From f78e81f96229ea01c2496e4a82aefc7958fc3327 Mon Sep 17 00:00:00 2001
From: andybeet <22455149+andybeet@users.noreply.github.com>
Date: Wed, 9 Oct 2024 16:41:16 +0000
Subject: [PATCH] =?UTF-8?q?Deploying=20to=20gh-pages=20from=20@=20andybeet?=
 =?UTF-8?q?/arfit@d3453c4d315d95a99cfcdd4ee354b3aed8e3d0cf=20=F0=9F=9A=80?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 404.html                                |  2 +-
 LICENSE-text.html                       |  2 +-
 articles/Equations.html                 |  2 +-
 articles/arfit.html                     |  4 +-
 articles/background_work.html           |  2 +-
 articles/index.html                     |  2 +-
 authors.html                            |  2 +-
 index.html                              |  2 +-
 pkgdown.js                              |  8 ++
 pkgdown.yml                             |  4 +-
 reference/arfit-package.html            | 11 ++-
 reference/check_data_validation.html    | 69 ++++++++++++++++++
 reference/est_beta_given_rho.html       |  2 +-
 reference/example_cedar_rapids.html     |  2 +-
 reference/fit_ar1_opt.html              |  2 +-
 reference/fit_to_data.html              | 82 +++++++++++++++++++++
 reference/index.html                    | 97 +++----------------------
 reference/likelihood_ar1.html           |  2 +-
 reference/sim_study_opt_ar1.html        | 10 ++-
 reference/simulate_ar1.html             |  8 +-
 reference/simulation_study_opt_ar1.html |  9 ++-
 search.json                             |  2 +-
 sitemap.xml                             | 15 +---
 23 files changed, 212 insertions(+), 129 deletions(-)
 create mode 100644 reference/check_data_validation.html
 create mode 100644 reference/fit_to_data.html

diff --git a/404.html b/404.html
index 1940dfd..76f127b 100644
--- a/404.html
+++ b/404.html
@@ -68,7 +68,7 @@ <h1>Page not found (404)</h1>
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer>
diff --git a/LICENSE-text.html b/LICENSE-text.html
index 731cd5e..b29a0fd 100644
--- a/LICENSE-text.html
+++ b/LICENSE-text.html
@@ -55,7 +55,7 @@ <h1>License</h1>
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/articles/Equations.html b/articles/Equations.html
index a844724..e4023a1 100644
--- a/articles/Equations.html
+++ b/articles/Equations.html
@@ -155,7 +155,7 @@ <h2 id="eq-6">Eq 6<a class="anchor" aria-label="anchor" href="#eq-6"></a>
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer>
diff --git a/articles/arfit.html b/articles/arfit.html
index 5eba513..0ddc16f 100644
--- a/articles/arfit.html
+++ b/articles/arfit.html
@@ -113,7 +113,7 @@ <h1>arfit</h1>
 performed.</p>
 <div class="sourceCode" id="cb2"><pre class="downlit sourceCode r">
 <code class="sourceCode R"><span><span class="va">fits</span> <span class="op">&lt;-</span> <span class="fu"><a href="../reference/example_cedar_rapids.html">example_cedar_rapids</a></span><span class="op">(</span>nBootSims <span class="op">=</span> <span class="fl">999</span><span class="op">)</span></span></code></pre></div>
-<p><img src="figures/Figure1.png"><!-- --></p>
+<p><img src="figures%2FFigure1.png"><!-- --></p>
   </main>
 </div>
 
@@ -124,7 +124,7 @@ <h1>arfit</h1>
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer>
diff --git a/articles/background_work.html b/articles/background_work.html
index d437e9d..7d16961 100644
--- a/articles/background_work.html
+++ b/articles/background_work.html
@@ -500,7 +500,7 @@ <h2 class="unnumbered" id="references">References<a class="anchor" aria-label="a
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer>
diff --git a/articles/index.html b/articles/index.html
index f81d3ff..eb883eb 100644
--- a/articles/index.html
+++ b/articles/index.html
@@ -55,7 +55,7 @@ <h3>All vignettes</h3>
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/authors.html b/authors.html
index 76e5d36..b191b8d 100644
--- a/authors.html
+++ b/authors.html
@@ -70,7 +70,7 @@ <h2 id="citation">Citation</h2>
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/index.html b/index.html
index 63fac01..8bc0b6b 100644
--- a/index.html
+++ b/index.html
@@ -118,7 +118,7 @@ <h2 data-toc-skip>Developers</h2>
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer>
diff --git a/pkgdown.js b/pkgdown.js
index 9757bf9..1a99c65 100644
--- a/pkgdown.js
+++ b/pkgdown.js
@@ -152,3 +152,11 @@ async function searchFuse(query, callback) {
   });
   });
 })(window.jQuery || window.$)
+
+document.addEventListener('keydown', function(event) {
+  // Check if the pressed key is '/'
+  if (event.key === '/') {
+    event.preventDefault();  // Prevent any default action associated with the '/' key
+    document.getElementById('search-input').focus();  // Set focus to the search input
+  }
+});
diff --git a/pkgdown.yml b/pkgdown.yml
index a5a5ead..3bf2b0a 100644
--- a/pkgdown.yml
+++ b/pkgdown.yml
@@ -1,11 +1,11 @@
 pandoc: 3.1.11
-pkgdown: 2.1.0
+pkgdown: 2.1.1
 pkgdown_sha: ~
 articles:
   arfit: arfit.html
   background_work: background_work.html
   Equations: Equations.html
-last_built: 2024-08-15T17:45Z
+last_built: 2024-10-09T16:41Z
 urls:
   reference: https://andybeet.github.io/arfit/reference
   article: https://andybeet.github.io/arfit/articles
diff --git a/reference/arfit-package.html b/reference/arfit-package.html
index e78b37f..9c4b35e 100644
--- a/reference/arfit-package.html
+++ b/reference/arfit-package.html
@@ -36,7 +36,7 @@
   <main id="main" class="col-md-9"><div class="page-header">
 
       <h1>arfit: Test for a trend in presence of autocorrelation</h1>
-      <small class="dont-index">Source: <a href="https://github.com/andybeet/arfit/blob/main/R/arfit.R" class="external-link"><code>R/arfit.R</code></a></small>
+      <small class="dont-index">Source: <a href="https://github.com/andybeet/arfit/blob/main/R/arfit-package.R" class="external-link"><code>R/arfit-package.R</code></a></small>
       <div class="d-none name"><code>arfit-package.Rd</code></div>
     </div>
 
@@ -53,8 +53,8 @@ <h2 id="details">Details<a class="anchor" aria-label="anchor" href="#details"></
 <li><p>Evaluates likelihood</p></li>
 <li><p>Fits likelihood</p></li>
 <li><p>Estimates MLE for model</p></li>
-<li><p>Example provides</p></li>
-<li><p>Tools for su=imulation study</p></li>
+<li><p>Example provided</p></li>
+<li><p>Tools for simulation study</p></li>
 </ul></div>
     <div class="section level2">
     <h2 id="references">References<a class="anchor" aria-label="anchor" href="#references"></a></h2>
@@ -65,8 +65,7 @@ <h2 id="references">References<a class="anchor" aria-label="anchor" href="#refer
     </div>
     <div class="section level2">
     <h2 id="see-also">See also<a class="anchor" aria-label="anchor" href="#see-also"></a></h2>
-    <div class="dont-index"><p>Useful links:</p><ul><li><p><a href="https://github.com/andybeet/arfit" class="external-link">https://github.com/andybeet/arfit</a></p></li>
-<li><p>Report bugs at <a href="https://github.com/andybeet/arfit/issues" class="external-link">https://github.com/andybeet/arfit/issues</a></p></li>
+    <div class="dont-index"><p>Useful links:</p><ul><li><p>Report bugs at <a href="https://github.com/andybeet/arfit/issues" class="external-link">https://github.com/andybeet/arfit/issues</a></p></li>
 </ul></div>
     </div>
     <div class="section level2">
@@ -83,7 +82,7 @@ <h2 id="author">Author<a class="anchor" aria-label="anchor" href="#author"></a><
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/reference/check_data_validation.html b/reference/check_data_validation.html
new file mode 100644
index 0000000..b8167c7
--- /dev/null
+++ b/reference/check_data_validation.html
@@ -0,0 +1,69 @@
+<!DOCTYPE html>
+<!-- Generated by pkgdown: do not edit by hand --><html lang="en"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><meta charset="utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge"><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"><title>Checks data set for misspecification — check_data_validation • arfit</title><script src="../deps/jquery-3.6.0/jquery-3.6.0.min.js"></script><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"><link href="../deps/bootstrap-5.3.1/bootstrap.min.css" rel="stylesheet"><script src="../deps/bootstrap-5.3.1/bootstrap.bundle.min.js"></script><link href="../deps/font-awesome-6.4.2/css/all.min.css" rel="stylesheet"><link href="../deps/font-awesome-6.4.2/css/v4-shims.min.css" rel="stylesheet"><script src="../deps/headroom-0.11.0/headroom.min.js"></script><script src="../deps/headroom-0.11.0/jQuery.headroom.min.js"></script><script src="../deps/bootstrap-toc-1.0.1/bootstrap-toc.min.js"></script><script src="../deps/clipboard.js-2.0.11/clipboard.min.js"></script><script src="../deps/search-1.0.0/autocomplete.jquery.min.js"></script><script src="../deps/search-1.0.0/fuse.min.js"></script><script src="../deps/search-1.0.0/mark.min.js"></script><!-- pkgdown --><script src="../pkgdown.js"></script><meta property="og:title" content="Checks data set for misspecification — check_data_validation"><meta name="description" content="Check column names, missing data, consecutive equally spaced time units"><meta property="og:description" content="Check column names, missing data, consecutive equally spaced time units"></head><body>
+    <a href="#main" class="visually-hidden-focusable">Skip to contents</a>
+
+
+    <nav class="navbar navbar-expand-lg fixed-top bg-light" data-bs-theme="light" aria-label="Site navigation"><div class="container">
+
+    <a class="navbar-brand me-2" href="../index.html">arfit</a>
+
+    <small class="nav-text text-muted me-auto" data-bs-toggle="tooltip" data-bs-placement="bottom" title="">0.0.0.9000</small>
+
+
+    <button class="navbar-toggler" type="button" data-bs-toggle="collapse" data-bs-target="#navbar" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
+      <span class="navbar-toggler-icon"></span>
+    </button>
+
+    <div id="navbar" class="collapse navbar-collapse ms-3">
+      <ul class="navbar-nav me-auto"><li class="nav-item"><a class="nav-link" href="../articles/arfit.html">Get started</a></li>
+<li class="active nav-item"><a class="nav-link" href="../reference/index.html">Reference</a></li>
+<li class="nav-item dropdown">
+  <button class="nav-link dropdown-toggle" type="button" id="dropdown-articles" data-bs-toggle="dropdown" aria-expanded="false" aria-haspopup="true">Articles</button>
+  <ul class="dropdown-menu" aria-labelledby="dropdown-articles"><li><a class="dropdown-item" href="../articles/background_work.html">Background</a></li>
+    <li><a class="dropdown-item" href="../articles/Equations.html">Equations</a></li>
+  </ul></li>
+      </ul><ul class="navbar-nav"><li class="nav-item"><form class="form-inline" role="search">
+ <input class="form-control" type="search" name="search-input" id="search-input" autocomplete="off" aria-label="Search site" placeholder="Search for" data-search-index="../search.json"></form></li>
+<li class="nav-item"><a class="external-link nav-link" href="https://github.com/andybeet/arfit/" aria-label="GitHub"><span class="fa fab fa-github fa-lg"></span></a></li>
+      </ul></div>
+
+
+  </div>
+</nav><div class="container template-reference-topic">
+<div class="row">
+  <main id="main" class="col-md-9"><div class="page-header">
+
+      <h1>Checks data set for misspecification</h1>
+      <small class="dont-index">Source: <a href="https://github.com/andybeet/arfit/blob/main/R/check_data_validation.R" class="external-link"><code>R/check_data_validation.R</code></a></small>
+      <div class="d-none name"><code>check_data_validation.Rd</code></div>
+    </div>
+
+    <div class="ref-description section level2">
+    <p>Check column names, missing data, consecutive equally spaced time units</p>
+    </div>
+
+    <div class="section level2">
+    <h2 id="ref-usage">Usage<a class="anchor" aria-label="anchor" href="#ref-usage"></a></h2>
+    <div class="sourceCode"><pre class="sourceCode r"><code><span><span class="fu">check_data_validation</span><span class="op">(</span><span class="va">dataSet</span><span class="op">)</span></span></code></pre></div>
+    </div>
+
+
+  </main></div>
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Andy Beet.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
+</div>
+
+    </footer></div>
+
+
+
+
+
+  </body></html>
+
diff --git a/reference/est_beta_given_rho.html b/reference/est_beta_given_rho.html
index 14b0695..d530ad1 100644
--- a/reference/est_beta_given_rho.html
+++ b/reference/est_beta_given_rho.html
@@ -73,7 +73,7 @@ <h2 id="arguments">Arguments<a class="anchor" aria-label="anchor" href="#argumen
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/reference/example_cedar_rapids.html b/reference/example_cedar_rapids.html
index 99d6ea5..1863176 100644
--- a/reference/example_cedar_rapids.html
+++ b/reference/example_cedar_rapids.html
@@ -69,7 +69,7 @@ <h2 id="arguments">Arguments<a class="anchor" aria-label="anchor" href="#argumen
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/reference/fit_ar1_opt.html b/reference/fit_ar1_opt.html
index 2da4d0e..d6fab85 100644
--- a/reference/fit_ar1_opt.html
+++ b/reference/fit_ar1_opt.html
@@ -56,7 +56,7 @@ <h2 id="ref-usage">Usage<a class="anchor" aria-label="anchor" href="#ref-usage">
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/reference/fit_to_data.html b/reference/fit_to_data.html
new file mode 100644
index 0000000..c6d6d3a
--- /dev/null
+++ b/reference/fit_to_data.html
@@ -0,0 +1,82 @@
+<!DOCTYPE html>
+<!-- Generated by pkgdown: do not edit by hand --><html lang="en"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><meta charset="utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge"><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"><title>Run test on user data set — fit_to_data • arfit</title><script src="../deps/jquery-3.6.0/jquery-3.6.0.min.js"></script><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"><link href="../deps/bootstrap-5.3.1/bootstrap.min.css" rel="stylesheet"><script src="../deps/bootstrap-5.3.1/bootstrap.bundle.min.js"></script><link href="../deps/font-awesome-6.4.2/css/all.min.css" rel="stylesheet"><link href="../deps/font-awesome-6.4.2/css/v4-shims.min.css" rel="stylesheet"><script src="../deps/headroom-0.11.0/headroom.min.js"></script><script src="../deps/headroom-0.11.0/jQuery.headroom.min.js"></script><script src="../deps/bootstrap-toc-1.0.1/bootstrap-toc.min.js"></script><script src="../deps/clipboard.js-2.0.11/clipboard.min.js"></script><script src="../deps/search-1.0.0/autocomplete.jquery.min.js"></script><script src="../deps/search-1.0.0/fuse.min.js"></script><script src="../deps/search-1.0.0/mark.min.js"></script><!-- pkgdown --><script src="../pkgdown.js"></script><meta property="og:title" content="Run test on user data set — fit_to_data"><meta name="description" content="Fits a linear model to data set assuming AR1 error"><meta property="og:description" content="Fits a linear model to data set assuming AR1 error"></head><body>
+    <a href="#main" class="visually-hidden-focusable">Skip to contents</a>
+
+
+    <nav class="navbar navbar-expand-lg fixed-top bg-light" data-bs-theme="light" aria-label="Site navigation"><div class="container">
+
+    <a class="navbar-brand me-2" href="../index.html">arfit</a>
+
+    <small class="nav-text text-muted me-auto" data-bs-toggle="tooltip" data-bs-placement="bottom" title="">0.0.0.9000</small>
+
+
+    <button class="navbar-toggler" type="button" data-bs-toggle="collapse" data-bs-target="#navbar" aria-controls="navbar" aria-expanded="false" aria-label="Toggle navigation">
+      <span class="navbar-toggler-icon"></span>
+    </button>
+
+    <div id="navbar" class="collapse navbar-collapse ms-3">
+      <ul class="navbar-nav me-auto"><li class="nav-item"><a class="nav-link" href="../articles/arfit.html">Get started</a></li>
+<li class="active nav-item"><a class="nav-link" href="../reference/index.html">Reference</a></li>
+<li class="nav-item dropdown">
+  <button class="nav-link dropdown-toggle" type="button" id="dropdown-articles" data-bs-toggle="dropdown" aria-expanded="false" aria-haspopup="true">Articles</button>
+  <ul class="dropdown-menu" aria-labelledby="dropdown-articles"><li><a class="dropdown-item" href="../articles/background_work.html">Background</a></li>
+    <li><a class="dropdown-item" href="../articles/Equations.html">Equations</a></li>
+  </ul></li>
+      </ul><ul class="navbar-nav"><li class="nav-item"><form class="form-inline" role="search">
+ <input class="form-control" type="search" name="search-input" id="search-input" autocomplete="off" aria-label="Search site" placeholder="Search for" data-search-index="../search.json"></form></li>
+<li class="nav-item"><a class="external-link nav-link" href="https://github.com/andybeet/arfit/" aria-label="GitHub"><span class="fa fab fa-github fa-lg"></span></a></li>
+      </ul></div>
+
+
+  </div>
+</nav><div class="container template-reference-topic">
+<div class="row">
+  <main id="main" class="col-md-9"><div class="page-header">
+
+      <h1>Run test on user data set</h1>
+      <small class="dont-index">Source: <a href="https://github.com/andybeet/arfit/blob/main/R/fit_to_data.R" class="external-link"><code>R/fit_to_data.R</code></a></small>
+      <div class="d-none name"><code>fit_to_data.Rd</code></div>
+    </div>
+
+    <div class="ref-description section level2">
+    <p>Fits a linear model to data set assuming AR1 error</p>
+    </div>
+
+    <div class="section level2">
+    <h2 id="ref-usage">Usage<a class="anchor" aria-label="anchor" href="#ref-usage"></a></h2>
+    <div class="sourceCode"><pre class="sourceCode r"><code><span><span class="fu">fit_to_data</span><span class="op">(</span>dataSet <span class="op">=</span> <span class="cn">NULL</span>, nBootSims <span class="op">=</span> <span class="fl">999</span><span class="op">)</span></span></code></pre></div>
+    </div>
+
+    <div class="section level2">
+    <h2 id="arguments">Arguments<a class="anchor" aria-label="anchor" href="#arguments"></a></h2>
+
+
+<dl><dt id="arg-dataset">dataSet<a class="anchor" aria-label="anchor" href="#arg-dataset"></a></dt>
+<dd><p>Data frame. Two columns (x,y)</p></dd>
+
+
+<dt id="arg-nbootsims">nBootSims<a class="anchor" aria-label="anchor" href="#arg-nbootsims"></a></dt>
+<dd><p>Numeric scalar. Number of bootstrap samples to perform</p></dd>
+
+</dl></div>
+
+  </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
+    </nav></aside></div>
+
+
+    <footer><div class="pkgdown-footer-left">
+  <p>Developed by Andy Beet.</p>
+</div>
+
+<div class="pkgdown-footer-right">
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
+</div>
+
+    </footer></div>
+
+
+
+
+
+  </body></html>
+
diff --git a/reference/index.html b/reference/index.html
index f677175..c9755d5 100644
--- a/reference/index.html
+++ b/reference/index.html
@@ -37,29 +37,6 @@ <h1>Package index</h1>
     </div>
 
     <div class="section level2">
-      <h2 id="data-simulation">Data simulation<a class="anchor" aria-label="anchor" href="#data-simulation"></a></h2>
-
-      <div class="section-desc"><p>simulate from AR1 and AR2 models</p></div>
-
-
-    </div><div class="section level2">
-
-
-
-
-      <dl><dt>
-
-          <code><a href="simulate_ar1.html">simulate_ar1()</a></code>
-
-        </dt>
-        <dd>Simulate data from an linear process with AR1 error</dd>
-      </dl><dl><dt>
-
-          <code><a href="simulate_ar2.html">simulate_ar2()</a></code>
-
-        </dt>
-        <dd>Simulate data from an linear process with AR2 error</dd>
-      </dl></div><div class="section level2">
       <h2 id="simulation-study">Simulation study<a class="anchor" aria-label="anchor" href="#simulation-study"></a></h2>
 
       <div class="section-desc"><p>Functions used in simulation study</p></div>
@@ -72,56 +49,26 @@ <h2 id="simulation-study">Simulation study<a class="anchor" aria-label="anchor"
 
       <dl><dt>
 
-          <code><a href="simulation_study_grid_ar1.html">simulation_study_grid_ar1()</a></code>
-
-        </dt>
-        <dd>Simulation study using grid search</dd>
-      </dl><dl><dt>
-
-          <code><a href="simulation_study_grid_ar2.html">simulation_study_grid_ar2()</a></code>
-
-        </dt>
-        <dd>Simulation study using optimization routine</dd>
-      </dl><dl><dt>
-
-          <code><a href="simulation_study_opt_ar1.html">simulation_study_opt_ar1()</a></code>
-
-        </dt>
-        <dd>Simulation study using optimization routine</dd>
-      </dl><dl><dt>
-
-          <code><a href="simulation_study_opt_ar2.html">simulation_study_opt_ar2()</a></code>
-
-        </dt>
-        <dd>Simulation study using optimization routine</dd>
-      </dl><dl><dt>
-
           <code><a href="sim_study_opt_ar1.html">sim_study_opt_ar1()</a></code>
 
         </dt>
         <dd>Simulation study using optimization routine, multiple cores</dd>
       </dl><dl><dt>
 
-          <code><a href="sim_single_opt_ar1.html">sim_single_opt_ar1()</a></code>
-
-        </dt>
-        <dd>Simulates n data sets and performs bootstrapping</dd>
-      </dl><dl><dt>
-
-          <code><a href="sim_study_opt_ar2.html">sim_study_opt_ar2()</a></code>
+          <code><a href="simulate_ar1.html">simulate_ar1()</a></code>
 
         </dt>
-        <dd>Simulation study using optimization routine, multiple cores</dd>
+        <dd>Simulate data from an linear process with AR1 error</dd>
       </dl><dl><dt>
 
-          <code><a href="sim_single_opt_ar2.html">sim_single_opt_ar2()</a></code>
+          <code><a href="simulation_study_opt_ar1.html">simulation_study_opt_ar1()</a></code>
 
         </dt>
-        <dd>Simulates n data sets and performs bootstrapping</dd>
+        <dd>Simulation study using optimization routine</dd>
       </dl></div><div class="section level2">
       <h2 id="likelihood-functions">Likelihood Functions<a class="anchor" aria-label="anchor" href="#likelihood-functions"></a></h2>
 
-      <div class="section-desc"><p>AR1 and AR2 likelihoods</p></div>
+      <div class="section-desc"><p>AR1 likelihood</p></div>
 
 
     </div><div class="section level2">
@@ -135,12 +82,6 @@ <h2 id="likelihood-functions">Likelihood Functions<a class="anchor" aria-label="
 
         </dt>
         <dd>Evaluate the likelihood of linear model with errors from an AR order 1 process</dd>
-      </dl><dl><dt>
-
-          <code><a href="likelihood_ar2.html">likelihood_ar2()</a></code>
-
-        </dt>
-        <dd>Evaluates the likelihood of AR2</dd>
       </dl></div><div class="section level2">
       <h2 id="estimation-functions">Estimation Functions<a class="anchor" aria-label="anchor" href="#estimation-functions"></a></h2>
 
@@ -154,28 +95,16 @@ <h2 id="estimation-functions">Estimation Functions<a class="anchor" aria-label="
 
       <dl><dt>
 
-          <code><a href="fit_ar1_grid.html">fit_ar1_grid()</a></code>
-
-        </dt>
-        <dd>Fit linear model with AR(1) error by GLS</dd>
-      </dl><dl><dt>
-
-          <code><a href="fit_ar2_grid.html">fit_ar2_grid()</a></code>
-
-        </dt>
-        <dd>Fit linear model with AR(1) error by GLS</dd>
-      </dl><dl><dt>
-
           <code><a href="fit_ar1_opt.html">fit_ar1_opt()</a></code>
 
         </dt>
         <dd>Fit linear model with trend using optimization routine</dd>
       </dl><dl><dt>
 
-          <code><a href="fit_ar2_opt.html">fit_ar2_opt()</a></code>
+          <code><a href="fit_to_data.html">fit_to_data()</a></code>
 
         </dt>
-        <dd>Fit linear model with trend using optimization routine</dd>
+        <dd>Run test on user data set</dd>
       </dl><dl><dt>
 
           <code><a href="est_beta_given_rho.html">est_beta_given_rho()</a></code>
@@ -184,16 +113,10 @@ <h2 id="estimation-functions">Estimation Functions<a class="anchor" aria-label="
         <dd>Estimate beta conditional on known AR1 parameter</dd>
       </dl><dl><dt>
 
-          <code><a href="est_beta_given_rho2.html">est_beta_given_rho2()</a></code>
-
-        </dt>
-        <dd>Estimate beta conditional on known AR1 parameter</dd>
-      </dl><dl><dt>
-
-          <code><a href="isStationary_ar2.html">isStationary_ar2()</a></code>
+          <code><a href="check_data_validation.html">check_data_validation()</a></code>
 
         </dt>
-        <dd>Determines stationarity of AR2</dd>
+        <dd>Checks data set for misspecification</dd>
       </dl></div><div class="section level2">
       <h2 id="examples">Examples<a class="anchor" aria-label="anchor" href="#examples"></a></h2>
 
@@ -238,7 +161,7 @@ <h2 id="package-info">Package Info<a class="anchor" aria-label="anchor" href="#p
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/reference/likelihood_ar1.html b/reference/likelihood_ar1.html
index fec7e3b..4012325 100644
--- a/reference/likelihood_ar1.html
+++ b/reference/likelihood_ar1.html
@@ -78,7 +78,7 @@ <h2 id="value">Value<a class="anchor" aria-label="anchor" href="#value"></a></h2
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/reference/sim_study_opt_ar1.html b/reference/sim_study_opt_ar1.html
index 9007bc1..d1441a8 100644
--- a/reference/sim_study_opt_ar1.html
+++ b/reference/sim_study_opt_ar1.html
@@ -56,7 +56,8 @@ <h2 id="ref-usage">Usage<a class="anchor" aria-label="anchor" href="#ref-usage">
 <span>  nSims <span class="op">=</span> <span class="fl">200</span>,</span>
 <span>  nBootSims <span class="op">=</span> <span class="fl">500</span>,</span>
 <span>  setSeed <span class="op">=</span> <span class="cn">NULL</span>,</span>
-<span>  nCores <span class="op">=</span> <span class="cn">NULL</span></span>
+<span>  nCores <span class="op">=</span> <span class="cn">NULL</span>,</span>
+<span>  missing <span class="op">=</span> <span class="cn">F</span></span>
 <span><span class="op">)</span></span></code></pre></div>
     </div>
 
@@ -96,6 +97,11 @@ <h2 id="arguments">Arguments<a class="anchor" aria-label="anchor" href="#argumen
 <dd><p>Numeric scalar. Specify the number of cores to utilize (Default = NULL, utilizes n-1 cores)</p></dd>
 
 
+<dt id="arg-missing">missing<a class="anchor" aria-label="anchor" href="#arg-missing"></a></dt>
+<dd><p>Boolean. Whether to simulate with missing data (Default = F).
+If T then a single missing value is added at random to the response</p></dd>
+
+
 <dt id="arg-nvec">nVec<a class="anchor" aria-label="anchor" href="#arg-nvec"></a></dt>
 <dd><p>Numeric vector. Values for the length of time series to simulate</p></dd>
 
@@ -122,7 +128,7 @@ <h2 id="ref-examples">Examples<a class="anchor" aria-label="anchor" href="#ref-e
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/reference/simulate_ar1.html b/reference/simulate_ar1.html
index add6f90..9be0c36 100644
--- a/reference/simulate_ar1.html
+++ b/reference/simulate_ar1.html
@@ -50,7 +50,7 @@ <h1>Simulate data from an linear process with AR1 error</h1>
 
     <div class="section level2">
     <h2 id="ref-usage">Usage<a class="anchor" aria-label="anchor" href="#ref-usage"></a></h2>
-    <div class="sourceCode"><pre class="sourceCode r"><code><span><span class="fu">simulate_ar1</span><span class="op">(</span><span class="va">alpha</span>, beta <span class="op">=</span> <span class="fl">0</span>, <span class="va">sigma</span>, <span class="va">rho</span>, <span class="va">n</span><span class="op">)</span></span></code></pre></div>
+    <div class="sourceCode"><pre class="sourceCode r"><code><span><span class="fu">simulate_ar1</span><span class="op">(</span><span class="va">alpha</span>, beta <span class="op">=</span> <span class="fl">0</span>, <span class="va">sigma</span>, <span class="va">rho</span>, <span class="va">n</span>, missingValues <span class="op">=</span> <span class="cn">NULL</span><span class="op">)</span></span></code></pre></div>
     </div>
 
     <div class="section level2">
@@ -76,6 +76,10 @@ <h2 id="arguments">Arguments<a class="anchor" aria-label="anchor" href="#argumen
 <dt id="arg-n">n<a class="anchor" aria-label="anchor" href="#arg-n"></a></dt>
 <dd><p>Numeric scalar. Length of time series</p></dd>
 
+
+<dt id="arg-missingvalues">missingValues<a class="anchor" aria-label="anchor" href="#arg-missingvalues"></a></dt>
+<dd><p>Numeric vector. Indices of y values for which there is missing data</p></dd>
+
 </dl></div>
 
   </main><aside class="col-md-3"><nav id="toc" aria-label="Table of contents"><h2>On this page</h2>
@@ -87,7 +91,7 @@ <h2 id="arguments">Arguments<a class="anchor" aria-label="anchor" href="#argumen
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/reference/simulation_study_opt_ar1.html b/reference/simulation_study_opt_ar1.html
index de7401d..7cfa534 100644
--- a/reference/simulation_study_opt_ar1.html
+++ b/reference/simulation_study_opt_ar1.html
@@ -1,5 +1,7 @@
 <!DOCTYPE html>
-<!-- Generated by pkgdown: do not edit by hand --><html lang="en"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><meta charset="utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge"><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"><title>Simulation study using optimization routine — simulation_study_opt_ar1 • arfit</title><script src="../deps/jquery-3.6.0/jquery-3.6.0.min.js"></script><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"><link href="../deps/bootstrap-5.3.1/bootstrap.min.css" rel="stylesheet"><script src="../deps/bootstrap-5.3.1/bootstrap.bundle.min.js"></script><link href="../deps/font-awesome-6.4.2/css/all.min.css" rel="stylesheet"><link href="../deps/font-awesome-6.4.2/css/v4-shims.min.css" rel="stylesheet"><script src="../deps/headroom-0.11.0/headroom.min.js"></script><script src="../deps/headroom-0.11.0/jQuery.headroom.min.js"></script><script src="../deps/bootstrap-toc-1.0.1/bootstrap-toc.min.js"></script><script src="../deps/clipboard.js-2.0.11/clipboard.min.js"></script><script src="../deps/search-1.0.0/autocomplete.jquery.min.js"></script><script src="../deps/search-1.0.0/fuse.min.js"></script><script src="../deps/search-1.0.0/mark.min.js"></script><!-- pkgdown --><script src="../pkgdown.js"></script><meta property="og:title" content="Simulation study using optimization routine — simulation_study_opt_ar1"><meta name="description" content="Performs a simulation study to assess the performance of the test."><meta property="og:description" content="Performs a simulation study to assess the performance of the test."></head><body>
+<!-- Generated by pkgdown: do not edit by hand --><html lang="en"><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"><meta charset="utf-8"><meta http-equiv="X-UA-Compatible" content="IE=edge"><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"><title>Simulation study using optimization routine — simulation_study_opt_ar1 • arfit</title><script src="../deps/jquery-3.6.0/jquery-3.6.0.min.js"></script><meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"><link href="../deps/bootstrap-5.3.1/bootstrap.min.css" rel="stylesheet"><script src="../deps/bootstrap-5.3.1/bootstrap.bundle.min.js"></script><link href="../deps/font-awesome-6.4.2/css/all.min.css" rel="stylesheet"><link href="../deps/font-awesome-6.4.2/css/v4-shims.min.css" rel="stylesheet"><script src="../deps/headroom-0.11.0/headroom.min.js"></script><script src="../deps/headroom-0.11.0/jQuery.headroom.min.js"></script><script src="../deps/bootstrap-toc-1.0.1/bootstrap-toc.min.js"></script><script src="../deps/clipboard.js-2.0.11/clipboard.min.js"></script><script src="../deps/search-1.0.0/autocomplete.jquery.min.js"></script><script src="../deps/search-1.0.0/fuse.min.js"></script><script src="../deps/search-1.0.0/mark.min.js"></script><!-- pkgdown --><script src="../pkgdown.js"></script><meta property="og:title" content="Simulation study using optimization routine — simulation_study_opt_ar1"><meta name="description" content="Uses single Core.
+Performs a simulation study to assess the performance of the test."><meta property="og:description" content="Uses single Core.
+Performs a simulation study to assess the performance of the test."></head><body>
     <a href="#main" class="visually-hidden-focusable">Skip to contents</a>
 
 
@@ -39,7 +41,8 @@ <h1>Simulation study using optimization routine</h1>
     </div>
 
     <div class="ref-description section level2">
-    <p>Performs a simulation study to assess the performance of the test.</p>
+    <p>Uses single Core.
+Performs a simulation study to assess the performance of the test.</p>
     </div>
 
     <div class="section level2">
@@ -110,7 +113,7 @@ <h2 id="ref-examples">Examples<a class="anchor" aria-label="anchor" href="#ref-e
 </div>
 
 <div class="pkgdown-footer-right">
-  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.0.</p>
+  <p>Site built with <a href="https://pkgdown.r-lib.org/" class="external-link">pkgdown</a> 2.1.1.</p>
 </div>
 
     </footer></div>
diff --git a/search.json b/search.json
index 37751af..be6962c 100644
--- a/search.json
+++ b/search.json
@@ -1 +1 @@
-[{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-1","dir":"Articles","previous_headings":"","what":"Eq 1","title":"Equations","text":"Yt=β0+β1t+ϵt\\begin{align} \\tag{1}   Y_t = \\beta_0 + \\beta_1 t + \\epsilon_t   \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-2","dir":"Articles","previous_headings":"","what":"Eq 2","title":"Equations","text":"L(θ_;y_)=∏t=2np(Yt=yt|Yt−1=yt−1)p(Y1=y1)\\begin{align} \\tag{2} \\mathrm{L}\\left( \\underline{\\theta}; \\underline{y} \\right )= \\prod^n_{t=2} p\\left(Y_t = y_t | Y_{t-1}=y_{t-1}\\right)  p\\left(Y_1=y_1 \\right)  \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-3","dir":"Articles","previous_headings":"","what":"Eq 3","title":"Equations","text":"logL(θ_;y_)=−n2log2π−nlogσ+12log(1−ϕ2)−12σ2((1−ϕ2)(y1−β0−β1)2+∑t=2n(yt−ϕyt−1−β0(1−ϕ2)−tβ1+ϕ(t−1)β1)2)\\begin{align} \\tag{3} logL\\left( \\underline{\\theta}; \\underline{y} \\right ) = & -\\frac{n}{2}log2\\pi - nlog\\sigma + \\frac{1}{2}log(1-\\phi^2) \\\\ &-\\frac{1}{2\\sigma^2}\\left( (1-\\phi^2)(y_1-\\beta_0-\\beta_1)^2 + \\sum^n_{t=2}(y_t - \\phi y_{t-1}-\\beta_0(1-\\phi^2) -t\\beta_1 + \\phi(t-1)\\beta_1)^2 \\right)   \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-4","dir":"Articles","previous_headings":"","what":"Eq 4","title":"Equations","text":"σ̂2=1n((1−ϕ2)(y1−β0−β1)2+∑t=2n(yt−ϕyt−1−β0(1−ϕ2)−tβ1+ϕ(t−1)β1)2)\\begin{align} \\tag{4} \\hat\\sigma^2 = \\frac{1}{n}\\left( (1-\\phi^2)(y_1-\\beta_0-\\beta_1)^2 + \\sum^n_{t=2}(y_t - \\phi y_{t-1}-\\beta_0(1-\\phi^2) - t\\beta_1 + \\phi(t-1)\\beta_1)^2 \\right)  \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-5","dir":"Articles","previous_headings":"","what":"Eq 5","title":"Equations","text":"logL(β_,ϕ;y_)=const.+12log(1−ϕ2)−n2log((1−ϕ2)(y1−β0−β1)2+∑t=2n(yt−ϕyt−1−β0(1−ϕ2)−tβ1+ϕ(t−1)β1)2)=const.+12log(1−ϕ2)−n2log((1−ϕ2)(y1−X1β_)2+∑t=2n(yt−ϕyt−1−Xtβ_+ϕXt−1β_)2)\\begin{align} \\tag{5} logL\\left( \\underline{\\beta}, \\phi; \\underline{y} \\right ) &= const. + \\frac{1}{2}log(1-\\phi^2) \\\\ &-\\frac{n}{2}log\\left( (1-\\phi^2)(y_1-\\beta_0-\\beta_1)^2 + \\sum^n_{t=2}(y_t - \\phi y_{t-1}-\\beta_0(1-\\phi^2)-t\\beta_1 + \\phi(t-1)\\beta_1)^2 \\right) \\\\   &= const. + \\frac{1}{2}log(1-\\phi^2) \\\\ &-\\frac{n}{2}log\\left( (1-\\phi^2)(y_1-X_1\\underline{\\beta})^2 + \\sum^n_{t=2}(y_t - \\phi y_{t-1}-X_t\\underline{\\beta} + \\phi X_{t-1}\\underline{\\beta})^2 \\right)  \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-6","dir":"Articles","previous_headings":"","what":"Eq 6","title":"Equations","text":"logL(β_,ϕ_,σ;y_)=−n2log(2π)−n2log(σ2)+12log|Vp−1|−12σ2(yp_−μp_)TVp−1(yp_−μp_)−12σ2∑t=p+1n(yt−c−ϕ1yt−1−...−ϕpyt−p)2\\begin{align} \\tag{6} logL\\left( \\underline{\\beta}, \\underline{\\phi},\\sigma; \\underline{y} \\right ) &= -\\frac{n}{2}log(2\\pi) -\\frac{n}{2}log(\\sigma^2) +\\frac{1}{2}log \\left|V_p^{-1} \\right| \\\\ &-\\frac{1 }{2 \\sigma^2} (\\underline{y_p}-\\underline{\\mu_p})^T V_p^{-1}(\\underline{y_p}-\\underline{\\mu_p}) \\\\  &- \\frac{1}{2\\sigma^2}\\sum^n_{t=p+1} (y_t - c - \\phi_1y_{t-1} - ... - \\phi_p y_{t-p})^2   \\end{align} |Vp−1|\\left|V_p^{-1} \\right| determinant inverted matrix VpV_p, σ2Vp\\sigma^2V_p = variance-covariance matrix order p, μp_=Xpβ_\\underline{\\mu_p} = X_p\\underline{\\beta}, XpX_p pthp_{th} row design matrix corresponding time t = p cc = function fitted terms Xtβ_X_t\\underline{\\beta}","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"likelihood-for-arp-errors","dir":"Articles","previous_headings":"","what":"Likelihood for AR(p) errors","title":"Background","text":"Hamilton (1994) logL(β_,ϕ_,σ;y_)=−n2log(2π)−n2log(σ2)+12log|Vp−1|−12σ2(yp_−μp_)TVp−1(yp_−μp_)−12σ2∑t=p+1n(yt−c−ϕ1yt−1−...−ϕpyt−p)2\\begin{align} logL\\left( \\underline{\\beta}, \\underline{\\phi},\\sigma; \\underline{y} \\right ) &= -\\frac{n}{2}log(2\\pi) -\\frac{n}{2}log(\\sigma^2) +\\frac{1}{2}log \\left|V_p^{-1} \\right| \\\\ &-\\frac{1 }{2 \\sigma^2} (\\underline{y_p}-\\underline{\\mu_p})^T V_p^{-1}(\\underline{y_p}-\\underline{\\mu_p}) \\\\  &- \\frac{1}{2\\sigma^2}\\sum^n_{t=p+1} (y_t - c - \\phi_1y_{t-1} - ... - \\phi_p y_{t-p})^2 \\\\  \\end{align} |Vp−1|\\left|V_p^{-1} \\right| determinant inverted matrix VpV_p, σ2Vp\\sigma^2V_p = variance-covariance matrix order p, μp_=Xpβ_\\underline{\\mu_p} = X_p\\underline{\\beta}, XpX_p pthp_{th} row design matrix corresponding time t = p cc = function fitted terms Xtβ_X_t\\underline{\\beta}","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"likelihood-for-ar1-errors","dir":"Articles","previous_headings":"","what":"Likelihood for AR(1) errors","title":"Background","text":"Setting p = 1, σ2V1=σ21−ϕ12\\sigma^2V_1 = \\frac{\\sigma^2}{1-\\phi_1^2} = variance process, y, |V1−1|=1−ϕ12\\left|V_1^{-1}\\right| = 1-\\phi_1^2 μ1=X1β_=β0+β1\\mu_1 = X_1\\underline{\\beta} = \\beta_0 + \\beta_1 c=Xtβ_−ϕ1Xt−1β_c = X_t\\underline{\\beta} - \\phi_1 X_{t-1}\\underline{\\beta} y1y_1 first observation log likelihood, logL(β_,ϕ1,σ;y_)=−n2log(2π)−n2log(σ2)+12log(1−ϕ12)−12σ2(y1−X1β_)2(1−ϕ12)−12σ2∑t=2n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_))2\\begin{align}  logL\\left( \\underline{\\beta}, \\phi_1, \\sigma; \\underline{y} \\right ) &= -\\frac{n}{2}log(2\\pi) -\\frac{n}{2}log(\\sigma^2) +\\frac{1}{2}log(1-\\phi_1^2)  \\\\ &-\\frac{1 }{2 \\sigma^2} ({y_1}-X_1\\underline{\\beta})^2 (1-\\phi_1^2) \\\\ &- \\frac{1}{2\\sigma^2}\\sum^n_{t=2} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}))^2 \\\\  \\end{align} Differentiating logL(β_,ϕ1,σ;y_)logL\\left( \\underline{\\beta}, \\phi_1, \\sigma; \\underline{y} \\right ) respect σ\\sigma equating zero yields maximum likelihood estimator, σ̂2=1n[(y1−X1β_)2(1−ϕ12)+∑t=2n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_))2]\\begin{align} \\hat{\\sigma}^2 = \\frac{1}{n}\\left[ (y_1-X_1\\underline{\\beta})^2(1-\\phi_1^2) + \\sum^n_{t=2} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}))^2 \\right]  \\end{align} Substituting σ̂2\\hat{\\sigma}^2 back log likelihood yields (Beach MacKinnon (1978a)) logL(β_,ϕ1;y_)=const.+12log(1−ϕ2)−n2log((y1−X1β_)2(1−ϕ12)+∑t=2n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_))2)\\begin{align} logL\\left( \\underline{\\beta}, \\phi_1; \\underline{y} \\right ) &= const. +\\frac{1}{2}log(1-\\phi^2)  \\\\ &-\\frac{n}{2}log\\left( (y_1-X_1\\underline{\\beta})^2(1-\\phi_1^2) + \\sum^n_{t=2} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}))^2 \\right) \\\\ \\end{align} Two additional terms exist likelihood appear conditional likelihood used GLS procedures. term (y1−X1β_)2(1−ϕ12)(y_1-X_1\\underline{\\beta})^2(1-\\phi_1^2) ensures initial value effect estimates 12log(1−ϕ2)\\frac{1}{2}log(1-\\phi^2) constrains stationary condition hold.","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"maximization-of-the-likelihood","dir":"Articles","previous_headings":"Likelihood for AR(1) errors","what":"Maximization of the likelihood","title":"Background","text":"Maximization likelihood requires iterative numerical procedures. estimate β_\\underline{\\beta} maximizes log-likelihood conditional ϕ\\phi β̂_=(X*TX*)−1X*Ty*\\begin{align} \\underline{\\hat{\\beta}} = (X^{*T}X^*)^{-1}X^{*T}y^* \\end{align} X*=QXX^*=QX y*=Qyy^* = Qy QQ Prais Winsten transformation Q=[(1−ϕ12)1/20...−ϕ110......0−ϕ11]  \\begin{matrix} Q = \\end{matrix} \\begin{bmatrix} (1-\\phi_1^2)^{1/2} & 0 & ... \\\\ -\\phi_1 & 1 & 0 & ... \\\\ & & ... & 0 & -\\phi_1 & 1\\\\ \\end{bmatrix}  procedure searches ϕ1\\phi_1 required find maximum likelihood estimates β_\\underline\\beta, σ\\sigma, ϕ1\\phi_1","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"likelihood-for-ar2-errors","dir":"Articles","previous_headings":"","what":"Likelihood for AR(2) errors","title":"Background","text":"Following method AR(1) errors. Set p = 2, |V2−1|=(1+ϕ22)[(1−ϕ2)2−ϕ12]\\left|V_2^{-1}\\right| = (1+\\phi_2^2)\\left[(1-\\phi_2)^2 -\\phi_1^2 \\right] μ2_=(μ1,μ2)=(X1β_,X2β_)\\underline{\\mu_2} = (\\mu_1, \\mu_2) = (X_1\\underline{\\beta},X_2\\underline{\\beta}) vector means t = 1,2 y2_=(y1,y2)\\underline{y_2} = (y_1,y_2) corresponding vector observations [μ1μ2]=[1112][β0β1] \\begin{bmatrix}  \\mu_1 \\\\ \\mu_2 \\\\ \\end{bmatrix} = \\begin{bmatrix}  1 & 1 \\\\ 1 & 2 \\\\ \\end{bmatrix} \\begin{bmatrix}  \\beta_0 \\\\ \\beta_1 \\\\ \\end{bmatrix} c=Xtβ_−ϕ1Xt−1β_−ϕ2Xt−2β_c = X_t\\underline{\\beta} - \\phi_1 X_{t-1}\\underline{\\beta}- \\phi_2 X_{t-2}\\underline{\\beta} log likelihood, logL(β_,ϕ_,σ;y_)=−n2log(2π)−n2log(σ2)+12log((1+ϕ22)[(1−ϕ2)2−ϕ12])−12σ2(y2_−μ2_)TV2−1(y2_−μ2_)−12σ2∑t=3n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_)−ϕ2(yt−2−Xt−2β_))2\\begin{align}  logL\\left( \\underline{\\beta}, \\underline{\\phi}, \\sigma; \\underline{y} \\right ) &= -\\frac{n}{2}log(2\\pi) -\\frac{n}{2}log(\\sigma^2) +\\frac{1}{2}log((1+\\phi_2^2)\\left[(1-\\phi_2)^2 -\\phi_1^2 \\right])  \\\\ &-\\frac{1 }{2 \\sigma^2} (\\underline{y_2}-\\underline{\\mu_2})^T V_2^{-1}(\\underline{y_2}-\\underline{\\mu_2}) \\\\ &- \\frac{1}{2\\sigma^2}\\sum^n_{t=3} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}) - \\phi_2 (y_{t-2} -X_{t-2}\\underline{\\beta}))^2 \\\\  \\end{align} Differentiating logL(β_,ϕ_,σ;y_)logL\\left( \\underline{\\beta}, \\underline{\\phi}, \\sigma; \\underline{y} \\right ) respect σ\\sigma equating zero yields maximum likelihood estimator, σ̂2=1n[(y2_−μ2_)TV2−1(y2_−μ2_)+∑t=3n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_)−ϕ2(yt−2−Xt−2β_))2]\\begin{align} \\hat{\\sigma}^2 = \\frac{1}{n}\\left[ (\\underline{y_2}-\\underline{\\mu_2})^T V_2^{-1}(\\underline{y_2}-\\underline{\\mu_2}) + \\sum^n_{t=3} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}) - \\phi_2 (y_{t-2} -X_{t-2}\\underline{\\beta}))^2\\right]  \\end{align} Substituting σ̂2\\hat{\\sigma}^2 back log likelihood yields logL(β_,ϕ_;y_)=const.+12log((1+ϕ22)[(1−ϕ2)2−ϕ12])−n2log[(y2_−μ2_)TV2−1(y2_−μ2_)+∑t=3n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_)−ϕ2(yt−2−Xt−2β_))2]\\begin{align} logL\\left( \\underline{\\beta}, \\underline{\\phi}; \\underline{y} \\right) &= const. +\\frac{1}{2}log\\left((1+\\phi_2^2)\\left[(1-\\phi_2)^2 -\\phi_1^2 \\right]\\right)  \\\\ &-\\frac{n}{2}log\\left[ (\\underline{y_2}-\\underline{\\mu_2})^T V_2^{-1}(\\underline{y_2}-\\underline{\\mu_2}) + \\sum^n_{t=3} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}) - \\phi_2 (y_{t-2} -X_{t-2}\\underline{\\beta}))^2\\right]\\\\ \\end{align} Simplifying , 12log((1+ϕ22)[(1−ϕ2)2−ϕ12])=log(1+ϕ2)+12log(1−ϕ1−ϕ2)+12log(1+ϕ1−ϕ2)\\begin{align} \\frac{1}{2}log\\left((1+\\phi_2^2)\\left[(1-\\phi_2)^2 -\\phi_1^2 \\right]\\right) = log(1+\\phi_2) + \\frac{1}{2}log(1-\\phi_1-\\phi_2)+\\frac{1}{2}log(1+\\phi_1-\\phi_2) \\end{align} (y2_−μ2_)TV2−1(y2_−μ2_)=[(y1−μ1)(y2−μ2)][(1−ϕ22)−(ϕ1+ϕ1ϕ2)−(ϕ1+ϕ1ϕ2)(1−ϕ22)][(y1−μ1)(y2−μ2)]\\begin{align} (\\underline{y_2}-\\underline{\\mu_2})^T V_2^{-1}(\\underline{y_2}-\\underline{\\mu_2}) = & \\begin{bmatrix} (y_1 - \\mu_1) & (y_2 -\\mu_2) \\\\ \\end{bmatrix}   \\begin{bmatrix} (1-\\phi_2^2) & -(\\phi_1 + \\phi_1\\phi_2) \\\\ -(\\phi_1 + \\phi_1\\phi_2) & (1-\\phi_2^2)  \\\\ \\end{bmatrix}  \\begin{bmatrix} (y_1 - \\mu_1) \\\\ (y_2 - \\mu_2) \\\\ \\end{bmatrix} \\end{align} =(y1−μ1)2(1−ϕ22)−2(y1−μ1)(y2−μ2)(ϕ1+ϕ1ϕ2)+(y2−μ2)2(1−ϕ22)=(y1−μ1)2(1−ϕ22)−2(y1−μ1)(y2−μ2)ϕ1(1+ϕ2)+(y2−μ2)2(1−ϕ22)=(y1−X1β_)2(1−ϕ22)−2(y1−X1β_)(y2−X2β_)ϕ1(1+ϕ2)+(y2−X2β_)2(1−ϕ22)\\begin{align} = & (y_1 - \\mu_1)^2(1-\\phi_2^2)-2(y_1 - \\mu_1)(y_2 - \\mu_2)(\\phi_1 + \\phi_1\\phi_2)+(y_2 - \\mu_2)^2(1-\\phi_2^2)\\\\ = & (y_1 - \\mu_1)^2(1-\\phi_2^2)-2(y_1 - \\mu_1)(y_2 - \\mu_2)\\phi_1(1 + \\phi_2)+(y_2 - \\mu_2)^2(1-\\phi_2^2) \\\\ = & (y_1 - X_1\\underline{\\beta})^2(1-\\phi_2^2)-2(y_1-X_1\\underline{\\beta})(y_2-X_2\\underline{\\beta})\\phi_1(1 + \\phi_2)+(y_2 -X_2\\underline{\\beta})^2(1-\\phi_2^2) \\end{align} likelihood now form found Beach MacKinnon (1978b) $$\\begin{align} logL\\left( \\underline{\\beta}, \\underline{\\phi}; \\underline{y} \\right) &= const. +log(1+\\phi_2) + \\frac{1}{2}log(1-\\phi_1-\\phi_2)+\\frac{1}{2}log(1+\\phi_1-\\phi_2) \\\\ &-\\frac{n}{2}log\\left[ (y_1 - X_1\\underline{\\beta})^2(1-\\phi_2^2)-2(y_1-X_1\\underline{\\beta})(y_2-X_2\\underline{\\beta})\\phi_1(1 + \\phi_2)+(y_2 -X_2\\underline{\\beta})^2(1-\\phi_2^2) \\\\ + \\sum^n_{t=3} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}) - \\phi_2 (y_{t-2} -X_{t-2}\\underline{\\beta}))^2\\right]\\\\ \\end{align}$$","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"maximization-of-the-likelihood-1","dir":"Articles","previous_headings":"Likelihood for AR(2) errors","what":"Maximization of the likelihood","title":"Background","text":"case AR(1) model maximizing likelihood requires numerical methods. estimate β_\\underline{\\beta} maximizes log-likelihood conditional ϕ\\phi (Beach MacKinnon (1978b)) β̂_=(X*TX*)−1X*Ty*\\begin{align} \\underline{\\hat{\\beta}} = (X^{*T}X^*)^{-1}X^{*T}y^* \\end{align} X*=QXX^*=QX y*=Qyy^* = Qy QQ Prais Winsten transformation Q=[[(1−ϕ22)−ϕ12(1+ϕ2)/(1−ϕ2)]1/20...[ϕ12(1+ϕ2)/(1−ϕ2)]1/2(1−ϕ22)1/20...−ϕ2−ϕ110...0−ϕ2−ϕ110......0...0−ϕ2−ϕ11]  \\begin{matrix} Q = \\end{matrix} \\begin{bmatrix} \\left[(1-\\phi_2^2)-\\phi_1^2(1+\\phi_2)/(1-\\phi_2)\\right]^{1/2} & 0 & ... \\\\ \\left[\\phi_1^2(1+\\phi_2)/(1-\\phi_2)\\right]^{1/2} & (1-\\phi_2^2)^{1/2} & 0 & ... \\\\ -\\phi_2 & -\\phi_1 & 1 & 0 & ...\\\\ 0 & -\\phi_2 & -\\phi_1 & 1 & 0 & ...\\\\ ... \\\\ 0 & ... &0  & -\\phi_2 & -\\phi_1 & 1\\\\ \\end{bmatrix} procedure searches ϕ1\\phi_1 & ϕ2\\phi_2 required find maximum likelihood estimates β_\\underline\\beta, σ\\sigma, ϕ_\\underline{\\phi}","code":""},{"path":[]},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"ar1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"AR(1)","title":"Background","text":"L(θ_;y_)=∏t=2np(Yt=yt│Yt−1=yt−1)×p(Y1=y1)L(\\underline{\\theta};\\underline{y})=\\prod_{t=2}^n p(Y_t = y_t │ Y_{t-1}=y_{t-1}) × p(Y_1=y_1) θ_=(β0,β1,ϕ,σ2)\\underline{\\theta} = (\\beta_0,\\beta_1,\\phi,\\sigma^2), p(Yt=yt│Yt−1=yt−1)p(Y_t = y_t │ Y_{t-1}=y_{t-1}) conditional distribution yty_t given yt−1y_{t-1} p(Y1=y1)p(Y_1=y_1) distribution first point. Exact likelihood AR(1) process. conditional likelihood multiplied marginal likelihood first point.","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"density-of-py_1y_1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Density of p(Y1=y1)p(Y_1=y_1)","title":"Background","text":"p(Y1=y1)p(Y_1=y_1) normally distributed mean = X1β_X_1\\underline{\\beta} equates β0+β1\\beta_0 + \\beta_1 variance = σ21−ϕ2\\frac{\\sigma^2}{1-\\phi^2} density: p(Y1=y1)=12πσ2/(1−ϕ2)exp(−12(y1−β0−β1σ/(1−ϕ2))2)p(Y_1=y_1) = \\frac{1}{\\sqrt{2\\pi}\\sqrt{\\sigma^2/(1-\\phi^2)}}exp \\left( -\\frac{1}{2}\\left( \\frac{ y_1 - \\beta_0-\\beta_1 }{\\sigma/\\sqrt(1-\\phi^2)}\\right)^2\\right)","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"density-of-py_t-y_t-y_t-1y_t-1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Density of p(Yt=yt│Yt−1=yt−1)p(Y_t = y_t │ Y_{t-1}=y_{t-1})","title":"Background","text":"conditional distribution p(Yt=yt│Yt−1=yt−1)p(Y_t = y_t │ Y_{t-1}=y_{t-1}) also normally distributed get round way: Recall yt=β0+β1t+uty_t = \\beta_0 + \\beta_1 t + u_t ϕyt−1=ϕβ0+ϕβ1(t−1)+ϕut−1\\phi y_{t-1} = \\phi \\beta_0 + \\phi \\beta_1 (t-1) + \\phi u_{t-1} yt−ϕyt−1=β0+β1t+ut−ϕβ0−ϕβ1(t−1)−ϕut−1y_t - \\phi y_{t-1} = \\beta_0 + \\beta_1 t + u_t - \\phi \\beta_0 - \\phi\\beta_1(t-1) -\\phi u_{t-1} yt=ϕyt−1+β0(1−ϕ)+β1(t−ϕt+ϕ)+ety_t   = \\phi y_{t-1} + \\beta_0(1-\\phi) + \\beta_1 (t - \\phi t+\\phi) + e_t Rearrange obtain yt−ϕyt−1−β0(1−ϕ)−β1(t−ϕt+ϕ)=ety_t   - \\phi y_{t-1} - \\beta_0(1-\\phi) - \\beta_1 (t - \\phi t+\\phi) = e_t normal distribution mean = 0 variance = σ2\\sigma^2 results p(Yt=yt│Yt−1=yt−1)=∏t=2n1σ2πexp(−12(yt−ϕyt−1−β0(1−ϕ)−β1(t−ϕt+ϕ)σ)2)p(Y_t = y_t │ Y_{t-1}=y_{t-1}) = \\prod_{t=2}^n \\frac{1}{\\sigma \\sqrt{2\\pi}}exp \\left( -\\frac{1}{2}\\left( \\frac{ y_t   - \\phi y_{t-1} - \\beta_0(1-\\phi) - \\beta_1 (t - \\phi t+\\phi)  }{\\sigma}\\right)^2\\right)","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"exact-likelihood-for-ar1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Exact likelihood for AR1","title":"Background","text":"likelihood therefore: L(θ_;y_)=∏t=2n1σ2πexp(−12(yt−ϕyt−1−β0(1−ϕ)−β1(t−ϕt+ϕ)σ)2)×12πσ2/(1−ϕ2)exp(−12(y1−β0−β1σ/(1−ϕ2))2)L(\\underline{\\theta};\\underline{y}) = \\prod_{t=2}^n \\frac{1}{\\sigma \\sqrt{2\\pi}}exp \\left( -\\frac{1}{2}\\left( \\frac{ y_t   - \\phi y_{t-1} - \\beta_0(1-\\phi) - \\beta_1 (t - \\phi t+\\phi)  }{\\sigma}\\right)^2\\right) \\times \\frac{1}{\\sqrt{2\\pi}\\sqrt{\\sigma^2/(1-\\phi^2)}}exp \\left( -\\frac{1}{2}\\left( \\frac{ y_1 - \\beta_0 -\\beta_1 }{\\sigma/\\sqrt(1-\\phi^2)}\\right)^2\\right) Taking logs simplifying results : logL(θ_;y_)=−nlogσ−n2log(2π)−12log(1−ϕ2)−12σ2((y1−β0−β1)2(1−ϕ2)+Σt=2n(yt−ϕyt−1−β0(1−ϕ)−β1(t−ϕt+ϕ))2) logL(\\underline{\\theta};\\underline{y}) = -nlog\\sigma -\\frac{n}{2}log(2\\pi)-\\frac{1}{2}log(1-\\phi^2) -\\frac{1}{2\\sigma^2} \\left( (y_1-\\beta_0 -\\beta_1)^2(1-\\phi^2) + \\Sigma_{t=2}^n (y_t-\\phi y_{t-1}-\\beta_0(1-\\phi)-\\beta_1(t-\\phi t + \\phi))^2 \\right) Note β0(1−ϕ)+β1(t−ϕt+ϕ)=β0−β0ϕ+β1t−β1ϕt+β1ϕ=β0+β1t−ϕ(β0+β1(t−1))=Xtβ_−ϕXt−1β_\\begin{align}  &\\beta_0(1-\\phi) + \\beta_1 (t - \\phi t+\\phi) \\\\ &= \\beta_0 -\\beta_0\\phi + \\beta_1t - \\beta_1\\phi t + \\beta_1\\phi \\\\ &= \\beta_0 + \\beta_1t - \\phi (\\beta_0 +  \\beta_1(t -1)) \\\\ &= X_t\\underline{\\beta} - \\phi X_{t-1}\\underline{\\beta}  \\end{align} Using notation Beach MacKinnon (1978a) can simplify log likelihood, logL(θ_;y_)=−nlogσ−n2log(2π)−12log(1−ϕ2)−12σ2((y1−X1β_)2(1−ϕ2)+Σt=2n(yt−Xtβ_−ϕ(yt−1−Xt−1β_)2)\\begin{align}  logL(\\underline{\\theta};\\underline{y}) &= -nlog\\sigma -\\frac{n}{2}log(2\\pi)-\\frac{1}{2}log(1-\\phi^2) \\\\ &-\\frac{1}{2\\sigma^2} \\left( (y_1-X_1\\underline{\\beta})^2(1-\\phi^2) + \\Sigma_{t=2}^n (y_t- X_t\\underline{\\beta} - \\phi( y_{t-1}- X_{t-1}\\underline{\\beta})^2 \\right)\\\\ \\end{align} substituting σ̂2\\hat\\sigma^2 get logL(β_,ϕ;y_)=const.+12log(1−ϕ2)−n2log((y1−X1β_)2(1−ϕ2)+∑t=2n(yt−Xtβ_−ϕ(yt−1−Xt−1β_))2)\\begin{align} logL\\left( \\underline{\\beta}, \\phi; \\underline{y} \\right ) &= const. +\\frac{1}{2}log(1-\\phi^2)  \\\\ &-\\frac{n}{2}log\\left( (y_1-X_1\\underline{\\beta})^2(1-\\phi^2) + \\sum^n_{t=2} (y_t - X_t\\underline{\\beta} - \\phi (y_{t-1} -X_{t-1}\\underline{\\beta}))^2 \\right) \\\\ \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"ar2","dir":"Articles","previous_headings":"Likelihood from first principles","what":"AR(2)","title":"Background","text":"L(θ_;y_)=∏t=2np(Yt=yt│Yt−1=yt−1,Yt−2=yt−2)×p(Y2=y2|Y1=y1)×p(Y1=y1)L(\\underline{\\theta};\\underline{y})=\\prod_{t=2}^n p(Y_t=y_t│Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2}) \\times p(Y_2=y_2 | Y_1=y_1) \\times p(Y_1=y_1) Following reasoning can obtain densities three components likelihood. exact likelihood product conditional likelihood, conditional distribution y2|y1y_2 | y_1 marginal distribution y1y_1.","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"density-of-py_1y_1-1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Density of p(Y1=y1)p(Y_1=y_1)","title":"Background","text":"p(Y1=y1)p(Y_1=y_1) normally distributed mean = X1β_X_1\\underline{\\beta} equates β0+β1\\beta_0 + \\beta_1 variance = σ21−ϕ12−ϕ22\\frac{\\sigma^2}{1-\\phi_1^2 - \\phi_2^2}","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"density-of-py_2y_2y_1y_1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Density of p(Y2=y2│Y1=y1)p(Y_2=y_2│Y_1=y_1)","title":"Background","text":"p(Y2=y2│Y1=y1)p(Y_2=y_2│Y_1=y_1) complicated. Need work ","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"density-of-py_t-y_t-y_t-1y_t-1y_t-2y_t-2","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Density of p(Yt=yt│Yt−1=yt−1,Yt−2=yt−2)p(Y_t = y_t │ Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2})","title":"Background","text":"Recall yt=β0+β1t+uty_t = \\beta_0 + \\beta_1 t + u_t ϕ1yt−1=ϕ1β0+ϕ1β1(t−1)+ϕ1ut−1\\phi_1 y_{t-1} = \\phi_1 \\beta_0 + \\phi_1 \\beta_1 (t-1) + \\phi_1 u_{t-1} ϕ2yt−2=ϕ2β0+ϕ2β1(t−2)+ϕ2ut−2\\phi_2 y_{t-2} = \\phi_2 \\beta_0 + \\phi_2 \\beta_1 (t-2) + \\phi_2 u_{t-2} yt−ϕ1yt−1−ϕ2yt−2=β0+β1t+ut−ϕ1β0−ϕ1β1(t−1)−ϕ1ut−1−ϕ2β0−ϕ2β1(t−2)−ϕ2ut−2y_t - \\phi_1 y_{t-1} - \\phi_2 y_{t-2} = \\beta_0 + \\beta_1 t + u_t - \\phi_1 \\beta_0 - \\phi_1\\beta_1(t-1) -\\phi_1 u_{t-1} - \\phi_2 \\beta_0 - \\phi_2\\beta_1(t-2) -\\phi_2 u_{t-2} simplifies yt−ϕ1yt−1−ϕ2yt−2−β0(1−ϕ1−ϕ2)−β1(t−ϕ1(t−1)−ϕ2(t−2))=ety_t - \\phi_1 y_{t-1} - \\phi_2 y_{t-2} - \\beta_0 (1-\\phi_1-\\phi_2) - \\beta_1 (t - \\phi_1(t-1)- \\phi_2(t-2))  = e_t results p(Yt=yt│Yt−1=yt−1,Yt−2=yt−2)=∏t=3n1σ2πexp(−12(yt−ϕ1yt−1−ϕ2yt−2−β0(1−ϕ1−ϕ2)−β1(t−ϕ1(t−1)−ϕ2(t−2))σ)2)p(Y_t = y_t │ Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2}) = \\prod_{t=3}^n \\frac{1}{\\sigma \\sqrt{2\\pi}}exp \\left( -\\frac{1}{2}\\left( \\frac{ y_t - \\phi_1 y_{t-1} - \\phi_2 y_{t-2} - \\beta_0 (1-\\phi_1-\\phi_2) - \\beta_1 (t - \\phi_1(t-1)- \\phi_2(t-2))  }{\\sigma}\\right)^2\\right)","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"exact-likelihood-for-ar2","dir":"Articles","previous_headings":"","what":"Exact likelihood for AR2","title":"Background","text":"likelihood therefore:","code":""},{"path":[]},{"path":"https://andybeet.github.io/arfit/authors.html","id":null,"dir":"","previous_headings":"","what":"Authors","title":"Authors and Citation","text":"Andy Beet. Author, maintainer.","code":""},{"path":"https://andybeet.github.io/arfit/authors.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Authors and Citation","text":"Beet (2024). arfit: Fits full AR1 model. R package version 0.0.0.9000.","code":"@Manual{,   title = {arfit: Fits full AR1 model},   author = {Andy Beet},   year = {2024},   note = {R package version 0.0.0.9000}, }"},{"path":[]},{"path":[]},{"path":"https://andybeet.github.io/arfit/index.html","id":"legal-disclaimer","dir":"","previous_headings":"Contact","what":"Legal disclaimer","title":"Fits full AR1 model","text":"repository scientific product official communication National Oceanic Atmospheric Administration, United States Department Commerce. NOAA GitHub project code provided ‘’ basis user assumes responsibility use. claims Department Commerce Department Commerce bureaus stemming use GitHub project governed applicable Federal law. reference specific commercial products, processes, services service mark, trademark, manufacturer, otherwise, constitute imply endorsement, recommendation favoring Department Commerce. Department Commerce seal logo, seal logo DOC bureau, shall used manner imply endorsement commercial product activity DOC United States Government.","code":""},{"path":"https://andybeet.github.io/arfit/reference/arfit-package.html","id":null,"dir":"Reference","previous_headings":"","what":"arfit: Test for a trend in presence of autocorrelation — arfit-package","title":"arfit: Test for a trend in presence of autocorrelation — arfit-package","text":"Tools test trend presence autocorrelation. tools reproduce results publication","code":""},{"path":"https://andybeet.github.io/arfit/reference/arfit-package.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"arfit: Test for a trend in presence of autocorrelation — arfit-package","text":"Simulates data Evaluates likelihood Fits likelihood Estimates MLE model Example provides Tools su=imulation study","code":""},{"path":"https://andybeet.github.io/arfit/reference/arfit-package.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"arfit: Test for a trend in presence of autocorrelation — arfit-package","text":"learn using arfit, start vignette: browseVignettes(package=\"arfit\") click index ","code":""},{"path":[]},{"path":"https://andybeet.github.io/arfit/reference/arfit-package.html","id":"author","dir":"Reference","previous_headings":"","what":"Author","title":"arfit: Test for a trend in presence of autocorrelation — arfit-package","text":"Maintainer: Andy Beet andrew.beet@noaa.gov (ORCID)","code":""},{"path":"https://andybeet.github.io/arfit/reference/est_beta_given_rho.html","id":null,"dir":"Reference","previous_headings":"","what":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho","title":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho","text":"Uses Prais Winsten transformation utilizes first observation","code":""},{"path":"https://andybeet.github.io/arfit/reference/est_beta_given_rho.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho","text":"","code":"est_beta_given_rho(xt, yt, rho)"},{"path":"https://andybeet.github.io/arfit/reference/est_beta_given_rho.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho","text":"xt Numeric vector. Time series time points (1, ..., n) yt Numeric vector. Time series observations (length = n) rho Numeric scalar. AR1 parameter","code":""},{"path":"https://andybeet.github.io/arfit/reference/est_beta_given_rho2.html","id":null,"dir":"Reference","previous_headings":"","what":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho2","title":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho2","text":"Uses Prais Winsten transformation utilizes first observation","code":""},{"path":"https://andybeet.github.io/arfit/reference/est_beta_given_rho2.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho2","text":"","code":"est_beta_given_rho2(xt, yt, rho1, rho2 = 0)"},{"path":"https://andybeet.github.io/arfit/reference/est_beta_given_rho2.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho2","text":"xt Numeric vector. Time series time points (1, ..., n) yt Numeric vector. Time series observations (length = n) rho1 Numeric scalar. AR1 parameter rho2 Numeric scalar. AR2 parameter","code":""},{"path":"https://andybeet.github.io/arfit/reference/example_cedar_rapids.html","id":null,"dir":"Reference","previous_headings":"","what":"Run test on example data set — example_cedar_rapids","title":"Run test on example data set — example_cedar_rapids","text":"Use data USGS 05464500 Cedar River Cedar Rapids, IA","code":""},{"path":"https://andybeet.github.io/arfit/reference/example_cedar_rapids.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Run test on example data set — example_cedar_rapids","text":"","code":"example_cedar_rapids(dataSet = arfit::cedar_rapids, nBootSims = 999)"},{"path":"https://andybeet.github.io/arfit/reference/example_cedar_rapids.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Run test on example data set — example_cedar_rapids","text":"dataSet Data frame. Two columns (year, riverflow) nBootSims Numeric scalar. Number bootstrap samples perform","code":""},{"path":"https://andybeet.github.io/arfit/reference/fit_ar1_grid.html","id":null,"dir":"Reference","previous_headings":"","what":"Fit linear model with AR(1) error by GLS — fit_ar1_grid","title":"Fit linear model with AR(1) error by GLS — fit_ar1_grid","text":"Fit linear model AR(1) error GLS","code":""},{"path":"https://andybeet.github.io/arfit/reference/fit_ar1_grid.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Fit linear model with AR(1) error by GLS — fit_ar1_grid","text":"","code":"fit_ar1_grid(data, rhoVec = NULL, hypothesis)"},{"path":"https://andybeet.github.io/arfit/reference/fit_ar1_opt.html","id":null,"dir":"Reference","previous_headings":"","what":"Fit linear model with trend using optimization routine — fit_ar1_opt","title":"Fit linear model with trend using optimization routine — fit_ar1_opt","text":"Fit linear model trend using optimization routine","code":""},{"path":"https://andybeet.github.io/arfit/reference/fit_ar1_opt.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Fit linear model with trend using optimization routine — fit_ar1_opt","text":"","code":"fit_ar1_opt(data, rho, hypothesis)"},{"path":"https://andybeet.github.io/arfit/reference/fit_ar2_grid.html","id":null,"dir":"Reference","previous_headings":"","what":"Fit linear model with AR(1) error by GLS — fit_ar2_grid","title":"Fit linear model with AR(1) error by GLS — fit_ar2_grid","text":"Fit linear model AR(1) error GLS","code":""},{"path":"https://andybeet.github.io/arfit/reference/fit_ar2_grid.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Fit linear model with AR(1) error by GLS — fit_ar2_grid","text":"","code":"fit_ar2_grid(data, rhoVec = NULL, hypothesis)"},{"path":"https://andybeet.github.io/arfit/reference/fit_ar2_opt.html","id":null,"dir":"Reference","previous_headings":"","what":"Fit linear model with trend using optimization routine — fit_ar2_opt","title":"Fit linear model with trend using optimization routine — fit_ar2_opt","text":"Fit linear model trend using optimization routine","code":""},{"path":"https://andybeet.github.io/arfit/reference/fit_ar2_opt.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Fit linear model with trend using optimization routine — fit_ar2_opt","text":"","code":"fit_ar2_opt(data, rho, hypothesis)"},{"path":"https://andybeet.github.io/arfit/reference/isStationary_ar2.html","id":null,"dir":"Reference","previous_headings":"","what":"Determines stationarity of AR2 — isStationary_ar2","title":"Determines stationarity of AR2 — isStationary_ar2","text":"Given parameters AR2 determines process stationary","code":""},{"path":"https://andybeet.github.io/arfit/reference/isStationary_ar2.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Determines stationarity of AR2 — isStationary_ar2","text":"","code":"isStationary_ar2(rhos)"},{"path":"https://andybeet.github.io/arfit/reference/isStationary_ar2.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Determines stationarity of AR2 — isStationary_ar2","text":"rhos Numeric vector. AR2 parameters","code":""},{"path":"https://andybeet.github.io/arfit/reference/isStationary_ar2.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Determines stationarity of AR2 — isStationary_ar2","text":"Boolean. Process stationary ","code":""},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar1.html","id":null,"dir":"Reference","previous_headings":"","what":"Evaluate the likelihood of linear model with errors from an AR order 1 process — likelihood_ar1","title":"Evaluate the likelihood of linear model with errors from an AR order 1 process — likelihood_ar1","text":"Evaluate likelihood linear model errors AR order 1 process","code":""},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar1.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Evaluate the likelihood of linear model with errors from an AR order 1 process — likelihood_ar1","text":"","code":"likelihood_ar1(rho, dataf, hypothesis = \"null\")"},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar1.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Evaluate the likelihood of linear model with errors from an AR order 1 process — likelihood_ar1","text":"rho Numeric scalar. Autoregressive parameter dataf Data frame. (nx2). Defining xt yt values hypothesis Character string. \"null\" \"alt\" determines whether evaluate null alternative hypothesis. \"null\" fixes linear component, beta = 0.","code":""},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar1.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Evaluate the likelihood of linear model with errors from an AR order 1 process — likelihood_ar1","text":"likelihod value","code":""},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar2.html","id":null,"dir":"Reference","previous_headings":"","what":"Evaluates the likelihood of AR2 — likelihood_ar2","title":"Evaluates the likelihood of AR2 — likelihood_ar2","text":"Uses full likelihood explained Beach MacKinnon first two data points used likelihood","code":""},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar2.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Evaluates the likelihood of AR2 — likelihood_ar2","text":"","code":"likelihood_ar2(rhos, data, hypothesis = \"null\")"},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar2.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Evaluates the likelihood of AR2 — likelihood_ar2","text":"rhos Numeric vector (1x2). 1st 2nd order Autoregressive parameters data dd hypothesis Character string. \"null\" \"alt\" determines whether evaluate null alternative hypothesis. \"null\" fixes linear component, beta = 0.","code":""},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar2.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Evaluates the likelihood of AR2 — likelihood_ar2","text":"-ve Log likelihood","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_single_opt_ar1.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar1","title":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar1","text":"Performs n tests","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_single_opt_ar1.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar1","text":"","code":"sim_single_opt_ar1(   beta = 0,   rho = 0,   sigma = 0.25,   nT = 10,   nSims = 200,   nBootSims = 500 )"},{"path":"https://andybeet.github.io/arfit/reference/sim_single_opt_ar1.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar1","text":"beta Numeric.  Values beta_1 (slope/trend parameter) rho Numeric  Values autoregressive parameter sigma Numeric.  Values standard deviation noise nT Numeric.  Values length time series simulate nSims Numeric.  Number time series simulate nBootSims Numeric.  Number bootstrap data sets","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_single_opt_ar1.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar1","text":"convenience intercept, beta_0 set zero","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_single_opt_ar1.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar1","text":"","code":"if (FALSE) { # \\dontrun{ sim_single_opt(beta = 0.5,rho =0.5, sigma = 0.25, nT = 10, nSims = 200, nBootSims = 500) } # }"},{"path":"https://andybeet.github.io/arfit/reference/sim_single_opt_ar2.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar2","title":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar2","text":"Performs n tests","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_single_opt_ar2.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar2","text":"","code":"sim_single_opt_ar2(   beta = 0,   rho1 = 0,   rho2 = 0,   sigma = 0.25,   nT = 10,   nSims = 200,   nBootSims = 500 )"},{"path":"https://andybeet.github.io/arfit/reference/sim_single_opt_ar2.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar2","text":"beta Numeric.  Values beta_1 (slope/trend parameter) sigma Numeric.  Values standard deviation noise nT Numeric.  Values length time series simulate nSims Numeric.  Number time series simulate nBootSims Numeric.  Number bootstrap data sets rho Numeric  Values autoregressive parameter","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_single_opt_ar2.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar2","text":"convenience intercept, beta_0 set zero","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_single_opt_ar2.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Simulates n data sets and performs bootstrapping — sim_single_opt_ar2","text":"","code":"if (FALSE) { # \\dontrun{ sim_single_opt(beta = 0.5,rho =0.5, sigma = 0.25, nT = 10, nSims = 200, nBootSims = 500) } # }"},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar1.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","text":"Performs simulation study assess performance test. Utilizes multiple cores spread bootstrap samples multiple cores","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar1.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","text":"","code":"sim_study_opt_ar1(   outDir = here::here(\"out.txt\"),   betaVec = c(0, 0.12, 0.25, 0.5),   rhoVec = c(0, 0.25, 0.5, 0.75, 0.95),   sigmaVec = c(0.25, 0.5, 0.75),   nTVec = c(10, 25, 50),   nSims = 200,   nBootSims = 500,   setSeed = NULL,   nCores = NULL )"},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar1.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","text":"outDir Character string. Path output file betaVec Numeric vector. Values beta_1 (slope/trend parameter) rhoVec Numeric vector. Values autoregressive parameter sigmaVec Numeric vector. Values standard deviation noise nSims Numeric scalar. Number time series simulate nBootSims Numeric scalar. Number bootstrap data sets setSeed Numeric scalar. Value seed simulations. (Default = NULL, random number 1-e7 selected) nCores Numeric scalar. Specify number cores utilize (Default = NULL, utilizes n-1 cores) nVec Numeric vector. Values length time series simulate","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar1.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","text":"convenience intercept, beta_0 set zero","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar1.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","text":"","code":"if (FALSE) { # \\dontrun{ } # }"},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar2.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar2","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar2","text":"Performs simulation study assess performance test. Utilizes multiple cores spread bootstrap samples multiple cores","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar2.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar2","text":"","code":"sim_study_opt_ar2(   outDir = here::here(\"out.txt\"),   betaVec = c(0, 0.12, 0.25, 0.5),   rhoVec1 = c(0, 0.25, 0.5, 0.75, 0.95),   rhoVec2 = c(0, 0.25, 0.5, 0.75, 0.95),   sigmaVec = c(0.25, 0.5, 0.75),   nTVec = c(10, 25, 50),   nSims = 200,   nBootSims = 500,   setSeed = NULL,   nCores = NULL )"},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar2.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar2","text":"outDir Character string. Path output file betaVec Numeric vector. Values beta_1 (slope/trend parameter) rhoVec1 Numeric vector. Values autoregressive parameter order 1 rhoVec2 Numeric vector. Values autoregressive parameter order 2 sigmaVec Numeric vector. Values standard deviation noise nSims Numeric scalar. Number time series simulate nBootSims Numeric scalar. Number bootstrap data sets setSeed Numeric scalar. Value seed simulations. (Default = NULL, random number 1-e7 selected) nCores Numeric scalar. Specify number cores utilize (Default = NULL, utilizes n-1 cores) nVec Numeric vector. Values length time series simulate","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar2.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar2","text":"convenience intercept, beta_0 set zero","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar2.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar2","text":"","code":"if (FALSE) { # \\dontrun{ } # }"},{"path":"https://andybeet.github.io/arfit/reference/simulate_ar1.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulate data from an linear process with AR1 error — simulate_ar1","title":"Simulate data from an linear process with AR1 error — simulate_ar1","text":"model=   y_t = + bt + z_t z_t = rho z_t-1 + e_t   (e_t ~ N(0,sigma^2)) Uses unconditional mean variance simulate first data point","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulate_ar1.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulate data from an linear process with AR1 error — simulate_ar1","text":"","code":"simulate_ar1(alpha, beta = 0, sigma, rho, n)"},{"path":"https://andybeet.github.io/arfit/reference/simulate_ar1.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulate data from an linear process with AR1 error — simulate_ar1","text":"alpha Numeric scalar. Intercept beta Numeric scalar. Slope sigma Numeric scalar. Standard deviation error term rho Numeric scalar. Auto regressive parameter n Numeric scalar. Length time series","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulate_ar2.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulate data from an linear process with AR2 error — simulate_ar2","title":"Simulate data from an linear process with AR2 error — simulate_ar2","text":"#' model=   y_t = + b.t + z_t z_t = rho1. z_t-1 + rho2. z_t-2 + e_t   (e_t ~ N(0,sigma^2))","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulate_ar2.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulate data from an linear process with AR2 error — simulate_ar2","text":"","code":"simulate_ar2(alpha, beta = 0, sigma, rhos, nT)"},{"path":"https://andybeet.github.io/arfit/reference/simulate_ar2.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulate data from an linear process with AR2 error — simulate_ar2","text":"alpha Numeric scalar. Intercept beta Numeric Scalar. Slope sigma Numeric scalar. Variance error process rhos Numeric vector (1x2). AR2 parameters nT Numeric scalar. Length time series","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_grid_ar1.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulation study using grid search — simulation_study_grid_ar1","title":"Simulation study using grid search — simulation_study_grid_ar1","text":"Performs simulation study assess performance test. Depending user inputs take since Rho estimated using grid search -.99:.99:.01","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_grid_ar1.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulation study using grid search — simulation_study_grid_ar1","text":"","code":"simulation_study_grid_ar1(   outDir = here::here(\"out.txt\"),   betaVec = c(0.12, 0.25, 0.5),   rhoVec = c(0, 0.25, 0.5, 0.75, 0.95),   sigmaVec = c(0.25, 0.5, 0.75),   nTVec = c(10),   nSims = 200,   nBootSims = 500,   rhoSearch = NULL )"},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_grid_ar1.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulation study using grid search — simulation_study_grid_ar1","text":"outDir Character string. Path output file betaVec Numeric vector. Values beta_1 (slope/trend parameter) rhoVec Numeric vector. Values autoregressive parameter sigmaVec Numeric vector. Values standard deviation noise nSims Numeric scalar. Number time series simulate nBootSims Numeric scalar. Number bootstrap data sets rhoSearch Numeric Vector. Values use rho grid search eg (seq(-.99,.99,.01)) nVec Numeric vector. Values length time series simulate","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_grid_ar1.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Simulation study using grid search — simulation_study_grid_ar1","text":"convenience intercept, beta_0 set zero","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_grid_ar1.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Simulation study using grid search — simulation_study_grid_ar1","text":"","code":"if (FALSE) { # \\dontrun{ simulation_study_grid(here::here(\"output.txt\"),betaVec = 0.5,rhovec =0.5, sigmaVec = 0.25, nTVec = 10, nSims = 200, nBootSims = 500) } # }"},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_grid_ar2.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulation study using optimization routine — simulation_study_grid_ar2","title":"Simulation study using optimization routine — simulation_study_grid_ar2","text":"Performs simulation study assess performance test.","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_grid_ar2.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulation study using optimization routine — simulation_study_grid_ar2","text":"","code":"simulation_study_grid_ar2(   outDir = here::here(\"out_ar2.txt\"),   betaVec = c(0),   rho1Vec = c(0),   rho2Vec = c(0),   sigmaVec = c(0.25),   nTVec = c(10),   nSims = 200,   nBootSims = 500,   rhoSearch = NULL )"},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_grid_ar2.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulation study using optimization routine — simulation_study_grid_ar2","text":"outDir Character string. Path output file betaVec Numeric vector. Values beta_1 (slope/trend parameter) rho1Vec Numeric vector. Values autoregressive parameter, rho_! rho2Vec Numeric vector. Values autoregressive parameter, rho_2 sigmaVec Numeric vector. Values standard deviation noise nSims Numeric scalar. Number time series simulate nBootSims Numeric scalar. Number bootstrap data sets nVec Numeric vector. Values length time series simulate","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_grid_ar2.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Simulation study using optimization routine — simulation_study_grid_ar2","text":"convenience intercept, beta_0 set zero","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_grid_ar2.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Simulation study using optimization routine — simulation_study_grid_ar2","text":"","code":"if (FALSE) { # \\dontrun{ simulation_study_opt(here::here(\"output.txt\"),betaVec = 0.5,rhovec =0.5, sigmaVec = 0.25, nTVec = 10, nSims = 200, nBootSims = 500) } # }"},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar1.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulation study using optimization routine — simulation_study_opt_ar1","title":"Simulation study using optimization routine — simulation_study_opt_ar1","text":"Performs simulation study assess performance test.","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar1.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulation study using optimization routine — simulation_study_opt_ar1","text":"","code":"simulation_study_opt_ar1(   outDir = here::here(\"out.txt\"),   betaVec = c(0.12, 0.25, 0.5),   rhoVec = c(0, 0.25, 0.5, 0.75, 0.95),   sigmaVec = c(0.25, 0.5, 0.75),   nTVec = c(10),   nSims = 200,   nBootSims = 500 )"},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar1.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulation study using optimization routine — simulation_study_opt_ar1","text":"outDir Character string. Path output file betaVec Numeric vector. Values beta_1 (slope/trend parameter) rhoVec Numeric vector. Values autoregressive parameter sigmaVec Numeric vector. Values standard deviation noise nSims Numeric scalar. Number time series simulate nBootSims Numeric scalar. Number bootstrap data sets nVec Numeric vector. Values length time series simulate","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar1.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Simulation study using optimization routine — simulation_study_opt_ar1","text":"convenience intercept, beta_0 set zero","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar1.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Simulation study using optimization routine — simulation_study_opt_ar1","text":"","code":"if (FALSE) { # \\dontrun{ simulation_study_opt(here::here(\"output.txt\"),betaVec = 0.5,rhovec =0.5, sigmaVec = 0.25, nTVec = 10, nSims = 200, nBootSims = 500) } # }"},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar2.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulation study using optimization routine — simulation_study_opt_ar2","title":"Simulation study using optimization routine — simulation_study_opt_ar2","text":"Performs simulation study assess performance test.","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar2.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulation study using optimization routine — simulation_study_opt_ar2","text":"","code":"simulation_study_opt_ar2(   outDir = here::here(\"out_ar2.txt\"),   betaVec = c(0),   rho1Vec = c(0),   rho2Vec = c(0),   sigmaVec = c(0.25),   nTVec = c(10),   nSims = 200,   nBootSims = 500,   rhoSearch = NULL )"},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar2.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulation study using optimization routine — simulation_study_opt_ar2","text":"outDir Character string. Path output file betaVec Numeric vector. Values beta_1 (slope/trend parameter) rho1Vec Numeric vector. Values autoregressive parameter, rho_! rho2Vec Numeric vector. Values autoregressive parameter, rho_2 sigmaVec Numeric vector. Values standard deviation noise nSims Numeric scalar. Number time series simulate nBootSims Numeric scalar. Number bootstrap data sets nVec Numeric vector. Values length time series simulate","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar2.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Simulation study using optimization routine — simulation_study_opt_ar2","text":"convenience intercept, beta_0 set zero","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar2.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Simulation study using optimization routine — simulation_study_opt_ar2","text":"","code":"if (FALSE) { # \\dontrun{ simulation_study_opt(here::here(\"output.txt\"),betaVec = 0.5,rhovec =0.5, sigmaVec = 0.25, nTVec = 10, nSims = 200, nBootSims = 500) } # }"}]
+[{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-1","dir":"Articles","previous_headings":"","what":"Eq 1","title":"Equations","text":"Yt=β0+β1t+ϵt\\begin{align} \\tag{1}   Y_t = \\beta_0 + \\beta_1 t + \\epsilon_t   \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-2","dir":"Articles","previous_headings":"","what":"Eq 2","title":"Equations","text":"L(θ_;y_)=∏t=2np(Yt=yt|Yt−1=yt−1)p(Y1=y1)\\begin{align} \\tag{2} \\mathrm{L}\\left( \\underline{\\theta}; \\underline{y} \\right )= \\prod^n_{t=2} p\\left(Y_t = y_t | Y_{t-1}=y_{t-1}\\right)  p\\left(Y_1=y_1 \\right)  \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-3","dir":"Articles","previous_headings":"","what":"Eq 3","title":"Equations","text":"logL(θ_;y_)=−n2log2π−nlogσ+12log(1−ϕ2)−12σ2((1−ϕ2)(y1−β0−β1)2+∑t=2n(yt−ϕyt−1−β0(1−ϕ2)−tβ1+ϕ(t−1)β1)2)\\begin{align} \\tag{3} logL\\left( \\underline{\\theta}; \\underline{y} \\right ) = & -\\frac{n}{2}log2\\pi - nlog\\sigma + \\frac{1}{2}log(1-\\phi^2) \\\\ &-\\frac{1}{2\\sigma^2}\\left( (1-\\phi^2)(y_1-\\beta_0-\\beta_1)^2 + \\sum^n_{t=2}(y_t - \\phi y_{t-1}-\\beta_0(1-\\phi^2) -t\\beta_1 + \\phi(t-1)\\beta_1)^2 \\right)   \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-4","dir":"Articles","previous_headings":"","what":"Eq 4","title":"Equations","text":"σ̂2=1n((1−ϕ2)(y1−β0−β1)2+∑t=2n(yt−ϕyt−1−β0(1−ϕ2)−tβ1+ϕ(t−1)β1)2)\\begin{align} \\tag{4} \\hat\\sigma^2 = \\frac{1}{n}\\left( (1-\\phi^2)(y_1-\\beta_0-\\beta_1)^2 + \\sum^n_{t=2}(y_t - \\phi y_{t-1}-\\beta_0(1-\\phi^2) - t\\beta_1 + \\phi(t-1)\\beta_1)^2 \\right)  \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-5","dir":"Articles","previous_headings":"","what":"Eq 5","title":"Equations","text":"logL(β_,ϕ;y_)=const.+12log(1−ϕ2)−n2log((1−ϕ2)(y1−β0−β1)2+∑t=2n(yt−ϕyt−1−β0(1−ϕ2)−tβ1+ϕ(t−1)β1)2)=const.+12log(1−ϕ2)−n2log((1−ϕ2)(y1−X1β_)2+∑t=2n(yt−ϕyt−1−Xtβ_+ϕXt−1β_)2)\\begin{align} \\tag{5} logL\\left( \\underline{\\beta}, \\phi; \\underline{y} \\right ) &= const. + \\frac{1}{2}log(1-\\phi^2) \\\\ &-\\frac{n}{2}log\\left( (1-\\phi^2)(y_1-\\beta_0-\\beta_1)^2 + \\sum^n_{t=2}(y_t - \\phi y_{t-1}-\\beta_0(1-\\phi^2)-t\\beta_1 + \\phi(t-1)\\beta_1)^2 \\right) \\\\   &= const. + \\frac{1}{2}log(1-\\phi^2) \\\\ &-\\frac{n}{2}log\\left( (1-\\phi^2)(y_1-X_1\\underline{\\beta})^2 + \\sum^n_{t=2}(y_t - \\phi y_{t-1}-X_t\\underline{\\beta} + \\phi X_{t-1}\\underline{\\beta})^2 \\right)  \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/Equations.html","id":"eq-6","dir":"Articles","previous_headings":"","what":"Eq 6","title":"Equations","text":"logL(β_,ϕ_,σ;y_)=−n2log(2π)−n2log(σ2)+12log|Vp−1|−12σ2(yp_−μp_)TVp−1(yp_−μp_)−12σ2∑t=p+1n(yt−c−ϕ1yt−1−...−ϕpyt−p)2\\begin{align} \\tag{6} logL\\left( \\underline{\\beta}, \\underline{\\phi},\\sigma; \\underline{y} \\right ) &= -\\frac{n}{2}log(2\\pi) -\\frac{n}{2}log(\\sigma^2) +\\frac{1}{2}log \\left|V_p^{-1} \\right| \\\\ &-\\frac{1 }{2 \\sigma^2} (\\underline{y_p}-\\underline{\\mu_p})^T V_p^{-1}(\\underline{y_p}-\\underline{\\mu_p}) \\\\  &- \\frac{1}{2\\sigma^2}\\sum^n_{t=p+1} (y_t - c - \\phi_1y_{t-1} - ... - \\phi_p y_{t-p})^2   \\end{align} |Vp−1|\\left|V_p^{-1} \\right| determinant inverted matrix VpV_p, σ2Vp\\sigma^2V_p = variance-covariance matrix order p, μp_=Xpβ_\\underline{\\mu_p} = X_p\\underline{\\beta}, XpX_p pthp_{th} row design matrix corresponding time t = p cc = function fitted terms Xtβ_X_t\\underline{\\beta}","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"likelihood-for-arp-errors","dir":"Articles","previous_headings":"","what":"Likelihood for AR(p) errors","title":"Background","text":"Hamilton (1994) logL(β_,ϕ_,σ;y_)=−n2log(2π)−n2log(σ2)+12log|Vp−1|−12σ2(yp_−μp_)TVp−1(yp_−μp_)−12σ2∑t=p+1n(yt−c−ϕ1yt−1−...−ϕpyt−p)2\\begin{align} logL\\left( \\underline{\\beta}, \\underline{\\phi},\\sigma; \\underline{y} \\right ) &= -\\frac{n}{2}log(2\\pi) -\\frac{n}{2}log(\\sigma^2) +\\frac{1}{2}log \\left|V_p^{-1} \\right| \\\\ &-\\frac{1 }{2 \\sigma^2} (\\underline{y_p}-\\underline{\\mu_p})^T V_p^{-1}(\\underline{y_p}-\\underline{\\mu_p}) \\\\  &- \\frac{1}{2\\sigma^2}\\sum^n_{t=p+1} (y_t - c - \\phi_1y_{t-1} - ... - \\phi_p y_{t-p})^2 \\\\  \\end{align} |Vp−1|\\left|V_p^{-1} \\right| determinant inverted matrix VpV_p, σ2Vp\\sigma^2V_p = variance-covariance matrix order p, μp_=Xpβ_\\underline{\\mu_p} = X_p\\underline{\\beta}, XpX_p pthp_{th} row design matrix corresponding time t = p cc = function fitted terms Xtβ_X_t\\underline{\\beta}","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"likelihood-for-ar1-errors","dir":"Articles","previous_headings":"","what":"Likelihood for AR(1) errors","title":"Background","text":"Setting p = 1, σ2V1=σ21−ϕ12\\sigma^2V_1 = \\frac{\\sigma^2}{1-\\phi_1^2} = variance process, y, |V1−1|=1−ϕ12\\left|V_1^{-1}\\right| = 1-\\phi_1^2 μ1=X1β_=β0+β1\\mu_1 = X_1\\underline{\\beta} = \\beta_0 + \\beta_1 c=Xtβ_−ϕ1Xt−1β_c = X_t\\underline{\\beta} - \\phi_1 X_{t-1}\\underline{\\beta} y1y_1 first observation log likelihood, logL(β_,ϕ1,σ;y_)=−n2log(2π)−n2log(σ2)+12log(1−ϕ12)−12σ2(y1−X1β_)2(1−ϕ12)−12σ2∑t=2n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_))2\\begin{align}  logL\\left( \\underline{\\beta}, \\phi_1, \\sigma; \\underline{y} \\right ) &= -\\frac{n}{2}log(2\\pi) -\\frac{n}{2}log(\\sigma^2) +\\frac{1}{2}log(1-\\phi_1^2)  \\\\ &-\\frac{1 }{2 \\sigma^2} ({y_1}-X_1\\underline{\\beta})^2 (1-\\phi_1^2) \\\\ &- \\frac{1}{2\\sigma^2}\\sum^n_{t=2} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}))^2 \\\\  \\end{align} Differentiating logL(β_,ϕ1,σ;y_)logL\\left( \\underline{\\beta}, \\phi_1, \\sigma; \\underline{y} \\right ) respect σ\\sigma equating zero yields maximum likelihood estimator, σ̂2=1n[(y1−X1β_)2(1−ϕ12)+∑t=2n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_))2]\\begin{align} \\hat{\\sigma}^2 = \\frac{1}{n}\\left[ (y_1-X_1\\underline{\\beta})^2(1-\\phi_1^2) + \\sum^n_{t=2} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}))^2 \\right]  \\end{align} Substituting σ̂2\\hat{\\sigma}^2 back log likelihood yields (Beach MacKinnon (1978a)) logL(β_,ϕ1;y_)=const.+12log(1−ϕ2)−n2log((y1−X1β_)2(1−ϕ12)+∑t=2n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_))2)\\begin{align} logL\\left( \\underline{\\beta}, \\phi_1; \\underline{y} \\right ) &= const. +\\frac{1}{2}log(1-\\phi^2)  \\\\ &-\\frac{n}{2}log\\left( (y_1-X_1\\underline{\\beta})^2(1-\\phi_1^2) + \\sum^n_{t=2} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}))^2 \\right) \\\\ \\end{align} Two additional terms exist likelihood appear conditional likelihood used GLS procedures. term (y1−X1β_)2(1−ϕ12)(y_1-X_1\\underline{\\beta})^2(1-\\phi_1^2) ensures initial value effect estimates 12log(1−ϕ2)\\frac{1}{2}log(1-\\phi^2) constrains stationary condition hold.","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"maximization-of-the-likelihood","dir":"Articles","previous_headings":"Likelihood for AR(1) errors","what":"Maximization of the likelihood","title":"Background","text":"Maximization likelihood requires iterative numerical procedures. estimate β_\\underline{\\beta} maximizes log-likelihood conditional ϕ\\phi β̂_=(X*TX*)−1X*Ty*\\begin{align} \\underline{\\hat{\\beta}} = (X^{*T}X^*)^{-1}X^{*T}y^* \\end{align} X*=QXX^*=QX y*=Qyy^* = Qy QQ Prais Winsten transformation Q=[(1−ϕ12)1/20...−ϕ110......0−ϕ11]  \\begin{matrix} Q = \\end{matrix} \\begin{bmatrix} (1-\\phi_1^2)^{1/2} & 0 & ... \\\\ -\\phi_1 & 1 & 0 & ... \\\\ & & ... & 0 & -\\phi_1 & 1\\\\ \\end{bmatrix}  procedure searches ϕ1\\phi_1 required find maximum likelihood estimates β_\\underline\\beta, σ\\sigma, ϕ1\\phi_1","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"likelihood-for-ar2-errors","dir":"Articles","previous_headings":"","what":"Likelihood for AR(2) errors","title":"Background","text":"Following method AR(1) errors. Set p = 2, |V2−1|=(1+ϕ22)[(1−ϕ2)2−ϕ12]\\left|V_2^{-1}\\right| = (1+\\phi_2^2)\\left[(1-\\phi_2)^2 -\\phi_1^2 \\right] μ2_=(μ1,μ2)=(X1β_,X2β_)\\underline{\\mu_2} = (\\mu_1, \\mu_2) = (X_1\\underline{\\beta},X_2\\underline{\\beta}) vector means t = 1,2 y2_=(y1,y2)\\underline{y_2} = (y_1,y_2) corresponding vector observations [μ1μ2]=[1112][β0β1] \\begin{bmatrix}  \\mu_1 \\\\ \\mu_2 \\\\ \\end{bmatrix} = \\begin{bmatrix}  1 & 1 \\\\ 1 & 2 \\\\ \\end{bmatrix} \\begin{bmatrix}  \\beta_0 \\\\ \\beta_1 \\\\ \\end{bmatrix} c=Xtβ_−ϕ1Xt−1β_−ϕ2Xt−2β_c = X_t\\underline{\\beta} - \\phi_1 X_{t-1}\\underline{\\beta}- \\phi_2 X_{t-2}\\underline{\\beta} log likelihood, logL(β_,ϕ_,σ;y_)=−n2log(2π)−n2log(σ2)+12log((1+ϕ22)[(1−ϕ2)2−ϕ12])−12σ2(y2_−μ2_)TV2−1(y2_−μ2_)−12σ2∑t=3n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_)−ϕ2(yt−2−Xt−2β_))2\\begin{align}  logL\\left( \\underline{\\beta}, \\underline{\\phi}, \\sigma; \\underline{y} \\right ) &= -\\frac{n}{2}log(2\\pi) -\\frac{n}{2}log(\\sigma^2) +\\frac{1}{2}log((1+\\phi_2^2)\\left[(1-\\phi_2)^2 -\\phi_1^2 \\right])  \\\\ &-\\frac{1 }{2 \\sigma^2} (\\underline{y_2}-\\underline{\\mu_2})^T V_2^{-1}(\\underline{y_2}-\\underline{\\mu_2}) \\\\ &- \\frac{1}{2\\sigma^2}\\sum^n_{t=3} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}) - \\phi_2 (y_{t-2} -X_{t-2}\\underline{\\beta}))^2 \\\\  \\end{align} Differentiating logL(β_,ϕ_,σ;y_)logL\\left( \\underline{\\beta}, \\underline{\\phi}, \\sigma; \\underline{y} \\right ) respect σ\\sigma equating zero yields maximum likelihood estimator, σ̂2=1n[(y2_−μ2_)TV2−1(y2_−μ2_)+∑t=3n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_)−ϕ2(yt−2−Xt−2β_))2]\\begin{align} \\hat{\\sigma}^2 = \\frac{1}{n}\\left[ (\\underline{y_2}-\\underline{\\mu_2})^T V_2^{-1}(\\underline{y_2}-\\underline{\\mu_2}) + \\sum^n_{t=3} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}) - \\phi_2 (y_{t-2} -X_{t-2}\\underline{\\beta}))^2\\right]  \\end{align} Substituting σ̂2\\hat{\\sigma}^2 back log likelihood yields logL(β_,ϕ_;y_)=const.+12log((1+ϕ22)[(1−ϕ2)2−ϕ12])−n2log[(y2_−μ2_)TV2−1(y2_−μ2_)+∑t=3n(yt−Xtβ_−ϕ1(yt−1−Xt−1β_)−ϕ2(yt−2−Xt−2β_))2]\\begin{align} logL\\left( \\underline{\\beta}, \\underline{\\phi}; \\underline{y} \\right) &= const. +\\frac{1}{2}log\\left((1+\\phi_2^2)\\left[(1-\\phi_2)^2 -\\phi_1^2 \\right]\\right)  \\\\ &-\\frac{n}{2}log\\left[ (\\underline{y_2}-\\underline{\\mu_2})^T V_2^{-1}(\\underline{y_2}-\\underline{\\mu_2}) + \\sum^n_{t=3} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}) - \\phi_2 (y_{t-2} -X_{t-2}\\underline{\\beta}))^2\\right]\\\\ \\end{align} Simplifying , 12log((1+ϕ22)[(1−ϕ2)2−ϕ12])=log(1+ϕ2)+12log(1−ϕ1−ϕ2)+12log(1+ϕ1−ϕ2)\\begin{align} \\frac{1}{2}log\\left((1+\\phi_2^2)\\left[(1-\\phi_2)^2 -\\phi_1^2 \\right]\\right) = log(1+\\phi_2) + \\frac{1}{2}log(1-\\phi_1-\\phi_2)+\\frac{1}{2}log(1+\\phi_1-\\phi_2) \\end{align} (y2_−μ2_)TV2−1(y2_−μ2_)=[(y1−μ1)(y2−μ2)][(1−ϕ22)−(ϕ1+ϕ1ϕ2)−(ϕ1+ϕ1ϕ2)(1−ϕ22)][(y1−μ1)(y2−μ2)]\\begin{align} (\\underline{y_2}-\\underline{\\mu_2})^T V_2^{-1}(\\underline{y_2}-\\underline{\\mu_2}) = & \\begin{bmatrix} (y_1 - \\mu_1) & (y_2 -\\mu_2) \\\\ \\end{bmatrix}   \\begin{bmatrix} (1-\\phi_2^2) & -(\\phi_1 + \\phi_1\\phi_2) \\\\ -(\\phi_1 + \\phi_1\\phi_2) & (1-\\phi_2^2)  \\\\ \\end{bmatrix}  \\begin{bmatrix} (y_1 - \\mu_1) \\\\ (y_2 - \\mu_2) \\\\ \\end{bmatrix} \\end{align} =(y1−μ1)2(1−ϕ22)−2(y1−μ1)(y2−μ2)(ϕ1+ϕ1ϕ2)+(y2−μ2)2(1−ϕ22)=(y1−μ1)2(1−ϕ22)−2(y1−μ1)(y2−μ2)ϕ1(1+ϕ2)+(y2−μ2)2(1−ϕ22)=(y1−X1β_)2(1−ϕ22)−2(y1−X1β_)(y2−X2β_)ϕ1(1+ϕ2)+(y2−X2β_)2(1−ϕ22)\\begin{align} = & (y_1 - \\mu_1)^2(1-\\phi_2^2)-2(y_1 - \\mu_1)(y_2 - \\mu_2)(\\phi_1 + \\phi_1\\phi_2)+(y_2 - \\mu_2)^2(1-\\phi_2^2)\\\\ = & (y_1 - \\mu_1)^2(1-\\phi_2^2)-2(y_1 - \\mu_1)(y_2 - \\mu_2)\\phi_1(1 + \\phi_2)+(y_2 - \\mu_2)^2(1-\\phi_2^2) \\\\ = & (y_1 - X_1\\underline{\\beta})^2(1-\\phi_2^2)-2(y_1-X_1\\underline{\\beta})(y_2-X_2\\underline{\\beta})\\phi_1(1 + \\phi_2)+(y_2 -X_2\\underline{\\beta})^2(1-\\phi_2^2) \\end{align} likelihood now form found Beach MacKinnon (1978b) $$\\begin{align} logL\\left( \\underline{\\beta}, \\underline{\\phi}; \\underline{y} \\right) &= const. +log(1+\\phi_2) + \\frac{1}{2}log(1-\\phi_1-\\phi_2)+\\frac{1}{2}log(1+\\phi_1-\\phi_2) \\\\ &-\\frac{n}{2}log\\left[ (y_1 - X_1\\underline{\\beta})^2(1-\\phi_2^2)-2(y_1-X_1\\underline{\\beta})(y_2-X_2\\underline{\\beta})\\phi_1(1 + \\phi_2)+(y_2 -X_2\\underline{\\beta})^2(1-\\phi_2^2) \\\\ + \\sum^n_{t=3} (y_t - X_t\\underline{\\beta} - \\phi_1 (y_{t-1} -X_{t-1}\\underline{\\beta}) - \\phi_2 (y_{t-2} -X_{t-2}\\underline{\\beta}))^2\\right]\\\\ \\end{align}$$","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"maximization-of-the-likelihood-1","dir":"Articles","previous_headings":"Likelihood for AR(2) errors","what":"Maximization of the likelihood","title":"Background","text":"case AR(1) model maximizing likelihood requires numerical methods. estimate β_\\underline{\\beta} maximizes log-likelihood conditional ϕ\\phi (Beach MacKinnon (1978b)) β̂_=(X*TX*)−1X*Ty*\\begin{align} \\underline{\\hat{\\beta}} = (X^{*T}X^*)^{-1}X^{*T}y^* \\end{align} X*=QXX^*=QX y*=Qyy^* = Qy QQ Prais Winsten transformation Q=[[(1−ϕ22)−ϕ12(1+ϕ2)/(1−ϕ2)]1/20...[ϕ12(1+ϕ2)/(1−ϕ2)]1/2(1−ϕ22)1/20...−ϕ2−ϕ110...0−ϕ2−ϕ110......0...0−ϕ2−ϕ11]  \\begin{matrix} Q = \\end{matrix} \\begin{bmatrix} \\left[(1-\\phi_2^2)-\\phi_1^2(1+\\phi_2)/(1-\\phi_2)\\right]^{1/2} & 0 & ... \\\\ \\left[\\phi_1^2(1+\\phi_2)/(1-\\phi_2)\\right]^{1/2} & (1-\\phi_2^2)^{1/2} & 0 & ... \\\\ -\\phi_2 & -\\phi_1 & 1 & 0 & ...\\\\ 0 & -\\phi_2 & -\\phi_1 & 1 & 0 & ...\\\\ ... \\\\ 0 & ... &0  & -\\phi_2 & -\\phi_1 & 1\\\\ \\end{bmatrix} procedure searches ϕ1\\phi_1 & ϕ2\\phi_2 required find maximum likelihood estimates β_\\underline\\beta, σ\\sigma, ϕ_\\underline{\\phi}","code":""},{"path":[]},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"ar1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"AR(1)","title":"Background","text":"L(θ_;y_)=∏t=2np(Yt=yt│Yt−1=yt−1)×p(Y1=y1)L(\\underline{\\theta};\\underline{y})=\\prod_{t=2}^n p(Y_t = y_t │ Y_{t-1}=y_{t-1}) × p(Y_1=y_1) θ_=(β0,β1,ϕ,σ2)\\underline{\\theta} = (\\beta_0,\\beta_1,\\phi,\\sigma^2), p(Yt=yt│Yt−1=yt−1)p(Y_t = y_t │ Y_{t-1}=y_{t-1}) conditional distribution yty_t given yt−1y_{t-1} p(Y1=y1)p(Y_1=y_1) distribution first point. Exact likelihood AR(1) process. conditional likelihood multiplied marginal likelihood first point.","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"density-of-py_1y_1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Density of p(Y1=y1)p(Y_1=y_1)","title":"Background","text":"p(Y1=y1)p(Y_1=y_1) normally distributed mean = X1β_X_1\\underline{\\beta} equates β0+β1\\beta_0 + \\beta_1 variance = σ21−ϕ2\\frac{\\sigma^2}{1-\\phi^2} density: p(Y1=y1)=12πσ2/(1−ϕ2)exp(−12(y1−β0−β1σ/(1−ϕ2))2)p(Y_1=y_1) = \\frac{1}{\\sqrt{2\\pi}\\sqrt{\\sigma^2/(1-\\phi^2)}}exp \\left( -\\frac{1}{2}\\left( \\frac{ y_1 - \\beta_0-\\beta_1 }{\\sigma/\\sqrt(1-\\phi^2)}\\right)^2\\right)","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"density-of-py_t-y_t-y_t-1y_t-1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Density of p(Yt=yt│Yt−1=yt−1)p(Y_t = y_t │ Y_{t-1}=y_{t-1})","title":"Background","text":"conditional distribution p(Yt=yt│Yt−1=yt−1)p(Y_t = y_t │ Y_{t-1}=y_{t-1}) also normally distributed get round way: Recall yt=β0+β1t+uty_t = \\beta_0 + \\beta_1 t + u_t ϕyt−1=ϕβ0+ϕβ1(t−1)+ϕut−1\\phi y_{t-1} = \\phi \\beta_0 + \\phi \\beta_1 (t-1) + \\phi u_{t-1} yt−ϕyt−1=β0+β1t+ut−ϕβ0−ϕβ1(t−1)−ϕut−1y_t - \\phi y_{t-1} = \\beta_0 + \\beta_1 t + u_t - \\phi \\beta_0 - \\phi\\beta_1(t-1) -\\phi u_{t-1} yt=ϕyt−1+β0(1−ϕ)+β1(t−ϕt+ϕ)+ety_t   = \\phi y_{t-1} + \\beta_0(1-\\phi) + \\beta_1 (t - \\phi t+\\phi) + e_t Rearrange obtain yt−ϕyt−1−β0(1−ϕ)−β1(t−ϕt+ϕ)=ety_t   - \\phi y_{t-1} - \\beta_0(1-\\phi) - \\beta_1 (t - \\phi t+\\phi) = e_t normal distribution mean = 0 variance = σ2\\sigma^2 results p(Yt=yt│Yt−1=yt−1)=∏t=2n1σ2πexp(−12(yt−ϕyt−1−β0(1−ϕ)−β1(t−ϕt+ϕ)σ)2)p(Y_t = y_t │ Y_{t-1}=y_{t-1}) = \\prod_{t=2}^n \\frac{1}{\\sigma \\sqrt{2\\pi}}exp \\left( -\\frac{1}{2}\\left( \\frac{ y_t   - \\phi y_{t-1} - \\beta_0(1-\\phi) - \\beta_1 (t - \\phi t+\\phi)  }{\\sigma}\\right)^2\\right)","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"exact-likelihood-for-ar1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Exact likelihood for AR1","title":"Background","text":"likelihood therefore: L(θ_;y_)=∏t=2n1σ2πexp(−12(yt−ϕyt−1−β0(1−ϕ)−β1(t−ϕt+ϕ)σ)2)×12πσ2/(1−ϕ2)exp(−12(y1−β0−β1σ/(1−ϕ2))2)L(\\underline{\\theta};\\underline{y}) = \\prod_{t=2}^n \\frac{1}{\\sigma \\sqrt{2\\pi}}exp \\left( -\\frac{1}{2}\\left( \\frac{ y_t   - \\phi y_{t-1} - \\beta_0(1-\\phi) - \\beta_1 (t - \\phi t+\\phi)  }{\\sigma}\\right)^2\\right) \\times \\frac{1}{\\sqrt{2\\pi}\\sqrt{\\sigma^2/(1-\\phi^2)}}exp \\left( -\\frac{1}{2}\\left( \\frac{ y_1 - \\beta_0 -\\beta_1 }{\\sigma/\\sqrt(1-\\phi^2)}\\right)^2\\right) Taking logs simplifying results : logL(θ_;y_)=−nlogσ−n2log(2π)−12log(1−ϕ2)−12σ2((y1−β0−β1)2(1−ϕ2)+Σt=2n(yt−ϕyt−1−β0(1−ϕ)−β1(t−ϕt+ϕ))2) logL(\\underline{\\theta};\\underline{y}) = -nlog\\sigma -\\frac{n}{2}log(2\\pi)-\\frac{1}{2}log(1-\\phi^2) -\\frac{1}{2\\sigma^2} \\left( (y_1-\\beta_0 -\\beta_1)^2(1-\\phi^2) + \\Sigma_{t=2}^n (y_t-\\phi y_{t-1}-\\beta_0(1-\\phi)-\\beta_1(t-\\phi t + \\phi))^2 \\right) Note β0(1−ϕ)+β1(t−ϕt+ϕ)=β0−β0ϕ+β1t−β1ϕt+β1ϕ=β0+β1t−ϕ(β0+β1(t−1))=Xtβ_−ϕXt−1β_\\begin{align}  &\\beta_0(1-\\phi) + \\beta_1 (t - \\phi t+\\phi) \\\\ &= \\beta_0 -\\beta_0\\phi + \\beta_1t - \\beta_1\\phi t + \\beta_1\\phi \\\\ &= \\beta_0 + \\beta_1t - \\phi (\\beta_0 +  \\beta_1(t -1)) \\\\ &= X_t\\underline{\\beta} - \\phi X_{t-1}\\underline{\\beta}  \\end{align} Using notation Beach MacKinnon (1978a) can simplify log likelihood, logL(θ_;y_)=−nlogσ−n2log(2π)−12log(1−ϕ2)−12σ2((y1−X1β_)2(1−ϕ2)+Σt=2n(yt−Xtβ_−ϕ(yt−1−Xt−1β_)2)\\begin{align}  logL(\\underline{\\theta};\\underline{y}) &= -nlog\\sigma -\\frac{n}{2}log(2\\pi)-\\frac{1}{2}log(1-\\phi^2) \\\\ &-\\frac{1}{2\\sigma^2} \\left( (y_1-X_1\\underline{\\beta})^2(1-\\phi^2) + \\Sigma_{t=2}^n (y_t- X_t\\underline{\\beta} - \\phi( y_{t-1}- X_{t-1}\\underline{\\beta})^2 \\right)\\\\ \\end{align} substituting σ̂2\\hat\\sigma^2 get logL(β_,ϕ;y_)=const.+12log(1−ϕ2)−n2log((y1−X1β_)2(1−ϕ2)+∑t=2n(yt−Xtβ_−ϕ(yt−1−Xt−1β_))2)\\begin{align} logL\\left( \\underline{\\beta}, \\phi; \\underline{y} \\right ) &= const. +\\frac{1}{2}log(1-\\phi^2)  \\\\ &-\\frac{n}{2}log\\left( (y_1-X_1\\underline{\\beta})^2(1-\\phi^2) + \\sum^n_{t=2} (y_t - X_t\\underline{\\beta} - \\phi (y_{t-1} -X_{t-1}\\underline{\\beta}))^2 \\right) \\\\ \\end{align}","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"ar2","dir":"Articles","previous_headings":"Likelihood from first principles","what":"AR(2)","title":"Background","text":"L(θ_;y_)=∏t=2np(Yt=yt│Yt−1=yt−1,Yt−2=yt−2)×p(Y2=y2|Y1=y1)×p(Y1=y1)L(\\underline{\\theta};\\underline{y})=\\prod_{t=2}^n p(Y_t=y_t│Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2}) \\times p(Y_2=y_2 | Y_1=y_1) \\times p(Y_1=y_1) Following reasoning can obtain densities three components likelihood. exact likelihood product conditional likelihood, conditional distribution y2|y1y_2 | y_1 marginal distribution y1y_1.","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"density-of-py_1y_1-1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Density of p(Y1=y1)p(Y_1=y_1)","title":"Background","text":"p(Y1=y1)p(Y_1=y_1) normally distributed mean = X1β_X_1\\underline{\\beta} equates β0+β1\\beta_0 + \\beta_1 variance = σ21−ϕ12−ϕ22\\frac{\\sigma^2}{1-\\phi_1^2 - \\phi_2^2}","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"density-of-py_2y_2y_1y_1","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Density of p(Y2=y2│Y1=y1)p(Y_2=y_2│Y_1=y_1)","title":"Background","text":"p(Y2=y2│Y1=y1)p(Y_2=y_2│Y_1=y_1) complicated. Need work ","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"density-of-py_t-y_t-y_t-1y_t-1y_t-2y_t-2","dir":"Articles","previous_headings":"Likelihood from first principles","what":"Density of p(Yt=yt│Yt−1=yt−1,Yt−2=yt−2)p(Y_t = y_t │ Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2})","title":"Background","text":"Recall yt=β0+β1t+uty_t = \\beta_0 + \\beta_1 t + u_t ϕ1yt−1=ϕ1β0+ϕ1β1(t−1)+ϕ1ut−1\\phi_1 y_{t-1} = \\phi_1 \\beta_0 + \\phi_1 \\beta_1 (t-1) + \\phi_1 u_{t-1} ϕ2yt−2=ϕ2β0+ϕ2β1(t−2)+ϕ2ut−2\\phi_2 y_{t-2} = \\phi_2 \\beta_0 + \\phi_2 \\beta_1 (t-2) + \\phi_2 u_{t-2} yt−ϕ1yt−1−ϕ2yt−2=β0+β1t+ut−ϕ1β0−ϕ1β1(t−1)−ϕ1ut−1−ϕ2β0−ϕ2β1(t−2)−ϕ2ut−2y_t - \\phi_1 y_{t-1} - \\phi_2 y_{t-2} = \\beta_0 + \\beta_1 t + u_t - \\phi_1 \\beta_0 - \\phi_1\\beta_1(t-1) -\\phi_1 u_{t-1} - \\phi_2 \\beta_0 - \\phi_2\\beta_1(t-2) -\\phi_2 u_{t-2} simplifies yt−ϕ1yt−1−ϕ2yt−2−β0(1−ϕ1−ϕ2)−β1(t−ϕ1(t−1)−ϕ2(t−2))=ety_t - \\phi_1 y_{t-1} - \\phi_2 y_{t-2} - \\beta_0 (1-\\phi_1-\\phi_2) - \\beta_1 (t - \\phi_1(t-1)- \\phi_2(t-2))  = e_t results p(Yt=yt│Yt−1=yt−1,Yt−2=yt−2)=∏t=3n1σ2πexp(−12(yt−ϕ1yt−1−ϕ2yt−2−β0(1−ϕ1−ϕ2)−β1(t−ϕ1(t−1)−ϕ2(t−2))σ)2)p(Y_t = y_t │ Y_{t-1}=y_{t-1},Y_{t-2}=y_{t-2}) = \\prod_{t=3}^n \\frac{1}{\\sigma \\sqrt{2\\pi}}exp \\left( -\\frac{1}{2}\\left( \\frac{ y_t - \\phi_1 y_{t-1} - \\phi_2 y_{t-2} - \\beta_0 (1-\\phi_1-\\phi_2) - \\beta_1 (t - \\phi_1(t-1)- \\phi_2(t-2))  }{\\sigma}\\right)^2\\right)","code":""},{"path":"https://andybeet.github.io/arfit/articles/background_work.html","id":"exact-likelihood-for-ar2","dir":"Articles","previous_headings":"","what":"Exact likelihood for AR2","title":"Background","text":"likelihood therefore:","code":""},{"path":[]},{"path":"https://andybeet.github.io/arfit/authors.html","id":null,"dir":"","previous_headings":"","what":"Authors","title":"Authors and Citation","text":"Andy Beet. Author, maintainer.","code":""},{"path":"https://andybeet.github.io/arfit/authors.html","id":"citation","dir":"","previous_headings":"","what":"Citation","title":"Authors and Citation","text":"Beet (2024). arfit: Fits full AR1 model. R package version 0.0.0.9000.","code":"@Manual{,   title = {arfit: Fits full AR1 model},   author = {Andy Beet},   year = {2024},   note = {R package version 0.0.0.9000}, }"},{"path":[]},{"path":[]},{"path":"https://andybeet.github.io/arfit/index.html","id":"legal-disclaimer","dir":"","previous_headings":"Contact","what":"Legal disclaimer","title":"Fits full AR1 model","text":"repository scientific product official communication National Oceanic Atmospheric Administration, United States Department Commerce. NOAA GitHub project code provided ‘’ basis user assumes responsibility use. claims Department Commerce Department Commerce bureaus stemming use GitHub project governed applicable Federal law. reference specific commercial products, processes, services service mark, trademark, manufacturer, otherwise, constitute imply endorsement, recommendation favoring Department Commerce. Department Commerce seal logo, seal logo DOC bureau, shall used manner imply endorsement commercial product activity DOC United States Government.","code":""},{"path":"https://andybeet.github.io/arfit/reference/arfit-package.html","id":null,"dir":"Reference","previous_headings":"","what":"arfit: Test for a trend in presence of autocorrelation — arfit-package","title":"arfit: Test for a trend in presence of autocorrelation — arfit-package","text":"Tools test trend presence autocorrelation. tools reproduce results publication","code":""},{"path":"https://andybeet.github.io/arfit/reference/arfit-package.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"arfit: Test for a trend in presence of autocorrelation — arfit-package","text":"Simulates data Evaluates likelihood Fits likelihood Estimates MLE model Example provided Tools simulation study","code":""},{"path":"https://andybeet.github.io/arfit/reference/arfit-package.html","id":"references","dir":"Reference","previous_headings":"","what":"References","title":"arfit: Test for a trend in presence of autocorrelation — arfit-package","text":"learn using arfit, start vignette: browseVignettes(package=\"arfit\") click index ","code":""},{"path":[]},{"path":"https://andybeet.github.io/arfit/reference/arfit-package.html","id":"author","dir":"Reference","previous_headings":"","what":"Author","title":"arfit: Test for a trend in presence of autocorrelation — arfit-package","text":"Maintainer: Andy Beet andrew.beet@noaa.gov (ORCID)","code":""},{"path":"https://andybeet.github.io/arfit/reference/check_data_validation.html","id":null,"dir":"Reference","previous_headings":"","what":"Checks data set for misspecification — check_data_validation","title":"Checks data set for misspecification — check_data_validation","text":"Check column names, missing data, consecutive equally spaced time units","code":""},{"path":"https://andybeet.github.io/arfit/reference/check_data_validation.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Checks data set for misspecification — check_data_validation","text":"","code":"check_data_validation(dataSet)"},{"path":"https://andybeet.github.io/arfit/reference/est_beta_given_rho.html","id":null,"dir":"Reference","previous_headings":"","what":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho","title":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho","text":"Uses Prais Winsten transformation utilizes first observation","code":""},{"path":"https://andybeet.github.io/arfit/reference/est_beta_given_rho.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho","text":"","code":"est_beta_given_rho(xt, yt, rho)"},{"path":"https://andybeet.github.io/arfit/reference/est_beta_given_rho.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Estimate beta conditional on known AR1 parameter — est_beta_given_rho","text":"xt Numeric vector. Time series time points (1, ..., n) yt Numeric vector. Time series observations (length = n) rho Numeric scalar. AR1 parameter","code":""},{"path":"https://andybeet.github.io/arfit/reference/example_cedar_rapids.html","id":null,"dir":"Reference","previous_headings":"","what":"Run test on example data set — example_cedar_rapids","title":"Run test on example data set — example_cedar_rapids","text":"Use data USGS 05464500 Cedar River Cedar Rapids, IA","code":""},{"path":"https://andybeet.github.io/arfit/reference/example_cedar_rapids.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Run test on example data set — example_cedar_rapids","text":"","code":"example_cedar_rapids(dataSet = arfit::cedar_rapids, nBootSims = 999)"},{"path":"https://andybeet.github.io/arfit/reference/example_cedar_rapids.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Run test on example data set — example_cedar_rapids","text":"dataSet Data frame. Two columns (year, riverflow) nBootSims Numeric scalar. Number bootstrap samples perform","code":""},{"path":"https://andybeet.github.io/arfit/reference/fit_ar1_opt.html","id":null,"dir":"Reference","previous_headings":"","what":"Fit linear model with trend using optimization routine — fit_ar1_opt","title":"Fit linear model with trend using optimization routine — fit_ar1_opt","text":"Fit linear model trend using optimization routine","code":""},{"path":"https://andybeet.github.io/arfit/reference/fit_ar1_opt.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Fit linear model with trend using optimization routine — fit_ar1_opt","text":"","code":"fit_ar1_opt(data, rho, hypothesis)"},{"path":"https://andybeet.github.io/arfit/reference/fit_to_data.html","id":null,"dir":"Reference","previous_headings":"","what":"Run test on user data set — fit_to_data","title":"Run test on user data set — fit_to_data","text":"Fits linear model data set assuming AR1 error","code":""},{"path":"https://andybeet.github.io/arfit/reference/fit_to_data.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Run test on user data set — fit_to_data","text":"","code":"fit_to_data(dataSet = NULL, nBootSims = 999)"},{"path":"https://andybeet.github.io/arfit/reference/fit_to_data.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Run test on user data set — fit_to_data","text":"dataSet Data frame. Two columns (x,y) nBootSims Numeric scalar. Number bootstrap samples perform","code":""},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar1.html","id":null,"dir":"Reference","previous_headings":"","what":"Evaluate the likelihood of linear model with errors from an AR order 1 process — likelihood_ar1","title":"Evaluate the likelihood of linear model with errors from an AR order 1 process — likelihood_ar1","text":"Evaluate likelihood linear model errors AR order 1 process","code":""},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar1.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Evaluate the likelihood of linear model with errors from an AR order 1 process — likelihood_ar1","text":"","code":"likelihood_ar1(rho, dataf, hypothesis = \"null\")"},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar1.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Evaluate the likelihood of linear model with errors from an AR order 1 process — likelihood_ar1","text":"rho Numeric scalar. Autoregressive parameter dataf Data frame. (nx2). Defining xt yt values hypothesis Character string. \"null\" \"alt\" determines whether evaluate null alternative hypothesis. \"null\" fixes linear component, beta = 0.","code":""},{"path":"https://andybeet.github.io/arfit/reference/likelihood_ar1.html","id":"value","dir":"Reference","previous_headings":"","what":"Value","title":"Evaluate the likelihood of linear model with errors from an AR order 1 process — likelihood_ar1","text":"likelihod value","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar1.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","text":"Performs simulation study assess performance test. Utilizes multiple cores spread bootstrap samples multiple cores","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar1.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","text":"","code":"sim_study_opt_ar1(   outDir = here::here(\"out.txt\"),   betaVec = c(0, 0.12, 0.25, 0.5),   rhoVec = c(0, 0.25, 0.5, 0.75, 0.95),   sigmaVec = c(0.25, 0.5, 0.75),   nTVec = c(10, 25, 50),   nSims = 200,   nBootSims = 500,   setSeed = NULL,   nCores = NULL,   missing = F )"},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar1.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","text":"outDir Character string. Path output file betaVec Numeric vector. Values beta_1 (slope/trend parameter) rhoVec Numeric vector. Values autoregressive parameter sigmaVec Numeric vector. Values standard deviation noise nSims Numeric scalar. Number time series simulate nBootSims Numeric scalar. Number bootstrap data sets setSeed Numeric scalar. Value seed simulations. (Default = NULL, random number 1-e7 selected) nCores Numeric scalar. Specify number cores utilize (Default = NULL, utilizes n-1 cores) missing Boolean. Whether simulate missing data (Default = F). T single missing value added random response nVec Numeric vector. Values length time series simulate","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar1.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","text":"convenience intercept, beta_0 set zero","code":""},{"path":"https://andybeet.github.io/arfit/reference/sim_study_opt_ar1.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Simulation study using optimization routine, multiple cores — sim_study_opt_ar1","text":"","code":"if (FALSE) { # \\dontrun{ } # }"},{"path":"https://andybeet.github.io/arfit/reference/simulate_ar1.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulate data from an linear process with AR1 error — simulate_ar1","title":"Simulate data from an linear process with AR1 error — simulate_ar1","text":"model=   y_t = + bt + z_t z_t = rho z_t-1 + e_t   (e_t ~ N(0,sigma^2)) Uses unconditional mean variance simulate first data point","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulate_ar1.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulate data from an linear process with AR1 error — simulate_ar1","text":"","code":"simulate_ar1(alpha, beta = 0, sigma, rho, n, missingValues = NULL)"},{"path":"https://andybeet.github.io/arfit/reference/simulate_ar1.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulate data from an linear process with AR1 error — simulate_ar1","text":"alpha Numeric scalar. Intercept beta Numeric scalar. Slope sigma Numeric scalar. Standard deviation error term rho Numeric scalar. Auto regressive parameter n Numeric scalar. Length time series missingValues Numeric vector. Indices y values missing data","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar1.html","id":null,"dir":"Reference","previous_headings":"","what":"Simulation study using optimization routine — simulation_study_opt_ar1","title":"Simulation study using optimization routine — simulation_study_opt_ar1","text":"Uses single Core. Performs simulation study assess performance test.","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar1.html","id":"ref-usage","dir":"Reference","previous_headings":"","what":"Usage","title":"Simulation study using optimization routine — simulation_study_opt_ar1","text":"","code":"simulation_study_opt_ar1(   outDir = here::here(\"out.txt\"),   betaVec = c(0.12, 0.25, 0.5),   rhoVec = c(0, 0.25, 0.5, 0.75, 0.95),   sigmaVec = c(0.25, 0.5, 0.75),   nTVec = c(10),   nSims = 200,   nBootSims = 500 )"},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar1.html","id":"arguments","dir":"Reference","previous_headings":"","what":"Arguments","title":"Simulation study using optimization routine — simulation_study_opt_ar1","text":"outDir Character string. Path output file betaVec Numeric vector. Values beta_1 (slope/trend parameter) rhoVec Numeric vector. Values autoregressive parameter sigmaVec Numeric vector. Values standard deviation noise nSims Numeric scalar. Number time series simulate nBootSims Numeric scalar. Number bootstrap data sets nVec Numeric vector. Values length time series simulate","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar1.html","id":"details","dir":"Reference","previous_headings":"","what":"Details","title":"Simulation study using optimization routine — simulation_study_opt_ar1","text":"convenience intercept, beta_0 set zero","code":""},{"path":"https://andybeet.github.io/arfit/reference/simulation_study_opt_ar1.html","id":"ref-examples","dir":"Reference","previous_headings":"","what":"Examples","title":"Simulation study using optimization routine — simulation_study_opt_ar1","text":"","code":"if (FALSE) { # \\dontrun{ simulation_study_opt(here::here(\"output.txt\"),betaVec = 0.5,rhovec =0.5, sigmaVec = 0.25, nTVec = 10, nSims = 200, nBootSims = 500) } # }"}]
diff --git a/sitemap.xml b/sitemap.xml
index 4d38cb9..3806e74 100644
--- a/sitemap.xml
+++ b/sitemap.xml
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