data Daily closing price of IBM stock, Jan 1, 1980 to Oct 8, 1992 64 37 62 5 63 5 63 37 63 12 67 37 65 37 67 5 67 66 87 70 12 70 69 25 69 62 69 69 71 25 70 62 70 12 71 70 37 70 37 68 62 69 37 69 12 69 62 68 37 67 12 67 25 66 75 68 87 69 37 68 12 67 62 67 62 67 67 25 66 25 66 65 12 65 62 63 37 63 5 62 75 63 25 62 61 12 61 61 62 62 25 61 37 60 12 59 87 59 12 59 62 58 87 58 25 56 75 54 12 54 75 53 57 5 55 87 55 75 54 87 55 5 54 87 54 87 53 37 53 5 54 62 54 25 53 5 53 52 51 12 51 25 51 25 51 25 53 87 53 5 53 75 55 37 54 5 54 87 54 87 54 37 54 55 37 54 87 55 12 53 87 52 12 52 25 53 25 53 52 5 53 53 37 52 75 52 87 53 75 54 75 54 75 55 25 56 37 54 37 55 37 56 56 37 58 12 57 57 12 56 75 57 5 58 58 58 87 60 37 58 87 59 62 57 87 57 87 58 87 59 37 59 75 59 25 59 75 58 75 59 5 60 62 61 12 61 12 62 61 75 61 5 61 62 62 75 65 63 75 64 25 65 62 65 12 66 65 12 64 87 64 75 64 37 65 65 12 65 5 65 25 65 12 64 75 64 25 65 62 65 75 65 25 66 75 66 75 66 75 68 87 68 75 66 37 66 66 87 67 5 67 37 67 12 66 75 65 87 65 65 5 65 5 66 12 67 5 66 62 66 5 64 25 66 12 65 75 66 37 65 87 66 12 65 62 67 25 66 5 67 68 12 66 87 67 5 66 12 64 62 63 75 64 12 65 62 65 37 66 25 68 67 75 70 70 12 69 25 70 5 69 75 70 37 68 62 68 68 5 68 67 75 65 87 66 62 66 12 66 62 65 5 65 12 66 62 67 5 67 5 68 5 66 5 67 12 67 25 67 12 70 87 71 25 71 62 71 62 72 71 37 72 70 37 70 37 69 5 68 75 68 75 68 12 66 5 67 87 68 68 37 67 5 66 37 66 12 63 75 64 5 65 75 64 5 66 25 65 25 63 62 64 62 67 12 67 25 68 5 68 5 69 68 68 62 67 87 67 87 69 12 70 25 71 5 69 5 67 62 67 37 66 75 66 5 66 75 65 75 66 87 66 62 65 65 12 65 25 65 65 65 5 64 62 64 87 64 37 63 37 64 25 64 5 64 64 25 63 5 63 62 62 61 75 61 75 61 75 62 62 61 25 60 5 62 62 12 64 25 64 75 64 25 63 62 62 5 61 87 62 61 87 63 62 5 62 25 64 63 25 64 62 64 25 64 75 63 75 63 12 62 87 62 25 63 87 63 12 61 75 61 5 62 37 63 62 12 61 12 60 37 60 60 61 75 60 75 60 12 59 12 59 62 60 60 61 62 60 5 60 37 60 37 60 62 61 12 60 37 59 62 58 62 59 12 58 37 57 75 57 87 57 37 57 87 56 56 75 56 75 56 62 56 37 56 75 56 62 56 37 55 62 55 5 55 5 56 12 57 87 58 58 75 60 25 59 87 59 12 58 58 37 58 25 57 37 57 75 59 59 37 59 87 58 87 58 5 57 62 57 5 58 12 59 37 59 5 59 58 75 57 87 57 87 56 87 56 87 56 87 57 57 56 62 56 62 56 37 55 87 56 87 56 62 56 62 56 12 55 37 55 5 55 12 54 87 55 55 62 55 37 55 37 56 37 56 12 56 5 58 25 58 37 56 87 57 57 87 58 75 57 87 58 25 58 56 75 56 62 57 57 25 56 87 55 87 55 87 55 5 55 25 55 87 55 12 56 55 75 54 62 54 54 54 54 12 54 75 55 75 55 87 55 5 54 87 54 12 54 54 87 54 37 54 5 54 62 53 62 54 87 53 12 54 12 54 55 55 12 54 25 55 55 5 55 87 55 5 53 75 52 5 52 25 51 25 51 25 51 62 51 37 51 12 51 50 62 50 75 49 5 49 51 5 53 53 52 37 51 5 50 87 51 62 52 25 52 5 52 75 51 62 51 51 50 37 50 37 50 62 50 5 52 25 52 62 52 62 54 5 54 5 54 12 53 54 54 62 53 5 53 25 53 87 54 62 54 37 53 37 53 62 54 12 55 75 56 75 56 37 56 87 56 12 57 57 56 5 56 57 25 56 87 56 87 58 25 57 56 75 56 75 55 62 56 75 57 5 57 37 58 87 59 75 61 87 61 61 5 62 5 61 62 62 61 75 62 62 64 37 63 62 62 62 63 62 75 63 25 63 25 61 5 61 37 61 87 61 25 61 61 62 25 62 25 62 75 62 37 60 37 60 87 62 61 62 61 87 62 12 61 59 87 59 12 58 5 57 5 58 62 59 58 25 57 87 58 87 57 37 57 37 57 87 57 62 59 12 59 5 59 12 59 75 59 59 87 60 37 59 75 61 61 5 61 5 61 75 61 75 62 25 62 25 62 12 62 63 62 64 62 64 5 63 75 63 25 63 87 64 37 65 75 66 65 12 64 87 64 12 64 25 64 5 64 75 65 12 64 87 64 75 63 87 64 37 63 87 63 62 63 62 62 87 62 75 62 5 63 62 62 62 25 61 87 62 62 61 5 61 5 61 75 62 61 5 60 25 60 58 25 58 12 58 75 59 62 59 12 59 75 59 5 58 5 58 37 58 62 59 87 61 12 60 5 60 25 61 12 61 12 60 62 60 62 60 60 60 37 61 61 5 62 5 63 5 64 5 66 87 66 75 66 75 65 87 67 37 67 5 66 62 66 5 65 87 65 5 64 87 65 5 65 62 66 75 66 65 62 63 87 63 12 63 25 63 62 37 62 25 62 87 62 87 66 65 75 66 68 25 69 37 68 37 69 12 69 12 69 69 87 70 5 69 75 71 37 72 12 72 12 72 5 72 5 71 87 71 37 72 87 72 5 73 37 74 12 74 12 74 75 76 75 75 5 76 37 76 76 62 76 25 74 75 73 37 74 12 74 5 75 12 79 80 37 81 120 83 500 82 75 83 500 80 120 79 5 83 000 82 12 84 5 83 75 82 75 80 82 81 62 79 62 79 87 81 5 81 870 85 25 84 84 250 83 37 85 75 84 370 85 37 83 37 82 250 82 12 84 83 75 83 500 80 500 81 25 82 75 82 75 83 37 82 5 86 5 86 5 88 25 88 25 92 75 95 120 94 5 93 620 92 75 93 870 91 000 89 370 90 75 93 37 92 5 95 75 93 500 93 620 93 620 96 370 95 120 96 62 96 120 96 25 93 000 95 620 96 000 97 620 96 62 98 000 98 000 98 120 98 620 99 12 99 250 99 62 98 500 97 87 94 62 93 870 96 000 95 120 97 620 97 370 98 870 97 96 000 95 500 97 370 97 12 95 120 94 5 96 370 96 370 99 98 870 98 37 98 37 98 620 98 620 97 12 98 75 100 87 100 37 99 101 75 102 12 101 87 102 25 103 100 87 102 25 100 25 100 75 100 5 100 87 98 870 98 870 99 870 100 75 100 12 102 12 102 25 102 12 102 37 102 87 104 25 101 75 101 75 102 37 102 102 62 103 25 103 87 106 62 107 108 75 109 5 110 12 112 62 111 5 115 12 115 117 25 114 5 117 12 116 117 25 117 115 115 5 116 5 115 75 117 62 117 5 116 87 115 5 115 5 116 5 115 25 114 111 75 111 25 110 75 112 5 114 62 115 62 114 37 113 113 111 25 111 87 113 5 114 115 75 113 12 113 37 113 75 114 25 117 25 118 121 121 75 121 12 122 123 12 123 5 123 25 123 120 87 118 120 120 25 121 121 119 25 121 62 119 75 121 5 123 25 120 87 121 25 122 120 12 119 87 120 87 125 37 125 5 124 37 125 87 125 75 123 37 122 120 37 120 75 120 5 121 25 119 5 119 75 119 118 12 118 12 117 75 118 5 120 5 121 62 123 121 62 122 5 121 25 119 37 116 5 116 25 117 87 119 117 37 119 5 118 62 119 75 119 75 121 87 122 37 122 25 122 25 121 5 122 122 87 121 75 123 12 123 75 123 87 123 124 62 126 62 128 62 127 87 128 75 127 87 126 87 128 12 129 87 131 12 132 12 132 25 134 25 133 132 62 133 62 131 75 131 75 128 75 130 5 129 75 127 128 25 128 5 128 129 37 128 126 75 126 75 126 5 123 87 122 25 122 12 122 25 123 62 125 25 126 87 127 5 124 5 124 75 123 37 123 5 125 25 123 120 87 120 87 121 119 25 121 62 117 37 118 87 117 87 118 62 117 62 119 25 119 122 25 124 122 12 120 87 119 5 120 87 121 25 121 75 123 62 122 87 123 62 123 62 124 62 124 25 123 12 122 122 121 75 123 87 124 25 123 5 123 37 122 25 122 25 121 25 119 120 5 121 120 5 118 87 117 116 12 116 75 115 37 114 62 114 87 113 87 114 12 113 75 114 37 110 75 108 75 110 5 107 75 109 109 87 109 110 87 109 62 109 62 109 75 109 75 109 37 109 12 109 25 111 25 112 87 110 110 25 111 5 112 111 109 25 107 62 108 5 108 110 25 110 62 111 112 113 75 114 114 75 114 112 75 112 25 112 25 113 87 115 37 113 75 114 112 111 12 111 108 75 109 75 109 62 110 108 87 112 5 111 25 112 75 112 25 111 12 110 75 110 75 109 87 110 75 111 87 113 25 113 37 113 75 116 5 116 25 115 62 112 75 113 25 114 25 112 87 112 37 111 25 112 25 113 112 110 37 109 37 108 62 108 107 62 106 75 107 25 107 25 106 62 107 25 107 75 107 87 108 25 105 37 105 75 105 62 105 75 104 25 104 5 104 37 100 75 99 5 101 75 102 75 108 12 105 87 105 25 105 5 104 12 103 12 105 25 105 75 105 87 107 12 107 12 106 37 105 75 107 25 107 12 105 25 104 37 105 25 107 25 108 12 107 5 107 106 5 106 12 105 37 107 12 108 25 109 108 37 110 75 112 87 114 87 120 25 121 75 121 37 118 62 122 75 121 122 25 121 25 122 123 12 123 123 25 125 87 124 25 125 125 62 124 75 125 37 124 25 123 62 123 75 123 75 122 5 122 37 123 37 121 62 122 75 122 122 87 126 12 126 75 127 62 126 62 125 87 126 124 25 123 75 124 124 62 125 87 124 25 122 62 121 75 121 25 121 87 121 25 121 12 120 87 121 87 120 25 121 87 122 5 121 25 121 87 126 125 126 126 62 126 75 125 12 124 12 124 62 126 75 124 62 125 75 125 87 126 75 127 5 125 75 124 87 123 25 124 122 75 123 12 122 75 121 5 120 121 25 121 121 122 62 122 87 124 75 123 12 122 121 75 120 120 25 117 87 118 5 117 12 117 87 119 12 118 75 117 75 118 75 119 123 12 123 122 37 123 75 124 25 124 25 123 75 123 12 123 5 123 12 123 12 121 120 119 75 120 37 119 75 120 37 123 75 122 75 124 75 124 62 123 87 123 62 124 12 128 25 129 62 132 5 133 12 133 25 134 62 137 136 62 136 37 135 62 137 62 136 5 135 37 137 12 137 25 133 5 131 75 133 75 132 131 37 131 37 131 87 133 75 133 75 132 87 133 87 134 5 133 12 134 135 87 135 135 132 87 130 12 129 62 131 5 131 87 130 128 87 128 62 128 37 130 5 129 37 128 75 127 75 124 5 125 37 126 37 126 62 127 128 12 127 75 126 5 127 127 123 87 124 62 125 87 127 128 129 25 129 75 129 62 127 5 127 62 127 75 129 128 62 128 87 127 12 126 25 126 5 125 62 125 37 125 124 75 125 75 127 12 127 87 130 12 130 25 128 128 5 129 130 75 133 133 132 37 131 37 130 87 130 87 130 37 129 75 128 128 62 129 12 129 75 128 5 129 75 127 37 127 62 126 120 75 118 62 121 12 120 12 119 87 119 37 118 75 119 37 120 87 122 123 12 124 123 75 124 87 124 37 123 75 123 75 124 5 123 5 121 25 123 123 75 124 25 125 5 128 5 129 37 128 75 129 25 129 87 129 37 129 12 130 25 132 129 62 130 5 131 37 132 5 131 62 131 25 129 37 129 25 128 37 127 37 127 12 126 62 126 75 126 37 125 75 125 87 127 5 128 37 126 62 126 87 128 25 128 25 128 127 5 126 62 126 62 127 12 127 25 128 37 129 12 129 37 127 87 127 25 127 12 127 5 128 37 126 75 127 5 128 126 75 128 12 126 75 124 123 75 123 5 123 87 126 62 124 37 124 37 123 5 124 25 124 37 124 25 124 37 125 62 128 128 129 37 128 5 127 62 127 5 129 12 130 5 128 62 128 129 37 130 5 131 12 129 87 130 87 132 25 133 37 132 87 132 12 132 5 134 87 135 37 135 136 62 136 75 138 5 138 25 139 140 25 139 5 139 139 140 25 140 25 139 75 137 87 138 62 141 87 140 75 141 75 144 62 146 62 149 148 62 150 25 152 25 152 5 152 87 153 75 154 25 154 152 75 152 75 153 25 155 5 158 25 155 5 155 5 152 154 37 154 37 155 62 148 87 150 37 148 5 149 149 25 152 62 156 150 87 149 37 149 5 144 25 148 150 149 62 151 75 151 149 5 151 5 154 62 154 154 87 155 155 75 157 25 156 75 155 155 62 156 25 156 25 158 87 157 5 159 12 159 75 158 87 157 5 158 25 155 87 150 87 151 25 149 75 148 37 146 75 146 148 5 152 149 150 5 150 37 150 87 152 37 151 62 150 12 149 148 87 148 147 62 149 12 149 12 151 5 149 12 151 149 62 148 25 149 5 152 149 25 150 25 149 75 152 37 152 62 154 75 154 62 152 62 154 87 153 155 25 157 62 159 5 161 37 159 37 156 25 156 87 154 12 155 62 154 25 151 25 149 149 75 150 148 87 148 12 145 37 144 87 144 12 146 25 144 5 143 37 143 87 143 87 147 151 37 152 87 152 37 152 25 152 75 150 150 75 149 87 146 75 148 37 149 12 148 149 62 148 5 146 87 146 37 145 146 87 145 12 146 5 147 5 146 75 147 146 5 148 62 149 5 149 149 145 25 144 62 144 25 145 75 143 12 139 25 135 75 133 12 132 131 75 131 87 133 62 133 12 133 12 134 12 132 37 131 25 132 25 132 5 131 37 131 62 130 87 131 12 131 130 131 132 5 134 134 25 133 12 135 37 136 75 139 25 138 138 139 141 12 141 12 140 138 75 138 75 136 136 12 139 37 140 37 143 75 143 87 144 12 139 25 137 37 137 138 25 137 62 138 37 136 87 138 62 139 25 137 5 135 75 135 5 134 25 134 5 133 133 130 87 133 5 128 12 127 5 122 123 62 122 120 87 123 121 12 121 87 120 25 121 37 120 37 121 5 121 37 122 12 120 75 120 87 122 123 62 126 37 125 87 123 12 121 37 121 5 123 12 123 75 122 75 119 87 121 37 122 120 5 123 123 5 123 25 124 62 127 12 126 126 127 12 127 37 129 87 128 62 127 75 126 62 127 75 126 87 128 128 126 5 127 62 128 12 126 75 125 37 125 12 123 12 120 5 122 122 122 120 62 120 87 120 120 122 123 62 123 123 5 122 87 122 25 120 37 116 5 118 62 120 120 125 25 122 75 122 75 127 37 126 25 127 25 129 37 130 128 62 128 75 132 25 133 135 12 136 87 135 75 133 75 133 37 134 132 62 134 5 134 5 138 37 138 139 12 139 62 143 37 143 141 75 139 37 139 5 138 5 138 12 139 87 140 37 139 37 138 87 142 87 141 75 144 25 144 87 144 62 147 12 146 25 147 148 62 149 75 152 87 155 62 154 150 75 152 37 150 12 151 12 148 149 5 149 37 146 5 147 62 144 145 147 62 148 12 150 87 150 150 150 157 154 25 154 25 151 75 154 25 155 25 158 62 160 12 160 62 163 166 87 166 75 164 87 163 75 163 165 37 166 25 165 12 160 62 161 156 75 156 62 157 25 156 62 156 62 160 62 161 162 62 160 159 62 157 25 160 25 161 37 160 161 25 159 87 157 75 157 62 156 62 159 62 162 161 37 160 87 161 37 163 25 164 87 166 12 167 87 166 5 165 37 162 5 164 164 25 164 25 165 12 163 87 166 75 166 25 167 169 87 167 5 168 168 12 167 164 62 162 75 161 37 160 12 161 25 162 12 161 25 161 5 162 37 161 159 25 159 37 160 25 163 164 37 166 75 169 170 37 173 37 173 25 174 5 172 87 172 62 174 75 174 75 174 62 172 167 87 166 25 166 5 168 37 162 87 162 5 161 62 160 75 160 75 157 75 157 25 157 75 161 12 162 5 157 5 156 62 157 87 155 37 150 5 155 75 154 25 155 87 156 152 75 150 5 150 75 154 62 155 25 156 5 151 152 75 151 5 147 37 149 5 148 75 145 25 140 12 135 104 115 122 75 120 120 75 112 118 118 120 25 122 5 124 5 121 87 120 62 123 75 119 5 118 12 117 75 119 25 122 25 121 62 120 117 75 118 37 115 62 117 75 117 5 118 5 117 62 117 62 114 87 110 75 111 25 110 5 106 5 107 75 109 5 111 5 114 111 110 12 115 12 116 118 87 115 5 117 25 118 25 118 25 119 5 119 87 119 87 115 75 116 117 115 5 115 5 120 75 121 87 122 87 123 37 114 87 117 75 115 25 116 12 115 87 119 117 75 111 75 110 37 111 110 5 113 25 111 87 112 75 113 75 112 37 110 109 87 109 108 62 108 107 75 109 12 111 62 111 25 112 112 113 75 112 75 112 5 113 37 115 25 115 37 116 12 114 5 115 75 117 5 117 117 25 116 5 116 87 117 25 117 62 116 62 113 75 115 5 115 5 114 115 25 114 87 114 25 113 75 113 12 111 62 110 107 108 12 105 87 105 12 107 62 107 62 107 5 107 5 109 62 109 25 111 62 113 37 113 62 116 12 111 37 114 12 114 113 12 112 87 112 37 113 75 113 87 114 25 114 113 12 113 37 114 12 114 62 113 25 111 25 110 5 110 5 110 87 108 75 108 75 110 37 111 62 110 12 108 87 110 5 109 5 108 75 109 25 108 5 108 25 108 108 112 5 113 12 112 62 113 25 114 37 113 75 116 62 115 37 116 117 118 75 118 87 117 12 117 5 117 12 120 25 123 125 75 125 12 125 127 127 127 37 126 62 126 62 129 25 126 87 127 62 126 37 126 75 125 25 125 25 124 75 125 5 125 12 124 124 121 75 120 12 122 37 122 62 121 12 123 75 125 75 126 25 125 75 125 75 124 25 123 87 123 12 121 62 118 87 119 118 115 12 115 75 114 62 114 62 112 75 110 75 110 87 112 75 111 62 112 25 113 37 112 87 111 5 110 25 114 114 112 87 111 75 112 12 114 37 113 62 115 114 62 113 75 114 25 112 75 112 75 113 25 112 112 87 112 25 113 25 113 87 115 37 115 37 114 87 114 5 113 87 114 116 116 12 118 75 117 5 118 62 119 75 121 12 124 37 122 37 124 87 124 37 123 12 122 87 121 62 120 25 120 75 122 62 124 122 87 122 25 120 25 119 12 119 75 119 87 119 75 116 62 117 12 117 5 115 5 115 5 115 75 115 62 117 118 118 116 118 87 119 37 118 5 118 25 119 25 120 87 122 12 121 119 37 120 37 120 5 120 75 120 5 121 121 25 122 75 123 25 123 122 5 123 37 123 37 123 12 122 5 123 25 121 87 121 87 121 122 5 122 25 121 87 122 121 12 122 25 123 12 123 37 124 25 123 87 125 124 25 123 62 122 37 124 25 123 25 126 126 87 128 75 130 62 129 62 128 37 127 75 126 62 127 62 127 126 37 125 126 124 75 127 125 87 125 25 125 25 125 5 123 37 123 37 121 12 121 62 121 5 119 87 120 5 120 87 121 5 119 87 119 62 118 62 118 25 118 87 117 87 117 118 12 112 5 109 5 110 12 109 25 109 62 109 62 110 12 110 109 62 108 75 109 12 108 5 107 12 107 75 108 109 75 110 111 12 111 12 109 25 110 87 111 37 113 112 37 112 37 113 5 113 5 114 5 114 5 114 62 114 113 37 112 37 111 5 110 62 109 37 109 5 109 75 109 62 109 12 110 75 112 75 111 37 111 5 112 111 12 109 87 109 5 109 75 109 62 109 5 109 5 108 25 109 62 110 110 12 109 109 62 109 5 109 25 108 37 108 87 111 110 37 109 37 109 5 110 37 109 5 109 75 109 12 110 87 112 114 37 113 87 111 5 111 87 112 25 112 25 112 111 62 112 12 114 62 113 87 114 75 114 87 114 75 116 114 75 114 37 112 75 114 12 113 5 112 75 112 5 113 113 37 115 114 62 114 113 75 115 5 117 37 118 25 117 12 117 5 115 5 116 12 116 25 115 5 114 75 115 12 113 113 62 114 116 116 12 118 116 37 116 87 117 12 117 37 117 37 117 75 116 75 116 25 116 116 25 116 87 115 37 115 75 115 5 116 12 116 116 116 12 116 5 116 25 117 5 111 5 109 109 25 108 62 106 108 25 107 5 107 5 109 25 108 25 107 75 107 62 102 103 102 25 101 75 104 12 104 25 103 5 103 101 87 100 75 99 870 99 5 100 25 100 97 87 98 500 96 62 97 370 96 750 97 370 98 120 98 25 98 000 97 87 97 12 99 12 98 75 99 5 100 12 100 12 100 37 99 370 98 870 97 75 97 620 97 12 99 250 99 62 98 620 98 000 96 870 96 120 97 95 87 95 120 95 93 620 94 750 94 62 95 120 95 37 95 37 94 5 94 750 94 12 94 12 94 12 98 000 98 870 100 99 750 100 37 99 370 99 99 870 97 87 98 120 100 12 98 870 99 5 98 620 96 62 96 750 97 5 95 87 96 870 97 250 97 250 98 620 97 75 98 37 99 62 101 103 12 102 75 102 5 101 12 103 25 103 75 103 75 103 5 103 5 102 87 104 87 103 25 102 62 104 103 75 103 87 104 104 87 105 105 75 105 5 107 106 5 107 75 106 25 106 25 107 108 62 109 108 107 25 106 5 105 104 12 104 5 105 87 106 106 12 105 87 106 62 106 5 106 12 105 87 105 5 105 87 106 37 107 12 107 12 110 87 111 12 109 87 109 12 109 37 109 37 108 87 109 37 108 75 107 37 109 108 109 12 109 25 110 5 110 5 110 75 111 25 112 37 114 114 25 115 62 115 37 116 12 115 37 117 117 75 118 75 118 75 116 25 116 25 119 75 120 87 120 119 5 120 37 120 75 120 62 120 62 118 75 119 87 120 25 121 37 120 87 120 37 118 75 118 75 118 119 12 116 25 116 62 116 75 118 117 62 117 5 118 5 117 62 117 62 117 37 118 119 117 75 119 12 120 121 122 37 120 75 119 87 120 117 62 115 87 114 75 114 75 113 12 111 37 112 12 111 5 111 62 109 75 108 12 103 87 103 103 12 103 25 102 103 75 104 37 105 103 75 101 37 102 25 100 5 99 96 870 100 37 104 37 103 62 103 87 101 75 101 87 101 87 102 5 102 37 102 37 105 5 105 87 107 5 107 87 105 104 37 105 12 107 25 108 37 107 87 106 75 105 106 37 105 37 104 37 106 37 109 108 25 107 37 107 87 108 25 109 5 105 37 103 5 101 100 25 99 250 99 370 100 75 105 37 107 62 107 75 106 75 108 62 108 12 106 62 106 106 37 105 5 107 25 108 37 107 87 107 87 106 5 107 5 110 25 113 112 113 5 112 62 113 62 114 75 113 37 114 12 114 12 112 62 113 87 113 5 112 37 112 113 62 113 37 114 75 114 62 111 5 112 5 113 37 112 87 114 37 112 87 111 25 111 5 113 5 112 75 113 75 113 87 113 87 113 87 113 5 113 62 113 37 113 113 112 12 112 5 112 12 110 25 109 106 87 108 37 108 12 106 75 107 5 109 12 115 75 117 62 119 62 118 119 25 121 122 62 124 25 124 5 127 126 75 126 87 127 37 128 5 129 87 128 5 129 5 132 37 132 75 134 62 135 12 137 37 137 37 139 5 137 87 135 5 133 25 133 62 129 75 130 87 128 75 131 130 62 133 5 132 87 132 5 131 25 129 12 127 129 12 128 12 126 62 127 87 115 12 114 12 111 87 111 62 113 5 113 87 113 87 113 87 113 87 112 25 113 25 113 12 113 5 112 62 113 111 25 111 25 110 62 108 5 106 87 109 25 109 87 109 75 109 5 108 5 108 37 108 25 108 107 37 104 87 103 104 105 87 103 62 103 37 101 87 103 75 105 87 103 37 105 87 106 102 75 104 12 103 75 103 25 101 5 104 62 104 62 105 105 105 12 104 12 104 37 106 12 106 37 105 103 25 103 102 102 37 102 25 101 5 100 62 100 37 99 5 100 87 101 99 12 99 62 98 37 99 370 98 620 97 620 97 12 98 37 99 12 98 120 98 120 98 620 100 5 99 5 98 75 98 25 99 370 99 250 96 62 96 25 98 25 100 5 102 100 62 100 75 100 5 100 5 101 37 101 62 101 25 101 25 100 25 101 100 37 99 99 250 98 870 98 620 99 98 500 98 37 96 5 95 95 75 95 620 94 370 94 870 95 120 95 25 94 5 95 500 96 870 96 870 98 000 98 620 98 25 99 370 101 99 62 101 37 104 102 87 103 62 104 87 105 37 104 87 104 25 104 62 105 87 105 104 87 102 25 103 62 102 62 101 12 99 98 25 98 870 98 000 97 250 99 750 99 870 101 25 104 25 101 62 99 870 100 37 100 25 98 500 98 500 98 25 98 000 98 25 98 870 100 25 98 25 98 37 97 96 62 96 62 99 870 100 25 100 12 99 370 98 870 100 96 25 97 250 96 5 95 500 95 37 94 750 95 120 97 87 94 250 94 250 92 5 92 250 91 25 90 90 25 89 85 120 86 25 87 75 88 620 88 000 86 870 85 500 86 25 84 870 85 75 88 25 88 120 88 120 88 25 89 370 90 37 89 89 90 25 90 37 92 250 94 62 92 370 91 25 90 87 90 25 92 370 95 37 95 500 96 370 95 25 92 87 95 620 93 500 93 120 93 75 93 25 91 870 91 750 90 91 370 92 250 91 25 89 870 89 250 89 750 90 25 92 12 90 75 89 750 89 750 89 62 89 62 89 750 89 870 89 370 88 25 88 25 87 75 86 870 87 5 88 25 87 370 86 62 86 370 87 370 87 87 87 75 89 12 89 62 88 75 87 620 87 5 85 87 86 120 85 620 85 25 85 120 84 12 83 25 83 37 83 500 82 87 82 250 81 750 83 120 82 370 84 870 85 120 86 000 87 5 88 25 88 500 88 620 88 620 89 5 90 90 120 89 250 88 120 88 120 88 500 88 120 90 75 90 620 92 5 93 870 93 37 93 75 93 120 93 870 93 500 92 370 91 750 91 62 90 75 92 5 91 62 91 5 91 750 91 750 90 500 90 87 91 120 90 75 90 620 90 620 89 750 89 90 37 91 000 90 620 90 120 92 93 000 94 12 93 120 92 620 94 5 95 95 25 95 87 97 620 97 12 97 250 98 620 97 87 98 25 96 870 96 870 98 120 96 750 97 87 97 5 97 620 97 87 97 87 99 250 100 25 95 92 87 93 75 92 5 92 75 92 75 92 370 93 75 94 370 95 37 94 750 94 870 94 250 91 62 88 25 87 12 88 500 88 000 88 37 88 120 88 37 88 620 88 000 86 62 85 87 86 000 85 120 87 12 86 370 87 370 87 250 86 62 87 620 87 5 86 62 86 000 86 000 86 120 86 870 87 5 88 000 88 37 85 500 83 120 83 75 83 500 84 370 83 25 83 620 82 5 82 250 82 75 81 000 80 75 80 120 78 5 78 87 79 75 78 37 79 37 5 Download the data dailyibm dat from Canvas and save it to your working directory in R Then, read it in using the command ibm scan( dailyibm dat , skip 1) This series is the daily closing price of IBM stock from Jan 1, 1980 to Oct 8, 1992 (a) (b) (C) (d) (E) (f) Make a plot of the data and an ACF plot of the data Does the time series appear to be stationary Explain Interpret the ACF plot in this situation Difference the data Plot this di'erenced data, and make an ACF plot What is your opinion of whether the series is stationary after differencing Another option for attempting to obtain stationary data when there is something similar to an exponential trend is to take the logarithm Use the R command 1030 to take the logarithm of the data Plot this transformed data Does the transformed data appear stationary Explain Perhaps some combination of differencing and the logarithmic transform will give us stationary data Why would log (diff (113m)) not be a very good idea Try the opposite, difference the log transformed data difflogiban diff(log(ibm)) Except for a few extreme outliers, does this transformation succeed in creating stationary data Delete the extreme outliers using the following command difflogibm difflogibm difflogibm c 1 Plot this data and the ACF for this data Sometimes with very long time series like this one, portions of the series exhibit different behavior than other portions Break the series into two parts using the following commands difflogibm1 difflogimelz500 difflogibm2 difflogimeSOl length(difflogibm) Plot both of these and create ACF plots of each Do you notice a difference between these two sections of the larger time series Assume the model for the data that we have called difflogibm2 is of the following form (it 6 E where e1, ,e,, are normal white noise with variance 0' Is this reasonable from what you now know of this time series How would you estimate 6 and 0 Give the estimates

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data Daily closing price of IBM stock, Jan 1, 1980 to Oct. 8, 1992 64.37 62.5 63.5 63.37 63.12 67.37 65.37 67.5 67. 66.87 70.12

data

Daily closing price of IBM stock, Jan 1, 1980 to Oct. 8, 1992

64.37

62.5

63.5

63.37

63.12

67.37

65.37

67.5

67.

66.87

70.12

70.

69.25

69.62

69.

71.25

70.62

70.12

71.

70.37

68.62

69.37

69.12

69.62

68.37

67.12

67.25

66.75

68.87

69.37

68.12

67.62

67.

67.25

66.25

66.

65.12

65.

62.

63.37

63.5

62.75

63.25

62.

61.12

61.

61.62

62.25

61.37

60.12

59.87

59.12

59.62

58.87

58.25

56.75

54.12

54.75

53.

57.5

55.87

55.75

54.87

55.5

54.87

53.37

53.5

54.62

54.25

53.5

53.

52.

51.12

51.25

53.87

53.5

53.75

55.37

54.5

54.87

54.37

54.

55.37

54.87

55.12

53.87

52.12

52.25

53.25

53.

52.5

53.

53.37

52.75

52.87

53.75

54.75

55.25

56.37

54.37

55.37

56.

56.37

58.12

57.

57.12

56.75

57.5

58.

58.87

60.37

58.87

59.62

57.87

58.87

59.37

59.75

59.25

59.75

58.75

59.5

60.62

61.12

62.

61.75

61.5

61.62

62.75

65.

63.75

64.25

65.62

65.12

66.

65.12

64.87

64.75

64.37

65.

65.12

65.5

65.25

65.12

64.75

64.25

65.62

65.75

65.25

66.75

68.87

68.75

66.37

66.

66.87

67.5

67.37

67.12

66.75

65.87

65.

65.5

66.12

67.5

66.62

66.5

64.25

66.12

65.75

66.37

65.87

66.12

65.62

67.25

66.5

67.

68.12

66.87

67.5

66.12

64.62

63.75

64.12

65.62

65.37

66.25

68.

67.75

70.

70.12

69.25

70.5

69.75

70.37

68.62

68.

68.5

68.

67.75

65.87

66.62

66.12

66.62

65.5

65.12

66.62

67.5

68.5

66.5

67.12

67.25

67.12

70.87

71.25

71.62

72.

71.37

72.

70.37

69.5

68.75

68.12

66.5

67.87

68.

68.37

67.5

66.37

66.12

63.75

64.5

65.75

64.5

66.25

65.25

63.62

64.62

67.12

67.25

68.5

69.

68.

68.62

67.87

69.12

70.25

71.5

69.5

67.62

67.37

66.75

66.5

66.75

65.75

66.87

66.62

65.

65.12

65.25

65.

65.5

64.62

64.87

64.37

63.37

64.25

64.5

64.

64.25

63.5

63.

62.62

61.75

62.

61.25

60.5

62.

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56.

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56.62

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58.37

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69.37

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80.37

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94.5

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96.62

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100.37

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101.75

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99.870

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102.12

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102.37

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102.37

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102.62

103.25

103.87

106.62

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108.75

109.5

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111.5

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114.5

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115.5

116.5

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117.62

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112.5

114.62

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114.37

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123.5

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120.25

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119.75

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126.62

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129.87

131.12

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133.62

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128.75

130.5

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123.62

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123.87

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116.75

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114.62

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108.75

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112.37

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110.37

109.37

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105.37

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105.62

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104.5

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105.87

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105.87

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107.5

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118.62

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162.87

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157.75

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82.5

82.250

82.75

81.000

80.75

80.120

78.5

78.87

79.75

78.37

79.37

5. Download the data \"dailyibm.dat\" from Canvas and save it to your working directory in R. Then, read it in using the command ibm=scan( " dailyibm . dat " , skip= 1) This series is the daily closing price of IBM stock from Jan 1, 1980 to Oct 8, 1992. (a) (b) (C) (d) (E) (f) Make a plot of the data and an ACF plot of the data Does the time series appear to be stationary? Explain. Interpret the ACF plot in this situation. Difference the data Plot this di'erenced data, and make an ACF plot. What is your opinion of whether the series is stationary after differencing? Another option for attempting to obtain stationary data when there is something similar to an exponential trend is to take the logarithm. Use the R command 1030 to take the logarithm of the data. Plot this transformed data Does the transformed data appear stationary? Explain. Perhaps some combination of differencing and the logarithmic transform will give us stationary data. Why would log (diff (113m)) not be a very good idea? Try the opposite, difference the log transformed data difflogiban=diff(log(ibm)). Except for a few extreme outliers, does this transformation succeed in creating stationary data? Delete the extreme outliers using the following command: difflogibm=difflogibm[difflogibm > -c 1] Plot this data and the ACF for this data Sometimes with very long time series like this one, portions of the series exhibit different behavior than other portions. Break the series into two parts using the following commands: difflogibm1= difflogimelz500] difflogibm2= difflogimeSOl:length(difflogibm)] Plot both of these and create ACF plots of each. Do you notice a difference between these two sections of the larger time series? Assume the model for the data that we have called difflogibm2 is of the following form: (it = 6 + E: where e1, . .. ,e,, are normal white noise with variance 0'". Is this reasonable from what you now know of this time series? How would you estimate 6 and 0? Give the estimates