03.05.2024 OLS Distribution and Tests#

Sampling Distribution of OLS-Estimator#

Assumption:

  • error \(\epsilon\)​​ = independent of explanatory variables

  • normally distributed with mean = 0 and sd = \(\sigma^2\)

=> depends on empirical topic

e.g Wage regression

  • wage > 0 \(\neq\) normal distributed

  • minimum wage => population eran exaclty minimum

=> solve with log

Theorem

\[\begin{split} \implies \hat{ \beta_j } \sim Normal[\beta_j, Var(\beta_j)] \\ \end{split}\]

t-statistic#

Normal Model with CLM Assumptions

\[ y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + ...+ \beta_k x_k + u \]

Distribution of Beta => follows T-distribution

\[ (\hat{ \beta_j }- \beta_j) / se(\hat{ \beta_j }) \sim t_{n-k-1} \]

Test the Null Hypothesis (\(H_0 : \beta_j \le 0\))

\[ t_{\beta_j} = \hat{ \beta_j } / se(\beta_j) \]

Example: hourly wage equation for n=526

\[ ln \widehat{ \text{wage} } = \underset{(.104)}{.284} + \underset{(.007)}{.092 \cdot educ} + \underset{(.0017)}{.0041\cdot exper}+ \underset{(.003)}{.022 \cdot tenure} \]

  • \(H_0\): \(\beta_{exper} \le 0\) vs \(H_1: \beta_{exper} > 0\)​

  • Test Statistic: \(t_{\beta_j} = \hat{ \beta_j } / se(\beta_j)\)

  • Rejection Rule:

    • calculate distribution value

      • use normal distribution, because df>120

      • choose significance level = 5%

    • \(N(0.95) = 1.645\)

  • Calculate \(t_{exper} = \frac{ 0.0041 }{0.0017}= 2.41\)

  • 2.41 > 1.645 => \(H_1\) accepted

=> variable is different from 0 => good

Confidence Intervals#

= interval estimates

\[ \hat{ \beta_j } \pm c \cdot se(\hat{ \beta_j }) \]

Example:

  • given: degrees of freedom = 25, Confidence = 95%

  • c = 2.06 (look here)

  • \([\hat{ \beta_j }-2.06 \cdot se(\hat{ \beta_j }),\hat{ \beta_j }+2.06 \cdot se(\hat{ \beta_j })]\)

Test Hypotheses: Multiple Parameters#

Test a single hypothesis with more than one \(\beta\)

Example: Returns to Education at junior colleges vs. four year colleges

\[ \ln wage = \beta_0+ \beta_1 junior + \beta_2 univ+ \beta_3 exper + u \]

Result:

\[ \ln wage = \underset{.021}{1.472} + \underset{.0069}{.0667} junior + \underset{.0023}{.0769} univ + \underset{.0002}{.0049} exper \]

Numerator: \(\hat{ \beta_1 }-\hat{ \beta_2 } = 0.0067-0.0769 = -0.0102\)

Denominator: fucking complicated?!!!!!

F-Test#

probably (hopefully) not relevant

to test mutliple hypothesis at the same time

Example: Salary of a Baseball Player

\[ \ln salary = \beta_0+\beta_1 years + \beta_2 gamesyr + \beta_3 bavg + \beta_4 hrunsyr + \beta_5 rbisyr + u \]

Hypothesis: after controlling for years and games/yr, the other performance statistics do not have any effect

\[ H_0: \beta_3=0, \beta_4 = 0, \beta_5 = 0 \]

Results in unrestricted Model:

\[\begin{split} \ln \widehat{salary} = \underset{0.29}{11.19} + \underset{0.0121}{0.0689} years + \underset{0.0026}{0.0126} gamesyr + \\ \underset{0.0011}{0.0009} bavg + \underset{0.0161}{0.0144} hrunsyr + \underset{0.0072}{0.0108} rbisyr \\ SSR = 183.186, R^2 = 0.6728 \end{split}\]

Result in restricted Model

\[\begin{split} \ln \widehat{salary} = \underset{.11}{11.22} + \underset{.0125}{.0713} years + \underset{.0013}{.0202} gamesyr \\ SSR = 198.311, R^2 = .5971 \end{split}\]

F-Test

\[ F = \frac{ (SSR_r -SSR_{ur})/q}{SSR_{ur}/(n-k-1)} \]

  • Question: increase in SSR large enough to consider dropping?

  • F = always nonnegative

  • q = \(df_r-df_{ur}\)

  • \(n-k-1 = df_{ur}\)

Rejection Rule: c is taken from F distribution

\[ F > c, \text{ with c =} F_{q,n-k-1}(1-\alpha) \]

also often used: F-Statistic to test all variables (confirm that they are different from 0)