TESTING FOR THE SIGNIFICANCE OF A SUBSET OF COEFFICIENTS-THE WALD TEST

(UNRESTRICTED) MODEL

MODEL A

reg / price sqft / bedrms baths

Source

/ SS / df MS / Number of obs / 14
F (3, 10) / 16.99
Model / 85114.94 / 3. 28371.6473 / Prob > F / 0.0003
Residual / 16700.07 / 10.1670.00687 / R-squared / 0.836
Adj R-squared / 0.7868
Total / 101815 / 13 7831.9239 / Root MSE / 40.866
price / Coef. / Std. Err. t / P>t / [95% Conf. / Interval]
sqft / 0.1548 / .0319404 4.85 / 0.001 / 0.083632 / 0.225968
bedrms / -21.5875 / 27.02933 -0.80 / 0.443 / -81.8126 / 38.63758
baths / -12.1928 / 43.25 -0.28 / 0.784 / -108.56 / 84.17425
_cons / 129.0616 / 88.30326 1.46 / 0.175 / -67.6903 / 325.8136

Wald Test:

test bedrms baths

( 1) bedrms = 0

( 2) baths = 0

F( 2, 10) = 0.47

Prob > F = 0.6375

EXPLANATION OF THE TEST: Wald test here is used to test the joint significance of a subset of coefficients, namely, bedrms and baths. These two variables are individually insignificant based on t-tests with very high p values. But before dropping them together, we may want to test the joint significance of them using Wald test.

The command test bedrms baths tests whether baths and bedrms are insignificant jointly. Since the null says they are, and F-stat’s p-value=0.6375, then we cannot reject the null. DROP both baths and bedrms from the regression equation. They don’t belong to the model.

SPECIAL WALD TEST: This is an F-test for the significance of all variables in the model, i.e. sqft, bedrms and baths. Hence, the null states betas of all variables in the model are set equal to zero.

Null :

Alternative : At least some are non zero.

test bedrms baths sqft

( 1) bedrms = 0

( 2) baths = 0

( 3) sqft = 0

F( 3, 10) = 16.99

Prob > F = 0.0003

Notice that F-test p-value is 0.0003, which is lower than 1% of . Hence, we can reject the null and at least some variables in this trio is significant. This variable is SQFT! (based on the t-test).

RESTRICTED MODEL

MODEL B

reg / price sqft
Source / SS / df MS / Number of obs / 14
F( 1, 12) / 54.86

Model

/ 83541.4429 / 1 83541.4429 / Prob > F / 0
Residual / 18273.5678 / 12 1522.79731 / R-squared / 0.8205
Adj R-squared / 0.8056
Total / 101815.011 / 13 7831.9239 / Root MSE / 39.023
price / Coef. / Std. Err. t / P>t / [95% Conf. / Interval]
sqft / 0.1387503 / .0187329 7.41 / 0 / 0.0979349 / 0.179566
cons / 52.3509 / 37.28549 1.40 / 0.186 / -28.88719 / 133.589