Hypothesis Testing

1
Hypothesis Testing

• Model Design Choosing the appropriate test
  Interpreting the results.

2
Examples

• Does Air Conditioning affect monthly rent?
• Does being at the beach increase the apartments
  rent by more than 200?
• Do amenities affect rent?
• Does a production function exhibit constant
  returns to scale?

3
Hypothesis Testing Tests Involving A Single
Parameter

• Set Up the Null and Alternative Hypotheses
• H0 b0
• HA b?0
• Choose a Level of Significance and Critical
  t-value
• Run the regression and obtain estimated t.
• Apply the decision rule by comparing the
  calculated t and the critical t-value in order to
  reject or accept the null hypothesis.

4
Test Statistic
5
Example -- Air Conditioning

• LARENT a1a2SQFTa3ACe
• Null Hypothesis?
• H0 a40
• HA a4 gt0
• Appropriate Test?
• Test Involving a Single Parameter
• Two-sided Test

6
Limitations of Significance Tests

• The t-test does not test theoretical validity
• A statistically insignificant result need not
  mean that a regressor has little or no effect on
  the dependent variable.
• The t-test does not test importance.
• The t-test is not intended for tests of the
  entire population.

7
Limitations of Significance Tests

• The t-test does not test theoretical validity
• P 10.9 - 3.2C 0.39C2
• (-13.9) (19.5)
• R2 .982, n21
• where P is consumer price index in Great Britain
  and C is cumulative rainfall.

8
Limitations of Significance Tests

• A statistically insignificant result need not
  mean that a regressor has little or no effect on
  the dependent variable.
• This is an example of Type II Error
• This problem is likely to arise where
• (i) The sample is too small.
• (ii) There is too little variance in the
  observed values of the explanatory variable.
• (iii) There is a high variance in the effects of
  X on the dependent variable.

9
Limitations of Significance Tests

• The t-statistic does not measure importance.
• How Important Is the SAT Score in Determining
  Class Rank?
• The test statistic t5.929/0.11850.25
• A 100 point increase in SAT score improves class
  rank by 6 percentile points

10
Does Being Near Beach Increase Rent by More than
200?

• H0 bgt200HA b0
• Type I error -reject H0 when it is true and lose
  moneyType II error - fail to reject H0 when it
  is false and lose money
• One Sided Test

11
Joint Tests on Several Regression Coefficients

• Set Up the Null and Alternative Hypotheses
• Choose level of significance and critical F value
• Run two regressions and obtain estimated F-value
• Restricted model assumes bs0
• Unrestricted model assumes bs ? 0
• Apply the decision rule

12
F Statistic
13
Do Amenities Matter?

• H0 bac0 bdw0 bpool0 bgym0 bfp0
  balarm0HA bac?0 or bdw?0 or bpool?0 or bgym?0
  or bfp?0 or balarm?0

14
Using F-Test

• Estimate model without ac,dw,fp, pool, gym, alarm
• Use residual (error) sum of squares from this
  model and residual (error) sum of squares from
  original model to calculate F-statistic.
• Degrees of freedom, numeratorq6
• Degrees of freedom,denominator153-10143

15
Testing a Linear Combination of Coefficients

• Does A Cobb Douglas production function exhibit
  constant returns to scale?
• QALaKb
• log Q g a ln L b ln K where gln A
• Set Up Null and Alternative Hypothesis
• H0 ab1
• HA ab?1 (or we may be interested in one tail
  test of economies of scale, abgt1)
• Choose level of significance and critical F

16
Testing a Linear Combination, continued

• Using the restriction, solve for one of three
  coefficients in terms of the others and
  substitute that in the unrestricted model to
  obtain the restricted model.
• Unrestricted model ln Q galn L b ln K
• Restricted model ln Q-ln K galn L - ln K
• Calculate the F-Statistic.