In this Comment, let Y denote time interval to the next eruption, and X denote duration of an eruption. We separate out or unstack the two years of data, letting Y1 denote the 107 values of Y from 1978, Y2 the 115 values of Y from 1979, deriving X1 and X2 from X in a similar manner.
A scatter plot of Y versus X appears in the case, "Eruptions of the 'Old Faithful' Geyser". Plots of Y1 versus X1 and Y2 versus X2, as shown in the Appendix below, are similar to the plot of Y versus X, but the two new plots display a difference: namely, the vertical spread of points is larger in the second plot than in the first. That is, the relationship between Y2 and X2 is stronger than that between Y1 and X1.
This can be seen another way. If we regress Y1 on X1, we find R-square = 73.7% and S = 6.68. But if we regress Y2 on X2, we find R-square = 81.8% and S = 5.44, a substantial improvement.
Suppose now, as in the case, we introduce lagged variables and drop the first observation of each day. Let the first order lag of Y2 be denoted by Y2_1, and other lags denoted similarly. Using the first year of data only, we can get a good model with R-square = 79.7% and S = 6.07. Using the data from both years along with the dummy variable YEAR, we can get a better model with R-square = 81.0% and S = 5.67. But using the second year of data only, we can get a still better model with R-square = 83.5% and S = 5.18.
In the case, a model is derived from the 1978 and 1979 data and applied to the year 1985. The tacit assumption is made that the data are stable from year to year. This assumption may be true only approximately.
Scatter Plot of Y1 vs X1
Y1 +---------------------------------------------------+
100 + |
| |
| ++ |
| + + + |
| + + + + + + |
| + + ++ 2 2++ 3 |
80 + + + + + + ++ 2 + |
| + + + + 2 + + + |
| ++ 23 2+ + ++ 2+ |
| ++ + 2 |
| + + + + + |
| |
60 + ++ + + |
| 2 + + + |
| + 2+ 2 + |
| 2 23 2 + |
| + |
| 3 |
40 + |
++---------+---------+---------+---------+---------++
1.6 2.3 3.0 3.7 4.4 5.1
X1
Scatter Plot of Y2 vs X2
Y2 +---------------------------------------------------+
100 + |
| |
| + |
| + 2 + |
| + +2 |
| + 2+ 2+22 + |
80 + 2+2 + 23 2 |
| 2 222 +2+ + |
| 2 2 +35 +2 |
| 3 + 2+ |
| + + + |
| ++ + + |
60 + + + 2 2 |
| ++ 2 |
| 22 3 + |
| ++2 3+ |
| + + |
| + + |
40 + |
++---------+---------+---------+---------+---------++
1.5 2.3 3.1 3.9 4.7 5.5
X2
LINEAR REGRESSION OF Y1
PREDICTOR
VARIABLES COEFFICIENT STD ERROR STUDENT'S T P
--------- ----------- --------- ----------- ------
CONSTANT 33.8282 2.26182 14.96 0.0000
X1 10.7410 0.62634 17.15 0.0000
R-SQUARED 0.7369 RESIDUAL MEAN SQUARE (MSE) 44.6573
ADJUSTED R-SQUARED 0.7344 STANDARD ERROR OF ESTIMATE 6.68261
LINEAR REGRESSION OF Y2
PREDICTOR
VARIABLES COEFFICIENT STD ERROR STUDENT'S T P
--------- ----------- --------- ----------- ------
CONSTANT 33.2572 1.75087 18.99 0.0000
X2 10.2512 0.45493 22.53 0.0000
R-SQUARED 0.8180 RESIDUAL MEAN SQUARE (MSE) 29.6118
ADJUSTED R-SQUARED 0.8164 STANDARD ERROR OF ESTIMATE 5.44167
LINEAR REGRESSION OF Y1
PREDICTOR
VARIABLES COEFFICIENT STD ERROR STUDENT'S T P VIF
--------- ----------- --------- ----------- ------ -----
CONSTANT 58.1498 7.12580 8.16 0.0000
X1 8.82446 0.84058 10.50 0.0000 1.8
Y1_1 -0.24920 0.06800 -3.66 0.0004 1.8
R-SQUARED 0.7711 RESIDUAL MEAN SQUARE (MSE) 41.1431
ADJUSTED R-SQUARED 0.7663 STANDARD ERROR OF ESTIMATE 6.41429
LINEAR REGRESSION OF Y1
PREDICTOR
VARIABLES COEFFICIENT STD ERROR STUDENT'S T P VIF
--------- ----------- --------- ----------- ------ -----
CONSTANT 64.1548 6.96458 9.21 0.0000
X1 8.85425 0.79593 11.12 0.0000 1.8
Y1_1 -0.53582 0.10461 -5.12 0.0000 4.9
X1_1 4.07700 1.17284 3.48 0.0008 4.1
R-SQUARED 0.7969 RESIDUAL MEAN SQUARE (MSE) 36.8845
ADJUSTED R-SQUARED 0.7905 STANDARD ERROR OF ESTIMATE 6.07326
LINEAR REGRESSION OF Y
PREDICTOR
VARIABLES COEFFICIENT STD ERROR STUDENT'S T P VIF
--------- ----------- --------- ----------- ------ -----
CONSTANT 55.8712 4.22566 13.22 0.0000
X 9.31683 0.47029 19.81 0.0000 1.7
YEAR -2.99949 0.81605 -3.68 0.0003 1.1
Y_1 -0.37154 0.07021 -5.29 0.0000 5.2
X_1 2.65669 0.79853 3.33 0.0010 4.8
R-SQUARED 0.8103 RESIDUAL MEAN SQUARE (MSE) 32.2007
ADJUSTED R-SQUARED 0.8065 STANDARD ERROR OF ESTIMATE 5.67457
LINEAR REGRESSION OF Y2
PREDICTOR
VARIABLES COEFFICIENT STD ERROR STUDENT'S T P VIF
--------- ----------- --------- ----------- ------ -----
CONSTANT 44.5381 4.92005 9.05 0.0000
X2 9.39219 0.55085 17.05 0.0000 1.5
Y2_1 -0.12033 0.04826 -2.49 0.0142 1.5
R-SQUARED 0.8348 RESIDUAL MEAN SQUARE (MSE) 26.7848
ADJUSTED R-SQUARED 0.8316 STANDARD ERROR OF ESTIMATE 5.17541