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Transcript of Part IV Statistical Inference - sjp/Teaching/ecm/lectures/ecmc4.pdfآ Part IV Statistical...
Statistical Inference
Part IV
Statistical Inference
As of Oct 2, 2019 Seppo Pynnönen Econometrics I
Statistical Inference
Sampling Distributions of the OLS Estimator
1 Statistical Inference
Sampling Distributions of the OLS Estimator
Testing for single population coefficients, the t-test
Testing Against One-Sided Alternatives
Two-Sided Alternatives
Other Hypotheses About βj
p-values
Economic, or Practical, versus Statistical Significance
Confidence Intervals for the Coefficients
Simultaneous Testing for Multiple Coefficients, The F -test
Testing General Linear Restrictions
On Reporting the Regression Results
Seppo Pynnönen Econometrics I
Statistical Inference
Sampling Distributions of the OLS Estimator
Regression model
yi = β0 + β1xi1 + · · ·+ βkxik + ui . (1)
To the assumptions 1–5 we add
Assumption 6: The error component u is independent of x1, . . . xk and
u ∼ N(0, σ2u). (2)
Seppo Pynnönen Econometrics I
Statistical Inference
Sampling Distributions of the OLS Estimator
Exploring further 4.1 (W 5e, p. 113)
Suppose that u is independent of the explanatory variables, and it takes on the values
−2, −1, 0, 1, and 2 with probability of 1/5. Does this violate assumptions 1–5 (Gauss-Markov assumptions)? Does this violate assumption 6?
Seppo Pynnönen Econometrics I
Statistical Inference
Sampling Distributions of the OLS Estimator
Remark 1
Assumption 6 implies E[u|x1, . . . , xk ] = E[u] = 0 and var[u|x1, . . . , xk ] = var[u] = σ2u, i.e Assumptions 1 and 2.
Remark 2
Assumption 3, i.e., cov[ui , uj ] = 0 together with Assumption 6 implies
that u1, . . . , un are independent.
Remark 3
Under assumptions 1–6 the OLS estimators β̂1, . . . , β̂k are Minimum
Variance Unbiased Estimators (MVUE). That is they are best among all
unbiased estimators (not only linear).
Seppo Pynnönen Econometrics I
Statistical Inference
Sampling Distributions of the OLS Estimator
Remark 4
y |x ∼ N(β0 + β1x1 + · · ·+ βkxk , σ2u), (3)
where x = (x1, . . . , xk)′ and y |x means ”conditional on x”.
Theorem 1
Under the assumptions 1–6, conditional on the sample values of the explanatory variables
β̂j ∼ N(βj , σ2β̂j ) (4)
and therefore β̂j − βj σβ̂j
∼ N(0, 1), (5)
where σ2 β̂j
= var [ β̂j
] and σβ̂j =
√ var
[ β̂j
] .
Seppo Pynnönen Econometrics I
Statistical Inference
Testing for single population coefficients, the t-test
1 Statistical Inference
Sampling Distributions of the OLS Estimator
Testing for single population coefficients, the t-test
Testing Against One-Sided Alternatives
Two-Sided Alternatives
Other Hypotheses About βj
p-values
Economic, or Practical, versus Statistical Significance
Confidence Intervals for the Coefficients
Simultaneous Testing for Multiple Coefficients, The F -test
Testing General Linear Restrictions
On Reporting the Regression Results
Seppo Pynnönen Econometrics I
Statistical Inference
Testing for single population coefficients, the t-test
Theorem 2
Under the assumptions 1–6
β̂j − βj sβ̂j
∼ tn−k−1, (6)
(the t-distribution with n − k − 1 degrees of freedom) where sβ̂j = se(β̂j) and k + 1 is the number of estimated regression coefficients.
Remark 5
The only difference between (5) and (6) is that in the latter the standard
deviation parameter σβ̂j is replaced by its estimator sβ̂j .
Seppo Pynnönen Econometrics I
Statistical Inference
Testing for single population coefficients, the t-test
In most applications the interest lies in testing the null hypothesis:
H0 : βj = 0. (7)
The t-test statistic is
tβ̂j = β̂j sβ̂j , (8)
which is t-distributed with n − k − 1 degrees of freedom if the null hypothesis is true.
These ”t-ratios” are printed in standard computer output in regression applications.
Seppo Pynnönen Econometrics I
Statistical Inference
Testing for single population coefficients, the t-test
Example 1
Wage example computer output.
Dependent variable: log(wage)
Residuals:
Min 1Q Median 3Q Max
-2.21158 -0.36393 -0.07263 0.29712 1.52339
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.583773 0.097336 5.998 3.74e-09 ***
educ 0.082744 0.007567 10.935 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.4801 on 524 degrees of freedom
Multiple R-squared: 0.1858,Adjusted R-squared: 0.1843
F-statistic: 119.6 on 1 and 524 DF, p-value: < 2.2e-16
Seppo Pynnönen Econometrics I
Statistical Inference
Testing for single population coefficients, the t-test
1 Statistical Inference
Sampling Distributions of the OLS Estimator
Testing for single population coefficients, the t-test
Testing Against One-Sided Alternatives
Two-Sided Alternatives
Other Hypotheses About βj
p-values
Economic, or Practical, versus Statistical Significance
Confidence Intervals for the Coefficients
Simultaneous Testing for Multiple Coefficients, The F -test
Testing General Linear Restrictions
On Reporting the Regression Results
Seppo Pynnönen Econometrics I
Statistical Inference
Testing for single population coefficients, the t-test
One-sided alternatives H1 : βj > 0 (9)
or H1 : βj < 0. (10)
In the former the rejection rule is to reject the null hypothesis at the chosen significance level, α
tβ̂j > cα, (11)
where cα is the 1− α fractile (or percentile) from the t-distribution with n − k − 1 degrees of freedom, such that P(tβ̂j > cα|H0 is true) = α. α is called the significance level of the test. Typically α is 0.05 or 0.01, i.e., 5% or 1%.
In the case of (10) the H0 is rejected if
tβ̂j < −cα. (12)
These tests are one-tailed test.
Seppo Pynnönen Econometrics I
Statistical Inference
Testing for single population coefficients, the t-test
Exploring Further 4.2 (W 5ed, p. 118) (adapted)
Community loan approval rates, apprate, are modeled as
apprate = β0 + β1percmin + β3avginc + β4avgwlth + β5avgdebt + u,
where percmin is the percentage minority in the community, avginc is average income, avgwlth is average wealth, and avgdept is average dept obligations.
Suppose we are interested to investigate whether there is difference in loan approval rates across neighborhoods due to racial and ethnic composition, when average income, average wealth, and average dept have been controlled for. We are particularly interested in whether there is discrimination against minorities in loan approval rates.
How would you state the null hypothesis in terms of the above model? How do you state the alternative hypothesis?
Seppo Pynnönen Econometrics I
Statistical Inference
Testing for single population coefficients, the t-test
Example 2
In the wage example, test
H0 : βexper = 0 against H1 : βexper > 0.
Dependent variable: log(wage)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.284360 0.104190 2.729 0.00656 **
educ 0.092029 0.007330 12.555 < 2e-16 ***
exper 0.004121 0.001723 2.391 0.01714 *
tenure 0.022067 0.003094 7.133 3.29e-12 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.4409 on 522 degrees of freedom
Multiple R-squared: 0.316,Adjusted R-squared: 0.3121
F-statistic: 80.39 on 3 and 522 DF, p-value: < 2.2e-16
Seppo Pynnönen Econometrics I
Statistical Inference
Testing for single population coefficients, the t-test
Example 2 continues
Thus, β̂exper = 0.004121, sβ̂exper = 0.001723 and
tβ̂exper = β̂exper sβ̂exper
= 0.004121
0.001723 ≈ 2.39,
and because 2.39 > 2.33 = c.01 (from t-distribution with 522 df), we
reject the null hypothesis and infer that experience affects positively on
wages.
Seppo Pynnönen Econometrics I
Statistical Inference
Testing for single population coefficients, the t-test
1 Statistical Inference
Sampling Distributions of the OLS Estimator
Testing for single population coefficients, the t-test
Testing Against One-Sided Alternatives
Two-Sided Alternatives
Other Hypotheses About βj
p-values
Economic, or Practical, versus Statistical Significance
Confidence Intervals for the Coefficients
Simultaneous Testing for Multiple Coefficients, The F -test
Testing General Linear Restrictions
On Reporting t
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