Exercise 3: Bivariate Regression Analysis and Model Evaluation

Exercise 3: Bivariate regression analysis and model evaluation

a)Reasonable assumptions:

• Zero conditional mean:
• Constant variance:
• Autocorrelation:
• Normal distributed:

b)

EQ( 1) Modelling Y by OLS

The dataset is: M:\ECON 4160\data\Sp8101.xls

The estimation sample is: 1980(1) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Constant 23.8431 7.113 3.35 0.0012 0.1245

X 0.574115 0.01996 28.8 0.0000 0.9128

sigma 26.6013 RSS 55902.4978

R^2 0.91284 F(1,79) = 827.4 [0.000]**

Adj.R^2 0.911737 log-likelihood -379.679

no. of observations 81 no. of parameters 2

mean(Y) 209.948 se(Y) 89.5389

AR 1-5 test: F(5,74) = 1473.0 [0.0000]**

ARCH 1-4 test: F(4,73) = 4905.3 [0.0000]**

Normality test: Chi^2(2) = 6.8954 [0.0318]*

Hetero test: F(2,78) = 27.065 [0.0000]**

Hetero-X test: F(2,78) = 27.065 [0.0000]**

RESET23 test: F(2,77) = 901.70 [0.0000]**

Interpretingtheresult:

• We have to perform a t-test to say something about the .

t-test:

To conclude that is not 1, the t-value in absolute value has to be larger than a critical value.

From this result we can conclude that is significantly different from 1.

High t-value gives very low p-value, which means

• The model’s standard error is very low, and also from this we can say that the model have a big certain. This is thus also the reason for the big t-value in absolute terms.

The t-value can though indicate statistical significance either because the estimated is “large” or the standard error is “small”. Too much focus on statistical significance can thus lead to the false conclusion that a variable is important for explaining Y even though its estimated effect is modest.

• this means that the variation in X explains approximately 90 % of the variation in Y. This is a very high number.

c)Is this model an adequate conditional model of given?

Even though, as we stated in b), the is large and the are significantly different from 1, the test fail. We have autocorrelation, heteroskedasticity and the model is not normally distributed. The assumptions we made in a) is not valid. We can’t perform a the t-test correctly. So we can’t say for sure that the is significant.

When we correct for autocorrelation and heteroskedasticity in the model, we get robust standard errors.

Robust standard errors

Coefficients SE HACSE HCSE JHCSE

Constant 23.843 7.1132 11.795 6.2624 6.4130

X 0.57411 0.019959 0.042387 0.022755 0.023236

Coefficients t-SE t-HACSE t-HCSE t-JHCSE

Constant 23.843 3.3520 2.0214 3.8074 3.7180

X 0.57411 28.764 13.545 25.231 24.708

The result is quite similar as in b). We have same values on the , which is still significantly different from 1, the standard error is still very small and there is no heteroskedasticity and autocorrelation. But: it is still not normallydistributed

The importance of normal distribution when we have robust standard error is a matter of preferences to the person who is making the model.

Some would say that the modell is still not valid, and some would think that this is an ok model.

d)The criteria we use to evaluate this model:

• R-squared
• T-test/significance level
• The test: AR, ARCH, normality test, hetero test,

e)The inverted model:

EQ( 2) Modelling X by OLS

The dataset is: M:\ECON 4160\data\Sp8101.xls

The estimation sample is: 1980(1) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Constant -9.65664 12.60 -0.766 0.4459 0.0074

Y 1.59000 0.05528 28.8 0.0000 0.9128

sigma 44.2691 RSS 154820.562

R^2 0.91284 F(1,79) = 827.4 [0.000]**

Adj.R^2 0.911737 log-likelihood -420.935

no. of observations 81 no. of parameters 2

mean(X) 324.16 se(X) 149.008

AR 1-5 test: F(5,74) = 7880.4 [0.0000]**

ARCH 1-4 test: F(4,73) = 8133.5 [0.0000]**

Normality test: Chi^2(2) = 29.254 [0.0000]**

Hetero test: F(2,78) = 6.2583 [0.0030]**

Hetero-X test: F(2,78) = 6.2583 [0.0030]**

RESET23 test: F(2,77) = 2.9361 [0.0590]

Robust standard errors

Coefficients SE HACSE HCSE JHCSE

Constant -9.6566 12.605 9.3902 4.8329 4.8894

Y 1.5900 0.055277 0.078550 0.040839 0.040972

Coefficients t-SE t-HACSE t-HCSE t-JHCSE

Constant -9.6566 -0.76612 -1.0284 -1.9981 -1.9750

Y 1.5900 28.764 20.242 38.934 38.807

If we have causality, which means that we don’t know if X is explained by Y or if Y is explained by X.

Have to run a t-test:

The coefficient is significant different from 1

The standard error is very small and the R-squared is very high, 0.91.

The tests still fail: the is autocorrelation, heteroskedasticity and not normally distributed.

f)Evaluation criteria:

Reasonable: the test still fail, the t-test haven’t changes, except the t-value of the constant is no longer significant.

g)

EQ( 3) Modelling Y by OLS

The dataset is: M:\ECON 4160\data\Sp8101.xls

The estimation sample is: 1980(2) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Y_1 1.05790 0.006588 161. 0.0000 0.9971

Constant -0.611499 0.8666 -0.706 0.4826 0.0065

X 0.886688 0.08276 10.7 0.0000 0.6016

X_1 -0.930794 0.08366 -11.1 0.0000 0.6196

sigma 1.37508 RSS 143.704417

R^2 0.999764 F(3,76) = 1.072e+005 [0.000]**

Adj.R^2 0.999754 log-likelihood -136.944

no. of observations 80 no. of parameters 4

mean(Y) 212.206 se(Y) 87.7537

AR 1-5 test: F(5,71) = 6.0258 [0.0001]**

ARCH 1-4 test: F(4,72) = 1.9184 [0.1166]

Normality test: Chi^2(2) = 1.1521 [0.5621]

Hetero test: F(6,73) = 1.3187 [0.2598]

Hetero-X test: F(9,70) = 1.5844 [0.1369]

RESET23 test: F(2,74) = 21.630 [0.0000]**

R-squared adjusted is very high

is not significantly different from 1. Can not reject that this value I 1.

We see from the test that there is still autocorrelation. The other tests are ok. There is no longer heteroskedasticity, and the normal distribution is valid.

i)Strategy for evaluating the tree models against each other: model 2 explains more than model 1 because the R-squared is bigger.

“simple-to-general”: you go from a model with few variables to many variables

“general-to-simple”:the opposite.

From (6) to (8), simple to general.

From (8) to (6) general to simple.

Exercise 4:Bivariate regressions with autocorrelated errors

Model 1:

EQ( 4) Modelling Y by OLS

The dataset is: M:\ECON 4160\data\Sp8101.xls

The estimation sample is: 1980(2) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Constant 25.1914 7.340 3.43 0.0010 0.1312

X 0.570766 0.02047 27.9 0.0000 0.9088

sigma 26.6688 RSS 55475.3342

R^2 0.908811 F(1,78) = 777.4 [0.000]**

Adj.R^2 0.907642 log-likelihood -375.182

no. of observations 80 no. of parameters 2

mean(Y) 212.206 se(Y) 87.7537

AR 1-5 test: F(5,73) = 1361.1 [0.0000]**

ARCH 1-4 test: F(4,72) = 4733.9 [0.0000]**

Normality test: Chi^2(2) = 6.9911 [0.0303]*

Hetero test: F(2,77) = 26.463 [0.0000]**

Hetero-X test: F(2,77) = 26.463 [0.0000]**

RESET23 test: F(2,76) = 882.91 [0.0000]**

EQ( 5) Modelling Y by OLS

The dataset is: M:\ECON 4160\data\Sp8102.xls

The estimation sample is: 1980(1) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Constant -3.29461 0.8385 -3.93 0.0002 0.1635

X 0.987515 0.1237 7.98 0.0000 0.4466

sigma 2.49631 RSS 492.294488

R^2 0.446567 F(1,79) = 63.75 [0.000]**

Adj.R^2 0.439562 log-likelihood -188.021

no. of observations 81 no. of parameters 2

mean(Y) 3.02321 se(Y) 3.33453

AR 1-5 test: F(5,74) = 41.391 [0.0000]**

ARCH 1-4 test: F(4,73) = 24.357 [0.0000]**

Normality test: Chi^2(2) = 16.133 [0.0003]**

Hetero test: F(2,78) = 1.6170 [0.2051]

Hetero-X test: F(2,78) = 1.6170 [0.2051]

RESET23 test: F(2,77) = 1.3818 [0.2573]

EQ( 6) Modelling Y by OLS

The dataset is: M:\ECON 4160\data\Sp8103.xls

The estimation sample is: 1980(1) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Constant -0.0514192 0.09848 -0.522 0.6030 0.0034

X 0.00987416 0.09443 0.105 0.9170 0.0001

sigma 0.885816 RSS 61.9889045

R^2 0.000138391 F(1,79) = 0.01093 [0.917]

Adj.R^2 -0.0125181 log-likelihood -104.101

no. of observations 81 no. of parameters 2

mean(Y) -0.0510664 se(Y) 0.880323

AR 1-5 test: F(5,74) = 0.68673 [0.6350]

ARCH 1-4 test: F(4,73) = 0.69229 [0.5997]

Normality test: Chi^2(2) = 1.9979 [0.3683]

Hetero test: F(2,78) = 0.068104 [0.9342]

Hetero-X test: F(2,78) = 0.068104 [0.9342]

RESET23 test: F(2,77) = 0.30157 [0.7405]

• The first data-set is the same as in 3a)
• Sp8102:
• R-squared are about 0,44, so the variables explains less here than in the previous data set.
• The standard error is a bit bigger, more variation in the observations
• the coefficient on X is not significantly different from 1.
• The test show that there is less probability for heteroskedasticity, but there is still autocorrelation.
• Sp8103:
• R-squared is zero and the coefficient is almost zero.
• The test are here positive though: no autocorrelation, no heteroskedastisity and there is normality.

Model 2:

EQ( 7) Modelling Y by OLS

The dataset is: M:\ECON 4160\data\Sp8101.xls

The estimation sample is: 1980(1) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Constant 23.8431 7.113 3.35 0.0012 0.1245

X 0.574115 0.01996 28.8 0.0000 0.9128

sigma 26.6013 RSS 55902.4978

R^2 0.91284 F(1,79) = 827.4 [0.000]**

Adj.R^2 0.911737 log-likelihood -379.679

no. of observations 81 no. of parameters 2

mean(Y) 209.948 se(Y) 89.5389

AR 1-5 test: F(5,74) = 1473.0 [0.0000]**

ARCH 1-4 test: F(4,73) = 4905.3 [0.0000]**

Normality test: Chi^2(2) = 6.8954 [0.0318]*

Hetero test: F(2,78) = 27.065 [0.0000]**

Hetero-X test: F(2,78) = 27.065 [0.0000]**

RESET23 test: F(2,77) = 901.70 [0.0000]**

EQ( 8) Modelling Y by OLS

The dataset is: M:\ECON 4160\data\Sp8102.xls

The estimation sample is: 1980(2) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Y_1 0.910625 0.04064 22.4 0.0000 0.8685

Constant -0.698079 0.3294 -2.12 0.0374 0.0558

X 0.0476839 0.09402 0.507 0.6135 0.0034

X_1 0.0943174 0.1007 0.937 0.3517 0.0114

sigma 0.85749 RSS 55.8820059

R^2 0.93712 F(3,76) = 377.5 [0.000]**

Adj.R^2 0.934638 log-likelihood -99.1637

no. of observations 80 no. of parameters 4

mean(Y) 3.01193 se(Y) 3.35402

AR 1-5 test: F(5,71) = 0.33846 [0.8880]

ARCH 1-4 test: F(4,72) = 0.58851 [0.6720]

Normality test: Chi^2(2) = 1.3638 [0.5057]

Hetero test: F(6,73) = 0.73998 [0.6192]

Hetero-X test: F(9,70) = 1.0028 [0.4462]

RESET23 test: F(2,74) = 2.2272 [0.1150]

EQ( 9) Modelling Y by OLS

The dataset is: M:\ECON 4160\data\Sp8103.xls

The estimation sample is: 1980(2) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Y_1 0.0395480 0.1093 0.362 0.7185 0.0017

Constant -0.0732441 0.09597 -0.763 0.4477 0.0076

X 0.0208488 0.09170 0.227 0.8208 0.0007

X_1 0.227191 0.09151 2.48 0.0152 0.0750

sigma 0.856199 RSS 55.7138403

R^2 0.0769963 F(3,76) = 2.113 [0.106]

Adj.R^2 0.040562 log-likelihood -99.0432

no. of observations 80 no. of parameters 4

mean(Y) -0.0669558 se(Y) 0.87411

AR 1-5 test: F(5,71) = 0.62517 [0.6811]

ARCH 1-4 test: F(4,72) = 0.70692 [0.5898]

Normality test: Chi^2(2) = 0.35271 [0.8383]

Hetero test: F(6,73) = 0.68527 [0.6620]

Hetero-X test: F(9,70) = 0.77278 [0.6417]

RESET23 test: F(2,74) = 0.99447 [0.3748]

Sp8102:

• R-squaredhigh
• The coefficients to the X’s is not significant
• Coefficient to the lagged Y is significant
• The tests are ok

Sp8103:

• R-squared low
• The coefficients to the X and lagged Y are not significant
• The coefficient to lagged X is significant
• The tests are ok

Model 3

RALS

EQ(10) Modelling Y by RALS

The dataset is: M:\ECON 4160\data\Sp8101.xls

The estimation sample is: 1980(2) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Constant -6.92801 4.495 -1.54 0.1274 0.0299

X 0.792874 0.02379 33.3 0.0000 0.9352

Uhat_1 1.05356 0.005487 192. 0.0000 0.9979

sigma 1.37865 RSS 146.35158

no. of observations 80 no. of parameters 3

mean(Y) 212.206 se(Y) 87.7537

NLS using analytical derivatives (eps1=0.0001; eps2=0.005):

Strong convergence

Roots of error polynomial:

real imag modulus

1.0536 0.00000 1.0536

ARCH 1-4 test: F(4,72) = 2.5384 [0.0472]*

Normality test: Chi^2(2) = 1.5593 [0.4586]

Hetero test: F(2,77) = 0.56747 [0.5693]

Hetero-X test: F(2,77) = 0.56747 [0.5693]

EQ(11) Modelling Y by RALS

The dataset is: M:\ECON 4160\data\Sp8102.xls

The estimation sample is: 1980(2) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Constant 0.186602 5.575 0.0335 0.9734 0.0000

X 0.00911751 0.09517 0.0958 0.9239 0.0001

Uhat_1 0.976263 0.03010 32.4 0.0000 0.9318

sigma 0.881768 RSS 59.8686801

no. of observations 80 no. of parameters 3

mean(Y) 3.01193 se(Y) 3.35402

NLS using analytical derivatives (eps1=0.0001; eps2=0.005):

Strong convergence

Roots of error polynomial:

real imag modulus

0.97626 0.00000 0.97626

ARCH 1-4 test: F(4,72) = 0.99624 [0.4153]

Normality test: Chi^2(2) = 1.0621 [0.5880]

Hetero test: F(2,77) = 0.23177 [0.7937]

Hetero-X test: F(2,77) = 0.23177 [0.7937]

EQ(12) Modelling Y by RALS

The dataset is: M:\ECON 4160\data\Sp8103.xls

The estimation sample is: 1980(2) - 2000(1)

Coefficient Std.Error t-value t-prob Part.R^2

Constant -0.0679270 0.1038 -0.654 0.5150 0.0055

X -0.00438685 0.09414 -0.0466 0.9630 0.0000

Uhat_1 0.0469083 0.1129 0.416 0.6789 0.0022

sigma 0.884456 RSS 60.2342082

no. of observations 80 no. of parameters 3

mean(Y) -0.0669558 se(Y) 0.87411

NLS using analytical derivatives (eps1=0.0001; eps2=0.005):

Strong convergence

Roots of error polynomial:

real imag modulus

0.046908 0.00000 0.046908

ARCH 1-4 test: F(4,72) = 0.61528 [0.6530]

Normality test: Chi^2(2) = 1.7531 [0.4162]

Hetero test: F(2,77) = 0.12131 [0.8859]

Hetero-X test: F(2,77) = 0.12131 [0.8859]

Sp8101

• The autocorrelation-coefficient is bigger than 1. This means that the model is explosive and therefore unstable.
• The tests are ok. Which is logical because Rals-estimation take into account the autocorrelation.

Sp8102

• The autocorrelation-coefficient is less than 1, which means that the model is stable. But since its value is 0,97, the speed of convergence is low.
• The tests satisfy the classical assumptions

Sp8103:

• The autocorrelation-coefficient is very low, 0,0469, which reflects a high speed of convergence. This means that the model is very stable.

Exercise 10)

a)Using the equations given in this exercise;

pt = aqt+ bst+ ut1

qt = cpt+ edt+ ut2

X1 = qt = quantum

X2 = pt = pris

X3 = st = supply

X4 = dt = demand

Equation 1 consists of the variable, st, which is not represented in the demand equation. On the other hand, we see that the demand function also consists of a variable, dt, that is not represented in the supply equation. This is the requirement of exact identification of both equations in this system.

Ox Professional version 6.00 (Windows_64/U/MT) (C) J.A. Doornik, 1994-2009

---- PcGive 13.0 session started at 10:24:16 on 8-10-2010 ----

SYS( 1) Estimating the system by OLS (RF)

The dataset is: M:\ECON 4160\data\bfm101.xls

The estimation sample is: 1 - 400

URF equation for: X1

Coefficient Std.Error t-value t-prob

X3 0.376430 0.04132 9.11 0.0000

X4 0.146745 0.04260 3.44 0.0006

Constant U 0.103629 0.06013 1.72 0.0856

sigma = 1.19601 RSS = 567.8859679

URF equation for: X2

Coefficient Std.Error t-value t-prob

X3 -0.219346 0.01676 -13.1 0.0000

X4 0.378187 0.01728 21.9 0.0000

Constant U 0.0579151 0.02440 2.37 0.0181

sigma = 0.485192 RSS = 93.4582914

log-likelihood -729.665772 -T/2log|Omega| 405.485055

|Omega| 0.13167411 log|Y'Y/T| -0.0451642009

R^2(LR) 0.862243 R^2(LM) 0.49697

no. of observations 400 no. of parameters 6

F-test on regressors except unrestricted: F(4,792) = 335.467 [0.0000] **

F-tests on retained regressors, F(2,396) =

X3 552.144 [0.000]** X4 469.487 [0.000]**

Constant U 2.82915 [0.060]

correlation of URF residuals (standard deviations on diagonal)

X1 X2

X1 1.1960 0.77656

X2 0.77656 0.48519

correlation between actual and fitted

X1 X2

0.43589 0.79185

Single-equation diagnostics using reduced-form residuals:

X1 : Portmanteau(12): Chi^2(12) = 19.394 [0.0795]

X1 : AR 1-2 test: F(2,395) = 3.0443 [0.0487]*

X1 : ARCH 1-1 test: F(1,398) = 1.3745 [0.2417]

X1 : Normality test: Chi^2(2) = 0.48019 [0.7866]

X1 : Hetero test: F(4,395) = 1.1420 [0.3363]

X1 : Hetero-X test: F(5,394) = 1.1468 [0.3350]

X2 : Portmanteau(12): Chi^2(12) = 16.886 [0.1539]

X2 : AR 1-2 test: F(2,395) = 1.6114 [0.2009]

X2 : ARCH 1-1 test: F(1,398) = 0.11376 [0.7361]

X2 : Normality test: Chi^2(2) = 3.2973 [0.1923]

X2 : Hetero test: F(4,395) = 1.1115 [0.3506]

X2 : Hetero-X test: F(5,394) = 0.95466 [0.4455]

Vector Portmanteau(12): Chi^2(48) = 47.614 [0.4886]

Vector AR 1-2 test: F(8,784) = 1.0564 [0.3919]

Vector Normality test: Chi^2(4) = 5.9089 [0.2061]

Vector Hetero test: F(12,1040)= 0.80499 [0.6456]

Vector Hetero-X test: F(15,1082)= 0.85315 [0.6179]

Vector RESET23 test: F(8,784) = 0.82584 [0.5799]

MOD( 2) Estimating the model by 1SLS (SF)

The dataset is: M:\ECON 4160\data\bfm101.xls

The estimation sample is: 1 - 400

Equation for: X1 (quantum)

Coefficient Std.Error t-value t-prob

X2 0.820108 0.1060 7.74 0.0000

X4 -0.180153 0.05974 -3.02 0.0027

Constant U -0.00314324 0.06207 -0.0506 0.9596

sigma = 1.22593

Equation for: X2 (price)

Coefficient Std.Error t-value t-prob

X3 -0.371772 0.02091 -17.8 0.0000

X1 0.380681 0.02287 16.6 0.0000

Constant U -0.0168808 0.02780 -0.607 0.5440

sigma = 0.553067

log-likelihood -1126.41198 -T/2log|Omega| 8.7388452

no. of observations 400 no. of parameters 6

No restrictions imposed

correlation of structural residuals (standard deviations on diagonal)

X1 X2

X1 1.2259 0.00000

X2 0.00000 0.55307

Single-equation diagnostics using reduced-form residuals:

X1 : AR 1-2 test: F(2,395) = 238.62 [0.0000]**

X1 : ARCH 1-1 test: F(1,398) = 2.0339 [0.1546]

X1 : Normality test: Chi^2(2) = 4.3366 [0.1144]

X1 : Hetero test: F(4,395) = 26.184 [0.0000]**

X1 : Hetero-X test: F(5,394) = 34.393 [0.0000]**

X2 : AR 1-2 test: F(2,395) = 559.43 [0.0000]**

X2 : ARCH 1-1 test: F(1,398) = 0.075181 [0.7841]

X2 : Normality test: Chi^2(2) = 0.36136 [0.8347]

X2 : Hetero test: F(4,395) = 40.440 [0.0000]**

X2 : Hetero-X test: F(5,394) = 97.155 [0.0000]**

Vector Normality test: Chi^2(4) = 2.2586 [0.6883]

Vector Hetero test: F(12,1040)= 41.372 [0.0000]**

Vector Hetero-X test: F(15,1082)= 56.822 [0.0000]**

MOD( 3) Estimating the model by 2SLS

The dataset is: M:\ECON 4160\data\bfm101.xls

The estimation sample is: 1 - 400

Equation for: X1

Coefficient Std.Error t-value t-prob

X2 -1.71615 0.3017 -5.69 0.0000

X4 0.795769 0.1347 5.91 0.0000

Constant U 0.203020 0.09915 2.05 0.0413

sigma = 1.91585

Equation for: X2

Coefficient Std.Error t-value t-prob

X3 -1.18947 0.2634 -4.52 0.0000

X1 2.57716 0.6609 3.90 0.0001

Constant U -0.209154 0.1482 -1.41 0.1590

sigma = 2.72275

log-likelihood -729.665772 -T/2log|Omega| 405.485055

no. of observations 400 no. of parameters 6

No restrictions imposed

correlation of structural residuals (standard deviations on diagonal)

X1 X2

X1 1.9158 -0.92495

X2 -0.92495 2.7228

Single-equation diagnostics using reduced-form residuals:

X1 : AR 1-2 test: F(2,395) = 3.0443 [0.0487]*

X1 : ARCH 1-1 test: F(1,398) = 1.3745 [0.2417]

X1 : Normality test: Chi^2(2) = 0.48019 [0.7866]

X1 : Hetero test: F(4,395) = 1.1420 [0.3363]

X1 : Hetero-X test: F(5,394) = 1.1468 [0.3350]

X2 : AR 1-2 test: F(2,395) = 1.6114 [0.2009]

X2 : ARCH 1-1 test: F(1,398) = 0.11376 [0.7361]

X2 : Normality test: Chi^2(2) = 3.2973 [0.1923]

X2 : Hetero test: F(4,395) = 1.1115 [0.3506]

X2 : Hetero-X test: F(5,394) = 0.95466 [0.4455]

Vector Normality test: Chi^2(4) = 5.9089 [0.2061]

Vector Hetero test: F(12,1040)= 0.80499 [0.6456]

Vector Hetero-X test: F(15,1082)= 0.85315 [0.6179]

• Each equation in the system has an endogenous variable along with the other exogenous variables on the RHS. This violates the classical assumption about zero conditional mean in the disturbance term, since we condition on all the RHS variables . OLS on the system gives biased and inconsistent estimators.
• OLS regression result show that the estimate of X2 is positive on the demand function. This contradict the economic theory about that an increase in price should dampen the demand. The other estimates correspond to economic theory.
• 2SLS, ILS and IV gives the same result, when it is exact identification. This can be shown by theory.