# REN 576 by ajizai

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```									REN 576                Lab 5: Non-Spherical Disturbances
Create a data set in Excel

We are going to create a data set of 50 observations in Excel to use in SAS.
The data set will have two independent variables, two random errors and two dependent
variables. The first random error will be autocorrelated. The second will be heteroskedastic.

Use these formulas to create the independent variables and errors:
For X1(all observations): ‘=5*rand( )’.
For X2 (all observations): ‘=10*rand( )’
For e1 (first observation only) : ‘=3*rand( )’
(for obs 2): ‘=0.7*D2+3*rand( )’
(for the rest of the observations copy second observation down).
For e2 (first 25 observations): ‘=2*rand( )’
(for last 25 observations): ‘=4*rand( )’

Each dependent variable will be created as a linear combination of the independent variables and
one of the random errors.

Use these formulas to create the dependent variables:
For Y1 (OBS=1): ‘=1+2*B2+3*C2+D2
For Y1 (for the rest of the observations copy the first observation down)
For Y2 (OBS=1): ‘=4+5*B2+6*C2+E2
For Y2 (for the rest of the observations copy the first observation down)
Import the Data into a SAS file
You should know how to do this by now.

In SAS Analyst estimate the regression equations by OLS and test for autocorrelation and
heteroskedasticity

In the SAS Program Editor write a program to invoke the procedure that will test for
autocorrelation or heteroskedasticity and estimate the models more efficiently

This part of this exercise involves using www.uri.edu/SASDOC to find the appropriate SAS
procedure and programming codes for this problem

Write the SAS Program codes you use here:

Finally, fill in the blanks of the table below and answer the questions.
Model 1                          Model 2
OLS      EGLS                    OLS      EGLS

Intercept
(s.e.)

X1
(s.e.)

X2
(s.e)

F

R-Square

R-Bar Square

DW/ARCHTest

DW/LM/Q Prob

1. What are the true models for Y1 and Y2?

2. Which estimated coefficients are unbiased?

3. Which estimated coefficients are efficient?

4. Which estimated standard errors are unbiased?
08:03 Wednesday, October 27, 2004   1
The REG Procedure
Model: MODEL1
Dependent Variable: Y2 Y2

Analysis of Variance

Sum of             Mean
Source                  DF            Squares           Square     F Value    Pr > F

Model                    2              20788             10394    8178.04    <.0001
Error                   47           59.73512           1.27096
Corrected Total         49              20848

Root MSE                 1.12737    R-Square         0.9971
Dependent Mean          50.19484    Adj R-Sq         0.9970
Coeff Var                2.24599

Parameter Estimates

Parameter      Standard
Variable    Label       DF            Estimate         Error       t Value    Pr > |t|

Intercept   Intercept    1            5.53455           0.43295      12.78      <.0001
X1          X1           1            5.03422           0.11163      45.10      <.0001
X2          X2           1            5.97855           0.05185     115.30      <.0001
08:03 Wednesday, October 27, 2004   2

The REG Procedure
Model: MODEL1
Dependent Variable: Y1 Y1

Analysis of Variance

Sum of             Mean
Source                  DF            Squares           Square     F Value    Pr > F

Model                    2         5136.92132     2568.46066       1594.86    <.0001
Error                   47           75.69184        1.61046
Corrected Total         49         5212.61316

Root MSE                 1.26904    R-Square         0.9855
Dependent Mean          26.92552    Adj R-Sq         0.9849
Coeff Var                4.71315

Parameter Estimates

Parameter      Standard
Variable    Label       DF            Estimate         Error       t Value    Pr > |t|

Intercept   Intercept    1            5.46508           0.48735      11.21      <.0001
X1          X1           1            2.13025           0.12566      16.95      <.0001
X2          X2           1            3.04649           0.05837      52.20      <.0001
08:03 Wednesday, October 27, 2004   3

The REG Procedure
Model: MODEL1
Dependent Variable: Y2 Y2

Test of First and Second
Moment Specification

DF     Chi-Square     Pr > ChiSq

5           3.55           0.6162
Durbin-Watson D                   1.791
Number of Observations               50
1st Order Autocorrelation         0.069
08:03 Wednesday, October 27, 2004   4

The REG Procedure
Model: MODEL1
Dependent Variable: Y1 Y1

Test of First and Second
Moment Specification

DF    Chi-Square       Pr > ChiSq

5         6.05            0.3013

Durbin-Watson D                  0.559
Number of Observations              50
1st Order Autocorrelation        0.670
http://www.uri.edu/sasdoc/ets/chap8/sect3.htm#idxaut0010

Autoregressive Error Model

The following statements regress Y on TIME with the errors assumed to follow a second-order
autoregressive process. The order of the autoregressive model is specified by the NLAG=2
option. The Yule-Walker estimation method is used by default. The example uses the
METHOD=ML option to specify the exact maximum likelihood method instead.
proc autoreg data=a;
model y = time / nlag=2 method=ml;
run;

AUTOCORRELATION

Our code***

options pageno=1;
proc reg data=Work.Lab5;
model Y1 = X1 X2 / dwprob nlag=1 ;
run;
quit;
08:03 Wednesday, October 27, 2004          1

The AUTOREG Procedure

Dependent Variable      Y1
Y1

Ordinary Least Squares Estimates

SSE                 75.6918351    DFE                       47
MSE                    1.61046    Root MSE             1.26904
SBC                 174.362287    AIC               168.626218
Regress R-Square        0.9855    Total R-Square        0.9855
Durbin-Watson           0.5586    Pr < DW               <.0001
Pr > DW                 1.0000
NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for
testing negative autocorrelation.

Standard                Approx          Variable
Variable         DF      Estimate                Error   t Value    Pr > |t|          Label

Intercept         1          5.4651            0.4874      11.21        <.0001
X1                1          2.1302            0.1257      16.95        <.0001        X1
X2                1          3.0465            0.0584      52.20        <.0001        X2

Estimates of Autocorrelations

Lag      Covariance       Correlation          -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

0         1.5138              1.000000      |                      |********************|
1         1.0149              0.670441      |                      |*************       |

Preliminary MSE        0.8334

Estimates of Autoregressive Parameters

Standard
Lag          Coefficient              Error   t Value

1         -0.670441           0.109396         -6.13

Heteroskedasticity
LINEAR
specifies the linear function; that is, the HETERO statement variables predict the error
variance linearly. The following model is estimated when you specify LINK=LINEAR:
*** Linear Regression ***;
options pageno=1;
proc autoreg data=Work.Lab5;
model Y2 = X1 X2 / archtest;
run;
quit;

08:03 Wednesday, October 27, 2004   1

The AUTOREG Procedure

Dependent Variable       Y1
Y1

Ordinary Least Squares Estimates

SSE                 75.6918351    DFE                       47
MSE                    1.61046    Root MSE             1.26904
SBC                 174.362287    AIC               168.626218
Regress R-Square        0.9855    Total R-Square        0.9855
Durbin-Watson           0.5586    Pr < DW               <.0001
Pr > DW                 1.0000
NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for
testing negative autocorrelation.

Standard                  Approx   Variable
Variable         DF      Estimate              Error   t Value      Pr > |t|   Label

Intercept         1           5.4651         0.4874      11.21       <.0001
X1                1           2.1302         0.1257      16.95       <.0001    X1
X2                1           3.0465         0.0584      52.20       <.0001    X2

Estimates of Autocorrelations

Lag      Covariance        Correlation       -1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1

0         1.5138            1.000000      |                     |********************|
1         1.0149            0.670441      |                     |*************       |

Preliminary MSE         0.8334

Estimates of Autoregressive Parameters

Standard
Lag        Coefficient              Error    t Value

1         -0.670441           0.109396       -6.13
08:03 Wednesday, October 27, 2004   2

The AUTOREG Procedure

Yule-Walker Estimates

SSE                 34.5762675    DFE                       46
MSE                    0.75166    Root MSE             0.86698
SBC                 139.696084    AIC               132.047992
Regress R-Square        0.9960    Total R-Square        0.9934
Durbin-Watson           1.5521    Pr < DW               0.0765
Pr > DW                 0.9235
NOTE: Pr<DW is the p-value for testing positive autocorrelation, and Pr>DW is the p-value for
testing negative autocorrelation.

Standard                  Approx    Variable
Variable       DF     Estimate          Error    t Value     Pr > |t|    Label

Intercept       1       5.9214         0.4306      13.75       <.0001
X1              1       2.0107         0.0644      31.22       <.0001    X1
X2              1       3.0138         0.0302      99.88       <.0001    X2
08:03 Wednesday, October 27, 2004   1

The AUTOREG Procedure

Dependent Variable      Y2
Y2

Ordinary Least Squares Estimates

SSE                    59.7351193       DFE                          47
MSE                       1.27096       Root MSE                1.12737
SBC                    162.524778       AIC                  156.788709
Regress R-Square           0.9971       Total R-Square           0.9971
Durbin-Watson              1.7908

Q and LM Tests for ARCH Disturbances

Order                Q    Pr > Q                 LM     Pr > LM

1         2.2946       0.1298          1.9805         0.1593
2         5.1794       0.0750          4.0953         0.1290
3         6.0210       0.1106          4.2986         0.2310
4         6.4619       0.1672          5.5172         0.2382
5         6.8542       0.2317          6.2022         0.2870
6         7.0237       0.3187          6.8490         0.3350
7         7.3808       0.3903          7.0916         0.4194
8        10.9244       0.2060          8.7188         0.3666
9        11.3836       0.2503          8.7346         0.4621
10        11.5477       0.3165          8.7437         0.5566
11        16.2716       0.1313         14.0898         0.2281
12        17.2841       0.1392         15.3651         0.2221

Standard                       Approx    Variable
Variable         DF      Estimate           Error      t Value        Pr > |t|    Label

Intercept         1       5.5345          0.4329         12.78         <.0001
X1                1       5.0342          0.1116         45.10         <.0001     X1
X2                1       5.9785          0.0519        115.30         <.0001     X2

Algorithm converged.
08:03 Wednesday, October 27, 2004   2

The AUTOREG Procedure

Linear Heteroscedasticity Estimates

SSE              70.6339075      Observations                50
MSE                 1.41268      Root MSE               1.18856
Log Likelihood   -68.485113      Total R-Square          0.9966
SBC              156.530342      AIC                 146.970227
Normality Test       4.4414      Pr > ChiSq              0.1085
Hetero Test         15.4045      Pr > ChiSq              <.0001

Standard                   Approx   Variable
Variable       DF     Estimate          Error    t Value      Pr > |t|   Label

Intercept       1       5.1175         0.2879      17.78       <.0001
X1              1       4.9615         0.0943      52.60       <.0001    X1
X2              1       6.0174         0.0353     170.41       <.0001    X2
HET0            1       0.5736         0.1415       4.05       <.0001
HET1            1       6.5885         4.9216       1.34       0.1807

```
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