lecture 11 preview: hypothesis testing and the wald test wald test let statistical software do the...

Lecture 11 Preview: Hypothesis Testing and the Wald Test

Wald Test

Let Statistical Software Do the Work

Testing the Significance of the “Entire” Model

No Money Illusion Theory: Calculating Prob[Results IF H0 True]

Clever Algebraic Manipulation

Equivalence of Two-Tailed Student-t Tests and Wald Tests (F-Tests)

Three Important Distributions: Normal, Student-t, and F

Restricted Regression Reflects H0

Unrestricted Regression Reflects H1

Comparing the Restricted Sum of Squared Residuals and the Unrestricted Sum of Squared Residuals: The F-statistic

Two-Tailed t-Test

Wald Test

Dependent Variable: LogQExplanatory Variable(s):

Estimate SE t-Statistic Prob

LogP 0.411812 0.093532 -4.402905 0.0003LogI 0.508061 0.266583 1.905829 0.0711LogChickP 0.124724 0.071415 1.746465 0.0961Const 9.499258 2.348619 4.044615 0.0006

Number of Observations 24 Degrees of Freedom 20

Step 1: Collect data, run the regression, and interpret the estimates

Model: No Money Illusion Theory: P + I + CP = 0

Taking logs: log(Qt) = log(Const) + Plog(Pt) + Ilog(It) + CPlog(ChickPt) + et

Interpreting the coefficient estimates: bI = Income Elasticity of Demand = .51

bP = (Own) Price Elasticity of Demand = .41 bCP = Cross Price Elasticity of Demand = .12

What does the sum of the elasticity estimates equal? bP + bI + bCP = .41 + .51 + .12

Critical Regression Result: The sum of the elasticity estimates is .22; the sum is .22 from 0.A 1 percent increase in all prices and income increases the quantity demanded by .22 percent

Step 2: Play the cynic and challenge the results; construct the null and alternative hypotheses:

H0: P + I + CP = 0 Cynic’s view is correct: No money illusion

H1: P + I + CP 0 Cynic’s view is incorrect: Money illusion present

Cynic’s view: Despite the results, money illusion is not present.

Step 0: Construct a model reflecting the theory to be tested.

No Money Illusion Theory: Increasing all prices and income by the same proportion does not affect the quantity of goods demanded.

=.22

EViews

The evidence suggests that the no money illusion theory is incorrect.

http://www3.amherst.edu/~fwesthoff/MITLinks/PP-BeefDemand-1985-1986.wf1

Step 3: Formulate the question to assess the cynic’s view.

Specific Question: Critical regression result: The sum of elasticity estimates equals .22, the sum is .22 from 0. What is the probability that the sum of elasticity estimates in one regression would be at least .22 from 0, if H0 were true (if the sum of the actual elasticities equaled 0)?

Question: How can we calculate Prob[Results IF H0 True]?

H0: P + I + CP = 0 Cynic is correct: No money illusionH1: P + I + CP ≠ 0 Cynic is incorrect: Money illusion present

Generic Question: What is the probability that the results would be like those we actually obtained (or even stronger), if the cynic were correct and H0 actually true?

Answer: There are three ways:Clever Algebraic Manipulation:

Wald Test

Let Statistical Software Do the Work

Step 4: Use the properties of the estimation procedure to calculate Prob[Results IF H0 True].

Prob[Results IF H0 True] =.43

Dependent Variable: LogQExplanatory Variable(s): Estimate SE t-Statistic Prob

LogPLessLogChickP 0.467358 0.062229 -7.510284 0.0000LogILessLogChickP 0.301906 0.068903 4.381606 0.0003Const 11.36876 0.260482 43.64516 0.0000

Number of Observations 24 Degrees of Freedom 21Sum of Squared Residuals .004825

Calculating Prob[Results IF H0 True] – Method 2: Wald Test

log(Qt) = log(Const) + Plog(Pt) + Ilog(It) + CPlog(ChickPt) + et

Model and Hypotheses:

H0: P + I + CP = 0 No money illusion.H1: P + I + CP 0 Money illusion present

First, consider the restricted model, the model that is consistent with the null hypothesis:H0: P + I + CP = 0 log(Qt) = log(Const) + Plog(Pt) + Ilog(It) + CPlog(ChickPt) + et CP = (P + I) = log(Const) + Plog(Pt) + Ilog(It) (P + I) log(ChickPt) + et

= log(Const) + Plog(Pt) + Ilog(It) Plog(ChickPt) Ilog(ChickPt) + et = log(Const) + Plog(Pt) Plog(ChickPt) + Ilog(It) Ilog(ChickPt) + et

= log(Const) + P[log(Pt) log(ChickPt)] + I[log(It) log(ChickPt)] + et

SSRR = .004825 DFR = 24 3 = 21

Restricted Regression:

EViews

log(Qt)= log(Const) + PLogPLessLogChickPt + ILogILessLogChickPt + et

Two Regressions: A Restricted Regression and an Unrestricted

Regression






Model and Hypotheses:

H0: P + I + CP = 0 H1: P + I + CP 0

Next, consider the unrestricted model, the model that is consistent with the alternative hypothesis. Here we allow the estimates to take on any values; their values are unrestricted:

Unrestricted Regression:

SSRU = .004675 DFU = 24 4 = 20

Compare the sum of squared residuals from the restricted and unrestricted regressions:

SSRR = .004825

In general, SSRR SSRU

Is this a coincidence? No.

The OLS estimation procedure chooses the estimates of the constant and coefficients so as to minimize the sum of squared residuals.The unrestricted regression places no restrictions on the estimates. Imposing a restriction limits the ability to make the sum of squared residuals as small as possible.

Note that SSRR > SSRU

A restriction can never decrease the sum of squared residuals; a restriction can only increase the sum.

No money illusion.Money illusion present

EViews

Lab 11.1


http://www3.amherst.edu/~fwesthoff/MITLinks/MIT-Lab-11-01.html

SSRR will be larger than SSRU.

Restriction not actually true

Restriction should cost much in terms of SSR

Expect SSRR to be

much larger than SSRU

Expect F-statistic to be large

Depends on whether or not H0 is actually true.

If H0 is not actually true

Restriction actually true

Restriction should cost little in terms of SSR

Expect SSRR to be

only a little larger than SSRU

Expect F-statistic to be small

If H0 is actually true

But, by how much?

Restricted regressionUnrestricted regression

How do we decide if the SSRR is much or just a little larger than SSRU?

Question: Can the F-statistic be negative? No. Because SSRR SSRU

H0: P + I + CP = 0 H1: P + I + CP 0

No money illusion.Money illusion present

Calculating the F-Statistic:

SSRR = .004875 DFR = 21

SSRU = .004625 DFU = 20SSRR SSRU = .000150 DFR DFU = 1

H0: P + I + CP = 0 No money illusion

H1: P + I + CP 0 Money illusion present

SSRR = .004875 DFR = 21 SSRU = .004625 DFU = 20

Restricted regression Unrestricted regression

=.000150/1

.004675/20=

.000150

.000234= .64

Cynic’s ViewSure, SSRR is larger than SSRU and the F-statistic is not 0, but this is just the “luck of the draw.” In fact, H0 is true, money illusion is not present, the restriction is true.

H0: P + I + CP = 0 No money illusion Restriction actually true

H1: P + I + CP 0 Money illusion present Restriction actually not trueQuestion Assess the Cynic’s View

Generic Question: What is the probability that the results would be like those we actually obtained (or even stronger), if the cynic were correct and H0 actually true?Specific Question: Regression results: F-statistic equals .64. What is the probability that the F-statistic from one pair of regressions would be .64 or more, if H0 were true (if no money illusion were present or equivalently if the restriction were true)?Answer: Prob[Results IF Cynic Correct] or equivalently Prob[Results IF H0 True]

H0: P + I + CP = 0 No money illusion Restriction actually trueH1: P + I + CP 0 Money illusion present Restriction actually not true

= .64.000150

.004675=

/ 1

/ 20

Prob[Results IF H0 True]

Lab11.3

.43Calculating Prob[Results IF H0 True] Using a Simulation

Prob[Results IF H0 True] small

Unlikely that H0 is true

Prob[Results IF H0 True] large

Likely that H0 is true

Do not reject H0

Reject H0

Specific Question: Regression results: F-statistic equals .64. What is the probability that the F-statistic from one pair of regressions would be .64 or more, if H0 were true (if money illusion is not present or equivalently if the restriction is true)? Answer: Prob[Results IF H0 True]

SSRDem = .004675

SSRNum = .000150

DFDem = 20

DFNum = 1

Calculating Prob[Results IF H0 True] Using the F-Distribution


DFNum = 1

DFDem = 20

.64

.43The F-distribution is described by the degrees of freedom of the numerator and the degrees of freedom of the denominator:

Question: Using the “traditional” significance levels, would you reject H0?Answer: No.

Lab 11.4

But, what if we could not use a simulation?

= .43



Wald Test

Degrees of FreedomValue Num Dem Prob

F-statistic 0.641644 1 20 0.4325





Method 1 – Clever Algebra Approach: Prob[Results IF H0 True]What is the probability that the coefficient estimate from one regression would be at least .22 from 0, if H0 were true (if there were no money illusion, if Clever actually equaled 0)?

= .43

Method 2 – Wald Approach: Prob[Results IF H0 True]What is the probability that the F-statistic from one pair of regressions would be .64 or more, if H0 were true (if there were no money illusion, if actual elasticity sum to 0)?

= .43

Conclusion: While the approaches differ, the two methods produce identical results.In fact, it can be shown rigorously that methods 1 and 2 are equivalent. Calculating Prob[Results IF H0 True] – Method 3: Let statistical software do the work

Coefficient restriction: C(1) + C(2) + C(3) = 0

The F-statistic equals .64. If the restriction we specified were actually true, the probability that the F-statistic from one repetition of the experiment would be .64 or more equals .43.

The probabilities in all three methods are identical.

Run the unrestricted regression:

Note: The F-statistics in the second and third methods are identical.

C(1)C(2)C(3)C(4)

EViews Numbering

Convention:

EViews


Comparing methods 1, 2, and 3:

Method 1 – Clever Algebra Approach: Prob[Results IF H0 True]What is the probability that the coefficient estimate from one regression would be at least .22 from 0, if H0 were true (if there were no money illusion, if Clever actually equaled 0)?

= .43

Method 2 – Wald Approach: Prob[Results IF H0 True]What is the probability that the F-statistic from one pair of regressions would be .64 or more, if H0 were true (if there were no money illusion, if actual elasticity sum to 0)?

= .43

Method 3 – Statistical Software Approach: Prob[Results IF H0 True] = .43

Conclusion: While the approaches differ, the methods produce identical results.





Const 12.30576 0.005752 2139.539 0.0000Number of Observations 24 Degrees of Freedom 23Sum of Squared Residuals .018261

Testing the “Entire” Model

Model: log(Qt) = log(Const) + Plog(Pt) + Ilog(It) + CPlog(ChickPt) + e

H0: P = 0 and I = 0 and CP = 0 The explanatory variables have no effect on the dependent variable.

H1: P 0 and/or I 0 and/or CP 0 At least one explanatory variable has an effect on the dependent variable.

Restricted Regression. H0: P = 0 and I = 0 and CP = 0

Unrestricted Regression. H1: P 0 and/or I 0 and/or CP 0

NB: If the null hypothesis is actually true, we have a good or bad model?log(Qt) = log(Const)

SSRU = .004675 DFU = 24 4 = 20

DFR = 24 1 = 23 SSRR = .018261


EViews

Very bad!


Prob[Results IF H0 True]: The F-statistic equals 19.4. What is the probability that the F-statistic from one pair of regressions would be 19.4 or more, if H0 were true (if prices and income actually had no effect on the quantity demanded, if P, I, and CP actually equaled 0)?

Prob[Results IF H0 True] small Prob[Results IF H0 True] large



SSRR = . 018261 DFR = 24 1 = 23

SSRU = .004675 DFU = 24 – 4 = 20SSRR SSRU = . 013586 DFR DFU = 3

=. 013586 /3

.004675/20=

.004529

.000234= 19.4

Calculate the F-statistic:

Unlikely H0 is true

Reject H0

Likely H0 is true

Do not reject H0

SSRDem = .000234 DFDem

SSRNum = .013586 DFNum

= 20

= 3 Calculating

Prob[Results IF H0 True] Lab 11.5

Prob[Results IF H0 True] < .0001


Wald TestDegrees of Freedom

Value Num Dem ProbF-statistic 19.37223 3 20 0.0000

Ordinary Least Squares (OLS)Dependent Variable: LogQExplanatory Variable(s): Estimate SE t-Statistic Prob


Number of Observations 24 Degrees of Freedom 20Sum of Squared Residuals .004675F-Statistic 19.3722 Prob[F-Statistic] .000004

Prob[Results IF H0 True]: What is the probability that the F-statistic from one pair of regressions would be 19.4 or more, if H0 were true (if prices and income actually have no effect on the quantity demanded, if P, I, and CP actually equaled 0)?

In fact, statistical software calculates this F-statistic and probability automatically:

Prob[Results IF H0 True] < .0001

Prob[Results IF H0 True] small



Unlikely H0 is true

Reject H0

Coefficient restriction: C(1) = C(2) = C(3) = 0Using the “traditional” significance levels

would you reject H0? EViews

Could we let statistical software do the work?

C(1)C(2)C(3)C(4)

EViews Numbering

Convention:

Aside: Can a probability equal precisely 0? No.

EViews







Equivalence of Two-Tailed t-Tests and Wald TestsClaim: A two-tailed t-test is a special case of a Wald Testlog(Qt) = log(Const) + Plog(Pt) + Ilog(It) + CPlog(ChickPt) + et We shall illustrate this by

considering the constant elasticity demand model.

H0: CP = 0H1: CP 0

Price of chicken does not affect the quantity of beef demandedPrice of chicken does affect the quantity of beef demanded

To justify the claim, focus attention on the coefficient of the price of chicken, CP, and consider the following “two-tail” hypotheses:

Prob[Results IF H0 True]: What is the probability that the coefficient estimate in one regression would be at least .12 from 0, if H0 were true (if the actual coefficient, CP, equaled 0; that is, if the price of chicken had no effect on the quantity demanded)? First, we shall calculate the probability using a Student t-test and then a Wald test.

Critical Regression Result: The LogChickP coefficient estimate equals .12. The estimate does not equal 0; the estimate is .12 from 0. This evidence suggests that the chicken price does affect the quantity of beef demanded.

EViews





Use the estimation procedure’s general properties to calculate Prob[Results IF H0 True].

H0: CP = 0 Price of chicken has no effect on the quantity of beef demandedH1: CP 0 Price of chicken has an effect on the quantity of beef demanded

bCP

t-distribution

OLS estimation procedure unbiased

Mean[bCP] = CP SE[bCP]

If H0 were true

Number of observations

Number of parameters

StandardError

DF = 24 4 = 20= 0 = .0714


Tails Probability: Probability that a coefficient estimate, bCP, resulting from one regression would will lie at least .12 from 0, if the coefficient, CP, actually equaled 0.

Estimate was .12: What is the probability that the coefficient estimate in one regression would be at least .12 from 0, if H0 were true (if the coefficient, CP actually equaled 0)?

.0961/2

Mean = 0SE = .0714DF = 20

.120

Tails Probability = .0961.0961/2

= .0961

.12.12


LogP 0.305725 0.074513 -4.102963 0.0005LogI 0.869706 0.175895 4.944466 0.0001Const 6.407302 1.616841 3.962852 0.0007





Wald TestModel: log(Qt) = log(Const) + Plog(Pt) + Ilog(It) + CPlog(ChickPt) + et

“Two-tail” hypothesis:H0: CP = 0 Price of chicken does not affect the quantity of beef demanded H1: CP 0 Price of chicken does affect the quantity of beef demanded

Restricted Regression – the null hypothesis: CP = 0

Unrestricted Regression – the alternative hypothesis: CP 0

SSRR = .005388

SSRU = .004675

DFR = 24 3 = 21

DFU = 24 4 = 20

log(Qt) = log(Const) + Plog(Pt) + Ilog(It) + et

EViews



Wald Test

Degrees of FreedomValue Num Dem Prob

F-statistic 3.050139 1 20 .0961

SSRR = .005388 DFR = 24 3 = 21

SSRU = .004675 DFU = 24 – 4 = 20SSRR SSRU = .000713 DFR DFU = 1

=. 000713 /1

.004675/20= 3.05

H0: CP = 0 Price of chicken has no effect on the quantity demanded. Restricted regression H1: CP 0 Price of chicken has an effect on the quantity demanded. Unrestricted regression

Calculate the F-statistic:

DFNum = 1

DFDem = 20

3.05

.0961

Wald Test: Prob[Results IF H0 True] = .0961The two probabilities are identical.

It can be shown rigorously that the two-tailed Student-t test is a special case of the Wald test in which the restriction requires the coefficient to equal 0.

Let software calculate Prob[Results IF H0 True]:

Prob[Results IF H0 True]: The F-statistic equals 3.05. What is the probability that the F-statistic from one pair of regressions would be 3.05 or more, if the H0 were true (if the actual coefficient, CP, equaled 0; that is, if the price of chicken actually has no effect on the quantity demanded)?

Two-tailed Student-t Test: Prob[Results IF H0 True] = .0961Summary of Prob[Results IF H0 True] Calculations

Prob[Results IF H0 True] = .0961

EViews


Three Important Distributions: Normal, Student-t, and F

Theories Involving a Single Variable

Student-t distribution

Theories InvolvingTwo or More Variables

F distribution

which is described by the

Mean SE DF

DF of the Numerator

DF of the Denominator

If SD is known

Normal distribution

which is described

by the

If SD is not known

which is described

by the

Mean SD

lecture 11 preview: hypothesis testing and the wald test wald test let statistical software do the...

Documents