ECON4150 - Introductory Econometrics Lecture 7: OLS with ...

ECON4150 - Introductory Econometrics Lecture 7: OLS with Multiple Regressors

?Hypotheses tests?

Monique de Haan (moniqued@econ.uio.no)

Stock and Watson Chapter 7

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Lecture outline

? Hypothesis test for single coefficient in multiple regression analysis ? Confidence interval for single coefficient in multiple regression ? Testing hypotheses on 2 or more coefficients

? The F-statistic ? The overall regression F-statistic ? Testing single restrictions involving multiple coefficients ? Measures of fit in multiple regression model ? SER, R2 and R2 ? Relation between (homoskedasticity-only) F-statistic and the R2 ? Interpreting measures of fit ? Interpreting "stars" in a table with regression output

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Hypothesis test for single coefficient in multiple regression

analysis Thursday February 2 14:18:25 2017 Page 1

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Statistics/Data Analysis

1 . regress test_score class_size el_pct, robust

Linear regression

Number of obs F(2, 417) Prob > F R-squared Root MSE

= = = = =

420 223.82 0.0000 0.4264 14.464

test_score

class_size el_pct _cons

Robust Coef. Std. Err.

-1.101296 -.6497768

686.0322

.4328472 .0310318 8.728224

t P>|t|

[95% Conf. Interval]

-2.54 -20.94

78.60

0.011 0.000 0.000

-1.95213 -.710775 668.8754

-.2504616 -.5887786

703.189

Does changing class size,while holding the percentage of English learners constant, have a statistically significant effect on test scores? (using a 5% significance level).

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Hypothesis test for single coefficient in multiple regression analysis

Under the 4 Least Squares assumptions of the multiple regression model:

Assumption 1: E (ui |X1i , ..., Xki ) = 0 Assumption 2: (Yi , X1i , ..., Xki ) for i = 1, ..., n are (i.i.d) Assumption 3: Large outliers are unlikely Assumption 4: No perfect multicollinearity

The OLS estimators j for j = 1, .., k are approximately normally distributed in large samples

In addition

t = j - j0 N (0, 1) SE j

We can thus perform, hypothesis tests in same way as in regression model with 1 regressor.

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Hypothesis test for single coefficient in multiple regression analysis

H0 : j = j,0 H1 : j = j,0

Step 1: Estimate Yi = 0 + 1X1i + ... + j Xji + ... + 1Xki + ui by OLS to obtain j

Step 2: Compute the standard error of j (requires matrix algebra) Step 3: Compute the t-statistic

t act = j - j,0 SE j

Step 4: Reject the null hypothesis if ? |tact | > critical value ? or if p - value < significance level

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