forcasting recurrence

need to forecast a recurrence:
attached is a graph that we are suppose to use to answer the questions. The questions are also attached to this post. I am soo confused about this exercise!. SO if someone could do this that would be GREAT!!!.

Sheet
1

cst

Period

0.2 2

0.2 3

0.2 4

0.2 5

0.2 6

0.2 7

8 1

0.2 8

0.2 9

119

0.2 10

0.2 11

0.2 12

0.2 13

0.2 14

0.2 15

0.2 16

0.2 17

0.2 18

0.2 19

0.2 20

0.2 21

.58

0.2 22 2.6

0.2 23

0.2 24

0.2 25

0.2 26 2.6

0.2 27

0.2 28

.99

0.2 29

0.2 30

.39

0.2 31

32 199

0.2 32 3.1

.29

0.2 33

045

0.2 34

0.2 35

.90

0.2 36

0.2 37

0.2 38

0.2 39

0.2 40

0.2 41

0.2 42

0.2 43

0.2 44

0.2 45

0.2 46

47 172

0.2 47

48 194

0.2 48

49 196

.74

0.2 49 0.91

50 196

0.2 50

51 236

588

0.2 51 2.44

0.2 52

.094

0.2 53

0.2 54

55 264

0.2 55

0.2 56 2.7

0.2 57 2.8

0.2 58

59 180

0.2 59

0.2 60 3.21

61 204

.45

0.2 61 3.2

.76

0.2 62 3.38

63 235

0.2 63

0.2 64

234

031

0.2 65

66 264

0.2 66

0.2 67

0.2 68

69 259

0.2 69 3.31

70 229

0.2 70

71 203

0.2 71 3.29

72 229

.2178

027

0.2 72

73 242

0.2 73 2.3

0.2 74 2.3

0.2 75

0.2 76 2.3

77 270

0.2 77

0.2 78 2.25

0.2 79 2.24

0.2 80

81 312

.43

0.2 81

82 274

.347

0.2 82

83 237

0.2 83 2.29

84 278

229

0.2 84

85 284

0.2 85 2.24

0.2 86 2.39

0.2 87

.64

0.2 88 2.3

0.2 89 2.48

90 374

0284

0.2 90 2.6

91 413

0.2 91

0.2 92

93 355

0.2 93 2.72

.19

0.2 94 2.73

95 271

0.2 95 2.69

96 306

.64

0.2 96 2.73

97 315

0.2 97

98 301

.0112

0.2 98 2.82

99 356

0.2 99

100

0.2 100

355

0.2 101

422

.55

0.2 102 2.87

103 465

0.2 103

104 467

0.2 104 2.92

0.2 105

347

0.2 106 2.9

107

297

553

0.2 107

108

0.2 108 2.99

109 340

0.2 109

110 318

.2424

011

0.2 110

362

0.2 111 3.38

112 348

0.2 112

0.2 113

114 435

0.2 114 3.52

115

2404

0.2 115 3.52

116

23566

0.2 116 3.53

117 404

0.2 117

359

0.2 118

119

.96

0.2 119

120 337

0.2 120 3.48

121

20394

0.2 121

122 342

315

0.2 122 3.5

.93

0.2 123 3.53

124

288

0.2 124

125

073

0.2 125

126

0.2 126

548

0.2 127

128

871

0.2 128

129 463

0.2 129 3.93

407

0.2 130 3.93

362

0.2 131

132 405

103

0.2 132 3.8

133 417

.97

0.2 133 3.84

134

0.2 134 3.84

135

0.2 135

136

0.2 136 4.03

137 472

0.2 137

138 535

0.2 138

622

0.2 139

140

0.2 140 4.65

141 508

.21

25304

0.2 141 4.59

142 461

.3712

243

0.2 142

390

195

0.2 143

144 432

0.2 144 4.5

145 4.8
146

149

150 4.96
151

152

153

3.84

155 3.6
156 3.54
157

158

159

160 4.56
161

162

163 5
165

166

167

168

170

171

172 5.35

5.45

174

175

176

177

178

179

180

181 7
182

184 7
185 7.24
188

189

191

192 6.68
193

194

195 6.12
196

198

199

200 3.7
201 3.38
202

203

204

205 5.4
207

208

209

210

211 3.38
212 3.2
214

217

218

219

220

221

222 5.07
224 5.6
225

226

227

228

229

230

231

233 7.83
234

235

236

237

238

239

240 7.9
242

243

246

248 5.5
249

253

254 6.44
255

256

257

259 4.87
260

261 5
263 5.2
264 5.41
265 5.23
266

267

269 4.75
270

271 4.62
272

4.6

274

4.96

277 5.19
278 5.49
280

281 6.1
282

283 6.44
284 6.45
285

286 6.29

6.41

288

289

292

293

296

297

298

299

300

301

302

303

304

305

306

307 12
310

312

313 8.06

315

317

318

319

321

322

326

327 14.7
329

330

333

334

336

337

338

339

340

341

342

347

348

349 9.08
350

351 9
352 8.64
353 8.76
354 9
355 8.9
356

9.52

358

359

360

362

363

364

365

8.06

369 8.52
370

371

373 7.08
374

375 7.1
376

377 7.24

7.1

379 7.07
380

382

383

384

385

388

389 5.35
390 5.53
391

393 5.59
394

5.66

396

397

398 6.04
399 6.4
400 6.13

5.69

402

403 5.81
404 5.66
405 5.7
406 5.91
407 6.26
408 6.46
409 6.73
410 7.06

7.24

413 7.76

8.07

8.27

416 8.53
417

418

419

420

422 7.9
424

425

427 7.64
428

7.9

430 7.77
431 7.74
432

7.62

435

437 7.06
439

441 5.91
445

448

449 4.56

3.8

452 3.84
453 4.04
456

3.21

458

2.91

460 2.86
461 3.13
463 3

2.93

465 2.95

2.87

467

3.07

3.04

2.95

472 3.02
473 3.1
474 3.06

3.5

479 4.14
480 4.14
481

4.62

484 4.95
485

5.6

487

5.77

489

490 5.65
491 5.67
493

5.4

495 5.28
496 5.28
497

498 5.14
499 5
501 4.96

4.95

5.02

5.09

505

506

5.09

508 4.99
511 5.03
512

5.14

514

515 5.05
516

5.05

5.14

519 4.95

4.97

5.14

5.16

5.09

5.03

4.95

4.98

4.96

530 4.9

3.96

535

4.44

4.5

4.72

4.86

5.07

5.2

5.32

548

5.69

5.66

5.69

553 5.96

6.09

556 6.11
557

558 5.77
559 5.15
560 4.88

4.42

564

566 3.36
567

568

570 1.69

1.79

1.73

576 1.7
577

578 1.62
579

581

1.6

1.72

587

588

589

590

592 2.37

2.37

594 2.39
596 2.72
597 2.69
599 2.73

2.78

2.67

2.89

604

605 3.08
606 2.95
607 3.06

3.5

610 3.4

3.57

3.46

615 3.47
616 3.53
617

618 3.8

3.96

4.02

621

622 4.09

3.7

2.78

2.4

2.07

2.42

3.06

3.52

3.6

3.91

3.92

639

4.03

4.6

644 4.6

4.97

4.78

4.99

4.73

4.29

4.27

4.46

4.01

3.63

3.53

3.6

3.42

661 3.39

3.46

663

3.23

3.63

3.76

3.73

3.62

671 3.61
672 3.72

3.76

3.69

3.45

3.38

3.4

3.39

679 3.62

3.46

3.41

3.33

683 3.4

3.37

3.4

3.39

3.41

3.5

3.54

3.61

3.76

693

3.87

3.95

4.01

697 4.04
698 4
700 4.15
701 4.04

4.01

3.93

705 4.01

4.02

4.04

4.03

710 4.08
712 4.12

4.08

4.09

717

4.33

720

4.9

5.02

4.97

4.89

5.05

5.79

5.48

5.53

732 5.19

4.75

4.73

4.38

4.61

4.89

740 5.23

5.41

742 5.52

5.71

5.53

5.61

5.8

748

6.11

5.42

5.52

5.57

6.16

6.22

7.77

8.1

770

7.15

7.94

774

7.46

778

779 6.25

5.75

5.61

5.07

4.5

5.22

6.32

787 6.74

5.35

5.14

5.56

796

797 5.55

5.64

799

5.79

6.06

803 5.94

6.83

6.81

6.96

8.27

822 8.15

8.41

7.24

6.81

7.76

7.39

7.17

835

8.29

7.46

842 7.06

7.13

6.84

7.31

7.12

6.86

6.09

5.68

853 6.22

6.44

855

6.51

6.84

7.19

7.22

7.3

7.61

7.7

868 7.85

8.07

8.3

871

8.33

8.41

877 9.5

9.69

887

889

892

900

901

13.65

15.5

911

14.73

917

928

10.9

932 11.3
939

941

944

946

10.9

951 11.05

8.4

8.41

8.1

7.3

6.86

7.27

966

6.86

970 6.56

6.46

972

6.41

6.56

6.58

7.82

7.74

980

8.67

7.99

7.87

987 7.5

7.83

8.24

990

8.77

993

9.2

9.32

999 9.61

9.4

7.83

8.13

8.02

7.8

7.77

8.13

8.4

8.26

8.22

8.27

8.07

7.74

7.47

7.38

7.08

7.35

7.23

7.12

7.39

7.38

6.8

6.5

5.9

5.39

5.4

5.81

5.6

4.91

4.72

4.42

4.64

5.14

5.21

4.93

4.4

4.3

4.4

4.18

4.5

4.54

4.48

4.83

5.4

6.27

6.5

7.71

7.25

6.68

6.27

5.8

5.89

6.1

5.89

5.77

5.57

5.39

5.2

5.14

5.79

6.11

6.27

6.49

6.45

6.21

6.41

5.82

5.91

6.16

6.38

6.42

6.24

6.06

5.38

5.43

5.57

5.58

5.61

5.52

5.47

4.62

4.18

4.57

4.48

4.61

4.9

5.03

5.33

5.7

5.77

5.75

5.94

6.49

6.65

6.43

6.17

6.02

5.79

5.26

4.43

4.42

4.35

4.04

3.45

3.22

3.62

4.14

4.01

3.8

3.49

3.01

2.25

2.32

2.23

2.3

2.48

2.51

2.61

2.68

2.75

2.78

2.9

2.97

2.88

2.89

2.96

2.9

2.84

2.96

3.18

3.07

3.11

3.33

3.38

3.49

3.59

3.46

3.34

3.41

3.48

3.6

3.8

3.93

3.92

3.97

3.72

3.21

2.98

2.88

2.92

2.97

3.2

3.54

3.76

3.8

3.86

4.02

3.96

4.12

4.31

4.34

4.4

4.43

4.68

4.53

4.69

4.72

4.25

4.35

4.15

3.9

3.8

3.89

3.93

3.84

3.74

3.78

3.71

3.92

4.04

3.98

3.92

4.06

4.08

4.04

3.93

3.84

3.87

3.91

4.01

3.98

3.93

3.92

3.86

3.92

3.93

3.97

3.93

3.99

4.02

4.08

4.11

4.12

4.13

4.17

4.15

4.22

4.2

4.17

4.19

4.2

4.19

4.15

4.18

4.19

1292 4.21

4.21

4.2

4.21

4.2

4.25

4.29

4.35

4.45

4.62

4.61

4.83

4.87

4.75

4.78

5.02

5.22

5.18

5.01

5.16

4.58

4.54

4.59

4.85

5.02

5.16

5.28

5.3

5.48

5.75

5.7

5.53

5.56

5.74

5.64

5.72

5.5

5.42

5.46

5.58

5.7

6.03

6.04

6.3

6.17

6.32

6.69

7.16

7.1

7.14

7.65

7.79

7.24

7.07

7.39

7.84

7.46

7.39

6.84

6.39

6.24

6.11

5.7

5.83

6.39

6.73

6.58

6.14

5.93

5.81

5.93

6.08

6.07

6.19

6.13

6.11

6.21

6.55

6.48

6.28

6.36

6.46

6.9

7.13

7.4

7.09

6.79

6.73

6.74

6.99

6.96

7.54

7.81

8.04

7.9

7.43

7.5

7.39

7.73

8.23

8.06

7.86

8.06

8.4

8.43

8.05

7.74

7.79

7.73

7.56

7.9

7.86

7.83

7.77

7.41

7.29

7.21

7.39

7.46

7.33

7.4

7.58

7.69

7.96

8.03

8.04

8.15

8.35

8.64

8.41

8.64

9.1

9.18

9.25

8.91

8.95

9.33

10.3

10.8

11.1

12.84

14.1

15.15

12.34

10.55

10.4

10.85

11.85

11.57

11.5

11.38

11.51

10.85

9.78

8.7

7.3

7.71

7.8

7.3

7.17

7.45

7.43

7.25

7.11

7.08

7.25

8.02

8.61

8.4

8.76

9.42

9.52

8.67

8.21

8.37

8.72

9.09

9.06

9.26

8.98

8.8

8.96

9.11

9.09

9.18

8.86

8.02

1587 8.19

8.01

7.87

7.84

8.21

8.79

8.76

8.47

8.75

8.72

8.39

7.85

8.11

8.04

8.07

8.28

8.27

7.9

7.65

7.53

7.09

7.34

7.54

7.48

7.39

6.84

6.42

6.59

6.87

6.77

6.6

6.26

5.98

5.97

6.04

5.96

5.81

5.68

5.36

5.33

5.72

5.77

5.75

5.97

6.48

7.1

7.3

7.24

7.46

7.74

7.96

7.81

7.78

7.47

7.2

7.06

6.63

6.17

6.28

6.49

6.2

6.04

5.93

5.71

5.65

5.81

6.27

6.51

6.74

6.87

6.64

6.83

6.53

6.2

6.3

6.58

6.42

6.69

6.89

6.71

6.49

6.22

6.3

6.21

6.03

5.88

5.81

5.54

5.57

5.65

5.64

5.65

5.5

5.46

5.34

4.81

4.53

4.83

4.65

4.72

5.23

5.18

5.54

5.9

5.79

5.94

5.92

6.11

6.03

6.28

6.66

6.52

6.26

5.99

6.44

6.1

6.05

5.83

5.8

5.74

5.72

5.24

5.16

5.1

4.89

5.14

5.39

5.28

5.24

4.97

4.73

4.57

4.65

5.09

5.04

4.91

5.28

5.21

5.16

4.93

4.65

3.87

3.94

4.05

4.03

																						Period	Airline Sales	Airline	F	Alpha	Interest Rates
			11																0.2	0.	7
11				8			12	0.				9
			3			13	11			3.2	1.01
			4	1		29	11				6		96	0.		98
							5	1		21		1		19		36	0.	93
	1		35			119		69		44	1.		15
				14	1		22		55		52	1.22
	48	1		27		80			16	1.		17
		136	1		31		43		53		28	1.28
	10	13			2.6		74		26		24	1.		59
		104			129		39		60		92	1.		45
1		18			124		51		88		79	1.		41
		115	1		23		40		110		34		1.6
		116			121		72		20		82	1.9
		141			120		57		67		62		2.07
	135	12			4.6		61		73		49		2.23
1		25			126	292		58		1		99			2.24
	149	126.		38		340		71		359	2.		54
	1		70	1		30		90	257		87		2.4
170			138	253		56	2.		32
	158				144	980	304	453			2.25
1		33	1		47		42		356
		114	144.		66		94		50		2.61
		140	13	8.53		91	588	2.49
	145	13	8.8		271		270						2.3
	150	140.0	617	1		63
	1		78				142	493		64	307		2.8
163	149.	239		194				2.9
		172		151	159			355		2.99
178		155	993		274		84				3.21
199		160		46	198		75			3.1
1		68	156		95	900
	184		174	255		204		3.08
	162	1		76		234		37			3.07
	146	17						3.38		236		301				3.06
	166		167		97		89	408		3.29
	171	16		7.5		278		312	327	3.16
		180		168	222			264		86		3.37
	193	170.5		77		81	889						3.5
	181		175		622	495	911	3.58
1		83		176	249	799	672		3.31
	218		177	599	397	383		3.04
	230	1		85	679	179		2.44
		242	19				4.5	389		432	1.53
	209	20				4.03		117	946	1.3
	191		205	280		435	1.13
	202	224		548		0.91
			196	779	388	0.83
	195	238		107
19					5.7		390	448		1.69
19			5.8	351		259
		235			203	868		100	671	2.63
			229		210	480		137		2.67
	243	21								3.8	558	449		2.7
	219	700		467	592		2.82
	272		228	560		374	742
		237	23							7.2	299	594
	211	23					7.1	639		407				2.95
	231	589	152		2.84
	201		221	567	1292	208
	217	370		337
	188		214	296	701
20	9.4		103	161								4.04
	227	214.5	282		128		4.05
	65	217.0	226	369					4.15
	220	418		109	225				4.4
	302	229.		134	487	618			4.3
	293	24		3.7	590	944					3.9
253.	566	207
25				4.65	285	604		3.23
24			9.5	286	283
	240	289	2.46
23						7.9	263		122
	233	23		8.7	410	497
	267	23	7.62	352	398			2.48
	269	243.	498	822	185
	248	990	581	748			2.37
		315	25		2.87		465	399
	364		265	303	972	319
		347	285.0	427	778		2.42
	297		417		284			2.39
	300	339	382		2.29
29			5.07	787	506
28					3.4	720	2.33
282.3	698	377
	277	28		2.69	587		112
	317		281	556	616	2.28
	313		288		535	693
		318	293.	516	855	481
	298		413	385
31			3.53		422		508			2.72
		405		333	424	338	200				2.73
34	7.7	394	705	605
		306		349	157	484
340.	553	261	1587
	326	260	892
	322	514	871	416			2.92
	321	697		133
317.00	901	577				2.78
		348	32	4.8	212	165	2.74
	101		329	445	770	932	2.83
	102		334	661	607
352.0	452	928	596		2.91
374.6			362		342	877
	105			404		393		108	987	430			2.89
	106	39				5.2	189	441
	305	38			5.6	519		2.93
	336	369.50	380	564
362.	803		125		3.18
	358	330	3.32
	111		350	939	400
352.55	515	712			3.45
	113		363	351.	644	256	966				3.52
	353	153	557
	491	370.		132	458
	505	394.30	579
41				6.4		463	853				3.54
	118	41	3.95	570	710		3.47
	310		402	456	568			3.48
	384	371	254
	360	37			4.89	732				3.46
371.	917	576
	123		406		365	428	610
	396		373	474	842		3.57
	420	37		8.3	431					3.6
		472	38			6.6	354	485								3.84
	127		403	490	835	887	3.81
	559	432.599	266												3.93
45				7.8	941	496
	130	45	8.9	530	797
	131	44	8.52	246	379		3.89
431.2	182	597
	425	460	683
	391	424.1	796	621
	419	417.5	437	489	717							3.92
		461	41				7.83	499	774
42			6.46	999	341			4.09
43			5.5	439	473		4.38
	139	45					5.4	951	578			4.59
	606	48			8.76	615	663
	512	409
	511	740					4.62
		143		501	970	192		4.64
	479	376	375
							4.9
147		5.3
148				5.35
	5.32
				4.72
		4.56
				4.2
154
				4.21
	4.27
			4.42
			4.73
				4.97
164		4.98
	5.1
	5.38
			5.66
			5.52
169	5.31
				5.09
		5.19
173
				5.9
			6.1
	6.12
	6.02
							6.11
				6.04
	6.44
	6.9
183			7.09
186		7.82
187	7.87
		7.13
	6.63
190					6.5
				6.8
		6.45
			6.41
			5.91
197					5.28
		4.87
		4.44
		3.86
			4.14
			4.75
206	4.94
	4.69
	4.46
	4.22
						4.01
213		3.73
	3.71
215		3.69
216			3.91
			3.98
				4.02
4.66
4.74
		4.78
223			5.41
		6.09
					6.2
				6.3
7.19
	8.01
		8.67
	8.29
232		7.22
			7.4
				7.77
		7.12
		7.96
	8.33
	8.23
241	7.55
	8.96
				8.06
244							7.46
245			7.47
	7.15
247	6.26
	5.49
250				5.61
251				5.23
252		5.34
		6.13
			6.42
5.96
		5.48
258	5.44
	4.88
262		4.86
								5.14
5.08
268	4.92
		4.35
4.67
273
		4.54
275
276					5.02
279							5.81
		6.16
	6.07
	6.29
287
			6.73
7.01
290				7.08
291			7.85
	7.99
			8.64
294		9.08
295	9.35
	9.32
9.48
9.46
	9.61
	9.06
	9.2
9.52
10.26
1	1.7
1	1.79
12.04
308	1	2.86
309	15.2
13.2
311	8.58
		7.07
314			9.1
10.27
316	1	1.62
13.73
15.49
15.02
320		14.7
1	3.36
13.69
323	16.3
324		14.73
325	1				4.95
	15.5
328	13.54
	10.8
		10.85
331	12.28
332	13.48
1	2.68
12.7
335	12.09
12.47
	11.3
8.68
7.92
		7.71
				8.07
	7.94
343			7.86
344			8.1
345	8.35
346			8.21
	8.19
	8.79
9.34
		9.09
357
	9.69
9.83
9.87
361	10.12
	10.4
	10.3
9.74
	8.61
366
367			7.76
368					8.27
7.95
	7.48
372	6.95
	7.14
	7.16
378
				7.06
381			6.56
		6.06
6.15
			6.21
		5.83
386					5.53
387			5.21
		5.18
	5.43
392		5.59
			5.64
395
	5.67
			5.69
401
					5.77
411
412							7.3
414
415
8.82
8.65
				8.4
		8.15
421	7.88
423	7.75
	7.64
	7.69
426	7.63
					7.74
429
		7.73
433
434	7.45
7.36
436			7.17
438				6.74
			6.22
440				5.94
442					5.65
443			5.46
444					5.57
		5.58
446			5.33
447			5.22
		4.99
450	4.07
451
454	3.75
455			3.63
3.66
457
	3.13
459
462		3.22
464
466
		2.96
468
469
470		3.02
471
475		2.98
476	3.25
477
478	3.68
	4.33
482			4.48
483
5.29
486
			5.71
488
5.73
492		5.47
		5.42
494
	5.36
500				4.83
502
503
504
	5.15
			5.05
507
509				5.03
510			4.91
	5.01
513
					5.16
		4.93
517
518
520
521
522
523		5.04
524
525
526
527
528
529
531				4.61
532
533	4.41
534	4.39
	4.34
536
537
538			4.29
539
540			4.57
541	4.55
542
543		4.68
544
545
546
547
	5.55
549
550
551						5.79
552
554
555
			6.17
561
562				3.87
563				3.62
		3.49
565	3.51
2.64
2.16
569	1.87
571	1.65
572		1.73
573
574		1.72
575
1.68
1.63
580	1.58
1.23
582	1.19
583	1.47
584	1.49
585
586
1.75
1.81
1.99
2.12
591	2.26
593
595		2.51
598	2.62
600			2.88
601
602
603
3.19
608				3.39
609
611
612					3.76
613
614		3.42
		3.61
619
620
	4.06
623
624		3.11
625
626	2.58
627
628	2.15
629
630	2.08
631
632
633
634	3.67
635
636			3.72
637
638
		3.97
640
641			4.25
642	4.52
643
645
646
647		4.85
648	5.12
649
650
651
652
653
654
655
656				3.41
657	3.43
658
659
660
662
3.35
664			3.33
665
666
667		3.59
668
669
670
673
674
675
676
677
678
680
681
682
684
685
686
687
688
689
690
691
692	3.77
3.82
694
695
696
699				4.12
702
703
704
706
707
708					4.08
709
711		4.13
713		4.11
714
715
716	4.16
4.24
718
719		4.45
4.79
721
722
723
724
725
726
727		5.26
728			5.68
729
730
731
733
734
735	4.47
736
737
738
739	5.06
741
743
744
745
746
747
				5.75
749
750		5.82
751			5.54
752
753				5.39
754
755
756
757
758			6.32
759		6.38
760		6.25
761	6.51
762			6.83
763		7.31
764		7.29
765
766	7.57
767			7.65
768
769		8.24
		7.79
771
772								7.39
773
		7.84
775		7.56
776
777		7.11
		6.96
780
781
782
783
784
785			5.97
786
788	6.35
789			5.89
790			5.56
791
792	5.27
793
794	5.25
795
	5.88
798
5.78
800
801
802		6.05
804	6.01
805					6.27
806				6.58
807				6.86
808	6.78
809
810		6.84
811			7.54
812	7.89
813					7.25
814			6.81
815
816
817
818	6.76
819	7.35
820		8.05
821
823					8.41
824	8.66
825
826
827		7.61
828
829		7.23
830		6.65
831
832
833
834
7.72
836	8.16
837
838			7.81
839
840			7.43
841		6.99
843
844
845		7.27
846
847
848
849		6.66
850			6.24
851
852
854
6.47
856	6.31
857		6.55
858			6.39
859
860		6.79
861
862
863
864
865
866	7.67
867
869
870
8.54
872
873
874	8.62
875	9.04
876		9.33
878	9.29
879	9.38
880	9.43
881		9.42
882		8.95
883	8.94
884	9.14
885
886			10.9
	11.1
888	10.71
10.88
890		12.84
891	14.05
12.02
893	9.44
894		8.91
895	9.27
896	10.63
897			11.5
898	12.01
899	13.31
	13.65
1	3.01
902
903	13.51
904	14.09
905	15.08
906	14.29
907		15.15
908
909	16.22
910
13.11
912	13.66
913	14.64
914
915		14.1
916	1			4.18
13.77
918	14.48
919
920	12.62
921	12.03
922	10.62
923	9.98
924	9.88
925	9.64
926	9.91
927	9.84
9.76
929	9.66
930	10.32
931
933	11.07
934	10.87
935	10.96
936	11.13
937	10.93
938		11.05
11.59
940	11.98
1	2.75
942	13.18
943	13.08
12.5
945		12.34
	11.85
947
948	10.56
949	10.43
950		10.55
952	10.49
953	9.75
954	9.05
955			9.18
956	9.31
957	9.37
958		9.25
959	8.88
960
961
962
963
964
965
	7.41
967
968					6.49
969	6.62
971
	6.43
973
974
975
976	7.32
977				8.02
978
979
	8.03
981
982		8.75
983
984			8.13
985
986			7.38
988
989
	8.22
991	8.44
992
8.57
994		8.43
995			8.72
996		9.11
997
998
1000
1001		8.98
1002		8.37
1003
1004
1005		8.26
1006
1007
1008
1009
1010		8.39
1011	8.63
1012	8.78
1013	8.69
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030	6.23
1031
1032
1033
1034				5.72
1035	6.18
1036				5.93
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046		4.58
1047
1048
1049
1050				4.53
1051			4.43
1052	4.36
1053			4.17
1054
1055
1056
1057
1058
1059
1060		5.99
1061	6.34
1062
1063			6.48
1064
1065			6.69
1066	7.04
1067	7.44
1068
1069	7.66
1070
1071		6.89
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090		6.08
1091
1092
1093
1094				6.03
1095
1096	6.61
1097
1098
1099
1100
1101		5.98
1102	5.84
1103	5.76
1104			5.74
1105
1106
1107
1108
1109
1110
1111
1112			5.24
1113
1114
1115
1116
1117
1118
1119	5.11
1120
1121
1122
1123	5.62
1124
1125
1126
1127		5.92
1128		6.14
1129
1130
1131		6.53
1132	6.36
1133		6.77
1134
1135				6.28
1136
1137
1138	5.85
1139
1140
1141	4.77
1142	4.71
1143
1144
1145	4.51
1146
1147		4.31
1148
1149
1150	3.14
1151
1152
1153	3.56
1154	3.55
1155
1156
1157
1158
1159
1160	2.52
1161		2.32
1162
1163
1164
1165
1166	2.36
1167	2.38
1168	2.43
1169
1170
1171
1172	2.65
1173
1174
1175	2.76
1176
1177
1178			2.97
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192		3.34
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207	3.09
1208	3.05
1209
1210
1211
1212
1213
1214
1215
1216
1217		3.74
1218
1219
1220
1221		3.99
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232	4.49
1233
1234	4.28
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244		3.78
1245
1246
1247
1248	3.88
1249
1250
1251
1252
1253		3.94
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267	3.83
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282	4.23
1283
1284
1285				4.19
1286
1287
1288
1289
1290
1291
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308		4.81
1309
1310
1311
1312
1313
1314	4.84
1315
1316	4.63
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331	5.87
1332
1333
1334
1335
1336
1337
1338
1339
1340		6.19
1341
1342
1343
1344	6.57
1345	6.72
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355	7.91
1356
1357
1358		7.53
1359
1360		7.33
1361
1362
1363
1364
1365
1366
1367
1368		6.52
1369
1370
1371
1372
1373
1374
1375	5.95
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388		6.64
1389		6.71
1390	6.67
1391	6.85
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401		7.21
1402	7.51
1403		7.58
1404
1405
1406				8.04
1407
1408
1409	7.68
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420	8.14
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431	7.59
1432
1433
1434			6.87
1435
1436
1437
1438	7.37
1439
1440	7.28
1441
1442
1443		7.34
1444	7.52
1445
1446
1447
1448
1449
1450
1451
1452	8.46
1453
1454
1455	8.42
1456
1457	8.81
1458	9.01
1459
1460
1461	9.12
1462
1463
1464
1465
1466	9.03
1467
1468
1469	10.65
1470	10.39
1471
1472	12.41
1473	12.75
1474	11.47
1475	10.18
1476		9.78
1477	10.25
1478
1479		11.51
1480	11.75
1481	12.68
1482
1483	12.57
1484	13.19
1485	13.12
1486	13.68
1487
1488	13.47
1489	14.28
1490	14.94
1491	15.32
1492
1493	13.39
1494	13.72
1495	14.59
1496	14.43
1497	13.86
1498	13.87
1499	13.62
1500	14.3
1501	13.95
1502	13.06
1503
1504	10.91
1505
1506	10.54
1507	10.46
1508	10.72
1509	10.51
1510
1511	10.38
1512
1513		11.38
1514
1515	11.65
1516	11.54
1517	11.69
1518	11.83
1519	11.67
1520	11.84
1521	12.32
1522	12.63
1523	13.41
1524	13.56
1525	13.36
1526	12.72
1527	12.52
1528	12.16
1529
1530
1531
1532
1533	11.86
1534	11.43
1535
1536	10.16
1537	10.31
1538	10.33
1539	10.37
1540	10.24
1541
1542		9.26
1543	9.19
1544
1545		7.78
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561	8.45
1562
1563
1564
1565		8.86
1566	8.99
1567
1568
1569
1570
1571
1572	8.92
1573
1574
1575
1576
1577
1578
1579
1580	9.17
1581	9.36
1582
1583
1584		8.28
1585
1586		8.11
1588
1589
1590
1591
1592		8.47
1593	8.59
1594
1595
1596	8.48
1597
1598
1599	8.89
1600
1601
1602	8.08
1603	8.09
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613	7.42
1614
1615	7.03
1616
1617
1618
1619
1620	7.26
1621
1622		6.59
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642	6.97
1643	7.18
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668	6.91
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742	4.26
1743
1744
1745
1746

&C&A
&CPage &P

Sheet1

Month
Airline Sales
Airline Sales/Fcst vs Time (Months)

Sheet2

Time (Months)
Interest Rate (%)
US Interest Rates vs. Time

Statistics

144

Regression 1

1759935.3217999

142

143

Standard Error

72.1572204126 102.6685371631

3.6660430975003E-61

2.4769884434 2.842081567

SUMMARY OUTPUT
	Regression
Multiple R	0.9240281154
R Square	0.8538279581
Adjusted R Square	0.8527985775
	Standard Error	46.0628868742
Observations
ANOVA
df	SS	MS	Significance F
	1759935.3217999	829.4580035686		3.6660430975003E-61
Residual	301294.115700105	2121.7895471838
Total	2061229.4375
Coefficients	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	87.4128787879	7.7173075344	11.3268621729	1.36527204108949E-21		72.1572204126		102.6685371631
X Variable 1	2.6595350052	0.0923439633	28.8003125603		2.4769884434		2.842081567

IST 230 Exercise on Forecasting Recurrences

Use Excel or OpenOffice to forecast the Airline Sales and US Interest Rate data in the data spreadsheet (data.xls or data.ods). You’ll see that I started this for you for the airline data. There’s also a tutorial in the exercise directory which you may find helpful.

Smooth the data using exponential smoothing. If you think the time series does not have a long-term trend up or down, use simple exponential smoothing. If you think the time series does have a long-term trend up or down, use trend-adjusted exponential smoothing for each of these time series. Note that this requires that you first do the simple exponential smoothing, and then compute the trend as the smoothed difference between successive forecasts. Finally, compute the trend-adjusted forecast as the sum of these two. Your objective should be to get a fairly smooth (hence the term “smoothing”) curve that follows the trend and runs as closely as possible through the middle of the data. This will usually give an RMS error as small as possible without following the cycles.

Add some type of cyclical adjustment. My usual approach is to compute a ratio the actual data and the exponentially smoothed forecast – a so-called “cyclical index” (or if it follows a calendar-year pattern, it’s called a “seasonal index”.) You’ll find additional info in the tutorial.

For both time series (airline-sales and interest-rates), forecast 12 periods ahead (h=12), starting from the beginning of your data, and extrapolate your forecast 12 periods past where you have actual data, based on your latest trend estimate and your cyclical index. Note you won’t be able to assess the forecast for these, since you don’t have actual data.

Use the exponential smoothing formulae given below.

Compute the mean-squared error by summing the squared differences between the forecast values and the actual values.

Your deliverable a one or two page executive summary of your results in a PDF document entitled exercise8-flastname , where flastname is your first initial and last name (e.g., for me, dmudgett). In your summary, present

– A brief one-sentence statement of your problem plus an explanation of how you
selected alpha, beta, and figured out the cyclical pattern. (1 paragraph)

– Graphs of your forecast and actual data for your final choices of alpha and beta
(should be just 2 graphs)

– A table of RMS forecast error for at least 3 choices of alpha and beta, for each time
series.

Here are the exponential smoothing formulae (carefully read the explanation of n-periods ahead forecasting).

Simple Exponential Smoothing:

The simple exponential smoothing forecast is given by:

F(t+1) = F(t) + alpha*(A(t) – F(t)) = (1-alpha)*F(t) + alpha*A(t) (1)

where

F(t+1) = the one-period-ahead forecast for period t+1

F(t) = the forecast for period t

A(t) = the actual value in period t

alpha is known as the smoothing constant, a constant in the range (0,1)

Trend-Adjusted Exponential Smoothing:

The trend-adjusted exponentially smoothed forecast is given by:

FT(t+1) = F(t+1) + T(t) (2)

where F(t+1) = the one-period-ahead exponentially smoothed forecast for period t+1

T(t) = the trend correction in period t, which is given by:

T(t) = beta*(F(t)-F(t-1)) + (1-beta)*T(t-1) (3)

and beta is the smoothing constant for the trend estimation, also a constant in the range (0,1).

NOTE: Equation (2) is *not* a recurrence. Compute the recurrences (1) and (3) first, then use equation (2).

You can think of beta exactly the same way as you think about alpha for simple exponential smoothing. In fact, it’s probably a good idea to set beta = alpha, unless you have a reason to think it should be different.

Comment: alpha and beta are usually determined by experimentation, to see what works best on the time-series you’re forecasting. To give you some idea what the affect is, think about it this way:

alpha/beta equivalent n-period moving average

0.1 about 12 periods

0.2 about 6 periods

0.3 about 4 periods

0.5 about 3 periods

0.7 about 2 periods

These are just rough equivalences, but give you an idea how fast the smoother reacts to a major change in the data pattern.

Final point: Extrapolating the smoothed forecast more than one time-period ahead.

For simple exponential smoothing, forecasting more than one period ahead is no big deal. Just use the h-period-ahead formula (h is called the forecasting horizon):

F(t+h) = F(t+1) = (1-alpha)*F(t) + alpha*A(t) (4)

You also compute the trend exactly as before – use equation (3) completely unmodified. But equation (2) may change, depending on how many periods you want to extrapolate the trend. For example, if you want to forecast h-periods ahead and also extrapolate the trend h periods ahead, then change equation (2) to

FT(t+h) = F(t+h) + h*T(t) (5)

As a practical matter, a forecaster may not want to extrapolate the trend all the way through the forecast horizon. Let’s say, for example that you want to forecast 6 periods ahead, and aren’t sure the trend will continue all 6 periods. Let’s say a retailer wants to use a forecast to decide how much of a product they needed 6 months down the line. If they extrapolate the trend all 6 months and it doesn’t happen, they’ll wind up with too much stock. In this case it may make sense to be more conservative and extrapolate the trend *less then* the total number of periods in the forecast horizon.

Either way – once you decide how many periods you want to extrapolate the forecast – call that integer m – then the trend-adjusted forecast is given by this slightly modified equation 5:

FT(t+k) = F(t+k) + min(k,m)*T(t) (6)

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