-Average with eXogenous regressors (VARMAX)<a class="headerlink" href="#vector-autoregressive-moving-average-with-exogenous-regressors-varmax" title="Link to this heading">¶</a></h3>
<p>The <cite>VARMAX</cite> class is an example of a multivariate statespace model.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.varmax.VARMAX.html#statsmodels.tsa.statespace.varmax.VARMAX" title="statsmodels.tsa.statespace.varmax.VARMAX (Python class) — Vector Autoregressive Moving Average with eXogenous regressors model"><code class="xref py py-obj docutils literal notranslate"><span class="pre">varmax.VARMAX</span></code></a>(endog[, exog, order, trend, ...])</p></td>
<td><p>Vector Autoregressive Moving Average with eXogenous regressors model</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.varmax.VARMAXResults.html#statsmodels.tsa.statespace.varmax.VARMAXResults" title="statsmodels.tsa.statespace.varmax.VARMAXResults (Python class) — Class to hold results from fitting an VARMAX model."><code class="xref py py-obj docutils literal notranslate"><span class="pre">varmax.VARMAXResults</span></code></a>(model, params, ...[, ...])</p></td>
<td><p>Class to hold results from fitting an VARMAX model.</p></td>
</tr>
</tbody>
</table>
<p>For an example of the use of this model, see the <a class="reference internal" href="examples/notebooks/generated/statespace_varmax.html"><span class="doc">VARMAX example notebook</span></a> or the very brief code snippet below:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><code><span class="c1"># Load the statsmodels api</span>
<span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>

<span class="c1"># Load your (multivariate) dataset</span>
<span class="n">endog</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;your/dataset/here.csv&#39;</span><span class="p">)</span>

<span class="c1"># Fit a local level model</span>
<span class="n">mod_var1</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">VARMAX</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">))</span>
<span class="c1"># Note that mod_var1 is an instance of the VARMAX class</span>

<span class="c1"># Fit the model via maximum likelihood</span>
<span class="n">res_var1</span> <span class="o">=</span> <span class="n">mod_var1</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="c1"># Note that res_var1 is an instance of the VARMAXResults class</span>

<span class="c1"># Show the summary of results</span>
<span class="nb">print</span><span class="p">(</span><span class="n">res_var1</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>

<span class="c1"># Construct impulse responses</span>
<span class="n">irfs</span> <span class="o">=</span> <span class="n">res_ll</span><span class="o">.</span><span class="n">impulse_responses</span><span class="p">(</span><span class="n">steps</span><span class="o">=</span><span class="mi">10</span><span class="p">)</span>
</code></pre></div>
</div>
<h3 id="dynamic-factor-models">Dynamic Factor Models<a class="headerlink" href="#dynamic-factor-models" title="Link to this heading">¶</a></h3>
<p>Statsmodels has two classes that support dynamic factor models:
<cite>DynamicFactorMQ</cite> and <cite>DynamicFactor</cite>. Each of these models has strengths, but
in general the <cite>DynamicFactorMQ</cite> class is recommended. This is because it fits
parameters using the Expectation-Maximization (EM) algorithm, which is more
robust and can handle including hundreds of observed series. In addition, it
allows customization of which variables load on which factors. However, it does
not yet support including exogenous variables, while <cite>DynamicFactor</cite> does
support that feature.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.dynamic_factor_mq.DynamicFactorMQ.html#statsmodels.tsa.statespace.dynamic_factor_mq.DynamicFactorMQ" title="statsmodels.tsa.statespace.dynamic_factor_mq.DynamicFactorMQ (Python class) — Dynamic factor model with EM algorithm; option for monthly/quarterly data."><code class="xref py py-obj docutils literal notranslate"><span class="pre">dynamic_factor_mq.DynamicFactorMQ</span></code></a>(endog[, ...])</p></td>
<td><p>Dynamic factor model with EM algorithm; option for monthly/quarterly data.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.dynamic_factor_mq.DynamicFactorMQResults.html#statsmodels.tsa.statespace.dynamic_factor_mq.DynamicFactorMQResults" title="statsmodels.tsa.statespace.dynamic_factor_mq.DynamicFactorMQResults (Python class) — Results from fitting a dynamic factor model"><code class="xref py py-obj docutils literal notranslate"><span class="pre">dynamic_factor_mq.DynamicFactorMQResults</span></code></a>(...)</p></td>
<td><p>Results from fitting a dynamic factor model</p></td>
</tr>
</tbody>
</table>
<p>For an example of the <cite>DynamicFactorMQ</cite> class, see the very brief code snippet below:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><code><span class="c1"># Load the statsmodels api</span>
<span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>

<span class="c1"># Load your dataset</span>
<span class="n">endog</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;your/dataset/here.csv&#39;</span><span class="p">)</span>

<span class="c1"># Create a dynamic factor model</span>
<span class="n">mod_dfm</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">DynamicFactorMQ</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">k_factors</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">factor_order</span><span class="o">=</span><span class="mi">2</span><span class="p">)</span>
<span class="c1"># Note that mod_dfm is an instance of the DynamicFactorMQ class</span>

<span class="c1"># Fit the model via maximum likelihood, using the EM algorithm</span>
<span class="n">res_dfm</span> <span class="o">=</span> <span class="n">mod_dfm</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="c1"># Note that res_dfm is an instance of the DynamicFactorMQResults class</span>

<span class="c1"># Show the summary of results</span>
<span class="nb">print</span><span class="p">(</span><span class="n">res_ll</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>

<span class="c1"># Show a plot of the r^2 values from regressions of</span>
<span class="c1"># individual estimated factors on endogenous variables.</span>
<span class="n">fig_dfm</span> <span class="o">=</span> <span class="n">res_ll</span><span class="o">.</span><span class="n">plot_coefficients_of_determination</span><span class="p">()</span>
</code></pre></div>
</div>
<p>The <cite>DynamicFactor</cite> class is suitable for models with a smaller number of
observed variables</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.dynamic_factor.DynamicFactor.html#statsmodels.tsa.statespace.dynamic_factor.DynamicFactor" title="statsmodels.tsa.statespace.dynamic_factor.DynamicFactor (Python class) — Dynamic factor model"><code class="xref py py-obj docutils literal notranslate"><span class="pre">dynamic_factor.DynamicFactor</span></code></a>(endog, ...[, ...])</p></td>
<td><p>Dynamic factor model</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.dynamic_factor.DynamicFactorResults.html#statsmodels.tsa.statespace.dynamic_factor.DynamicFactorResults" title="statsmodels.tsa.statespace.dynamic_factor.DynamicFactorResults (Python class) — Class to hold results from fitting an DynamicFactor model."><code class="xref py py-obj docutils literal notranslate"><span class="pre">dynamic_factor.DynamicFactorResults</span></code></a>(model, ...)</p></td>
<td><p>Class to hold results from fitting an DynamicFactor model.</p></td>
</tr>
</tbody>
</table>
<p>For an example of the use of the <cite>DynamicFactor</cite> model, see the
<a class="reference internal" href="examples/notebooks/generated/statespace_dfm_coincident.html"><span class="doc">Dynamic Factor example notebook</span></a></p>
<h3 id="linear-exponential-smoothing-models">Linear Exponential Smoothing Models<a class="headerlink" href="#linear-exponential-smoothing-models" title="Link to this heading">¶</a></h3>
<p>The <cite>ExponentialSmoothing</cite> class is an implementation of linear exponential
smoothing models using a state space approach.</p>
<p><strong>Note</strong>: this model is available at <cite>sm.tsa.statespace.ExponentialSmoothing</cite>;
it is not the same as the model available at <cite>sm.tsa.ExponentialSmoothing</cite>.
See below for details of the differences between these classes.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.exponential_smoothing.ExponentialSmoothing.html#statsmodels.tsa.statespace.exponential_smoothing.ExponentialSmoothing" title="statsmodels.tsa.statespace.exponential_smoothing.ExponentialSmoothing (Python class) — Linear exponential smoothing models"><code class="xref py py-obj docutils literal notranslate"><span class="pre">exponential_smoothing.ExponentialSmoothing</span></code></a>(endog)</p></td>
<td><p>Linear exponential smoothing models</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.exponential_smoothing.ExponentialSmoothingResults.html#statsmodels.tsa.statespace.exponential_smoothing.ExponentialSmoothingResults" title="statsmodels.tsa.statespace.exponential_smoothing.ExponentialSmoothingResults (Python class) — Results from fitting a linear exponential smoothing model"><code class="xref py py-obj docutils literal notranslate"><span class="pre">exponential_smoothing.ExponentialSmoothingResults</span></code></a>(...)</p></td>
<td><p>Results from fitting a linear exponential smoothing model</p></td>
</tr>
</tbody>
</table>
<p>A very brief code snippet follows:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><code><span class="c1"># Load the statsmodels api</span>
<span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>

<span class="c1"># Load your dataset</span>
<span class="n">endog</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_csv</span><span class="p">(</span><span class="s1">&#39;your/dataset/here.csv&#39;</span><span class="p">)</span>

<span class="c1"># Simple exponential smoothing, denoted (A,N,N)</span>
<span class="n">mod_ses</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">statespace</span><span class="o">.</span><span class="n">ExponentialSmoothing</span><span class="p">(</span><span class="n">endog</span><span class="p">)</span>
<span class="n">res_ses</span> <span class="o">=</span> <span class="n">mod_ses</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>

<span class="c1"># Holt&#39;s linear method, denoted (A,A,N)</span>
<span class="n">mod_h</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">statespace</span><span class="o">.</span><span class="n">ExponentialSmoothing</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">trend</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">res_h</span> <span class="o">=</span> <span class="n">mod_h</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>

<span class="c1"># Damped trend model, denoted (A,Ad,N)</span>
<span class="n">mod_dt</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">statespace</span><span class="o">.</span><span class="n">ExponentialSmoothing</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">trend</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
                                                <span class="n">damped_trend</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">res_dt</span> <span class="o">=</span> <span class="n">mod_dt</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>

<span class="c1"># Holt-Winters&#39; trend and seasonality method, denoted (A,A,A)</span>
<span class="c1"># (assuming that `endog` has a seasonal periodicity of 4, for example if it</span>
<span class="c1"># is quarterly data).</span>
<span class="n">mod_hw</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">statespace</span><span class="o">.</span><span class="n">ExponentialSmoothing</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">trend</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
                                                <span class="n">seasonal</span><span class="o">=</span><span class="mi">4</span><span class="p">)</span>
<span class="n">res_hw</span> <span class="o">=</span> <span class="n">mod_hw</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
</code></pre></div>
</div>
<p><strong>Differences between Statsmodels’ exponential smoothing model classes</strong></p>
<p>There are several differences between this model class, available at
<cite>sm.tsa.statespace.ExponentialSmoothing</cite>, and the model class available at
<cite>sm.tsa.ExponentialSmoothing</cite>.</p>
<ul class="simple">
<li><p>This model class only supports <em>linear</em> exponential smoothing models, while
<cite>sm.tsa.ExponentialSmoothing</cite> also supports multiplicative models.</p></li>
<li><p>This model class puts the exponential smoothing models into state space form
and then applies the Kalman filter to estimate the states, while
<cite>sm.tsa.ExponentialSmoothing</cite> is based on exponential smoothing recursions.
In some cases, this can mean that estimating parameters with this model class
will be somewhat slower than with <cite>sm.tsa.ExponentialSmoothing</cite>.</p></li>
<li><p>This model class can produce confidence intervals for forecasts, based on an
assumption of Gaussian errors, while <cite>sm.tsa.ExponentialSmoothing</cite> does not
support confidence intervals.</p></li>
<li><p>This model class supports concentrating initial values out of the objective
function, which can improve performance when there are many initial states to
estimate (for example when the seasonal periodicity is large).</p></li>
<li><p>This model class supports many advanced features available to state space
models, such as diagnostics and fixed parameters.</p></li>
</ul>
<p><strong>Note</strong>: this class is based on a “multiple sources of error” (MSOE) state
space formulation and not a “single source of error” (SSOE) formulation.</p>
<h3 id="custom-state-space-models">Custom state space models<a class="headerlink" href="#custom-state-space-models" title="Link to this heading">¶</a></h3>
<p>The true power of the state space model is to allow the creation and estimation
of custom models. Usually that is done by extending the following two classes,
which bundle all of state space representation, Kalman filtering, and maximum
likelihood fitting functionality for estimation and results output.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEModel.html#statsmodels.tsa.statespace.mlemodel.MLEModel" title="statsmodels.tsa.statespace.mlemodel.MLEModel (Python class) — State space model for maximum likelihood estimation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">mlemodel.MLEModel</span></code></a>(endog, k_states[, exog, ...])</p></td>
<td><p>State space model for maximum likelihood estimation</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.html#statsmodels.tsa.statespace.mlemodel.MLEResults" title="statsmodels.tsa.statespace.mlemodel.MLEResults (Python class) — Class to hold results from fitting a state space model."><code class="xref py py-obj docutils literal notranslate"><span class="pre">mlemodel.MLEResults</span></code></a>(model, params, results)</p></td>
<td><p>Class to hold results from fitting a state space model.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.PredictionResults.html#statsmodels.tsa.statespace.mlemodel.PredictionResults" title="statsmodels.tsa.statespace.mlemodel.PredictionResults (Python class) — Prediction result from MLE models"><code class="xref py py-obj docutils literal notranslate"><span class="pre">mlemodel.PredictionResults</span></code></a>(model, ...[, ...])</p></td>
<td><p>Prediction result from MLE models</p></td>
</tr>
</tbody>
</table>
<p>For a basic example demonstrating creating and estimating a custom state space
model, see the <a class="reference external" href="examples/notebooks/generated/statespace_local_linear_trend.ipynb">Local Linear Trend example notebook</a>.
For a more sophisticated example, see the source code for the <cite>SARIMAX</cite> and
<cite>SARIMAXResults</cite> classes, which are built by extending <cite>MLEModel</cite> and
<cite>MLEResults</cite>.</p>
<p>In simple cases, the model can be constructed entirely using the MLEModel
class. For example, the AR(2) model from above could be constructed and
estimated using only the following code:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><code><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="kn">from</span><span class="w"> </span><span class="nn">scipy.signal</span><span class="w"> </span><span class="kn">import</span> <span class="n">lfilter</span>
<span class="kn">import</span><span class="w"> </span><span class="nn">statsmodels.api</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">sm</span>

<span class="c1"># True model parameters</span>
<span class="n">nobs</span> <span class="o">=</span> <span class="nb">int</span><span class="p">(</span><span class="mf">1e3</span><span class="p">)</span>
<span class="n">true_phi</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2</span><span class="p">]</span>
<span class="n">true_sigma</span> <span class="o">=</span> <span class="mi">1</span><span class="o">**</span><span class="mf">0.5</span>

<span class="c1"># Simulate a time series</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">1234</span><span class="p">)</span>
<span class="n">disturbances</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">true_sigma</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">nobs</span><span class="p">,))</span>
<span class="n">endog</span> <span class="o">=</span> <span class="n">lfilter</span><span class="p">([</span><span class="mi">1</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="n">true_phi</span><span class="p">],</span> <span class="n">disturbances</span><span class="p">)</span>

<span class="c1"># Construct the model</span>
<span class="k">class</span><span class="w"> </span><span class="nc">AR2</span><span class="p">(</span><span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">statespace</span><span class="o">.</span><span class="n">MLEModel</span><span class="p">):</span>
    <span class="k">def</span><span class="w"> </span><span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">endog</span><span class="p">):</span>
        <span class="c1"># Initialize the state space model</span>
        <span class="nb">super</span><span class="p">(</span><span class="n">AR2</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="fm">__init__</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">k_states</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">k_posdef</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
                                  <span class="n">initialization</span><span class="o">=</span><span class="s1">&#39;stationary&#39;</span><span class="p">)</span>

        <span class="c1"># Setup the fixed components of the state space representation</span>
        <span class="bp">self</span><span class="p">[</span><span class="s1">&#39;design&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span>
        <span class="bp">self</span><span class="p">[</span><span class="s1">&#39;transition&#39;</span><span class="p">]</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
                                  <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">]]</span>
        <span class="bp">self</span><span class="p">[</span><span class="s1">&#39;selection&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>

    <span class="c1"># Describe how parameters enter the model</span>
    <span class="k">def</span><span class="w"> </span><span class="nf">update</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">params</span><span class="p">,</span> <span class="n">transformed</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">):</span>
        <span class="n">params</span> <span class="o">=</span> <span class="nb">super</span><span class="p">(</span><span class="n">AR2</span><span class="p">,</span> <span class="bp">self</span><span class="p">)</span><span class="o">.</span><span class="n">update</span><span class="p">(</span><span class="n">params</span><span class="p">,</span> <span class="n">transformed</span><span class="p">,</span> <span class="o">**</span><span class="n">kwargs</span><span class="p">)</span>

        <span class="bp">self</span><span class="p">[</span><span class="s1">&#39;transition&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="p">:]</span> <span class="o">=</span> <span class="n">params</span><span class="p">[:</span><span class="mi">2</span><span class="p">]</span>
        <span class="bp">self</span><span class="p">[</span><span class="s1">&#39;state_cov&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">params</span><span class="p">[</span><span class="mi">2</span><span class="p">]</span>

    <span class="c1"># Specify start parameters and parameter names</span>
    <span class="nd">@property</span>
    <span class="k">def</span><span class="w"> </span><span class="nf">start_params</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span>
        <span class="k">return</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>  <span class="c1"># these are very simple</span>

<span class="c1"># Create and fit the model</span>
<span class="n">mod</span> <span class="o">=</span> <span class="n">AR2</span><span class="p">(</span><span class="n">endog</span><span class="p">)</span>
<span class="n">res</span> <span class="o">=</span> <span class="n">mod</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>
</code></pre></div>
</div>
<p>This results in the following summary table:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><code>                           <span class="n">Statespace</span> <span class="n">Model</span> <span class="n">Results</span>
<span class="o">==============================================================================</span>
<span class="n">Dep</span><span class="o">.</span> <span class="n">Variable</span><span class="p">:</span>                      <span class="n">y</span>   <span class="n">No</span><span class="o">.</span> <span class="n">Observations</span><span class="p">:</span>                 <span class="mi">1000</span>
<span class="n">Model</span><span class="p">:</span>                            <span class="n">AR2</span>   <span class="n">Log</span> <span class="n">Likelihood</span>               <span class="o">-</span><span class="mf">1389.437</span>
<span class="n">Date</span><span class="p">:</span>                <span class="n">Wed</span><span class="p">,</span> <span class="mi">26</span> <span class="n">Oct</span> <span class="mi">2016</span>   <span class="n">AIC</span>                           <span class="mf">2784.874</span>
<span class="n">Time</span><span class="p">:</span>                        <span class="mi">00</span><span class="p">:</span><span class="mi">42</span><span class="p">:</span><span class="mi">03</span>   <span class="n">BIC</span>                           <span class="mf">2799.598</span>
<span class="n">Sample</span><span class="p">:</span>                             <span class="mi">0</span>   <span class="n">HQIC</span>                          <span class="mf">2790.470</span>
                               <span class="o">-</span> <span class="mi">1000</span>
<span class="n">Covariance</span> <span class="n">Type</span><span class="p">:</span>                  <span class="n">opg</span>
<span class="o">==============================================================================</span>
                 <span class="n">coef</span>    <span class="n">std</span> <span class="n">err</span>          <span class="n">z</span>      <span class="n">P</span><span class="o">&gt;|</span><span class="n">z</span><span class="o">|</span>      <span class="p">[</span><span class="mf">0.025</span>      <span class="mf">0.975</span><span class="p">]</span>
<span class="o">------------------------------------------------------------------------------</span>
<span class="n">param</span><span class="mf">.0</span>        <span class="mf">0.4395</span>      <span class="mf">0.030</span>     <span class="mf">14.730</span>      <span class="mf">0.000</span>       <span class="mf">0.381</span>       <span class="mf">0.498</span>
<span class="n">param</span><span class="mf">.1</span>       <span class="o">-</span><span class="mf">0.2055</span>      <span class="mf">0.032</span>     <span class="o">-</span><span class="mf">6.523</span>      <span class="mf">0.000</span>      <span class="o">-</span><span class="mf">0.267</span>      <span class="o">-</span><span class="mf">0.144</span>
<span class="n">param</span><span class="mf">.2</span>        <span class="mf">0.9425</span>      <span class="mf">0.042</span>     <span class="mf">22.413</span>      <span class="mf">0.000</span>       <span class="mf">0.860</span>       <span class="mf">1.025</span>
<span class="o">===================================================================================</span>
<span class="n">Ljung</span><span class="o">-</span><span class="n">Box</span> <span class="p">(</span><span class="n">Q</span><span class="p">):</span>                       <span class="mf">24.25</span>   <span class="n">Jarque</span><span class="o">-</span><span class="n">Bera</span> <span class="p">(</span><span class="n">JB</span><span class="p">):</span>                 <span class="mf">0.22</span>
<span class="n">Prob</span><span class="p">(</span><span class="n">Q</span><span class="p">):</span>                              <span class="mf">0.98</span>   <span class="n">Prob</span><span class="p">(</span><span class="n">JB</span><span class="p">):</span>                         <span class="mf">0.90</span>
<span class="n">Heteroskedasticity</span> <span class="p">(</span><span class="n">H</span><span class="p">):</span>               <span class="mf">1.05</span>   <span class="n">Skew</span><span class="p">:</span>                            <span class="o">-</span><span class="mf">0.04</span>
<span class="n">Prob</span><span class="p">(</span><span class="n">H</span><span class="p">)</span> <span class="p">(</span><span class="n">two</span><span class="o">-</span><span class="n">sided</span><span class="p">):</span>                  <span class="mf">0.66</span>   <span class="n">Kurtosis</span><span class="p">:</span>                         <span class="mf">3.02</span>
<span class="o">===================================================================================</span>

<span class="n">Warnings</span><span class="p">:</span>
<span class="p">[</span><span class="mi">1</span><span class="p">]</span> <span class="n">Covariance</span> <span class="n">matrix</span> <span class="n">calculated</span> <span class="n">using</span> <span class="n">the</span> <span class="n">outer</span> <span class="n">product</span> <span class="n">of</span> <span class="n">gradients</span> <span class="p">(</span><span class="nb">complex</span><span class="o">-</span><span class="n">step</span><span class="p">)</span><span class="o">.</span>
</code></pre></div>
</div>
<p>The results object has many of the attributes and methods you would expect from
other statsmodels results objects, including standard errors, z-statistics,
and prediction / forecasting.</p>
<p>More advanced usage is possible, including specifying parameter
transformations, and specifying names for parameters for a more informative
output summary.</p>
<h2 id="overview-of-usage">Overview of usage<a class="headerlink" href="#overview-of-usage" title="Link to this heading">¶</a></h2>
<p>All state space models follow the typical Statsmodels pattern:</p>
<ol class="arabic simple">
<li><p>Construct a <strong>model instance</strong> with an input dataset</p></li>
<li><p>Apply parameters to the model (for example, using <cite>fit</cite>) to construct a <strong>results instance</strong></p></li>
<li><p>Interact with the results instance to examine the estimated parameters, explore residual diagnostics, and produce forecasts, simulations, or impulse responses.</p></li>
</ol>
<p>An example of this pattern is as follows:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><code><span class="c1"># Load in the example macroeconomic dataset</span>
<span class="n">dta</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">macrodata</span><span class="o">.</span><span class="n">load_pandas</span><span class="p">()</span><span class="o">.</span><span class="n">data</span>
<span class="c1"># Make sure we have an index with an associated frequency, so that</span>
<span class="c1"># we can refer to time periods with date strings or timestamps</span>
<span class="n">dta</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="s1">&#39;1959Q1&#39;</span><span class="p">,</span> <span class="s1">&#39;2009Q3&#39;</span><span class="p">,</span> <span class="n">freq</span><span class="o">=</span><span class="s1">&#39;QS&#39;</span><span class="p">)</span>

<span class="c1"># Step 1: construct an SARIMAX model for US inflation data</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">SARIMAX</span><span class="p">(</span><span class="n">dta</span><span class="o">.</span><span class="n">infl</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">),</span> <span class="n">trend</span><span class="o">=</span><span class="s1">&#39;c&#39;</span><span class="p">)</span>

<span class="c1"># Step 2: fit the model&#39;s parameters by maximum likelihood</span>
<span class="n">results</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>

<span class="c1"># Step 3: explore / use results</span>

<span class="c1"># - Print a table summarizing estimation results</span>
<span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">summary</span><span class="p">())</span>

<span class="c1"># - Print only the estimated parameters</span>
<span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>

<span class="c1"># - Create diagnostic figures based on standardized residuals:</span>
<span class="c1">#   (1) time series graph</span>
<span class="c1">#   (2) histogram</span>
<span class="c1">#   (3) Q-Q plot</span>
<span class="c1">#   (4) correlogram</span>
<span class="n">results</span><span class="o">.</span><span class="n">plot_diagnostics</span><span class="p">()</span>

<span class="c1"># - Examine diagnostic hypothesis tests</span>
<span class="c1"># Jarque-Bera: [test_statistic, pvalue, skewness, kurtosis]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">test_normality</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s1">&#39;jarquebera&#39;</span><span class="p">))</span>
<span class="c1"># Goldfeld-Quandt type test: [test_statistic, pvalue]</span>
<span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">test_heteroskedasticity</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s1">&#39;breakvar&#39;</span><span class="p">))</span>
<span class="c1"># Ljung-Box test: [test_statistic, pvalue] for each lag</span>
<span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">test_serial_correlation</span><span class="p">(</span><span class="n">method</span><span class="o">=</span><span class="s1">&#39;ljungbox&#39;</span><span class="p">))</span>

<span class="c1"># - Forecast the next 4 values</span>
<span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">forecast</span><span class="p">(</span><span class="mi">4</span><span class="p">))</span>

<span class="c1"># - Forecast until 2020Q4</span>
<span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">forecast</span><span class="p">(</span><span class="s1">&#39;2020Q4&#39;</span><span class="p">))</span>

<span class="c1"># - Plot in-sample dynamic prediction starting in 2005Q1</span>
<span class="c1">#   and out-of-sample forecasts until 2010Q4 along with</span>
<span class="c1">#   90% confidence intervals</span>
<span class="n">predict_results</span> <span class="o">=</span> <span class="n">results</span><span class="o">.</span><span class="n">get_prediction</span><span class="p">(</span><span class="n">start</span><span class="o">=</span><span class="s1">&#39;2005Q1&#39;</span><span class="p">,</span> <span class="n">end</span><span class="o">=</span><span class="s1">&#39;2010Q4&#39;</span><span class="p">,</span> <span class="n">dynamic</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">predict_df</span> <span class="o">=</span> <span class="n">predict_results</span><span class="o">.</span><span class="n">summary_frame</span><span class="p">(</span><span class="n">alpha</span><span class="o">=</span><span class="mf">0.10</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">()</span>
<span class="n">predict_df</span><span class="p">[</span><span class="s1">&#39;mean&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">ax</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">fill_between</span><span class="p">(</span><span class="n">predict_df</span><span class="o">.</span><span class="n">index</span><span class="p">,</span> <span class="n">predict_df</span><span class="p">[</span><span class="s1">&#39;mean_ci_lower&#39;</span><span class="p">],</span>
                <span class="n">predict_df</span><span class="p">[</span><span class="s1">&#39;mean_ci_upper&#39;</span><span class="p">],</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>

<span class="c1"># - Simulate two years of new data after the end of the sample</span>
<span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">simulate</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="n">anchor</span><span class="o">=</span><span class="s1">&#39;end&#39;</span><span class="p">))</span>

<span class="c1"># - Impulse responses for two years</span>
<span class="nb">print</span><span class="p">(</span><span class="n">results</span><span class="o">.</span><span class="n">impulse_responses</span><span class="p">(</span><span class="mi">8</span><span class="p">))</span>
</code></pre></div>
</div>
<h2 id="basic-methods-and-attributes-for-estimation-filtering-smoothing">Basic methods and attributes for estimation / filtering / smoothing<a class="headerlink" href="#basic-methods-and-attributes-for-estimation-filtering-smoothing" title="Link to this heading">¶</a></h2>
<p>The most-used methods for a state space model are:</p>
<ul class="simple">
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEModel.fit.html#statsmodels.tsa.statespace.mlemodel.MLEModel.fit" title="statsmodels.tsa.statespace.mlemodel.MLEModel.fit (Python method) — Fits the model by maximum likelihood via Kalman filter."><code class="xref py py-meth docutils literal notranslate"><span class="pre">fit</span></code></a> - estimate parameters via maximum
likelihood and return a results object (this object will have also performed
Kalman filtering and smoothing at the estimated parameters). This is the most
commonly used method.</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEModel.smooth.html#statsmodels.tsa.statespace.mlemodel.MLEModel.smooth" title="statsmodels.tsa.statespace.mlemodel.MLEModel.smooth (Python method) — Kalman smoothing"><code class="xref py py-meth docutils literal notranslate"><span class="pre">smooth</span></code></a> - return a results object
associated with a given vector of parameters after performing Kalman
filtering and smoothing</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEModel.loglike.html#statsmodels.tsa.statespace.mlemodel.MLEModel.loglike" title="statsmodels.tsa.statespace.mlemodel.MLEModel.loglike (Python method) — Loglikelihood evaluation"><code class="xref py py-meth docutils literal notranslate"><span class="pre">loglike</span></code></a> - compute the log-likelihood
of the data using a given vector of parameters</p></li>
</ul>
<p>Some useful attributes of a state space model are:</p>
<ul class="simple">
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEModel.param_names.html#statsmodels.tsa.statespace.mlemodel.MLEModel.param_names" title="statsmodels.tsa.statespace.mlemodel.MLEModel.param_names (Python property) — (list of str) List of human readable parameter names (for parameters actually included in the model)."><code class="xref py py-meth docutils literal notranslate"><span class="pre">param_names</span></code></a> - names of the
parameters used by the model</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEModel.state_names.html#statsmodels.tsa.statespace.mlemodel.MLEModel.state_names" title="statsmodels.tsa.statespace.mlemodel.MLEModel.state_names (Python property) — (list of str) List of human readable names for unobserved states."><code class="xref py py-meth docutils literal notranslate"><span class="pre">state_names</span></code></a> - names of the
elements of the (unobserved) state vector</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEModel.start_params.html#statsmodels.tsa.statespace.mlemodel.MLEModel.start_params" title="statsmodels.tsa.statespace.mlemodel.MLEModel.start_params (Python property) — (array) Starting parameters for maximum likelihood estimation."><code class="xref py py-meth docutils literal notranslate"><span class="pre">start_params</span></code></a> - initial parameter
estimates used a starting values for numerical maximum likelihood
optimization</p></li>
</ul>
<p>Other methods that are used less often are:</p>
<ul class="simple">
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEModel.filter.html#statsmodels.tsa.statespace.mlemodel.MLEModel.filter" title="statsmodels.tsa.statespace.mlemodel.MLEModel.filter (Python method) — Kalman filtering"><code class="xref py py-meth docutils literal notranslate"><span class="pre">filter</span></code></a> - return a results object
associated with a given vector of parameters after only performing Kalman
filtering (but not smoothing)</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEModel.simulation_smoother.html#statsmodels.tsa.statespace.mlemodel.MLEModel.simulation_smoother" title="statsmodels.tsa.statespace.mlemodel.MLEModel.simulation_smoother (Python method) — Retrieve a simulation smoother for the state space model."><code class="xref py py-meth docutils literal notranslate"><span class="pre">simulation_smoother</span></code></a> -
return an object that can perform simulation smoothing</p></li>
</ul>
<h2 id="output-and-postestimation-methods-and-attributes">Output and postestimation methods and attributes<a class="headerlink" href="#output-and-postestimation-methods-and-attributes" title="Link to this heading">¶</a></h2>
<p>Commonly used methods include:</p>
<ul class="simple">
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.summary.html#statsmodels.tsa.statespace.mlemodel.MLEResults.summary" title="statsmodels.tsa.statespace.mlemodel.MLEResults.summary (Python method) — Summarize the Model"><code class="xref py py-meth docutils literal notranslate"><span class="pre">summary</span></code></a> - construct a table that
presents model fit statistics, estimated parameters, and other summary output</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.predict.html#statsmodels.tsa.statespace.mlemodel.MLEResults.predict" title="statsmodels.tsa.statespace.mlemodel.MLEResults.predict (Python method) — In-sample prediction and out-of-sample forecasting"><code class="xref py py-meth docutils literal notranslate"><span class="pre">predict</span></code></a> - compute in-sample
predictions and out-of-sample forecasts (point estimates only)</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.get_prediction.html#statsmodels.tsa.statespace.mlemodel.MLEResults.get_prediction" title="statsmodels.tsa.statespace.mlemodel.MLEResults.get_prediction (Python method) — In-sample prediction and out-of-sample forecasting"><code class="xref py py-meth docutils literal notranslate"><span class="pre">get_prediction</span></code></a> - compute
in-sample predictions and out-of-sample forecasts, including confidence
intervals</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.forecast.html#statsmodels.tsa.statespace.mlemodel.MLEResults.forecast" title="statsmodels.tsa.statespace.mlemodel.MLEResults.forecast (Python method) — Out-of-sample forecasts"><code class="xref py py-meth docutils literal notranslate"><span class="pre">forecast</span></code></a> - compute out-of-sample
forecasts (point estimates only) (this is a convenience wrapper around
<cite>predict</cite>)</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.get_forecast.html#statsmodels.tsa.statespace.mlemodel.MLEResults.get_forecast" title="statsmodels.tsa.statespace.mlemodel.MLEResults.get_forecast (Python method) — Out-of-sample forecasts and prediction intervals"><code class="xref py py-meth docutils literal notranslate"><span class="pre">get_forecast</span></code></a> - compute
out-of-sample forecasts, including confidence intervals (this is a
convenience wrapper around <cite>get_prediction</cite>)</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.simulate.html#statsmodels.tsa.statespace.mlemodel.MLEResults.simulate" title="statsmodels.tsa.statespace.mlemodel.MLEResults.simulate (Python method) — Simulate a new time series following the state space model"><code class="xref py py-meth docutils literal notranslate"><span class="pre">simulate</span></code></a> - simulate new data
according to the state space model</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.impulse_responses.html#statsmodels.tsa.statespace.mlemodel.MLEResults.impulse_responses" title="statsmodels.tsa.statespace.mlemodel.MLEResults.impulse_responses (Python method) — Impulse response function"><code class="xref py py-meth docutils literal notranslate"><span class="pre">impulse_responses</span></code></a> -
compute impulse responses from the state space model</p></li>
</ul>
<p>Commonly used attributes include:</p>
<ul class="simple">
<li><p><code class="xref py py-meth docutils literal notranslate"><span class="pre">params</span></code> - estimated parameters</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.bse.html#statsmodels.tsa.statespace.mlemodel.MLEResults.bse" title="statsmodels.tsa.statespace.mlemodel.MLEResults.bse (Python attribute) — The standard errors of the parameter estimates."><code class="xref py py-meth docutils literal notranslate"><span class="pre">bse</span></code></a> - standard errors of estimated
parameters</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.pvalues.html#statsmodels.tsa.statespace.mlemodel.MLEResults.pvalues" title="statsmodels.tsa.statespace.mlemodel.MLEResults.pvalues (Python attribute) — (array) The p-values associated with the z-statistics of the coefficients. Note that the coefficients are assumed to have a Normal distribution."><code class="xref py py-meth docutils literal notranslate"><span class="pre">pvalues</span></code></a> - p-values associated with
estimated parameters</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.llf.html#statsmodels.tsa.statespace.mlemodel.MLEResults.llf" title="statsmodels.tsa.statespace.mlemodel.MLEResults.llf (Python attribute) — (float) The value of the log-likelihood function evaluated at params."><code class="xref py py-meth docutils literal notranslate"><span class="pre">llf</span></code></a> - log-likelihood of the data at
the estimated parameters</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.sse.html#statsmodels.tsa.statespace.mlemodel.MLEResults.sse" title="statsmodels.tsa.statespace.mlemodel.MLEResults.sse (Python attribute) — (float) Sum of squared errors"><code class="xref py py-meth docutils literal notranslate"><span class="pre">sse</span></code></a>,
<a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.mse.html#statsmodels.tsa.statespace.mlemodel.MLEResults.mse" title="statsmodels.tsa.statespace.mlemodel.MLEResults.mse (Python attribute) — (float) Mean squared error"><code class="xref py py-meth docutils literal notranslate"><span class="pre">mse</span></code></a>, and
<a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.mae.html#statsmodels.tsa.statespace.mlemodel.MLEResults.mae" title="statsmodels.tsa.statespace.mlemodel.MLEResults.mae (Python attribute) — (float) Mean absolute error"><code class="xref py py-meth docutils literal notranslate"><span class="pre">mae</span></code></a> - sum of squared errors,
mean square error, and mean absolute error</p></li>
<li><p>Information criteria, including: <a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.aic.html#statsmodels.tsa.statespace.mlemodel.MLEResults.aic" title="statsmodels.tsa.statespace.mlemodel.MLEResults.aic (Python attribute) — (float) Akaike Information Criterion"><code class="xref py py-meth docutils literal notranslate"><span class="pre">aic</span></code></a>,
<a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.aicc.html#statsmodels.tsa.statespace.mlemodel.MLEResults.aicc" title="statsmodels.tsa.statespace.mlemodel.MLEResults.aicc (Python attribute) — (float) Akaike Information Criterion with small sample correction"><code class="xref py py-meth docutils literal notranslate"><span class="pre">aicc</span></code></a>,
<a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.bic.html#statsmodels.tsa.statespace.mlemodel.MLEResults.bic" title="statsmodels.tsa.statespace.mlemodel.MLEResults.bic (Python attribute) — (float) Bayes Information Criterion"><code class="xref py py-meth docutils literal notranslate"><span class="pre">bic</span></code></a>, and
<a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.hqic.html#statsmodels.tsa.statespace.mlemodel.MLEResults.hqic" title="statsmodels.tsa.statespace.mlemodel.MLEResults.hqic (Python attribute) — (float) Hannan-Quinn Information Criterion"><code class="xref py py-meth docutils literal notranslate"><span class="pre">hquc</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.fittedvalues.html#statsmodels.tsa.statespace.mlemodel.MLEResults.fittedvalues" title="statsmodels.tsa.statespace.mlemodel.MLEResults.fittedvalues (Python attribute) — (array) The predicted values of the model. An (nobs x k_endog) array."><code class="xref py py-meth docutils literal notranslate"><span class="pre">fittedvalues</span></code></a> - fitted values
from the model (note that these are one-step-ahead predictions)</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.resid.html#statsmodels.tsa.statespace.mlemodel.MLEResults.resid" title="statsmodels.tsa.statespace.mlemodel.MLEResults.resid (Python attribute) — (array) The model residuals. An (nobs x k_endog) array."><code class="xref py py-meth docutils literal notranslate"><span class="pre">resid</span></code></a> - residuals from the model (note
that these are one-step-ahead prediction errors)</p></li>
</ul>
<h3 id="estimates-and-covariances-of-the-unobserved-state">Estimates and covariances of the unobserved state<a class="headerlink" href="#estimates-and-covariances-of-the-unobserved-state" title="Link to this heading">¶</a></h3>
<p>It can be useful to compute estimates of the unobserved state vector
conditional on the observed data. These are available in the results object
<a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.states.html#statsmodels.tsa.statespace.mlemodel.MLEResults.states" title="statsmodels.tsa.statespace.mlemodel.MLEResults.states (Python property) — !! processed by numpydoc !!"><code class="xref py py-meth docutils literal notranslate"><span class="pre">states</span></code></a>, which contains the following
elements:</p>
<ul class="simple">
<li><p><cite>states.filtered</cite> - filtered (one-sided) estimates of the state vector. The
estimate of the state vector at time <cite>t</cite> is based on the observed data up
to and including time <cite>t</cite>.</p></li>
<li><p><cite>states.smoothed</cite> - smoothed (two-sided) estimates of the state vector. The
estimate of the state vector at time <cite>t</cite> is based on all observed data in
the sample.</p></li>
<li><p><cite>states.filtered_cov</cite> - filtered (one-sided) covariance of the state vector</p></li>
<li><p><cite>states.smoothed_cov</cite> - smoothed (two-sided) covariance of the state vector</p></li>
</ul>
<p>Each of these elements are Pandas <cite>DataFrame</cite> objects.</p>
<p>As an example, in a “local level + seasonal” model estimated via the
<cite>UnobservedComponents</cite> components class we can get an estimates of the
underlying level and seasonal movements of a series over time.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><code><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>

<span class="c1"># Retrieve monthly retail sales for clothing</span>
<span class="kn">from</span><span class="w"> </span><span class="nn">pandas_datareader.data</span><span class="w"> </span><span class="kn">import</span> <span class="n">DataReader</span>
<span class="n">clothing</span> <span class="o">=</span> <span class="n">DataReader</span><span class="p">(</span><span class="s1">&#39;MRTSSM4481USN&#39;</span><span class="p">,</span> <span class="s1">&#39;fred&#39;</span><span class="p">,</span> <span class="n">start</span><span class="o">=</span><span class="s1">&#39;1992&#39;</span><span class="p">)</span><span class="o">.</span><span class="n">asfreq</span><span class="p">(</span><span class="s1">&#39;MS&#39;</span><span class="p">)[</span><span class="s1">&#39;MRTSSM4481USN&#39;</span><span class="p">]</span>

<span class="c1"># Construct a local level + seasonal model</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">UnobservedComponents</span><span class="p">(</span><span class="n">clothing</span><span class="p">,</span> <span class="s1">&#39;llevel&#39;</span><span class="p">,</span> <span class="n">seasonal</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
<span class="n">results</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>

<span class="c1"># Plot the data, the level, and seasonal</span>
<span class="n">clothing</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">results</span><span class="o">.</span><span class="n">states</span><span class="o">.</span><span class="n">smoothed</span><span class="p">[</span><span class="s1">&#39;level&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
<span class="n">results</span><span class="o">.</span><span class="n">states</span><span class="o">.</span><span class="n">smoothed</span><span class="p">[</span><span class="s1">&#39;seasonal&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">2</span><span class="p">])</span>
</code></pre></div>
</div>
<h3 id="residual-diagnostics">Residual diagnostics<a class="headerlink" href="#residual-diagnostics" title="Link to this heading">¶</a></h3>
<p>Three diagnostic tests are available after estimation of any statespace model,
whether built in or custom, to help assess whether the model conforms to the
underlying statistical assumptions. These tests are:</p>
<ul class="simple">
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.test_normality.html#statsmodels.tsa.statespace.mlemodel.MLEResults.test_normality" title="statsmodels.tsa.statespace.mlemodel.MLEResults.test_normality (Python method) — Test for normality of standardized residuals."><code class="xref py py-meth docutils literal notranslate"><span class="pre">test_normality</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.test_heteroskedasticity.html#statsmodels.tsa.statespace.mlemodel.MLEResults.test_heteroskedasticity" title="statsmodels.tsa.statespace.mlemodel.MLEResults.test_heteroskedasticity (Python method) — Test for heteroskedasticity of standardized residuals"><code class="xref py py-meth docutils literal notranslate"><span class="pre">test_heteroskedasticity</span></code></a></p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.test_serial_correlation.html#statsmodels.tsa.statespace.mlemodel.MLEResults.test_serial_correlation" title="statsmodels.tsa.statespace.mlemodel.MLEResults.test_serial_correlation (Python method) — Ljung-Box test for no serial correlation of standardized residuals"><code class="xref py py-meth docutils literal notranslate"><span class="pre">test_serial_correlation</span></code></a></p></li>
</ul>
<p>A number of standard plots of regression residuals are available for the same
purpose. These can be produced using the command
<a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.plot_diagnostics.html#statsmodels.tsa.statespace.mlemodel.MLEResults.plot_diagnostics" title="statsmodels.tsa.statespace.mlemodel.MLEResults.plot_diagnostics (Python method) — Diagnostic plots for standardized residuals of one endogenous variable"><code class="xref py py-meth docutils literal notranslate"><span class="pre">plot_diagnostics</span></code></a>.</p>
<h3 id="applying-estimated-parameters-to-an-updated-or-different-dataset">Applying estimated parameters to an updated or different dataset<a class="headerlink" href="#applying-estimated-parameters-to-an-updated-or-different-dataset" title="Link to this heading">¶</a></h3>
<p>There are three methods that can be used to apply estimated parameters from a
results object to an updated or different dataset:</p>
<ul class="simple">
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.append.html#statsmodels.tsa.statespace.mlemodel.MLEResults.append" title="statsmodels.tsa.statespace.mlemodel.MLEResults.append (Python method) — Recreate the results object with new data appended to the original data"><code class="xref py py-meth docutils literal notranslate"><span class="pre">append</span></code></a> - retrieve a new results
object with additional observations that follow after the end of the current
sample appended to it (so the new results object contains both the current
sample and the additional observations)</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.extend.html#statsmodels.tsa.statespace.mlemodel.MLEResults.extend" title="statsmodels.tsa.statespace.mlemodel.MLEResults.extend (Python method) — Recreate the results object for new data that extends the original data"><code class="xref py py-meth docutils literal notranslate"><span class="pre">extend</span></code></a> - retrieve a new results
object for additional observations that follow after end of the current
sample (so the new results object contains only the new observations but NOT
the current sample)</p></li>
<li><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEResults.apply.html#statsmodels.tsa.statespace.mlemodel.MLEResults.apply" title="statsmodels.tsa.statespace.mlemodel.MLEResults.apply (Python method) — Apply the fitted parameters to new data unrelated to the original data"><code class="xref py py-meth docutils literal notranslate"><span class="pre">apply</span></code></a> - retrieve a new results object
for a completely different dataset</p></li>
</ul>
<p>One cross-validation exercise on time-series data involves fitting a model’s
parameters based on a training sample (observations through time <cite>t</cite>) and
then evaluating the fit of the model using a test sample (observations <cite>t+1</cite>,
<cite>t+2</cite>, …). This can be conveniently done using either <cite>apply</cite> or <cite>extend</cite>. In
the example below, we use the <cite>extend</cite> method.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><code><span class="c1"># Load in the example macroeconomic dataset</span>
<span class="n">dta</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">datasets</span><span class="o">.</span><span class="n">macrodata</span><span class="o">.</span><span class="n">load_pandas</span><span class="p">()</span><span class="o">.</span><span class="n">data</span>
<span class="c1"># Make sure we have an index with an associated frequency, so that</span>
<span class="c1"># we can refer to time periods with date strings or timestamps</span>
<span class="n">dta</span><span class="o">.</span><span class="n">index</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">date_range</span><span class="p">(</span><span class="s1">&#39;1959Q1&#39;</span><span class="p">,</span> <span class="s1">&#39;2009Q3&#39;</span><span class="p">,</span> <span class="n">freq</span><span class="o">=</span><span class="s1">&#39;QS&#39;</span><span class="p">)</span>

<span class="c1"># Separate inflation data into a training and test dataset</span>
<span class="n">training_endog</span> <span class="o">=</span> <span class="n">dta</span><span class="p">[</span><span class="s1">&#39;infl&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">iloc</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span>
<span class="n">test_endog</span> <span class="o">=</span> <span class="n">dta</span><span class="p">[</span><span class="s1">&#39;infl&#39;</span><span class="p">]</span><span class="o">.</span><span class="n">iloc</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">:]</span>

<span class="c1"># Fit an SARIMAX model for inflation</span>
<span class="n">training_model</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">SARIMAX</span><span class="p">(</span><span class="n">training_endog</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>
<span class="n">training_results</span> <span class="o">=</span> <span class="n">training_model</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>

<span class="c1"># Extend the results to the test observations</span>
<span class="n">test_results</span> <span class="o">=</span> <span class="n">training_results</span><span class="o">.</span><span class="n">extend</span><span class="p">(</span><span class="n">test_endog</span><span class="p">)</span>

<span class="c1"># Print the sum of squared errors in the test sample,</span>
<span class="c1"># based on parameters computed using only the training sample</span>
<span class="nb">print</span><span class="p">(</span><span class="n">test_results</span><span class="o">.</span><span class="n">sse</span><span class="p">)</span>
</code></pre></div>
</div>
<h3 id="understanding-the-impact-of-data-revisions">Understanding the Impact of Data Revisions<a class="headerlink" href="#understanding-the-impact-of-data-revisions" title="Link to this heading">¶</a></h3>
<p>Statespace model results expose a <code class="xref py py-meth docutils literal notranslate"><span class="pre">news</span></code> method that
can be used to understand the impact of data revisions – news – on model
parameters.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.news.NewsResults.html#statsmodels.tsa.statespace.news.NewsResults" title="statsmodels.tsa.statespace.news.NewsResults (Python class) — Impacts of data revisions and news on estimates of variables of interest"><code class="xref py py-obj docutils literal notranslate"><span class="pre">news.NewsResults</span></code></a>(news_results, model, ...[, ...])</p></td>
<td><p>Impacts of data revisions and news on estimates of variables of interest</p></td>
</tr>
</tbody>
</table>
<h2 id="additional-options-and-tools">Additional options and tools<a class="headerlink" href="#additional-options-and-tools" title="Link to this heading">¶</a></h2>
<p>All state space models have the following options and tools:</p>
<h3 id="holding-some-parameters-fixed-and-estimating-the-rest">Holding some parameters fixed and estimating the rest<a class="headerlink" href="#holding-some-parameters-fixed-and-estimating-the-rest" title="Link to this heading">¶</a></h3>
<p>The <a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEModel.fit_constrained.html#statsmodels.tsa.statespace.mlemodel.MLEModel.fit_constrained" title="statsmodels.tsa.statespace.mlemodel.MLEModel.fit_constrained (Python method) — Fit the model with some parameters subject to equality constraints."><code class="xref py py-meth docutils literal notranslate"><span class="pre">fit_constrained</span></code></a> method
allows fixing some parameters to known values and then estimating the rest via
maximum likelihood. An example of this is:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><code><span class="c1"># Construct a model</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">SARIMAX</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>

<span class="c1"># To find out the parameter names, use:</span>
<span class="nb">print</span><span class="p">(</span><span class="n">model</span><span class="o">.</span><span class="n">param_names</span><span class="p">)</span>

<span class="c1"># Fit the model with a fixed value for the AR(1) coefficient:</span>
<span class="n">results</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit_constrained</span><span class="p">({</span><span class="s1">&#39;ar.L1&#39;</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">})</span>
</code></pre></div>
</div>
<p>Alternatively, you can use the
<a class="reference internal" href="generated/statsmodels.tsa.statespace.mlemodel.MLEModel.fix_params.html#statsmodels.tsa.statespace.mlemodel.MLEModel.fix_params" title="statsmodels.tsa.statespace.mlemodel.MLEModel.fix_params (Python method) — Fix parameters to specific values (context manager)"><code class="xref py py-meth docutils literal notranslate"><span class="pre">fix_params</span></code></a> context manager:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><code><span class="c1"># Construct a model</span>
<span class="n">model</span> <span class="o">=</span> <span class="n">sm</span><span class="o">.</span><span class="n">tsa</span><span class="o">.</span><span class="n">SARIMAX</span><span class="p">(</span><span class="n">endog</span><span class="p">,</span> <span class="n">order</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">))</span>

<span class="c1"># Fit the model with a fixed value for the AR(1) coefficient using the</span>
<span class="c1"># context manager</span>
<span class="k">with</span> <span class="n">model</span><span class="o">.</span><span class="n">fix_params</span><span class="p">({</span><span class="s1">&#39;ar.L1&#39;</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">}):</span>
    <span class="n">results</span> <span class="o">=</span> <span class="n">model</span><span class="o">.</span><span class="n">fit</span><span class="p">()</span>
</code></pre></div>
</div>
<h3 id="low-memory-options">Low memory options<a class="headerlink" href="#low-memory-options" title="Link to this heading">¶</a></h3>
<p>When the observed dataset is very large and / or the state vector of the model
is high-dimensional (for example when considering long seasonal effects), the
default memory requirements can be too large. For this reason, the <cite>fit</cite>,
<cite>filter</cite>, and <cite>smooth</cite> methods accept an optional <cite>low_memory=True</cite> argument,
which can considerably reduce memory requirements and speed up model fitting.</p>
<p>Note that when using <cite>low_memory=True</cite>, not all results objects will be
available. However, residual diagnostics, in-sample (non-dynamic) prediction,
and out-of-sample forecasting are all still available.</p>
<h2 id="low-level-state-space-representation-and-kalman-filtering">Low-level state space representation and Kalman filtering<a class="headerlink" href="#low-level-state-space-representation-and-kalman-filtering" title="Link to this heading">¶</a></h2>
<p>While creation of custom models will almost always be done by extending
<cite>MLEModel</cite> and <cite>MLEResults</cite>, it can be useful to understand the superstructure
behind those classes.</p>
<p>Maximum likelihood estimation requires evaluating the likelihood function of
the model, and for models in state space form the likelihood function is
evaluated as a byproduct of running the Kalman filter.</p>
<p>There are two classes used by <cite>MLEModel</cite> that facilitate specification of the
state space model and Kalman filtering: <cite>Representation</cite> and <cite>KalmanFilter</cite>.</p>
<p>The <cite>Representation</cite> class is the piece where the state space model
representation is defined. In simple terms, it holds the state space matrices
(<cite>design</cite>, <cite>obs_intercept</cite>, etc.; see the introduction to state space models,
above) and allows their manipulation.</p>
<p><cite>FrozenRepresentation</cite> is the most basic results-type class, in that it takes a
“snapshot” of the state space representation at any given time. See the class
documentation for the full list of available attributes.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.representation.Representation.html#statsmodels.tsa.statespace.representation.Representation" title="statsmodels.tsa.statespace.representation.Representation (Python class) — State space representation of a time series process"><code class="xref py py-obj docutils literal notranslate"><span class="pre">representation.Representation</span></code></a>(k_endog, k_states)</p></td>
<td><p>State space representation of a time series process</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.representation.FrozenRepresentation.html#statsmodels.tsa.statespace.representation.FrozenRepresentation" title="statsmodels.tsa.statespace.representation.FrozenRepresentation (Python class) — Frozen Statespace Model"><code class="xref py py-obj docutils literal notranslate"><span class="pre">representation.FrozenRepresentation</span></code></a>(model)</p></td>
<td><p>Frozen Statespace Model</p></td>
</tr>
</tbody>
</table>
<p>The <cite>KalmanFilter</cite> class is a subclass of Representation that provides
filtering capabilities. Once the state space representation matrices have been
constructed, the <a class="reference internal" href="generated/statsmodels.tsa.statespace.kalman_filter.KalmanFilter.filter.html#statsmodels.tsa.statespace.kalman_filter.KalmanFilter.filter" title="statsmodels.tsa.statespace.kalman_filter.KalmanFilter.filter (Python method) — Apply the Kalman filter to the statespace model."><code class="xref py py-meth docutils literal notranslate"><span class="pre">filter</span></code></a>
method can be called, producing a <cite>FilterResults</cite> instance; <cite>FilterResults</cite> is
a subclass of <cite>FrozenRepresentation</cite>.</p>
<p>The <cite>FilterResults</cite> class not only holds a frozen representation of the state
space model (the design, transition, etc. matrices, as well as model
dimensions, etc.) but it also holds the filtering output, including the
<code class="xref py py-attr docutils literal notranslate"><span class="pre">filtered</span> <span class="pre">state</span></code> and
loglikelihood (see the class documentation for the full list of available
results). It also provides a
<a class="reference internal" href="generated/statsmodels.tsa.statespace.kalman_filter.FilterResults.predict.html#statsmodels.tsa.statespace.kalman_filter.FilterResults.predict" title="statsmodels.tsa.statespace.kalman_filter.FilterResults.predict (Python method) — In-sample and out-of-sample prediction for state space models generally"><code class="xref py py-meth docutils literal notranslate"><span class="pre">predict</span></code></a> method, which allows
in-sample prediction or out-of-sample forecasting. A similar method,
<code class="xref py py-meth docutils literal notranslate"><span class="pre">predict</span></code>, provides
additional prediction or forecasting results, including confidence intervals.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.kalman_filter.KalmanFilter.html#statsmodels.tsa.statespace.kalman_filter.KalmanFilter" title="statsmodels.tsa.statespace.kalman_filter.KalmanFilter (Python class) — State space representation of a time series process, with Kalman filter"><code class="xref py py-obj docutils literal notranslate"><span class="pre">kalman_filter.KalmanFilter</span></code></a>(k_endog, k_states)</p></td>
<td><p>State space representation of a time series process, with Kalman filter</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.kalman_filter.FilterResults.html#statsmodels.tsa.statespace.kalman_filter.FilterResults" title="statsmodels.tsa.statespace.kalman_filter.FilterResults (Python class) — Results from applying the Kalman filter to a state space model."><code class="xref py py-obj docutils literal notranslate"><span class="pre">kalman_filter.FilterResults</span></code></a>(model)</p></td>
<td><p>Results from applying the Kalman filter to a state space model.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.kalman_filter.PredictionResults.html#statsmodels.tsa.statespace.kalman_filter.PredictionResults" title="statsmodels.tsa.statespace.kalman_filter.PredictionResults (Python class) — Results of in-sample and out-of-sample prediction for state space models generally"><code class="xref py py-obj docutils literal notranslate"><span class="pre">kalman_filter.PredictionResults</span></code></a>(results, ...)</p></td>
<td><p>Results of in-sample and out-of-sample prediction for state space models generally</p></td>
</tr>
</tbody>
</table>
<p>The <cite>KalmanSmoother</cite> class is a subclass of <cite>KalmanFilter</cite> that provides
smoothing capabilities. Once the state space representation matrices have been
constructed, the <a class="reference internal" href="generated/statsmodels.tsa.statespace.kalman_smoother.KalmanSmoother.smooth.html#statsmodels.tsa.statespace.kalman_smoother.KalmanSmoother.smooth" title="statsmodels.tsa.statespace.kalman_smoother.KalmanSmoother.smooth (Python method) — Apply the Kalman smoother to the statespace model."><code class="xref py py-meth docutils literal notranslate"><span class="pre">filter</span></code></a>
method can be called, producing a <cite>SmootherResults</cite> instance; <cite>SmootherResults</cite>
is a subclass of <cite>FilterResults</cite>.</p>
<p>The <cite>SmootherResults</cite> class holds all the output from <cite>FilterResults</cite>, but
also includes smoothing output, including the
<code class="xref py py-attr docutils literal notranslate"><span class="pre">smoothed</span> <span class="pre">state</span></code> and
loglikelihood (see the class documentation for the full list of available
results). Whereas “filtered” output at time <cite>t</cite> refers to estimates conditional
on observations up through time <cite>t</cite>, “smoothed” output refers to estimates
conditional on the entire set of observations in the dataset.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.kalman_smoother.KalmanSmoother.html#statsmodels.tsa.statespace.kalman_smoother.KalmanSmoother" title="statsmodels.tsa.statespace.kalman_smoother.KalmanSmoother (Python class) — State space representation of a time series process, with Kalman filter and smoother."><code class="xref py py-obj docutils literal notranslate"><span class="pre">kalman_smoother.KalmanSmoother</span></code></a>(k_endog, k_states)</p></td>
<td><p>State space representation of a time series process, with Kalman filter and smoother.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.kalman_smoother.SmootherResults.html#statsmodels.tsa.statespace.kalman_smoother.SmootherResults" title="statsmodels.tsa.statespace.kalman_smoother.SmootherResults (Python class) — Results from applying the Kalman smoother and/or filter to a state space model."><code class="xref py py-obj docutils literal notranslate"><span class="pre">kalman_smoother.SmootherResults</span></code></a>(model)</p></td>
<td><p>Results from applying the Kalman smoother and/or filter to a state space model.</p></td>
</tr>
</tbody>
</table>
<p>The <cite>SimulationSmoother</cite> class is a subclass of <cite>KalmanSmoother</cite> that further
provides simulation and simulation smoothing capabilities. The
<a class="reference internal" href="generated/statsmodels.tsa.statespace.simulation_smoother.SimulationSmoother.simulation_smoother.html#statsmodels.tsa.statespace.simulation_smoother.SimulationSmoother.simulation_smoother" title="statsmodels.tsa.statespace.simulation_smoother.SimulationSmoother.simulation_smoother (Python method) — Retrieve a simulation smoother for the statespace model."><code class="xref py py-meth docutils literal notranslate"><span class="pre">simulation_smoother</span></code></a>
method can be called, producing a <cite>SimulationSmoothResults</cite> instance.</p>
<p>The <cite>SimulationSmoothResults</cite> class has a <cite>simulate</cite> method, that allows
performing simulation smoothing to draw from the joint posterior of the state
vector. This is useful for Bayesian estimation of state space models via Gibbs
sampling.</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.simulation_smoother.SimulationSmoother.html#statsmodels.tsa.statespace.simulation_smoother.SimulationSmoother" title="statsmodels.tsa.statespace.simulation_smoother.SimulationSmoother (Python class) — State space representation of a time series process, with Kalman filter and smoother, and with simulation smoother."><code class="xref py py-obj docutils literal notranslate"><span class="pre">simulation_smoother.SimulationSmoother</span></code></a>(...)</p></td>
<td><p>State space representation of a time series process, with Kalman filter and smoother, and with simulation smoother.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.simulation_smoother.SimulationSmoothResults.html#statsmodels.tsa.statespace.simulation_smoother.SimulationSmoothResults" title="statsmodels.tsa.statespace.simulation_smoother.SimulationSmoothResults (Python class) — Results from applying the Kalman smoother and/or filter to a state space model."><code class="xref py py-obj docutils literal notranslate"><span class="pre">simulation_smoother.SimulationSmoothResults</span></code></a>(...)</p></td>
<td><p>Results from applying the Kalman smoother and/or filter to a state space model.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.cfa_simulation_smoother.CFASimulationSmoother.html#statsmodels.tsa.statespace.cfa_simulation_smoother.CFASimulationSmoother" title="statsmodels.tsa.statespace.cfa_simulation_smoother.CFASimulationSmoother (Python class) — &quot;Cholesky Factor Algorithm&quot; (CFA) simulation smoother"><code class="xref py py-obj docutils literal notranslate"><span class="pre">cfa_simulation_smoother.CFASimulationSmoother</span></code></a>(model)</p></td>
<td><p>&quot;Cholesky Factor Algorithm&quot; (CFA) simulation smoother</p></td>
</tr>
</tbody>
</table>
<h2 id="statespace-tools">Statespace Tools<a class="headerlink" href="#statespace-tools" title="Link to this heading">¶</a></h2>
<p>There are a variety of tools used for state space modeling or by the SARIMAX
class:</p>
<table class="autosummary longtable docutils data align-default">
<tbody>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.tools.companion_matrix.html#statsmodels.tsa.statespace.tools.companion_matrix" title="statsmodels.tsa.statespace.tools.companion_matrix (Python function) — Create a companion matrix"><code class="xref py py-obj docutils literal notranslate"><span class="pre">tools.companion_matrix</span></code></a>(polynomial)</p></td>
<td><p>Create a companion matrix</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.tools.diff.html#statsmodels.tsa.statespace.tools.diff" title="statsmodels.tsa.statespace.tools.diff (Python function) — Difference a series simply and/or seasonally along the zero-th axis."><code class="xref py py-obj docutils literal notranslate"><span class="pre">tools.diff</span></code></a>(series[, k_diff, ...])</p></td>
<td><p>Difference a series simply and/or seasonally along the zero-th axis.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.tools.is_invertible.html#statsmodels.tsa.statespace.tools.is_invertible" title="statsmodels.tsa.statespace.tools.is_invertible (Python function) — Determine if a polynomial is invertible."><code class="xref py py-obj docutils literal notranslate"><span class="pre">tools.is_invertible</span></code></a>(polynomial[, threshold])</p></td>
<td><p>Determine if a polynomial is invertible.</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.tools.constrain_stationary_univariate.html#statsmodels.tsa.statespace.tools.constrain_stationary_univariate" title="statsmodels.tsa.statespace.tools.constrain_stationary_univariate (Python function) — Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation"><code class="xref py py-obj docutils literal notranslate"><span class="pre">tools.constrain_stationary_univariate</span></code></a>(...)</p></td>
<td><p>Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.tools.unconstrain_stationary_univariate.html#statsmodels.tsa.statespace.tools.unconstrain_stationary_univariate" title="statsmodels.tsa.statespace.tools.unconstrain_stationary_univariate (Python function) — Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer"><code class="xref py py-obj docutils literal notranslate"><span class="pre">tools.unconstrain_stationary_univariate</span></code></a>(...)</p></td>
<td><p>Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.tools.constrain_stationary_multivariate.html#statsmodels.tsa.statespace.tools.constrain_stationary_multivariate" title="statsmodels.tsa.statespace.tools.constrain_stationary_multivariate (Python function) — Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation for a vector autoregression."><code class="xref py py-obj docutils literal notranslate"><span class="pre">tools.constrain_stationary_multivariate</span></code></a>(...)</p></td>
<td><p>Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation for a vector autoregression.</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.tools.unconstrain_stationary_multivariate.html#statsmodels.tsa.statespace.tools.unconstrain_stationary_multivariate" title="statsmodels.tsa.statespace.tools.unconstrain_stationary_multivariate (Python function) — Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer"><code class="xref py py-obj docutils literal notranslate"><span class="pre">tools.unconstrain_stationary_multivariate</span></code></a>(...)</p></td>
<td><p>Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.tools.validate_matrix_shape.html#statsmodels.tsa.statespace.tools.validate_matrix_shape" title="statsmodels.tsa.statespace.tools.validate_matrix_shape (Python function) — Validate the shape of a possibly time-varying matrix, or raise an exception"><code class="xref py py-obj docutils literal notranslate"><span class="pre">tools.validate_matrix_shape</span></code></a>(name, shape, ...)</p></td>
<td><p>Validate the shape of a possibly time-varying matrix, or raise an exception</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="generated/statsmodels.tsa.statespace.tools.validate_vector_shape.html#statsmodels.tsa.statespace.tools.validate_vector_shape" title="statsmodels.tsa.statespace.tools.validate_vector_shape (Python function) — Validate the shape of a possibly time-varying vector, or raise an exception"><code class="xref py py-obj docutils literal notranslate"><span class="pre">tools.validate_vector_shape</span></code></a>(name, shape, ...)</p></td>
<td><p>Validate the shape of a possibly time-varying vector, or raise an exception</p></td>
</tr>
</tbody>
</table>


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