// Numbas version: exam_results_page_options
{"name": "Interpret Minitab output of AR(1) model", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"fo": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(fo1=me, fo1+random(40..60),fo1)", "description": "", "name": "fo"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0", "description": "", "name": "tol"}, "fo1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "(u*(me-2*mese)+(100-u)*(me+2*mese))/100", "description": "", "name": "fo1"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(2.18*sea,2)", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(-0.12*sed,2)", "description": "", "name": "c"}, "sea": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.05..0.45#0.05)", "description": "", "name": "sea"}, "fo3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(afo3,2)", "description": "", "name": "fo3"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..100)", "description": "", "name": "u"}, "afo4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "me+a*(fo3-me)", "description": "", "name": "afo4"}, "sed": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(10..150#5)", "description": "", "name": "sed"}, "per": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(25..70)", "description": "", "name": "per"}, "afo3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "me+a*(fo2-me)", "description": "", "name": "afo3"}, "me": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(50..150#10)", "description": "", "name": "me"}, "fo2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(afo2,2)", "description": "", "name": "fo2"}, "fo4": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(afo4,2)", "description": "", "name": "fo4"}, "afo2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "me+a*(fo -me)", "description": "", "name": "afo2"}, "mese": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(100..300#10)", "description": "", "name": "mese"}}, "ungrouped_variables": ["me", "a", "c", "fo1", "afo2", "fo2", "fo3", "per", "afo4", "sed", "u", "mese", "tol", "afo3", "fo4", "sea", "fo"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Interpret Minitab output of AR(1) model", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "a+tol", "minValue": "a-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "c+tol", "minValue": "c-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Input the values of X and Y:</p>\n<p>X= [[0]] &nbsp;(to 2 decimal places).</p>\n<p>Y= [[1]] &nbsp;(to 2 decimal places).</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me", "minValue": "me", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "a", "minValue": "a", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me", "minValue": "me", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>The estimated model has the following form:</p>\n<p>\\[Y_t=a+b(Y_{t-1}-c)+\\epsilon_t\\]</p>\n<p>Find $a=\\;$[[0]] &nbsp; &nbsp;$b=\\;$[[1]] &nbsp; &nbsp;$c=\\;$[[2]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "fo2", "minValue": "fo2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "fo3", "minValue": "fo3", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "fo4", "minValue": "fo4", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Use your model above to complete the following (edited ) output from Minitab:</p>\n<p><em>(Calculate each forecast to 2 decimal places, then use this value to 2 decimal places &nbsp;to calculate the next forecast.)</em></p>\n<p><em><br/></em>Forecast from period {per}:</p>\n<p></p>\n<table border=\"1\" class=\"minitab\">\n<tbody>\n<tr>\n<td>Period</td>\n<td>Forecast</td>\n</tr>\n<tr>\n<td>{per+1}</td>\n<td>{fo}</td>\n</tr>\n<tr>\n<td>{per+2}</td>\n<td>[[0]]</td>\n</tr>\n<tr>\n<td>{per+3}</td>\n<td>[[1]]</td>\n</tr>\n<tr>\n<td>{per+4}</td>\n<td>[[2]]</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Minitab was used to fit an AR(1) model to a stationary series. The following output was obtained:</p>\n<p></p>\n<p>FINAL ESTIMATE OF PARAMETERS</p>\n<p><tablebody></tablebody></p>\n<table border=\"1\" class=\"minitab\">\n<tbody>\n<tr>\n<td>Type</td>\n<td>Coef</td>\n<td>SE Coef</td>\n<td>T</td>\n<td>P</td>\n</tr>\n<tr>\n<td>AR 1</td>\n<td>X</td>\n<td>{sea}</td>\n<td>2.18</td>\n<td>0.034</td>\n</tr>\n<tr>\n<td>Constant</td>\n<td>Y</td>\n<td>{sed}</td>\n<td>-0.12</td>\n<td>0.902</td>\n</tr>\n<tr>\n<td>Mean</td>\n<td>{me}</td>\n<td>{mese}</td>\n<td></td>\n<td></td>\n</tr>\n</tbody>\n</table>", "tags": ["ACE2013", "AR(1)", "AR(1) model", "ARMA", "autoregression", "autoregressive", "checked2015", "forecast", "forecasting", "Minitab output", "minitab output", "models", "stationary series", "stationary time series", "statistics", "time series"], "rulesets": {}, "preamble": {"css": ".minitab {\nfont-family: 'Courier', monospace;\n}", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>22/03/2014:</strong></p>\n<p>Created.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Minitab was used to fit an AR(1) model to a stationary time series. Given the output answer the following questions about the model and use the model to make forecasts.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n<p>From the table we have $\\displaystyle \\frac{X}{\\var{sea}}=2.18$.</p>\n<p>Hence we find $X=\\var{sea}\\times 2.18=\\var{a}$ to 2 decimal places.</p>\n<p>Similarly we have &nbsp;$\\displaystyle \\frac{Y}{\\var{sed}}=-0.12$</p>\n<p>Hence we find $Y=\\var{sed}\\times -0.12=\\var{c}$ to 2 decimal places.</p>\n<p>b)</p>\n<p>The model is&nbsp;</p>\n<p>\\[Y_t= \\var{me}+\\var{a}(Y_{t-1}-\\var{me})+\\epsilon_t.\\]</p>\n<p>So $a=\\var{me},\\;\\;b=\\var{a},\\;\\;c=\\var{me}$.</p>\n<p>c)</p>\n<p>Putting $Y_{\\var{per+1}}=\\var{fo1}$ we find:</p>\n<p>\\[Y_{\\var {per + 2}} = \\var{me} + \\var{a} &nbsp;(Y_{\\var{per + 1}} -\\var{me}) = \\var{me} + \\var{a} \\times (\\var{fo1} -\\var{me}) = \\var{fo2}\\] to 2 decimal places.</p>\n<p>Using this value to 2dps we have:</p>\n<p>\\[Y_{\\var {per + 3}} = \\var{me} + \\var{a} &nbsp;(Y_{\\var{per + 2}} -\\var{me}) = \\var{me} + \\var{a} \\times (\\var{fo2} -\\var{me}) = \\var{fo3}\\] to 2 decimal places.</p>\n<p>Finally using this last value we have:</p>\n<p>\\[Y_{\\var {per + 4}} = \\var{me} + \\var{a} &nbsp;(Y_{\\var{per + 3}} -\\var{me}) = \\var{me} + \\var{a} \\times (\\var{fo3} -\\var{me}) = \\var{fo4}\\] to 2 decimal places.</p>\n<p>In summary we have.</p>\n<p>Forecast from period {per}:</p>\n<p></p>\n<table border=\"1\" class=\"minitab\">\n<tbody>\n<tr>\n<td>Period</td>\n<td>Forecast</td>\n</tr>\n<tr>\n<td>{per+1}</td>\n<td>{fo1}</td>\n</tr>\n<tr>\n<td>{per+2}</td>\n<td>{fo2}</td>\n</tr>\n<tr>\n<td>{per+3}</td>\n<td>{fo3}</td>\n</tr>\n<tr>\n<td>{per+4}</td>\n<td>{fo4}</td>\n</tr>\n</tbody>\n</table>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}