BS2247 Introduction to Econometrics Lecture 7: The multiple - - PowerPoint PPT Presentation

BS2247 Introduction to Econometrics Lecture 7: The multiple regression model

Testing single linear hypothesis (i.e., t statistic, p-value, confidence intervals, etc.)

• Dr. Kai Sun

Aston Business School

1 / 28

Why testing for hypothesis?

◮ OLS estimates, ˆ

β’s, are random variables - they are estimated from data, which are random draws from population.

◮ Intuitively, because ˆ

β’s are random, we want to test if they are close enough to a particular value.

◮ For example, if we obtained OLS estimate ˆ

β1 = 1 (i.e., the estimated β1 is 1), we must keep in mind that the true β1 may not necessarily be 1, although the true β1 should be close to 1, such as 1.2, etc.

◮ Because we never know the true β1, we want to test if β1 = 1

(or some other values) or not.

2 / 28

Why testing for hypothesis?

◮ OLS estimates, ˆ

β’s, are random variables - they are estimated from data, which are random draws from population.

◮ Intuitively, because ˆ

β’s are random, we want to test if they are close enough to a particular value.

◮ For example, if we obtained OLS estimate ˆ

β1 = 1 (i.e., the estimated β1 is 1), we must keep in mind that the true β1 may not necessarily be 1, although the true β1 should be close to 1, such as 1.2, etc.

◮ Because we never know the true β1, we want to test if β1 = 1

(or some other values) or not.

2 / 28

Why testing for hypothesis?

◮ OLS estimates, ˆ

β’s, are random variables - they are estimated from data, which are random draws from population.

◮ Intuitively, because ˆ

β’s are random, we want to test if they are close enough to a particular value.

◮ For example, if we obtained OLS estimate ˆ

β1 = 1 (i.e., the estimated β1 is 1), we must keep in mind that the true β1 may not necessarily be 1, although the true β1 should be close to 1, such as 1.2, etc.

◮ Because we never know the true β1, we want to test if β1 = 1

(or some other values) or not.

2 / 28

Why testing for hypothesis?

◮ OLS estimates, ˆ

β’s, are random variables - they are estimated from data, which are random draws from population.

◮ Intuitively, because ˆ

β’s are random, we want to test if they are close enough to a particular value.

◮ For example, if we obtained OLS estimate ˆ

β1 = 1 (i.e., the estimated β1 is 1), we must keep in mind that the true β1 may not necessarily be 1, although the true β1 should be close to 1, such as 1.2, etc.

◮ Because we never know the true β1, we want to test if β1 = 1

(or some other values) or not.

2 / 28

Assumptions of the Classical Linear Model (CLM)

◮ So far, we know that

OLS estimator is unbiased under MLR.1-MLR.4 OLS estimator is BLUE under MLR.1-MLR.5 (i.e., the Gauss-Markov assumptions)

◮ In order to do classical hypothesis testing, we need to add

another assumption: Assumption MLR.6: u is independent of x1, x2, . . . , xk, and u is normally distributed with zero mean and variance σ2: u ∼ N(0, σ2)

3 / 28

Assumptions of the Classical Linear Model (CLM)

◮ So far, we know that

OLS estimator is unbiased under MLR.1-MLR.4 OLS estimator is BLUE under MLR.1-MLR.5 (i.e., the Gauss-Markov assumptions)

◮ In order to do classical hypothesis testing, we need to add

another assumption: Assumption MLR.6: u is independent of x1, x2, . . . , xk, and u is normally distributed with zero mean and variance σ2: u ∼ N(0, σ2)

3 / 28

◮ MLR.6 also means that u|x1, . . . , xk ∼ N(0, σ2), or

y|x1, . . . , xk ∼ N(β0 + β1x1 + . . . + βkxk, σ2)

◮ Assumption MLR.1-MLR.6:

Classical Linear Model (CLM) assumptions

◮ Under the CLM assumptions, OLS estimator is not only

BLUE, but is the minimum variance unbiased estimator (MVUE) - the model may not necessarily be linear in parameter!

◮ For large sample, the normality assumption of u becomes less

• important. But here we just make the normality assumption.

4 / 28

◮ MLR.6 also means that u|x1, . . . , xk ∼ N(0, σ2), or

y|x1, . . . , xk ∼ N(β0 + β1x1 + . . . + βkxk, σ2)

◮ Assumption MLR.1-MLR.6:

Classical Linear Model (CLM) assumptions

◮ Under the CLM assumptions, OLS estimator is not only

BLUE, but is the minimum variance unbiased estimator (MVUE) - the model may not necessarily be linear in parameter!

◮ For large sample, the normality assumption of u becomes less

• important. But here we just make the normality assumption.

4 / 28

◮ MLR.6 also means that u|x1, . . . , xk ∼ N(0, σ2), or

y|x1, . . . , xk ∼ N(β0 + β1x1 + . . . + βkxk, σ2)

◮ Assumption MLR.1-MLR.6:

Classical Linear Model (CLM) assumptions

◮ Under the CLM assumptions, OLS estimator is not only

BLUE, but is the minimum variance unbiased estimator (MVUE) - the model may not necessarily be linear in parameter!

◮ For large sample, the normality assumption of u becomes less

• important. But here we just make the normality assumption.

4 / 28

◮ MLR.6 also means that u|x1, . . . , xk ∼ N(0, σ2), or

y|x1, . . . , xk ∼ N(β0 + β1x1 + . . . + βkxk, σ2)

◮ Assumption MLR.1-MLR.6:

Classical Linear Model (CLM) assumptions

◮ Under the CLM assumptions, OLS estimator is not only

BLUE, but is the minimum variance unbiased estimator (MVUE) - the model may not necessarily be linear in parameter!

◮ For large sample, the normality assumption of u becomes less

• important. But here we just make the normality assumption.

4 / 28

Distribution of ˆ β

◮ Under the CLM assumptions, it can be shown that

ˆ β|x ∼ N(β, Var(ˆ β|x))

◮ In the simple regression case, we have

ˆ β1 =

i xi−¯ x

• i(xi−¯

x)2 yi = i aiyi

ˆ β1 is a linear combination of yi “If a random variable is a linear combination of a normally distributed random variable, then this random variable is also normally distributed.”

5 / 28

Distribution of ˆ β

◮ Under the CLM assumptions, it can be shown that

ˆ β|x ∼ N(β, Var(ˆ β|x))

◮ In the simple regression case, we have

ˆ β1 =

i xi−¯ x

• i(xi−¯

x)2 yi = i aiyi

ˆ β1 is a linear combination of yi “If a random variable is a linear combination of a normally distributed random variable, then this random variable is also normally distributed.”

5 / 28

Distribution of ˆ β

◮ Under the CLM assumptions, it can be shown that

ˆ β|x ∼ N(β, Var(ˆ β|x))

◮ In the simple regression case, we have

ˆ β1 =

i xi−¯ x

• i(xi−¯

x)2 yi = i aiyi

ˆ β1 is a linear combination of yi “If a random variable is a linear combination of a normally distributed random variable, then this random variable is also normally distributed.”

5 / 28

◮ Similarly, in the multiple regression case, we have

ˆ βj|x1, . . . , xk ∼ N(βj, Var(ˆ βj|x1, . . . , xk))

◮ Standardize ˆ

βj, ˆ βj − βj sd(ˆ βj) ∼ N(0, 1)

6 / 28

The t Test

◮ Replace sd(ˆ

βj) =

• Var(ˆ

βj) (which depends on true σ2) with se(ˆ βj) =

• Var(ˆ

βj) (which depends on the estimated σ2, ˆ σ2), we can get a calculated t statistic: tCALC =

ˆ βj−βj se(ˆ βj) ∼ tn−k−1 ◮ We used se(·) instead of sd(·) is because σ2 is unknown, and

we have to estimate it.

◮ Degrees of freedom (DF) = n-(k+1)

= the number of observations - the number of parameters (i.e., number of “free” observations)

7 / 28

The t Test

◮ Replace sd(ˆ

βj) =

• Var(ˆ

βj) (which depends on true σ2) with se(ˆ βj) =

• Var(ˆ

βj) (which depends on the estimated σ2, ˆ σ2), we can get a calculated t statistic: tCALC =

ˆ βj−βj se(ˆ βj) ∼ tn−k−1 ◮ We used se(·) instead of sd(·) is because σ2 is unknown, and

we have to estimate it.

◮ Degrees of freedom (DF) = n-(k+1)

= the number of observations - the number of parameters (i.e., number of “free” observations)

7 / 28

The t Test

◮ Replace sd(ˆ

βj) =

• Var(ˆ

βj) (which depends on true σ2) with se(ˆ βj) =

• Var(ˆ

βj) (which depends on the estimated σ2, ˆ σ2), we can get a calculated t statistic: tCALC =

ˆ βj−βj se(ˆ βj) ∼ tn−k−1 ◮ We used se(·) instead of sd(·) is because σ2 is unknown, and

we have to estimate it.

◮ Degrees of freedom (DF) = n-(k+1)

= the number of observations - the number of parameters (i.e., number of “free” observations)

7 / 28

◮ Knowing the sampling distribution for the standardized

estimator allows us to carry out hypothesis tests

◮ First, state a null hypothesis, for example,

H0 : βj = 0 (CANNOT write as ˆ βj = 0) If accept null, then accept that xj has no effect on y, controlling for other x’s

◮ Second, form the calculated t statistic:

tCALC =

ˆ βj−0 se(ˆ βj)

8 / 28

◮ Knowing the sampling distribution for the standardized

estimator allows us to carry out hypothesis tests

◮ First, state a null hypothesis, for example,

H0 : βj = 0 (CANNOT write as ˆ βj = 0) If accept null, then accept that xj has no effect on y, controlling for other x’s

◮ Second, form the calculated t statistic:

tCALC =

ˆ βj−0 se(ˆ βj)

8 / 28

◮ Knowing the sampling distribution for the standardized

estimator allows us to carry out hypothesis tests

◮ First, state a null hypothesis, for example,

H0 : βj = 0 (CANNOT write as ˆ βj = 0) If accept null, then accept that xj has no effect on y, controlling for other x’s

◮ Second, form the calculated t statistic:

tCALC =

ˆ βj−0 se(ˆ βj)

8 / 28

◮ Third, besides our null, H0, we need

(1) an alternative hypothesis, H1, which may be one-sided, or two-sided H1 : βj > 0 or H1 : βj < 0 is one-sided alternative H1 : βj = 0 is a two-sided alternative (2) a significance level, α (can be 1%, 5%, or 10%) It is a threshold value to determine if the probability of rejecting the null when it is true (i.e., committing type I error) is large or

• not. Alternatively, it answers the question: how confidently

can I reject the null? If α = 5%, then the confidence level that the null is not rejected when it is true is 1 − α = 95%.

9 / 28

◮ Third, besides our null, H0, we need

(1) an alternative hypothesis, H1, which may be one-sided, or two-sided H1 : βj > 0 or H1 : βj < 0 is one-sided alternative H1 : βj = 0 is a two-sided alternative (2) a significance level, α (can be 1%, 5%, or 10%) It is a threshold value to determine if the probability of rejecting the null when it is true (i.e., committing type I error) is large or

• not. Alternatively, it answers the question: how confidently

can I reject the null? If α = 5%, then the confidence level that the null is not rejected when it is true is 1 − α = 95%.

9 / 28

One-Sided Alternatives: H1 : βj > 0

H1 : βj > 0

◮ Based on the significance level, say, α = 0.05, and degrees of

freedom, n − k − 1, we can look up the (1 − α)-th percentile in a t distribution, call this tCritical, the critical t statistic. Refer to Appendix G.

◮ Rejection rule:

(1) Reject the null hypothesis if the tCALC > tCritical (2) Cannot reject the null hypothesis if the tCALC ≤ tCritical

10 / 28

One-Sided Alternatives: H1 : βj > 0

H1 : βj > 0

◮ Based on the significance level, say, α = 0.05, and degrees of

freedom, n − k − 1, we can look up the (1 − α)-th percentile in a t distribution, call this tCritical, the critical t statistic. Refer to Appendix G.

◮ Rejection rule:

(1) Reject the null hypothesis if the tCALC > tCritical (2) Cannot reject the null hypothesis if the tCALC ≤ tCritical

10 / 28

One-sided t test graph for H1 : βj > 0

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• r

!"#$ %

11 / 28

One-Sided Alternatives: H1 : βj < 0

If the alternative hypothesis is: H1 : βj < 0 Rejection rule: (1) Reject the null hypothesis if the tCALC < −tCritical (2) Cannot reject the null hypothesis if the tCALC ≥ −tCritical

12 / 28

Two-Sided Alternatives: H1 : βj = 0

H1 : βj = 0 Rejection rule: (1) Reject the null hypothesis if the |tCALC| > tCritical (2) Cannot reject the null hypothesis if the |tCALC| ≤ tCritical

13 / 28

Two-sided t test graph

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• r

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14 / 28

Summary for H0 : βj = 0

◮ Unless otherwise stated, the alternative is assumed to be

two-sided

◮ If we reject the null, we typically say “xj is statistically

significant at the α level”

◮ If we cannot reject the null, we typically say “xj is statistically

insignificant at the α level”

15 / 28

Summary for H0 : βj = 0

◮ Unless otherwise stated, the alternative is assumed to be

two-sided

◮ If we reject the null, we typically say “xj is statistically

significant at the α level”

◮ If we cannot reject the null, we typically say “xj is statistically

insignificant at the α level”

15 / 28

Summary for H0 : βj = 0

◮ Unless otherwise stated, the alternative is assumed to be

two-sided

◮ If we reject the null, we typically say “xj is statistically

significant at the α level”

◮ If we cannot reject the null, we typically say “xj is statistically

insignificant at the α level”

15 / 28

Testing other hypotheses

◮ A more general form of the t statistic recognizes that we may

want to test something like H0 : βj = aj

◮ The appropriate calculated t statistic is:

tCALC =

ˆ βj−aj se(ˆ βj) ∼ tn−k−1 ◮ aj = 0 is just a special case. However, most softwares, like

Stata, report tCALC based on aj = 0, i.e., we are most often interested in if xj is statistically significant or not.

16 / 28

Testing other hypotheses

◮ A more general form of the t statistic recognizes that we may

want to test something like H0 : βj = aj

◮ The appropriate calculated t statistic is:

tCALC =

ˆ βj−aj se(ˆ βj) ∼ tn−k−1 ◮ aj = 0 is just a special case. However, most softwares, like

Stata, report tCALC based on aj = 0, i.e., we are most often interested in if xj is statistically significant or not.

16 / 28

Testing other hypotheses

◮ A more general form of the t statistic recognizes that we may

want to test something like H0 : βj = aj

◮ The appropriate calculated t statistic is:

tCALC =

ˆ βj−aj se(ˆ βj) ∼ tn−k−1 ◮ aj = 0 is just a special case. However, most softwares, like

Stata, report tCALC based on aj = 0, i.e., we are most often interested in if xj is statistically significant or not.

16 / 28

Confidence Intervals

◮ ˆ

βj is sometimes called point estimate.

◮ But actually we can use classical hypothesis testing to

construct a confidence interval of the true βj, within which we are “confident” that the true βj should lie.

◮ The interval’s lower and upper bound gives interval estimate. ◮ Recall tCALC = ˆ βj−βj se(ˆ βj) and we cannot reject the null hypothesis

if the |tCALC| ≤ tCritical, so −tCritical ≤ tCALC ≤ tCritical can be thought of as the confidence interval of tCALC.

17 / 28

Confidence Intervals

◮ ˆ

βj is sometimes called point estimate.

◮ But actually we can use classical hypothesis testing to

construct a confidence interval of the true βj, within which we are “confident” that the true βj should lie.

◮ The interval’s lower and upper bound gives interval estimate. ◮ Recall tCALC = ˆ βj−βj se(ˆ βj) and we cannot reject the null hypothesis

if the |tCALC| ≤ tCritical, so −tCritical ≤ tCALC ≤ tCritical can be thought of as the confidence interval of tCALC.

17 / 28

Confidence Intervals

◮ ˆ

βj is sometimes called point estimate.

◮ But actually we can use classical hypothesis testing to

construct a confidence interval of the true βj, within which we are “confident” that the true βj should lie.

◮ The interval’s lower and upper bound gives interval estimate. ◮ Recall tCALC = ˆ βj−βj se(ˆ βj) and we cannot reject the null hypothesis

if the |tCALC| ≤ tCritical, so −tCritical ≤ tCALC ≤ tCritical can be thought of as the confidence interval of tCALC.

17 / 28

Confidence Intervals

◮ ˆ

βj is sometimes called point estimate.

◮ But actually we can use classical hypothesis testing to

construct a confidence interval of the true βj, within which we are “confident” that the true βj should lie.

◮ The interval’s lower and upper bound gives interval estimate. ◮ Recall tCALC = ˆ βj−βj se(ˆ βj) and we cannot reject the null hypothesis

if the |tCALC| ≤ tCritical, so −tCritical ≤ tCALC ≤ tCritical can be thought of as the confidence interval of tCALC.

17 / 28

Confidence Intervals

◮ But we are ultimately interested in βj.

Since tCALC is a function of βj, we can solve for βj as ˆ βj − tCritical · se(ˆ βj) ≤ βj ≤ ˆ βj + tCritical · se(ˆ βj)

◮ Recall that tCritical is determined by α. Say,

if α = 0.05, then we are (1 − α) × 100% = 95% confident that the true βj lies in the interval, ˆ βj ± tCritical · se(ˆ βj).

18 / 28

Confidence Intervals

◮ But we are ultimately interested in βj.

Since tCALC is a function of βj, we can solve for βj as ˆ βj − tCritical · se(ˆ βj) ≤ βj ≤ ˆ βj + tCritical · se(ˆ βj)

◮ Recall that tCritical is determined by α. Say,

if α = 0.05, then we are (1 − α) × 100% = 95% confident that the true βj lies in the interval, ˆ βj ± tCritical · se(ˆ βj).

18 / 28

Computing p-values for t statistics

◮ The p-value provides answer to the question

“What is the smallest significance level at which the null would be rejected?”

◮ This indicates that we can find the p-value based on tCALC. ◮ In fact, we were trying to find tCritical based on α. ◮ Technically,

for one-sided test, p = P(t > tCALC) or p = P(t < −tCALC) for two-sided test, p = P(|t| > tCALC) = 2P(t > tCALC)

◮ Rejection rule based on p-value:

Reject the null hypothesis if the p < α

19 / 28

Computing p-values for t statistics

◮ The p-value provides answer to the question

“What is the smallest significance level at which the null would be rejected?”

◮ This indicates that we can find the p-value based on tCALC. ◮ In fact, we were trying to find tCritical based on α. ◮ Technically,

for one-sided test, p = P(t > tCALC) or p = P(t < −tCALC) for two-sided test, p = P(|t| > tCALC) = 2P(t > tCALC)

◮ Rejection rule based on p-value:

Reject the null hypothesis if the p < α

19 / 28

Computing p-values for t statistics

◮ The p-value provides answer to the question

“What is the smallest significance level at which the null would be rejected?”

◮ This indicates that we can find the p-value based on tCALC. ◮ In fact, we were trying to find tCritical based on α. ◮ Technically,

for one-sided test, p = P(t > tCALC) or p = P(t < −tCALC) for two-sided test, p = P(|t| > tCALC) = 2P(t > tCALC)

◮ Rejection rule based on p-value:

Reject the null hypothesis if the p < α

19 / 28

Computing p-values for t statistics

◮ The p-value provides answer to the question

“What is the smallest significance level at which the null would be rejected?”

◮ This indicates that we can find the p-value based on tCALC. ◮ In fact, we were trying to find tCritical based on α. ◮ Technically,

for one-sided test, p = P(t > tCALC) or p = P(t < −tCALC) for two-sided test, p = P(|t| > tCALC) = 2P(t > tCALC)

◮ Rejection rule based on p-value:

Reject the null hypothesis if the p < α

19 / 28

Computing p-values for t statistics

◮ The p-value provides answer to the question

“What is the smallest significance level at which the null would be rejected?”

◮ This indicates that we can find the p-value based on tCALC. ◮ In fact, we were trying to find tCritical based on α. ◮ Technically,

for one-sided test, p = P(t > tCALC) or p = P(t < −tCALC) for two-sided test, p = P(|t| > tCALC) = 2P(t > tCALC)

◮ Rejection rule based on p-value:

Reject the null hypothesis if the p < α

19 / 28

◮ Most computer packages, like Stata, will compute the p-value

for you, assuming a two-sided test

◮ If you really want a one-sided alternative, just divide the

two-sided p-value by 2

◮ Stata provides the calculated t statistic, p-value, and 95%

confidence interval for H0 : βj = 0 for you, in columns labeled “t”, “p > |t|” and “[95% Conf. Interval]”, respectively.

20 / 28

◮ Most computer packages, like Stata, will compute the p-value

for you, assuming a two-sided test

◮ If you really want a one-sided alternative, just divide the

two-sided p-value by 2

◮ Stata provides the calculated t statistic, p-value, and 95%

confidence interval for H0 : βj = 0 for you, in columns labeled “t”, “p > |t|” and “[95% Conf. Interval]”, respectively.

20 / 28

◮ Most computer packages, like Stata, will compute the p-value

for you, assuming a two-sided test

◮ If you really want a one-sided alternative, just divide the

two-sided p-value by 2

◮ Stata provides the calculated t statistic, p-value, and 95%

confidence interval for H0 : βj = 0 for you, in columns labeled “t”, “p > |t|” and “[95% Conf. Interval]”, respectively.

20 / 28

Testing a Linear Combination

◮ Suppose instead of testing whether β1 is equal to a constant,

you want to test if it is equal to another parameter, that is H0 : β1 = β2

◮ Use same basic procedure for forming a t statistic

tCALC =

ˆ β1− ˆ β2 se(ˆ β1− ˆ β2)

where se(ˆ β1 − ˆ β2) =

• Var(ˆ

β1 − ˆ β2) =

• Var(ˆ

β1) + Var(ˆ β2) − 2 Cov(ˆ β1, ˆ β2)

◮ Need

Cov(ˆ β1, ˆ β2) to calculate the t statistic.

21 / 28

Testing a Linear Combination

◮ Suppose instead of testing whether β1 is equal to a constant,

you want to test if it is equal to another parameter, that is H0 : β1 = β2

◮ Use same basic procedure for forming a t statistic

tCALC =

ˆ β1− ˆ β2 se(ˆ β1− ˆ β2)

where se(ˆ β1 − ˆ β2) =

• Var(ˆ

β1 − ˆ β2) =

• Var(ˆ

β1) + Var(ˆ β2) − 2 Cov(ˆ β1, ˆ β2)

◮ Need

Cov(ˆ β1, ˆ β2) to calculate the t statistic.

21 / 28

Testing a Linear Combination

◮ Suppose instead of testing whether β1 is equal to a constant,

you want to test if it is equal to another parameter, that is H0 : β1 = β2

◮ Use same basic procedure for forming a t statistic

tCALC =

ˆ β1− ˆ β2 se(ˆ β1− ˆ β2)

where se(ˆ β1 − ˆ β2) =

• Var(ˆ

β1 − ˆ β2) =

• Var(ˆ

β1) + Var(ˆ β2) − 2 Cov(ˆ β1, ˆ β2)

◮ Need

Cov(ˆ β1, ˆ β2) to calculate the t statistic.

21 / 28

Testing a Linear Combination

◮ Many packages, like Stata, will have an option to get it, or

will just perform the test for you

◮ In Stata, after

reg y x1 x2 . . . xk you would type test x1 = x2 to get a p-value for the test

22 / 28

Testing a Linear Combination

◮ Many packages, like Stata, will have an option to get it, or

will just perform the test for you

◮ In Stata, after

reg y x1 x2 . . . xk you would type test x1 = x2 to get a p-value for the test

22 / 28

◮ Alternatively, you can restate the problem to get the test you

• want. Call it “reparameterization.”

◮ For example, we estimated the model

y = β0 + β1x1 + β2x2 + u, and want to test the null hypothesis H0 : β1 = β2

◮ We can restate the null as H0 : θ = β1 − β2 = 0 ◮ Solve for β1 = θ + β2, and plug it into the estimated equation,

y = β0 + (θ + β2)x1 + β2x2 + u = β0 + θx1 + β2(x1 + x2) + u

◮ Then use the simple t statistic as tCALC = ˆ θ−0 se(ˆ θ) ◮ Exercise: what if we want to test H0 : β1 + β2 = 1 using this

reparameterization method?

23 / 28

◮ Alternatively, you can restate the problem to get the test you

• want. Call it “reparameterization.”

◮ For example, we estimated the model

y = β0 + β1x1 + β2x2 + u, and want to test the null hypothesis H0 : β1 = β2

◮ We can restate the null as H0 : θ = β1 − β2 = 0 ◮ Solve for β1 = θ + β2, and plug it into the estimated equation,

y = β0 + (θ + β2)x1 + β2x2 + u = β0 + θx1 + β2(x1 + x2) + u

◮ Then use the simple t statistic as tCALC = ˆ θ−0 se(ˆ θ) ◮ Exercise: what if we want to test H0 : β1 + β2 = 1 using this

reparameterization method?

23 / 28

◮ Alternatively, you can restate the problem to get the test you

• want. Call it “reparameterization.”

◮ For example, we estimated the model

y = β0 + β1x1 + β2x2 + u, and want to test the null hypothesis H0 : β1 = β2

◮ We can restate the null as H0 : θ = β1 − β2 = 0 ◮ Solve for β1 = θ + β2, and plug it into the estimated equation,

y = β0 + (θ + β2)x1 + β2x2 + u = β0 + θx1 + β2(x1 + x2) + u

◮ Then use the simple t statistic as tCALC = ˆ θ−0 se(ˆ θ) ◮ Exercise: what if we want to test H0 : β1 + β2 = 1 using this

reparameterization method?

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◮ Alternatively, you can restate the problem to get the test you

• want. Call it “reparameterization.”

◮ For example, we estimated the model

y = β0 + β1x1 + β2x2 + u, and want to test the null hypothesis H0 : β1 = β2

◮ We can restate the null as H0 : θ = β1 − β2 = 0 ◮ Solve for β1 = θ + β2, and plug it into the estimated equation,

y = β0 + (θ + β2)x1 + β2x2 + u = β0 + θx1 + β2(x1 + x2) + u

◮ Then use the simple t statistic as tCALC = ˆ θ−0 se(ˆ θ) ◮ Exercise: what if we want to test H0 : β1 + β2 = 1 using this

reparameterization method?

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◮ Alternatively, you can restate the problem to get the test you

• want. Call it “reparameterization.”

◮ For example, we estimated the model

y = β0 + β1x1 + β2x2 + u, and want to test the null hypothesis H0 : β1 = β2

◮ We can restate the null as H0 : θ = β1 − β2 = 0 ◮ Solve for β1 = θ + β2, and plug it into the estimated equation,

y = β0 + (θ + β2)x1 + β2x2 + u = β0 + θx1 + β2(x1 + x2) + u

◮ Then use the simple t statistic as tCALC = ˆ θ−0 se(ˆ θ) ◮ Exercise: what if we want to test H0 : β1 + β2 = 1 using this

reparameterization method?

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◮ Alternatively, you can restate the problem to get the test you

• want. Call it “reparameterization.”

◮ For example, we estimated the model

y = β0 + β1x1 + β2x2 + u, and want to test the null hypothesis H0 : β1 = β2

◮ We can restate the null as H0 : θ = β1 − β2 = 0 ◮ Solve for β1 = θ + β2, and plug it into the estimated equation,

y = β0 + (θ + β2)x1 + β2x2 + u = β0 + θx1 + β2(x1 + x2) + u

◮ Then use the simple t statistic as tCALC = ˆ θ−0 se(ˆ θ) ◮ Exercise: what if we want to test H0 : β1 + β2 = 1 using this

reparameterization method?

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Empirical Application

Using the data in CEOSAL1.CSV, estimate an equation to explain salaries of CEOs in terms of annual firm sales, return on equity, (roe, in percentage form), and return on the firm’s stock (ros, in percentage form): log(salary) = β0 + β1 log(sales) + β2roe + β3ros + u.

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The estimated equation is: (1) Find by what percentage is salary predicted to increase if ros increases by 50 points? Answer: The proportionate effect on salary is .00024(50) = .012. To obtain the percentage effect, we multiply this by 100: 1.2%.

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The estimated equation is: (1) Find by what percentage is salary predicted to increase if ros increases by 50 points? Answer: The proportionate effect on salary is .00024(50) = .012. To obtain the percentage effect, we multiply this by 100: 1.2%.

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(2) State and test the null hypothesis that, after controlling for sales and roe, ros has no effect on CEO salary. State the alternative that better stock market performance increases a CEO’s salary. Carry out the test at the 10% significance level. Answer: H0 : β3 = 0, H1 : β3 > 0 The t statistic on ros is .00024/.00054 ≈ .44, and the 10% critical value for a one-tailed test, using df = ∞, is obtained from Table G.2 as 1.282. tCALC = .44 < tCritical = 1.282, and therefore, we fail to reject H0 at the 10% level.

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(2) State and test the null hypothesis that, after controlling for sales and roe, ros has no effect on CEO salary. State the alternative that better stock market performance increases a CEO’s salary. Carry out the test at the 10% significance level. Answer: H0 : β3 = 0, H1 : β3 > 0 The t statistic on ros is .00024/.00054 ≈ .44, and the 10% critical value for a one-tailed test, using df = ∞, is obtained from Table G.2 as 1.282. tCALC = .44 < tCritical = 1.282, and therefore, we fail to reject H0 at the 10% level.

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(3) State and test the null hypothesis that, roe has no ceteris paribus effect on CEO salary. Test this against a two-sided alternative at the 1% level. Answer: H0 : β2 = 0, H1 : β2 = 0 The t statistic on roe is .0174/.0041 ≈ 4.24, and the 1% critical value for a two-tailed test, using df = ∞, is obtained from Table G.2 as 2.576. tCALC = 4.24 > tCritical = 2.576, and therefore, we reject H0 at the 1% level.

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(3) State and test the null hypothesis that, roe has no ceteris paribus effect on CEO salary. Test this against a two-sided alternative at the 1% level. Answer: H0 : β2 = 0, H1 : β2 = 0 The t statistic on roe is .0174/.0041 ≈ 4.24, and the 1% critical value for a two-tailed test, using df = ∞, is obtained from Table G.2 as 2.576. tCALC = 4.24 > tCritical = 2.576, and therefore, we reject H0 at the 1% level.

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Reading

Chapter 4, Introductory Econometrics - A Modern Approach, 4th Edition, J. Wooldridge

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