class: center, middle, inverse, title-slide .title[ # Analiza przestrzenna dróg rowerowych z podziałem na województwa w 2021 r. ] .institute[ ### <small> Karolina Czarnecka, Sebastian Siwak, Alicja Kuban </small> ] .date[ ### 2024/06/22 ] --- ## Wybór zjawiska i określenie jego determinantu <small> Celem projektu jest sprawdzenie czy długość dróg rowerowych rozkłada się przestrzennie, jeśli tak to dobrany zostanie model przestrzenny, który w najlepszy sposób wyjaśni to zjawisko. Zdecydowaliśmy się analizę dróg rowerowych, ponieważ jesteśmy fanami kolarstwa szosowego i czujemy potrzebę lepszego poznania rozkładu przestrzennego dróg, aby w przyszłości stworzyć atlas z poradami dla rowerzystów ekonomistów. <p> Zmienne objaśniające, czyli te które mogą determinować długość dróg rowerowych w poszczególnych województwach: -- <br> - `zaludnienie` - Obszary o większej liczbie zaludnienia mają tendencje do posiadania lepszej infrastruktury dla rowerzystów -- <br> <br> - `samochody` - Jako ze samochody są droższe w utrzymaniu i nie każdego może stać na utrzymanie może to spowodować większą skłonność do podróżowania rowerem -- <br> <br> - `dochody województw` - Czym bogatsze województwo tym więcej może przeznaczyć środków na budowę ścieżek rowerowych </small> --- ## Mapa i dane statystyczne .pull-left[ <br> **Prezentowane dane:** <small> - Dane pochodzą z Główego Urzędu Statystycznego. - Badane zjawisko analizowane z podziałem na województwa. - Analiza przestrzenna dotyczy 2021 roku. </small> <br> **Stworzenie macierzy sąsiedztwa** <small> - Do badania przestrzennej zależności długości tras rowerowych w poszczególnych województwach, wykorzystana zostanie macierz sąsiedztwa: `królowa I rzędu` </small> ] .pull-right[ <br> <br> <img src="projekt2.0_files/figure-html/unnamed-chunk-6-1.png" style="display: block; margin: auto;" /> ] --- ## Wizualizacja badanego zjawiska <img src="projekt2.0_files/figure-html/unnamed-chunk-8-1.png" style="display: block; margin: auto;" /> <br> <small> **Wnioski:** Najwięcej kilometrów dróg rowerowych jest w województwie mazowieckim. Drugie miejsce zajmuje województwo wielkopolskie, trzecie- pomorskie. Najmniej kilometrów dróg rowerowych jest w województwie świętokrzyskim. </small> --- ## Badanie zależności przestrzennych zmiennej objaśniającej .panelset[ .panel[.panel-name[Hipoteza] <br> <small> **Test Morana**- sprawdzamy czy istnieją zależności przestrzenne i czy są one istotne oraz czy istnieje związek między składnikiem losowym modelu, a resztami z opóźnienia przestrzennego. </small> <br> <small> **Hipoteza:** </p> H0: Brak zależności przestrzennych <p> HA: Istnieją zależności przestrzenne </small> ] .panel[.panel-name[Test Morana] <br> <br> <br> <div class="tabwid"><style>.cl-f50f60d6{}.cl-f4f39090{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-f4fcf644{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 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0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f4fd2ca4{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f4fd2ca5{width:0.897in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f4fd2cae{width:2.918in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f4fd2cb8{width:0.99in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-f50f60d6'><thead><tr style="overflow-wrap:break-word;"><th class="cl-f4fd2c4a"><p class="cl-f4fcf644"><span class="cl-f4f39090">Moran I statistic</span></p></th><th class="cl-f4fd2c4a"><p class="cl-f4fcf644"><span class="cl-f4f39090">Expectation</span></p></th><th class="cl-f4fd2c4a"><p class="cl-f4fcf644"><span class="cl-f4f39090">Variance</span></p></th><th class="cl-f4fd2c54"><p class="cl-f4fcf644"><span class="cl-f4f39090">Standard deviate</span></p></th><th class="cl-f4fd2c5e"><p class="cl-f4fcf644"><span class="cl-f4f39090">p.value</span></p></th><th class="cl-f4fd2c68"><p class="cl-f4fcf658"><span class="cl-f4f39090">method</span></p></th><th class="cl-f4fd2c72"><p class="cl-f4fcf658"><span class="cl-f4f39090">alternative</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-f4fd2c73"><p class="cl-f4fcf644"><span class="cl-f4f39090">0.2</span></p></td><td class="cl-f4fd2c73"><p class="cl-f4fcf644"><span class="cl-f4f39090">-0.1</span></p></td><td class="cl-f4fd2c73"><p class="cl-f4fcf644"><span class="cl-f4f39090">0.0</span></p></td><td class="cl-f4fd2c86"><p class="cl-f4fcf644"><span class="cl-f4f39090">1.8</span></p></td><td class="cl-f4fd2c87"><p class="cl-f4fcf644"><span class="cl-f4f39090">0.0362</span><span class="cl-f4f39090"> *</span></p></td><td class="cl-f4fd2c90"><p class="cl-f4fcf658"><span class="cl-f4f39090">Global Moran I for regression residuals</span></p></td><td class="cl-f4fd2c9a"><p class="cl-f4fcf658"><span class="cl-f4f39090">greater</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="7"class="cl-f4fd2c9b"><p class="cl-f4fcf658"><span class="cl-f4f39090">Signif. codes: 0 &lt;= '***' &lt; 0.001 &lt; '**' &lt; 0.01 &lt; '*' &lt; 0.05</span></p></td></tr></tfoot></table></div> ] .panel[.panel-name[Wizualizacja testu Morana] <img src="projekt2.0_files/figure-html/unnamed-chunk-11-1.png" style="display: block; margin: auto;" /> ] .panel[.panel-name[Wnioski] <br> <br> <small> **Wnioski:** p-value < ⍺ (na poziomie istotności 0.05), należy zatem odrzucić H0 na korzyść HA, oznacza, to że model jest przestrzenny. Graficzną prezentacją autokorelacji przestrzennej jest wykres rozrzutu Morana. Największe skupienia występują w III i I ćwiartce, co wskazuje na podobnych sąsiadów. Obiekty o niskich wartościach otaczane są o niskich wartościach, zaś obiekty o wysokich wartościach są otaczane przez obiekty o wysokich wartościach. Kierunek linii regresji jest >0 (dodatnia korelacja), co oznacza, występowanie klastrów o podobnych wartościach. </small> ] ] --- class: center, middle ## Budowa modelu Szukanie modelu, który najlepiej wyjaśni badane zjawisko --- ### Model OLS - Model przyczynowo-skutkowy bez interakcji przestrzennych .panelset[ .panel[.panel-name[Opis modelu] <br> <br> <br> <small> **Regresja metodą najmniejszych kwadratów** (**OLS**) jest powszechną techniką szacowania współczynników równań regresji liniowej, które opisują związek pomiędzy jedną lub większą liczbą niezależnych zmiennych ilościowych, a zmienną zależną (prosta lub wielokrotna regresja liniowa). </small> ] .panel[.panel-name[Parametry modelu] .pull-left[ <small> <small> <table style="border-collapse:collapse; border:none;"> <tr> <th style="border-top: double; text-align:center; font-style:normal; font-weight:bold; padding:0.2cm; text-align:left; ">&nbsp;</th> <th colspan="3" style="border-top: double; text-align:center; font-style:normal; font-weight:bold; padding:0.2cm; ">Dependent variable</th> </tr> <tr> <td style=" text-align:center; border-bottom:1px solid; font-style:italic; font-weight:normal; text-align:left; ">Predictors</td> <td style=" text-align:center; border-bottom:1px solid; font-style:italic; font-weight:normal; ">Estimates</td> <td style=" text-align:center; border-bottom:1px solid; font-style:italic; font-weight:normal; ">CI</td> <td style=" text-align:center; border-bottom:1px solid; font-style:italic; font-weight:normal; ">p</td> </tr> <tr> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:left; ">(Intercept)</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; ">&#45;861.52</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; ">&#45;4286.19&nbsp;&ndash;&nbsp;2563.14</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; ">0.594</td> </tr> <tr> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:left; ">zaludnienie</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; ">&#45;0.33</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; ">&#45;0.89&nbsp;&ndash;&nbsp;0.23</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; ">0.226</td> </tr> <tr> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:left; ">samochody</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; ">2496.92</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; ">&#45;2071.96&nbsp;&ndash;&nbsp;7065.79</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; ">0.257</td> </tr> <tr> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:left; ">dochody woj</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; ">0.69</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; ">0.36&nbsp;&ndash;&nbsp;1.02</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:center; "><strong>0.001</strong></td> </tr> <tr> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:left; padding-top:0.1cm; padding-bottom:0.1cm; border-top:1px solid;">Observations</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; padding-top:0.1cm; padding-bottom:0.1cm; text-align:left; border-top:1px solid;" colspan="3">16</td> </tr> <tr> <td style=" padding:0.2cm; text-align:left; vertical-align:top; text-align:left; padding-top:0.1cm; padding-bottom:0.1cm;">R² / R² adjusted</td> <td style=" padding:0.2cm; text-align:left; vertical-align:top; padding-top:0.1cm; padding-bottom:0.1cm; text-align:left;" colspan="3">0.764 / 0.705</td> </tr> </table> <div class="tabwid"><style>.cl-f5c85654{}.cl-f5b024c6{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-f5b024da{font-family:'Arial';font-size:7.3pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(153, 153, 153, 1.00);background-color:transparent;}.cl-f5b9e5ce{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f5b9e5ec{margin:0;text-align:center;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f5ba271e{width:0.65in;background-color:transparent;vertical-align: top;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f5ba2732{width:1.262in;background-color:transparent;vertical-align: top;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-f5c85654'><tbody><tr style="overflow-wrap:break-word;"><td class="cl-f5ba271e"><p class="cl-f5b9e5ce"><span class="cl-f5b024c6">value</span><br><span class="cl-f5b024da">numeric</span></p></td><td class="cl-f5ba2732"><p class="cl-f5b9e5ec"><span class="cl-f5b024c6">0.0004500615</span></p></td></tr></tbody></table></div> </small> </small> ] .pull-right[ <br> <br> <small> <small> **Wnioski:** Zmienna, które okazały się istotne, dla modelu OLS to: `dochody województwa` <p> Zmienne: `zaludnienie` i `samochody` są nieistotne, jednak test Morana wskazuje na zależności przestrzenne, a więc pozostałe zmienne nie zostały usunięte z modelu, dodatkowo taki dobór zapewnia wysoki poziom dopasowania modelu (R-kwadrat równy 0,764) </small> </small> ] ] .panel[.panel-name[Test Morana dla modelu OLS] <small> <small>  <br> <br> </small> </small> ] ] --- ### Testowanie model OLS .panelset[ .panel[.panel-name[Test Morana] <br> <small> **Test Morana dla modelu OLS** bada czy mamy do czynienia z czynnikiem przestrzennym, przyjmujemy ⍺ na poziomie 0.05 <br> <br> **Hipoteza:** <p> H0: Brak zależności przestrzennej HA: Istnieją zależności przestrzenne, model przestrzenny w lepszy sposób opiszę zjawisko niż OLS </small> <br> <div class="tabwid"><style>.cl-f6357ac2{}.cl-f6205ae8{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-f627f2da{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f627f2ee{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f6281a80{width:0.948in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid 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1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6281ab3{width:0.897in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6281ac6{width:2.918in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6281ac7{width:0.99in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6281ada{width:0.948in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6281ae4{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6281ae5{width:0.897in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6281af8{width:2.918in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6281af9{width:0.99in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-f6357ac2'><thead><tr style="overflow-wrap:break-word;"><th class="cl-f6281a80"><p class="cl-f627f2da"><span class="cl-f6205ae8">estimate1</span></p></th><th class="cl-f6281a80"><p class="cl-f627f2da"><span class="cl-f6205ae8">estimate2</span></p></th><th class="cl-f6281a80"><p class="cl-f627f2da"><span class="cl-f6205ae8">estimate3</span></p></th><th class="cl-f6281a8a"><p class="cl-f627f2da"><span class="cl-f6205ae8">statistic</span></p></th><th class="cl-f6281a94"><p class="cl-f627f2da"><span class="cl-f6205ae8">p.value</span></p></th><th class="cl-f6281a9e"><p class="cl-f627f2ee"><span class="cl-f6205ae8">method</span></p></th><th class="cl-f6281a9f"><p class="cl-f627f2ee"><span class="cl-f6205ae8">alternative</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-f6281aa8"><p class="cl-f627f2da"><span class="cl-f6205ae8">0.2</span></p></td><td class="cl-f6281aa8"><p class="cl-f627f2da"><span class="cl-f6205ae8">-0.1</span></p></td><td class="cl-f6281aa8"><p class="cl-f627f2da"><span class="cl-f6205ae8">0.0</span></p></td><td class="cl-f6281ab2"><p class="cl-f627f2da"><span class="cl-f6205ae8">1.8</span></p></td><td class="cl-f6281ab3"><p class="cl-f627f2da"><span class="cl-f6205ae8">0.0362</span><span class="cl-f6205ae8"> *</span></p></td><td class="cl-f6281ac6"><p class="cl-f627f2ee"><span class="cl-f6205ae8">Global Moran I for regression residuals</span></p></td><td class="cl-f6281ac7"><p class="cl-f627f2ee"><span class="cl-f6205ae8">greater</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="7"class="cl-f6281ada"><p class="cl-f627f2ee"><span class="cl-f6205ae8">Signif. codes: 0 &lt;= '***' &lt; 0.001 &lt; '**' &lt; 0.01 &lt; '*' &lt; 0.05</span></p></td></tr></tfoot></table></div> <br> <small> **Wnioski:** p-value < ⍺ , zatem należy odrzucić H0, na korzyść HA, co oznacza, że należy znaleźć model przestrzenny, który lepiej wyjaśni badane zjawisko niż OLS </small> ] .panel[.panel-name[Test Jarque Bera] <small> <br> **Test Jarque Bera** mówi o normalności rozkładu, przyjmujemy ⍺ na poziomie 0.05 <p> <br> **Hipoteza:** <p> H0: Składnik losowy w zadanym modelu ma rozkład normalny HA: Składnik losowy nie ma rozkładu normalnego </small> <small> <div class="tabwid"><style>.cl-f69415aa{}.cl-f67ae38c{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-f68638cc{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f68638e0{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f6866aa4{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6866ac2{width:0.88in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6866acc{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6866ad6{width:1.457in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6866ad7{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6866aea{width:0.88in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6866af4{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6866afe{width:1.457in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6866aff{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6866b08{width:0.88in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6866b12{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6866b1c{width:1.457in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-f69415aa'><thead><tr style="overflow-wrap:break-word;"><th class="cl-f6866aa4"><p class="cl-f68638cc"><span class="cl-f67ae38c">statistic</span></p></th><th class="cl-f6866ac2"><p class="cl-f68638cc"><span class="cl-f67ae38c">p.value</span></p></th><th class="cl-f6866acc"><p class="cl-f68638cc"><span class="cl-f67ae38c">parameter</span></p></th><th class="cl-f6866ad6"><p class="cl-f68638e0"><span class="cl-f67ae38c">method</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-f6866ad7"><p class="cl-f68638cc"><span class="cl-f67ae38c">4.6</span></p></td><td class="cl-f6866aea"><p class="cl-f68638cc"><span class="cl-f67ae38c">0.0998</span><span class="cl-f67ae38c"> .</span></p></td><td class="cl-f6866af4"><p class="cl-f68638cc"><span class="cl-f67ae38c">2.0</span></p></td><td class="cl-f6866afe"><p class="cl-f68638e0"><span class="cl-f67ae38c">Jarque Bera Test</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="4"class="cl-f6866aff"><p class="cl-f68638e0"><span class="cl-f67ae38c">Signif. codes: 0 &lt;= '***' &lt; 0.001 &lt; '**' &lt; 0.01 &lt; '*' &lt; 0.05</span></p></td></tr></tfoot></table></div> </small> <br> <small> **Wnioski:** p-value > ⍺ , nie ma podstaw do odrzucenia H0, co oznacza, że reszty mają rozkład normalny. </small> ] .panel[.panel-name[Test Breusha-Pagana] <br> <small> **Test Breusha-Pagana** bada homoskedastyczność, przyjmujemy ⍺ na poziomie 0.05 <p> <br> **Hipoteza:** <p> H0: Składnik losowy jest homoskedastyczny HA: Składnik losowy jest heteroskedastyczny <br> </small> <br> <div class="tabwid"><style>.cl-f6ddd140{}.cl-f6c655d8{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-f6d1538e{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f6d15398{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f6d17ce2{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6d17cec{width:0.88in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6d17cf6{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6d17d00{width:1.627in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6d17d14{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6d17d1e{width:0.88in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6d17d28{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6d17d29{width:1.627in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6d17d32{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6d17d3c{width:0.88in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6d17d46{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f6d17d47{width:1.627in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-f6ddd140'><thead><tr style="overflow-wrap:break-word;"><th class="cl-f6d17ce2"><p class="cl-f6d1538e"><span class="cl-f6c655d8">statistic</span></p></th><th class="cl-f6d17cec"><p class="cl-f6d1538e"><span class="cl-f6c655d8">p.value</span></p></th><th class="cl-f6d17cf6"><p class="cl-f6d1538e"><span class="cl-f6c655d8">parameter</span></p></th><th class="cl-f6d17d00"><p class="cl-f6d15398"><span class="cl-f6c655d8">method</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-f6d17d14"><p class="cl-f6d1538e"><span class="cl-f6c655d8">2.4</span></p></td><td class="cl-f6d17d1e"><p class="cl-f6d1538e"><span class="cl-f6c655d8">0.4912</span><span class="cl-f6c655d8"> </span></p></td><td class="cl-f6d17d28"><p class="cl-f6d1538e"><span class="cl-f6c655d8">3.0</span></p></td><td class="cl-f6d17d29"><p class="cl-f6d15398"><span class="cl-f6c655d8">Breusch-Pagan test</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="4"class="cl-f6d17d32"><p class="cl-f6d15398"><span class="cl-f6c655d8">Signif. codes: 0 &lt;= '***' &lt; 0.001 &lt; '**' &lt; 0.01 &lt; '*' &lt; 0.05</span></p></td></tr></tfoot></table></div> <small> <br> **Wnioski:** Wartości p-value > ⍺ wskazują na brak podstaw do odrzucenia H0, składnik losowy jest homoskedastyczny. </small> ] ] --- ### Model SAR .panelset[ .panel[.panel-name[Opis modelu] .pull-left[ <small> <br> **Model autoregresji przestrzennej (SAR - Spatial Autoregressive Model)** Jest to jeden z podstawowych modeli stosowanych w ekonometrii przestrzennej, używany do analizy danych przestrzennych, gdzie obserwacje są powiązane nie tylko ze względu na swoje wartości, ale także ze względu na ich położenie geograficzne. <p> <br> W Modelu SAR zmienne zaludnienie i dochody województw są istotne statystycznie. </small> ] .pull-right[ <small> <small> <br> <br> <div class="tabwid"><style>.cl-f7c53936{}.cl-f7a91a3a{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-f7a91a58{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(153, 153, 153, 1.00);background-color:transparent;}.cl-f7b1e34a{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7b20bd6{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7b20bd7{width:1.177in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7b20be0{width:0.88in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7b20be1{width:0.939in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7b20bea{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-f7c53936'><thead><tr style="overflow-wrap:break-word;"><th class="cl-f7b20b18"><p class="cl-f7b1e34a"><span class="cl-f7a91a3a">Name</span></p></th><th class="cl-f7b20b22"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">Estimate</span></p></th><th class="cl-f7b20b2c"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">Std..Error</span></p></th><th class="cl-f7b20b36"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">z.value</span></p></th><th class="cl-f7b20b36"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">Pr...z..</span></p></th></tr><tr style="overflow-wrap:break-word;"><th class="cl-f7b20b40"><p class="cl-f7b1e34a"><span class="cl-f7a91a58">character</span></p></th><th class="cl-f7b20b41"><p class="cl-f7b1e35e"><span class="cl-f7a91a58">numeric</span></p></th><th class="cl-f7b20b4a"><p class="cl-f7b1e35e"><span class="cl-f7a91a58">numeric</span></p></th><th class="cl-f7b20b54"><p class="cl-f7b1e35e"><span class="cl-f7a91a58">numeric</span></p></th><th class="cl-f7b20b54"><p class="cl-f7b1e35e"><span class="cl-f7a91a58">numeric</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-f7b20b5e"><p class="cl-f7b1e34a"><span class="cl-f7a91a3a">Intercept</span></p></td><td class="cl-f7b20b68"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">-485.9</span></p></td><td class="cl-f7b20b69"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">1,153.8</span></p></td><td class="cl-f7b20b72"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">-0.4</span></p></td><td class="cl-f7b20b72"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">0.7</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-f7b20b73"><p class="cl-f7b1e34a"><span class="cl-f7a91a3a">zaludnienie</span></p></td><td class="cl-f7b20b7c"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">-0.4</span></p></td><td class="cl-f7b20b86"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">0.2</span></p></td><td class="cl-f7b20b90"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">-2.3</span></p></td><td class="cl-f7b20b90"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">0.0</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-f7b20b9a"><p class="cl-f7b1e34a"><span class="cl-f7a91a3a">samochody</span></p></td><td class="cl-f7b20ba4"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">1,514.3</span></p></td><td class="cl-f7b20bae"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">1,576.6</span></p></td><td class="cl-f7b20baf"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">1.0</span></p></td><td class="cl-f7b20baf"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">0.3</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-f7b20bb8"><p class="cl-f7b1e34a"><span class="cl-f7a91a3a">dochody_woj</span></p></td><td class="cl-f7b20bc2"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">0.7</span></p></td><td class="cl-f7b20bcc"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">0.1</span></p></td><td class="cl-f7b20bd6"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">6.4</span></p></td><td class="cl-f7b20bd6"><p class="cl-f7b1e35e"><span class="cl-f7a91a3a">0.0</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="5"class="cl-f7b20bd7"><p class="cl-f7b1e34a"><span class="cl-f7a91a3a">n: 4</span></p></td></tr></tfoot></table></div> </small> </small> ] ] .panel[.panel-name[Rho] .pull-left[ <small> <br> <br> **Interpretacja** <p> Parametr przestrzenny Rho,jest przestrzennie opóźnionym mnożnikiem y, mierzy średni wpływ na obserwacje sąsiadujących obserwacji (parametr błędu opóźnienia), w przypadku zadanego modelu wynosi 0.442, i jest istotny statystycznie na poziomie alfa <0.05. Oznacza to, że model SAR wyjaśni badane zjawisko lepiej niż OLS. </small> ] .pull-right[ <br> <br> <br> <div class="tabwid"><style>.cl-f80b4138{}.cl-f7f38606{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-f7f3861a{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(153, 153, 153, 1.00);background-color:transparent;}.cl-f7fc56aa{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f7fc56c8{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f7fc9598{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7fc95ac{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7fc95b6{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7fc95b7{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7fc95c0{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7fc95ca{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7fc95cb{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7fc95d4{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7fc95d5{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7fc95de{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7fc95e8{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f7fc95f2{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-f80b4138'><thead><tr style="overflow-wrap:break-word;"><th class="cl-f7fc9598"><p class="cl-f7fc56aa"><span class="cl-f7f38606">Rho</span></p></th><th class="cl-f7fc95ac"><p class="cl-f7fc56c8"><span class="cl-f7f38606">Value</span></p></th></tr><tr style="overflow-wrap:break-word;"><th class="cl-f7fc95b6"><p class="cl-f7fc56aa"><span class="cl-f7f3861a">character</span></p></th><th class="cl-f7fc95b7"><p class="cl-f7fc56c8"><span class="cl-f7f3861a">numeric</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-f7fc95c0"><p class="cl-f7fc56aa"><span class="cl-f7f38606">value</span></p></td><td class="cl-f7fc95ca"><p class="cl-f7fc56c8"><span class="cl-f7f38606">0.4</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-f7fc95cb"><p class="cl-f7fc56aa"><span class="cl-f7f38606">Std.Err</span></p></td><td class="cl-f7fc95d4"><p class="cl-f7fc56c8"><span class="cl-f7f38606">0.2</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-f7fc95c0"><p class="cl-f7fc56aa"><span class="cl-f7f38606">z-value</span></p></td><td class="cl-f7fc95ca"><p class="cl-f7fc56c8"><span class="cl-f7f38606">2.6</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-f7fc95d5"><p class="cl-f7fc56aa"><span class="cl-f7f38606">p-value</span></p></td><td class="cl-f7fc95de"><p class="cl-f7fc56c8"><span class="cl-f7f38606">0.0</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="2"class="cl-f7fc95e8"><p class="cl-f7fc56aa"><span class="cl-f7f38606">n: 4</span></p></td></tr></tfoot></table></div> ] ] .panel[.panel-name[Wald statistic] **Wald Statistic** <p> <small> **Hipoteza:** <p> H0: Nie ma przestrzennej autoregresji w modelu. (Należy wrócić do OLS) <p> HA: Istnieje istotna przestrzenna autoregresja w modelu.(Model SAR) <br> <small> <small> <small> <div class="tabwid"><style>.cl-f85dd452{}.cl-f847452a{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-f84fb408{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f84fb426{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f84fdc76{width:0.863in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdc8a{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdc94{width:0.897in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdc9e{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdc9f{width:3.028in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdca8{width:0.863in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdcb2{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdcbc{width:0.897in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdcc6{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdcd0{width:3.028in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdcda{width:0.863in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdce4{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdce5{width:0.897in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdcee{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f84fdcf8{width:3.028in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-f85dd452'><thead><tr style="overflow-wrap:break-word;"><th class="cl-f84fdc76"><p class="cl-f84fb408"><span class="cl-f847452a">estimate</span></p></th><th class="cl-f84fdc8a"><p class="cl-f84fb408"><span class="cl-f847452a">statistic</span></p></th><th class="cl-f84fdc94"><p class="cl-f84fb408"><span class="cl-f847452a">p.value</span></p></th><th class="cl-f84fdc9e"><p class="cl-f84fb408"><span class="cl-f847452a">parameter</span></p></th><th class="cl-f84fdc9f"><p class="cl-f84fb426"><span class="cl-f847452a">method</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-f84fdca8"><p class="cl-f84fb408"><span class="cl-f847452a">0.4</span></p></td><td class="cl-f84fdcb2"><p class="cl-f84fb408"><span class="cl-f847452a">6.6</span></p></td><td class="cl-f84fdcbc"><p class="cl-f84fb408"><span class="cl-f847452a">0.0105</span><span class="cl-f847452a"> *</span></p></td><td class="cl-f84fdcc6"><p class="cl-f84fb408"><span class="cl-f847452a">1.0</span></p></td><td class="cl-f84fdcd0"><p class="cl-f84fb426"><span class="cl-f847452a">Wald diagnostics for spatial dependence</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="5"class="cl-f84fdcda"><p class="cl-f84fb426"><span class="cl-f847452a">Signif. codes: 0 &lt;= '***' &lt; 0.001 &lt; '**' &lt; 0.01 &lt; '*' &lt; 0.05</span></p></td></tr></tfoot></table></div> </small> </small> </small> **Wnioski:** P-value < ⍺, zatem należy odrzucić H0, na rzecz HA, istnieje autoregresja w zadanym modelu. Parametr w modelu jest istotny statystycznie na poziomie ufności 95%. </small> ] .panel[.panel-name[LR test] **LR test** <p> <small> **Hipoteza:** <p> H0: Różnica między oszacowniami MNW i MNK jest nieistotna, model bez interakcji przestrzennej <p> HA: Różnica między oszacowniami MNW i MNK jest istotna, wybieramy model SAR <p> <br> <small> <small> <small> <div class="tabwid"><style>.cl-f8c74b76{}.cl-f8a54d00{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-f8b23c5e{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f8b23c86{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 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1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f8b2973a{width:3.767in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f8b2974e{width:0.948in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f8b29758{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f8b29762{width:0.897in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f8b2976c{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f8b29776{width:3.767in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-f8c74b76'><thead><tr style="overflow-wrap:break-word;"><th class="cl-f8b296b8"><p class="cl-f8b23c5e"><span class="cl-f8a54d00">estimate1</span></p></th><th class="cl-f8b296b8"><p class="cl-f8b23c5e"><span class="cl-f8a54d00">estimate2</span></p></th><th class="cl-f8b296d6"><p class="cl-f8b23c5e"><span class="cl-f8a54d00">statistic</span></p></th><th class="cl-f8b296f4"><p class="cl-f8b23c5e"><span class="cl-f8a54d00">p.value</span></p></th><th class="cl-f8b296fe"><p class="cl-f8b23c5e"><span class="cl-f8a54d00">parameter</span></p></th><th class="cl-f8b29708"><p class="cl-f8b23c86"><span class="cl-f8a54d00">method</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-f8b29712"><p class="cl-f8b23c5e"><span class="cl-f8a54d00">-111.0</span></p></td><td class="cl-f8b29712"><p class="cl-f8b23c5e"><span class="cl-f8a54d00">-113.5</span></p></td><td class="cl-f8b2971c"><p class="cl-f8b23c5e"><span class="cl-f8a54d00">4.8</span></p></td><td class="cl-f8b29726"><p class="cl-f8b23c5e"><span class="cl-f8a54d00">0.0283</span><span class="cl-f8a54d00"> *</span></p></td><td class="cl-f8b29730"><p class="cl-f8b23c5e"><span class="cl-f8a54d00">1.0</span></p></td><td class="cl-f8b2973a"><p class="cl-f8b23c86"><span class="cl-f8a54d00">Likelihood Ratio diagnostics for spatial dependence</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-f8b2974e"><p class="cl-f8b23c86"><span class="cl-f8a54d00">Signif. codes: 0 &lt;= '***' &lt; 0.001 &lt; '**' &lt; 0.01 &lt; '*' &lt; 0.05</span></p></td></tr></tfoot></table></div> </small> </small> **Wnioski:** P-value < ⍺, zatem należy odrzucić H0, na rzecz HA, wybieramy model SAR, ponieważ różnica między oszacowniami MNW i MNK jest istotna </small> </small> ] ] --- ### Model SEM .panelset[ .panel[.panel-name[Opis modelu] .pull-left[ <br> <br> <small> **Autokorelacja przestrzenna w składniku losowym (SEM- Spatial Error Model)** Jest to jeden z podstawowych modeli stosowanych w ekonometrii przestrzennej. Jest on używany do analizy danych przestrzennych, gdzie błędy w modelu mogą być przestrzennie skorelowane. W Modelu SEM zmienna dochody województw jest statystycznie istotna. </small> ] .pull-right[ <br> <small> <small> <div class="tabwid"><style>.cl-f9966d66{}.cl-f981f958{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-f981f976{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(153, 153, 153, 1.00);background-color:transparent;}.cl-f9897bf6{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-f9897c0a{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 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1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f989a496{width:0.939in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f989a4a0{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f989a4aa{width:1.177in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f989a4b4{width:0.88in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f989a4b5{width:0.939in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f989a4be{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f989a4bf{width:1.177in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f989a4c8{width:0.88in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f989a4d2{width:0.939in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f989a4dc{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-f9966d66'><thead><tr style="overflow-wrap:break-word;"><th class="cl-f989a3f6"><p class="cl-f9897bf6"><span class="cl-f981f958">Name</span></p></th><th class="cl-f989a400"><p class="cl-f9897c0a"><span class="cl-f981f958">Estimate</span></p></th><th class="cl-f989a40a"><p class="cl-f9897c0a"><span class="cl-f981f958">Std..Error</span></p></th><th class="cl-f989a414"><p class="cl-f9897c0a"><span class="cl-f981f958">z.value</span></p></th><th class="cl-f989a414"><p class="cl-f9897c0a"><span class="cl-f981f958">Pr...z..</span></p></th></tr><tr style="overflow-wrap:break-word;"><th class="cl-f989a41e"><p class="cl-f9897bf6"><span class="cl-f981f976">character</span></p></th><th class="cl-f989a428"><p class="cl-f9897c0a"><span class="cl-f981f976">numeric</span></p></th><th class="cl-f989a432"><p class="cl-f9897c0a"><span class="cl-f981f976">numeric</span></p></th><th class="cl-f989a43c"><p class="cl-f9897c0a"><span class="cl-f981f976">numeric</span></p></th><th class="cl-f989a43c"><p class="cl-f9897c0a"><span class="cl-f981f976">numeric</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-f989a43d"><p class="cl-f9897bf6"><span class="cl-f981f958">Intercept</span></p></td><td class="cl-f989a446"><p class="cl-f9897c0a"><span class="cl-f981f958">-400.6</span></p></td><td class="cl-f989a450"><p class="cl-f9897c0a"><span class="cl-f981f958">1,259.6</span></p></td><td class="cl-f989a451"><p class="cl-f9897c0a"><span class="cl-f981f958">-0.3</span></p></td><td class="cl-f989a451"><p class="cl-f9897c0a"><span class="cl-f981f958">0.8</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-f989a45a"><p class="cl-f9897bf6"><span class="cl-f981f958">zaludnienie</span></p></td><td class="cl-f989a46e"><p class="cl-f9897c0a"><span class="cl-f981f958">-0.3</span></p></td><td class="cl-f989a46f"><p class="cl-f9897c0a"><span class="cl-f981f958">0.2</span></p></td><td class="cl-f989a478"><p class="cl-f9897c0a"><span class="cl-f981f958">-1.5</span></p></td><td class="cl-f989a478"><p class="cl-f9897c0a"><span class="cl-f981f958">0.1</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-f989a482"><p class="cl-f9897bf6"><span class="cl-f981f958">samochody</span></p></td><td class="cl-f989a48c"><p class="cl-f9897c0a"><span class="cl-f981f958">1,912.8</span></p></td><td class="cl-f989a496"><p class="cl-f9897c0a"><span class="cl-f981f958">1,594.5</span></p></td><td class="cl-f989a4a0"><p class="cl-f9897c0a"><span class="cl-f981f958">1.2</span></p></td><td class="cl-f989a4a0"><p class="cl-f9897c0a"><span class="cl-f981f958">0.2</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-f989a4aa"><p class="cl-f9897bf6"><span class="cl-f981f958">dochody_woj</span></p></td><td class="cl-f989a4b4"><p class="cl-f9897c0a"><span class="cl-f981f958">0.6</span></p></td><td class="cl-f989a4b5"><p class="cl-f9897c0a"><span class="cl-f981f958">0.1</span></p></td><td class="cl-f989a4be"><p class="cl-f9897c0a"><span class="cl-f981f958">5.7</span></p></td><td class="cl-f989a4be"><p class="cl-f9897c0a"><span class="cl-f981f958">0.0</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="5"class="cl-f989a4bf"><p class="cl-f9897bf6"><span class="cl-f981f958">n: 4</span></p></td></tr></tfoot></table></div><div class="tabwid"><style>.cl-f9d9d29a{}.cl-f9b259a4{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 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rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-f9bec9fa{width:0.922in;background-color:transparent;vertical-align: top;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-f9d9d29a'><tbody><tr style="overflow-wrap:break-word;"><td class="cl-f9bec9dc"><p class="cl-f9be8a76"><span class="cl-f9b259a4">summary.modelSEM..lambda</span><br><span class="cl-f9b259c2">numeric</span></p></td><td class="cl-f9bec9fa"><p class="cl-f9be8a94"><span class="cl-f9b259a4">0.538173</span></p></td></tr></tbody></table></div> </small> </small> ] ] .panel[.panel-name[Lambda] .pull-left[ <small> <br> <br> **Interpretacja** <p> Parametr przestrzenny lambda wiąże składniki przestrzenne, w przypadku zadanego modelu wynosi 0.538, i jest istotny statystycznie (p-value wynosi 0.028) na poziomie alfa <0.05. Oznacza to, że model SEM wyjaśni badane zjawisko lepiej niż OLS. </small> ] .pull-right[ <br> <br> <div class="tabwid"><style>.cl-fa332d36{}.cl-fa166e26{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-fa166e44{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(153, 153, 153, 1.00);background-color:transparent;}.cl-fa2130a4{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-fa2130b8{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-fa2159b2{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa2159c6{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa2159d0{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa2159da{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa2159db{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa2159e4{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa2159e5{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa2159ee{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa2159ef{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa2159f8{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa215a02{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa215a0c{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa215a0d{width:0.922in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa215a20{width:0.829in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-fa332d36'><thead><tr style="overflow-wrap:break-word;"><th class="cl-fa2159b2"><p class="cl-fa2130a4"><span class="cl-fa166e26">Lambda</span></p></th><th class="cl-fa2159c6"><p class="cl-fa2130b8"><span class="cl-fa166e26">Value</span></p></th></tr><tr style="overflow-wrap:break-word;"><th class="cl-fa2159d0"><p class="cl-fa2130a4"><span class="cl-fa166e44">character</span></p></th><th class="cl-fa2159da"><p class="cl-fa2130b8"><span class="cl-fa166e44">numeric</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-fa2159db"><p class="cl-fa2130a4"><span class="cl-fa166e26">value</span></p></td><td class="cl-fa2159e4"><p class="cl-fa2130b8"><span class="cl-fa166e26">0.5</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fa2159e5"><p class="cl-fa2130a4"><span class="cl-fa166e26">Std.Err</span></p></td><td class="cl-fa2159ee"><p class="cl-fa2130b8"><span class="cl-fa166e26">0.2</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fa2159ef"><p class="cl-fa2130a4"><span class="cl-fa166e26">z-value</span></p></td><td class="cl-fa2159f8"><p class="cl-fa2130b8"><span class="cl-fa166e26">2.3</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-fa215a02"><p class="cl-fa2130a4"><span class="cl-fa166e26">p-value</span></p></td><td class="cl-fa215a0c"><p class="cl-fa2130b8"><span class="cl-fa166e26">0.0</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="2"class="cl-fa215a0d"><p class="cl-fa2130a4"><span class="cl-fa166e26">n: 4</span></p></td></tr></tfoot></table></div> ] ] .panel[.panel-name[Wald statistic] **Wald Statistic** <p> <small> **Hipoteza:** <p> H0: Nie ma przestrzennej autokorelacji w błędach modelu.λ = 0 (Należy wrócić do OLS) <p> HA: Istnieje istotna przestrzenna autokorelacja w błędach modelu.λ ≠ 0 (Model SEM) <br> <small> <small> <small> <div class="tabwid"><style>.cl-fa8721ac{}.cl-fa6d0740{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-fa751840{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-fa75185e{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-fa753e7e{width:0.863in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753e92{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753e93{width:0.897in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753e9c{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753ea6{width:3.028in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753eb0{width:0.863in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753eba{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753ebb{width:0.897in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753ec4{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753ece{width:3.028in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753ed8{width:0.863in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753ed9{width:0.795in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753ee2{width:0.897in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753eec{width:0.981in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-fa753ef6{width:3.028in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(255, 255, 255, 0.00);border-top: 0 solid rgba(255, 255, 255, 0.00);border-left: 0 solid rgba(255, 255, 255, 0.00);border-right: 0 solid rgba(255, 255, 255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-fa8721ac'><thead><tr style="overflow-wrap:break-word;"><th class="cl-fa753e7e"><p class="cl-fa751840"><span class="cl-fa6d0740">estimate</span></p></th><th class="cl-fa753e92"><p class="cl-fa751840"><span class="cl-fa6d0740">statistic</span></p></th><th class="cl-fa753e93"><p class="cl-fa751840"><span class="cl-fa6d0740">p.value</span></p></th><th class="cl-fa753e9c"><p class="cl-fa751840"><span class="cl-fa6d0740">parameter</span></p></th><th class="cl-fa753ea6"><p class="cl-fa75185e"><span class="cl-fa6d0740">method</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-fa753eb0"><p class="cl-fa751840"><span class="cl-fa6d0740">0.5</span></p></td><td class="cl-fa753eba"><p class="cl-fa751840"><span class="cl-fa6d0740">5.2</span></p></td><td class="cl-fa753ebb"><p class="cl-fa751840"><span class="cl-fa6d0740">0.0228</span><span class="cl-fa6d0740"> *</span></p></td><td class="cl-fa753ec4"><p class="cl-fa751840"><span class="cl-fa6d0740">1.0</span></p></td><td class="cl-fa753ece"><p class="cl-fa75185e"><span class="cl-fa6d0740">Wald diagnostics for spatial dependence</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="5"class="cl-fa753ed8"><p class="cl-fa75185e"><span class="cl-fa6d0740">Signif. codes: 0 &lt;= '***' &lt; 0.001 &lt; '**' &lt; 0.01 &lt; '*' &lt; 0.05</span></p></td></tr></tfoot></table></div> </small> </small> </small> <small> **Wnioski:** p-value < ⍺ należy zatem odrzucić hipotezę zerową na korzyść hipotezy alternatywej. Hipoteza alternatywna zakłada, że istnieje autokorelacja przestrzenna. </small> </small> ] .panel[.panel-name[LR test] **LR test** <p> <small> **Hipoteza:** <p> H0: Różnica między oszacowniami MNW i MNK jest nieistotna, wybieramy model bez interakcji przestrzennej <p> HA: Różnica między oszacowniami MNW i MNK jest istotna, wybieramy model SEM <p> <br> <small> <small> <small> <div class="tabwid"><style>.cl-fae87e98{}.cl-facb4850{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-fad6f038{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-fad6f04c{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 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255, 0.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-fae87e98'><thead><tr style="overflow-wrap:break-word;"><th class="cl-fad71b44"><p class="cl-fad6f038"><span class="cl-facb4850">estimate1</span></p></th><th class="cl-fad71b44"><p class="cl-fad6f038"><span class="cl-facb4850">estimate2</span></p></th><th class="cl-fad71b4e"><p class="cl-fad6f038"><span class="cl-facb4850">statistic</span></p></th><th class="cl-fad71b58"><p class="cl-fad6f038"><span class="cl-facb4850">p.value</span></p></th><th class="cl-fad71b62"><p class="cl-fad6f038"><span class="cl-facb4850">parameter</span></p></th><th class="cl-fad71b6c"><p class="cl-fad6f04c"><span class="cl-facb4850">method</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-fad71b80"><p class="cl-fad6f038"><span class="cl-facb4850">-112.5</span></p></td><td class="cl-fad71b80"><p class="cl-fad6f038"><span class="cl-facb4850">-113.5</span></p></td><td class="cl-fad71b81"><p class="cl-fad6f038"><span class="cl-facb4850">1.9</span></p></td><td class="cl-fad71b8a"><p class="cl-fad6f038"><span class="cl-facb4850">0.1720</span><span class="cl-facb4850"> </span></p></td><td class="cl-fad71b94"><p class="cl-fad6f038"><span class="cl-facb4850">1.0</span></p></td><td class="cl-fad71b9e"><p class="cl-fad6f04c"><span class="cl-facb4850">Likelihood Ratio diagnostics for spatial dependence</span></p></td></tr></tbody><tfoot><tr style="overflow-wrap:break-word;"><td colspan="6"class="cl-fad71b9f"><p class="cl-fad6f04c"><span class="cl-facb4850">Signif. codes: 0 &lt;= '***' &lt; 0.001 &lt; '**' &lt; 0.01 &lt; '*' &lt; 0.05</span></p></td></tr></tfoot></table></div> </small> </small> </small> <small> **Wnioski:** p-value > ⍺, zatem różnica oszacowań MNW i MNK jest nieistotna, należy zrezygnować z modelu SEM, na korzyść modelu OLS. </small> </small> ] ] --- ### Model SARMA/SARAR .panelset[ .panel[.panel-name[Opis modelu] .pull-left[ <small> <small> <br> **SARMA (Spatial Autoregressive Moving Average) i SARAR (Spatial Autoregressive Autoregressive)** są zaawansowanymi modelami stosowanymi w ekonometrii przestrzennej. Umożliwiają one jednoczesne uwzględnienie przestrzennej autokorelacji zarówno w zmiennej zależnej, jak i w składnikach losowych. Poniżej znajduje się szczegółowy opis tych modeli. - `Model SARMA` łączy w sobie elementy przestrzennej autoregresji oraz przestrzennego modelu średniej ruchomej. - `Model SARAR` również łączy w sobie elementy przestrzennej autoregresji, ale uwzględnia autoregresję zarówno w zmiennej zależnej, jak i w błędach. </small> </small> ] .pull-right[ <br>  ] ] .panel[.panel-name[Testowanie i wnioski] .pull-left[ <br> <br> <small> **WNIOSKI:** <p> - Parametr Rho w zadanym modelu wynosi 0.45 i jest istotny statystycznie na poziomie 0.05. - Parametr lambda równy jest -0.05 i nie jest istotny statystycznie na zadanym poziomie ⍺ wynoszącym 0.05. - Test LR, na poziomie istotności 0.09 wskazuje, że model SARMA nie wyjaśnia badanego zjawiska i należy wrócić do modelu OLS. - Błąd AIC wynosi 236.09. </small> ] .pull-right[ <br> <br>  ] ] ] --- ### Spatial Durbin Model .panelset[ .panel[.panel-name[Opis modelu] .pull-left[ <small> <br> <br> **Spatial Durbin Model (SDM)** To zaawansowany model ekonometryczny stosowany do analizy danych przestrzennych, który uwzględnia zarówno bezpośrednie, jak i pośrednie efekty przestrzenne w zmiennej zależnej oraz zmiennych niezależnych. Poniżej znajduje się szczegółowy opis tego modelu. Dla badanego modelu istotne statystycznie okazały się zmienne zaludnienie i dochody województw. </small> ] .pull-right[ <br>  ] </small> ] .panel[.panel-name[Testowanie i wnioski] .pull-left[ <br> <small> **WNIOSKI** <p> - Parametr Rho wynosi 0.28 i jest nieistotny statystycznie, ponieważ jego p-value wynosi 0.362 a tym samym, wskazuje, że przestrznny mnożnik y nie ma wpływu na obserwacje sąsiadujące, czyli model nie jest w stanie wyjaśnić zjawiska, należy zatem wrócić do modelu OLS. - Również wartości testów: LR, Wald i LM są wysokie co również wskazuje na konieczność powrotu do modelu OLS. </small> ] .pull-right[ <br> <br>  ] ] ] --- <br> ### SLX model/SCM - Spatial Cross-Regressive Model .pull-left[ <small> **Model SLX (Spatial Lag of X) lub SCM (Spatial Cross-Regressive Model)** to modele ekonometryczne stosowane do analizy danych przestrzennych, które uwzględniają wpływ przestrzennych opóźnień zmiennych niezależnych na zmienną zależną. <br> <br> <br> **WNIOSKI**: <p> - Istotna statystycznie jest jedynie zmienna dochody województw, co wskazuje na to, że model nie opisuje dobrze badanego przez nas zjawiska. </small> ] .pull-right[ <br> <br>  </small> ] --- ### SDEM - Spatial Durbin Error Model .panelset[ .panel[.panel-name[Opis modelu] .pull-left[ <small> <br> **Spatial Durbin Error Model (SDEM)** to zaawansowany model ekonometryczny stosowany w analizie danych przestrzennych. Uwzględnia on zarówno przestrzenne opóźnienia zmiennych objaśniających, jak i przestrzenną autokorelację w składnikach losowych. Poniżej znajduje się szczegółowy opis tego modelu. Zmienne, które okazały się być istotne dla zadanego modelu to: zaludnienie, dochody województw, oraz logarytm zmiennej dochody województw. </small> ] .pull-right[  ] ] .panel[.panel-name[Testowanie i wnioski] .pull-left[ <small> <br> <br> **WNIOSKI:** - Parametr lambda wynosi 0.235 i jest nieistotny statystycznie ponieważ p-value 0.461 jest większe od alfy na poziomie 0.05. - Test LR i Wald również wskazują na słabe dopasowanie modelu, a tym samym model OLS jest lepiej dopasowany niż SEDM - AIC wynosi 239.47 </small> ] .pull-right[  ] ] ] --- <br> ### Wybór modelu i uzasadnienie .panelset[ .panel[.panel-name[Podsumowanie] <small> .pull-left[ <small> <br> **Wnioski:** <p> - Kryterium informacyjne Akaikego jest najniższe dla modelu SAR zaś najwyższe dla modelu SEDM - Loglikehood jest największe dla SEDM, a najniższe dla modelu SARMA - `Jedynie model SAR dla testu LR ma value mniejsze od 0.05, więc jako jedyny jest lepszy od modelu OLS` - Ponadto model SAR jako jedyny przeszedł przez wszystkie testy, ze wskazaniem, że jest on lepszy niż OLS. </small> ] .pull-right[  ] ] .panel[.panel-name[Najlepszy model - SAR] .pull-left[ <small> <small> **Uzasadnienie:** <p> Po szczegółowej analizie i przejściu przez wszystkie modele, modelem, który najlepiej wyjaśnia opisane zjawisko jest model SAR. Długość dróg rowerowych w Polsce z podziałem na wojewódźtwa jest wyjaśniany przez przestrzennie opóźniony mnożnik y, to oznacza, że zmienne, które okazały się być istotne: zaludnienie i dochody województw, wpływają na długość tras rowerowych w jednostkach sąsiadujących, a te następnie oddziałują na dane województwo. Ponadto model autoregresji przestrzennej wykazuje zarówno w teście Wald jak i LR p-value < α na poziomie istotności 0.05, co wskazuje, że model SAR jest lepszy od modelu OLS. </small> </small> ] .pull-right[  ] ] ]