# Data set C

Document Sample

```					Seasonality
Often times the observations in a time series data set represent less than a year. In such situations it is useful
to control for seasonal effects in order to understand how the data varies over time.
Time series data often is organized into less than year long intervals. When working with such data it is worth
knowing what seasonality exists within the year. For example, the seasonality in sales across the year would be
very helpful for managing inventory requirements. Regression analysis easily calculates seasonal variations
using dummy variables. This tutorial introduces how to set up and interpret these variables.
This tutorial employs three data sets that have been created to show seasonality in linear and constant growth
settings. (These data sets have been created in a fashion similar to the data set in the growth modeling tutorial.
The setup for creating each data set has been hidden in rows 1-5 and columns B-E on each data sheet.) Data
set A has a linear trend and the data shows no seasonality (and is thus denoted DA-L,non-s). Data set B has
linear trend and has seasonality (DB-L,s). Data set C has nonlinear trend and seasonality (DC-NL,s). The thee
data worksheets have been reduced so that you do not have to scroll down when running regressions. Each
includes data worksheet includes a graph of the data.
For each data set regressions are run with and without seasonal dummy variables. In each data set, three
seasonal dummy variables are created using Q3 as the base. Seasonal dummy variables are easy to construct if
you use the repetitive aspect of Excel to advantage. Code only one year's worth of dummy variables (for example
see cells I7:K10 in the DA-L,non-s worksheet). At the start of the second year, reference the first (note the
equations in cells I11:K11), then drag these equations to the end of the data set.
The difference between data set A and B is that B is seasonal while A is not. This is easily seen by comparing
the two graphs on pages DA-L,non-s and DA-L,s. B's graph has regular peaks and valleys, while A's does not.
Note that this same regularity of peaks and valleys occurs with C but C is nonlinear in nature (it is based on a
constant growth model). Two regressions are shown for A and B (the first letter R stands for regression (and D
stands for data set) and the second letter stands for the data set that the model is estimated from), and four
regressions are shown for C. The discussion in each regression (and across regressions) focus on the line fit
plot and residual plot together with a few numbers that have been highlighted and formatted to ease comparison.
Coefficient interpretation is put off until data set C when linear and nonlinear models can be directly compared.
2
Data set A, DA-L,non-s) R always increases with additional variables so the equation with seasonal
2                2
dummies necessarily has higher R but adjusted R declines and Standard Error of Estimate (SEE) increases
with the addition of the seasonal dummies. The increased explanatory power from the extra independent
variables did not overcome the harm caused by loss of degrees of freedom. The t-statistics for each of the
seasonal dummy variables is not statistically significant. Finally, there is no significant change in residual plot
between equations, neither exhibits a pattern. The best fit equation in this instance is RA,L.
2                 2
Data set B, DB-L,s) The seasonality in data set B dominates the data set. As a result, R and adjusted R
increase dramatically and SEE decreases dramatically. T-statistics for each coefficient are strongly significant in
the seasonal equation and the strong seasonal pattern of the residuals in the RB,L equation vanishes in the
RB,L+s equation. The best fit equation is RB,L+s.
Before examining data set C it is worth considering why the Durbin-Watson statistic in RB,L did not show that
we had serial correlation. The Durbin-Watson test is for first order serial correlation (i.e.. consecutive quarters'
errors are correlated). In this instance, we do not have first order serial correlation, we have 4th order serial
correlation because we have quarterly seasonal data. For further help in understanding time series correlations,
consult an econometrics text.
Data set C, DC-NL,s) The seasonality in data set C is also clear, as a result the model without seasonal
dummies, RC,L, does not provide a very effective fit. The residual plot has a strong regular saw-tooth pattern.
Heteroskedasticity is also present here (and in RC,L+s) as the spread of the residual plot is an increasing
function of time. (This was not present in the first two residual plots because the random error terms that were
introduced into those data sets was additive in nature; in data set C, the random error term is multiplicative.
(Compare the equation in cell F7 of DB-L,s and DC-NL,S. The random error component is in the (hidden)
column E.)) The seasonal dummies take care of the saw-tooth pattern in RC,L+s but the nonlinearity is visible in
the residual plot. The Durbin-Watson statistic also supports this view as it exhibits positive serial correlation
(dlower=1.08=dw).
By contrast, the Durbin-Watson statistic for the RC,NL+s equation exhibits no serial correlation (dw=2.10 in
cell D28). (A second nonlinear equation, RC,NL+s123, is shown that only differs in the choice of base quarter.
This allows us to more finely understand the interpretation of coefficients within the nonlinear model.) Before we
discuss the interpretation of coefficients in this model it is useful to consider coefficient interpretation in the linear
Coefficient Interpretation
The coefficients in these basic time series models are readily interpretable if you remember a simple rule. If
the dependent variable is Q, then coefficients represent absolute levels of change in Q. By contrast, if the
dependent variable is ln(Q), then coefficients represent relative levels of change in Q.
Consider the quarterly dummy variable coefficients in the RB,L+s worksheet. Each coefficient represents the
best guess estimate of how much larger or smaller this quarter is than the base quarter, ceteris paribus. That is,
the first quarter is 67 smaller than the third, the second is 50 smaller, and the fourth is 54 larger, ceteris paribus.
(Ceteris paribus here refers to holding the trend constant (i.e.. the time variable) but of course, as we move from
one quarter to the next this does not happen.)
The coefficient of time in the linear model provides a best guess estimate to the increase (or decrease) in
number of units per time period, ceteris paribus. The notion of ceteris paribus is very important in this context.
Using the equation from the RB,L+s worksheet, the coefficient of time is 1.1. This says that each quarter, expect
an increase in Q of just over one unit, ceteris paribus. This is not the same as saying that in moving from quarter
1 to quarter 2 you increase Q by 1.1 units because as you move from quarter 1 to quarter 2 two other parts of the
equation change; the Q1 dummy variable turns off and the Q2 dummy variable turns on (notice it is easiest to
think of dummy variables as light switched that are either on or off). As a result, the correct statement for moving
from quarter 1 to quarter 2 is: we expect Q to increase by 18.2 units in moving from the first to the second
quarter of a year (18.2=-(-67.4)+1.1+(-50.3) = Q1turnedoff+1quarter+Q2turnedon). The coefficients in this model
refer to absolute units of Q; the same is not true in the logarithmic model.
The time trend in the nonlinear model represents the deseasonalized growth rate per period. As discussed in
c
the growth rate tutorial, the growth rate in Q per unit of time is given by e - 1 where c is the coefficient of time in
the ln(Q) equation. Since we are interested in annual growth rates, we need to annualize the quarterly
information given in this model. The table in cells C1:J8 of the RC,NL+s123 page provide various interpretations
of quarterly and annual growth rates based on different notions of compounding.
The coefficient of a quarterly dummy variable shows the difference in size of that quarter in relative terms
relative to the base quarter. (With the linear model the coefficient of a quarterly dummy variable shows the
difference in size of that quarter in absolute terms relative to the base quarter). Two interpretations for each
coefficient are given in the table in cells A23:G26 of the RC,NL+s page. Both are based on notions of relative
The two nonlinear seasonal regressions only differ in their choice of base quarter. RC,NL+s uses Q3 and
RC,NL+s123 uses Q4. By clicking back and forth between the pages you can see that many aspects of the two
regressions are identical. This should not be surprising in light of the discussion in the dummy variable tutorial
that compared two equations for Canadian bond sales that only differed in their choice of base for their dummy
variable (peace versus war). Both equations had identical summary statistics and ANOVA tables. The only
difference lay in the magnitudes and signs of the dummy variables. The same is true here.
The reason RC,NL+s123 was included in this tutorial is to provide a second way of interpreting seasonal
dummy variables. Since the 4th quarter is the peak quarter, the dummy variables for the other quarters can be
interpreted as how much slack there is relative to peak in each quarter (see the interpretation of these coefficients
in the table in cells A23:G26 of the RC,NL+s123 page). Such information would be useful for inventory
managers, marketing managers, or for a maintenance manager wishing to coordinate scheduled maintanence
time
Quarter Quantity (quarters)   Q1       Q2       Q4
90.01       97          0        1        0        0
90.02       88          1        0        1        0   160
90.03     104           2        0        0        0
90.04     116           3        0        0        1
91.01     116           4        1        0        0
91.02     122           5        0        1        0
140
91.03       84          6        0        0        0
91.04     105           7        0        0        1
92.01     119           8        1        0        0   120
92.02       98          9        0        1        0
92.03     103          10        0        0        0
92.04     128          11        0        0        1   100
93.01     104          12        1        0        0
93.02     103          13        0        1        0
93.03     106          14        0        0        0
93.04       94         15        0        0        1   80
94.01     110          16        1        0        0
94.02     131          17        0        1        0
94.03     119          18        0        0        0   60
94.04     105          19        0        0        1
95.01     117          20        1        0        0         0   4   8
95.02     117          21        0        1        0
95.03     135          22        0        0        0
95.04     122          23        0        0        1
96.01     122          24        1        0        0
96.02     120          25        0        1        0
96.03     146          26        0        0        0
96.04     136          27        0        0        1
97.01     152          28        1        0        0
97.02     122          29        0        1        0
97.03     147          30        0        0        0
97.04     115          31        0        0        1
Quantity

12   16     20   24   28   32
SUMMARY OUTPUT                                                          Quantity                      time (quarters) Line Fit P
Regression Statistics                                                   160
Multiple R 0.661874                                                     140
R Square     0.4381                                                     120
SEE           12.52                                                      80
Observations       32                                                    60
40
ANOVA                                                                    20
df        SS       MS        F   Significance F            0
Regression           1 3666.638 3666.638 23.38815   3.7E-05                 0         4       8        12
Residual            30 4703.199 156.7733                                                               time (quarters)
Total               31 8369.837
Residuals                   time (quarters) Residual P
Coefficients
Standard Error t Stat    P-value Lower 95%Upper 95% 30 95.0%
Lower     Upper 95.0%
Intercept     97.79     4.325032     22.6     2.13E-20 88.9545 106.6203 20 88.9545 106.6203
time (quarters)1.16     0.239725      4.8      3.7E-05 0.669759 1.648927 0.669759 1.648927
10
0
0         4       8     12
-10
RESIDUAL OUTPUT                                                           -20
2.04 dw                            -30
time (quarters)
Predicted Quantity
Observation          Residuals     9574.879 numerator
1 97.78739 -0.78971
2 98.94673 -10.7236       98.68143
3 100.1061       4.3365   226.8055
4 101.2654 14.49932       103.2829
5 102.4248 13.55874       0.884683
6 103.5841 18.74723       26.92039
7 104.7434 -20.5793       1546.578
8 105.9028      -1.2446   373.8314
9 107.0621 11.8881        172.4677
10 108.2215 -10.0885       482.9698
11 109.3808 -6.28286       14.48276
12 110.5402 17.36417       559.1817
13 111.6995 -8.16861       651.9226
14 112.8588 -9.63514       2.150706
15 114.0182 -7.75326       3.541479
16 115.1775 -21.3568       185.0575
17 116.3369 -6.01612       235.3377
18 117.4962 13.54426       382.6085
19 118.6556 0.692975       165.1556
20 119.8149 -14.4698       229.9106
21 120.9742 -4.24415       104.5645
22 122.1336 -4.83599       0.350276
23 123.2929 12.13349       287.9632
24 124.4523 -2.30511       208.4732
25 125.6116 -3.47319       1.364398
26 126.771 -6.90303         11.7638
27 127.9303 17.79182       609.8357
28 129.0896 6.56709        125.9946
29 130.249 21.50757 223.2178
30 131.4083 -8.95739 928.1136
31 132.5677 14.04689 529.197
32 133.727   -18.851 1082.27
time (quarters) Line Fit Plot

12     16    20     24   28    32
time (quarters)

time (quarters) Residual Plot
Upper 95.0%

12    16    20    24   28    32

time (quarters)
SUMMARY OUTPUT                                                               Quantity                      time (quarters) Line Fit P
Regression Statistics                                                        160
Multiple R 0.679139                                                          140
R Square     0.4612                                                          120
SEE           12.92                                                           80
Observations       32                                                         60
ANOVA                                                                         40
df        SS       MS                 F   Significance F       20
Regression           4 3860.417 965.1043           5.77853 0.00171             0
Residual            27 4509.419 167.0155                                           0       4       8       12
Total               31 8369.837                                                                            time (quarters)

Standard Error
Coefficients                t Stat                               Lower 95.0%
P-value Lower 95%Upper 95% Residuals Upper 95.0% (quarters) Residual P
time
Intercept     99.29     6.064914        16.4      1.52E-15 86.84704 111.7354 86.84704 111.7354
30
time (quarters)1.18     0.249266         4.7      6.38E-05 0.666295 1.689199 0.666295 1.689199
Q1             1.27     6.480927        0.20      0.845947 -12.0264 14.56911 20
-12.0264 14.56911
Q2            -4.14      6.46653       -0.64      0.527784 -17.4046 9.131827 10
-17.4046 9.131827
Q4            -4.29      6.46653       -0.66      0.512564 -17.5595 8.976932 -17.5595 8.976932
0
0       4       8     12
-10
-20
-30
time (quarters)
RESIDUAL OUTPUT                      1.992763 dw
Predicted Quantity
Observation          Residuals        8986.203 numerator
1 100.5626 -3.56488
2 96.33256 -8.10939          20.65259
3 101.6467 2.795875          118.9249
4 98.53316 17.23157          208.3894
5 105.2735 10.70995          42.53153
6 101.0435 21.28778          111.8904
7 106.3577 -22.1936          1890.627
8 103.2441 1.41404           557.3188
9 109.9845 8.965691          57.02743
10 105.7545 -7.62154          275.1363
11 111.0687 -7.97071           0.12192
12 107.9551 19.94919          779.5209
13 114.6955 -11.1646          968.0697
14 110.4655 -7.24182          15.38845
15 115.7797 -9.51473          5.166123
16 112.6661 -18.8454          87.06205
17 119.4065 -9.08576          95.25129
18 115.1765 15.86397          622.4887
19 120.4906 -1.14211          289.2068
20 117.3771      -12.032      118.5904
21 124.1175      -7.3874      21.57265
22 119.8875      -2.5899         23.016
23 125.2016 10.22478          164.2162
24 122.0881 0.059061          103.3419
25 128.8285 -6.69006           45.5506
26 124.5985 -4.73056          3.839643
27 129.9126 15.8095            421.894
28   126.7991 8.857649 48.32826
29   133.5395 18.21708 87.59895
30   129.3095 -6.85854 628.7865
31   134.6236 11.99095 355.3033
32   131.5101   -16.634 819.3905
time (quarters) Line Fit Plot
Q1 Residual Plot

Residuals
40                                              Q2 Residual Plot
20
0                                                         Q4 Residual Plot
-20 0                40
Residuals
0.5                1                                    1.5
-40                  20
0                   40

Residuals
-20 0                 20       Q1
0.5                          1                                      1.5
-40                    0                                                             Q1 Line Fit Plot
-20 0        0.2            0.4             0.6               0.8                    1      1.2
12     16   20          24         28               32                   -40                    Q2
time (quarters)                                                                                                                                                   Q2 Line Fit Plot

Quantity
200       Q4
150
100
Upper 95.0% (quarters) Residual Plot
time                                                                                                         50                                                              Quantity
0                                                             Q4 Line Fit Plot

Quantity
200
0                150
0.5      1        1.5                    Predicted Quantity
100
50
0

Quantity
Q1 200
0    150
0.5        1                        1.5
100
50
0    Q2
0        0.5                       1
12     16     20           24     28               32
Q4

time (quarters)
Quantity
Q4 Line Fit Plot
Predicted Quantity
Quantity
Predicted Quantity
Quantity
1.5        Predicted Quantity
time
Quarter    Quantity                Q1       Q2       Q4
(quarters)
90.01         37            0        1        0        0
90.02         48            1        0        1        0   240
90.03
90.04
104
167
2
3
0
0
0
0
0
1
220
91.01         54            4        1        0        0   200
91.02
91.03
80
84
5
6
0
0
1
0
0
0
180
91.04        158            7        0        0        1   160
92.01
92.02
54
55
8
9
1
0
0
1
0
0
140
92.03        103           10        0        0        0   120
92.04        183           11        0        0        1
93.01         36           12        1        0        0
100
93.02         58           13        0        1        0    80
93.03        106           14        0        0        0
93.04        151           15        0        0        1    60
94.01         41           16        1        0        0    40
94.02         84           17        0        1        0
94.03        119           18        0        0        0    20
94.04
95.01
165
45
19
20
0
1
0
0
1
0
0
95.02         69           21        0        1        0         0   4   8
95.03        135           22        0        0        0
95.04        184           23        0        0        1
96.01         48           24        1        0        0
96.02         70           25        0        1        0
96.03        146           26        0        0        0
96.04        199           27        0        0        1
97.01         75           28        1        0        0
97.02         71           29        0        1        0
97.03        147           30        0        0        0
97.04        180           31        0        0        1
Quantity

8   12    16    20   24   28   32
SUMMARY OUTPUT                                                           Quantity                      time (quarters) Line Fit
250
Regression Statistics
Multiple R 0.300881                                                      200
R Square     0.0905
150
SEE           50.23                                                      100
Observations       32
50

ANOVA                                                                      0
df        SS       MS        F   Significance F                 0       4       8        12
time (quarters)
Regression           1 7533.908 7533.908 2.986226 0.094258
Residual            30 75686.59 2522.886                                  Residuals                    time (quarters) Residual
Total               31 83220.5
100

Standard Error t Stat
Coefficients                       P-value Lower 95%Upper 95%          Upper 95.0%
Lower 95.0%
50
Intercept     76.02     17.35011     4.38     0.000133 40.59014 111.4574 40.59014 111.4574
time (quarters)1.66     0.961671     1.73     0.094258 -0.30216 3.625828 -0.30216 3.625828
0
0       4       8     12
-50
RESIDUAL OUTPUT
2.13 dw                            -100
time (quarters)
Predicted Quantity
Observation          Residuals     161175.5 numerator
1 76.02375 -39.0261
2 77.68559 -29.8624       83.97257
3 79.34742 25.09515       3020.334
4 81.00926 86.25548       3740.586
5 82.67109 -29.0876       13304.02
6 84.33293      -4.0016   629.3071
7 85.99476 -1.83064       4.713055
8 87.6566 70.50159        5231.951
9 89.31843 -35.1682       11166.11
10 90.98027 -36.4473       1.636013
11 92.6421 10.45585        2199.904
12 94.30394 89.10038       6184.962
13 95.96578 -59.6349       22122.18
14 97.62761 -39.6039       401.2401
15 99.28945 6.975484       2169.639
16 100.9513 50.36941       1883.033
17 102.6131 -61.8924       12602.71
18 104.275 -20.0345        1752.083
19 105.9368 13.41174        1118.65
20 107.5986 57.24645       1921.481
21 109.2605 -64.5304       14829.59
22 110.9223 -42.0247        506.505
23 112.5841 22.84229       4207.726
24 114.246 69.40119        2167.732
25 115.9078 -68.1694       18925.66
26 117.5696 -47.7017       418.9254
27 119.2315 26.49065       5504.506
28   120.8933 78.26342 2680.42
29   122.5551 -47.5986 15841.25
30    124.217  -53.366 33.26347
31   125.8788 20.73575 5491.075
32   127.5406 52.83538 1030.386
time (quarters) Line Fit Plot

12    16    20     24   28     32
time (quarters)

time (quarters) Residual Plot

Upper 95.0%

12    16   20     24   28     32

time (quarters)
SUMMARY OUTPUT                                                           Quantity                      time (quarters) Line Fit Plo
Regression Statistics                                                   250
Multiple R 0.97217
R Square 0.9451                                                         200
SEE          13.01
Observations      32                                                    100
ANOVA                                                                    50
df       SS       MS       F   Significance F
0
Regression          4 78652.4 19663.1 116.2199 1.32E-16
0       4       8        12    16    20
Residual          27 4568.095 169.1887
time (quarters)
Total             31 83220.5
Standard Error t Stat
Coefficients                                                  Lower 95.0%
P-value Lower 95%Upper 95% Residual Upper 95.0% (quarters) Residual Pl
time
Intercept    101.29 6.104244        16.6                                  30 s
1.09E-15 88.76634 113.8161 88.76634 113.8161
time (quarters)1.1     0.250883      4.2      0.000263 0.537978 1.567515 0.537978 1.567515
20
Q1            -67.4    6.522955    -10.3      7.05E-11 -80.7626 -53.9947 -80.7626 -53.9947
Q2            -50.3    6.508465     -7.7      2.64E-08 -63.6156 -36.9071 10
-63.6156 -36.9071
Q4             54.3    6.508465      8.3      5.87E-09 40.97946 67.68797 40.97946 67.68797
0
0       4       8      12    16    20
-10
18.2
-20
-30
RESIDUAL OUTPUT                    1.92   dw                                                             time (quarters)
Predicted Quantity
Observation         Residuals     8760.612 numerator
1 33.9126 3.085118
2 52.0826 -4.25939       53.94186
3 103.397 1.045875       28.14589
4 158.783 8.481575       55.28962
5 38.1235 15.45995       48.69779
6 56.2935 24.03778       73.57912
7 107.608 -23.4436       2254.478
8 162.994 -4.83596       346.2428
9 42.3345 11.81569       277.2775
10 60.5045 -5.97154       316.3857
11 111.819 -8.72071       7.557937
12 167.205 16.19919       621.0015
13 46.5455 -10.2146       697.6898
14 64.7155 -6.69182        12.4102
15 116.03 -9.76473         9.44278
16 171.416 -20.0954       106.7235
17 50.7565 -10.0358       101.1971
18 68.9265 15.31397       642.6084
19 120.241 -0.89211        262.637
20 175.627     -10.782    97.81058
21 54.9675 -10.2374       0.296629
22 73.1375     -4.2399        35.97
23 124.452 10.97478       231.4867
24 179.838 3.809061       51.34759
25 59.1785 -11.4401       232.5356
26 77.3485 -7.48056       15.67765
27 128.663 17.0595        602.2145
28   184.049 15.10765 3.80973
29   63.3895 11.56708 12.53563
30   81.5595 -10.7085 496.2031
31   132.874 13.74095 597.7776
32    188.26 -7.88404 467.6405
me (quarters) Line Fit Plot
Q1 Residual Plot

40
Q2 Residual Plot

Residuals
20
0
Plot
-20 0 Q4 Residual 0.4
40 0.2

Residuals
0.6             0.8        1      1.2
-40    20
0
Residuals    40                                                                     Q1
20                       -20 0    0.2                                0.4              0.6       0.8     1          1.2
0                       -40                                           Q1 Line Fit Plot
20     24   28    32               -20 0               0.2                0.4             0.6            0.8              1
Q2        1.2
-40
ime (quarters)                                                                                                                              Q2 Line Fit Plot

Quantity
300       Q4
200
me (quarters) Residual Plot                                                                 100                                                        Quantity
0                                                       Q4 Line Fit Plot

Quantity
300
0               200
0.5      1        1.5               Predicted Quantity
100
0

Quantity
Q1 300
0    200
0.5        1                   1.5
100
0    Q2
0        0.5                  1
20    24   28    32
Q4

time (quarters)
1.2

Quantity
Q4 Line Fit Plot
Predicted Quantity
Quantity
Predicted Quantity
Quantity
1.5        Predicted Quantity
time
Quarter   Quantity   ln(Q)                Q1   Q2   Q4
(quarters)
90.01       52      3.946         0      1    0    0
90.02       74      4.310         1      0    1    0
90.03       83      4.414         2      0    0    0
600
90.04      147      4.987         3      0    0    1
91.01       57      4.034         4      1    0    0
91.02       79      4.372         5      0    1    0
500
91.03      116      4.750         6      0    0    0
91.04
92.01
161
77
5.080
4.340
7
8
0
1
0
0
1
0
400
92.02       75      4.316         9      0    1    0
92.03
92.04
133
190
4.891
5.247
10
11
0
0
0
0
0
1
300
93.01       68      4.226        12      1    0    0
93.02
93.03
120
134
4.790
4.899
13
14
0
0
1
0
0
0
200
93.04      213      5.361        15      0    0    1
94.01       98      4.581        16      1    0    0    100
94.02      141      4.951        17      0    1    0
94.03      164      5.097        18      0    0    0
94.04      268      5.590        19      0    0    1      0
95.01       98      4.580        20      1    0    0
95.02      153      5.027        21      0    1    0          0   4   8
95.03      264      5.575        22      0    0    0
95.04      308      5.731        23      0    0    1
96.01      162      5.088        24      1    0    0
96.02      222      5.404        25      0    1    0
96.03      257      5.551        26      0    0    0
96.04      438      6.082        27      0    0    1
97.01      167      5.117        28      1    0    0
97.02      278      5.628        29      0    1    0
97.03      367      5.905        30      0    0    0
97.04      535      6.283        31      0    0    1
Quantity

12    16    20   24   28   32
SUMMARY OUTPUT                                                           Quantity           time (quarters) Line Fit Plo
600
Regression Statistics                                                    500
Multiple R 0.762725
400
R Square     0.5817
300
200
SEE           75.05
Observations       32                                                    100
0
0    4     8      12
ANOVA
time (quarters)
df        SS       MS            F   Significance F
Regression           1 235046.8 235046.8      41.7273 3.89E-07             Residuals         time (quarters) Residual Pl
Residual            30 168987.8 5632.927
Total               31 404034.6                                            250
200
Coefficients
Standard Error t Stat    P-value Lower 95%Upper 95% 150 95.0%
Lower      Upper 95.0%
Intercept    34.14964 25.9251 1.317243         0.19773 -18.7964 87.09569 100
-18.7964 87.09569
50
9.282287 1.436961 6.459667
time (quarters)                               3.89E-07 6.347625 12.21695 6.347625 12.21695
0
-50 0   4    8    12
-100
-150
RESIDUAL OUTPUT                                                                                 time (quarters)
1.79 dw
Predicted Quantity
Observation          Residuals     301661.8 numerator
1 34.14964 17.56501
2 43.43193 31.03957       181.5637
3 52.71421 29.88319       1.337227
4 61.9965 84.55195        2988.674
5 71.27879 -14.7754       9865.919
6 80.56108 -1.39791       178.9567
7 89.84336 25.77934       738.6033
8 99.12565 61.68121       1288.944
9 108.4079 -31.6996       8719.983
10 117.6902 -42.8308       123.9037
11 126.9725 6.088941       2393.145
12 136.2548 53.81941       2278.197
13 145.5371 -77.0645       17130.59
14 154.8194 -34.5446       1807.937
15 164.1017 -29.9891       20.75283
16 173.384 39.47225        4824.879
17 182.6662 -85.0459       15504.77
18 191.9485 -50.6154        1185.46
19 201.2308 -37.6434       168.2735
20 210.5131 57.14769       8985.346
21 219.7954 -122.243       32180.92
22 229.0777      -76.572   2085.818
23     238.36 25.3796      10394.12
24 247.6423 60.71729       1248.752
25 256.9245 -94.8915       24214.09
26 266.2068 -43.8907       2601.076
27 275.4891 -18.0232       669.1293
28   284.7714 153.0329 29260.2
29   294.0537 -127.247 78556.66
30    303.336 -25.3544 10382.03
31   312.6183 54.22911 6333.541
32   321.9005 213.4404 25348.24
time (quarters) Line Fit Plot

12    16    20     24   28    32
time (quarters)

time (quarters) Residual Plot

Upper 95.0%

12    16    20    24   28    32

time (quarters)
SUMMARY OUTPUT                                                             Quantity      time (quarters) Line Fit Plot
Regression Statistics                                                      600
Multiple R 0.925081
500
R Square     0.8558
400
SEE           46.46                                                        300
Observations       32                                                      200
ANOVA                                                                      100
df       SS       MS        F   Significance F                0
Regression          4 345762.3 86440.58 40.05156 5.57E-11                        0       4       8       12
Residual           27 58272.29 2158.233                                                                  time (quarters)
Total              31 404034.6
Residuals        time (quarters) Residual Plo
Standard Error t Stat
Coefficients                       P-value Lower 95%Upper 95%Lower 95.0% Upper 95.0%
Intercept      53.25    21.80195      2.4                                 150
0.021418 8.514727 97.98249 8.514727 97.98249
time (quarters)8.52     0.896055      9.5      4.1E-10 6.685242 10.36234 6.685242 10.36234
100
Q1            -75.41    23.29742     -3.2     0.003192 -123.208 -27.6029 -123.208 -27.6029
Q2            -38.24    23.24567    -1.65     0.111535 -85.9385 9.453791 -85.9385 9.453791
50
Q4             84.28    23.24567      3.6     0.001181 36.58218 131.9745 36.58218 131.9745
0
0       4       8    12
-50

-100
time (quarters)
RESIDUAL OUTPUT                     1.08   dw
Predicted Quantity
Observation          Residuals     62981.71 numerator
1 -22.1566 73.87126
2 23.53005 50.94145       525.7761
3 70.29619 12.30121       1493.068
4 163.0983 -16.5499       832.3839
5 11.93857 44.56484       3735.006
6 57.62522 21.53794       530.2379
7 104.3914 11.23134        106.226
8 197.1935 -36.3866       2267.471
9 46.03374 30.67455       4497.202
10 91.7204       -16.861    2259.63
11 138.4865 -5.42509       130.7805
12 231.2887 -41.2145       1280.879
13 80.12892 -11.6563       873.6844
14 125.8156 -5.54082       37.39906
15 172.5817 -38.4692       1084.275
16 265.3838 -52.5276       197.6408
17 114.2241 -16.6038       1290.524
18 159.9107 -18.5776       3.896124
19 206.6769 -43.0895       600.8297
20 299.479 -31.8182        127.0407
21 148.3193 -50.7666       359.0408
22 194.0059 -41.5002       85.86594
23 240.7721 22.9675        4156.085
24 333.5742 -25.2146       2321.519
25 182.4144 -20.3814       23.36031
26 228.1011       -5.785   213.0544
27 274.8672 -17.4013        134.939
28   367.6694   70.13498   7662.605
29   216.5096   -49.7026   14361.05
30   262.1963   15.78527   4288.663
31   308.9624   57.88496   1772.384
32   401.7645   133.5764     5729.2
time (quarters) Line Fit Plot                                                  Q1 Residual Plot

200

Residuals
100                                            Q2 Residual Plot
0
-100 0                 Q4 0.5
200 Residual Plot    1                                      1.5

Residuals
100         Q1
200

Residuals
0
100
12     16    20     24   28   32                                     -100 0             0.5Q1                                     1
Line Fit Plot
0
time (quarters)                                 -100 0   0.2                      0.4   0.6            0.8               Q2
1       1.2
600 Q4                                   Q2 Line Fit Plot

Quantity
time (quarters) Residual Plot                                                           400
Upper 95.0%                                                                                    200
0             600

Quantity
-200 0           400
0.5      1       1.5
200
0   Q1  600

Quantity
-200 0     400
0.5       1
200
0  Q2
-200 0         0.5
12    16    20    24   28   32

time (quarters)
1.5

Q2 Line Fit Plot
Quantity
Q4 Line Fit Plot
Predicted Quantity
Quantity
Predicted Quantity
1           1.5                      Quantity
Predicted Quantity
0.5           1       1.5
Q4
SUMMARY OUTPUT                                                                                            time (quarters) Line Fit P
Regression Statistics
ln(q)
Multiple R   0.9839537                                                                 8.0
R Square       0.9682                                                                  6.0
SEE            0.1148                                                                  4.0
Observations           32                                                              2.0
ANOVA
0.0
df          SS       MS     F Significance F
Regression               4   10.82449 2.70612 205.3 8.7E-20
0   4   8    12
Residual                27   0.355931 0.01318                                                         time (quarters)
Total                   31   11.18042
S
Coefficients tandard Error   t Stat   P-valueLower 95%    Upper 95% Lower 95.0%Upper 95.0%(quarters) Residual Plo
Residuals       time
Intercept        4.377       0.053882       81.2     8E-34 4.26673      4.487848174 4.266733 4.487848
0.2
time (quarters) 0.047        0.002215       21.4     2E-18 0.04283      0.051915911 0.042828 0.051916
Q1              -0.551       0.057578       -9.6     4E-10 -0.66956                  0.1
-0.433282652 -0.66956 -0.43328
Q2              -0.238       0.057451       -4.1     3E-04 -0.35609    -0.120330568 -0.35609 -0.12033
0
Q4               0.363       0.057451        6.3     9E-07 0.24466      0.480420389 0.244663 4 0.48042
0         8    12
-0.1
=exp(Qcoef)   Understanding the seasonal dummies                      =exp(Qcoef)-1
57.6% Q1 as a percent of Q3            Q1 less than Q3 by             -42.4%        -0.2
78.8% Q2 as a percent of Q3            Q2 less than Q3 by             -21.2%        -0.3
143.7% Q4 as a percent of Q3            Q4 more than Q3 by             43.7%                            time (quarters)

ObservationPredicted ln(q)Residuals    0.74646 numerator
1 3.8258669 0.119874            2.10 dw
2 4.1864535 0.123963        1.7E-05
3 4.4720347 -0.058056       0.03313
4 4.8819485 0.105408        0.02672
5 4.015355 0.018946         0.00748
6 4.3759416 -0.00443        0.00055
7 4.6615228      0.08881    0.00869
8 5.0714366 0.008767        0.00641
9 4.2048431 0.135167        0.01598
10 4.5654296 -0.249818        0.14821
11 4.8510109        0.0398    0.08388
12 5.2609247 -0.01351         0.00284
13 4.3943312 -0.167897        0.02384
14 4.7549177 0.034861         0.04111
15 5.040499 -0.141819         0.03122
16 5.4504127 -0.089796        0.00271
17 4.5838193 -0.002734        0.00758
18 4.9444058 0.006714         8.9E-05
19 5.2299871 -0.132639        0.01942
20 5.6399008 -0.05018          0.0068
21 4.7733074 -0.192915        0.02037
22 5.1338939 -0.106692        0.00743
23 5.4194751 0.155487         0.06874
24 5.8293889 -0.098122        0.06432
25 4.9627955 0.125005         0.04979
26 5.323382 0.080718          0.00196
27 5.6089632 -0.058076        0.01926
28    6.018877 0.062895 0.01463
29   5.1522836 -0.035446 0.00967
30   5.5128701 0.114685 0.02254
31   5.7984513 0.106495 6.7E-05
32   6.2083651 0.074539 0.00102
time (quarters) Line Fit Plot                                             Q1 Residual Plot

0.2           Q2 Residual Plot

Residuals
0
-0.2 0
0.2                         0.5                       1                     1.5

Residuals
-0.4           Q4 Residual Plot
0
-0.2 0                     0.5             Q1    1                        1.5
0.2

Residuals
-0.4
0
12 16 20 24 28 32                                                                        Q2           Q1         Line Fit Plot
-0.2 0         0.2        0.4         0.6          0.8             1      1.2
time (quarters)                                     -0.4
Q2 Line Fit Plot
8.000 Q4
6.000

ln(q)
4.000
Upper 95.0%(quarters) Residual Plot
time                                                                                 2.000                                                       ln(q)
0.000         8.000
6.000                                     Q4 Line Fit Plot

ln(q)
0      4.000 0.5              1            1.5         Predicted ln(q)
2.000
0.000
0          Q18.000
0.5
6.000                 1           1.5

ln(q)
4.000
2.000      Q2
0.000
12   16    20    24   28       32                                                                                         0              0.5         1

Q4

time (quarters)
ln(q)
Q4 Line Fit Plot
Predicted ln(q) ln(q)
1.5       Predicted ln(q)
ln(q)
1          1.5         Predicted ln(q)
SUMMARY OUTPUT                            Examining different notions of compounding
Regression Statistics       Formula          Rate          Assumption implicit in using this formula
Multiple R     0.983954           coef time    4.74% quarterly growth rate, no compounding
R Square        0.9682     exp(coef time)-1    4.85% quarterly growth rate, continuous compounding
Adj. R^2        0.9634          4*coef time   18.95% APR, no compounding
SEE             0.1148    (1+coef time)^4-1   20.34% APR, compounded quarterly
Observations           32exp(coef time)^4-1   20.86% APR, continuously compounded
ANOVA              df          SS         MS          F     Significance F
Regression              4 10.8244898 2.706122     205.279466 8.6607E-20
Residual               27 0.35593091 0.013183
Total                  31 11.1804207
Standard Error t Stat
Coefficients                         P-value     Lower 95%      Upper 95%
Intercept        4.740     0.05536386    85.6     2.0246E-34    4.62623522     4.85342957
time (quarters) 0.0474     0.00221455    21.4     1.8396E-18    0.04282814     0.05191591
Q1              -0.914     0.05779096   -15.8     3.5514E-15   -1.03254265    -0.79538831
Q2              -0.601     0.05757841   -10.4     5.6672E-11   -0.71889202    -0.48260989
Q3              -0.363     0.05745051    -6.3     9.3742E-07   -0.48042039    -0.24466314
=exp(Qcoef)    Understanding the seasonal dummies relative to peak           =exp(Qcoef)-1
40.1%      Q1 as a percent of peak (Q4)         Q1 less than peak (Q4) by     -59.9%
54.8%      Q2 as a percent of peak (Q4)         Q2 less than peak (Q4) by     -45.2%
69.6%      Q3 as a percent of peak (Q4)         Q3 less than peak (Q4) by     -30.4%
RESIDUAL OUTPUT                          2.10     dw
ObservationPredicted ln(q) Residuals   0.746465    numerator
1 3.825867 0.11987427
2 4.186453 0.12396302 1.67E-05
3 4.472035 -0.0580565 0.033131
4 4.881948 0.10540765 0.026721
5 4.015355 0.01894592 0.007476
6 4.375942 -0.0044305 0.000546
7 4.661523 0.08880961 0.008694
8 5.071437 0.00876746 0.006407
9 4.204843 0.13516679 0.015977
10 4.56543 -0.2498182 0.148213
11 4.851011 0.03980023 0.083879
12 5.260925 -0.0135101 0.002842
13 4.394331 -0.1678973 0.023835
14 4.754918 0.03486095 0.041111
15 5.040499 -0.1418195 0.031216
16 5.450413 -0.0897959 0.002706
17 4.583819 -0.0027335 0.00758
18 4.944406 0.00671383 8.93E-05
19 5.229987 -0.1326394 0.019419
20 5.639901 -0.0501804 0.006799
21 4.773307 -0.1929149 0.020373
22 5.133894 -0.1066919 0.007434
23 5.419475 0.15548696 0.068738
24 5.829389 -0.0981225 0.064318
25 4.962795 0.12500485 0.049786
26 5.323382 0.08071815 0.001961
27   5.608963   -0.0580759   0.019264
28   6.018877   0.06289505   0.014634
29   5.152284   -0.0354461   0.009671
30    5.51287   0.11468457   0.022539
31   5.798451   0.10649453   6.71E-05
32   6.208365   0.07453874   0.001021
mpounding                                                           time (quarters) Residual Plot
licit in using this formula                                                                   Q1 Residual Plot
no compounding                              0.2

Residuals
0
continuous compounding                                           0.2

Residuals
-0.2 0                                  10              Q2 Residual Plot
20         30                                       40
-0.4
0
-0.2 0                 0.2             0.4             0.6          0.8            1               1.2
0.2                             time (quarters) Q3 Residual Plot

Residuals
-0.4
0
-0.2 0                 0.2                 0.4       Q1
0.6          0.8               1                 1.2

Residuals
0.15
0.1                                        time (quarters) Line Fit Plot Fit
Q1 Line                           Plot
-0.4
0.05
0                             Q2

ln(q)
0              10.000
5.000
0.2             0.4 8.000 0.6               0.8                 1
0.000
6.000 Q3

ln(q)
0             10
4.000    20         30
2.000
8.000
0.000 (quarters)

ln(q)
time  6.000
0  4.000 0.5        1
2.000
0.000
0     Q1
4.000
3.950

ln(q)
3.900
3.850
3.800
) Line Fit Plot Fit
Q1 Line           Plot

1       Q2 Line Fit Plot
1.2
ln(q)
40
ln(q)
Predicted ln(q)
Q3 Line Fit Plot
1             1.5         Predicted ln(q) ln(q)
0.5
4.000                     1           1.5       Predicted ln(q)
3.950
3.900          Q2
3.850                                                       ln(q)
3.800
0              0.5         1          1.5        Predicted ln(q)

Q3

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