// Numbas version: exam_results_page_options
{"name": "Choose most appropriate model from Minitab output of AR(1) model and AR(2) stationary series", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"w": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2)", "description": "", "name": "w"}, "est22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,random(0.075..0.110#0.005),random(0.305..0.455#0.005))", "description": "", "name": "est22"}, "thismany": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(250..500#10)", "description": "", "name": "thismany"}, "thatmany": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,me,thismany+random(20..50))", "description": "", "name": "thatmany"}, "flse": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=2, \"AR(1)\", \"AR(2)\")", "description": "", "name": "flse"}, "flser22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=2,\"retain this hypothesis\",\"reject this hypothesis\")", "description": "", "name": "flser22"}, "flse221": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"slight evidence against\"", "description": "", "name": "flse221"}, "est11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.255..0.455#0.005)", "description": "", "name": "est11"}, "correct22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "If(t=1,if(w=1,\"slight evidence against\", \"no evidence against\"),\"moderate evidence against\")", "description": "", "name": "correct22"}, "me1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(50..250#10)", "description": "", "name": "me1"}, "thisno": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,\"\",\"two\")", "description": "", "name": "thisno"}, "forc1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround((me1+est11*(thismany-me1)),2)", "description": "", "name": "forc1"}, "p11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=1, random(0.005..0.009#0.001),random(0.015..0.045#0.005))", "description": "", "name": "p11"}, "flse321": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"no evidence against\"", "description": "", "name": "flse321"}, "q1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,0.01,0.05)", "description": "", "name": "q1"}, "me": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,me1,me2)", "description": "", "name": "me"}, "ine": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1, \"\\\\>\",\"\\\\<\")", "description": "", "name": "ine"}, "est12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0.255..0.455#0.005)", "description": "", "name": "est12"}, "that": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,\"\",thatmany)", "description": "", "name": "that"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2)", "description": "", "name": "t"}, "p22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,if(w=1,random(0.06..0.095#0.005),random(0.15..0.555#0.005)),random(0.015..0.045#0.005))", "description": "", "name": "p22"}, "tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.01", "description": "", "name": "tol"}, "v": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2)", "description": "", "name": "v"}, "p12": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1, random(0.005..0.009#0.001),random(0.015..0.045#0.005))", "description": "", "name": "p12"}, "flse222": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,if(w=1,\"no evidence against\",\"moderate evidence against\"), \"slight evidence against\")", "description": "", "name": "flse222"}, "u": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,2)", "description": "", "name": "u"}, "correct11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=1, \"strong evidence against\",\"moderate evidence against\")", "description": "", "name": "correct11"}, "an": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,\"\",\"and the one before that \")", "description": "", "name": "an"}, "correct": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1, \"AR(1)\", \"AR(2)\")", "description": "", "name": "correct"}, "me2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "me1+random(-20,-10,10,20)", "description": "", "name": "me2"}, "p": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,est11,est12)", "description": "", "name": "p"}, "correct21": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1, \"strong evidence against\", \"moderate evidence against\")", "description": "", "name": "correct21"}, "flse211": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"slight evidence against\"", "description": "", "name": "flse211"}, "flse111": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"no evidence against\"", "description": "", "name": "flse111"}, "flse322": {"templateType": "anything", "group": "Ungrouped variables", "definition": "If(t=1,if(w=1,\"moderate evidence against\", \"slight evidence against\"),\"no evidence against\")", "description": "", "name": "flse322"}, "flse311": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=2, \"strong evidence against\",\"moderate evidence against\")", "description": "", "name": "flse311"}, "correctr22": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,\"retain this hypothesis\",\"reject this hypothesis\")", "description": "", "name": "correctr22"}, "q3": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(v=1,0.01,0.05)", "description": "", "name": "q3"}, "y": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,0,est22)", "description": "", "name": "y"}, "s": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,\"\",\"s\")", "description": "", "name": "s"}, "q2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,if(w=1,0.05,0.1),0.05)", "description": "", "name": "q2"}, "flse122": {"templateType": "anything", "group": "Ungrouped variables", "definition": "\"strong evidence against\"", "description": "", "name": "flse122"}, "flse121": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(u=1,\"moderate evidence against\",\"strong evidence against\")", "description": "", "name": "flse121"}, "forc": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,forc1,precround((me2+est12*(thismany-me2)+est22*(thatmany-me2)),2))", "description": "", "name": "forc"}}, "ungrouped_variables": ["est12", "est11", "flse122", "p22", "an", "me1", "me2", "ine", "flse", "flse121", "q1", "q3", "q2", "correct11", "flse111", "flse221", "flse222", "flse321", "flse322", "tol", "correct", "est22", "that", "forc", "correctr22", "thisno", "forc1", "p11", "p12", "me", "thismany", "flse311", "thatmany", "flser22", "correct21", "flse211", "correct22", "p", "s", "u", "t", "w", "v", "y"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Choose most appropriate model from Minitab output of AR(1) model and AR(2) stationary series", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me1", "minValue": "me1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "est11", "minValue": "est11", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me1", "minValue": "me1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me2", "minValue": "me2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "est12", "minValue": "est12", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me2", "minValue": "me2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "est22", "minValue": "est22", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "me2", "minValue": "me2", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Write down the full estimated AR(1) and AR(2) models:</p>\n<p>$\\text{AR(1)}:\\;\\;Y_t=a_1+b_1(Y_{t-1}-c_1)+\\epsilon_t$</p>\n<p>Find $a_1=\\;$[[0]] &nbsp; &nbsp;$b_1=\\;$[[1]] &nbsp; &nbsp;$c_1=\\;$[[2]]</p>\n<p>$\\text{AR(2)}:\\;\\;Y_t=a_2+b_2(Y_{t-1}-c_2)+d_2(Y_{t-2}-e_2)+\\epsilon_t$</p>\n<p>Find $a_2=\\;$[[3]], &nbsp; &nbsp;$b_2=\\;$[[4]], &nbsp; $c_2=\\;$[[5]], &nbsp; $d_2=\\;$[[6]], &nbsp; $e_2=\\;$[[7]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["<p>{Correct21}</p>", "<p>{Flse121}</p>", "<p>{Flse221}</p>", "<p>{Flse321}</p>"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0], "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>retain this hypothesis</p>", "<p>reject this hypothesis</p>"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [0, 1], "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>{Correct22}</p>", "<p>{Flse122}</p>", "<p>{Flse222}</p>", "<p>{Flse322}</p>"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0], "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>{Correctr22}</p>", "<p>{Flser22}</p>"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0], "marks": 0}], "type": "gapfill", "prompt": "<p>For the AR(2) fit, choose one of the following:</p>\n<p>1) For the null hypothesis $\\text{H}_0:\\;\\alpha_1=0$ &nbsp;there is:</p>\n<p>[[0]]</p>\n<p>So we:</p>\n<p>[[1]]</p>\n<p>2) For the null hypothesis $\\text{H}_0:\\;\\alpha_2=0$ &nbsp;there is:</p>\n<p>[[2]]</p>\n<p>So we:</p>\n<p>[[3]]</p>\n<p></p>\n<p></p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"displayType": "radiogroup", "choices": ["<p>{Correct11}</p>", "<p>{Flse111}</p>", "<p>{Flse211}</p>", "<p>{Flse311}</p>"], "displayColumns": 0, "distractors": ["", "", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0, 0, 0], "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>retain this hypothesis</p>", "<p>reject this hypothesis</p>"], "displayColumns": 0, "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [0, 1], "marks": 0}], "type": "gapfill", "prompt": "<p>For the AR(1) fit, choose one of the following:</p>\n<p>For the null hypothesis $\\text{H}_0:\\;\\alpha_1=0$ &nbsp;there is:</p>\n<p>[[0]]</p>\n<p>So we:</p>\n<p>[[1]]</p>\n<p></p>", "showCorrectAnswer": true, "marks": 0}, {"displayType": "radiogroup", "choices": ["<p>{Correct}</p>", "<p>{Flse}</p>"], "displayColumns": 0, "prompt": "<p>Which of the two models seems most appropriate here?</p>", "distractors": ["", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [1, 0], "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "forc+tol", "minValue": "forc-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Using the most appropriate model as identified in the last part, forecast the next value in the series if the last value was $\\var{thismany}$ {an} $\\var{that}$.</p>\n<p>Forecast= [[0]] (enter to 2 decimal places.)</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>Minitab was used to fit both an AR(1) model and an AR(2) to a stationary series. The following table summarises the results obtained:</p>\n<p></p>\n<p></p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td></td>\n<th colspan=\"2\">AR(1) Model</th><th colspan=\"2\">AR(2) Model</th></tr>\n<tr>\n<td>Parameter</td>\n<td>Estimate</td>\n<td>p-Value</td>\n<td>Estimate</td>\n<td>p-Value</td>\n</tr>\n<tr>\n<td>$\\alpha_1$</td>\n<td>{est11}</td>\n<td>{p11}</td>\n<td>{est12}</td>\n<td>{p12}</td>\n</tr>\n<tr>\n<td>$\\alpha_2$</td>\n<td>$\\ldots$</td>\n<td>$\\ldots$</td>\n<td>{est22}</td>\n<td>{p22}</td>\n</tr>\n<tr>\n<td>$\\mu$</td>\n<td>{me1}</td>\n<td>$\\ldots$</td>\n<td>{me2}</td>\n<td>$\\ldots$</td>\n</tr>\n</tbody>\n</table>", "tags": ["ACE2013", "AR(1)", "AR(1) model", "AR(2)", "AR(2) model", "ARMA", "autoregression", "autoregressive", "checked2015", "forecast", "forecasting", "Minitab output", "minitab output", "models", "p values", "stationary series", "stationary time series", "statistics", "time series"], "rulesets": {}, "preamble": {"css": ".minitab {\nfont-family: 'Courier', monospace;\n}", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>23/03/2014:</strong></p>\n<p>Created.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Minitab was used to fit both an AR(1) model and an AR(2) to a stationary series. A &nbsp;table is given summarising the results obtained from Minitab. Choose the most appropriate model and make a forecast based on that model.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a) Using the data given we have the models are given by:</p>\n<p><strong>AR(1):&nbsp;</strong></p>\n<p>\\[Y_t=a_1+b_1(Y_{t-1}-c_1)+\\epsilon_t\\]</p>\n<p>where $a_1=\\var{me1},\\;\\;b_1=\\alpha_1=\\var{est11},\\;\\;c_1=\\var{me1}$</p>\n<p><strong>AR(2):</strong></p>\n<p>\\[Y_t=a_2+b_2(Y_{t-1}-c_2)+d_2(Y_{t-2}-e_2)+\\epsilon_t\\]</p>\n<p>where $a_2=\\var{me2},\\;\\;b_2=\\alpha_1=\\var{est12},\\;\\;c_2=\\var{me2},\\;\\;d_2=\\alpha_2=\\var{est22},\\;\\;e_2=\\var{me2}$</p>\n<p>b)</p>\n<p>Using the above diagram for making decisions on retaining or rejecting the hypotheses on the coefficients we see that:</p>\n<p><strong>For the AR(2) fit:</strong></p>\n<p>$\\var{est12}$ has a p-value of $\\var{p12}$ and so there is {Correct21} the hypothesis that $\\text{H}_0:\\;\\alpha_1=0$ . So we reject this hypothesis.</p>\n<p>&nbsp;$\\var{est22}$ has a p-value of $\\var{p22}$ and so there is {Correct22} the hypothesis that $\\text{H}_0:\\;\\alpha_2=0$ . So we {Correctr22}.</p>\n<p>c)</p>\n<p><strong>For the AR(1) fit:</strong></p>\n<p>&nbsp;$\\alpha_1=\\var{est11}$ has a p-value of $\\var{p11} \\lt \\var{q3}$ and so there is {Correct11} the hypothesis that&nbsp;&nbsp;$\\text{H}_0:\\;\\alpha_1=0$&nbsp;.</p>\n<p>So we reject this hypothesis.</p>\n<p>d)</p>\n<p>We see that since in either case we reject the hypothesis that $\\alpha_1=0$ , but that for the AR(2) model we {Correctr22} &nbsp;that $\\alpha_2=0$ &nbsp;, so we choose the {Correct} model as the most appropriate one.</p>\n<p>e)</p>\n<p>Using the {Correct} model we have the forecast for the next period:</p>\n<p>Forecast =$ \\simplify[all,!collectNumbers]{{me}+{p}({thismany}-{me})+{y}({thatmany}-{me})}=\\var{forc}$ to 2 decimal places.</p>\n<p></p>\n<p></p>\n<p></p>", "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/"}]}