Part IV Statistical Inference - sjp/Teaching/ecm/lectures/ecmc4.pdfآ  Part IV Statistical...

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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