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### 45-734 PROBABILITY AND STATISTICS II Homework Answers #1 (4th Mini AY1997-98)

1. The model:

GNP = b 0 + b 1 MILMOB

where GNP is the % change in GNP and MILMOB is the change in military mobilization is fitted below: (Note: One did not need to transform the given data.)

```============================================================
LS // Dependent Variable is GNP
Date: 02/28/98   Time: 17:54
Sample: 1915 1988
Included observations: 74
============================================================
Variable      CoefficienStd. Errort-Statistic  Prob.
============================================================
C           3.025348   0.547696   5.523772   0.0000
MILMOB        3.697863   0.547363   6.755782   0.0000
============================================================
R-squared            0.387966    Mean dependent var 3.061841
Adjusted R-squared   0.379466    S.D. dependent var 5.980692
S.E. of regression   4.711230    Akaike info criter 3.126553
Sum squared resid    1598.089    Schwarz criterion  3.188825
Log likelihood      -218.6839    F-statistic        45.64060
Durbin-Watson stat   1.266900    Prob(F-statistic)  0.000000
============================================================
```
1. The estimates for the slope and intercept coefficients are: b1 = 3.698 and b0 = 3.025 respectively. This means that for each unit increase in the variable change in military mobilization the variable change in GNP increases by 3.698 times the change in military mobilization. The 2-tailed observed significance (P-value) of both coefficients is 0, implying that at any reasonable significance level one would reject the null hypotheses Ho: b0 =0 and Ho: b1 =0. In the latter case the null hypothesis for the MILMOB variable specifies that the variable MILMOB has a coefficient 0 in the model and hence has no effect on the variable GNP on the right hand side. The same is true for the constant term. Since we reject both of these hypotheses we can assume that both coefficients are significantly different from 0 which means they are meaningful in the model.

The residuals of a regression are saved in the variable name by the reserved word RESID after every regression. The EVIEWS command GENR (generate) can be used to save the residuals into a variable called RES, say, as follows:

GENR RES=RESID

We can then look at the residuals by typing

SHOW RES

in EVIEWS. This display is below.

The residuals that have absolute values over 9 are in bold: (note there are others with values near 9; 9 is an arbitrary cutoff point.)

```                               RESID
===================================================================
Last updated: 02/28/98 - 17:54

1915    3.590703    4.243941   -5.229786   -6.045869    4.284621
1920   -1.212403   -5.540562    3.094281    9.495950    0.136206
1925   -0.560048    3.289923   -2.058828   -1.831165    2.819159
1930   -12.90689   -11.90410   -17.40243   -5.141860    4.353963
1935    4.758457    10.08610    1.796987   -7.625791    4.544439
1940    4.199656    9.527733    8.606488   -0.239746   -1.300503
1945   -6.350234   -0.043979   -1.939323    1.251375   -3.354685
1950    5.612538    2.589378    0.021078    1.231581   -3.645527
1955    3.335193   -0.593930   -1.236150   -3.282037    2.967425
1960   -0.688404   -0.385380    1.580729    1.290056    2.273051
1965    2.730666    1.827576   -0.671611    0.785562   -0.391364
1970   -2.535118    0.468764    2.575059    2.203462   -3.373036
1975   -4.195150    1.858301    1.582365    2.190687   -0.482279
1980   -3.190959   -1.124885   -5.614083    0.495162    3.541703
1985    0.281572   -0.214503    0.285960    0.777179
===================================================================
```

The largest residuals are the following:

```        1928         9.495950
1930        12.90689
1931       -11.90410
1932       -17.40243
1942         9.52773
```
The model works poorly during the early part of the great depression and in 1942. There were other factors affecting the GNP during the depression besides military mobilization. In 1942, one might surmise that the war was having a positive effect on the economy in the United States, but no real build up of the military had yet occurred as the United States had just entered the war. The residual graph is shown below.

2. The fact that there was a large military mobilization during World War II and that the GNP also grew quickly at that point in time explains why the actual and predicted values track well at that time. The down "spike" in lines in 1946 is due to the end of world war II. At this point in United States history there was both a large decrease in GNP and a corresponding negative change in military mobilization. This can be checked by using the show command. (The "spike" is due to the large decrease in GNP which is the dependent variable in this problem.)

```========================================================
OBS       YEAR       MILMOB       GNP         RESID
========================================================
2      1946.000   -6.516724   -21.11658   -0.043979
```
3. The following regressions were run after using the SMPL command:

```============================================================
LS // Dependent Variable is GNP
Date: 02/28/98   Time: 18:10
Sample: 1915 1945
Included observations: 31
============================================================
Variable      CoefficienStd. Errort-Statistic  Prob.
============================================================
C           2.834469   1.281825   2.211276   0.0351
MILMOB        3.594665   1.330917   2.700893   0.0114
============================================================
R-squared            0.200988    Mean dependent var 3.819669
Adjusted R-squared   0.173436    S.D. dependent var 7.525475
S.E. of regression   6.841829    Akaike info criter 3.908451
Sum squared resid    1357.508    Schwarz criterion  4.000966
Log likelihood      -102.5681    F-statistic        7.294821
Durbin-Watson stat   1.101443    Prob(F-statistic)  0.011426
============================================================

============================================================
LS // Dependent Variable is GNP
Date: 02/28/98   Time: 18:11
Sample: 1947 1988
Included observations: 42
============================================================
Variable      CoefficienStd. Errort-Statistic  Prob.
============================================================
C           3.235515   0.372849   8.677815   0.0000
MILMOB        5.290072   1.427013   3.707093   0.0006
============================================================
R-squared            0.255711    Mean dependent var 3.078169
Adjusted R-squared   0.237103    S.D. dependent var 2.748479
S.E. of regression   2.400630    Akaike info criter 1.797910
Sum squared resid    230.5210    Schwarz criterion  1.880657
Log likelihood      -95.35153    F-statistic        13.74254
Durbin-Watson stat   2.112150    Prob(F-statistic)  0.000635
============================================================
```
In both regressions the slope coefficient of MILMOB is significant, i.e. significantly different from 0. However, after World War II the effect of MILMOB on the change in GNP is even greater. It is 5.29*change in MILMOB after 1945 as compare to 3.59*change in MILMOB before 1945. The value of this coefficient in the original (whole data set) regression was very close to that in the pre 1945 regression, but somewhat different from the post 1945 regression results. The R coefficients are not high in either case, which indicates that the data do not fit the linear model all that well. However, assuming these models do work one might say the effect of military mobilization on the economy is even greater since World War II.

2. Problem 11.2 MWS Working Directly With a Calculator

åi=1,10 Xi = 720,    åi=1,10 Xi2 = 106554,
åi=1,10 Yi = 721,    åi=1,10 Yi2 = 105817,    åi=1,10 XiYi = 106155

```        ^    106155 - (1/10)*720*721    54243
Hence   b1 = ----------------------- = ------- = .9913916
106554 - (1/10)*720*720    54714
^
and   b0 = 721/10 - .9913916*(720/10) = .7198048

The least squares straight line is:
^   ^    ^
and   y = b0 + b1x = .7198048 + .9913916*x

The expected change in audited value (y) for a one-unit change in
^
book value (x), is b1 = .9913916

When x = 100, the best estimate of y is
^
and   y = .7198048 + .9913916*100 = 99.858965
```

3. Problem 11.2 MWS EVIEWS

Below is the EVIEWS output for problem 11.2:
```
============================================================
LS // Dependent Variable is Y
Date: 02/28/98   Time: 18:21
Sample: 1 10
Included observations: 10
============================================================
Variable      CoefficienStd. Errort-Statistic  Prob.
============================================================
C           0.719805   1.176355   0.611894   0.5576
X           0.991392   0.011396   86.99447   0.0000
============================================================
R-squared            0.998944    Mean dependent var 72.10000
Adjusted R-squared   0.998812    S.D. dependent var 77.33973
S.E. of regression   2.665648    Akaike info criter 2.137751
Sum squared resid    56.84544    Schwarz criterion  2.198268
Log likelihood      -22.87814    F-statistic        7568.037
Durbin-Watson stat   2.455810    Prob(F-statistic)  0.000000
============================================================
```